536 
OPTICS. 
Cor. i. Since the altitude of an arc of any colour in the 
how is equal to the radius of this arc diminifhed by the 
fun's altitude, when the fun is in the horizon, the alti¬ 
tude of the arc is equal to its radius. 
Cor. 2. The radius of any arc in the rainbow is equal to 
the altitude of the arc above the horizon, together with 
the fun’s altitude. 
Cor. 3. When the fun’s altitude above the horizon is 
equal to, or exceeds, 42 0 2', the primary bow cannot be 
feen ; nor the fecondary, when his altitude is equal to, or 
exceeds, 54° 10'. 
Prop. VI. Having given the radius of an arc of any co¬ 
lour in the primary rainbow, to find the ratio of the 
fine of incidence to the fine of refradtion when rays of 
that colour pals out of air into water. 
If A be the angle of incidence of the effedtive rays, B 
the angle of refraction, the radius of the arc is 4B—2A, 
(Prop. III.) Let the tangent of 2B—A, half this angle, to 
the radius unity, be a ; z the tangent of B. Then 22= 
tang. A, (Prop. IV.) All’o, from the principles of trigono¬ 
metry, tang. 2B : 2X tang. B(2 2-) :: i 2 : i 2 — z 2 -, therefore 
tang. 2B — ~~: 2 - Again, tang. 2B—A (a) : tang. 2B— 
tang. A -2 2) :: I 2 : hence, 
A. Cl Z z • 
— zz~a-\-- —— ; and by redudhon, 22 s —3«s 2 — a — o. 
1— z 2 
The value of z being obtained from this equation, the 
angles B and A, and confequently their fines, may be 
found from the tables'. _ . 
Cor. In the fame manner, if p be the number of reflec¬ 
tions within the drop, 2 the tangent of B, Q the tange nt of 
p-j-i . B, a the tangen t of p-]-i . B—A, th en p-f i . 2— 
tang. A; and a : Q —p+i . z i 2 : i 2 4 y>+i , Qr; 
therefore Q—p + i ■ z=a+p+i . aQz. From which 
equation, the value of 2 being found, the angles A and B, 
and confequently their fines, may be determined by the 
Tables. 
Of CAUSTICS. 
A cauftic curve is a curve formed by the concourfe, or 
coincidence, of the rays of light refledted or refradted 
from fome other curve. 
Every curve has its twofold cauftic ; accordingly, Cau- 
flics are divided into Catacaujiics and Diacaujiics; the one 
formed by refledlion, the angle of refledtion being equal to 
that of incidence, the other by refradtion. 
M. Bouguer obferves, that there are two cauftics formed 
at the fame time by convex and concave furfaces ; and 
that they occafion two different images of objedts feen by 
refledtion from them. 
Cauftic curves have this remarkable property, that, 
when the curves that produce them are geometrical, they 
are equal to known right lines. Thus, the cauftic formed 
by refledted rays from a quadrant of a circle, which 
came at firft parallel to the diameter, is equal to three- 
fourths of the diameter; which is a fort of redtification of 
curves that preceded the invention of the new' dodtrine of 
infinites, on which moft of our rectifications are built. 
Cauftic curves are ufually luppofed to be the invention 
of M.Tfchirnhaufen ; but it is only the name he invented. 
The firft mention he made of them was in the year 1682, 
when he produced no inftance but that of the cauftic in a 
circle, which he might have learned from Dr. Barrow’s 
Ledtiones Opticse, publifhed in 1669. It would have been 
eafy for him to have done the fame for any curve, by the 
help of the radius of curvature publifhed by Huygens in 
his Horologium Ofcillatorium, in 1673. It is certain 
this had been done by fir Iiaac Newton as early as the year 
x66y, as appears from his Ledfiones Opticas, which were 
read that year at Cambridge, though not published till 
after his death, viz. in 1728. 
Prop. I. When a fmall pencil of diverging or converg¬ 
ing rays is incident obliquely upon afpherical refledfor, 
in a plane which palfes through its centre, to find the 
geometrical focus of refledted rays. 
