Part 5

_Exper._ 12. In the middle of a black Paper I made a round Hole about a fifth or sixth Part of an Inch in diameter. Upon this Paper I caused the Spectrum of homogeneal Light described in the former Proposition, so to fall, that some part of the Light might pass through the Hole of the Paper. This transmitted part of the Light I refracted with a Prism placed behind the Paper, and letting this refracted Light fall perpendicularly upon a white Paper two or three Feet distant from the Prism, I found that the Spectrum formed on the Paper by this Light was not oblong, as when 'tis made (in the third Experiment) by refracting the Sun's compound Light, but was (so far as I could judge by my Eye) perfectly circular, the Length being no greater than the Breadth. Which shews, that this Light is refracted regularly without any Dilatation of the Rays.

_Exper._ 13. In the homogeneal Light I placed a Paper Circle of a quarter of an Inch in diameter, and in the Sun's unrefracted heterogeneal white Light I placed another Paper Circle of the same Bigness. And going from the Papers to the distance of some Feet, I viewed both Circles through a Prism. The Circle illuminated by the Sun's heterogeneal Light appeared very oblong, as in the fourth Experiment, the Length being many times greater than the Breadth; but the other Circle, illuminated with homogeneal Light, appeared circular and distinctly defined, as when 'tis view'd with the naked Eye. Which proves the whole Proposition.

_Exper._ 14. In the homogeneal Light I placed Flies, and such-like minute Objects, and viewing them through a Prism, I saw their Parts as distinctly defined, as if I had viewed them with the naked Eye. The same Objects placed in the Sun's unrefracted heterogeneal Light, which was white, I viewed also through a Prism, and saw them most confusedly defined, so that I could not distinguish their smaller Parts from one another. I placed also the Letters of a small print, one while in the homogeneal Light, and then in the heterogeneal, and viewing them through a Prism, they appeared in the latter Case so confused and indistinct, that I could not read them; but in the former they appeared so distinct, that I could read readily, and thought I saw them as distinct, as when I view'd them with my naked Eye. In both Cases I view'd the same Objects, through the same Prism at the same distance from me, and in the same Situation. There was no difference, but in the Light by which the Objects were illuminated, and which in one Case was simple, and in the other compound; and therefore, the distinct Vision in the former Case, and confused in the latter, could arise from nothing else than from that difference of the Lights. Which proves the whole Proposition.

And in these three Experiments it is farther very remarkable, that the Colour of homogeneal Light was never changed by the Refraction.

_PROP._ VI. THEOR. V.

_The Sine of Incidence of every Ray considered apart, is to its Sine of Refraction in a given Ratio._

That every Ray consider'd apart, is constant to it self in some degree of Refrangibility, is sufficiently manifest out of what has been said. Those Rays, which in the first Refraction, are at equal Incidences most refracted, are also in the following Refractions at equal Incidences most refracted; and so of the least refrangible, and the rest which have any mean Degree of Refrangibility, as is manifest by the fifth, sixth, seventh, eighth, and ninth Experiments. And those which the first Time at like Incidences are equally refracted, are again at like Incidences equally and uniformly refracted, and that whether they be refracted before they be separated from one another, as in the fifth Experiment, or whether they be refracted apart, as in the twelfth, thirteenth and fourteenth Experiments. The Refraction therefore of every Ray apart is regular, and what Rule that Refraction observes we are now to shew.[E]

The late Writers in Opticks teach, that the Sines of Incidence are in a given Proportion to the Sines of Refraction, as was explained in the fifth Axiom, and some by Instruments fitted for measuring of Refractions, or otherwise experimentally examining this Proportion, do acquaint us that they have found it accurate. But whilst they, not understanding the different Refrangibility of several Rays, conceived them all to be refracted according to one and the same Proportion, 'tis to be presumed that they adapted their Measures only to the middle of the refracted Light; so that from their Measures we may conclude only that the Rays which have a mean Degree of Refrangibility, that is, those which when separated from the rest appear green, are refracted according to a given Proportion of their Sines. And therefore we are now to shew, that the like given Proportions obtain in all the rest. That it should be so is very reasonable, Nature being ever conformable to her self; but an experimental Proof is desired. And such a Proof will be had, if we can shew that the Sines of Refraction of Rays differently refrangible are one to another in a given Proportion when their Sines of Incidence are equal. For, if the Sines of Refraction of all the Rays are in given Proportions to the Sine of Refractions of a Ray which has a mean Degree of Refrangibility, and this Sine is in a given Proportion to the equal Sines of Incidence, those other Sines of Refraction will also be in given Proportions to the equal Sines of Incidence. Now, when the Sines of Incidence are equal, it will appear by the following Experiment, that the Sines of Refraction are in a given Proportion to one another.

