OPT 
Afterwards the fame De Dominis obferved, that all the 
rays of the fame colour mult leave the drop of water in a 
part fimilarly lituated with refpeft to the eye, in order 
that each of the colours may appear in a circle, thecentre 
of which is a point of the heavens in a line drawn from 
the fun through the eye of the fpedtator. The red rays, 
he obferved, mud iffue from the drop neareft to the bot¬ 
tom of it, in order that the circle of red may be the outer- 
moft, and the moll elevated in the bow. 
Though De Dominis conceived fo juftly the manner in 
which the inner rainbow is formed, he was far from having 
as juft an idea of the caufe of the exterior bow. This he 
endeavoured to explain in the very fame manner as the 
interior, viz. by one refledtion of the light within the 
drop, preceded and followed by a refradtion; fuppoling 
only that the rays which formed the exterior bow were 
returned to the eye by a part of the drop lower than that 
which tranfmitted the red of the interior bow. He alfo 
fuppofed that the rays which formed one of the bows came 
from the upper limb of the fun, and thofe which formed 
the others from the lower limb, without confidering that 
the bows ought thus to have been contiguous ; or rather, 
that an indefinite number of bows would have had their 
colours all intermixed. 
When fir Ifaac Newton difcovered the different refran- 
gibility of the rays of light, he immediately applied the 
difcovery to the phenomena of the rainbow, taking up 
the fubjedl where De Dominis and DesCartes were obliged 
to leave their inveltigations imperfedt. 
To underftand how the interior bow is formed, let O 
(fig. 5,) be the eye of a fpedlator ; SOP a line palling 
through the eye and the fun. At the point O, in the 
line PO, make the angle POE=42° 2'; then, when a drop 
of rain, FE, is in fuch a fituation that the angle which 
OE makes with a perpendicular to its furface at E is 59 0 
23', a fmall pencil of parallel red rays will emerge from 
it at E, and enter the eye in the diredlion EO. For, if 
EO be cofidered as the incident pencil, it will emerge, 
after two refradtions and one refledtion, in the diredtion 
FS, which makes an angle of 137 0 58' with OE produced, 
(Prop. IV. Ex. 1.) or an angle of 52 0 2' with OE, and is 
therefore parallel to OS ; thus FS will pafs through the 
fun. Converfely, out of the beam of light proceeding 
from S, which falls upon the drop, the red rays incident 
at, and near to, F, will, after two refradtions and one re¬ 
fledtion, emerge parallel, and, entering the eye in the di¬ 
redtion EO, will excite the fenfation of their proper co¬ 
lour. The diflance of the fun is fo great, that two lines, 
drawn from any points upon the furface of the earth to 
a point in his difc, may be confidered as parallel, in thefe 
calculations. 
In the fame manner, if OE revolve about the axis OP, 
every drop of water in the furface of the cone thus de- 
fcribed, will tranfmit to the eye a fmall parallel pencil of 
red rays; and thus a red arc, whofe radius, meafured by 
the angle which it fubtends at the eye , is 42 0 2' will appear 
in the falling rain, oppofite to the fun. The other red 
rays of the beam which falls upon the drop FE, will, at 
their emergence, be inclined at different angles to the di¬ 
redtion of the incident rays, and be fo much difperfed 
before they reach the eye, and enter it in fo weak a 
ftate, mixed with other rays, as to produce no diftindt 
effedt. 
The parallel pencils of red rays, which emerge from 
other drops, fall above or below the eye. If the angle 
POD be4o° 16', and OD revolve about the axis OP, every 
drop of rain in the furface of the cone thus defcribed, will 
tranfmit to the eye a parallel pencil of violet rays ; and 
thus a violet arc will be formed, whofe radius is 40 0 16'. 