Let BC (fig. 7, 8.) be a fpherical refledfor whofe centre 
is E; QA, QB, two rays of a fmall pencil incident ob¬ 
liquely upon it, on the plane QAE; AG, Bg, the refledted 
rays, or thole rays produced backwards ; q, their inter- 
fedtion. From E draw EDrl, EGg, at right angles to 
QA, AG ; and, when the arc AB is diminifhed without 
limit, they are alfo at right angles to QB, Bg. Join EA, 
AB ; from A draw A b, A a, at right angles to QB, q B, 
produced if neceffary; bifedt AD, AG, in F ,f. Then, 
the angles EAD, EAG, are the angles of incidence and 
refledtion of the ray QA, or equal to them ; therefore they 
are equal to each other; the angles EDA, EGA, are right 
angles; and the fide EA is common to the two triangles 
EAD, EAG ; confequently, ADzzzAG, and EDznEG. 
In the fame manner, Ed=Eg; whence, DcfczGg. Again, 
in the evanefcent triangles AB«, ABi>, the angles AB£>, 
AB«, are complements to the angles of incidence and re¬ 
fledtion of the ray QB, or equal to thofe complements ; 
therefore they are equal to each other; alfo, the angles 
A&B, AnB, are right angles ; and AB is common to the 
two triangles ; confequently, Ab—Aa. 
Now, in the fimilar triangles QD</, QA b, QA : QD :: 
A b : D d :: A a : Gg; and in the fimilar triangles qAa, 
qGg, Acr : Gg :: A q : qG~, therefore, QA : QD :: Aq 
: qG ; whence, by compofition, and divifion, QA + QD : 
QA^QD :: Aq-\-qG ; Aq^-qG ; that is, 2QF : aFA :: 
zAf : Af-ffq — 'Aj—fq (2 qf) j or, QF : FA :: Af : fq. 
This conclufion depends upon the fuppolition that, wdien 
QA and QD are meafured in the fame diredtion from Q, 
#A and qG are meafured in oppofite diredlions from q. If 
this be not the. cafe, Aq\qG—zqf; and Aq-^qG—?fAzz. 
2FA; therefore zqfzzzztfF, and qfzzzQF. Now let the 
rays be incident nearly perpendicularly upon the refledfor, 
and F and/coincide with the principal focus ; therefore 
Q and q are always equally diftant from the principal 
focus, which is ablurd. 
Cor. 1. In the cafe reprefented by fig. 7, QF and FA 
are meafured in oppofite diredlions from F; and fince 
QA : QD :: Aq : qG, and QA is greater than QD, Aq 
is greater than qG ; and therefore Af and fq are mea- 
fured in oppofite diredtions from f; hence it follows, that 
the equal rectangles QFxfq and FAX A/ have, in this 
cafe, the fame fign ; therefore they will always have the 
fame fign ; that is, whenever QF and FA are meafured in 
oppofite diredlions from F, Af and fq are meafured in 
oppofite diredtions from/; and the contrary. 
Cor. 2. When the incident rays are parallel, FA is eva- - 
nefcent with refpedl to QF; therefore fq is evanefcent 
with refpedl to Af; or, q coincides with/! Here, AQ= 
^AG=|AD. 
Cor. 3. If D be the focus of incident rays, G will be the 
focus of refledted rays. In this cafe, QF=FA; there¬ 
fore Af—fq; and, fince QF and FA are meafured in op¬ 
pofite directions from F, A/'and/? mult be meafured in 
oppofite diredtions from /; confequently, q coincides 
with G. 
Cor. 4. If Q be a point in the circumference of the cir¬ 
cle BC, FA=fQF; therefore fq—% Ay—fjQA; hence, 
A^eatJQA-f-^Q-A—i'QA- _ t 
Cor. 5. The fame propofitions are true of any other re¬ 
flecting curve, 1 if E be the centre of curvature of the 
evanefcent arc AB. 
If an indefinite number of fmall pencils belonging to 
the focus Q, be incident, in the fame manner, upon the 
reflecting furface BC, the curve which is the focus of the 
geometrical foci of reflected rays, is called the cauftic hy 
reflection. 
Prop. II. To determine the form of the cauftic, when 
the refledling curve is a circular arc, and parallel rays 
are incident in the plane of the circle. 
x Let 
