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amianos and Gr osseteste

BRUCE S EASTWOOD

Communica ted by M. CLAGETT

In a pec ul ia r sense the op t ica l t r ea t i ses of DAMIANOS ( four th c en t ury A.D. )1

an d ROBERT GROSSETESTE (ca. t 16 7- - t 253) rep rese nt the sam e leve l of achieve -

ment in the h i s tory of op t ic s . Al though separa ted by n ine cen tur ies , DAMIANOS

an d GROSSETESTE inco rpora te in th e i r r e spec t iv e wo rks on op t ic s s imi la r degrees

o f s oph i s t i c a t i on i n t he t r e a t m e n t o f r e f le c t ion a nd r e f r a ct i on . B o t h m a d e m uc h

of the equ a l i ty of angles of inc idence an d r e f lec t ion , bu t f a i l ed to dea l wi t h mo re

c om p l e x know n p he n om e na o f r e fl e c ti on . F o r i n s t a nc e , t he f oc a l po i n t o f a pa r a -

bo l i c m i r r o r , und i s c us s e d by e i t he r o f ou r a u t ho r s , wa s a c c u r a t e l y de s c r i be d by

APOLLONIUS2 an d ALHAZEN.3 L ikewise for r e f rac t ion , b r ie f ly cove red in the w orks

t o be a na l yz e d be l ow , t he r e i s on l y e l e m e n t a r y t r e a t m e n t . M os t i n t r i gu i ng i s t he

qu an t i t a t iv e l aw of r e f r ac t ion which the y s ta ted . I)AMIANOS and GROSSETESTE

pr e s e n t t o t he r e a de r t he a m a z i ng l y i na c c u r a t e no t i on t ha t t he a ng l e o f r e f r a c t ion

i s e xa c t l y ha l f t he a ng l e o f i nci de nc e . F u r t he r m or e , e a c h wr i t e r f i nds h is r a t i ona l e

f o r t h i s qu a n t i t a t i v e l a w in t he p r i nci p l e s o f e c on om y a nd un i f o r m i t y . Ap pe a l i ng

t o t he s e a pr ior i bases , each se ized on the same form, though incor r ec t , f o r the

l a w o f r e f ra c t i on . T he ba c kg r o unds a nd d e ve l op m e n t s o f t he r e l e va n t op t i c a l

t r e a t i s e s r e ve a l t he s i m i l ar na t u r e o f t he i r c onc e p t ua l f r a m e w or ks .

1 Da ted thus by GEORGE

SARTON,Introduction to the History o/Science (Baltim ore :

Wi lliam s a nd Wilkin s, 1927), I, 354, following FRIEDRICH HULTSCH, Dami anos (3) ,

Pa ul ys Real-Ency elopddie tier classischen A ltertumswissenscha/t, ed. G. WISSOWA (St utt -

ga rt: Metz lerscher, t90 Q, IV (8er Hlb bd .), 2055. HULTSCH note s th a t DAIVIIANOS s

be st place d in th e fou rth centu ry, if HELIODORUS (his father o r master) is older th an

THEON ALEXA NDRIN IYS (mid fourth century).

2 See,

e.g.

THOMA S L. HEATH , The fragment of Anthem ius on burning mirrors

a nd t he ' F r a gm e n t um m a t he m a t i c um ] 3ob i e ns e ' , Bibliotheca Mathematica, 3e Folge,

V II (1907), 232; CHRISTIAN BELGER, Ei n neues Frag me ntu m Ma them aticu m

Bobiense , Hermes, XVI (1881), 271--2.

Fo r me diev al L ati n version, tran slat ed b y GERARD of CREMONA or his school,

v. J. L. HEIBERG ~; E. WIEDEMANN, I bn al Hai ta ms Schr if t f iber parabol ische Hohl-

spiegel , Bibliotheca Mathematiea, 3e Folge, X (1909-- t0) , 20 t- -3 7; Engl ish t rans l .

of Arabic te xt in H . J . J . WINTER & W. ARAFAT, Ibn a l -Ha i than l on the Parabolo ida l

Focuss ing Mir ror , Journal o] the Royal Asiatic Society o/ Bengal: Science, 3rd set.,

XV (1949) , 25- -40.



D e r i v a t i o n s o f a L a w o f R e f r a c t i o n 5

°

T h e o p t i c a l t r e a t i s e a s c r i b e d t o D A M IA N OS h a s c o m e t o b e c o n s i d e r e d h i s o n l y

w i t h d i f f i c ul ty . N o w k n o w n o n l y f r o m G r e e k m a n u s c r i p t s o f t h e f o u r t e e n t h

c e n t u r y a n d l a te r , 4 t h e w o r k h a s a p p e a r e d u n d e r t it l es a s s i g n in g i t a l t e r n a t e l y

to DAM I ANOS a n d t o HE L IODOROS o f L ARIS S A.5 H UL TS CH n o t e s t h a t m a n u s c r i p t s

a s s i g n in g t h e w o r k t o D A M IA N O S c o n t a i n e d f o u r t e e n c h a p t e r s ( as e d i te d b y

S CH ~N E), w h i l e s u p e r s c r i p t i o n s i n d i c a t i n g H EL IO D OR A N a u t h o r s h i p w e r e f o l l o w e d

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

c h a p t e r s , i d e n t i c a l u n d e r b o t h t i t le s , t o H EL IO D OR O S, a n d t h e f o u r t e e n t h t o

D A M IA N O S. I n t h i s c a s e , D A M IA N O S, m e n t i o n e d a s s t u d e n t ( o r so n ) o f H EL IO D O R O S

i n t h e t i tl e, r e d a c t e d h i s m a s t e r ' s w o r k a n d a d d e d a f in a l c h a p t e r . ~ T o c o m p l i c a t e

t h e i ss u e t h e r e a p p e a r s a m u c h m o r e e x t e n d e d v e r si o n d r a w n u p b y A N GE L

V ER G ET IU S i ll t 6 5 7 ; ~ t h e m a t e r i a l b e y o n d t h e o r i g i n a l f o u r t e e n c h a p t e r s , h o w e v e r ,

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

b y G E O RG IU S P A CH Y M E RE S ( t 2 4 2 - - c a . t 3 1 0 ) , s V ER G ET IU S ' m o t i v e f o r t h i s a d -

d i t io n s e em s t o h a v e b e e n i n g e n u o u s e n o u g h - - t h e a d d i t io n a l m o n e y p a i d f o r

c o p y i n g a l o n g er w o r k 2

T h e p e r s o n a l i t y o f D A M IA N O S i s c l o a k e d i n f a r g r e a t e r o b s c u r i t y t h a n t h e

t r ea t is e a s c r i b e d t o h i m . I t h a s b e e n m a i n t a i n e d t h a t t h e n a m e D o M N I N O S

w a s a c o r r u p t i o n o f

D A M I A N O S .

T h i s a s s e r t i o n le d o n e sc h o l a r t o a t t r i b u t e t h e

o p t ic a l t r e a ti s e t o t h e w r i t e r o f a n a n t i - N i c h o m a c h a n b o o k o n a r i th m e t i c . H o w -

e v e r , D O M NIN O S o f L AR IS SA , t h e m a t h e m a t i c i a n , i s w e l l - k n o w n a s a n i n d e p e n d e n t

p e r s o n a l i t y v i a t h e a r t i c le o n h i m b y S UID A S, c o n f i r m e d b y D A M A SC IU S a n d

MARINUS.1° O n t h e o t h e r h a n d , n o t h i n g i s p r e s e n t l y c e r ti f ia b l e a b o u t t h e l if e

a n d w o r k o f D AM IA NO S b e y o n d t h e p r e s e n t t r e a ti s e a n d i t s a p p r o x i m a t e d a t e

( f o u r t h c e n t u r y ) .

