src/testdata/Isaac.Newton-Opticks.txt

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OPTICKS:

OR, A

TREATISE

OF THE

_Reflections_, _Refractions_,
_Inflections_ and _Colours_

OF

LIGHT.

_The_ FOURTH EDITION, _corrected_.

By Sir _ISAAC NEWTON_, Knt.

LONDON:

Printed for WILLIAM INNYS at the West-End of St. _Paul's_. MDCCXXX.

TITLE PAGE OF THE 1730 EDITION




SIR ISAAC NEWTON'S ADVERTISEMENTS




Advertisement I


_Part of the ensuing Discourse about Light was written at the Desire of
some Gentlemen of the_ Royal-Society, _in the Year 1675, and then sent
to their Secretary, and read at their Meetings, and the rest was added
about twelve Years after to complete the Theory; except the third Book,
and the last Proposition of the Second, which were since put together
out of scatter'd Papers. To avoid being engaged in Disputes about these
Matters, I have hitherto delayed the printing, and should still have
delayed it, had not the Importunity of Friends prevailed upon me. If any
other Papers writ on this Subject are got out of my Hands they are
imperfect, and were perhaps written before I had tried all the
Experiments here set down, and fully satisfied my self about the Laws of
Refractions and Composition of Colours. I have here publish'd what I
think proper to come abroad, wishing that it may not be translated into
another Language without my Consent._

_The Crowns of Colours, which sometimes appear about the Sun and Moon, I
have endeavoured to give an Account of; but for want of sufficient
Observations leave that Matter to be farther examined. The Subject of
the Third Book I have also left imperfect, not having tried all the
Experiments which I intended when I was about these Matters, nor
repeated some of those which I did try, until I had satisfied my self
about all their Circumstances. To communicate what I have tried, and
leave the rest to others for farther Enquiry, is all my Design in
publishing these Papers._

_In a Letter written to Mr._ Leibnitz _in the year 1679, and published
by Dr._ Wallis, _I mention'd a Method by which I had found some general
Theorems about squaring Curvilinear Figures, or comparing them with the
Conic Sections, or other the simplest Figures with which they may be
compared. And some Years ago I lent out a Manuscript containing such
Theorems, and having since met with some Things copied out of it, I have
on this Occasion made it publick, prefixing to it an_ Introduction, _and
subjoining a_ Scholium _concerning that Method. And I have joined with
it another small Tract concerning the Curvilinear Figures of the Second
Kind, which was also written many Years ago, and made known to some
Friends, who have solicited the making it publick._

                                        _I. N._

April 1, 1704.


Advertisement II

_In this Second Edition of these Opticks I have omitted the Mathematical
Tracts publish'd at the End of the former Edition, as not belonging to
the Subject. And at the End of the Third Book I have added some
Questions. And to shew that I do not take Gravity for an essential
Property of Bodies, I have added one Question concerning its Cause,
chusing to propose it by way of a Question, because I am not yet
satisfied about it for want of Experiments._

                                        _I. N._

July 16, 1717.


Advertisement to this Fourth Edition

_This new Edition of Sir_ Isaac Newton's Opticks _is carefully printed
from the Third Edition, as it was corrected by the Author's own Hand,
and left before his Death with the Bookseller. Since Sir_ Isaac's
Lectiones Opticæ, _which he publickly read in the University of_
Cambridge _in the Years 1669, 1670, and 1671, are lately printed, it has
been thought proper to make at the bottom of the Pages several Citations
from thence, where may be found the Demonstrations, which the Author
omitted in these_ Opticks.

       *       *       *       *       *

Transcriber's Note: There are several greek letters used in the
descriptions of the illustrations. They are signified by [Greek:
letter]. Square roots are noted by the letters sqrt before the equation.

       *       *       *       *       *

THE FIRST BOOK OF OPTICKS




_PART I._


My Design in this Book is not to explain the Properties of Light by
Hypotheses, but to propose and prove them by Reason and Experiments: In
order to which I shall premise the following Definitions and Axioms.




_DEFINITIONS_


DEFIN. I.

_By the Rays of Light I understand its least Parts, and those as well
Successive in the same Lines, as Contemporary in several Lines._ For it
is manifest that Light consists of Parts, both Successive and
Contemporary; because in the same place you may stop that which comes
one moment, and let pass that which comes presently after; and in the
same time you may stop it in any one place, and let it pass in any
other. For that part of Light which is stopp'd cannot be the same with
that which is let pass. The least Light or part of Light, which may be
stopp'd alone without the rest of the Light, or propagated alone, or do
or suffer any thing alone, which the rest of the Light doth not or
suffers not, I call a Ray of Light.


DEFIN. II.

_Refrangibility of the Rays of Light, is their Disposition to be
refracted or turned out of their Way in passing out of one transparent
Body or Medium into another. And a greater or less Refrangibility of
Rays, is their Disposition to be turned more or less out of their Way in
like Incidences on the same Medium._ Mathematicians usually consider the
Rays of Light to be Lines reaching from the luminous Body to the Body
illuminated, and the refraction of those Rays to be the bending or
breaking of those lines in their passing out of one Medium into another.
And thus may Rays and Refractions be considered, if Light be propagated
in an instant. But by an Argument taken from the Æquations of the times
of the Eclipses of _Jupiter's Satellites_, it seems that Light is
propagated in time, spending in its passage from the Sun to us about
seven Minutes of time: And therefore I have chosen to define Rays and
Refractions in such general terms as may agree to Light in both cases.


DEFIN. III.

_Reflexibility of Rays, is their Disposition to be reflected or turned
back into the same Medium from any other Medium upon whose Surface they
fall. And Rays are more or less reflexible, which are turned back more
or less easily._ As if Light pass out of a Glass into Air, and by being
inclined more and more to the common Surface of the Glass and Air,
begins at length to be totally reflected by that Surface; those sorts of
Rays which at like Incidences are reflected most copiously, or by
inclining the Rays begin soonest to be totally reflected, are most
reflexible.


DEFIN. IV.

_The Angle of Incidence is that Angle, which the Line described by the
incident Ray contains with the Perpendicular to the reflecting or
refracting Surface at the Point of Incidence._


DEFIN. V.

_The Angle of Reflexion or Refraction, is the Angle which the line
described by the reflected or refracted Ray containeth with the
Perpendicular to the reflecting or refracting Surface at the Point of
Incidence._


DEFIN. VI.

_The Sines of Incidence, Reflexion, and Refraction, are the Sines of the
Angles of Incidence, Reflexion, and Refraction._


DEFIN. VII

_The Light whose Rays are all alike Refrangible, I call Simple,
Homogeneal and Similar; and that whose Rays are some more Refrangible
than others, I call Compound, Heterogeneal and Dissimilar._ The former
Light I call Homogeneal, not because I would affirm it so in all
respects, but because the Rays which agree in Refrangibility, agree at
least in all those their other Properties which I consider in the
following Discourse.


DEFIN. VIII.

_The Colours of Homogeneal Lights, I call Primary, Homogeneal and
Simple; and those of Heterogeneal Lights, Heterogeneal and Compound._
For these are always compounded of the colours of Homogeneal Lights; as
will appear in the following Discourse.




_AXIOMS._


AX. I.

_The Angles of Reflexion and Refraction, lie in one and the same Plane
with the Angle of Incidence._


AX. II.

_The Angle of Reflexion is equal to the Angle of Incidence._


AX. III.

_If the refracted Ray be returned directly back to the Point of
Incidence, it shall be refracted into the Line before described by the
incident Ray._


AX. IV.

