rain, ag the axis or diameter of the drop, and sa a ray of light that comes from the fun and enters the drop at the point a. This ray sa, because it is perpendicular to both the furfaces, will país ftraight through the drop in the line agb without being refracted; but any collateral rays, fuch as thofe that fall about sb, as they pafs through the drop, will be made to converge to their axis, and passing out at ʼn will meet the axis at h: rays which fall farther from the axis then sb, fuch as those which fall about s c, will likewife be made to converge; but then their focus will be nearer to the drop than b. Suppofe therefore i to be the focus to which the rays that fall about se will converge, any ray sc, when it has defcribed the line co within the drop and is tending to the focus i, will pass out of the drop at the point o. The rays that fall upon the drop about s d, more remote ftill from the axis, will converve to a focus ftill nearer than i, as suppose at k. These rays, therefore, go out of the drop at p. The rays, that fall Atill more remote from the axis, as se, will converge to a focus nearer than k, as fuppofe at ; and the ray se when it has described the line eo within the drop, and is tending to 1, will pafs out at the point o. The rays that fall ftill more remote from the axis will converge to a focus ftill nearer. Thus the ray sƒwill after refraction converge to a focus at m, which is nearer then 1; and having defcribed the line fn. within the drop, it will país out to the point ". Now here we may obferve, that as any rays sb or sc, fall farther above the axis sa, the points n or o, where they pass out behind the drop, will be farther above g; or that, as the incident ray rifes from the axis sa, the arc gno increases till we come to fome ray sd, which paffes out of the drop at p; and is the highest point where any ray that falls upon the quadrant or quarter a x can pafs out: for any rays se, or sf, that fall higher than sd will not pafs out in any point above p, but at the points o or n, which are below it. Confequently, though the arc gnop increases, whilft the distance of the incident ray from the axis sa increafes, till we come to the ray sd; yet afterwards the higher the ray falls above the axis sa this arc pong will decreafe.

441. We have hitherto spoken of the points on the hinder part of the drop, where the rays pafs out of it; but this was for the fake of determining the points from whence thofe rays are reflected, which do not pafs out behind the drop. For, in explaining the rain-bow, we have no farther reafon to confider those rays which go through the drop; fince they can never come to the eye of a spectator placed any where in the lines or qt with his face towards the drop. Now, as there are many rays which país out of the drop between g and p, fo fome few rays will be reflected from thence; and confequently the several points between g and p, which are the points where fome of the rays pafs out of the drop, are like wife the points of reflection of the rest which do not pass out. Therefore, in respect of those rays which are reflected, we may call gp the arc of reflection; and may fay, that this arc of reflection increases, as the distance of the incident ray from the axis sa increafes, till we come to the ray sd; VOL. XVI. PART II.

the arc of reflection is gn for the rays b, it is go for the ray sc; and gp for the ray sd. But after this, as the diftance of the incident ray from the axis sa increafes, the arc of reflection decreases; for og less than pg is the arc of reflection for the ray so, and ng is the arc of reflection for the ray sf.

442. Hence it is obvious, that fome one ray, which falls above s d, may be reflected from the fame point with fome other ray which falls below sd. Thus, for inftance, the ray sb will be reflected from the point n, and the ray sfwill be reflect ed from the fame point; and confequently, when the reflected rays nr, nq, are refracted, as they pafs out of the drop at r and q, they will be pa rallel, by what has been shown in the former part of this propofition. But fince the intermediate rays, which enter the drop between sf and s by are not reflected from the fame point n, these two rays alone will be parallel to one another when they come out of the drop, and the intermediate rays will not be parallel to them. And confequently these rays rv, qt, though they are parallel after they emerge at and 4, will not be contiguous, and for that reafon will not be ef fectual; the ray sd is reflected from p, which has been fhown to be the limit of the arc of reflection; such rays as fall just above s d, and just be low sd, will be reflected from nearly the fame point p, as appears from what has been already fhown. These rays therefore will be parallel, because they are reflected from the fame point p; and they will likewise be contiguous, because they all of them enter the drop at one and the fame place, very near to d. Confequently, fuch rays as enter the drop at d, and are reflected from p the limit of the arc of reflection, will be effectual; fince, when they emerge at the fore part of the drop between a and y, they will be both parallel and contiguous. If we can make out that the rain. bow is produced by the rays of the fun which are thus reflected from drops of rain as they fall whilft the fun fhines upon them, this propofition may ferve to fhow us, that this appearance is not produced by any rays that fall upon any part, and are reflected from any part of thofe drops: fince this appearance cannot be produced by any rays but those which are effectual; and effectual rays must always enter each drop at one certain place in the fore part of it, and must likewife be reflected from one certain place in the hinder furface.

