Part 19

_Obs._ 8. I measured therefore the thickness of this concavo-convex Plate of Glass, and found it every where 1/4 of an Inch precisely. Now, by the sixth Observation of the first Part of this Book, a thin Plate of Air transmits the brightest Light of the first Ring, that is, the bright yellow, when its thickness is the 1/89000th part of an Inch; and by the tenth Observation of the same Part, a thin Plate of Glass transmits the same Light of the same Ring, when its thickness is less in proportion of the Sine of Refraction to the Sine of Incidence, that is, when its thickness is the 11/1513000th or 1/137545th part of an Inch, supposing the Sines are as 11 to 17. And if this thickness be doubled, it transmits the same bright Light of the second Ring; if tripled, it transmits that of the third, and so on; the bright yellow Light in all these cases being in its Fits of Transmission. And therefore if its thickness be multiplied 34386 times, so as to become 1/4 of an Inch, it transmits the same bright Light of the 34386th Ring. Suppose this be the bright yellow Light transmitted perpendicularly from the reflecting convex side of the Glass through the concave side to the white Spot in the center of the Rings of Colours on the Chart: And by a Rule in the 7th and 19th Observations in the first Part of this Book, and by the 15th and 20th Propositions of the third Part of this Book, if the Rays be made oblique to the Glass, the thickness of the Glass requisite to transmit the same bright Light of the same Ring in any obliquity, is to this thickness of 1/4 of an Inch, as the Secant of a certain Angle to the Radius, the Sine of which Angle is the first of an hundred and six arithmetical Means between the Sines of Incidence and Refraction, counted from the Sine of Incidence when the Refraction is made out of any plated Body into any Medium encompassing it; that is, in this case, out of Glass into Air. Now if the thickness of the Glass be increased by degrees, so as to bear to its first thickness, (_viz._ that of a quarter of an Inch,) the Proportions which 34386 (the number of Fits of the perpendicular Rays in going through the Glass towards the white Spot in the center of the Rings,) hath to 34385, 34384, 34383, and 34382, (the numbers of the Fits of the oblique Rays in going through the Glass towards the first, second, third, and fourth Rings of Colours,) and if the first thickness be divided into 100000000 equal parts, the increased thicknesses will be 100002908, 100005816, 100008725, and 100011633, and the Angles of which these thicknesses are Secants will be 26' 13'', 37' 5'', 45' 6'', and 52' 26'', the Radius being 100000000; and the Sines of these Angles are 762, 1079, 1321, and 1525, and the proportional Sines of Refraction 1172, 1659, 2031, and 2345, the Radius being 100000. For since the Sines of Incidence out of Glass into Air are to the Sines of Refraction as 11 to 17, and to the above-mentioned Secants as 11 to the first of 106 arithmetical Means between 11 and 17, that is, as 11 to 11-6/106, those Secants will be to the Sines of Refraction as 11-6/106, to 17, and by this Analogy will give these Sines. So then, if the obliquities of the Rays to the concave Surface of the Glass be such that the Sines of their Refraction in passing out of the Glass through that Surface into the Air be 1172, 1659, 2031, 2345, the bright Light of the 34386th Ring shall emerge at the thicknesses of the Glass, which are to 1/4 of an Inch as 34386 to 34385, 34384, 34383, 34382, respectively. And therefore, if the thickness in all these Cases be 1/4 of an Inch (as it is in the Glass of which the Speculum was made) the bright Light of the 34385th Ring shall emerge where the Sine of Refraction is 1172, and that of the 34384th, 34383th, and 34382th Ring where the Sine is 1659, 2031, and 2345 respectively. And in these Angles of Refraction the Light of these Rings shall be propagated from the Speculum to the Chart, and there paint Rings about the white central round Spot of Light which we said was the Light of the 34386th Ring. And the Semidiameters of these Rings shall subtend the Angles of Refraction made at the Concave-Surface of the Speculum, and by consequence their Diameters shall be to the distance of the Chart from the Speculum as those Sines of Refraction doubled are to the Radius, that is, as 1172, 1659, 2031, and 2345, doubled are to 100000. And therefore, if the distance of the Chart from the Concave-Surface of the Speculum be six Feet (as it was in the third of these Observations) the Diameters of the Rings of this bright yellow Light upon the Chart shall be 1'688, 2'389, 2'925, 3'375 Inches: For these Diameters are to six Feet, as the above-mention'd Sines doubled are to the Radius. Now, these Diameters of the bright yellow Rings, thus found by Computation are the very same with those found in the third of these Observations by measuring them, _viz._ with 1-11/16, 2-3/8, 2-11/12, and 3-3/8 Inches, and therefore the Theory of deriving these Rings from the thickness of the Plate of Glass of which the Speculum was made, and from the Obliquity of the emerging Rays agrees with the Observation. In this Computation I have equalled the Diameters of the bright Rings made by Light of all Colours, to the Diameters of the Rings made by the bright yellow. For this yellow makes the brightest Part of the Rings of all Colours. If you desire the Diameters of the Rings made by the Light of any other unmix'd Colour, you may find them readily by putting them to the Diameters of the bright yellow ones in a subduplicate Proportion of the Intervals of the Fits of the Rays of those Colours when equally inclined to the refracting or reflecting Surface which caused those Fits, that is, by putting the Diameters of the Rings made by the Rays in the Extremities and Limits of the seven Colours, red, orange, yellow, green, blue, indigo, violet, proportional to the Cube-roots of the Numbers, 1, 8/9, 5/6, 3/4, 2/3, 3/5, 9/16, 1/2, which express the Lengths of a Monochord sounding the Notes in an Eighth: For by this means the Diameters of the Rings of these Colours will be found pretty nearly in the same Proportion to one another, which they ought to have by the fifth of these Observations.

