528 PHILOSOPHICAL TRANSACTIONS. [aNNO 1700. 



the point opposite to the sun, when the number of reflections is odd. But if 

 that number be even, the double of that greatest angle is the distance of the 

 iris from the sun itself. 



Now that those greatest excesses may be had, the refraction of any liquid 

 being given, or the ratio of the sine of incidence to the sine of the refracted 

 angle ; we must take notice that the excess of two refracted angles above one 

 angle of incidence, is then greatest, when the momentary augment of the angle 

 of incidence is exactly double to the momentary augment of the refracted angle. 

 But, of three refracted angles, the excess is then greatest, when the momentary 

 augment of the angle of incidence is triple to the moment of the refracted 

 angle ; and so of the rest. And this is manifest of itself. Now we shall ob- 

 tain the angles themselves by premising the following lemma, which we must 

 demonstrate. 



Lemma. — The legs of any plain triangle continuing the same, if the vertical 

 angle be increased or diminished by any angle less than any given one, the 

 moments or instantaneous mutations of the angles at the base will be to one 

 another reciprocally as the segments of the base. 



Let ABC (fig. 17) pi- 12) be a triangle, whose vertex is a, the legs ab and ac; 

 and the base bc ; upon which let fall the perpendicular ad. Then let the angle 

 BAC be increased by any indivisible moment cac, and draw the lines scf/ and cd 

 which will differ from the lines bod and cd only intellectually. I say the mo- 

 ment of the angle abc, that is cbc, is to the moment of the angle acb, or acd, 

 as CD to bd, that is reciprocally as the segments of the base. For as the angle 

 ACD is the sum of the angles abc and bac, its moment will also be the sum of 

 the moments of those angles, or cac -f cbc. But cac is equal to the angle 

 CDc; for because of the right angle at d, the points a, d, c, c, are at the cir- 

 cumference of a circle whose diameter is ac, by Eucl. 3, g. And therefore the 

 sum of the angles cbc and cdc, that is the angle ncd, will be the moment of 

 the angle acd, or acb. But the angles cbc, and V)cd, being indefinitely little, 

 are to one another as their opposite sides, or as cd or cd to bd, that is, reci- 

 procally as the segments of the base. q. e. d. 



Now if either of the angles b or c be acute, changing what is to be changed 

 the lemma will be demonstrated as above. 



Carol. — Hence it follows, that the moments of the angles at the base are to 

 one another directly as the tangents of those angles. 



By the help of this lemma we may easily obtain the diameter of any rainbow, 

 either by a geometrical construction or a calculation. For assuming any right 

 line CA i,fig. 18), first let it be divided in d, so that ca may be to cd in the ratio 

 of refraction, which in water is as 250 to 187, or more accurately as 529 to 396, 



