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time at a loss when they endeavoured to assign reasons for all the particular colours, and for the order of them. Indeed nothing but the doctrine of the different refrangibility of the rays of light, which was a discovery reserved for the great sir Isaac Newton, could furnish a complete solution of this difficulty. De Dominis supposed that the red rays were those which had traversed the least space in the inside of a drop of water, and therefore retained more of their native force, and consequently, striking the eye more briskly, gave it a stronger sensation; that the green and blue colours were produced by those rays, the force of which had been, in some measure, obtunded in passing through a greater body of water; and that all the intermediate colours were composed (according to the hypothesis which generally prevailed at that time) of a mixture of these three primary ones. That the different colours were caused by some difference in the impulse of light upon the eye, and the greater or less impression that was thereby made upon it, was an opinion which had been adopted by many persons, who had ventured to depart from the authority of Aristotle.

Afterwards the same De Dominis observed, that all the rays of the same colour must leave the drop of water in a part similarly situated with respect to the eye, in order that each of the colours may appear in a circle, the centre of which is a point of the heavens, in a line drawn from the sun through the eye of the spectator. The red rays, he observed, must issue from the drop nearest to the bottom of it, in order that the circle of red may be the outermost, and therefore the most elevated in the bow.

Notwithstanding De Dominis conceived so justly of the manner in which the inner rainbow is formed, he was far from having as just an idea of the cause of the exterior bow. This he endeavoured to explain in the very same manner in which he had done the interior, viz. by one reflection of the light within the drop, preceded and followed by a refraction; supposing only that the rays which formed the exterior bow were returned to the eye by a part of the drop lower than that which transmitted the red of the interior bow. He also supposed that the rays which formed one of the bows came from the superior part of the sun's disk, and those which formed the other from the inferior part of it. He did not consider, that upon those principles the two bows ought to have been contiguous; or rather, that an indefinite number of bows would have had their colours all intermixed; which would have been no bow at all.

When sir Isaac Newton discovered the different refrangibility of the rays of light, he immediately applied his new theory of light and colours to the phenomena of the rainbow, taking this remarkable object of philosophical enquiry where De Dominis and Descartes, for want of this knowledge, were obliged to leave their investigations imperfect. For they could give no good reason why the bow should be coloured, and much less could they give any satisfactory account of the order in which the colours appear.

If different particles of light had not different degrees of refrangibility, on which the colours depend, the rainbow, besides being much narrower than it is, would be colourless; but the different refrangibility of differently coloured rays being admitted, the reason is obvious, both why the bow should be coloured, and also why

the colours should appear in the order in which they are observed. Let a (pl. 148, fig. 9.) be a drop of water, and S a pencil of light; which, on its leaving the drop of water, reaches the eye of the spectator. This ray, at its entrance into the drop, begins to be decomposed into its proper colours; and upon leaving the drop, after one reflection and a second refraction, it is farther decomposed into as many small differentlycoloured pencils as there are primitive colours in the light. Three of them only are drawn in this figure, of which the blue is the most, and the red the least refracted.

The doctrine of the different refrangibility of light enables us to give a reason for the size of a bow of each particular colour. Newton, having found that the sines of refraction of the most refrangible and least refrangible rays, in passing from rain-water into air, are in the proportion of 185 to 182, when the sine of incidence is 138, calculated the size of the bow; and he found, that if the sun was only a physical point, without sensible magnitude, the breadth of the inner bow would be two degrees; and if to this 30′ was added for the apparent diameter of the sun, the whole breadth would be 24 degrees. But as the outermost colours, especially the violet, are extremely faint, the breadth of the bow will not in reality appear to exceed two degrees. He finds, by the same principles, that the breadth of the exterior bow, if it was every where equally vivid, would be 4° 20′. But in this case there is a greater deduction to be made, on account of the faintness of the light of the exterior bow; so that, in fact, it will not appear to be more than three degrees broad.

The principal phenomena of the rainbow are all explained on`sir Isaac Newton's principles, in the following propositions:

When the rays of the sun fall upon a drop of rain and enter into it, some of them, after one reflection and two refractions, may come to the eye of a spectator who has his back towards the sun, and his face towards the drop.

If XY (fig. 10.) is a drop of rain, and the sun shines upon it in any lines sf, sd, sa, &c. most of the rays will enter into the drop; some few of them only will be reflected from the first surface; these rays which are reflected from thence do not come under our present consideration, because they are never refracted at all. The greatest part of the rays then enter the drop, and those passing on to the second surface, will most of them be transmitted through the drop; but neither do those rays which are thus transmitted fall under our present consideration, since they are not reflected. For the rays, which are described in the proposition, are such as are twice refracted and once reflected. However, at the second surface, or hinder part of the drop, at pg, some few rays will be reflected, whilst the rest are transmitted: those rays proceed in some such lines as nr, nq; and coming out of the drop in the lines r, qt, may fall upon the eye of a spectator, who is placed anywhere in those lines, with his face towards the drop, and consequently with his back towards the sun, which is supposed to shine upon the drop in the lines sf, sd, sa, &c. These rays are twice refracted and once reflected; they are refracted when they pass out of the air into the drop; they are reflected from the second surface, and are refracted again when they pass out of the drop into the air.