443. V. "When rays that are effectual emerge from a drop of rain after one reflection and two refractions, thofe which are moft refrangible will, at their emerfion, make a lefs angle with the incident rays than thofe do which are leaft refrangible; and by these means the rays of different colours will be separated from one another."

444. Let fh and gi (fig. 2, Plate CCLVII.) be effectual violet rays emerging from the drop at fg; and ƒn, gp, effectual red rays emerging from the fame drop at the fame place. Now, though all the violet rays are parallel to one another, becaufe they are fuppofed effectual, and though all the red rays are likewife parallel to one another for the fame reafon; yet the violet rays will not be parallel to the red rays. These rays, as they

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have different colours, and different degrees of refrangibility, will diverge from one another; any violet ray gi, which emerges at g, will diverge from any red ray gp, which emerges at the fame place. Now, both the violet ray gi, and the red ray gp, as they pafs out of the drop of water into the air, will be refracted from the perpendicular lo. But the violet ray is more refrangible than the red one; and for that reason gi, or the refracted violet ray, will make a greater angle with the perpendicular than go the refracted red ray; or the angle go will be greater than the angle #go. Suppofe the incident ray sb to be continued in the direction s k, and the violet ray ig to be continued backward in the direction ik, till it meets the incident ray at k; fuppofe likewife the red ray pg to be continued backwards in the fame manner, till it meets the incident ray at w: the angle iks is that which the violet ray,, or moft refrangible ray at its emerfion, makes with the incident ray; and the angle pas is that which the red ray, or leaft refrangible ray at its emerfion, makes with the incident ray. The angle iks is less than the angle paus, For, in the triangle, gwk, gws, or paus, is the external angle at the bafe, and g kw, or iks is one of the internal oppofite angles; and either internal oppofite angle is lefs than the external angle at the bafe. (Euc. b. I. prop. 16.) What has been shown to be true of the rays gi and gp might be shown in the fame manner of the rays ƒh and ƒn, or of any other rays that emerge refpectively parallel to gi and gp. But all the effectual violet rays are parallel to gi, and all the effectual red rays are parallel to gp. Therefore the effectual violet rays at their emerfion make a lefs angle with the incident ones than the effectual red ones. And for the fame reason, in all the other forts of rays, thofe which are moft refrangible, at their emerfion from a drop of rain after one reflection, will make a lefs angle with the incident rays tha thofe do which are lefs refrangible.

445. Or otherwife: When the rays gi and g emerge at the fame point g, as they both come out of water into air, and confequently are refracted from the perpendicular, instead of going ftraight forwards in the line eg continued, they will both be turned round upon the point g fron the perpendicular go. Now, it is eafy to conceive, that either of thefe lines might be turned in this manner upon the point g as upon a centre, till they became parallel to sb the incident ray. But if either of these lines or rays were refracted fo much from go as to become parallel to sb, the ray so much refracted would, after emersion,” make no angle with sk, because it would be parallel to it. And confequently that ray which is moft turned round upon the point g, or that ray which is most refrangible, will after emerfion be nearest parallel to the incident ray, or will make the leaft angle with it. The fame may be proved of all other rays emerging parallel to gi and gp refpectively, or of all effectual rays; thofe which are moft refrangible will after emerfion make a lefs angle with the incident rays than thofe do which are leaft refrangible. 1

446. But fince the effectual rays of different colours make different angles with sk at their

emerfion, they will be feparated from one another; fo that if the eye was placed in the beam fg his it would receive only rays of one colour from the drop xag v; and if it was placed in the beam fgnp, it would receive only rays of fome other colour. The angles wp, which the leaft refrangible or red rays make with the incident ones, when they emerge fo as to be effectual, is found by calculation to be 42° 2. And the angle ki, which the most refrangible rays make with the incident ones, when they emerge fo as to be effectual, is found to be 40° 17'. The rays, which have the intermediate degrees of refrangibility, make with the incident ones intermediate angles between 42° 2′ and 40° 17'.

