Opticks Part 4

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 CG and AD/EF 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 CG as above, then the perpendicular velocity of any other emerging Ray CE which was AD/EF CF, will be equal to the square Root of CD_q_ + (_MCq/NGq_ 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 a.s.suming 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 Gla.s.ses, 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 Gla.s.s 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 alt.i.tude 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 Gla.s.s was of a Colour inclining to green. The last of the Prisms mentioned in the third Experiment was of clear white Gla.s.s. 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 Gla.s.s, 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 s.p.a.ce, into which Object-gla.s.ses of Telescopes can collect all sorts of Parallel Rays, is about the 27-1/2th Part of half the Aperture of the Gla.s.s, or 55th Part of the whole Aperture; and that the Focus of the most refrangible Rays is nearer to the Object-gla.s.s than the Focus of the least refrangible ones, by about the 27-1/2th Part of the distance between the Object-gla.s.s 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-gla.s.ses of Telescopes and such-like Gla.s.ses, 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 Gla.s.s 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 Gla.s.s inclining to green, I took another Prism of clear white Gla.s.s; 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 Gla.s.s, which refracted the Light irregularly. Wherefore I covered that Part of the Gla.s.s with black Paper, and letting the Light pa.s.s 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 Gla.s.ses, 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.

When I observed and compared the deepest sensible red, and the Colour in the Confine of green and blue, which at the Rectilinear Sides of the Spectrum was distant from it half the Length of those Sides, the Focus where the Confine of green and blue cast the Species of the Lines distinctly on the Paper, was nearer to the Lens than the Focus, where the red cast those Lines distinctly on it by about 2-1/2 or 2-3/4 Inches. For sometimes the Measures were a little greater, sometimes a little less, but seldom varied from one another above 1/3 of an Inch.

For it was very difficult to define the Places of the Foci, without some little Errors. Now, if the Colours distant half the Length of the Image, (measured at its Rectilinear Sides) give 2-1/2 or 2-3/4 Difference of the distances of their Foci from the Lens, then the Colours distant the whole Length ought to give 5 or 5-1/2 Inches difference of those distances.

But here it's to be noted, that I could not see the red to the full end of the Spectrum, but only to the Center of the Semicircle which bounded that end, or a little farther; and therefore I compared this red not with that Colour which was exactly in the middle of the Spectrum, or Confine of green and blue, but with that which verged a little more to the blue than to the green: And as I reckoned the whole Length of the Colours not to be the whole Length of the Spectrum, but the Length of its Rectilinear Sides, so compleating the semicircular Ends into Circles, when either of the observed Colours fell within those Circles, I measured the distance of that Colour from the semicircular End of the Spectrum, and subducting half this distance from the measured distance of the two Colours, I took the Remainder for their corrected distance; and in these Observations set down this corrected distance for the difference of the distances of their Foci from the Lens. For, as the Length of the Rectilinear Sides of the Spectrum would be the whole Length of all the Colours, were the Circles of which (as we shewed) that Spectrum consists contracted and reduced to Physical Points, so in that Case this corrected distance would be the real distance of the two observed Colours.

When therefore I farther observed the deepest sensible red, and that blue whose corrected distance from it was 7/12 Parts of the Length of the Rectilinear Sides of the Spectrum, the difference of the distances of their Foci from the Lens was about 3-1/4 Inches, and as 7 to 12, so is 3-1/4 to 5-4/7.

When I observed the deepest sensible red, and that indigo whose corrected distance was 8/12 or 2/3 of the Length of the Rectilinear Sides of the Spectrum, the difference of the distances of their Foci from the Lens, was about 3-2/3 Inches, and as 2 to 3, so is 3-2/3 to 5-1/2.

When I observed the deepest sensible red, and that deep indigo whose corrected distance from one another was 9/12 or 3/4 of the Length of the Rectilinear Sides of the Spectrum, the difference of the distances of their Foci from the Lens was about 4 Inches; and as 3 to 4, so is 4 to 5-1/3.

