594 
OPTICS. 
The angles A and B may alfo be determined by the 
following conftruftion. In the ftraight line CEDA, 
(fig. 4 ) take CA to CD as m to n, and CA to CE as p+i 
to i ; with the centre C and radius CD, defcribe an arc 
DB, cutting the circle ABE, whofe diameter is AE, in 
B ; draw ABF; and join BC ; then, the fine of the Z. 
CBF will be to the fine of the £ CAF as m to n; and 
the tangent of CBF to the tangent of CAF as p-\- i to'i ; 
and confequently CBF, CAF, will be the angles required. 
Join BE, and complete the parallelogram CEBG; pro¬ 
duce CG till it meets ABF in F. Then, in the triangle 
CAB, fin. CBA (fin. CBF) : fin. CAB :: CA : CB :: 
CA : CD :: m : n. Again, fince CF is parallel to EB, 
the /_ BFG, is equal to the EBA, and is therefore a 
right angle ; confequently, the lines FC, FG, are tangents 
of the angles CBF, GBF (CAF), to the radius BF; and, in 
the fimilar triangles FCA, FGB, FC : FG :: CA : GB 
:: CA : CE :: p + 1 : I. 
Example x. If a finall pencil of parallel red rays be in¬ 
cident upon a fphere of water, at an angle of about 59 0 
23', and fuft’er two refractions and one reflection, the rays 
will emerge parallel. 
Here p — 1; and m : n :: 108 : 81 :: 4. : 3; there¬ 
fore, 
V 27 : V 7 i or 
a — J —; and the angle 
v 27 
whofe cofine, to the radius unity 
, is J —, 
“ 27 
59 
nearly. 
The angle of refraCtion B, whofe fine is to the fine of 
59 0 23' :: 3 : 4, is 40° 12'. Hence, the whole deviation, 
2R — 4B + 2A, (Prop. III.) is 137 0 58'; which.fub- 
traCted from 180°, gives the inclination of the incident 
to the emergent pencil 42 0 2'. When violet rays are thus 
incident and emergent, m : n :: 109 : 81 ; and, in this 
cafe, A 2= 58° 40'; B 22: 39° 24'; hence, 2R—4B + 2A 
is 139 0 44', and the inclination of the emergent to the 
incident pencil, 40 0 16'. 
Example 2. If parallel red rays fall upon a fphere of 
water, they will emerge parallel, after two refraCtions and 
two intermediate reflections, when the angle of incidence 
is about 71 0 50'. 
In this cafe, p — 2 ; and, 1 : x :: 4/ 72 : |/ 7 ; there¬ 
fore the cofine of the angle of incidence is ^J —» 
which correfponds to an angle of 71 0 50' nearly. Alfo, 
B 22:45 0 27'; and the whole deviation, 4R — 6B + 2A, 
=2230° 58'; hence, the inclination of the emergent to 
the incident pencil, which is the excels of the whole de¬ 
viation above 180 0 ,222 50° 58' nearly. When violet rays 
are thus incident and emergent, A 22: 71° 26'; 62=44° 
47'; 4R — 6B -j- 2A222 234° 10'; and the inclination of 
the emergent to the incident pencil 2= 54° 10' nearly. 
Of the Formation of the Rainbow. 
It has long been known that the rainbow is owing to 
the refraftion and reflection of the fun’s light by drops of 
rain. Antonio de Dominis firft difcovered that the in¬ 
terior or primary bow is caufed by two refraCtions of 
the rays of light at each drop of water, and one reflection 
between them ; and the exterior or fecondary bow, by 
two refraCtions and two reflections between them. This 
difcovery he confirmed by experiments, which have been 
fuccefsfully repeated by more modern writers. If glafs 
globes, filled with water, be placed in the fun’s light, 
they may be elevated or depreffed till they fucceflively 
tranfmit to the eye the colours of each bow, in their pro¬ 
per order. Newton's Optics, Book I. Prop. ix. 
Maurolycus was the firft who pretended to have mea- 
fured the diameters of the two rainbows with much ex- 
aCtnefs ; and he found that of the inner bow to be 45°, 
and that of the outer bow 56°; from which Des Cartes 
takes occalion to obferve how little we can depend upon 
the obfervations of thofe who were not acquainted with 
the caufe of the phenomena. 
