RAINBOW. 



magnitude. Newton's Optics, part ii. prop. 9. prob. 4. 



p. i47> ed. 3. 



Rainbow, Dimenfum of the. — Defcartcs firft determined 

 its diameter by a ttntnlive and indirett method ; laying it 

 down, that the magnitiulL- of the bow depends on the degree 

 of refraftion of the fluid ; and aduming the ratio of the fine 

 of incidence to tiiat of refratlicii to be in water as 250 to 



'87- 



But Dr. Halley has fince, in the Philofophical Tranfac- 



tions, N"^ 267, given us a fimple direft method of deter- 

 mining the diameter of the rainbow from the ratio of re- 

 fraftion of the fluid being given ; or, 'u'lce "verfi, the dia- 

 meter of tiie rainbow being given, to determine the refrac- 

 tive power ot the fluid. 



The principles of Dr. Halley's confirmation for this pur- 

 pofe, illullrated and tacihtated by Dr. Morgan, bifhop of 

 Ely, will be underllood from the following view of them. 

 Let SN, sn, {Piatt XVIII. Optics, f^\^. 5.) be two of the 

 efficacious rays incident upon a drop ol rain ; thefe, when 

 refrafted to the (ame point F, and thence reflefted to 

 G, g, will have the parts within the drop on one fide N F, 

 n F, equal to thofe on the other fide FG, F^, from the 

 nature of the circle, and becaufe the angles of incidence 

 CFN, CF«, are equal to the angles of refleftion CFG, 

 CF^. And fince the parti within the drop are equal and 

 alike fituated, they will be fimilarly fituated with regard 

 to the drop itfelf ; and, confequently, as the incident rays 

 SN, S n, arc fuppofed to be parallel, the emergent rays 

 G R, % ry vvill be aHo parallel. From C, the centre, draw 

 the radii CN, C«, C F, then will C N F -= C FN be the 

 angle of refraction, and the fmall are N;; is the nafcent in- 

 crement of the angle of incidence B C N ; and as it meafures 

 the angle at the centre NCn, it is double of the angle at the 

 circumference upon the fame arc, v'fz. NF«, which is the 

 nafcent increment of the angle of refraftion NFC. Far- 

 ther, let the ray S N [fig. 6. ) enter the lower part of the 

 drop, and be twice reflefted within the drop at F and G ; 

 then is the ray N F equal to the ray F G, and the arc N F 

 = the are FG. Draw fg parallel to FG, and it will be 

 the refletled part of fome ray s n, whofe obliquity to the 

 drop makes it crofs the ray NF in its refradlion ; tlien will 

 the part nf = fg, and the arc nf = fg, and the fmall arc 

 F/ - Gg. Therefore, 2 F/ = (F/ + Gg ^ the arc 

 FG — fg = NF — "f — ) Nn'— F/; confequently 

 N « = 3 F/, i. c. the nafcent increment of the angle of 

 incidence is equal to three times that of the angle of re- 

 fraftion. After a like manner it may be fhewn, that after 

 three, four, five, &c. refleftions, the increment of the 

 angle of incidence will be four, five, fix, &c. times greater 

 than that of the angle of refrattion. Hence, in order to 

 find the angle of incidence of an efficacious ray, after any 

 given number of refleftions, we are to find an angle vvhofe 

 nafcent increment has the fame ratio to the increment of 

 its correfponding angle of refraftion, generated in the fame 

 time, as the given number of refleftions (n) increafed by 

 unity has to unity ; i. f. as n + I to I. But thele incre- 

 ments are as the tangents of the refpeftive angles direftly. 