_Refraction out of the rarer Medium into the denser, is made towards the
Perpendicular; that is, so that the Angle of Refraction be less than the
Angle of Incidence._


AX. V.

_The Sine of Incidence is either accurately or very nearly in a given
Ratio to the Sine of Refraction._

Whence if that Proportion be known in any one Inclination of the
incident Ray, 'tis known in all the Inclinations, and thereby the
Refraction in all cases of Incidence on the same refracting Body may be
determined. Thus if the Refraction be made out of Air into Water, the
Sine of Incidence of the red Light is to the Sine of its Refraction as 4
to 3. If out of Air into Glass, the Sines are as 17 to 11. In Light of
other Colours the Sines have other Proportions: but the difference is so
little that it need seldom be considered.

[Illustration: FIG. 1]

Suppose therefore, that RS [in _Fig._ 1.] represents the Surface of
stagnating Water, and that C is the point of Incidence in which any Ray
coming in the Air from A in the Line AC is reflected or refracted, and I
would know whither this Ray shall go after Reflexion or Refraction: I
erect upon the Surface of the Water from the point of Incidence the
Perpendicular CP and produce it downwards to Q, and conclude by the
first Axiom, that the Ray after Reflexion and Refraction, shall be
found somewhere in the Plane of the Angle of Incidence ACP produced. I
let fall therefore upon the Perpendicular CP the Sine of Incidence AD;
and if the reflected Ray be desired, I produce AD to B so that DB be
equal to AD, and draw CB. For this Line CB shall be the reflected Ray;
the Angle of Reflexion BCP and its Sine BD being equal to the Angle and
Sine of Incidence, as they ought to be by the second Axiom, But if the
refracted Ray be desired, I produce AD to H, so that DH may be to AD as
the Sine of Refraction to the Sine of Incidence, that is, (if the Light
be red) as 3 to 4; and about the Center C and in the Plane ACP with the
Radius CA describing a Circle ABE, I draw a parallel to the
Perpendicular CPQ, the Line HE cutting the Circumference in E, and
joining CE, this Line CE shall be the Line of the refracted Ray. For if
EF be let fall perpendicularly on the Line PQ, this Line EF shall be the
Sine of Refraction of the Ray CE, the Angle of Refraction being ECQ; and
this Sine EF is equal to DH, and consequently in Proportion to the Sine
of Incidence AD as 3 to 4.

In like manner, if there be a Prism of Glass (that is, a Glass bounded
with two Equal and Parallel Triangular ends, and three plain and well
polished Sides, which meet in three Parallel Lines running from the
three Angles of one end to the three Angles of the other end) and if the
Refraction of the Light in passing cross this Prism be desired: Let ACB
[in _Fig._ 2.] represent a Plane cutting this Prism transversly to its
three Parallel lines or edges there where the Light passeth through it,
and let DE be the Ray incident upon the first side of the Prism AC where
the Light goes into the Glass; and by putting the Proportion of the Sine
of Incidence to the Sine of Refraction as 17 to 11 find EF the first
refracted Ray. Then taking this Ray for the Incident Ray upon the second
side of the Glass BC where the Light goes out, find the next refracted
Ray FG by putting the Proportion of the Sine of Incidence to the Sine of
Refraction as 11 to 17. For if the Sine of Incidence out of Air into
Glass be to the Sine of Refraction as 17 to 11, the Sine of Incidence
out of Glass into Air must on the contrary be to the Sine of Refraction
as 11 to 17, by the third Axiom.

[Illustration: FIG. 2.]

Much after the same manner, if ACBD [in _Fig._ 3.] represent a Glass
spherically convex on both sides (usually called a _Lens_, such as is a
Burning-glass, or Spectacle-glass, or an Object-glass of a Telescope)
and it be required to know how Light falling upon it from any lucid
point Q shall be refracted, let QM represent a Ray falling upon any
point M of its first spherical Surface ACB, and by erecting a
Perpendicular to the Glass at the point M, find the first refracted Ray
MN by the Proportion of the Sines 17 to 11. Let that Ray in going out of
the Glass be incident upon N, and then find the second refracted Ray
N_q_ by the Proportion of the Sines 11 to 17. And after the same manner
may the Refraction be found when the Lens is convex on one side and
plane or concave on the other, or concave on both sides.

[Illustration: FIG. 3.]


AX. VI.

_Homogeneal Rays which flow from several Points of any Object, and fall
perpendicularly or almost perpendicularly on any reflecting or
refracting Plane or spherical Surface, shall afterwards diverge from so
many other Points, or be parallel to so many other Lines, or converge to
so many other Points, either accurately or without any sensible Error.
And the same thing will happen, if the Rays be reflected or refracted
successively by two or three or more Plane or Spherical Surfaces._

The Point from which Rays diverge or to which they converge may be
called their _Focus_. And the Focus of the incident Rays being given,
that of the reflected or refracted ones may be found by finding the
Refraction of any two Rays, as above; or more readily thus.

_Cas._ 1. Let ACB [in _Fig._ 4.] be a reflecting or refracting Plane,
and Q the Focus of the incident Rays, and Q_q_C a Perpendicular to that
Plane. And if this Perpendicular be produced to _q_, so that _q_C be
equal to QC, the Point _q_ shall be the Focus of the reflected Rays: Or
if _q_C be taken on the same side of the Plane with QC, and in
proportion to QC as the Sine of Incidence to the Sine of Refraction, the
Point _q_ shall be the Focus of the refracted Rays.

[Illustration: FIG. 4.]

_Cas._ 2. Let ACB [in _Fig._ 5.] be the reflecting Surface of any Sphere
whose Centre is E. Bisect any Radius thereof, (suppose EC) in T, and if
in that Radius on the same side the Point T you take the Points Q and
_q_, so that TQ, TE, and T_q_, be continual Proportionals, and the Point
Q be the Focus of the incident Rays, the Point _q_ shall be the Focus of
the reflected ones.

[Illustration: FIG. 5.]

_Cas._ 3. Let ACB [in _Fig._ 6.] be the refracting Surface of any Sphere
whose Centre is E. In any Radius thereof EC produced both ways take ET
and C_t_ equal to one another and severally in such Proportion to that
Radius as the lesser of the Sines of Incidence and Refraction hath to
the difference of those Sines. And then if in the same Line you find any
two Points Q and _q_, so that TQ be to ET as E_t_ to _tq_, taking _tq_
the contrary way from _t_ which TQ lieth from T, and if the Point Q be
the Focus of any incident Rays, the Point _q_ shall be the Focus of the
refracted ones.

[Illustration: FIG. 6.]

And by the same means the Focus of the Rays after two or more Reflexions
or Refractions may be found.

[Illustration: FIG. 7.]