When I observed the deepest sensible red, and that Part of the violet next the indigo, whose corrected distance from the red was 10/12 or 5/6 of the Length of the Rectilinear Sides of the Spectrum, the difference of the distances of their Foci from the Lens was about 4-1/2 Inches, and as 5 to 6, so is 4-1/2 to 5-2/5. For sometimes, when the Lens was advantageously placed, so that its Axis respected the blue, and all Things else were well ordered, and the Sun shone clear, and I held my Eye very near to the Paper on which the Lens cast the Species of the Lines, I could see pretty distinctly the Species of those Lines by that Part of the violet which was next the indigo; and sometimes I could see them by above half the violet, For in making these Experiments I had observed, that the Species of those Colours only appear distinct, which were in or near the Axis of the Lens: So that if the blue or indigo were in the Axis, I could see their Species distinctly; and then the red appeared much less distinct than before. Wherefore I contrived to make the Spectrum of Colours shorter than before, so that both its Ends might be nearer to the Axis of the Lens. And now its Length was about 2-1/2 Inches, and Breadth about 1/5 or 1/6 of an Inch. Also instead of the black Lines on which the Spectrum was cast, I made one black Line broader than those, that I might see its Species more easily; and this Line I divided by short cross Lines into equal Parts, for measuring the distances of the observed Colours. And now I could sometimes see the Species of this Line with its Divisions almost as far as the Center of the semicircular violet End of the Spectrum, and made these farther Observations.

When I observed the deepest sensible red, and that Part of the violet, whose corrected distance from it was about 8/9 Parts of the Rectilinear Sides of the Spectrum, the Difference of the distances of the Foci of those Colours from the Lens, was one time 4-2/3, another time 4-3/4, another time 4-7/8 Inches; and as 8 to 9, so are 4-2/3, 4-3/4, 4-7/8, to 5-1/4, 5-11/32, 5-31/64 respectively.

When I observed the deepest sensible red, and deepest sensible violet, (the corrected distance of which Colours, when all Things were ordered to the best Advantage, and the Sun shone very clear, was about 11/12 or 15/16 Parts of the Length of the Rectilinear Sides of the coloured Spectrum) I found the Difference of the distances of their Foci from the Lens sometimes 4-3/4 sometimes 5-1/4, and for the most part 5 Inches or thereabouts; and as 11 to 12, or 15 to 16, so is five Inches to 5-2/2 or 5-1/3 Inches.

And by this Progression of Experiments I satisfied my self, that had the Light at the very Ends of the Spectrum been strong enough to make the Species of the black Lines appear plainly on the Paper, the Focus of the deepest violet would have been found nearer to the Lens, than the Focus of the deepest red, by about 5-1/3 Inches at least. And this is a farther Evidence, that the Sines of Incidence and Refraction of the several sorts of Rays, hold the same Proportion to one another in the smallest Refractions which they do in the greatest.

My Progress in making this nice and troublesome Experiment I have set down more at large, that they that shall try it after me may be aware of the Circ.u.mspection requisite to make it succeed well. And if they cannot make it succeed so well as I did, they may notwithstanding collect by the Proportion of the distance of the Colours of the Spectrum, to the Difference of the distances of their Foci from the Lens, what would be the Success in the more distant Colours by a better trial. And yet, if they use a broader Lens than I did, and fix it to a long strait Staff, by means of which it may be readily and truly directed to the Colour whose Focus is desired, I question not but the Experiment will succeed better with them than it did with me. For I directed the Axis as nearly as I could to the middle of the Colours, and then the faint Ends of the Spectrum being remote from the Axis, cast their Species less distinctly on the Paper than they would have done, had the Axis been successively directed to them.