D AM IA NO S h a d b e f o r e h i m a n a v a i l a b le o p t i c a l t r a d i t i o n o f g r e a t r i c h n e s s

G RO SSE TE ST E w a s l e ss f o r t u n a t e - - y e t s e e m s t o h a v e m a d e p a r s i m o n i o u s u s e

o f i t i n h i s o w n o p t i c s . I n c a t o p t r i c s a l o n e t h e r e w a s d e t a i l e d w o r k b y A I~ CH I-

4 DAM IANOS,

S c h r i / t ii b er o p t i k m i t A u s z i ~ g e n a u s G e m i n o s ,

ed . RICHARD SCHONE

( Be r l i n : Re i c h s d r u c k e r e i , 1 89 7) , p p . v i - - v i i i .

5 Ib id . , p . 5 ; PAUL TANNERY R a p p o r t s u r u n e m i s s i o n e n I t a l i e , M d m o i r e s

s c i e n t i / i q u e s

( P a r i s : Ga u th i e r - V i l l a r s , 1 9 t2 ) ,

I I

3 2 0 - - 1 ; H U L TS C H, P a u l y - W i s s o w a ,

I V , 2 0 5 4 - - 5 .

6 Ib id .

H o w e v e r , t h e r e r e m a i n s s o m e d o u b t o f t h i s a m o n g m o r e r e c e n t s c h ol ar s,

e.g. , i n P AUL VE R E E CKE'S e d . a n d t r a n s l , o f E UCL I DE , L ' O p t i q u e e l l a C a t o p t r i q u e

( P a r i s : De s c l6 e e t B r o u w e r , 1 93 8) , p . x l i i; c ]. S ART ON'S t a t e m e n t , T h e wh o le q u e s t i o n

i s v e r y o b s c u r e . i n

I n t r o d u c t i o n ,

I, 354.

7 D a m i a n i H e l i o d o r i d e o p t i c i s l i b r i d u o , ed . ERASMUS BARTHOLIN (Par i s : C ram oisy ,

1657) ; v. TANNERY R a p p o r t , p . 3 19.

s A l s o , t h e r e w a s a d d e d a l o n g f r a g m e n t f r o m H E R O'S Catop t r i cs a s a t e r m i n a t i o n

o f t h e f i r s t o f t h e two b o o k s i l l BART HOL I N ' Se d i t i o n . T h e m a te r i a l f r o m P ACHYM E RE S ,

c o n s t i t u t i n g t h e s e c o n d b o o k o f B A R TH O L IN , i s a n a b r i d g m e n t o f E U CL ID 'S @ t i c s ,

a p p a r e n t l y o r ig i n al w i t h PACHYMERES a n d f o u n d a s w el l i n a n i s ol a te d a n d a n o n y m o u s

s t a t e i n P a r i s m s . g r . 2 4 7 7 , wh ic h wa s c o p i e d b y VE RGE T IUS .

TANNERY

R a p p o r t ,

p . 320 ; VER EECKE,

Opt ique e t Ca top t r ique ,

p . x l ii i. F o r P ACHYM ERE S' t r e a tm e n t o f

op t ics ,

v. Quadrivium de Georges ]~achym~re,

edd . P . TANNERY & E . STEPHANOU

( Va t i c a n C i ty : B ib l . Ap o s t . Va t . , 1 9 4 0 ) , p p . 3 1 8 - - 2 8 .

a TANNERY R a p p o r t , p p . 3 2 2 -- 3 .

lo PAUL TANNERY

D o m n i n o s

d e L a r i s s a ,

M d m o i r e s s c i e n t i / i q u e s

( P a r i s : Ga u th i e r -

V i l l a r s , t 9 t 2 ) , I I , t 0 5 - - 6 .



226 13. S. EASTWOOD

MEDES,11 APOLLONIUS, 2 HERO,Is PTOLEMY,14 and a EUCLIDEAN au thor; 15 DAMIANOS

uses littl e fro m th em. Maki ng expli cit reference to the optic al works of HERO 16

and PTOLEMY, v he seems to have given no serious at ten ti on to their tre atm ent s

of refraction. It is even possible that he did not make first-hand use of them,

preferring more general, encyclopedic sources, is Or he ma y have ignored or mis-

understood the more difficult material such as PTOLEMY'S discussion of refractive

angl es; 19 he ce rt ai nl y c ont rad ic ts PTOLEMY on this. DAMIANOS th us seems to

remain on the most e lementary p lane in his h y p o t h e s e s on light.

The brief 2° treatise by DAMIANOS aims prim aril y to establish t he ide nti ty of

visual rays with light rays. The extram issi on theory of vision 21 is assume d from

n Tes tim ony to a treatis e by ARCHIMEDES on bu rni ng mirrors can be found i n

JOANNES TZETZES,His to r ia r u m v a r ia r u m c h i l i a d e s , ed. THEOPHILUS KIESSLING (Leip-

zig: Vogel, t826) , p. 479; also APULEIUS MADAURENSIS,

Opera ,

ed. R. HELM (Leipzig:

Teubner, t905), I I i 1 8 - - 9 . An excellent survey on the question of an ARCHIMEDEAN

Catop tr ics is A. ROME, N o t e s sur les passages des eatopt riques d'Archim~ de conserv4s

par Th6on d'Alexandrie, Annales de la Soc id td Sc ien t i / ique de Bruxe l le s , Ln (t932),

30--41. The treatise itself has not survived.

12 Su pr a n. 3. This work of APOLLONIUS is no t ex tant .

13 HERO of ALEXANDRIA, Op e r a o mn ia , edd. L. Nix & W. SCHMIDT (Leipzig:

Teubner, 1900), n t, pp. 303--65.

14 The rel evant por tions of PTOLEMY'S o p t i c s in L ' Op t iq u e d e C la u d e P to ld md e

dam la vers ion la t ine d 'apr~s l ' a rabe de l ' dmir Eugene de S ic i le , ed. ALBERT LEJEUNE

(Louvain: 13iblioth~que de l'Universit6, t 956).

15 Th at is, th e Catop tr ics ascribed to EUCLID. While clearly not by EUCLID, this

work may well be a product of TnEoN, whose redaction of EUCLID'S

o p t i c s

shows

inferi ority to the original, e.g., in prop. 22, just as the Catop tr ics is generally inferior

to the o p t i c s . See J. L. HEIBERG, Li t te rargesch ich tl iche S tud ien i~ber Eu k l id (Leipzig:

Teubner, t 882), pp. 7, t 50--2; EUCLID, Op e r a o mn ia , edd. J. L. HEIBERG & H. MENGE

(Leipzig: Teubner, 1895), v i i p. t (note comparis on of terms for r a y on p. xlix);

VER EECKE Opt ique e t Ca top tr ique , p. xxix. See esp. The arg ume nt s of ALBERT

LEJEUNE, Recherches sur la ca top tr ique grecque (13russels: Palais des Acad4mies, 1957),

pp . t t2 - -49 ; ib id . , pp. t45--6 argues for an original Catop tr ics by EUCLID as well.

xs DAMIANOS,o p t i h , p. 20. FRIEDRICH I~ULTSCH, in a review of SCHONE'S ed it io n

of DAMIAlVOS' Opt ics ( (Ber l iner Ph i lo log ische Woehenschr i] t , XVIII (t898), 1414))

mai ntai ns tha t HERO'S Ca t@ tr i e s contained details on refraction as well.

iv DAMIANOS, o p t i k , p . 4 .

is The eclectic, encyclopedic collecting of data on a topic was a n establi shed tren d

b e f o r e DAMIANOS, as is shown by WILLIAM H. S T A H L R o m a n S c i e n c e (Madison:

U. of Wisconsin, 1962).

19 Cer tai nly THEON ignored or mis unders tood PTOLEMY on refraction (ROME,

Catopt riques d'Archim~de , pp. 39--40). Th at a majo r work in optics may be

known but not widely understood or used is evidenced by the history of ALHAZEN'S

optics in med ieval I sla m; ALKINDI'S optics was co mmon ly used instead, and phen om-

ena of refraction c onti nued to be misu nders tood despite t he avai labi lity of ALHAZEN'S

w o r k s .