_Cas._ 4. Let ACBD [in _Fig._ 7.] be any refracting Lens, spherically
Convex or Concave or Plane on either side, and let CD be its Axis (that
is, the Line which cuts both its Surfaces perpendicularly, and passes
through the Centres of the Spheres,) and in this Axis produced let F and
_f_ be the Foci of the refracted Rays found as above, when the incident
Rays on both sides the Lens are parallel to the same Axis; and upon the
Diameter F_f_ bisected in E, describe a Circle. Suppose now that any
Point Q be the Focus of any incident Rays. Draw QE cutting the said
Circle in T and _t_, and therein take _tq_ in such proportion to _t_E as
_t_E or TE hath to TQ. Let _tq_ lie the contrary way from _t_ which TQ
doth from T, and _q_ shall be the Focus of the refracted Rays without
any sensible Error, provided the Point Q be not so remote from the Axis,
nor the Lens so broad as to make any of the Rays fall too obliquely on
the refracting Surfaces.[A]

And by the like Operations may the reflecting or refracting Surfaces be
found when the two Foci are given, and thereby a Lens be formed, which
shall make the Rays flow towards or from what Place you please.[B]

So then the Meaning of this Axiom is, that if Rays fall upon any Plane
or Spherical Surface or Lens, and before their Incidence flow from or
towards any Point Q, they shall after Reflexion or Refraction flow from
or towards the Point _q_ found by the foregoing Rules. And if the
incident Rays flow from or towards several points Q, the reflected or
refracted Rays shall flow from or towards so many other Points _q_
found by the same Rules. Whether the reflected and refracted Rays flow
from or towards the Point _q_ is easily known by the situation of that
Point. For if that Point be on the same side of the reflecting or
refracting Surface or Lens with the Point Q, and the incident Rays flow
from the Point Q, the reflected flow towards the Point _q_ and the
refracted from it; and if the incident Rays flow towards Q, the
reflected flow from _q_, and the refracted towards it. And the contrary
happens when _q_ is on the other side of the Surface.


AX. VII.

_Wherever the Rays which come from all the Points of any Object meet
again in so many Points after they have been made to converge by
Reflection or Refraction, there they will make a Picture of the Object
upon any white Body on which they fall._

So if PR [in _Fig._ 3.] represent any Object without Doors, and AB be a
Lens placed at a hole in the Window-shut of a dark Chamber, whereby the
Rays that come from any Point Q of that Object are made to converge and
meet again in the Point _q_; and if a Sheet of white Paper be held at
_q_ for the Light there to fall upon it, the Picture of that Object PR
will appear upon the Paper in its proper shape and Colours. For as the
Light which comes from the Point Q goes to the Point _q_, so the Light
which comes from other Points P and R of the Object, will go to so many
other correspondent Points _p_ and _r_ (as is manifest by the sixth
Axiom;) so that every Point of the Object shall illuminate a
correspondent Point of the Picture, and thereby make a Picture like the
Object in Shape and Colour, this only excepted, that the Picture shall
be inverted. And this is the Reason of that vulgar Experiment of casting
the Species of Objects from abroad upon a Wall or Sheet of white Paper
in a dark Room.

In like manner, when a Man views any Object PQR, [in _Fig._ 8.] the
Light which comes from the several Points of the Object is so refracted
by the transparent skins and humours of the Eye, (that is, by the
outward coat EFG, called the _Tunica Cornea_, and by the crystalline
humour AB which is beyond the Pupil _mk_) as to converge and meet again
in so many Points in the bottom of the Eye, and there to paint the
Picture of the Object upon that skin (called the _Tunica Retina_) with
which the bottom of the Eye is covered. For Anatomists, when they have
taken off from the bottom of the Eye that outward and most thick Coat
called the _Dura Mater_, can then see through the thinner Coats, the
Pictures of Objects lively painted thereon. And these Pictures,
propagated by Motion along the Fibres of the Optick Nerves into the
Brain, are the cause of Vision. For accordingly as these Pictures are
perfect or imperfect, the Object is seen perfectly or imperfectly. If
the Eye be tinged with any colour (as in the Disease of the _Jaundice_)
so as to tinge the Pictures in the bottom of the Eye with that Colour,
then all Objects appear tinged with the same Colour. If the Humours of
the Eye by old Age decay, so as by shrinking to make the _Cornea_ and
Coat of the _Crystalline Humour_ grow flatter than before, the Light
will not be refracted enough, and for want of a sufficient Refraction
will not converge to the bottom of the Eye but to some place beyond it,
and by consequence paint in the bottom of the Eye a confused Picture,
and according to the Indistinctness of this Picture the Object will
appear confused. This is the reason of the decay of sight in old Men,
and shews why their Sight is mended by Spectacles. For those Convex
glasses supply the defect of plumpness in the Eye, and by increasing the
Refraction make the Rays converge sooner, so as to convene distinctly at
the bottom of the Eye if the Glass have a due degree of convexity. And
the contrary happens in short-sighted Men whose Eyes are too plump. For
the Refraction being now too great, the Rays converge and convene in the
Eyes before they come at the bottom; and therefore the Picture made in
the bottom and the Vision caused thereby will not be distinct, unless
the Object be brought so near the Eye as that the place where the
converging Rays convene may be removed to the bottom, or that the
plumpness of the Eye be taken off and the Refractions diminished by a
Concave-glass of a due degree of Concavity, or lastly that by Age the
Eye grow flatter till it come to a due Figure: For short-sighted Men see
remote Objects best in Old Age, and therefore they are accounted to have
the most lasting Eyes.

[Illustration: FIG. 8.]


AX. VIII.

_An Object seen by Reflexion or Refraction, appears in that place from
whence the Rays after their last Reflexion or Refraction diverge in
falling on the Spectator's Eye._

[Illustration: FIG. 9.]

If the Object A [in FIG. 9.] be seen by Reflexion of a Looking-glass
_mn_, it shall appear, not in its proper place A, but behind the Glass
at _a_, from whence any Rays AB, AC, AD, which flow from one and the
same Point of the Object, do after their Reflexion made in the Points B,
C, D, diverge in going from the Glass to E, F, G, where they are
incident on the Spectator's Eyes. For these Rays do make the same
Picture in the bottom of the Eyes as if they had come from the Object
really placed at _a_ without the Interposition of the Looking-glass; and
all Vision is made according to the place and shape of that Picture.

In like manner the Object D [in FIG. 2.] seen through a Prism, appears
not in its proper place D, but is thence translated to some other place
_d_ situated in the last refracted Ray FG drawn backward from F to _d_.

[Illustration: FIG. 10.]

And so the Object Q [in FIG. 10.] seen through the Lens AB, appears at
the place _q_ from whence the Rays diverge in passing from the Lens to
the Eye. Now it is to be noted, that the Image of the Object at _q_ is
so much bigger or lesser than the Object it self at Q, as the distance
of the Image at _q_ from the Lens AB is bigger or less than the distance
of the Object at Q from the same Lens. And if the Object be seen through
two or more such Convex or Concave-glasses, every Glass shall make a new
Image, and the Object shall appear in the place of the bigness of the
last Image. Which consideration unfolds the Theory of Microscopes and
Telescopes. For that Theory consists in almost nothing else than the
describing such Glasses as shall make the last Image of any Object as
distinct and large and luminous as it can conveniently be made.

I have now given in Axioms and their Explications the sum of what hath
hitherto been treated of in Opticks. For what hath been generally
agreed on I content my self to assume under the notion of Principles, in
order to what I have farther to write. And this may suffice for an
Introduction to Readers of quick Wit and good Understanding not yet
versed in Opticks: Although those who are already acquainted with this
Science, and have handled Glasses, will more readily apprehend what
followeth.

FOOTNOTES:

[A] In our Author's _Lectiones Opticæ_, Part I. Sect. IV. Prop 29, 30,
there is an elegant Method of determining these _Foci_; not only in
spherical Surfaces, but likewise in any other curved Figure whatever:
And in Prop. 32, 33, the same thing is done for any Ray lying out of the
Axis.

[B] _Ibid._ Prop. 34.




_PROPOSITIONS._



_PROP._ I. THEOR. I.

_Lights which differ in Colour, differ also in Degrees of
Refrangibility._

The PROOF by Experiments.