Now by what has been said, it's certain that the Rays which differ in Refrangibility do not converge to the same Focus; but if they flow from a lucid Point, as far from the Lens on one side as their Foci are on the other, the Focus of the most refrangible Rays shall be nearer to the Lens than that of the least refrangible, by above the fourteenth Part of the whole distance; and if they flow from a lucid Point, so very remote from the Lens, that before their Incidence they may be accounted parallel, the Focus of the most refrangible Rays shall be nearer to the Lens than the Focus of the least refrangible, by about the 27th or 28th Part of their whole distance from it. And the Diameter of the Circle in the middle s.p.a.ce between those two Foci which they illuminate, when they fall there on any Plane, perpendicular to the Axis (which Circle is the least into which they can all be gathered) is about the 55th Part of the Diameter of the Aperture of the Gla.s.s. So that 'tis a wonder, that Telescopes represent Objects so distinct as they do. But were all the Rays of Light equally refrangible, the Error arising only from the Sphericalness of the Figures of Gla.s.ses would be many hundred times less. For, if the Object-gla.s.s of a Telescope be Plano-convex, and the Plane side be turned towards the Object, and the Diameter of the Sphere, whereof this Gla.s.s is a Segment, be called D, and the Semi-diameter of the Aperture of the Gla.s.s be called S, and the Sine of Incidence out of Gla.s.s into Air, be to the Sine of Refraction as I to R; the Rays which come parallel to the Axis of the Gla.s.s, shall in the Place where the Image of the Object is most distinctly made, be scattered all over a little Circle, whose Diameter is _(Rq/Iq) (S cub./D quad.)_ very nearly,[H] as I gather by computing the Errors of the Rays by the Method of infinite Series, and rejecting the Terms, whose Quant.i.ties are inconsiderable. As for instance, if the Sine of Incidence I, be to the Sine of Refraction R, as 20 to 31, and if D the Diameter of the Sphere, to which the Convex-side of the Gla.s.s is ground, be 100 Feet or 1200 Inches, and S the Semi-diameter of the Aperture be two Inches, the Diameter of the little Circle, (that is (_Rq S cub.)/(Iq D quad._)) will be (31 31 8)/(20 20 1200 1200) (or 961/72000000) Parts of an Inch. But the Diameter of the little Circle, through which these Rays are scattered by unequal Refrangibility, will be about the 55th Part of the Aperture of the Object-gla.s.s, which here is four Inches. And therefore, the Error arising from the Spherical Figure of the Gla.s.s, is to the Error arising from the different Refrangibility of the Rays, as 961/72000000 to 4/55, that is as 1 to 5449; and therefore being in comparison so very little, deserves not to be considered.

[Ill.u.s.tration: FIG. 27.]

But you will say, if the Errors caused by the different Refrangibility be so very great, how comes it to pa.s.s, that Objects appear through Telescopes so distinct as they do? I answer, 'tis because the erring Rays are not scattered uniformly over all that Circular s.p.a.ce, but collected infinitely more densely in the Center than in any other Part of the Circle, and in the Way from the Center to the Circ.u.mference, grow continually rarer and rarer, so as at the Circ.u.mference to become infinitely rare; and by reason of their Rarity are not strong enough to be visible, unless in the Center and very near it. Let ADE [in _Fig._ 27.] represent one of those Circles described with the Center C, and Semi-diameter AC, and let BFG be a smaller Circle concentrick to the former, cutting with its Circ.u.mference the Diameter AC in B, and bisect AC in N; and by my reckoning, the Density of the Light in any Place B, will be to its Density in N, as AB to BC; and the whole Light within the lesser Circle BFG, will be to the whole Light within the greater AED, as the Excess of the Square of AC above the Square of AB, is to the Square of AC. As if BC be the fifth Part of AC, the Light will be four times denser in B than in N, and the whole Light within the less Circle, will be to the whole Light within the greater, as nine to twenty-five. Whence it's evident, that the Light within the less Circle, must strike the Sense much more strongly, than that faint and dilated Light round about between it and the Circ.u.mference of the greater.