And thus I satisfy'd my self, that these Rings were of the same kind and Original with those of thin Plates, and by consequence that the Fits or alternate Dispositions of the Rays to be reflected and transmitted are propagated to great distances from every reflecting and refracting Surface. But yet to put the matter out of doubt, I added the following Observation.

_Obs._ 9. If these Rings thus depend on the thickness of the Plate of Glass, their Diameters at equal distances from several Speculums made of such concavo-convex Plates of Glass as are ground on the same Sphere, ought to be reciprocally in a subduplicate Proportion of the thicknesses of the Plates of Glass. And if this Proportion be found true by experience it will amount to a demonstration that these Rings (like those formed in thin Plates) do depend on the thickness of the Glass. I procured therefore another concavo-convex Plate of Glass ground on both sides to the same Sphere with the former Plate. Its thickness was 5/62 Parts of an Inch; and the Diameters of the three first bright Rings measured between the brightest Parts of their Orbits at the distance of six Feet from the Glass were 3.4-1/6.5-1/8. Inches. Now, the thickness of the other Glass being 1/4 of an Inch was to the thickness of this Glass as 1/4 to 5/62, that is as 31 to 10, or 310000000 to 100000000, and the Roots of these Numbers are 17607 and 10000, and in the Proportion of the first of these Roots to the second are the Diameters of the bright Rings made in this Observation by the thinner Glass, 3.4-1/6.5-1/8, to the Diameters of the same Rings made in the third of these Observations by the thicker Glass 1-11/16, 2-3/8. 2-11/12, that is, the Diameters of the Rings are reciprocally in a subduplicate Proportion of the thicknesses of the Plates of Glass.

So then in Plates of Glass which are alike concave on one side, and alike convex on the other side, and alike quick-silver'd on the convex sides, and differ in nothing but their thickness, the Diameters of the Rings are reciprocally in a subduplicate Proportion of the thicknesses of the Plates. And this shews sufficiently that the Rings depend on both the Surfaces of the Glass. They depend on the convex Surface, because they are more luminous when that Surface is quick-silver'd over than when it is without Quick-silver. They depend also upon the concave Surface, because without that Surface a Speculum makes them not. They depend on both Surfaces, and on the distances between them, because their bigness is varied by varying only that distance. And this dependence is of the same kind with that which the Colours of thin Plates have on the distance of the Surfaces of those Plates, because the bigness of the Rings, and their Proportion to one another, and the variation of their bigness arising from the variation of the thickness of the Glass, and the Orders of their Colours, is such as ought to result from the Propositions in the end of the third Part of this Book, derived from the Phaenomena of the Colours of thin Plates set down in the first Part.