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When rays of light reflected from a drop of rain come to the eye, those are called effectual which are able to excite a sensation. When rays of light come out of a drop of rain, they will not be effectual, unless they are parallel and contiguous.

There are but few rays that can come to the eye at all for since the greatest part of those rays which enter the drop XY (fig. 10.) between X and a, pass out of the drop through the hinder surface pg; only few are reflected from thence, and come out through the nearer surface between a and y. Now, such rays as emerge, or come out of the drop, between a and Y, will be ineffectual, unless they are parallel to one anether, as ro and q are; because such rays as eone out diverging from one another will be so far asunder when they come to the eye, that all of them cannot enter the pupil; and the very few that can enter it will not be sufficient to ex

cite any sensation. But even rays, which are parallel, as rv, 9 t, will not be effectual, unless there are several of them contiguous or very near to one another. The two rays rv and gt alone will not be perceived, though both of them eater the eye; for so very few rays are not sufficient to excite a sensation.

When rays of light come out of a drop of rain after one reflection, those will be effectual which are reflected from the same point, and

which entered the drop near to one another. Any rays, as s b and c d (pl. 148, fig. 11.), when they have passed out of the air into a drop of water, will be refracted towards the perpendiculars bi dl; and as the rays falls farther from the axis av than the ray cd, sb will be more refracted than cd; so that these rays, though parallel to one another at their incidence, may describe the lines be and de after refraction, and be both of them reflected from one and the same point e. Now all rays which are thus reflected from one and the same point, when they have described the lines ef, eg, and after reflection emerge at ƒ and g, will be so refracted, when they pass out of the drop into the air, as to describe the lines fh, gi, parallel to one another. If these rays were to return from e in the lines et, ed, and were to emerge at b and d, they would be refracted into the lines of their incidence bs, dr. But if these rays, instead of being returned in the lines el, ed, are reflected from the same point e in the lines eg, ef, the lines of reflection ee and ef will be inclined both to one another, and to the surface of the drop; just as much as the lines eb and ed are. First eb and eg make just the same angle with the surface of the drop: for the angle bes, which eb makes with the surface of the drop, is the complement of incidence, and the angle gev, which eg makes with the surface, is the complement of reflection; and these two are equal to one another. In the same manner we might prove, that ed and eƒ make equal angles with the surface of the drop. Secondly, the angle bed is equal to the angle feg; or the reflected rays eg, ef, and the incident rays be, de, are equally inclined to each other. For the angle of incidence bel is equal to the angle of reflection gel, and the angle of incidence del is equal to the angle of reflection fel; consequently the difference between the angles of insidence is equal to the difference between the an

gles of reflection, or bel-del-gel-fel, or bed-gef. Since therefore either the lines eg ef, or the lines e b, e d, are equally inclined both to one another and to the surface of the drop, the rays will be refracted in the same manner, whether they were to return in the lines eb, ed, or are reflected in the lines eg, ef. But if they were to return in the lines e b, e d, the refraction, when they emerge at b and d, would make them parallel. Therefore, if they are reflected from one and the same point e in the lines eg, ef, the refraction, when they emerge at g and ƒ, will likewise make them parallel.

But though such rays as are reflected from the same point in the hinder part of a drop of rain are parallel to one another when they emerge, and so have one condition that is requisite towards making them effectual, yet there is another condition necessary; for rays, that are effectual, must be contiguous as well as parallel. And though rays, which enter the drop in different places, may be parallel when they emerge, those only will be contiguous which enter it nearly at the same place.

Let XY (fig. 10.) be a drop of rain, ag the axis or diameter of the drop, and s & a ray of light that comes from the sun and enters the drop at the point a. This ray sa, because it is perpendicular to both the surfaces, will pass straight through the drop in the line agh without being that fall about sb, as they pass through the drop, refracted; but any collateral rays, such as those will be made to converge to their axis, and passing out at n will meet the axis at h: rays which fall farther from the axis than sh, such as those which fall about sc, will likewise be made to converge; but then their focus will be nearer to the drop than h. Suppose therefore i to be the focus to which the rays that fall about sc will converge, any ray, sc, when it has described 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 still from the axis, will converge to a focus still nearer than i, as suppose at . These rays therefore go out of the drop at p. The rays, that fall still more remote from the axis, as se,will converge to a focus nearer than k, as suppose at ; and the ray se, when it has described the line eo within the drop, and is tending to, will pass out at the point o. The rays that fall still more remote from the axis will converge to a focus still nearer. Thus the ray sf will, after refraction, converge to a focus at m, which is nearer than ; and having described the line fn within the drop, it will pass out to the point ". Now here we may observe, that as any rays s b, or se 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 rises from the axis sa, the arc gno increases, till we come to some ray s d, which passes out of the drop at p; and this is the highest point where any ray that falls upon the quadrant or quarter a x can pass out: for any rays se, or sf, that fall higher than sd, will not pass out in any point above p, but at the points o or n, which are below it. Consequently, though the arc gnop increases, whilst the distance of the incident ray from the axis sa increases, till we come to the ray sd; yet afterwards, the higher the ray falls above the axis s a, this are pong will decrease.