_Exper._ 15. The Sun shining into a dark Chamber through a little round Hole in the Window-shut, let S [in _Fig._ 26.] represent his round white Image painted on the opposite Wall by his direct Light, PT his oblong coloured Image made by refracting that Light with a Prism placed at the Window; and _pt_, or _2p 2t_, _3p 3t_, his oblong colour'd Image made by refracting again the same Light sideways with a second Prism placed immediately after the first in a cross Position to it, as was explained in the fifth Experiment; that is to say, _pt_ when the Refraction of the second Prism is small, _2p 2t_ when its Refraction is greater, and _3p 3t_ when it is greatest. For such will be the diversity of the Refractions, if the refracting Angle of the second Prism be of various Magnitudes; suppose of fifteen or twenty Degrees to make the Image _pt_, of thirty or forty to make the Image _2p 2t_, and of sixty to make the Image _3p 3t_. But for want of solid Glass Prisms with Angles of convenient Bignesses, there may be Vessels made of polished Plates of Glass cemented together in the form of Prisms and filled with Water. These things being thus ordered, I observed that all the solar Images or coloured Spectrums PT, _pt_, _2p 2t_, _3p 3t_ did very nearly converge to the place S on which the direct Light of the Sun fell and painted his white round Image when the Prisms were taken away. The Axis of the Spectrum PT, that is the Line drawn through the middle of it parallel to its rectilinear Sides, did when produced pass exactly through the middle of that white round Image S. And when the Refraction of the second Prism was equal to the Refraction of the first, the refracting Angles of them both being about 60 Degrees, the Axis of the Spectrum _3p 3t_ made by that Refraction, did when produced pass also through the middle of the same white round Image S. But when the Refraction of the second Prism was less than that of the first, the produced Axes of the Spectrums _tp_ or _2t 2p_ made by that Refraction did cut the produced Axis of the Spectrum TP in the points _m_ and _n_, a little beyond the Center of that white round Image S. Whence the proportion of the Line 3_t_T to the Line 3_p_P was a little greater than the Proportion of 2_t_T or 2_p_P, and this Proportion a little greater than that of _t_T to _p_P. Now when the Light of the Spectrum PT falls perpendicularly upon the Wall, those Lines 3_t_T, 3_p_P, and 2_t_T, and 2_p_P, and _t_T, _p_P, are the Tangents of the Refractions, and therefore by this Experiment the Proportions of the Tangents of the Refractions are obtained, from whence the Proportions of the Sines being derived, they come out equal, so far as by viewing the Spectrums, and using some mathematical Reasoning I could estimate. For I did not make an accurate Computation. So then the Proposition holds true in every Ray apart, so far as appears by Experiment. And that it is accurately true, may be demonstrated upon this Supposition. _That Bodies refract Light by acting upon its Rays in Lines perpendicular to their Surfaces._ But in order to this Demonstration, I must distinguish the Motion of every Ray into two Motions, the one perpendicular to the refracting Surface, the other parallel to it, and concerning the perpendicular Motion lay down the following Proposition.

If any Motion or moving thing whatsoever be incident with any Velocity on any broad and thin space terminated on both sides by two parallel Planes, and in its Passage through that space be urged perpendicularly towards the farther Plane by any force which at given distances from the Plane is of given Quantities; the perpendicular velocity of that Motion or Thing, at its emerging out of that space, shall be always equal to the square Root of the sum of the square of the perpendicular velocity of that Motion or Thing at its Incidence on that space; and of the square of the perpendicular velocity which that Motion or Thing would have at its Emergence, if at its Incidence its perpendicular velocity was infinitely little.