There are yet other Phaenomena of these Rings of Colours, but such as follow from the same Propositions, and therefore confirm both the Truth of those Propositions, and the Analogy between these Rings and the Rings of Colours made by very thin Plates. I shall subjoin some of them.

_Obs._ 10. When the beam of the Sun's Light was reflected back from the Speculum not directly to the hole in the Window, but to a place a little distant from it, the common center of that Spot, and of all the Rings of Colours fell in the middle way between the beam of the incident Light, and the beam of the reflected Light, and by consequence in the center of the spherical concavity of the Speculum, whenever the Chart on which the Rings of Colours fell was placed at that center. And as the beam of reflected Light by inclining the Speculum receded more and more from the beam of incident Light and from the common center of the colour'd Rings between them, those Rings grew bigger and bigger, and so also did the white round Spot, and new Rings of Colours emerged successively out of their common center, and the white Spot became a white Ring encompassing them; and the incident and reflected beams of Light always fell upon the opposite parts of this white Ring, illuminating its Perimeter like two mock Suns in the opposite parts of an Iris. So then the Diameter of this Ring, measured from the middle of its Light on one side to the middle of its Light on the other side, was always equal to the distance between the middle of the incident beam of Light, and the middle of the reflected beam measured at the Chart on which the Rings appeared: And the Rays which form'd this Ring were reflected by the Speculum in Angles equal to their Angles of Incidence, and by consequence to their Angles of Refraction at their entrance into the Glass, but yet their Angles of Reflexion were not in the same Planes with their Angles of Incidence.

_Obs._ 11. The Colours of the new Rings were in a contrary order to those of the former, and arose after this manner. The white round Spot of Light in the middle of the Rings continued white to the center till the distance of the incident and reflected beams at the Chart was about 7/8 parts of an Inch, and then it began to grow dark in the middle. And when that distance was about 1-3/16 of an Inch, the white Spot was become a Ring encompassing a dark round Spot which in the middle inclined to violet and indigo. And the luminous Rings encompassing it were grown equal to those dark ones which in the four first Observations encompassed them, that is to say, the white Spot was grown a white Ring equal to the first of those dark Rings, and the first of those luminous Rings was now grown equal to the second of those dark ones, and the second of those luminous ones to the third of those dark ones, and so on. For the Diameters of the luminous Rings were now 1-3/16, 2-1/16, 2-2/3, 3-3/20, &c. Inches.

When the distance between the incident and reflected beams of Light became a little bigger, there emerged out of the middle of the dark Spot after the indigo a blue, and then out of that blue a pale green, and soon after a yellow and red. And when the Colour at the center was brightest, being between yellow and red, the bright Rings were grown equal to those Rings which in the four first Observations next encompassed them; that is to say, the white Spot in the middle of those Rings was now become a white Ring equal to the first of those bright Rings, and the first of those bright ones was now become equal to the second of those, and so on. For the Diameters of the white Ring, and of the other luminous Rings encompassing it, were now 1-11/16, 2-3/8, 2-11/12, 3-3/8, &c. or thereabouts.

When the distance of the two beams of Light at the Chart was a little more increased, there emerged out of the middle in order after the red, a purple, a blue, a green, a yellow, and a red inclining much to purple, and when the Colour was brightest being between yellow and red, the former indigo, blue, green, yellow and red, were become an Iris or Ring of Colours equal to the first of those luminous Rings which appeared in the four first Observations, and the white Ring which was now become the second of the luminous Rings was grown equal to the second of those, and the first of those which was now become the third Ring was become equal to the third of those, and so on. For their Diameters were 1-11/16, 2-3/8, 2-11/12, 3-3/8 Inches, the distance of the two beams of Light, and the Diameter of the white Ring being 2-3/8 Inches.