447. VI. "If a line is fuppofed to be drawn from the centre of the fun through the eye of the fpectator, the angle which any effectual ray, after two refractions and one reflection, makes with the incident ray, will be equal to the angle which it makes-with that line."

448." Let the eye of the fpectator be at i (fig. 2. pl. CCLVII.) and let qt be the line fuppoled to be drawn from the centre of the fun through the eye of the fpectator; the angle git, which any effectual ray makes with this line, will be equal to the angle i ks, which the fame ray makes with the incident ray shorsk. If sb is a ray coming from the centre of the fun, then, fince qt is fuppof to be drawn from the fame point, these two lines, upon account of the remoteness of the point from whence they are drawn, may be looked upon as parallel to one another. But the right line ki croffing these two parallel lines will make the alternate angles equal. (Euc, b. I. prop. 29.) Therefore kit or git is equal to s ki.

449. VII. "When the fon fhines upon the drops of rain as they are falling, the rays that come from thofe drops to the eye of a spectator, after one reflection and two refractions, produce the primary rainbow."

450. If the fun' fhines upon the rain as it falls, there are commonly feen two bows, as AFB, CHD (fig. 3, plate CCLVII.) or if the cloud and rain do not reach over that whole fide of the fky where the bows appear, then only a part of one or both bows is feen in that place where the rain falls. Of thefe two bows, the innermoft AFB is the more vivid of the two, and this is called the primary bow. The outer part TFY of the primary bow is red, the inner part VEX is violet; the intermediate parts, reckoning from the red to the violet, are orange, yellow, green, blue, and indigo. Suppofe the fpectator's eye to be at O, and let LOP be an imaginary line drawa from the centre of the fun through the eye of the fpectator; if a beam of light S coming from the fun falls upon any drop F, and the rays that emerge at F in the line FO, fo as to be effectual, make an angle FOP of 42° 2′ with the line LP; then these effectual rays make an angle 42° 2' with the incident rays, by the preceding propofition, and confequently thefe rays will be red, fo that the drop F will appear red. All the other rays, which emerge at F, and would be effectual if they fell upon the eye, are refracted more than the red ones, and confequently will pafs above the eye. If a beam of light S falls upon the drop E; and

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the rays that emerge at E in the line EO, fo as to be effectual, make an angle EOP of 40° 17' with the line LP; then these effectual rays make like wife an angle of 40° 17' with the incident rays, and the drop E will appear of a violet Polour. All the other rays which emerge at E, and would be effectual if they came to the eye, are refracted less than the violet ones, and therefore pass below the eye. The intermediate drops between F and E will for the fame reasons be of the intermediate, colours.

451. Thus we have shown why a fet of drops 'from F to E, as they are falling, fhould appear of the primary colours, red, orange, yellow, green, blue, indigo, and violet. It is not necessary that the feveral drops which produce these colours fhould all of them fall at exactly the fame diftance from the eye. The angle FOP, for inftance, is the fame whether the distance of the drop from the eye is OF, or whether it is in any other part of the line OF fomething nearer to the eye. And whilft the angle FOP is the fame, the angle made by the emerging and incident rays, and confequently the colour of the drop will be the fame. This is equally true of any other drop; fo that, although in the figure the drops F and E are reprefented as falling perpendicularly one under the other, yet this is not neceffary in order to produce the bow. But the coloured line FE, which we have already accounted for, is only the breadth of the bow. It ftill remains to be shown, why not only the drop F thould appear red, but why all the other drops, quite from A to B in the arc ATFYB, fhould appear of the fame colour. Now it is evident, that wherever a drop of rain is placed, if the angle which the effectual rays make with the line LP is equal to the angle FOP, that is, if the angle which the effectual rays make with the incident rays is 42° 2′, any of thofe drops will be red, for the fame reafon that the drop F is of this colour.