_Exper._ 1.

I took a black oblong stiff Paper terminated by Parallel Sides, and with
a Perpendicular right Line drawn cross from one Side to the other,
distinguished it into two equal Parts. One of these parts I painted with
a red colour and the other with a blue. The Paper was very black, and
the Colours intense and thickly laid on, that the Phænomenon might be
more conspicuous. This Paper I view'd through a Prism of solid Glass,
whose two Sides through which the Light passed to the Eye were plane and
well polished, and contained an Angle of about sixty degrees; which
Angle I call the refracting Angle of the Prism. And whilst I view'd it,
I held it and the Prism before a Window in such manner that the Sides of
the Paper were parallel to the Prism, and both those Sides and the Prism
were parallel to the Horizon, and the cross Line was also parallel to
it: and that the Light which fell from the Window upon the Paper made an
Angle with the Paper, equal to that Angle which was made with the same
Paper by the Light reflected from it to the Eye. Beyond the Prism was
the Wall of the Chamber under the Window covered over with black Cloth,
and the Cloth was involved in Darkness that no Light might be reflected
from thence, which in passing by the Edges of the Paper to the Eye,
might mingle itself with the Light of the Paper, and obscure the
Phænomenon thereof. These things being thus ordered, I found that if the
refracting Angle of the Prism be turned upwards, so that the Paper may
seem to be lifted upwards by the Refraction, its blue half will be
lifted higher by the Refraction than its red half. But if the refracting
Angle of the Prism be turned downward, so that the Paper may seem to be
carried lower by the Refraction, its blue half will be carried something
lower thereby than its red half. Wherefore in both Cases the Light which
comes from the blue half of the Paper through the Prism to the Eye, does
in like Circumstances suffer a greater Refraction than the Light which
comes from the red half, and by consequence is more refrangible.

_Illustration._ In the eleventh Figure, MN represents the Window, and DE
the Paper terminated with parallel Sides DJ and HE, and by the
transverse Line FG distinguished into two halfs, the one DG of an
intensely blue Colour, the other FE of an intensely red. And BAC_cab_
represents the Prism whose refracting Planes AB_ba_ and AC_ca_ meet in
the Edge of the refracting Angle A_a_. This Edge A_a_ being upward, is
parallel both to the Horizon, and to the Parallel-Edges of the Paper DJ
and HE, and the transverse Line FG is perpendicular to the Plane of the
Window. And _de_ represents the Image of the Paper seen by Refraction
upwards in such manner, that the blue half DG is carried higher to _dg_
than the red half FE is to _fe_, and therefore suffers a greater
Refraction. If the Edge of the refracting Angle be turned downward, the
Image of the Paper will be refracted downward; suppose to [Greek: de],
and the blue half will be refracted lower to [Greek: dg] than the red
half is to [Greek: pe].

[Illustration: FIG. 11.]

_Exper._ 2. About the aforesaid Paper, whose two halfs were painted over
with red and blue, and which was stiff like thin Pasteboard, I lapped
several times a slender Thred of very black Silk, in such manner that
the several parts of the Thred might appear upon the Colours like so
many black Lines drawn over them, or like long and slender dark Shadows
cast upon them. I might have drawn black Lines with a Pen, but the
Threds were smaller and better defined. This Paper thus coloured and
lined I set against a Wall perpendicularly to the Horizon, so that one
of the Colours might stand to the Right Hand, and the other to the Left.
Close before the Paper, at the Confine of the Colours below, I placed a
Candle to illuminate the Paper strongly: For the Experiment was tried in
the Night. The Flame of the Candle reached up to the lower edge of the
Paper, or a very little higher. Then at the distance of six Feet, and
one or two Inches from the Paper upon the Floor I erected a Glass Lens
four Inches and a quarter broad, which might collect the Rays coming
from the several Points of the Paper, and make them converge towards so
many other Points at the same distance of six Feet, and one or two
Inches on the other side of the Lens, and so form the Image of the
coloured Paper upon a white Paper placed there, after the same manner
that a Lens at a Hole in a Window casts the Images of Objects abroad
upon a Sheet of white Paper in a dark Room. The aforesaid white Paper,
erected perpendicular to the Horizon, and to the Rays which fell upon it
from the Lens, I moved sometimes towards the Lens, sometimes from it, to
find the Places where the Images of the blue and red Parts of the
coloured Paper appeared most distinct. Those Places I easily knew by the
Images of the black Lines which I had made by winding the Silk about the
Paper. For the Images of those fine and slender Lines (which by reason
of their Blackness were like Shadows on the Colours) were confused and
scarce visible, unless when the Colours on either side of each Line were
terminated most distinctly, Noting therefore, as diligently as I could,
the Places where the Images of the red and blue halfs of the coloured
Paper appeared most distinct, I found that where the red half of the
Paper appeared distinct, the blue half appeared confused, so that the
black Lines drawn upon it could scarce be seen; and on the contrary,
where the blue half appeared most distinct, the red half appeared
confused, so that the black Lines upon it were scarce visible. And
between the two Places where these Images appeared distinct there was
the distance of an Inch and a half; the distance of the white Paper from
the Lens, when the Image of the red half of the coloured Paper appeared
most distinct, being greater by an Inch and an half than the distance of
the same white Paper from the Lens, when the Image of the blue half
appeared most distinct. In like Incidences therefore of the blue and red
upon the Lens, the blue was refracted more by the Lens than the red, so
as to converge sooner by an Inch and a half, and therefore is more
refrangible.

_Illustration._ In the twelfth Figure (p. 27), DE signifies the coloured
Paper, DG the blue half, FE the red half, MN the Lens, HJ the white
Paper in that Place where the red half with its black Lines appeared
distinct, and _hi_ the same Paper in that Place where the blue half
appeared distinct. The Place _hi_ was nearer to the Lens MN than the
Place HJ by an Inch and an half.

_Scholium._ The same Things succeed, notwithstanding that some of the
Circumstances be varied; as in the first Experiment when the Prism and
Paper are any ways inclined to the Horizon, and in both when coloured
Lines are drawn upon very black Paper. But in the Description of these
Experiments, I have set down such Circumstances, by which either the
Phænomenon might be render'd more conspicuous, or a Novice might more
easily try them, or by which I did try them only. The same Thing, I have
often done in the following Experiments: Concerning all which, this one
Admonition may suffice. Now from these Experiments it follows not, that
all the Light of the blue is more refrangible than all the Light of the
red: For both Lights are mixed of Rays differently refrangible, so that
in the red there are some Rays not less refrangible than those of the
blue, and in the blue there are some Rays not more refrangible than
those of the red: But these Rays, in proportion to the whole Light, are
but few, and serve to diminish the Event of the Experiment, but are not
able to destroy it. For, if the red and blue Colours were more dilute
and weak, the distance of the Images would be less than an Inch and a
half; and if they were more intense and full, that distance would be
greater, as will appear hereafter. These Experiments may suffice for the
Colours of Natural Bodies. For in the Colours made by the Refraction of
Prisms, this Proposition will appear by the Experiments which are now to
follow in the next Proposition.


_PROP._ II. THEOR. II.

_The Light of the Sun consists of Rays differently Refrangible._

The PROOF by Experiments.

[Illustration: FIG. 12.]

[Illustration: FIG. 13.]

_Exper._ 3.