When these two beams became more distant there emerged out of the middle of the purplish red, first a darker round Spot, and then out of the middle of that Spot a brighter. And now the former Colours (purple, blue, green, yellow, and purplish red) were become a Ring equal to the first of the bright Rings mentioned in the four first Observations, and the Rings about this Ring were grown equal to the Rings about that respectively; the distance between the two beams of Light and the Diameter of the white Ring (which was now become the third Ring) being about 3 Inches.

The Colours of the Rings in the middle began now to grow very dilute, and if the distance between the two Beams was increased half an Inch, or an Inch more, they vanish'd whilst the white Ring, with one or two of the Rings next it on either side, continued still visible. But if the distance of the two beams of Light was still more increased, these also vanished: For the Light which coming from several parts of the hole in the Window fell upon the Speculum in several Angles of Incidence, made Rings of several bignesses, which diluted and blotted out one another, as I knew by intercepting some part of that Light. For if I intercepted that part which was nearest to the Axis of the Speculum the Rings would be less, if the other part which was remotest from it they would be bigger.

_Obs._ 12. When the Colours of the Prism were cast successively on the Speculum, that Ring which in the two last Observations was white, was of the same bigness in all the Colours, but the Rings without it were greater in the green than in the blue, and still greater in the yellow, and greatest in the red. And, on the contrary, the Rings within that white Circle were less in the green than in the blue, and still less in the yellow, and least in the red. For the Angles of Reflexion of those Rays which made this Ring, being equal to their Angles of Incidence, the Fits of every reflected Ray within the Glass after Reflexion are equal in length and number to the Fits of the same Ray within the Glass before its Incidence on the reflecting Surface. And therefore since all the Rays of all sorts at their entrance into the Glass were in a Fit of Transmission, they were also in a Fit of Transmission at their returning to the same Surface after Reflexion; and by consequence were transmitted, and went out to the white Ring on the Chart. This is the reason why that Ring was of the same bigness in all the Colours, and why in a mixture of all it appears white. But in Rays which are reflected in other Angles, the Intervals of the Fits of the least refrangible being greatest, make the Rings of their Colour in their progress from this white Ring, either outwards or inwards, increase or decrease by the greatest steps; so that the Rings of this Colour without are greatest, and within least. And this is the reason why in the last Observation, when the Speculum was illuminated with white Light, the exterior Rings made by all Colours appeared red without and blue within, and the interior blue without and red within.

These are the Phaenomena of thick convexo-concave Plates of Glass, which are every where of the same thickness. There are yet other Phaenomena when these Plates are a little thicker on one side than on the other, and others when the Plates are more or less concave than convex, or plano-convex, or double-convex. For in all these cases the Plates make Rings of Colours, but after various manners; all which, so far as I have yet observed, follow from the Propositions in the end of the third part of this Book, and so conspire to confirm the truth of those Propositions. But the Phaenomena are too various, and the Calculations whereby they follow from those Propositions too intricate to be here prosecuted. I content my self with having prosecuted this kind of Phaenomena so far as to discover their Cause, and by discovering it to ratify the Propositions in the third Part of this Book.