20 In SCI-16NE'S edition, the Gr ee k t e x t i n c l u d i n g a s e p a r a t e l is t f c h a p t e r h e a d i n g s ,

fills l e s s t h a n e l e v e n p a g e s . T h e f ir s t p r i n t e d e d it i on , o n w h i c h t h e e d it i on s o f t h e

1 7 h a n d 1 8 h c e nt u ri e s w e r e b a s e d , r e qu i re s o n l y e l e v e n p a g e s f o r p a ra l le l c o l u m n s o f

t h e G r e e k t e x t w i t h L a t i n t r an sl at io n; v. H e l i o d o r i L a ri ss ae i C a p i t a O p t i c o r u m ,

second appendi x (unnumb ered pages) to La .Prospet t iva d i Eu c l ide . . . & . . . Specch i ,

transl . EGNATIO DANTI (Florence: Junt as , t 573).

21 In the history of optics this position was a major tradit ion in A nti qui ty and

the Middle Ages. A careful and brief surv ey of this t rad iti on can be found i n EDMUND

HOPPE, Geschichte d er op ti k (Leipzig: Weber, t926), pp. 5--25; for the Greco-Roman



Derivations of a Law of Refraction 227

the first chapter and seems to imply no problems for our author. Instead the

complete parallelism of visual and solar rays, whether direct, reflected, or re-

fracted, is shown. Acknowledging PLATO, s he too admit s the necessi ty of an

exterior as weU as all interior basis for vision. However, DAMIANOS ell us, there

are

some

people whose eyes emit strong enough light to see unaided in the dark;

such a persoI1 was the emperor TIBERIUS.~3

The general contents of this relatively unknown treatise are worth summariz ing

at least briefly, the fourteen chapters dealing with the following subjects.

t. Vision occurs by emanation from the eye.

2. What emanates from us is true light.

3. Visual rays are rectilinear and are propagated ill a right-angled cone.

4. Visual rays must emanate exclusively in the form of a cone.

5. The cone of rays must be right-angled, rather than acute or obtuse; this is

supported by both theory and observation.

6. Visual rays pass through invisible pores in the pupil, yet the rays irradiate

every point between the rays.

7. Ally object seen is viewed under a right angle or less. If the former, then

the object must be on the diameter of the base of the cone; otherwise, this may

or may not be the case.

8. An object seen under a larger angle appears larger because of the greater

number of rays striking it.

9. We see best and primarily along the axis of the cone of vision.

t 0. We are accustomed to seeing forward, and we therefore at tempt to see any

object directly ahead, or to interpret it as being ahead.

1 t. Tile apex of the visual cone is within the eye, and the pupil encloses the

COlle, which in turn intersects one-fourth of the surface of tile eye, a sphere.

t 2. Vision is completed either by direct rays or broken rays. Of broken rays

two sorts occur, reflected and refracted.

t3. In both rectilinear projection and reflection our visual rays act ider~tically

as do solar rays. Both visual and solar rays operate instantaneously over great

distances. Both visual and solar rays are reflected along the same lines.

t 4. Both reflection and refraction follow a definite law. This is the law of equal

angles, based on the principle of equality. The law is effective for both visual

and solar rays.

tradition a useful work is VASCO

RONCHI Histoire de la lumi~re

transl. J. TATON

(Paris: Colin, t956), ch. 1. The st rength of the emission theory was based upon its

early linkeage wi th geometrical optics. Aristotle s

Posterior Analytics

linked optics

directly to geometry. While EuDoxus was unwilling to play the metaphysical game in

geometrical astronomy, EUCLID seemed to find it useful in geometrical optics. The

great success of EUCLIDEANoptics made extramission an acceptable view for

GALEN

PTOLEMY, TI~EON, and, of course, DAMIANOS.

~ DAMIANOS optik p. 20 (ch. t 3).

2~ Probably on the basis of SUETONIUS

Tiberius

68, DAMIANOS asserts this in

ch. 2 (ed. cir., p. 4).



228 ]3. S. EASTWOOD

Clearly, for r efrac tion we must look a t DAMIANOS' last chapte r almos t ex-

clusively. Additional information from the earlier parts will serve, however, to

illuminate further the overall framework of the argument for refraction. The

whole of chapter 14 runs as follows.

One should recognize that reflection and refraction of our visual rays do not

occur by chance, without a definite law. Both occur without exception under

equal angles, based on the surface of the object at which our visual rays are

reflected or refracted. Hero the mechanician has proven in his catoptrics that

two points are connected more closely by reflection at equal angles than by

reflection at any other, unequal angles. He has proven, he sMd, that if Nature

does not wish to permit our visual ray to wander about fruitlessly, she will

let it break at equal angles. Likewise it can be shown that when our visual

ray penetrates an object, thus altering its direction, the refraction occurs at

equal angles (o#otcog ~

t)~Or, 'lOezc~t, 6 'v t ~od ¢1 dtd× ~czo t~ zq ~ 6~o eo¢ ~¢N ,~l~e~e'Qo~

~ d ~ 't 'Oo ~ ~s t tve2 e t' v ca yo ~ vkz~ ) . But hereupon it is clear that solar rays as

well are bent at equal angles. For one cannot maintain that the phenomenon

occurs according to the principle of equality with visual rays, but according

to inequality and chance with solar rays. And we have shown above that in

reflection, at least, both solar rays and visual rays bend at equal angles in

reflection. ~4

In the final chapter on refraction, as well as in the early chapters, on direct

vision, DAMIANOS searches out as simple a geometri cal si tua tion as possible to

account for the phenomena. In the first six chapters we find that he develops

not only the notion of vision as a truly illuminating process but the notion of a

right-angled cone of visual rays from the eye. The right-angled cone is chosen,

we learn in chapte r 5, ~5 because i t is the most determina te, most defined, form

of cone: Nature, being rational, prefers the definite to the indefinite, for the former

is better In ch apte r t4, the same sort of metaph ysical appeal suggests itself.

Equal angles are chosen for refraction as well as reflection, because an inequality

of angles (like a non-rectangular cone of rays) would be indefinite and a matter

of chance. Some sort of metaphysical principle of simplicity seems to operate

here, according to the text. An authoritative basis is claimed in the

C a t o p t r i c s

of HERO;

likewise, a tacit appeal to the

O p t i c s

of PTOLEMY s obvious in chapter 5

on the rec tangula r cone. 26 In both cases DAMIANOS goes beyond his authorities

in applying their conclusions to his situations. To reach a quantitative law of

refraction he draws on tile quantitative law of reflection. HERO showed that

24 o p t i k ,

pp. 20--2. It should be noted tha t the translation given here is not

verbatim, though strictly in accord with the sense of the original, and that both the

Greek text and Sc~I6I~B's German translation (pp. 21--3) have been used as a basis

for this English version. For I tal ian and Latin versions, see ])ANTI (ed. cir.

s u p r a n . 2 1 ) ,

n. pp. [f- 4v and If. 5v-- 6r of the two texts respectively].

25 DAMIANOS

o p t i k ,

pp. 6--8.

28 DAMIANOSclearly had in mind PTOLEMY, who experimentally determined the

rectangular shape (approximately) of the visual cone; v. ed. cir. (n. a 5

s u p r a ) ,

pp. 3 5--6.

A full discussion of the optical cone, or pyramid, ill PTOLEMYcan be found in ALBERT

LEJEUNE

E u d i d e e t P t o l d m d e : d e u x s t a d e s d e l ' o p t i q u e g d o m d t r i q u e g m c q u e (Louvain:

Bibl. de l'Universit6, 1948), pp. 42--55.



D e r i v a t i o n s o f a L a w o f R e f r a c t i o n 2 2 9

r e f l e c t i o n o c c u r s a t e q u a l a n g l e s , b e c a u s e t h i s i s t h e s h o r t e s t a n d q u i c k e s t p a t h . 27

H e , l ik e o th e r s , s m a k e s u s e o f t h e m e t a p h y s i c a l p r i n c ip l e o f e c o n o m y t o a r r i v e

a t h i s c o n c l u s i o n : h e i n s is t s t h a t N a t u r e d o e s n o t h i n g i n v a i n . T a k i n g u p w h e r e

H E R O le f t of f, D AM IA NO S a d d s a n o t h e r m e t a p h y s i c a l p r i n c i p l e , t h a t o f u n i f o r m i t y .