But it's farther to be noted, that the most luminous of the Prismatick Colours are the yellow and orange. These affect the Senses more strongly than all the rest together, and next to these in strength are the red and green. The blue compared with these is a faint and dark Colour, and the indigo and violet are much darker and fainter, so that these compared with the stronger Colours are little to be regarded. The Images of Objects are therefore to be placed, not in the Focus of the mean refrangible Rays, which are in the Confine of green and blue, but in the Focus of those Rays which are in the middle of the orange and yellow; there where the Colour is most luminous and fulgent, that is in the brightest yellow, that yellow which inclines more to orange than to green. And by the Refraction of these Rays (whose Sines of Incidence and Refraction in Gla.s.s are as 17 and 11) the Refraction of Gla.s.s and Crystal for Optical Uses is to be measured. Let us therefore place the Image of the Object in the Focus of these Rays, and all the yellow and orange will fall within a Circle, whose Diameter is about the 250th Part of the Diameter of the Aperture of the Gla.s.s. And if you add the brighter half of the red, (that half which is next the orange) and the brighter half of the green, (that half which is next the yellow) about three fifth Parts of the Light of these two Colours will fall within the same Circle, and two fifth Parts will fall without it round about; and that which falls without will be spread through almost as much more s.p.a.ce as that which falls within, and so in the gross be almost three times rarer. Of the other half of the red and green, (that is of the deep dark red and willow green) about one quarter will fall within this Circle, and three quarters without, and that which falls without will be spread through about four or five times more s.p.a.ce than that which falls within; and so in the gross be rarer, and if compared with the whole Light within it, will be about 25 times rarer than all that taken in the gross; or rather more than 30 or 40 times rarer, because the deep red in the end of the Spectrum of Colours made by a Prism is very thin and rare, and the willow green is something rarer than the orange and yellow. The Light of these Colours therefore being so very much rarer than that within the Circle, will scarce affect the Sense, especially since the deep red and willow green of this Light, are much darker Colours than the rest. And for the same reason the blue and violet being much darker Colours than these, and much more rarified, may be neglected. For the dense and bright Light of the Circle, will obscure the rare and weak Light of these dark Colours round about it, and render them almost insensible. The sensible Image of a lucid Point is therefore scarce broader than a Circle, whose Diameter is the 250th Part of the Diameter of the Aperture of the Object-gla.s.s of a good Telescope, or not much broader, if you except a faint and dark misty Light round about it, which a Spectator will scarce regard. And therefore in a Telescope, whose Aperture is four Inches, and Length an hundred Feet, it exceeds not 2'' 45''', or 3''. And in a Telescope whose Aperture is two Inches, and Length 20 or 30 Feet, it may be 5'' or 6'', and scarce above. And this answers well to Experience: For some Astronomers have found the Diameters of the fix'd Stars, in Telescopes of between 20 and 60 Feet in length, to be about 5'' or 6'', or at most 8'' or 10'' in diameter. But if the Eye-Gla.s.s be tincted faintly with the Smoak of a Lamp or Torch, to obscure the Light of the Star, the fainter Light in the Circ.u.mference of the Star ceases to be visible, and the Star (if the Gla.s.s be sufficiently soiled with Smoak) appears something more like a mathematical Point. And for the same Reason, the enormous Part of the Light in the Circ.u.mference of every lucid Point ought to be less discernible in shorter Telescopes than in longer, because the shorter transmit less Light to the Eye.

Now, that the fix'd Stars, by reason of their immense Distance, appear like Points, unless so far as their Light is dilated by Refraction, may appear from hence; that when the Moon pa.s.ses over them and eclipses them, their Light vanishes, not gradually like that of the Planets, but all at once; and in the end of the Eclipse it returns into Sight all at once, or certainly in less time than the second of a Minute; the Refraction of the Moon's Atmosphere a little protracting the time in which the Light of the Star first vanishes, and afterwards returns into Sight.

Now, if we suppose the sensible Image of a lucid Point, to be even 250 times narrower than the Aperture of the Gla.s.s; yet this Image would be still much greater than if it were only from the spherical Figure of the Gla.s.s. For were it not for the different Refrangibility of the Rays, its breadth in an 100 Foot Telescope whose aperture is 4 Inches, would be but 961/72000000 parts of an Inch, as is manifest by the foregoing Computation. And therefore in this case the greatest Errors arising from the spherical Figure of the Gla.s.s, would be to the greatest sensible Errors arising from the different Refrangibility of the Rays as 961/72000000 to 4/250 at most, that is only as 1 to 1200. And this sufficiently shews that it is not the spherical Figures of Gla.s.ses, but the different Refrangibility of the Rays which hinders the perfection of Telescopes.

There is another Argument by which it may appear that the different Refrangibility of Rays, is the true cause of the imperfection of Telescopes. For the Errors of the Rays arising from the spherical Figures of Object-gla.s.ses, are as the Cubes of the Apertures of the Object Gla.s.ses; and thence to make Telescopes of various Lengths magnify with equal distinctness, the Apertures of the Object-gla.s.ses, and the Charges or magnifying Powers ought to be as the Cubes of the square Roots of their lengths; which doth not answer to Experience. But the Errors of the Rays arising from the different Refrangibility, are as the Apertures of the Object-gla.s.ses; and thence to make Telescopes of various lengths, magnify with equal distinctness, their Apertures and Charges ought to be as the square Roots of their lengths; and this answers to Experience, as is well known. For Instance, a Telescope of 64 Feet in length, with an Aperture of 2-2/3 Inches, magnifies about 120 times, with as much distinctness as one of a Foot in length, with 1/3 of an Inch aperture, magnifies 15 times.