And the same Proposition holds true of any Motion or Thing perpendicularly retarded in its passage through that space, if instead of the sum of the two Squares you take their difference. The Demonstration Mathematicians will easily find out, and therefore I shall not trouble the Reader with it.

Suppose now that a Ray coming most obliquely in the Line MC [in _Fig._ 1.] be refracted at C by the Plane RS into the Line CN, and if it be required to find the Line CE, into which any other Ray AC shall be refracted; let MC, AD, be the Sines of Incidence of the two Rays, and NG, EF, their Sines of Refraction, and let the equal Motions of the incident Rays be represented by the equal Lines MC and AC, and the Motion MC being considered as parallel to the refracting Plane, let the other Motion AC be distinguished into two Motions AD and DC, one of which AD is parallel, and the other DC perpendicular to the refracting Surface. In like manner, let the Motions of the emerging Rays be distinguish'd into two, whereof the perpendicular ones are MC/NG x CG and AD/EF x CF. And if the force of the refracting Plane begins to act upon the Rays either in that Plane or at a certain distance from it on the one side, and ends at a certain distance from it on the other side, and in all places between those two limits acts upon the Rays in Lines perpendicular to that refracting Plane, and the Actions upon the Rays at equal distances from the refracting Plane be equal, and at unequal ones either equal or unequal according to any rate whatever; that Motion of the Ray which is parallel to the refracting Plane, will suffer no Alteration by that Force; and that Motion which is perpendicular to it will be altered according to the rule of the foregoing Proposition. If therefore for the perpendicular velocity of the emerging Ray CN you write MC/NG x CG as above, then the perpendicular velocity of any other emerging Ray CE which was AD/EF x CF, will be equal to the square Root of CD_q_ + (_MCq/NGq_ x CG_q_). And by squaring these Equals, and adding to them the Equals AD_q_ and MC_q_ - CD_q_, and dividing the Sums by the Equals CF_q_ + EF_q_ and CG_q_ + NG_q_, you will have _MCq/NGq_ equal to _ADq/EFq_. Whence AD, the Sine of Incidence, is to EF the Sine of Refraction, as MC to NG, that is, in a given _ratio_. And this Demonstration being general, without determining what Light is, or by what kind of Force it is refracted, or assuming any thing farther than that the refracting Body acts upon the Rays in Lines perpendicular to its Surface; I take it to be a very convincing Argument of the full truth of this Proposition.

So then, if the _ratio_ of the Sines of Incidence and Refraction of any sort of Rays be found in any one case, 'tis given in all cases; and this may be readily found by the Method in the following Proposition.

_PROP._ VII. THEOR. VI.

_The Perfection of Telescopes is impeded by the different Refrangibility of the Rays of Light._

The Imperfection of Telescopes is vulgarly attributed to the spherical Figures of the Glasses, and therefore Mathematicians have propounded to figure them by the conical Sections. To shew that they are mistaken, I have inserted this Proposition; the truth of which will appear by the measure of the Refractions of the several sorts of Rays; and these measures I thus determine.

In the third Experiment of this first Part, where the refracting Angle of the Prism was 62-1/2 Degrees, the half of that Angle 31 deg. 15 min. is the Angle of Incidence of the Rays at their going out of the Glass into the Air[F]; and the Sine of this Angle is 5188, the Radius being 10000. When the Axis of this Prism was parallel to the Horizon, and the Refraction of the Rays at their Incidence on this Prism equal to that at their Emergence out of it, I observed with a Quadrant the Angle which the mean refrangible Rays, (that is those which went to the middle of the Sun's coloured Image) made with the Horizon, and by this Angle and the Sun's altitude observed at the same time, I found the Angle which the emergent Rays contained with the incident to be 44 deg. and 40 min. and the half of this Angle added to the Angle of Incidence 31 deg. 15 min. makes the Angle of Refraction, which is therefore 53 deg. 35 min. and its Sine 8047. These are the Sines of Incidence and Refraction of the mean refrangible Rays, and their Proportion in round Numbers is 20 to 31. This Glass was of a Colour inclining to green. The last of the Prisms mentioned in the third Experiment was of clear white Glass. Its refracting Angle 63-1/2 Degrees. The Angle which the emergent Rays contained, with the incident 45 deg. 50 min. The Sine of half the first Angle 5262. The Sine of half the Sum of the Angles 8157. And their Proportion in round Numbers 20 to 31, as before.

