4. This disposition of the different coloured rays to be refracted some more than others our author calls their respective degrees of refrangibility. And since this difference of refrangibility discovers it self to be so regular, the next step is to find the rule it observes.
[5.] It is a common principle in optics, that the sine of the angle of incidence bears to the sine of the refracted angle a given proportion. If A B (in fig. 131, 132) represent the surface of any refracting substance, suppose of water or glass, and C D a ray of light incident upon that face in the point D, let D E be the ray, after it has passed the surface A B; if the ray pass out of the air into the substance whose surface is A B (as in fig. 131) it shall be turned from the surface, and if it pass out of that substance into air it shall be bent towards it (as in fig. 132) But if F G be drawn through the point D perpendicular to the surface A B, the angle under C D F made by the incident ray and this perpendicular is called the angle of incidence; and the angle under E D G, made by this perpendicular and the ray after refraction, is called the refracted angle. And if the circle H F I G be described with any interval cutting C D in H and D E in I, then the perpendiculars H K, I L being let fall upon F G, H K is called the sine of the angle under C D F the angle of incidence, and I L the sine of the angle under E D G the refracted angle. The first of these sines is called the sine of the angle of incidence, or more briefly the sine of incidence, the latter is the sine of the refracted angle, or the sine of refraction. And it has been found by numerous experiments that whatever proportion the sine of incidence H K bears to the sine of refraction I L in any one case, the same proportion shall hold in all cases; that is, the proportion between these sines will remain unalterably the same in the same refracting substance, whatever be the magnitude of the angle under C D F.
6. But now because optical writers did not observe that every beam of white light was divided by refraction, as has been here explained, this rule collected by them can only be understood in the gross of the whole beam after refraction, and not so much of any particular part of it, or at most only of the middle part of the beam. It therefore was incumbent upon our author to find by what law the rays were parted from each other; whether each ray apart obtained this property, and that the separation was made by the proportion between the sines of incidence and refraction being in each species of rays different; or whether the light was divided by some other rule. But he proves by a certain experiment that each ray has its sine of incidence proportional to its sine of refraction; and farther shews by mathematical reasoning, that it must be so upon condition only that bodies refract the light by acting upon it, in a direction perpendicular to the surface of the refracting body, and upon the same sort of rays always in an equal degree at the same distances[314].
7. Our great author teaches in the next place how from the refraction of the most refrangible and least refrangible rays to find the refraction of all the intermediate ones[315]. The method is this: if the sine of incidence be to the sine of refraction in the least refrangible rays as A to B C, (in fig. 133) and to the sine of refraction in the most refrangible as A to B D; if C E be taken equal to C D, and then E D be so divided in F, G, H, I, K, L, that E D, E F, E G, E H, E I, E K, E L, E C, shall be proportional to the eight lengths of musical chords, which found the notes in an octave, E D being the length of the key, E F the length of the tone above that key, E G the length of the lesser third, E H of the fourth, E I of the fifth, E K of the greater sixth, E L of the seventh, and E C of the octave above that key; that is if the lines E D, E F, E G, E H, E I, E K, E L, and E C bear the same proportion as the numbers, 1, 9/8, 5/6, ¾, ⅓, ¾, 9/61, ½, respectively then shall B D, B F, be the two limits of the sines of refraction of the violet-making rays, that is the violet-making rays shall not all of them have precisely the same sine of refraction, but none of them shall have a greater sine than B D, nor a less than B F, though there are violet-making rays which answer to any sine of refraction that can be taken between these two. In the same manner B F and B G are the limits of the sines of refraction of the indigo-making rays; B G, B H are the limits belonging to the blue-making rays; B H, B I the limits pertaining to the green-making rays, B I, B K the limits for the yellow-making rays; B K, B L the limits for the orange-making rays; and lastly, B L and B C the extreme limits of the sines of refraction belonging to the red-making rays. These are the proportions by which the heterogeneous rays of light are separated from each other in refraction.
8. When light passes out of glass into air, our author found A to B C as 50 to 77, and the same A to B D as 50 to 78. And when it goes out of any other refracting substance into air, the excess of the sine of refraction of any one species of rays above its sine of incidence bears a constant proportion, which holds the same in each species, to the excess of the sine of refraction of the same sort of rays above the sine of incidence into the air out of glass; provided the sines of incidence both in glass and the other substance are equal. This our author verified by transmitting the light through prisms of glass included within a prismatic vessel of water; and draws from those experiments the following observations: that whenever the light in passing through so many surfaces parting diverse transparent substances is by contrary refractions made to emerge into the air in a direction parallel to that of its incidence, it will appear afterwards white at any distance from the prisms, where you shall please to examine it; but if the direction of its emergence be oblique to its incidence, in receding from the place of emergence its edges shall appear tinged with colours: which proves that in the first case there is no inequality in the refractions of each species of rays, but that when any one species is so refracted as to emerge parallel to the incident rays, every sort of rays after refraction shall likewise be parallel to the same incident rays, and to each other; whereas on the contrary, if the rays of any one sort are oblique to the incident light, the several species shall be oblique to each other, and be gradually separated by that obliquity. From hence he deduces both the forementioned theorem, and also this other; that in each sort of rays the proportion of the sine of incidence to the sine of refraction, in the passage of the ray out of any refracting substance into another, is compounded of the proportion to which the sine of incidence would have to the sine of refraction in the passage of that ray out of the first substance into any third, and of the proportion which the sine of incidence would have to the sine of refraction in the passage of the ray out of that third substance into the second. From so simple and plain an experiment has our most judicious author deduced these important theorems, by which we may learn how very exact and circumspect he has been in this whole work of his optics; that notwithstanding his great particularity in explaining his doctrine, and the numerous collection of experiments he has made to clear up every doubt which could arise, yet at the same time he has used the greatest caution to make out every thing by the simplest and easiest means possible.