[Ill.u.s.tration: FIG. 28.]

Now were it not for this different Refrangibility of Rays, Telescopes might be brought to a greater perfection than we have yet describ'd, by composing the Object-gla.s.s of two Gla.s.ses with Water between them. Let ADFC [in _Fig._ 28.] represent the Object-gla.s.s composed of two Gla.s.ses ABED and BEFC, alike convex on the outsides AGD and CHF, and alike concave on the insides BME, BNE, with Water in the concavity BMEN. Let the Sine of Incidence out of Gla.s.s into Air be as I to R, and out of Water into Air, as K to R, and by consequence out of Gla.s.s into Water, as I to K: and let the Diameter of the Sphere to which the convex sides AGD and CHF are ground be D, and the Diameter of the Sphere to which the concave sides BME and BNE, are ground be to D, as the Cube Root of KK--KI to the Cube Root of RK--RI: and the Refractions on the concave sides of the Gla.s.ses, will very much correct the Errors of the Refractions on the convex sides, so far as they arise from the sphericalness of the Figure. And by this means might Telescopes be brought to sufficient perfection, were it not for the different Refrangibility of several sorts of Rays. But by reason of this different Refrangibility, I do not yet see any other means of improving Telescopes by Refractions alone, than that of increasing their lengths, for which end the late Contrivance of _Hugenius_ seems well accommodated. For very long Tubes are c.u.mbersome, and scarce to be readily managed, and by reason of their length are very apt to bend, and shake by bending, so as to cause a continual trembling in the Objects, whereby it becomes difficult to see them distinctly: whereas by his Contrivance the Gla.s.ses are readily manageable, and the Object-gla.s.s being fix'd upon a strong upright Pole becomes more steady.

Seeing therefore the Improvement of Telescopes of given lengths by Refractions is desperate; I contrived heretofore a Perspective by Reflexion, using instead of an Object-gla.s.s a concave Metal. The diameter of the Sphere to which the Metal was ground concave was about 25 _English_ Inches, and by consequence the length of the Instrument about six Inches and a quarter. The Eye-gla.s.s was Plano-convex, and the diameter of the Sphere to which the convex side was ground was about 1/5 of an Inch, or a little less, and by consequence it magnified between 30 and 40 times. By another way of measuring I found that it magnified about 35 times. The concave Metal bore an Aperture of an Inch and a third part; but the Aperture was limited not by an opake Circle, covering the Limb of the Metal round about, but by an opake Circle placed between the Eyegla.s.s and the Eye, and perforated in the middle with a little round hole for the Rays to pa.s.s through to the Eye. For this Circle by being placed here, stopp'd much of the erroneous Light, which otherwise would have disturbed the Vision. By comparing it with a pretty good Perspective of four Feet in length, made with a concave Eye-gla.s.s, I could read at a greater distance with my own Instrument than with the Gla.s.s. Yet Objects appeared much darker in it than in the Gla.s.s, and that partly because more Light was lost by Reflexion in the Metal, than by Refraction in the Gla.s.s, and partly because my Instrument was overcharged. Had it magnified but 30 or 25 times, it would have made the Object appear more brisk and pleasant. Two of these I made about 16 Years ago, and have one of them still by me, by which I can prove the truth of what I write. Yet it is not so good as at the first. For the concave has been divers times tarnished and cleared again, by rubbing it with very soft Leather. When I made these an Artist in _London_ undertook to imitate it; but using another way of polis.h.i.+ng them than I did, he fell much short of what I had attained to, as I afterwards understood by discoursing the Under-workman he had employed. The Polish I used was in this manner. I had two round Copper Plates, each six Inches in Diameter, the one convex, the other concave, ground very true to one another. On the convex I ground the Object-Metal or Concave which was to be polish'd, 'till it had taken the Figure of the Convex and was ready for a Polish. Then I pitched over the convex very thinly, by dropping melted Pitch upon it, and warming it to keep the Pitch soft, whilst I ground it with the concave Copper wetted to make it spread eavenly all over the convex. Thus by working it well I made it as thin as a Groat, and after the convex was cold I ground it again to give it as true a Figure as I could. Then I took Putty which I had made very fine by was.h.i.+ng it from all its grosser Particles, and laying a little of this upon the Pitch, I ground it upon the Pitch with the concave Copper, till it had done making a Noise; and then upon the Pitch I ground the Object-Metal with a brisk motion, for about two or three Minutes of time, leaning hard upon it. Then I put fresh Putty upon the Pitch, and ground it again till it had done making a noise, and afterwards ground the Object-Metal upon it as before. And this Work I repeated till the Metal was polished, grinding it the last time with all my strength for a good while together, and frequently breathing upon the Pitch, to keep it moist without laying on any more fresh Putty. The Object-Metal was two Inches broad, and about one third part of an Inch thick, to keep it from bending. I had two of these Metals, and when I had polished them both, I tried which was best, and ground the other again, to see if I could make it better than that which I kept. And thus by many Trials I learn'd the way of polis.h.i.+ng, till I made those two reflecting Perspectives I spake of above. For this Art of polis.h.i.+ng will be better learn'd by repeated Practice than by my Description. Before I ground the Object-Metal on the Pitch, I always ground the Putty on it with the concave Copper, till it had done making a noise, because if the Particles of the Putty were not by this means made to stick fast in the Pitch, they would by rolling up and down grate and fret the Object-Metal and fill it full of little holes.