_Obs._ 13. As Light reflected by a Lens quick-silver'd on the backside makes the Rings of Colours above described, so it ought to make the like Rings of Colours in passing through a drop of Water. At the first Reflexion of the Rays within the drop, some Colours ought to be transmitted, as in the case of a Lens, and others to be reflected back to the Eye. For instance, if the Diameter of a small drop or globule of Water be about the 500th part of an Inch, so that a red-making Ray in passing through the middle of this globule has 250 Fits of easy Transmission within the globule, and that all the red-making Rays which are at a certain distance from this middle Ray round about it have 249 Fits within the globule, and all the like Rays at a certain farther distance round about it have 248 Fits, and all those at a certain farther distance 247 Fits, and so on; these concentrick Circles of Rays after their transmission, falling on a white Paper, will make concentrick Rings of red upon the Paper, supposing the Light which passes through one single globule, strong enough to be sensible. And, in like manner, the Rays of other Colours will make Rings of other Colours. Suppose now that in a fair Day the Sun shines through a thin Cloud of such globules of Water or Hail, and that the globules are all of the same bigness; and the Sun seen through this Cloud shall appear encompassed with the like concentrick Rings of Colours, and the Diameter of the first Ring of red shall be 7-1/4 Degrees, that of the second 10-1/4 Degrees, that of the third 12 Degrees 33 Minutes. And accordingly as the Globules of Water are bigger or less, the Rings shall be less or bigger. This is the Theory, and Experience answers it. For in _June_ 1692, I saw by reflexion in a Vessel of stagnating Water three Halos, Crowns, or Rings of Colours about the Sun, like three little Rain-bows, concentrick to his Body. The Colours of the first or innermost Crown were blue next the Sun, red without, and white in the middle between the blue and red. Those of the second Crown were purple and blue within, and pale red without, and green in the middle. And those of the third were pale blue within, and pale red without; these Crowns enclosed one another immediately, so that their Colours proceeded in this continual order from the Sun outward: blue, white, red; purple, blue, green, pale yellow and red; pale blue, pale red. The Diameter of the second Crown measured from the middle of the yellow and red on one side of the Sun, to the middle of the same Colour on the other side was 9-1/3 Degrees, or thereabouts. The Diameters of the first and third I had not time to measure, but that of the first seemed to be about five or six Degrees, and that of the third about twelve. The like Crowns appear sometimes about the Moon; for in the beginning of the Year 1664, _Febr._ 19th at Night, I saw two such Crowns about her. The Diameter of the first or innermost was about three Degrees, and that of the second about five Degrees and an half. Next about the Moon was a Circle of white, and next about that the inner Crown, which was of a bluish green within next the white, and of a yellow and red without, and next about these Colours were blue and green on the inside of the outward Crown, and red on the outside of it. At the same time there appear'd a Halo about 22 Degrees 35' distant from the center of the Moon. It was elliptical, and its long Diameter was perpendicular to the Horizon, verging below farthest from the Moon. I am told that the Moon has sometimes three or more concentrick Crowns of Colours encompassing one another next about her Body. The more equal the globules of Water or Ice are to one another, the more Crowns of Colours will appear, and the Colours will be the more lively. The Halo at the distance of 22-1/2 Degrees from the Moon is of another sort. By its being oval and remoter from the Moon below than above, I conclude, that it was made by Refraction in some sort of Hail or Snow floating in the Air in an horizontal posture, the refracting Angle being about 58 or 60 Degrees.

THE

THIRD BOOK

OF

OPTICKS

_PART I._

_Observations concerning the Inflexions of the Rays of Light, and the Colours made thereby._

Grimaldo has inform'd us, that if a beam of the Sun's Light be let into a dark Room through a very small hole, the Shadows of things in this Light will be larger than they ought to be if the Rays went on by the Bodies in straight Lines, and that these Shadows have three parallel Fringes, Bands or Ranks of colour'd Light adjacent to them. But if the Hole be enlarged the Fringes grow broad and run into one another, so that they cannot be distinguish'd. These broad Shadows and Fringes have been reckon'd by some to proceed from the ordinary refraction of the Air, but without due examination of the Matter. For the circumstances of the Phaenomenon, so far as I have observed them, are as follows.

_Obs._ 1. I made in a piece of Lead a small Hole with a Pin, whose breadth was the 42d part of an Inch. For 21 of those Pins laid together took up the breadth of half an Inch. Through this Hole I let into my darken'd Chamber a beam of the Sun's Light, and found that the Shadows of Hairs, Thred, Pins, Straws, and such like slender Substances placed in this beam of Light, were considerably broader than they ought to be, if the Rays of Light passed on by these Bodies in right Lines. And particularly a Hair of a Man's Head, whose breadth was but the 280th part of an Inch, being held in this Light, at the distance of about twelve Feet from the Hole, did cast a Shadow which at the distance of four Inches from the Hair was the sixtieth part of an Inch broad, that is, above four times broader than the Hair, and at the distance of two Feet from the Hair was about the eight and twentieth part of an Inch broad, that is, ten times broader than the Hair, and at the distance of ten Feet was the eighth part of an Inch broad, that is 35 times broader.