568 
OPT 
If we fuppofe ABZY not to be a vacuum, but a rarer 
medium than the other, the cafe will ftill be the fame ; 
but the ray will not be fo much redrafted from its rectili¬ 
neal courfe, becaufe, the attraction of the particles of the 
upper medium being in a contrary direction to that of the 
attraction of thofe in the lower one, the attraction of the 
denfer medium will in fome meafure be deftroyed by that 
of the rarer. When a ray, on the contrary, palfes out of 
a denfer into a rarer medium, if its direction be perpen¬ 
dicular to the furface of the medium, it will only lofe 
fomevvhat of its velocity in palling through the fpaces of 
attraction of that medium (that is, the fpace wherein it 
is attracted more one way than it is another). If its di¬ 
rection be oblique, it will continually recede from the 
perpendicular during its paffage, and by that means have 
its obliquity increafed, juft as aftone thrown up obliquely 
from the furface of the earth increafes its obliquity all 
the time it rifes. Thus, fuppoling the ray TS palling out 
of the denfer medium ABCD into the rarer ABZY, when 
it arrives at S it will begin to be attrafled downwards, 
and fo will defcribe the curve SNL, and then proceed in 
the right line LK ; making a larger angle with the per¬ 
pendicular PR than the line TSX, in which it proceeded 
during its paffage through the other medium. 
We may here make a general obfervation on the forces 
which produce this deviation of the rays of light from 
their original path. They arife from the joint aCtion of 
all the particles of the body which are fufficien'tiy near 
the particle of light ; that is, whofe diftance from it is 
not greater than the line AE or GA; and, therefore, the 
whole force which aCfs on a particle, in its different fixa¬ 
tions between the planes GH and EF, follows a very dif¬ 
ferent law from the force exerted by one particle of the 
medium. 
The fpace through which the attraction of cohefion of 
the particles of matter is extended is fo very final), that, 
in confidering the progrefs of a ray of light out of one 
medium into another, the curvature it defcribes in pafling 
through the fpace of attraction is generally negleCted ; 
'and its path is fuppofed to be bent, or refraCted, only in 
the point where it enters the denfer medium. 
Mow the line which a ray defcribes before it enters a 
denfer or a rarer medium, is called the incident ray; and 
that which it defcribes after it has entered, is the re/ratted 
ray. The angle comprehended between the incident ray 
and the perpendicular, is the angle of incidence ; and that 
between the refraCted ray and the perpendicular, is the 
ang'lc of refraClion. 
There is a certain and immutable law, by which refrac¬ 
tion is always performed ; which is this : V/hatever in¬ 
clination a ray of light has to the furface of any medium 
before it enters it, the degree of refraCtion will always be 
fuch, that the fine of the angle of incidence, and that of 
the angle of refraCtion, will always have a conftant ratio 
to one another in that medium. 
To illuftrate this: Let us fuppofe ABCD, fig. z, to re- 
prefent a rarer, and ABEF a denfer, medium : let GH be 
a ray of light palling through the firft, and entering the 
fecond at H, and let HI be the refraCted ray ; then, l'up- 
pofing the perpendicular PR drawn through the point H, 
on the centre H, and with any radius, defcribe the circle 
ARPB ; and from G and I, where the incident and re¬ 
drafted rays cut the circle, let fall the lines GK and IL 
perpendicularly upon the line PR ; the former of thefe 
will be the fine of the angle of incidence, the latter of re¬ 
fraCtion. Now, if in this cafe the ray GH is fo refraCted 
at H, that GK is double or triple, See. of IL, then, what¬ 
ever other inclination the ray GH might have had, the 
fine of its angle of incidence would have been double or 
triple, See. to that of its angle of refraCtion. For inftance, 
had the ray palled in the line MH before refraCtion, it 
would have palled in fome line as HN afterwards, fo 
fituated, that MO lliould have been double or triple, Sec. 
of NQ. 
ICS. 
