RAINBOW. 



K A B : B F, and (o on for any given number 

 Wherefore, fuppofiiijr the ratio of refraAion 

 Ifa.ic Newlon has determined it, viz. as Io8 

 red rays, and as 109 to 8 1 in the violet rays ; 

 I (in the lead refrangible rays) = 1 08, 11 = 

 R = 81, and R R = 6561, and V — 



x/ 3 R R = 140.3, and v^ I' - R'^ = 71-4 

 As -/ 3 R R = 140.3, whofe logarithm 

 Is to V I' _ R' =71.4 

 So is radius 



To the cofme of the angle of incidence, 7 706868 



BAF = 30^37' - -\ . ^ 



Hence the angle of incidence A B F is 59° 23' m the 



red ravs. 



Secondly, Having given the angle of incidence, and the 

 ratio nfl toR, or of refradion; tofnd the angle which a ray 

 ef light emerging out of a refratiing fphere, after a given num- 

 ber '/ refraitions, malts with the axis of vifion, or an inci- 

 dent ray ; and, confequently, tofnd the diameter of the rainbow. 

 The angle of incidence, and the ratio of refraftion, being 

 given, the angle of refraftion is given ; which angle being 

 multiplied by double the number of refleclions increafed 

 by 2, and double the angle of incidence fubtrafted from the 

 produft, the angle remauiing is the angle fought. 

 In order to find the angle of refraftion, fay 

 As I — 108, whofe logarithm is - 2.033424 



Is to R = 81 - - - 1.908485 



So io the fine of incidence 59" 23' - 9-934798 



To the fine of the angle of refraftion 40" 1 2' 9.809859 

 Then, making the angle CNF (fg- 9-) = 4°" .'2', 

 N F will be the refrafted ray, which at F is reflefted into 

 F G, and at G emerges in G R. Produce the incident and 

 emergent rays S N and R G, till they interfeft each other 

 at X, and as C F bifefts the angle N F G, it will, if pro- 

 duced, bifeft the angle S X R. Then C F N = C X N 

 + F N X ; but F N X = C N X - C N F, or C F N ; 

 therefore CFN^-CXN +CNX-CFN; that is, 

 2CFN-CNX = CXN;or8o°24'-59°23'= 21° i' 

 = C X N ; therefore 2 C X N = S X R = 42^ 2', which 

 is.the meafure of the angle that the incident and emerging 

 rays, . which are the leaft refrangible, contain with each 

 other. If inftead of the ratio 108 to 81, we take that of 

 109 to 81, we (liall find the values of a' 3 R R. and 



v'l' — R , fuch as will give the angle of incidence B C N, 

 and the arc B N ^ 58" 40', and the angle S X R = 40° 17', 

 for the molt refrangible, or extreme violet rays. 



If the ray be twice reflected, vi-z. at F and G, as in the 

 produftion of the extreme bow, and emerges at H in the 

 direftion H A, intcrfefting the incident ray S N in Y, we 

 may find the angle AYS thus : produce A H till it meets 

 G X, produced in R ; then in the triangle H G R, the ex- 

 ternal angle H G X = H R G + G H R. But becaufe 

 of equal angles of refleftion at F and G, G H R = F G X, 

 therefore HGX - FGX = HGF ^ HRG = 

 3CGF, orCNF. ButSXR = 4CNF-2CNX; 

 therefore in the triangle Y X R we have the two internal 

 angles R + X = 6CNF- 2CNX--= the external 

 angle at Y, viz, A Y N. In this cafe, to find the angles 



of incidence and refraftion, we have \'' 8 R R : "v I — R' 

 :: radius : cofine of the angle of incidence; whence the 

 faid angle will be found 71° 50' = C N X. And as 108 : 

 gi :: tir.e of 71050' : fine of 45° 27' = C N F, the angle 



JO 



ofrefleftions. of refraftion ; therefore 6 x 45^27' — 2 x 71° 50' -::i 



to be, as fir 129° 3' = A Y N ; and, th»;refore, its complement AYS 



to 81 in the _; jq'"' 57', the angle required for the leaft refrangible rays. 