From the Length of the Image, which was about 9-3/4 or 10 Inches, subduct its Breadth, which was 2-1/8 Inches, and the Remainder 7-3/4 Inches would be the Length of the Image were the Sun but a Point, and therefore subtends the Angle which the most and least refrangible Rays, when incident on the Prism in the same Lines, do contain with one another after their Emergence. Whence this Angle is 2 deg. 0'. 7''. For the distance between the Image and the Prism where this Angle is made, was 18-1/2 Feet, and at that distance the Chord 7-3/4 Inches subtends an Angle of 2 deg. 0'. 7''. Now half this Angle is the Angle which these emergent Rays contain with the emergent mean refrangible Rays, and a quarter thereof, that is 30'. 2''. may be accounted the Angle which they would contain with the same emergent mean refrangible Rays, were they co-incident to them within the Glass, and suffered no other Refraction than that at their Emergence. For, if two equal Refractions, the one at the Incidence of the Rays on the Prism, the other at their Emergence, make half the Angle 2 deg. 0'. 7''. then one of those Refractions will make about a quarter of that Angle, and this quarter added to, and subducted from the Angle of Refraction of the mean refrangible Rays, which was 53 deg. 35', gives the Angles of Refraction of the most and least refrangible Rays 54 deg. 5' 2'', and 53 deg. 4' 58'', whose Sines are 8099 and 7995, the common Angle of Incidence being 31 deg. 15', and its Sine 5188; and these Sines in the least round Numbers are in proportion to one another, as 78 and 77 to 50.

Now, if you subduct the common Sine of Incidence 50 from the Sines of Refraction 77 and 78, the Remainders 27 and 28 shew, that in small Refractions the Refraction of the least refrangible Rays is to the Refraction of the most refrangible ones, as 27 to 28 very nearly, and that the difference of the Refractions of the least refrangible and most refrangible Rays is about the 27-1/2th Part of the whole Refraction of the mean refrangible Rays.

Whence they that are skilled in Opticks will easily understand,[G] that the Breadth of the least circular Space, into which Object-glasses of Telescopes can collect all sorts of Parallel Rays, is about the 27-1/2th Part of half the Aperture of the Glass, or 55th Part of the whole Aperture; and that the Focus of the most refrangible Rays is nearer to the Object-glass than the Focus of the least refrangible ones, by about the 27-1/2th Part of the distance between the Object-glass and the Focus of the mean refrangible ones.

And if Rays of all sorts, flowing from any one lucid Point in the Axis of any convex Lens, be made by the Refraction of the Lens to converge to Points not too remote from the Lens, the Focus of the most refrangible Rays shall be nearer to the Lens than the Focus of the least refrangible ones, by a distance which is to the 27-1/2th Part of the distance of the Focus of the mean refrangible Rays from the Lens, as the distance between that Focus and the lucid Point, from whence the Rays flow, is to the distance between that lucid Point and the Lens very nearly.

Now to examine whether the Difference between the Refractions, which the most refrangible and the least refrangible Rays flowing from the same Point suffer in the Object-glasses of Telescopes and such-like Glasses, be so great as is here described, I contrived the following Experiment.