 For, let A CD, ABD (fg. 7.) be the angles of inci- 

 dence and refraftion propofcd ; and if we fuppofe the line 

 AC to move about the point A in the plane of thofe angles, 

 the extremity of it, C, will defcribe the circular arc C c ; and 

 when AC is arrived at the fituatiou A c, the line BD will be 

 thereby removed into the fituation B (/. Draw r D, then is 

 the angle ACD = ABC ^ CAB, and the angle A <: ^ = 

 A B f -i- c A B ; therefore the excefs of A r </ above A C D, 

 or the increment of A C D, is equal to both the angles C B c 

 and C A :. But fince the angle A c C differs infinitely little 



from a right one, a circle defcribed on the diameter A C 

 fhall pafs through the points D and c ; and, therefore, the 

 angles C A f, C I) c, (infilling on the fame arc Cfofthe 

 faid circle) will be equal ; and, therefore, the increment of 

 the angle ACD is .equal to CBc -f- CDf = D c d. 

 But the nafcent angles \) c d and D B c are as their fines, 

 that is, as their oppofitc fides B D and D c -- D C, tlie 

 angle CD r being infinitely fmall ; but BD : CD :: DE 

 : Da (the line BE being parallel to AC) :: tangent of 

 the angle EBD — ACD : tangent of the angle ABD. 

 Therefore the increment D c</ of the angle ACD is to tiie 

 increment C B c of the angle ABD (generated in the fame 

 time) as the tangent of the former to the tangent of the 

 latter dire£tly. Hence the praxis is as follows : 



Firit, The ratio of the fine of incidence I, to the fine of re- 

 fradion R, being given; to find the angles of incidence, and 

 refradion of a ray, which becomes effectual after any given 

 number («) of nfidions. — Siippofe any given line, as A C 

 (fg. 8.) which divide in D ; ib as that AC : AD :: I : 

 R ; and again divide it in E, fo that AC may be to AE 

 as the given number of refleftions, increafed by unity, is to 

 unity; /. e. as « + 1 : i. Upon the diameter CE de- 

 fcribe a femicircle C B E, and from the centre A, v.ith the 

 radius A D, defcribe an arc D B interfecting the femicircle 

 ill B ; then drawing A B, C B, and letting fall the perpen- 

 dicular A F on C B produced ; A B C, or its complement 

 to two riglit angles, A B F, will be the angle of incidence ; 

 and A C B the angle of refrattioii required. For, draw- 

 ing B E parallel to A F, the triangles A C F and E C B 

 are fimilar. Now, the fine of the angle ABC, or A B F, 

 is to the fine of A C B as A C to A B = AD, /'. e. as I 

 to R ; therefore if A B F be the angle of incidence, A C F 

 will be the angle of refraction. Moreover, the nafcent in- 

 crement of A 15 F is to that of A C B (generated in the 

 fame time) as C F : B F :: C A : A E, on account of the 

 fimilar triagles ; i.e. as n + I to I, by conft;ruftion. The 

 ratio, therefore, of the nafcent increment of the angle of 

 incidence A B F to that of the angle of rcfraftion A C B, 

 is that ft'hich is required in the angles of incidence and re- 

 fraftion of an efficacious rav, after a given number of re- 

 flexions ; confequently the angles A B F and A C F are 

 thofe required. From this confl;ruftion we may eafily de- 

 duce fir Ifaac Newton's rule for finding the angle of inci- 

 dence A B F. See his Optics, p. 148, 149. 



For A C : A B :: I : R; whence A C = -=- x A B; 



and C F : B F :: n + 1 : I ; therefore C F = « + I x 

 B F ; or (putting n + i = »i)CF^»j x BF; and, on 

 account of the right angle at F, A C — C F" = A B" — 



B F' ; and, 



confequently, 



B F 



/ 1 1 - R R „ 



_ — \ / ;^^ i, • Hence, becaule 



A B V m' R' - R^ 



in the firft bow, the ray emerges after one refleftion, we have 

 n = I, m = 2, ni' = 4, and w" — 1=3; therefore, 



v^T^R R : v'll - RR :: A B : B F :: radius : cofine 

 of the angle of incidence. In the fecond bow, where there 



are two reflexions, ni' — 1 = 8; whence -v/ 8 R R : 



s' II _ R R :: A B : B F. In the third bow, after three 



refleatons, m' - i = 15 ; and \' 15 R R : \'' II — R R 



:: AB 