We have hitherto spoken of the points on the

hinder part of the drop, where the rays pass out of it; but this was for the sake of determining the points from whence those rays are reflected, which do not pass out behind the drop. For, in explaining the rainbow, we have no farther reason to consider those rays which go through the drop; since they can never come to the eye of a spectator placed any where in the lines rv or q t with his face towards the drop. Now, as there are many rays which pass out of the drop between g and p, so some few rays will be reflected from thence; and consequently the several points between g and p, which are the points where some of the rays pass out of the drop, are likewise the points of reflection for 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 say, that this are of reflection increases, as the distance of the incident ray from the axis sa increases, till we come to the ray sd; the arc of reflection is gn for the ray sb, it is go for the ray se; and gp for the ray sd. But after this, as the distance of the incident ray from the axis sa increases, the arc of reflection decreases; for og less than pg is the arc of reflection for the ray se, and ng is the arc of reflection for the ray sf.

From hence it is obvious, that some one ray, which falls above sd, may be reflected from the same point with some other ray which falls below sd. Thus, for instance, the ray sb will be reflected from the point, and the ray sƒ will be reflected from the same point; and consequently, when the reflected rays nr, nq, are refracted as they pass out of the drop at and q, they will be parallel, by what has been shown in the former part of this proposition. But since the intermediate rays, which enter the drop between sf and s, are not reflected from the same point n, these two rays alone will be the parallel to one another when they come out of the drop, and the intermediate rays will not be parallel to them. And consequently, these rays rv, 9t, though they are parallel after they emerge at r and q, will not be contiguous, and for that reason will not be effectual; the ray sd is reflected from p, which has been shewn to be the limit of the arc of reflection; such rays as fall just above sd, and just below sd, will be reflected from nearly the same point p, as appears from what has been already shewn. These rays therefore will be parallel, because they are reflected from the same point p; and they will likewise be contiguous, because they all of them enter the drop at one and the same place very near to d. Consequently, such rays as enter the drop at d, and are reflected from p the limit of the arc of reflection, will be effectual; since, 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 hereafter that the rainbow is produced by the rays of the sun which are thus reflected from drops of rain as they fall whilst the sun shines upon them, this proposition may serve to shew us, that this appearance is not produced by any rays that fall upon any part, and are reflected from any part of those drops: since 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 likewise be reflected from one certain place in the hinder surface.

When rays that are effectual emerge from a drop

of rain after one reflection and two refractions, those which are most refrangible will, at their emersion, make a less angle with the incident rays than those do which are least refrangible; and by this means the rays of different colours will be separated from one another.

Let fh and gi (pl. 148, fig. 11.) be effectual violet rays emerging from the drop at ƒg; and fn, gp, effectual red rays emerging from the same drop at the same place. Now, though all the violet rays are parallel to one another, because they are supposed effectual, and though all the red rays are likewise parallel to one another for the same reason; yet the violet rays will not be parallel to the red rays. These rays, as they 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 same place. Now, both the violet ray gi, and the red ray gp, as they pass 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 gp the refracted red ray; or the angle i go will be greater than the angle pgo. Suppose the incident ray s b to be continued in the direction sk, and the violet ray ig to be continued backward in the direction ik, till it meets the incident ray at k. Suppose likewise the red ray pg to be continued backwards in the same manner, till it meets the incident ray at w. The angle iks is that which the violet ray, or most refrangible ray at its emersion, makes with the incident ray; and the angle pws is that which the red ray, or least refrangible ray at its emersion, makes with the incident ray. The angle iks is less than the angle pws. For, in the triangle, gwk, gws, or pws, is the external angle at the base, and gk or iks is one of the internal opposite angles; and either internal opposite angle is less than the external angle at the base. (Euc. b. 1. prop. 16.) What has been shewn to be true of the rays gi and gp might be shewn in the same manner of the rays fh and ƒn, or of any other rays that emerge respectively 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 emersion make a less angle with the incident ones than the effectual red ones. And for the same reason, in all the other sorts of rays, those which are most refrangible, at their emersion from a drop of rain after one reflection, will make a less angle with the incident rays, than those do which are less refrangible.

Or otherwise: when the rays gi and gp emerge at the same point g, as they both come out of water into air, and consequently are refracted from the perpendicular, instead of going straight forwards in the line eg continued, they will both be turned round upon the point g, from the perpendicular go. Now it is easy to conceive, that either of these 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 so much from go as to become parallel to s, the ray so much refracted would, after emersion, make no augle with s✯, because it would be pa

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