In a very dark Chamber, at a round Hole, about one third Part of an Inch
broad, made in the Shut of a Window, I placed a Glass Prism, whereby the
Beam of the Sun's Light, which came in at that Hole, might be refracted
upwards toward the opposite Wall of the Chamber, and there form a
colour'd Image of the Sun. The Axis of the Prism (that is, the Line
passing through the middle of the Prism from one end of it to the other
end parallel to the edge of the Refracting Angle) was in this and the
following Experiments perpendicular to the incident Rays. About this
Axis I turned the Prism slowly, and saw the refracted Light on the Wall,
or coloured Image of the Sun, first to descend, and then to ascend.
Between the Descent and Ascent, when the Image seemed Stationary, I
stopp'd the Prism, and fix'd it in that Posture, that it should be moved
no more. For in that Posture the Refractions of the Light at the two
Sides of the refracting Angle, that is, at the Entrance of the Rays into
the Prism, and at their going out of it, were equal to one another.[C]
So also in other Experiments, as often as I would have the Refractions
on both sides the Prism to be equal to one another, I noted the Place
where the Image of the Sun formed by the refracted Light stood still
between its two contrary Motions, in the common Period of its Progress
and Regress; and when the Image fell upon that Place, I made fast the
Prism. And in this Posture, as the most convenient, it is to be
understood that all the Prisms are placed in the following Experiments,
unless where some other Posture is described. The Prism therefore being
placed in this Posture, I let the refracted Light fall perpendicularly
upon a Sheet of white Paper at the opposite Wall of the Chamber, and
observed the Figure and Dimensions of the Solar Image formed on the
Paper by that Light. This Image was Oblong and not Oval, but terminated
with two Rectilinear and Parallel Sides, and two Semicircular Ends. On
its Sides it was bounded pretty distinctly, but on its Ends very
confusedly and indistinctly, the Light there decaying and vanishing by
degrees. The Breadth of this Image answered to the Sun's Diameter, and
was about two Inches and the eighth Part of an Inch, including the
Penumbra. For the Image was eighteen Feet and an half distant from the
Prism, and at this distance that Breadth, if diminished by the Diameter
of the Hole in the Window-shut, that is by a quarter of an Inch,
subtended an Angle at the Prism of about half a Degree, which is the
Sun's apparent Diameter. But the Length of the Image was about ten
Inches and a quarter, and the Length of the Rectilinear Sides about
eight Inches; and the refracting Angle of the Prism, whereby so great a
Length was made, was 64 degrees. With a less Angle the Length of the
Image was less, the Breadth remaining the same. If the Prism was turned
about its Axis that way which made the Rays emerge more obliquely out of
the second refracting Surface of the Prism, the Image soon became an
Inch or two longer, or more; and if the Prism was turned about the
contrary way, so as to make the Rays fall more obliquely on the first
refracting Surface, the Image soon became an Inch or two shorter. And
therefore in trying this Experiment, I was as curious as I could be in
placing the Prism by the above-mention'd Rule exactly in such a Posture,
that the Refractions of the Rays at their Emergence out of the Prism
might be equal to that at their Incidence on it. This Prism had some
Veins running along within the Glass from one end to the other, which
scattered some of the Sun's Light irregularly, but had no sensible
Effect in increasing the Length of the coloured Spectrum. For I tried
the same Experiment with other Prisms with the same Success. And
particularly with a Prism which seemed free from such Veins, and whose
refracting Angle was 62-1/2 Degrees, I found the Length of the Image
9-3/4 or 10 Inches at the distance of 18-1/2 Feet from the Prism, the
Breadth of the Hole in the Window-shut being 1/4 of an Inch, as before.
And because it is easy to commit a Mistake in placing the Prism in its
due Posture, I repeated the Experiment four or five Times, and always
found the Length of the Image that which is set down above. With another
Prism of clearer Glass and better Polish, which seemed free from Veins,
and whose refracting Angle was 63-1/2 Degrees, the Length of this Image
at the same distance of 18-1/2 Feet was also about 10 Inches, or 10-1/8.
Beyond these Measures for about a 1/4 or 1/3 of an Inch at either end of
the Spectrum the Light of the Clouds seemed to be a little tinged with
red and violet, but so very faintly, that I suspected that Tincture
might either wholly, or in great Measure arise from some Rays of the
Spectrum scattered irregularly by some Inequalities in the Substance and
Polish of the Glass, and therefore I did not include it in these
Measures. Now the different Magnitude of the hole in the Window-shut,
and different thickness of the Prism where the Rays passed through it,
and different inclinations of the Prism to the Horizon, made no sensible
changes in the length of the Image. Neither did the different matter of
the Prisms make any: for in a Vessel made of polished Plates of Glass
cemented together in the shape of a Prism and filled with Water, there
is the like Success of the Experiment according to the quantity of the
Refraction. It is farther to be observed, that the Rays went on in right
Lines from the Prism to the Image, and therefore at their very going out
of the Prism had all that Inclination to one another from which the
length of the Image proceeded, that is, the Inclination of more than two
degrees and an half. And yet according to the Laws of Opticks vulgarly
received, they could not possibly be so much inclined to one another.[D]
For let EG [_Fig._ 13. (p. 27)] represent the Window-shut, F the hole
made therein through which a beam of the Sun's Light was transmitted
into the darkened Chamber, and ABC a Triangular Imaginary Plane whereby
the Prism is feigned to be cut transversely through the middle of the
Light. Or if you please, let ABC represent the Prism it self, looking
directly towards the Spectator's Eye with its nearer end: And let XY be
the Sun, MN the Paper upon which the Solar Image or Spectrum is cast,
and PT the Image it self whose sides towards _v_ and _w_ are Rectilinear
and Parallel, and ends towards P and T Semicircular. YKHP and XLJT are
two Rays, the first of which comes from the lower part of the Sun to the
higher part of the Image, and is refracted in the Prism at K and H, and
the latter comes from the higher part of the Sun to the lower part of
the Image, and is refracted at L and J. Since the Refractions on both
sides the Prism are equal to one another, that is, the Refraction at K
equal to the Refraction at J, and the Refraction at L equal to the
Refraction at H, so that the Refractions of the incident Rays at K and L
taken together, are equal to the Refractions of the emergent Rays at H
and J taken together: it follows by adding equal things to equal things,
that the Refractions at K and H taken together, are equal to the
Refractions at J and L taken together, and therefore the two Rays being
equally refracted, have the same Inclination to one another after
Refraction which they had before; that is, the Inclination of half a
Degree answering to the Sun's Diameter. For so great was the inclination
of the Rays to one another before Refraction. So then, the length of the
Image PT would by the Rules of Vulgar Opticks subtend an Angle of half a
Degree at the Prism, and by Consequence be equal to the breadth _vw_;
and therefore the Image would be round. Thus it would be were the two
Rays XLJT and YKHP, and all the rest which form the Image P_w_T_v_,
alike refrangible. And therefore seeing by Experience it is found that
the Image is not round, but about five times longer than broad, the Rays
which going to the upper end P of the Image suffer the greatest
Refraction, must be more refrangible than those which go to the lower
end T, unless the Inequality of Refraction be casual.

This Image or Spectrum PT was coloured, being red at its least refracted
end T, and violet at its most refracted end P, and yellow green and
blue in the intermediate Spaces. Which agrees with the first
Proposition, that Lights which differ in Colour, do also differ in
Refrangibility. The length of the Image in the foregoing Experiments, I
measured from the faintest and outmost red at one end, to the faintest
and outmost blue at the other end, excepting only a little Penumbra,
whose breadth scarce exceeded a quarter of an Inch, as was said above.