W i t h o u t e v e n b o t h e r i n g t o j u s t i f y h im s e l f , h e a s s u m e s ( in t h e p a s s a g e g i v e n a b o v e )

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

s i m i l a r l a w s : a u n i f o r m i t y e x i s t s b e t w e e n t h e l a w s o f re f l e c t i o n a n d r e f r a c t i o n . 29

I f t h e r e i s a n e q u a l i t y o f a n g le s in r e f le c t io n , t h e n t h e r e m u s t a l so b e a n e q u a l i t y

o f a n g l e s i n r e f r a c t i o n . 8° D AM IA NO S g o e s n o f u r t h e r i n d e s c r i b i n g j u s t w h e r e t h e s e

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

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

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

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

m e d i u m .

~

i s u a l r o y b e f o r e r e f r a c t i o n

c o m m o n s u r fa c e o f t w o m e d i a

f f

f

f f f 1 2

v i s u a l r o y a f t e r r e f r a c t i o n

a n g l e o a n g l e b

Fig . I

T h e f r u i t f u ln e s s o f t h e p r i n c i p le of u n i f o r m i t y i s n o t e x h a u s t e d h e r e b y a p -

p l i c a t i o n t o t h e p r o b l e m o f r e f r a c t i n g a n g l e s . DA M IA NO S a l s o w i e l d s a m e t h o d o l o g i -

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

t h e a s c e r t a i n e d l a w o f r e f ra c t i o n , a n d t h a t b o t h t y p e s o f r a y a r e t h e re f o r e o f

t h e s a m e e s s e n ce . 31 T o a r g u e t h a t v i s u a l r a y s d i f f e r i n r e f r a c t i o n f r o m s o l a r r a y s

37 E d. c i r . (n . t 4

supra),

p . 3 2 4 ; n o t e t h a t H E RO

assumes

t h e p a t h o f t h e r a y s

t o b e t h e s h o r t e s t p o s s i b le p a t h , a n d h e p r o v e s o n l y t h a t e q u a l a n g l e s g iv e t h i s s h o r t e s t

p a t h

3s E.g., ARISTOTLE, De caelo I , 4 , 27 t a .

39 P TO L EM Y c o n s i d e r e d r e f l e c t i o n a n d r e f r a c t i o n a s t w o c a s e s o f t h e s a m e p h e n o m -

e n o n L]EJEUNE,Euclide et Ptoldmde, p. 74, n . 4 ; e.g., optique de Ptoldmde, ed . LEJEUNE,

p. 246) . HERO (ed. c i r . n . 14

supra,

p . 3 22 ) u s e d f r a c t i o a s a g e n e r ic t e r m t o c o v e r

a l l t y p e s o f b e n t r a y s . I n m o s t R o m a n a u t h o r s t h e r e w a s n o c o n c e p t i o n o f t h e d i f f e re n c e

b e t w e e n r e f l e c t i o n a n d r e f r a c t i o n ,

e.g.,

i n

SENECA,Questiones naturales

I , v i i , 1 - - 2 .

3o W i t h o u t d o c u m e n t a r y e v id e n c e, i t h a s b e e n s u g g e s te d t h a t t h e s t a t e m e n t o f

e q u a l a n g l e s i n r e f r a c t i o n i s a n e r r o r o f a l a t e r c o p y i s t , n o t t o b e f o u n d i n D A M I A N O S '

though t ; v . JULIUS HIRSCHBERG,

Gesehiehte der Augenheilkunde im Al/erthum,

G r a e f e -

S a e m i s c h H a n d b u c h d e r g e s a m t e n A u g e n h e i l k u n d e , X l I ( L e ip z ig : B r e i t k o p f u n d

H~i r t e l , t 899) , p . 171 .

31 SIMEON SETH Conspectus return naturalium ch . 74 Anecdo/a Atheniensia et alia,

e d . A R M A N D

I)ELATTE,

L i e g e : V a i l l a n t - C a r m a n n e , 1 93 9,

I I

73) , r e f e r s t o PTOLEMY'S



2 3 0 B . S . E A ST W OO D :

w o u l d b e u n n e c e s s a r y , u n f o u n d e d , a n d u n e c o n o m i c a l i ll D A M IA N O S v i e w . T h u s

c h a p t e r 1 4 e x em p l i f ie s a p p e a l t o t h e p r i nc i p le s o f e c o n o m y a n d u n i f o r m i t y i n

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

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

I s th e r e a n y p o s s i b le a l t e r n a t i v e t o t h e i n t e r p r e t a t i o n w e h a v e g i v e n to

D A M IA N OS l a w o f r e f r a c t i o n ? T h e r e f e r e n c e t o e q u a l a n g l e s c a n n o t b e t o t h e

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

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

a p a r a l le l s i t u a t i o n ( t h e a c t u a l p a t h o f t h e r a y ) i s i n t e n d e d f o r r e f r a c t i o n . I f w e

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

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

HULTSCH32 h a s i n g e n i o u s l y s u g g e s t e d t h a t i n a p e c u l i a r s e n s e t h i s i s e x a c t l y w h a t

D AM IA NO S m e a n t . S t a t i n g t h a t t h e s a m e l a w w a s u n d o u b t e d l y i n H E R O S

Ca t -

o p t r i c s

i n a m o r e d e t a i l e d f o r m , H U L T S C H f e e ls t h a t t h e l a w e n v i s a g e s r e f r a c t i o n

tw i c e , t h r o u g h a g la s s p l a t e ( a p a r a l l e l e p ip e d ) a t a n a n g l e c l o se t o t h e p e r p e n d i c u l a r .

T h e r a y w o u l d t h e n l e a v e t h e p l a t e a t t h e s a m e a n g l e a s i t e n te r e d , g i v i n g e q u a l

a n g l e s o f i n c i d e n c e a n d r e f r a c t i o n . T h i s m e a n s t h a t D A MIA NO S m u s t h a v e r e c o g -

n i z e d t h e e x i s t e n c e o f r e c i p r o c a l a n g l e s i n r e f r a c t i o n f r o m a i r t o g l a s s a n d b a c k

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

r e q u ir e s u c h a r e c o g n i t io n - - u n l es s t h e b e n d i n g o f t h e r a y a t a n i n t e rf a c e b e

d e n i e d a l t o g e t h e r . T h a t k n o w l e d g e o f r e c ip r o c a l a n g l e s w a s a v a i l a b l e t o D A MIA NO S

i s c l e a r f r o m h i s r e f e r e n c e t o P T O LE M Y S 0 1 5 t i c s W h e t h e r t h i s i n f o r m a t i o n o n

r e f r a c t i o n w a s a s s i m i l a t e d b y D A M IA NO S r e m a i n s t o b e s e e n . T o p o s t u l a t e a l i k e l y

a n s w e r w e s h o u l d r e m e m b e r c e r t a i n t h i n g s a b o u t P TO L EM Y S d i s c u s s io n o f a n g l es

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

g l a ss , n o o b s e r v a t i o n s a r e r e c o r d e d f o r r a y s t r a v e l l i n g i n t h e r e v e r s e d i r e c ti o n .