Clichtovteus, who died in 1543, had maintained, that 
the fecond bow is the image of the firft, which he thought 
was evident from the inverted order of the colours.. For, 
faid he, when we look into the water, all the images that 
we fee reflected by it are inverted with refpeCt to the ob¬ 
jects themfelves ; the tops of the trees, for inftance, that 
Hand near the brink, appearing lower than the roots. 
As the rainbow' was oppofite to the fun, it was natural 
to imagine, that its colours were produced by fome kind 
of reflection of the rays of light from the drops of rain. 
No perfon feems to have thought of afcribing thefe co¬ 
lours to refraCtion, till one Fletcher of Breflaw, in a trea- 
tife publifhed in 1571, endeavoured to account for them by 
means of a double refraCtion and one reflection. But he 
imagined that a ray of light, after entering a drop of rain, 
and tufferinga refraCtion both at its entrance and exit, was 
afterwards reflected from another drop, before it reached 
the eye of the lpeCtator. He feems to have overlooked 
the reflection at the pofterior furface of the drop, or to 
have imagined that all the bendings of the light within 
the drop would not make a fuflicient curvature to bring 
the rays of the fun to the eye of the fpeCtator. That he 
ftiouid think of two refraCtions, was the neceffary confe- 
quence of his fuppofing that the ray entered the drop at 
all. This fuppofition, therefore, was all that he infti- 
tuted to explain the phenomena. Baptifta Porta fup- 
pofed that the rainbow is produced by the refraCtion of 
light in the whole body of rain or vapour, but not in the 
feparate drops. 
It is to a man who had no pretenfions to philofophy, 
that we are indebted for the true explanation. This was 
Antonio de Dominis, bifhop of Spalatro, whofe treatife 
De Radiis Vifus et Lucis, was publifhed by J. Bartolus 
i6ij. He firft maintained, that the double refraCting 
of Fletcher, with an intervening reflection, was fuflicient 
to produce the colours of the bow, and alfo to bring the 
rays that formed them to the eye of the fpeCtator, without 
any fubfequent reflection. He diftinCtly defcribes the pro- 
grefs of a ray of light entering the upper part of the drop, 
where it fuffers one refraCtion, and, after being thereby 
thrown upon the back part of the inner furface, is thence 
reflected to the lower part of the drop ; at which place 
undergoing a fecond refraCtion, it is thereby bent, fo as 
to come direCtly to the eye. To verify this hypothefis, 
De Dominis proceeded in a very fenfible and philofophi- 
cal manner. He procured a fmall globe of folid glafs, and 
viewing it when it was expofed to the rays of the fun, in 
the fame manner in which he had fuppofed that the drops 
of rain were fituated with refpeCt to them, he actually ob- 
ferved the fame colours which he had feen in the true 
rainbow, and in the fame order. 
Thus the circumftances in which the colours of the 
rainbow were formed, and the progrefs of a ray of light 
through a drop of water, were clearly underftood ; but 
philolophers were a long time at a lofs when they endea¬ 
voured to aflign’ reafons for all the particular colours, and 
for the order of them. Indeed nothing but the doCtrine 
of the different refrangibility of the rays of light, could 
furnifh a complete folution of this difficulty. De Dominis 
fuppofed that the red rays were thofe which had traverf’ed 
the leaft fpace in the infide of a drop of water, and there¬ 
fore retained more of their native force, and confequently, 
ftriking the eye more brilkly, gave it a ftronger fenfation ; 
that the green and blue colours were produced by thofe 
rays the force of which had been, in fome meafure, ob- 
tunded in pafling through a greater body of water; and 
that all the intermediate colours were compofed (accord¬ 
ing to the hypothefis which generally prevailed at that 
time) of a mixture of thefe three primary ones. That 
the different colours w-ere produced by fome difference in 
the impulfe of light upon the eye, was an opinion which 
had been adopted by many perfons who had ventured to 
depart from the authority of Ariftotle. 
Afterwards 