_Exper._ 4. In the Sun's Beam which was propagated into the Room through
the hole in the Window-shut, at the distance of some Feet from the hole,
I held the Prism in such a Posture, that its Axis might be perpendicular
to that Beam. Then I looked through the Prism upon the hole, and turning
the Prism to and fro about its Axis, to make the Image of the Hole
ascend and descend, when between its two contrary Motions it seemed
Stationary, I stopp'd the Prism, that the Refractions of both sides of
the refracting Angle might be equal to each other, as in the former
Experiment. In this situation of the Prism viewing through it the said
Hole, I observed the length of its refracted Image to be many times
greater than its breadth, and that the most refracted part thereof
appeared violet, the least refracted red, the middle parts blue, green
and yellow in order. The same thing happen'd when I removed the Prism
out of the Sun's Light, and looked through it upon the hole shining by
the Light of the Clouds beyond it. And yet if the Refraction were done
regularly according to one certain Proportion of the Sines of Incidence
and Refraction as is vulgarly supposed, the refracted Image ought to
have appeared round.

So then, by these two Experiments it appears, that in Equal Incidences
there is a considerable inequality of Refractions. But whence this
inequality arises, whether it be that some of the incident Rays are
refracted more, and others less, constantly, or by chance, or that one
and the same Ray is by Refraction disturbed, shatter'd, dilated, and as
it were split and spread into many diverging Rays, as _Grimaldo_
supposes, does not yet appear by these Experiments, but will appear by
those that follow.

_Exper._ 5. Considering therefore, that if in the third Experiment the
Image of the Sun should be drawn out into an oblong Form, either by a
Dilatation of every Ray, or by any other casual inequality of the
Refractions, the same oblong Image would by a second Refraction made
sideways be drawn out as much in breadth by the like Dilatation of the
Rays, or other casual inequality of the Refractions sideways, I tried
what would be the Effects of such a second Refraction. For this end I
ordered all things as in the third Experiment, and then placed a second
Prism immediately after the first in a cross Position to it, that it
might again refract the beam of the Sun's Light which came to it through
the first Prism. In the first Prism this beam was refracted upwards, and
in the second sideways. And I found that by the Refraction of the second
Prism, the breadth of the Image was not increased, but its superior
part, which in the first Prism suffered the greater Refraction, and
appeared violet and blue, did again in the second Prism suffer a greater
Refraction than its inferior part, which appeared red and yellow, and
this without any Dilatation of the Image in breadth.

[Illustration: FIG. 14]

_Illustration._ Let S [_Fig._ 14, 15.] represent the Sun, F the hole in
the Window, ABC the first Prism, DH the second Prism, Y the round Image
of the Sun made by a direct beam of Light when the Prisms are taken
away, PT the oblong Image of the Sun made by that beam passing through
the first Prism alone, when the second Prism is taken away, and _pt_ the
Image made by the cross Refractions of both Prisms together. Now if the
Rays which tend towards the several Points of the round Image Y were
dilated and spread by the Refraction of the first Prism, so that they
should not any longer go in single Lines to single Points, but that
every Ray being split, shattered, and changed from a Linear Ray to a
Superficies of Rays diverging from the Point of Refraction, and lying in
the Plane of the Angles of Incidence and Refraction, they should go in
those Planes to so many Lines reaching almost from one end of the Image
PT to the other, and if that Image should thence become oblong: those
Rays and their several parts tending towards the several Points of the
Image PT ought to be again dilated and spread sideways by the transverse
Refraction of the second Prism, so as to compose a four square Image,
such as is represented at [Greek: pt]. For the better understanding of
which, let the Image PT be distinguished into five equal parts PQK,
KQRL, LRSM, MSVN, NVT. And by the same irregularity that the orbicular
Light Y is by the Refraction of the first Prism dilated and drawn out
into a long Image PT, the Light PQK which takes up a space of the same
length and breadth with the Light Y ought to be by the Refraction of the
second Prism dilated and drawn out into the long Image _[Greek: p]qkp_,
and the Light KQRL into the long Image _kqrl_, and the Lights LRSM,
MSVN, NVT, into so many other long Images _lrsm_, _msvn_, _nvt[Greek:
t]_; and all these long Images would compose the four square Images
_[Greek: pt]_. Thus it ought to be were every Ray dilated by Refraction,
and spread into a triangular Superficies of Rays diverging from the
Point of Refraction. For the second Refraction would spread the Rays one
way as much as the first doth another, and so dilate the Image in
breadth as much as the first doth in length. And the same thing ought to
happen, were some rays casually refracted more than others. But the
Event is otherwise. The Image PT was not made broader by the Refraction
of the second Prism, but only became oblique, as 'tis represented at
_pt_, its upper end P being by the Refraction translated to a greater
distance than its lower end T. So then the Light which went towards the
upper end P of the Image, was (at equal Incidences) more refracted in
the second Prism, than the Light which tended towards the lower end T,
that is the blue and violet, than the red and yellow; and therefore was
more refrangible. The same Light was by the Refraction of the first
Prism translated farther from the place Y to which it tended before
Refraction; and therefore suffered as well in the first Prism as in the
second a greater Refraction than the rest of the Light, and by
consequence was more refrangible than the rest, even before its
incidence on the first Prism.

Sometimes I placed a third Prism after the second, and sometimes also a
fourth after the third, by all which the Image might be often refracted
sideways: but the Rays which were more refracted than the rest in the
first Prism were also more refracted in all the rest, and that without
any Dilatation of the Image sideways: and therefore those Rays for their
constancy of a greater Refraction are deservedly reputed more
refrangible.

[Illustration: FIG. 15]

But that the meaning of this Experiment may more clearly appear, it is
to be considered that the Rays which are equally refrangible do fall
upon a Circle answering to the Sun's Disque. For this was proved in the
third Experiment. By a Circle I understand not here a perfect
geometrical Circle, but any orbicular Figure whose length is equal to
its breadth, and which, as to Sense, may seem circular. Let therefore AG
[in _Fig._ 15.] represent the Circle which all the most refrangible Rays
propagated from the whole Disque of the Sun, would illuminate and paint
upon the opposite Wall if they were alone; EL the Circle which all the
least refrangible Rays would in like manner illuminate and paint if they
were alone; BH, CJ, DK, the Circles which so many intermediate sorts of
Rays would successively paint upon the Wall, if they were singly
propagated from the Sun in successive order, the rest being always
intercepted; and conceive that there are other intermediate Circles
without Number, which innumerable other intermediate sorts of Rays would
successively paint upon the Wall if the Sun should successively emit
every sort apart. And seeing the Sun emits all these sorts at once, they
must all together illuminate and paint innumerable equal Circles, of all
which, being according to their degrees of Refrangibility placed in
order in a continual Series, that oblong Spectrum PT is composed which I
described in the third Experiment. Now if the Sun's circular Image Y [in
_Fig._ 15.] which is made by an unrefracted beam of Light was by any
Dilation of the single Rays, or by any other irregularity in the
Refraction of the first Prism, converted into the oblong Spectrum, PT:
then ought every Circle AG, BH, CJ, &c. in that Spectrum, by the cross
Refraction of the second Prism again dilating or otherwise scattering
the Rays as before, to be in like manner drawn out and transformed into
an oblong Figure, and thereby the breadth of the Image PT would be now
as much augmented as the length of the Image Y was before by the
Refraction of the first Prism; and thus by the Refractions of both
Prisms together would be formed a four square Figure _p[Greek:
p]t[Greek: t]_, as I described above. Wherefore since the breadth of the
Spectrum PT is not increased by the Refraction sideways, it is certain
that the Rays are not split or dilated, or otherways irregularly
scatter'd by that Refraction, but that every Circle is by a regular and
uniform Refraction translated entire into another Place, as the Circle
AG by the greatest Refraction into the place _ag_, the Circle BH by a
less Refraction into the place _bh_, the Circle CJ by a Refraction still
less into the place _ci_, and so of the rest; by which means a new
Spectrum _pt_ inclined to the former PT is in like manner composed of
Circles lying in a right Line; and these Circles must be of the same
bigness with the former, because the breadths of all the Spectrums Y, PT
and _pt_ at equal distances from the Prisms are equal.