The following Table contains the refraCtive denfities 
of feveral bodies: 
Diamond - 2-500 Olive-oil - 1-469 
Flint-glafs - 1-585 Alcohol - - r 37 o 
Plate-glafs - - 1-502 Atmofpheric air 1000276 
Crown-glafs - 1525 Ice - - 1-31' 
Sulphuric acid 1-435 Water - - 1-336 
Solution of potafli 1-390 
This relation of the fine of the angle of incidence to 
that of refraCtion, which is a propolition of the molt ex- 
tenfive ufe in explaining the optical phenomena on phy- 
fical or mechanical principles, may be demonftrated in the 
following eafy and familiar manner. 
Lemma I. The augmentations or diminutions of the 
fquares of the velocities produced by the uniform ac¬ 
tion of accelerating or retarding forces, are proportional 
to. the forces, and to the fpaces along which they aft, 
jointly; or are proportional to the produftsof the forces 
multiplied by the fpaces. 
Let two bodies be uniformly accelerated from a ftate of 
reft in the points A a, along the fpaces AB, ub, fig. 3, 
by the accelerating forces F f; and let AC, ac, be fpaces 
deferibed in equal times : it is evident that, becaufe thefe 
fpaces are deferibed with motions uniformly accelerated, 
AC and ac are refpeftively the halves of the fpaces which 
would be uniformly deferibed during the fame time with 
the velocities acquired at C and c, and are therefore mea¬ 
sures of thefe velocities. And, as thefe velocities are 
uniformly acquired in equal times, they are meafures of 
the accelerating forces. Therefore, A C : ac=z F : f 
Alfo, from the nature of uniformly-accelerated motion, 
the fpaces are proportional to the fquares of the acquired 
velocities. Therefore, (ufing the fymbols V 2 C, i/ 2 e,- 
&c. to exprefs the fquares of the velocities at Cc, Sec.) we 
have 
l/ 2 B : V ' 2 C= 4 B : AC 
V 2 C : \/ 2 c= AC 2 : ac 2 
V 2 c : f 2 b~ac : ah 
Therefore, by equality of compound ratios 
: t/ 2 6=ABxAC : abxac, =z ABxF : al X f. 
And in like manner i/ 2 D : \/ 2( W ADxF : ady,f- 
and \/ 2 B — f 2 D : \/ 2 b —v/ 2 rA=BDxF : bdxf-Q.B.D. 
Corollary. If the forces are as the fpaces inverfely, the 
augmentations or diminutions of the fquares of the velo¬ 
cities are equal. If DB, db, be taken extremely fmall, the 
produfts BDxF and bdxf may be called the momentary 
aftions of the forces, or the momentary increments of the 
fquares of the velocities. It is ufually exprefled, by the 
writers on the higher mechanics, by the fyinbol/^, or fds, 
where f means the accelerating force, and s or ds means 
the indefinitely fmall fpace along which it is uniformly 
exerted. And the propofition is exprelfed by the flux¬ 
ionary equation fs=vu, becaufe vv is half the increment 
of v 2 , as is well known. 
Lemma II. If a particle of matter, moving with any ve¬ 
locity along the line AC, fig. 4, be impelled by an ac¬ 
celerating or retarding force, afting in the fame or in 
the oppofite direftion, and if the intenfity of the force 
in the different points B, F, H, C, Sec. be as the ordi¬ 
nates BD, FG, Sec. to the line DGE, the areas BFGD, 
BHKD, Sec. will be as the changes made on the fquare 
of the velocity at B, when the particle arrives at the 
points F, H, Sec. 
For, let BC be divided into innumerable fmall portions, 
of which let FH be one, and let the force be fuppofed to 
aft uniformly, or to be of invariable intenfity during the 
motion along FH ; draw GI perpendicular HK : it is 
evident that the rectangle FHIG will be as theproduftof 
the accelerating force by the fpace along which it afts, 
and will therefore exprefs the momentary increment of 
1 the 