_Exper._ 16. The Lens which I used in the second and eighth Experiments, being placed six Feet and an Inch distant from any Object, collected the Species of that Object by the mean refrangible Rays at the distance of six Feet and an Inch from the Lens on the other side. And therefore by the foregoing Rule, it ought to collect the Species of that Object by the least refrangible Rays at the distance of six Feet and 3-2/3 Inches from the Lens, and by the most refrangible ones at the distance of five Feet and 10-1/3 Inches from it: So that between the two Places, where these least and most refrangible Rays collect the Species, there may be the distance of about 5-1/3 Inches. For by that Rule, as six Feet and an Inch (the distance of the Lens from the lucid Object) is to twelve Feet and two Inches (the distance of the lucid Object from the Focus of the mean refrangible Rays) that is, as One is to Two; so is the 27-1/2th Part of six Feet and an Inch (the distance between the Lens and the same Focus) to the distance between the Focus of the most refrangible Rays and the Focus of the least refrangible ones, which is therefore 5-17/55 Inches, that is very nearly 5-1/3 Inches. Now to know whether this Measure was true, I repeated the second and eighth Experiment with coloured Light, which was less compounded than that I there made use of: For I now separated the heterogeneous Rays from one another by the Method I described in the eleventh Experiment, so as to make a coloured Spectrum about twelve or fifteen Times longer than broad. This Spectrum I cast on a printed Book, and placing the above-mentioned Lens at the distance of six Feet and an Inch from this Spectrum to collect the Species of the illuminated Letters at the same distance on the other side, I found that the Species of the Letters illuminated with blue were nearer to the Lens than those illuminated with deep red by about three Inches, or three and a quarter; but the Species of the Letters illuminated with indigo and violet appeared so confused and indistinct, that I could not read them: Whereupon viewing the Prism, I found it was full of Veins running from one end of the Glass to the other; so that the Refraction could not be regular. I took another Prism therefore which was free from Veins, and instead of the Letters I used two or three Parallel black Lines a little broader than the Strokes of the Letters, and casting the Colours upon these Lines in such manner, that the Lines ran along the Colours from one end of the Spectrum to the other, I found that the Focus where the indigo, or confine of this Colour and violet cast the Species of the black Lines most distinctly, to be about four Inches, or 4-1/4 nearer to the Lens than the Focus, where the deepest red cast the Species of the same black Lines most distinctly. The violet was so faint and dark, that I could not discern the Species of the Lines distinctly by that Colour; and therefore considering that the Prism was made of a dark coloured Glass inclining to green, I took another Prism of clear white Glass; but the Spectrum of Colours which this Prism made had long white Streams of faint Light shooting out from both ends of the Colours, which made me conclude that something was amiss; and viewing the Prism, I found two or three little Bubbles in the Glass, which refracted the Light irregularly. Wherefore I covered that Part of the Glass with black Paper, and letting the Light pass through another Part of it which was free from such Bubbles, the Spectrum of Colours became free from those irregular Streams of Light, and was now such as I desired. But still I found the violet so dark and faint, that I could scarce see the Species of the Lines by the violet, and not at all by the deepest Part of it, which was next the end of the Spectrum. I suspected therefore, that this faint and dark Colour might be allayed by that scattering Light which was refracted, and reflected irregularly, partly by some very small Bubbles in the Glasses, and partly by the Inequalities of their Polish; which Light, tho' it was but little, yet it being of a white Colour, might suffice to affect the Sense so strongly as to disturb the Phaenomena of that weak and dark Colour the violet, and therefore I tried, as in the 12th, 13th, and 14th Experiments, whether the Light of this Colour did not consist of a sensible Mixture of heterogeneous Rays, but found it did not. Nor did the Refractions cause any other sensible Colour than violet to emerge out of this Light, as they would have done out of white Light, and by consequence out of this violet Light had it been sensibly compounded with white Light. And therefore I concluded, that the reason why I could not see the Species of the Lines distinctly by this Colour, was only the Darkness of this Colour, and Thinness of its Light, and its distance from the Axis of the Lens; I divided therefore those Parallel black Lines into equal Parts, by which I might readily know the distances of the Colours in the Spectrum from one another, and noted the distances of the Lens from the Foci of such Colours, as cast the Species of the Lines distinctly, and then considered whether the difference of those distances bear such proportion to 5-1/3 Inches, the greatest Difference of the distances, which the Foci of the deepest red and violet ought to have from the Lens, as the distance of the observed Colours from one another in the Spectrum bear to the greatest distance of the deepest red and violet measured in the Rectilinear Sides of the Spectrum, that is, to the Length of those Sides, or Excess of the Length of the Spectrum above its Breadth. And my Observations were as follows.