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

r e f r a c ti o n b e t w e e n t w o m e d i a - - ev eI1 m a k i n g h i s o b s e r v a t i o n s f o r w a t e r - g l a s s

o n t h e b a s i s o f s u c h k n o w l e d g e - - P TO LE MY d e c li n e d t o t r e a t t h e q u e s t io n s y s t e m -

a t i c a ll y . 38 S e e m i n g l y , a n y s t u d e n t o f o p t ic s s h o u l d b e a b l e t o s e e th e r e c i p r o c a l

l a w i n P TO L EM Y a n d t o w o r k o u t i t s c o n s e q u e n c e s . Y e t a s a b l e a s t u d e n t a s

WITELO

i n t h e t h i r t e e n t h c e n t u r y , f a il e d c o m p l e t e l y t o c o m p r e h e n d P TO LE M Y S

m e a n i n g , b e c a u s e o f a g r e a t e r c o n c e r n f o r f ol lo w i n g t h e t e x t a s h e u n d e r s t o o d i t

r a t h e r t h a n f o r c o n d u c t i n g e x p e r i m e n t s 2 4 T h e d e g r a d e d s t a t e o f o p t i c a l s c ie n c e

b y t h e t i m e o f T HE ON c o u p l e d w i t h t h e s i m p l ic i st a n d n o n - e x p e r i m e n t a l t e n o r

v i e w o f t h e v i s u a l p n e u m a a p p e r t a i n i n g t o t h e q u i n te s s e nc e . S i m p l ic iu s ,

I n A r i s t o t e l i s

de caelo, ed . J . L . H EIBERG (Be r l i n : Re im er , 1894 ), p . 20 , sup por t s t he no t i on t h a t

PTOLEMY cons ide red v i sua l and i l l umina t i ng r ay s t o be qu in t e s sen t i a l ; S imp l i c iu s

c o m p a r e s P TO L EM Y S view w i th thos e of PLOTINUS an d XENARCHOS. See L E J E U N E ,

Eucl ide e t P lo ldmde , p p . 6 4 - - 6 .

32 O p. c ir. (n. 16 s u p r a , col . t4 t4 .

33 op tiq ue de Ptoldmde, p. 243, n. 31, and

L E J E U N E ,

Ca toptrique grecque, p. 158 for

the gene ra l s t a t em en t o f t h i s l aw . Re ason s fo r PTOLEMY S f a i l u r e t o p ro v ide r ec ip roca l

t ab l e s , a s we l l a s an en l i gh t en ing d i s cus s ion o f h i s who le t r e a tmen t o f ang l e s o f r e -

f r ac t i on , c an be found , i n t e r a l i a , i n

LEJEUNE

Cato ptrique grecque, pp . 155 - -66 .

s4WITELO

o p t io n , ed . FRIEDRICH RISNER (Base l : Episco pius , 1572) ,

p. 412.

W I TE L O S e r r o r, p a t e n t e n o u g h t o a m o d e r n e y e , h a s b e e n n o t e d m a n y t i m e s , e .g .,

b y E M I L W I L D E , Geschivhte t ier op ti k (Ber l in : R f icker und Pf ich ler , 1838) , I , 80- -2 .



D e r i v a t i o n s o f a L a w o f R e f r a c t i o n 2 3 i

o f D A M IA N OS ' t r e a t i s e , s t r o n g l y s u g g e s t t h a t D A M IA N OS w a s u n e n l i g h t e n e d a n d

u n c o n c e r n e d o v e r s o d e t a il e d a q u e s t i o n a s th e r e c i p r o c i t y o f a n gl e s o f r e f r a c ti o n .

T h e g e o m e t r y o f h is o p ti c s c a n o n l y b e d e sc r ib e d a s m e t a p h y s i c a l , n o t m a t h e -

m a t i c a l ; i n t h e o n l y p l a c e w h e r e a n y s e r io u s g e o m e t r y i s d i s cu s s ed , h e e rr s, a n d

h e e r r s d e s p i t e t h e a v a i l a b i l i t y o f a c o r r e c t s o l u t i o n s i n c e t h e t i m e o f A ~C H IM E DE S.

I n c h a p t e r t i D A M IA N OS n o t e s t h a t t h e p u p i l , w h i c h e n c l o s es t h e c o n e o f v i s u a l

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

s e e m s v e r y d e f i n i t e l y t o b e a i m e d a t d e t e r m i n i n g a n e x a c t p o i n t o f o ri g i n f o r

t h e c o n e o f v is i o n ,

i .e . ,

i n t h e m i d d l e o f t h e e y e ; 35 c e r t a i n l y h i s p e n c h a n t f o r

s i m p l i c i t y a n d e x a c t n e s s e x t e n d e d h e r e a s el s ew h e r e . T h e r e i s p o s i t e d a r i g h t -

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

t h e s p h e r i c a l s u r f a c e e q u a l t o r 2= . Y e t A R CH IM ED ES d e m o n s t r a t e d m u c h e a r li e r

t h a t t h e a r e a o f t h e i n t e r s e c t e d s e g m e n t w o u l d b e 3 / 5 r ~= . 38 N o r d o e s D A M IA N OS

s h o w i g n o r a n c e o f t h e t r e a t i s e 3~

O n t he Sphere and t he Cy l i nder ) ,

b u t r a t h e r s o m e

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

p e r t i n e n t p r o p o s i t i o n ( I, 3 4) t o r ef e r t o s u r fa c e a r e a s r a t h e r t h a n t o v o l u m e s a s

i t d o e s , f o r th e b a s i c p r o p o s i t i o n - - n o t i ts c o r o l l a r y , w h i c h D AM IA NO S s h o u l d

h a v e u s e d - - g i v e s t h e r a t i o o f t : 4 fo r t h e v o l u m e s o f a r i g h t a n g l e d c o n e ( w i th

h e i g h t e q u a l t o t h e s p h e r e ' s r a d i u s ) a n d a s p h e r e . 8s A n e v e n s i m p l e r p o s s i b l e b a s i s

f o r t h e e r r o r w o u l d b e e x t r a p o l a t i o n f r o m a p l a n a r ( t ri a n g le a n d c i rc le ) s i tu a t i o n ,

w h e r e D A M IA N OS ' e x p l i c i t r a t i o o f I : 4 w o u l d b e c o r r e c t , t o a s o l i d ( c o n e a n d

s p h e re ) s i t u a t i o n . T h e v e r y u n s o p h i s t i c a t e d n a t u r e o f s u c h a n e r r o r i s c o n s o n a n t

w i t h t h e w h o l e t o n e o f D A M IA N OS '

Optics.

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

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

a u r a o f e x a c t n e s s, w h e r e t h e r e a l l y s i g n i f ic a n t s t a t e m e n t s a r e in t h e r e a l m o f

m e t a p h y s i c s r a t h e r t h a n m a t h e m a t i c s . W i t h r e s p e ct t o th e n a t u r e o f l ig h t, o f

r e f le c t io n , a n d o f r e f r a c ti o n , t h e i m p o r t a n t s t a t e m e n t s h e m a k e s a r e t h o s e b a s e d

o n t h e p r i n c i p le s o f e c o n o m y a n d u n i f o r m i t y . D AM IA NO S' q u a n t i t a t i v e l a w o f

35 T h i s a p p e a r s t o b e P T OLE M Y'S v i e w LEJEUNE,Eucl ide et Ptoldmde, p . 55) and

i s c e r t a i n l y t h e w a y GALEN,HUN AYN IBN IStJ .XQ, ALHAZEN,a n d m o s t l a t e r m e d i e v a l

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

i n t u r n , i n f lu e n c e d o c u l a r a n a t o m y ; i n o r d e r t o a v o i d t h e p r o b l e m o f r e fr a c ti o n ,

G A L EN a n d m o s t o f h is m e d i e v a l s u c c e s so r s s a w t h e e y e a s a s e r i es o f c o n c e n t r i c

c i rc l es ( t h e tu n i c s a n d h u m o r s ) , s o t h a t r a y s f o r m i n g a c o n e w i t h i t s a p e x a t t h e c o m -

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

l a y e r .

36 D e sphaera et cylindro I,

3 4 ( c o ro l la r y ) p l u s s o m e e l e m e n t a r y E u c l i d e a n g e o m e t r y .