But because Metal is more difficult to polish than Gla.s.s, and is afterwards very apt to be spoiled by tarnis.h.i.+ng, and reflects not so much Light as Gla.s.s quick-silver'd over does: I would propound to use instead of the Metal, a Gla.s.s ground concave on the foreside, and as much convex on the backside, and quick-silver'd over on the convex side.

The Gla.s.s must be every where of the same thickness exactly. Otherwise it will make Objects look colour'd and indistinct. By such a Gla.s.s I tried about five or six Years ago to make a reflecting Telescope of four Feet in length to magnify about 150 times, and I satisfied my self that there wants nothing but a good Artist to bring the Design to perfection. For the Gla.s.s being wrought by one of our _London_ Artists after such a manner as they grind Gla.s.ses for Telescopes, though it seemed as well wrought as the Object-gla.s.ses use to be, yet when it was quick-silver'd, the Reflexion discovered innumerable Inequalities all over the Gla.s.s. And by reason of these Inequalities, Objects appeared indistinct in this Instrument. For the Errors of reflected Rays caused by any Inequality of the Gla.s.s, are about six times greater than the Errors of refracted Rays caused by the like Inequalities. Yet by this Experiment I satisfied my self that the Reflexion on the concave side of the Gla.s.s, which I feared would disturb the Vision, did no sensible prejudice to it, and by consequence that nothing is wanting to perfect these Telescopes, but good Workmen who can grind and polish Gla.s.ses truly spherical. An Object-gla.s.s of a fourteen Foot Telescope, made by an Artificer at _London_, I once mended considerably, by grinding it on Pitch with Putty, and leaning very easily on it in the grinding, lest the Putty should scratch it. Whether this way may not do well enough for polis.h.i.+ng these reflecting Gla.s.ses, I have not yet tried. But he that shall try either this or any other way of polis.h.i.+ng which he may think better, may do well to make his Gla.s.ses ready for polis.h.i.+ng, by grinding them without that Violence, wherewith our _London_ Workmen press their Gla.s.ses in grinding. For by such violent pressure, Gla.s.ses are apt to bend a little in the grinding, and such bending will certainly spoil their Figure. To recommend therefore the consideration of these reflecting Gla.s.ses to such Artists as are curious in figuring Gla.s.ses, I shall describe this optical Instrument in the following Proposition.

_PROP._ VIII. PROB. II.