452. If FOP was to turn round upon the line OP, so that one end of this line thould always be at the eye, and the other be at P oppofite to the fun; fuch a motion of this figure would be like that of a pair of compaffes turning round upon one of the legs OP with the opening FOP. In this revolution the drop F would defcribe a circle; P would be the centre, and ATFYB would be an arc in this circle. Now, fiuce, in this motion of the line and drop OF, the angle made by FO with OP, that is, the angle FOP, continues the fame; if the fun was to thine upon this drop as it revolves, the effectual rays would make the fame angle with the incident rays, in whatever part of the arc ATFYB the drop was to be. Therefore, whether the drop is at A, or at T, or at Y, or at B, or wherever elfe it is in this whole arc, it would appear red as it does at F. The drops of rain, as they fall, are not indeed turn⚫1 round in this manner; but then, as vait numbers of them are falling at once in right lines from the cloud, whilst one drop is at F, there will be others at Y, at T, at B, at A, and in every other part of the arc ATFYB; and all thefe drops will be red for the fame reafon that the drop F would have been red, if it had been in the fame place. Therefore,

when the fun fhines upon the rain as it falls, there will be a red arc ATFYB oppofite to the fun. In the fame manner, because the drop E is violet, we might prove that any other drep, which whilst it is falling is in any part of the arc AVEXB, will be violet; and confequently, at the fame time that the red arc ATFYB appears, there will likewife be a violet arc AVEXB below or within it. FE is the distance between these two coloured arcs; and from what has been faid, it follows, that the intermediate space between these two arcs will be filled up with arcs of the intermediate colours, orange, yellow, blue, green, and indigo. All these coloured arcs together make up the primary rainbow.

453. VIII. "The PRIMARY RAINBOW is never a greater arc than a femicircle."

454. SINCE the line LOP is drawn from the fun through the eye of the spectator, and fince P (fig. 1. pl. CCLVII.) is the centre of the rainbow, it follows, that the centre of the rainbow is always oppofite to the fun. The angle FOP is an angle of 42° 2, as was obferved, or F, the highest part of the bow, is 42° 2' from P the centre of it. If the fun is more than 42° 2' high, P the centre of the rainbow, which is oppofite to the fun, will be more than 42° 2' below the horizon; and confequently F, the top of the bow, which is only 42° 2' from P, will be below the horizon; that is, when the fun is more than 42° 2′ high, no primary rainbow will be feen. If the fun is fomething lefs than 42° 2' high, then P will be fomething lefs than 42° 2' below the horizon; and confequently F, which is only 42° 2' from P, will be juft above the horizon; that is, a small part of the bow at this height of the fun will appear clofe to the ground oppofite to the fun. If the fun is 20° high, then P will be 20° below the horizon; and F, the top of the bow, being 42° 2' from P, will be 22° 2' above the horizon; therefore, at this height of the fun, the bow will be an arc of a circle whofe centre is below the horizon; and confequently that arc of the circle which is above the horizon or the bow, will be lefs than a femicircle. If the fun is in the horizon, then P, the centre of the bow, will be in the oppofite part of the horizon; F, the top of the bow, will be 42° 2′ above the horizon; and the bow itself, because the horizon passes through the centre of it, will be a femicircle. More than a femicircle can never appear; becaufe, if the bow was more than a femicircie, P the centre of it must be above the horizon; but P is always opposite to the fun, therefore P cannot be above the horizon, unless the fun is below it; and when the fun is fet, or is below the horizon, it cannot shine upon the drops of rain as they fall; and confequently, when the fun is below the horizon, no bow at all can be feen.

455. IX. When the rays of the fun fall upon a drop of rain, fome of them, after two reflections and two refractions, may come to the eye of a fpectator, who has his back towards the fun, and his face towards the drop."

456. IF HGW (fig. 4. pl. CCLVII.) is a drop of rain, and parallel rays coming from the fun, as zv, yw, fall upon the lower part of it, they will

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