ARCHIMEDES,

Opera omnia,

e d . J . L . HE IBE RG ( L e ip z ig : T e u b n e r , t 9 1 0 ) , I , 13 2 ;

v . The Works o /Archimedes ,

e d . a n d t r a n s l . T . L . H E A TH ( C a m b r i d g e : C a m b r i d g e U . P . ,

t 897), p . 4 4. DA MIANO S'e r r o r o n t h i s p o in t wa s f i r s t n o t e d i n HI RS CHBE RG,

Augenhei l -

kunde i m A l t e r t hum ,

p p . t 6 9 - - t 7 0 .

87 DA MIAN OS,Optik, p . 4 q u o t e s b o o k 1 , p o s tu l a t e t o f t h e w o r k ( ed . c i r. n . 3 4 supra,

I , 8 ; Eng . t rans l , c i r . p . 3 ) .

38 W h e th e r o r n o t DAM IANOSh a d r e a d t h i s p a r t i c u l a r p r o p o s i t i o n i s n o t c e r t a i n ,

t h o u g h p r o b a b l e . A p a r a l l e l c i r c u m s t a n c e i s T H EO N 'S a p p a r e n t i g n o r a n c e o f PT OL EM Y 'S

e x p l a n a t i o n o f h o r i z o n m a g n i f i c a ti o n i n t h e opt ics , f o r T H E O N m a k e s u s e o f a c o n t r a -

d i c t o r y t h e o r y g i v e n e a r l ie r b y P TO L EM Y i n t h e Al m ages t ( R O M E , C a t o p t r i q u e

d ' A r c h i m ~ d e , p p . 39 - -- 4 0 ); T H E O N m a y h a v e r e a d o n l y p a r t o f PT O LE M Y 'S

Optics

o r h e m a y h a v e m i s u n d e r s t o o d t h e d i sc u s si o n o f r e f ra c t io n , f o r h e c e r t a i n l y w e a k e n s

par t s o f E ISCLII ) 'S

opt ics v .

VER EECKE,

Optique et Catoptrique,

p p . x x v i i - - x x v i i i ) .



232 B.S. EASTWOOD:

refraction is given on the basis of the principle of uniformity primarily, in order

to establish a parallel with reflection. Reflection, in turn, and therefore refraction

as well, accords with the principle of economy.

II.

The non-experimen tal natu re of DAMIANOS' law of refraction and its openly

metaphys ical foundat ions are mirr ored in the opt ical work of ROBERT GROSSETESTE

in the thirteenth century. To juxtapose and compare the refraction laws of these

two persons is not to claim a historical connection between them. In fact, what

makes the similarity interesting is the apparent lack of positive connection. There

seems to be absolutely no tradition of a Latin version or summary of DAMIANOS'

optics, not even references to the work, before the Renaissance. Yet GROSSETESTE

and DAMIANOS share a similar outlook on science. F rom an out look inspired b y

PLATONISM, each sees reference to authorities and experience as ancilla ry in-

formation, meaningless wi tho ut metaph ysical bases for synthesis. 89 While DAMIA-

NOS makes

o n l y

brief reference to these bases in his optics, GROSSETESTE develops

a veritable metaphysics of light, which he immediately applies to the study of

physical light. Th rough out his optics, metaphysical principles play an impo rtant

part. In what was probably the last of his significant works in science, GROSSE-

TESTE dealt with the prob lem of explaining the rainbow. In his treatise,

D e i r i d e

c a . 1 2 3 5 ) , * °

while making some use of empirical data, he relies ultimately upon

the principles of economy and uniformity in order to design a theory of the

rainbow. ~ The same principles lie behind his qua ntit ativ e law of refraction.

Ju st as DAMIANOS, GROSSETESTE seems to use only a small amount of the

optical work done before his time. But the lack of attention to available works

in the case of the earlier writer is replaced by a lack of availability of many

works for the later writer. While the optical treatises of such Greek writers as

&RCHIMEDES and APOLLONIUS seem to have vanishe d pe rmanently by the thi r-

teenth century, extant contributions by ANTHEMIUSa2 (fragmentary) and PTOLE-

MY *a were unavailable to GROSSETESTE. The pr imar y contributio n of the Islamic

world to this science, the D e a s p e c t i b u s of ALHAZEN, seems not to have been

89 The development of this point requires a separate paper in itself; such a paper,

enti tled Mediaeval Empiricism: the Case of GROSSETESTE'S Optics , appears in

S p e c u l u m , 43 (t968), 306--2t.

40 Text in

D ie ph i losoph isch en W erk e des Rober t Grossetes te , B ischo]s von L inco ln ,

ed. Ludwig 13aur (Miinster : Aschendorff, t 912), pp. 72--8 ; dated thus by RICHARD C.

DALES, Robe rt Grosseteste's Scientific Works,

I s i s ,

LI (1961), 402.

41 See BRUCE S. EASTWOOD, Robert Grosseteste's Theory of the Rainbow: A

Chapter in the History of Non-experimental Science,

Ar chi ves in t . d h is t , des sciences,

19 (1966), 313--32.

~3 ANTI~EMIUS is highly regarded by the Arabs,

e.g.,

in the Latin translation of

ALHAZEN'S

De s p e c u l i s c o mb u r e n t i b u s

(transl. late 12 h cent.), bu t WITELO alone

( P e r -

spec t iva

IX, 39--43) among the Latins shows knowledge of ANTItEMIUS. V. HEIBERG &

WIEOEMANN, Parabolische Hohlspiegel , p. 219; G.L. HUXLEY,

A n t h e m i u s o /

T r a l l e s : A S tu d y i n L a t e r Gr e e k Ge o m e t r y

(Cambridge, Mass., t959), pp. 3, 40--2.

43 PTOLEMY'S

OptiCS

appears to have become current in northwestern Europe only

after GROSSETESTE finished his scientific work,

viz.,

from ca. 1250; LEJEUNE,

Op t iq u e

de Ptoldmde,

p. 31.



Derivations of a Law of Refract ion 233

avai lable unt il aft er GROSSETESTE'S scienti fic period; 44 in an y case, it is ce rtainly

neve r used b y GROSSETESTE. The nat ure of his optics is large ly determine d b y

the elementary level of the treatises he clearly has before him. Little beyond the

EUCLIDEAN0 t~ ic s and Catoptrics, HEI~O'S Catoptrics (Liber Ptole m ei de specu lis),

and ALKINDI'S De asl~ectibus appears to constitute the optical tradition of the

first third of the thirteenth century. None of these discusses refraction, and only

tile simplest geometrical aspects of reflection and direct light are treated. With

these plus a few brief references to refract ion, 45 GROSSETESTE has a r ela tively

unlimited field for speculation and development of optical laws.

In his law of ref rac tion more thai1 a nywhere else GROSSETESTE SHOWS how

speculative, metap hysic al principles Call be applied in detail to the business of

science. In a notable passage of De ir ide he lays down the law that the angle of

refraction equals half the angle of incidence (when passing from a rarer to a

denser medium). This law is closely followed by still another optical law, one for

the location of all image in refraction, based on the same principle. Three con-

secutive pa rag rap hs by GROSSETESTE illuminate his whole appro ach towards

optics. 46 In tile firs* paragraph, containing tile law of refrac tion, he said.

The size of the angular declination of the refracted ray from a straight ingress

can be visualized as follows. First we conceive of a ray which passes from the

eye through the medium of tile air and meets a second transparent body; we

extend into the second transparency the straight line along which this ray

travels, and then from tile point at which the ray meets the transparency we

draw into the depth of that transparency a line which is perpendicular to its

surface. I say then that the path of the ray in the second transparency is

alone a line

div id ing equal ly the angle

[my italics~ which is formed by the

imaginary direct extension of the ray upon tile surface of the second trans-

parency into that mediumY

The passage clearly describes the case of refraction from a less to a more dense

medium. After constructing a straight line passing through both media and per-

44 GEORGE SARTON, The Tradition of the Optics of Ibn al Haitham ,

Is is ,

XXXIX (t938), 403--6; but c/. MARSHALLCLAGETT, Archimedes in the Middle Ages

(Madison: U. Wis. Press, 1964), I, 669. The issue is still very unclear. What seems to

be the earliest surviving Latin manuscript of this work is in the Crawford Libra ry of

tile Royal Observatory, Edinburgh, MS. 9---11--3 (20), dated t0 May t269; but the

colophon suggests working knowledge of the treatise by a Magister JOHANNES

LONDONIENSlS for some time previously.