_To shorten Telescopes._

Let ABCD [in _Fig._ 29.] represent a Gla.s.s spherically concave on the foreside AB, and as much convex on the backside CD, so that it be every where of an equal thickness. Let it not be thicker on one side than on the other, lest it make Objects appear colour'd and indistinct, and let it be very truly wrought and quick-silver'd over on the backside; and set in the Tube VXYZ which must be very black within. Let EFG represent a Prism of Gla.s.s or Crystal placed near the other end of the Tube, in the middle of it, by means of a handle of Bra.s.s or Iron FGK, to the end of which made flat it is cemented. Let this Prism be rectangular at E, and let the other two Angles at F and G be accurately equal to each other, and by consequence equal to half right ones, and let the plane sides FE and GE be square, and by consequence the third side FG a rectangular Parallelogram, whose length is to its breadth in a subduplicate proportion of two to one. Let it be so placed in the Tube, that the Axis of the Speculum may pa.s.s through the middle of the square side EF perpendicularly and by consequence through the middle of the side FG at an Angle of 45 Degrees, and let the side EF be turned towards the Speculum, and the distance of this Prism from the Speculum be such that the Rays of the Light PQ, RS, &c. which are incident upon the Speculum in Lines parallel to the Axis thereof, may enter the Prism at the side EF, and be reflected by the side FG, and thence go out of it through the side GE, to the Point T, which must be the common Focus of the Speculum ABDC, and of a Plano-convex Eye-gla.s.s H, through which those Rays must pa.s.s to the Eye. And let the Rays at their coming out of the Gla.s.s pa.s.s through a small round hole, or aperture made in a little plate of Lead, Bra.s.s, or Silver, wherewith the Gla.s.s is to be covered, which hole must be no bigger than is necessary for Light enough to pa.s.s through. For so it will render the Object distinct, the Plate in which 'tis made intercepting all the erroneous part of the Light which comes from the verges of the Speculum AB. Such an Instrument well made, if it be six Foot long, (reckoning the length from the Speculum to the Prism, and thence to the Focus T) will bear an aperture of six Inches at the Speculum, and magnify between two and three hundred times. But the hole H here limits the aperture with more advantage, than if the aperture was placed at the Speculum. If the Instrument be made longer or shorter, the aperture must be in proportion as the Cube of the square-square Root of the length, and the magnifying as the aperture. But it's convenient that the Speculum be an Inch or two broader than the aperture at the least, and that the Gla.s.s of the Speculum be thick, that it bend not in the working. The Prism EFG must be no bigger than is necessary, and its back side FG must not be quick-silver'd over. For without quicksilver it will reflect all the Light incident on it from the Speculum.

[Ill.u.s.tration: FIG. 29.]

In this Instrument the Object will be inverted, but may be erected by making the square sides FF and EG of the Prism EFG not plane but spherically convex, that the Rays may cross as well before they come at it as afterwards between it and the Eye-gla.s.s. If it be desired that the Instrument bear a larger aperture, that may be also done by composing the Speculum of two Gla.s.ses with Water between them.

If the Theory of making Telescopes could at length be fully brought into Practice, yet there would be certain Bounds beyond which Telescopes could not perform. For the Air through which we look upon the Stars, is in a perpetual Tremor; as may be seen by the tremulous Motion of Shadows cast from high Towers, and by the twinkling of the fix'd Stars. But these Stars do not twinkle when viewed through Telescopes which have large apertures. For the Rays of Light which pa.s.s through divers parts of the aperture, tremble each of them apart, and by means of their various and sometimes contrary Tremors, fall at one and the same time upon different points in the bottom of the Eye, and their trembling Motions are too quick and confused to be perceived severally. And all these illuminated Points const.i.tute one broad lucid Point, composed of those many trembling Points confusedly and insensibly mixed with one another by very short and swift Tremors, and thereby cause the Star to appear broader than it is, and without any trembling of the whole. Long Telescopes may cause Objects to appear brighter and larger than short ones can do, but they cannot be so formed as to take away that confusion of the Rays which arises from the Tremors of the Atmosphere. The only Remedy is a most serene and quiet Air, such as may perhaps be found on the tops of the highest Mountains above the grosser Clouds.

FOOTNOTES:

[C] _See our_ Author's Lectiones Opticae -- 10. _Sect. II. -- 29. and Sect.

III. Prop. 25._

[D] See our Author's _Lectiones Opticae_, Part. I. Sect. 1. --5.

[E] _This is very fully treated of in our_ Author's Lect. Optic. _Part_ I. _Sect._ II.

[F] _See our_ Author's Lect. Optic. Part I. Sect. II. -- 29.

[G] _This is demonstrated in our_ Author's Lect. Optic. _Part_ I.

_Sect._ IV. _Prop._ 37.

[H] _How to do this, is shewn in our_ Author's Lect. Optic. _Part_ I.