The drops between E and D will tranfmit to the eye pa¬ 
rallel pencils of rays of different colours, orange, yellow, 
green, blue, indigo, in the order which they have in the 
prifmatic fpedtrum; and the radii of the arcs of thefe re- 
fpedtive colours may be calculated by the method em¬ 
ployed in Prop. IV. Ex. 1. 
i c s. rm 
Again, let the angle POI (fig. 6.) — 50 0 58'; and the 
angle POL;=54 0 10'. Alfo, let OI, OL, revolve about 
the axis OP. Then, it may be fhown as in the preceding 
cafe, that every drop of rain in the conical furface gene¬ 
rated by OI will tranfmit to the eye a fmall parallel pen¬ 
cil of red rays, which has-buffered two refradtions and two 
refledtions, but fufficiently ftrong to excite the fenfation 
of its proper colour. Alfo, every drop in the conical fur¬ 
face generated by OL will tranfmit to the eye a fmall pencil 
of parallel violet rays ; and the intermediate drops, pa¬ 
rallel pencils of rays of the intermediate colours. Thus 
the exterior bow is formed, in which the radii of the red 
and violet arcs are, refpedti vely, 50 0 58', and 54 0 10'. The 
radii of the intermediate arcs may be determined by the 
method employed in Prop. IV. Ex. 2. 
Cur. 1. The colours in the two bows lie in a contrary 
order ; the. red forming the exterior ring of the primary, 
and the interior ring of the fecondary, bow. 
Cor. 2. Were the pencils fufficiently ftrong, a third 
bow, formed by two refradtions and tiiree refledtions of 
the fun’s rays in drops of rain, might be feen. But, 
when the rays which are refradted into a drop of water 
reach the farther furface, fome of them pafs out of the 
drop, and others are refledted within it. When thefe re- 
fiedted rays again meet the furface, fome of them pafs 
out of the drop, and others fuller another refledtion, and 
fo on. Thus the pencil becomes weaker at every refledtion; 
and at length it contains fo few rays as not tomakea diftindt 
impreftion upon the retina. This is a fadt, the caufe of 
which has not been fatisfaclorily explained. Sir Ifaac 
Newton fuppofes that rays of light, when they arrive 
at the furface of a medium, are fometimes in a ftate to be 
refledted, and fometimes to be tranfmitted; thefe ftates he 
calls fits of ealy refledtion and tranfmiffion ; and accounts 
for them in the following manner; “ Nothing more is 
requifite for putting the rays of light into fits of eafy 
refledtion and eafy tranfmiflion, than that they be fmall 
bodies, which, by their attradtive powers, or fome other 
force, ftir up vibrations in what they adt upon ; which vi¬ 
brations, being fwifter than the rays, overtake them fuc- 
ceflively, and agitate them, fo as by turns to increafe and 
diminilh their velocities, and thereby put them into thofe 
fits.” Opt. Query 29. 
Prob. V. To find the altitude of the higheft point of 
the rainbow above the horizon, and the breadth of the 
colours. 
The conftrudtion being made as at fig. 5. through O 
draw HOR parallel to the horizon. Then the angle ROS, 
or HOP, meafures the altitude of the point S above the 
horizon ; and the altitude of the higheft point of the red 
arc above the horizon, in the primary bow, is meafured 
by EOH, or EOP—HOP, which is equal to 42 0 2'—HOP. 
Alfo, the altitude of the higheft point of the violet arc is 
meafured by DOP—HOP, or 40 0 16'—HOP. Hence it 
follows, that the breadth of the bow, fuppoling it to be 
formed by the rays which come from one point S, in the 
fun’s difc, is 42 0 z‘ —40 0 16', or i° 46'. The breadth, 
thus determined, mull be increafed by 30', the fun’s ap¬ 
parent diameter; for, the higheft red arc is produced by 
the rays which flow from the loweft point in the fun’s 
difc, and if ROS, or HOP, meafure the altitude of the 
fun’s centre, the altitude of the higheft red. arc is 42 0 2' 
—HOP+13'; alfo, the lowelt violet arc is produced by 
the rays which flow from the higheft point in the fun’s 
difc, and therefore the altitude of this arc is 40 0 16'— 
HOP—15'; confequently, the breadth of the bow is i° 
46'-J~3 o', or 2 0 16'. 
In the fame manner it appears, that the altitude of the 
violet arc, in the exterior bow, is 54 0 10'—SOR, (fig. 6.) 
and the altitude of the red arc, 50 0 58'—SOR ; therefore, 
the breadth of the bow, formed by rays which proceed 
from any one point in the fun’s difc, is 3 0 12'. If to this 
we add 30', the fun’s apparent diameter, we have the ac¬ 
tual breadth of the exterior bow — 3 0 42'. 
Cor. 