~5 ]Being the only geometrical treatment clearly available, tile most important is

in a pseudo-Euclidean De speculis (not the Catoptr ica att ributed by PROCLUS to

EUCLID), appearing ill Alkindi, Tideus, und Pseudo-Euklid: drei optische Werke, edd.

A . A . B J O R N BO S . V O G L

(Leipzig: Teubner,

1 9 1 t ) , p p . 1 0 5 - - 6

(prop. t4 only).

4~ In more extended form the following analysis may be found in BRUCE S. EAST-

WOOD, Grosse tes te 's 'Quantitative' Law of Refraction: A Chapter in the History

of Non-experimental Science,

J. Hist . Ideas,

28 (1967), 403--14, Some new emphases

are made in the present paper, while the earlier article deals as well with tile following

points unmentioned here: (t) the epistemological basis of GROSSETESTE'S optics,

(2) scholarly confusions about the existence of a quant ita tive law of refraction in the

optics, and (3) speculation on the qualitative as opposed to the quantitative aspects

in the study of light.

~7 BAUR, Werke, p. 74.



2 3 4 B S ] ~A S T W O O D :

p e n d i c u l a r to t h e i n te r f ac e , h e p o i n ts o u t t h e d i r e c ti o n t a k e n b y a r a y i n t h e

d e n s e r m e d i u m i ll r e l a t i o n to t h e p a t h i t w o u l d h a v e f o l lo w e d , h a d i t c o n t i n u e d

d i r e c t l y a n d U l l r ef r ac t ed . T h a t p a t h i s i n d i c a t e d b y t h e p h r a s e

d i v i d e n t i a

p e r

a e q u a l ia a n g u l u m . " T h i s p e c u l ia r t e r m i n o l o g y h a s a n o b v i o u s m e a n i n g - - t h a t

t h e r a y i n t h e d e n s e r m e d i u m f ol lo w s a p a t h b i s ec t in g t h e a n g le f o r m e d b e t w e e n

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

t h e r a y f r o m t h e r a r e r m e d i u m i n t o t h e d e n s e r . H o w e v e r , t h e r e is a l es s o b v i o u s

m e a n i n g , o n l y p a r t i a l l y e x p l i c it , il l t h e p h r a s in g . T o s p e a k o f " d i v i s i o n o f a n

a n g le in t o e q u a l p a r t s " r a t h e r t h a n o f h a l v i n g t h a t a n g le m a y s e e m n o m o r e

t h a n a n a l t e r n a t i v e w a y o f sp e a k i n g . C e r t M n l y t h e r e s u l t fo r p h y s i c s i s t h e s a m e ,

i . e . ,

t h e o p t i c a l l a w t h a t r = ½ i ; b u t t h e r e s u l t f o r m e t a p h y s i c s i s q u i t e d i f f e r e n t .

H ad G RO SSETESTEs a i d e x p l i c i t ly t h a t r = ½ i, h e w o u l d s i m p l y h a v e b e e n w r o n g .

I n s t re s s in g t h e n o t i o n o f e q u a l i t y , h o w e v e r , l i k e D AM IA NO S b e f o r e h i m h e w a s

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

L e s t t h e c o n c l u s i o n s e e m t o o i n g e n i o u s a n e x e r c is e o f t o o l i tt l e e v i d e n c e , le t u s

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

a p p r e c i a t e t h e f u l l s i g n if i ca n c e o f t h e a p p e a l t o e q u a l i t y .

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

w h i c h t h e h a l f - a n g l e la w i s b a s e d . T h e t e x t r u n s a s f ol lo w s .

T h a t t h e q u a n t i t y o f t h e a n g l e i s s o d e t e r m i n e d i ll t h e r e f r a c t i o n of a r a y ,

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

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

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

i s b y t h e m o s t f i n it e , m o s t o r d e r e d , s h o r t e s t a n d b e s t m e a n s p o s si b le , ks

Here GROSSETESTE

h a s s t a t e d e x a c t l y t h e s a m e c o n c lu s io n as

DAMIABIOS

g a v e i n

t h e l a s t c h a p t e r o f h is

O p t i c s .

B e c a u s e r e f l e c ti o n o c c u rs a t e q u a l a n g le s , r e f r a c t i o n

a ls o m u s t o c c u r a t e q u a l a n g le s . W h y ? B e c a u s e o f t h e m e t a p h y s i c a l p r i n c ip l e o f

eco no m y, b o t h H El l O ( f o l l ow ed by D AlV ilA NO S) an d G RO SSETESTE a s se r t ed t he

n e c e s s i t y o f a n e q u a l i t y b e t w e e n t h e a n g l e s o f i n c id e n c e a n d r e f le c t io n . F i n d i n g

n o e s s e n t i a l d i f f e r e n c e b e t w e e n r e f l e c t i o n a n d r e f r a c t i o n , G R O S SE T ES T E ( li k e

DAMIANOS

e a s il y m a d e t h e t r a n s it i o n f r o m t h e f o r m e r to t h e l a t te r . H e s a y s

o p e n l y t h a t r e f r a c t i o n a n d r e fl e c ti o ll a r e

s i m i l a r

e x p e r i e n c e s . " C e r t a i n l y t h i s i n

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

o f p h e n o m e n a . 49 B u t G R O SS ET E ST E p r o c e e d s f r o m t h i s m e t h o d o l o g i c a l p r i n c i p l e

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

c o n s i d e r r e f r a c t i o n a n d r e f l e c ti o n a s s im i l ar , i t i s n e c e s s a r y t o d o s o U n i f o r m i t y

b e c o m e s a p r in c i p l e o f N a t u r e h e r e . A l l p h e n o m e n a o f l i g h t m u s t f o ll o w n a t u r a l

4s Ib id . , pp . 74 - - 5 .

4~ T he exam pl e o f the r a i nbow i s t he m o s t com m on , a s i t w as a l w ay s cons i de r ed

t he r e su l t o f r e f l e c t ion . SEN ECA r e f e r r ed t o m ag n i f i c a t ion by w a t e r a s a c a se o f r e -

f lect io n; v. S~N~QIJE,Q u e s t i o n s n a t u r e l l e s , ed. PAU L OLTRAMARE (Pa ris : ]Belles L et t r es ,

196t) , I , 33.

LEJEUNE,

E u c l i d e e t P t o l d m d e ,

p. 76 , n . 4 note s t h a t eve n PTOLEMY

cons i de r ed r e f l e c ti on and r e f r ac t i on a s tw o ca se s o f t he s am e phen om enon . T he open i ng

l ine of Pseudo-EUCLID,

D e s p e c u l i s ,

t4 (ar t . c i r . n. 45

s u p r a )

sugges t s t ha t r e f l e c t i on

a n d r e f r a c t i o n a r e e s s e n t i a l l y s i m i l a r . T h e m e d i e v a l p s e u d o A R I S T O T E L I A N D e p r o -

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

r e f l e c t i o n ( B . N . m s . f a t . 4 7 8 , f . 5 8 r ; V a t . m s . f a t . 2 0 8 3 , f . 2 0 9 r - - v ) .



De r iva t i ons o f a La w o f Re f r ac t i on 235

l a w s in a s im p l e m a n n e r , h e a p p e a r s t o s a y . B o t h r e f l e c ti o n a n d r e f r a c t i o n , d i f f e r e n t

a s p e c t s o f t h e s a m e s p ec ie s of p h e n o m e n o n , m u s t a c c o r d w i t h t h e l a w o f e q u a l

a n g l e s . E q u a l i t y i s t h e h i g h e s t d e g r e e o f o r d e r i n r e f l e c t i o n . W h e r e d o e s t h i s

o r d e r e d e q u a l i t y a p p e a r i n r e f ra c t i o n ? I t c a n n o lo n g e r b e a re l a t io n s h i p o f

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

f o r m e d w i t h i n t h e s e c o n d m e d i u m .