[9.] Our author adds but one remark more upon refraction, which is, that if refraction be performed in the manner he has supposed from the light’s being pressed by the refracting power perpendicularly toward the surface of the refracting body, and consequently be made to move swifter in the body than before its incidence; whether this power act equally at all distances or otherwise, provided only its power in the same body at the same distances remain without variation the same in one inclination of the incident rays as well as another; he observes that the refracting powers in different bodies will be in the duplicate proportion of the tangents of the lead angles, which the refracted light can make with the surfaces of the refracting bodies[316]. This observation may be explained thus. When the light passes into any refracting substance, it has been shewn above that the sine of incidence bears a constant proportion to the sine of refraction. Suppose the light to pass to the refracting body A B C D (in fig. 134) in the line E F, and to fall upon it at the point F, and then to proceed within the body in the line F G. Let H I be drawn through F perpendicular to the surface A B, and any circle K L M N be described to the center F. Then from the points O and P where this circle cuts the incident and refracted ray, the perpendiculars O Q, P R being drawn, the proportion of O Q to P R will remain the same in all the different obliquities, in which the same ray of light can fall on the surface A B. Now O Q is less than F L the semidiameter of the circle K L M N, but the more the ray E F is inclined down toward the surface A B, the greater will O Q be, and will approach nearer to the magnitude of F L. But the proportion of O Q to P R remaining always the same, when O Q, is largest, P R will also be greatest; so that the more the incident ray E F is inclined toward the surface A B, the more the ray F G after refraction will be inclined toward the same. Now if the line F S T be so drawn, that S V being perpendicular to F I shall be to F L the semidiameter of the circle in the constant proportion of P R to O Q; then the angle under N F T is that which I meant by the least of all that can be made by the refracted ray with this surface, for the ray after refraction would proceed in this line, if it were to come to the point F lying on the very surface A B; for if the incident ray came to the point F in any line between A F and F H, the ray after refraction would proceed forward in some line between F T and F I. Here if N W be drawn perpendicular to F N, this line N W in the circle K L M N is called the tangent of the angle under N F S. Thus much being premised, the sense of the forementioned proposition is this. Let there be two refracting substances (in fig. 135) A B C D, and E F G H. Take a point, as I, in the surface A B, and to the center I with any semidiameter describe the circle K L M. In like manner on the surface E F take some point N, as a center, and describe with the same semidiameter the circle O P Q. Let the angle under B I R be the least which the refracted light can make with the surface A B, and the angle under F N S the least which the refracted light can make with the surface E F. Then if L T be drawn perpendicular to A B, and P V perpendicular to E F; the whole power, wherewith the substance A B C D acts on the light, will bear to the whole power wherewith the substance E F G H acts on, the light, a proportion, which is duplicate of the proportion, which L T bears to P V.
[10.] Upon comparing according to this rule the refractive powers of a great many bodies it is found, that unctuous bodies which abound most with sulphureous parts refract the light two or three times more in proportion to their density than others: but that those bodies, which seem to receive in their composition like proportions of sulphureous parts, have their refractive powers proportional to their densities; as appears beyond contradiction by comparing the refractive power of so rare a substance as the air with that of common glass or rock crystal, though these substances are 2000 times denser than air; nay the same proportion is found to hold without sensible difference in comparing air with pseudo-topar and glass of antimony, though the pseudo-topar be 3500 times denser than air, and glass of antimony no less than 4400 times denser. This power in other substances, as salts, common water, spirit of wine, &c. seems to bear a greater proportion to their densities than these last named, according as they abound with sulphurs more than these; which makes our author conclude it probable, that bodies act upon the light chiefly, if not altogether, by means of the sulphurs in them; which kind of substances it is likely enters in some degree the composition of all bodies. Of all the substances examined by our author, none has so great a refractive power, in respect of its density, as a diamond.
[11.] Our author finishes these remarks, and all he offers relating to refraction, with observing, that the action between light and bodies is mutual, since sulphureous bodies, which are most readily set on fire by the sun’s light, when collected upon them with a burning glass, act more upon light in refracting it, than other bodies of the same density do. And farther, that the densest bodies, which have been now shewn to act most upon light, contract the greatest heat by being exposed to the summer sun.
12. Having thus dispatched what relates to refraction, we must address ourselves to discourse of the other operation of bodies upon light in reflecting it. When light passes through a surface, which divides two transparent bodies differing in density, part of it only is transmitted, another part being reflected. And if the light pass out of the denser body into the rarer, by being much inclined to the foresaid surface at length no part of it shall pass through, but be totally reflected. Now that part of the light, which suffers the greatest refraction, shall be wholly reflected with a less obliquity of the rays, than the parts of the light which undergo a less degree of refraction; as is evident from the last experiment recited in the first chapter; where, as the prisms D E F, G H I, (in fig. 129.) were turned about, the violet light was first totally reflected, and then the blue, next to that the green, and so of the rest. In consequence of which our author lays down this proportion; that the sun’s light differs in reflexibility, those rays being most reflexible, which are most refrangible. And collects from this, in conjunction with other arguments, that the refraction and reflection, of light are produced by the same cause, compassing those different effects only by the difference of circumstances with which it is attended. Another proof of this being taken by our author from what he has discovered of the passage of light through thin transparent plates, viz. that any particular species of light, suppose, for instance, the red-making rays, will enter and pass out of such a plate, if that plate be of some certain thicknesses; but if it be of other thicknesses, it will not break through it, but be reflected back: in which is seen, that the thickness of the plate determines whether the power, by which that plate acts upon the light, shall reflect it, or suffer it to pass through.