we (hall have gut for the moft refrangible rays, where I : R :: 109 : 81, 



- 11664; and we have the angle of incidence 71° 26', and the angle of 



R' = 5103, refraftion 44° 47'; and, therefore, the angle AYS - 



♦V, f i-n ?4° 7'- I" the fame manner the fame angles are calculated 



; inereiore, ^'t ' , - n :,■ 



alter three or tour retleCtions. 



2.147045 From the preceding problems we obtain the following re- 



1. 853913 fults, viz. 



10.000000 T D„:..K \ Ked 42" 



40 



50 



54 

 If the angle made by a ray, after three or four refleftioni, 

 were required, and therefore the diameters of the third and 

 fourth rainbow ( which are fcarcely ever feen, on account of 

 the great diminution of the rays, by fo many repeated re- 

 fleftions) they will be found. 



I. Rainbow, 



II. RainboW; 



f Red 



I Violet 

 y Red 

 I Violet 



2 

 17 

 57 



]\ 



The fpeftator's 

 back being turned 

 to the fun. 



LII. 



Rainbow, J vf^,^^ 



IV. Rainbow, J y'^jg^ 



The fpeftator 

 being turned to- 

 wards the fun. 



Dr. Morgan's Differtation upon the Rainbow, among 

 the notes upon Rohault's Syfteni of Philofophy, part iii. 

 chap. xvii. 



Hence, the breadth of the rainbows is eafily found : for 

 the greatelt femi-diameter of the firil bow, i. e. from red to 

 red, being 42° 2', and the leaft, viz. from violet to violet, 

 40^ 17', the breadth of the fafcia or bow, meafured acrofs 

 from red to violet, will be l°45' ; and the greatell diameter 

 of the fecond bow being 54" 7', and the leaft 50° 57', the 

 breadth of the fafcia will be 3= 10'. And hence the diftance 

 between the two will be found 8° 55'. 



In thefe mJafures the fun is only efteemed a point; where- 

 fore, as his diameter is really about 30', or 32', fo much 

 muft be added to the breadth of each fafcia or bow, from 

 red to violet, and fo much mull be fubtrafted from the 

 diftance betvreen them. 



This will leave the breadth of the primary bow, 2° 15' ; 

 that of the fecondary bow, 3° 40' ; and the interval be- 

 tween the bows, 8^ 25' ; which dimenfions, deduced by cal- 

 culation, fir Ifaac Newton ail'ures us from his own obierva- 

 tions, agree very exaftly with thofe found by aftual men- 

 furation in the heavens. Optics, p. 153. ed. 3. 



Rainbow, Particular Phenomena of the. From this 

 theory of the rainbow, all the particular phenomena of it 

 are eafily deduced : hence we fee why the iris is always of 

 the fame breadth ; becaufe the intermediate degrees of refran- 

 gibility of the rays between red and violet, which are its 

 extreme colours, are always the fame. See Apparent Mag- 

 nitude, 



Secondly, Why it is more diftinftly terminated on the 

 fide of the red than on that of the violet ? there being nc 

 efficacious rays in the fpace adjoining to the red drops, i. e. 

 to the fpace between the bows, whence it terminates ab- 

 ruptly ; whereas, in the fpace on the fide of the violet ones,, 

 there are fome rays emitted to the eye, which, though too 

 feeble to affeft it ftrongly, yet have this effeft, that they 

 foften the violet edge infenfibly, fo that it is difficult to de- 

 termine precifely where it terminates. 



Thirdly, Why the bow (hifts its fituation as the eye does; 

 and, as the popular phrafe has \\., jlies from thofe who follow 

 it; and follows' thofe that fy from if:' the coloured drops being 

 difpofed under a certain angle about the axis of vifion, which 