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

a g a i n a n d c o n c l u s i v e l y e s t a b l i s h e d i n G R OS SE TE ST E'S m i n d b y t h e l a w f o r t h e

l o c a t io n o f a n i m a g e i n r e fl e c ti o n a n d r e f ra c t io n . T h e t h i r d p a r a g r a p h f o r m u l a t e s

t h e l a w a s f o l l o w s .

A t h i n g w h i c h is s e en th r o u g h m a n y t r a n s p a r e n t m e d i a d o e s n o t a p p e a r t o

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

e x t e n d e d f r o m t h e e y e a l o n g a s tr a i g h t l in e , a n d o f a li ne d r a w n f r o m t h e

v i e w e d o b j e c t p e r p e n d i c u l a r t o t h e s u r f ac e n e a r e r t h e e y e o f t h e s e c o n d t r a n s -

p a r e n t m e d i u m . M o r e o v e r , t h is is s h o w n t o u s t h r o u g h t h a t e x p e r ie n c e , a n d

s i m i la r re a s o ns , b y w h i c h w e k n o w t h a t a t h i n g s e e n i n a m i r r o r a p p e a r s a t

t h e j u n c t u r e o f t h e e x t e n d e d l in e o f s i g ht a n d o f a li ne d r a w n p e r p e n d i c u l a r

t o t h e s u r f a c e o f t h e m i r r o r . 5°

H e r e a g a i n t h e i l lu m i n a t i n g p a r a l l e l of r e f le c t i o n a n d r e f r a c t i o n a p p e a r s . O n t h e

b a s i s o f t h e w e l l - k n o w n l a w o f i m a g e l o c a t i o n i n r e f l e c t i o n , G R O SS ET ES TE g i v e s

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

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

s t a t e d b y t h e f o l l o w i n g w o r d s ; t h e l o c a t i o n o f t h e i m a g e is d e f i n e d b y t h e i n t e r -

s e c t i o n of a p e r p e n d i c u l a r t o t h e s u r f a c e f r o m t h e v i e w e d o b j e c t a n d a d ir e c t

c o n t i n u a t i o n b e y o n d t h e s u r f a c e o f t h e l in e o f v i s io n f r o m t h e e y e to t h e m e d i u m .

T h e p h r a s i n g i s i d e n t i c a l f o r b o t h r e f l e c t io n a n d r e f r a c t i o n , b u t t h e a c t u a l g e o -

m e t r y i s q u i t e d i f fe r e nt . T h e v e r b a l s i m i la r it y , e v i d e n ce o f a m e t a p h y s i c a l u n i f o r m -

i t y p r i n c i p le , i s m o r e i m p o r t a n t t o G RO SS ET ES TE h a n a n y d i a g r a m m a t i c d i f f e re n c e .

L i k e I )A M IA N O S h e i s h a r d l y c o n c e r n e d f o r d e t a i l e d g e o m e t r i c a l t r e a t m e n t o f

o p t i c a l p r o p o s it i o n s. T h e s o r t o f g e o m e t r y u s e d i n o p t i c s b y PTOLEMY ALHAZEN

a n d W I TE L O i s o f l i t t le u s e t o G R O SS ET E ST E . F o r h i m t h e a p p l i c a b i l i t y , r a t h e r

t h a n t h e a p p l i c at i o n , o f g e o m e t r y t o o p t i c s is s uf fi c ie n t. G e o m e t r y r e p r e s e n t s

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

t r e a t m e n t o f l ig h t . A r e a d i n g o f h i s D e l i ne i s , a n g u li s , e t / i g u r i s w i l l m a k e s t h i s

a b u n d a n t l y c l e a r; o n e n e e d n o t g o a s fa r a s t h e c o s m o g o n y of D e l u c e t o f i n d

l i g h t m e t a p h y s i c s i n t h e w o r k o f G R O SS ET ES TE . 1

50 BA UR , We@e, p . 75 .

51 Fo r som e d i s cus s ion o f li gh t m e t ap hy s i c s i n Gros se t e s t e v . A .C . CROMBIE

Rober t Grossetes te an d the Or ig ins o[ Ex pe r im en ta l Sc ience , 11 00- -17 00 (Oxfo rd :

C laren do n Pre ss, 1962), chs. 5- -6 ; ROBER T GROSSETESTE,O n L i g h t (De luce) , t rans l .

CLARE C. RIEDL (Mi lwauk ee : M arq ue t te U. P . , 1942) ; LUDWIG

[BAuR

Die P h i lo s o p h i e

des Rober t Grossetes te , B isch o /s von L in co ln ( t 1253)

(Mi in s t e r : Aschendor f f , t 9 t7 ) ,

p p . 7 6---1 09 . N o n e o f t h e s e m a k e s m u c h o f th e g e o m e t r i c a l m e t a p h y s i c s i n v o l v e d

(c[.

BAUR,

Phi losoph ie ,

pp. 16 - -7 , wh ich m isses the po in t o f GROSSETESTE'Ss e p a r a t i o n

o f p h y s ic s a n d m a t h e m a t i c s ) ; b r i e f a t t e n t i o n i s g i v e n t o i t i n m y d i sc u s si o n i n op. cir.,

n. 39 supra .

17 a Arch. Hist. Exac t Set. Vol. 6



2 3 6 B . S . E AS TW O O D: D e r i v a t i o n s o f a L a w o f R e f r a c t i o n

U s i n g g e o m e t r y a s t h e h a n d m a i d e n o f p h y s i c s a n d m e t a p h y s i c s , D AM IA NO S

a n d G RO SSE TE ST E d e v e l o p o p t i c a l d i s c u s s i o n s w h i c h c o n t a i n m o r e s p e c u l a t i o n

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

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

r e f r a c t i o n o n t h e b a s i s o f r e f le c t io n . T h e i n i t i a l i d e n t i t y o f o u t l o o k f o u n d o n t h i s

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

e l e m e n t a r y t r e a t i s e s o n g e o m e t r i c a l o p t i c s c o m b i n e d w i t h a N e o p l a t o n i c e n v i ro n -

m e n t . F o r t h e s e t w o g e n e r i c e l e m e n t s a r e c o m m o n t o G RO SSE TE ST E a n d D A MIA NO S.

E a c h s e e m s t o h a v e m a d e u s e o f s i m p l e g e o m e t r i c a l o p t i c s , o v e r l a i d w i t h a N e o -

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

e n t i t i e s . 5~ O n t h e s e b a s e s t h e y l a i d d o w n t h e e a r l ie s t e x t a n t s t a t e m e n t s o f a

q u a n t i t a t i v e l a w o f r e f r a c t i o n . 5~

52 A h i g h l y s u g g e s t i v e w o r k o n t h i s t h e m e i s PH I L IP

~V~ERLAI~

F r o m P l a t o n i s m t o

N e o p l a t o n i s m ,

2 nd e d i t i o n ( T h e H a g u e : N i j h o f f , 1 96 0) .

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

l a w . B o t h P TO LE M Y a n d

ALHAZEN

k n o w i n g m u c h m o r e a b o u t r e f ra c t i o n a n d t a k i n g

e x p e r i m e n t s s e r i o u s l y , w e r e n o t s o r a s h a s t o f o r m u l a t e a l a w ; t h e f a i lu r e t o c a l c u l a t e

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

N o t a b l y , t h o s e w h o a p p r o a c h e d s c ie n c e w i t h a m o r e N e o p l a t o n i c b e n t s e e m e d i n t e n t

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

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

D e p a r t m e n t o f S o c i al S c ie n ce s

C l a r k s o n C o l le g e of T e c h n o l o g y

P o t s d a m , N e w Y o r k

R e c e i v e d N o v e m b e r 4 , 1 9 6 9 )

archive for history of exact sciences volume 6 issue 3 1970 [doi 10.1007%2fbf00327236] bruce s....

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