I considered farther, that by the breadth of the hole F through which
the Light enters into the dark Chamber, there is a Penumbra made in the
Circuit of the Spectrum Y, and that Penumbra remains in the rectilinear
Sides of the Spectrums PT and _pt_. I placed therefore at that hole a
Lens or Object-glass of a Telescope which might cast the Image of the
Sun distinctly on Y without any Penumbra at all, and found that the
Penumbra of the rectilinear Sides of the oblong Spectrums PT and _pt_
was also thereby taken away, so that those Sides appeared as distinctly
defined as did the Circumference of the first Image Y. Thus it happens
if the Glass of the Prisms be free from Veins, and their sides be
accurately plane and well polished without those numberless Waves or
Curles which usually arise from Sand-holes a little smoothed in
polishing with Putty. If the Glass be only well polished and free from
Veins, and the Sides not accurately plane, but a little Convex or
Concave, as it frequently happens; yet may the three Spectrums Y, PT and
_pt_ want Penumbras, but not in equal distances from the Prisms. Now
from this want of Penumbras, I knew more certainly that every one of the
Circles was refracted according to some most regular, uniform and
constant Law. For if there were any irregularity in the Refraction, the
right Lines AE and GL, which all the Circles in the Spectrum PT do
touch, could not by that Refraction be translated into the Lines _ae_
and _gl_ as distinct and straight as they were before, but there would
arise in those translated Lines some Penumbra or Crookedness or
Undulation, or other sensible Perturbation contrary to what is found by
Experience. Whatsoever Penumbra or Perturbation should be made in the
Circles by the cross Refraction of the second Prism, all that Penumbra
or Perturbation would be conspicuous in the right Lines _ae_ and _gl_
which touch those Circles. And therefore since there is no such Penumbra
or Perturbation in those right Lines, there must be none in the
Circles. Since the distance between those Tangents or breadth of the
Spectrum is not increased by the Refractions, the Diameters of the
Circles are not increased thereby. Since those Tangents continue to be
right Lines, every Circle which in the first Prism is more or less
refracted, is exactly in the same proportion more or less refracted in
the second. And seeing all these things continue to succeed after the
same manner when the Rays are again in a third Prism, and again in a
fourth refracted sideways, it is evident that the Rays of one and the
same Circle, as to their degree of Refrangibility, continue always
uniform and homogeneal to one another, and that those of several Circles
do differ in degree of Refrangibility, and that in some certain and
constant Proportion. Which is the thing I was to prove.

There is yet another Circumstance or two of this Experiment by which it
becomes still more plain and convincing. Let the second Prism DH [in
_Fig._ 16.] be placed not immediately after the first, but at some
distance from it; suppose in the mid-way between it and the Wall on
which the oblong Spectrum PT is cast, so that the Light from the first
Prism may fall upon it in the form of an oblong Spectrum [Greek: pt]
parallel to this second Prism, and be refracted sideways to form the
oblong Spectrum _pt_ upon the Wall. And you will find as before, that
this Spectrum _pt_ is inclined to that Spectrum PT, which the first
Prism forms alone without the second; the blue ends P and _p_ being
farther distant from one another than the red ones T and _t_, and by
consequence that the Rays which go to the blue end [Greek: p] of the
Image [Greek: pt], and which therefore suffer the greatest Refraction in
the first Prism, are again in the second Prism more refracted than the
rest.

[Illustration: FIG. 16.]

[Illustration: FIG. 17.]

The same thing I try'd also by letting the Sun's Light into a dark Room
through two little round holes F and [Greek: ph] [in _Fig._ 17.] made in
the Window, and with two parallel Prisms ABC and [Greek: abg] placed at
those holes (one at each) refracting those two beams of Light to the
opposite Wall of the Chamber, in such manner that the two colour'd
Images PT and MN which they there painted were joined end to end and lay
in one straight Line, the red end T of the one touching the blue end M
of the other. For if these two refracted Beams were again by a third
Prism DH placed cross to the two first, refracted sideways, and the
Spectrums thereby translated to some other part of the Wall of the
Chamber, suppose the Spectrum PT to _pt_ and the Spectrum MN to _mn_,
these translated Spectrums _pt_ and _mn_ would not lie in one straight
Line with their ends contiguous as before, but be broken off from one
another and become parallel, the blue end _m_ of the Image _mn_ being by
a greater Refraction translated farther from its former place MT, than
the red end _t_ of the other Image _pt_ from the same place MT; which
puts the Proposition past Dispute. And this happens whether the third
Prism DH be placed immediately after the two first, or at a great
distance from them, so that the Light refracted in the two first Prisms
be either white and circular, or coloured and oblong when it falls on
the third.

_Exper._ 6. In the middle of two thin Boards I made round holes a third
part of an Inch in diameter, and in the Window-shut a much broader hole
being made to let into my darkned Chamber a large Beam of the Sun's
Light; I placed a Prism behind the Shut in that beam to refract it
towards the opposite Wall, and close behind the Prism I fixed one of the
Boards, in such manner that the middle of the refracted Light might pass
through the hole made in it, and the rest be intercepted by the Board.
Then at the distance of about twelve Feet from the first Board I fixed
the other Board in such manner that the middle of the refracted Light
which came through the hole in the first Board, and fell upon the
opposite Wall, might pass through the hole in this other Board, and the
rest being intercepted by the Board might paint upon it the coloured
Spectrum of the Sun. And close behind this Board I fixed another Prism
to refract the Light which came through the hole. Then I returned
speedily to the first Prism, and by turning it slowly to and fro about
its Axis, I caused the Image which fell upon the second Board to move up
and down upon that Board, that all its parts might successively pass
through the hole in that Board and fall upon the Prism behind it. And in
the mean time, I noted the places on the opposite Wall to which that
Light after its Refraction in the second Prism did pass; and by the
difference of the places I found that the Light which being most
refracted in the first Prism did go to the blue end of the Image, was
again more refracted in the second Prism than the Light which went to
the red end of that Image, which proves as well the first Proposition as
the second. And this happened whether the Axis of the two Prisms were
parallel, or inclined to one another, and to the Horizon in any given
Angles.

_Illustration._ Let F [in _Fig._ 18.] be the wide hole in the
Window-shut, through which the Sun shines upon the first Prism ABC, and
let the refracted Light fall upon the middle of the Board DE, and the
middle part of that Light upon the hole G made in the middle part of
that Board. Let this trajected part of that Light fall again upon the
middle of the second Board _de_, and there paint such an oblong coloured
Image of the Sun as was described in the third Experiment. By turning
the Prism ABC slowly to and fro about its Axis, this Image will be made
to move up and down the Board _de_, and by this means all its parts from
one end to the other may be made to pass successively through the hole
_g_ which is made in the middle of that Board. In the mean while another
Prism _abc_ is to be fixed next after that hole _g_, to refract the
trajected Light a second time. And these things being thus ordered, I
marked the places M and N of the opposite Wall upon which the refracted
Light fell, and found that whilst the two Boards and second Prism
remained unmoved, those places by turning the first Prism about its Axis
were changed perpetually. For when the lower part of the Light which
fell upon the second Board _de_ was cast through the hole _g_, it went
to a lower place M on the Wall and when the higher part of that Light
was cast through the same hole _g_, it went to a higher place N on the
Wall, and when any intermediate part of the Light was cast through that
hole, it went to some place on the Wall between M and N. The unchanged
Position of the holes in the Boards, made the Incidence of the Rays upon
the second Prism to be the same in all cases. And yet in that common
Incidence some of the Rays were more refracted, and others less. And
those were more refracted in this Prism, which by a greater Refraction
in the first Prism were more turned out of the way, and therefore for
their Constancy of being more refracted are deservedly called more
refrangible.

[Illustration: FIG. 18.]

[Illustration: FIG. 20.]

_Exper._ 7. At two holes made near one another in my Window-shut I
placed two Prisms, one at each, which might cast upon the opposite Wall
(after the manner of the third Experiment) two oblong coloured Images of
the Sun. And at a little distance from the Wall I placed a long slender
Paper with straight and parallel edges, and ordered the Prisms and Paper
so, that the red Colour of one Image might fall directly upon one half
of the Paper, and the violet Colour of the other Image upon the other
half of the same Paper; so that the Paper appeared of two Colours, red
and violet, much after the manner of the painted Paper in the first and
second Experiments. Then with a black Cloth I covered the Wall behind
the Paper, that no Light might be reflected from it to disturb the
Experiment, and viewing the Paper through a third Prism held parallel
to it, I saw that half of it which was illuminated by the violet Light
to be divided from the other half by a greater Refraction, especially
when I went a good way off from the Paper. For when I viewed it too near
at hand, the two halfs of the Paper did not appear fully divided from
one another, but seemed contiguous at one of their Angles like the
painted Paper in the first Experiment. Which also happened when the
Paper was too broad.

[Illustration: FIG. 19.]

Sometimes instead of the Paper I used a white Thred, and this appeared
through the Prism divided into two parallel Threds as is represented in
the nineteenth Figure, where DG denotes the Thred illuminated with
violet Light from D to E and with red Light from F to G, and _defg_ are
the parts of the Thred seen by Refraction. If one half of the Thred be
constantly illuminated with red, and the other half be illuminated with
all the Colours successively, (which may be done by causing one of the
Prisms to be turned about its Axis whilst the other remains unmoved)
this other half in viewing the Thred through the Prism, will appear in
a continual right Line with the first half when illuminated with red,
and begin to be a little divided from it when illuminated with Orange,
and remove farther from it when illuminated with yellow, and still
farther when with green, and farther when with blue, and go yet farther
off when illuminated with Indigo, and farthest when with deep violet.
Which plainly shews, that the Lights of several Colours are more and
more refrangible one than another, in this Order of their Colours, red,
orange, yellow, green, blue, indigo, deep violet; and so proves as well
the first Proposition as the second.

I caused also the coloured Spectrums PT [in _Fig._ 17.] and MN made in a
dark Chamber by the Refractions of two Prisms to lie in a Right Line end
to end, as was described above in the fifth Experiment, and viewing them
through a third Prism held parallel to their Length, they appeared no
longer in a Right Line, but became broken from one another, as they are
represented at _pt_ and _mn_, the violet end _m_ of the Spectrum _mn_
being by a greater Refraction translated farther from its former Place
MT than the red end _t_ of the other Spectrum _pt_.

I farther caused those two Spectrums PT [in _Fig._ 20.] and MN to become
co-incident in an inverted Order of their Colours, the red end of each
falling on the violet end of the other, as they are represented in the
oblong Figure PTMN; and then viewing them through a Prism DH held
parallel to their Length, they appeared not co-incident, as when view'd
with the naked Eye, but in the form of two distinct Spectrums _pt_ and
_mn_ crossing one another in the middle after the manner of the Letter
X. Which shews that the red of the one Spectrum and violet of the other,
which were co-incident at PN and MT, being parted from one another by a
greater Refraction of the violet to _p_ and _m_ than of the red to _n_
and _t_, do differ in degrees of Refrangibility.

I illuminated also a little Circular Piece of white Paper all over with
the Lights of both Prisms intermixed, and when it was illuminated with
the red of one Spectrum, and deep violet of the other, so as by the
Mixture of those Colours to appear all over purple, I viewed the Paper,
first at a less distance, and then at a greater, through a third Prism;
and as I went from the Paper, the refracted Image thereof became more
and more divided by the unequal Refraction of the two mixed Colours, and
at length parted into two distinct Images, a red one and a violet one,
whereof the violet was farthest from the Paper, and therefore suffered
the greatest Refraction. And when that Prism at the Window, which cast
the violet on the Paper was taken away, the violet Image disappeared;
but when the other Prism was taken away the red vanished; which shews,
that these two Images were nothing else than the Lights of the two
Prisms, which had been intermixed on the purple Paper, but were parted
again by their unequal Refractions made in the third Prism, through
which the Paper was view'd. This also was observable, that if one of the
Prisms at the Window, suppose that which cast the violet on the Paper,
was turned about its Axis to make all the Colours in this order,
violet, indigo, blue, green, yellow, orange, red, fall successively on
the Paper from that Prism, the violet Image changed Colour accordingly,
turning successively to indigo, blue, green, yellow and red, and in
changing Colour came nearer and nearer to the red Image made by the
other Prism, until when it was also red both Images became fully
co-incident.

I placed also two Paper Circles very near one another, the one in the
red Light of one Prism, and the other in the violet Light of the other.
The Circles were each of them an Inch in diameter, and behind them the
Wall was dark, that the Experiment might not be disturbed by any Light
coming from thence. These Circles thus illuminated, I viewed through a
Prism, so held, that the Refraction might be made towards the red
Circle, and as I went from them they came nearer and nearer together,
and at length became co-incident; and afterwards when I went still
farther off, they parted again in a contrary Order, the violet by a
greater Refraction being carried beyond the red.

_Exper._ 8. In Summer, when the Sun's Light uses to be strongest, I
placed a Prism at the Hole of the Window-shut, as in the third
Experiment, yet so that its Axis might be parallel to the Axis of the
World, and at the opposite Wall in the Sun's refracted Light, I placed
an open Book. Then going six Feet and two Inches from the Book, I placed
there the above-mentioned Lens, by which the Light reflected from the
Book might be made to converge and meet again at the distance of six
Feet and two Inches behind the Lens, and there paint the Species of the
Book upon a Sheet of white Paper much after the manner of the second
Experiment. The Book and Lens being made fast, I noted the Place where
the Paper was, when the Letters of the Book, illuminated by the fullest
red Light of the Solar Image falling upon it, did cast their Species on
that Paper most distinctly: And then I stay'd till by the Motion of the
Sun, and consequent Motion of his Image on the Book, all the Colours
from that red to the middle of the blue pass'd over those Letters; and
when those Letters were illuminated by that blue, I noted again the
Place of the Paper when they cast their Species most distinctly upon it:
And I found that this last Place of the Paper was nearer to the Lens
than its former Place by about two Inches and an half, or two and three
quarters. So much sooner therefore did the Light in the violet end of
the Image by a greater Refraction converge and meet, than the Light in
the red end. But in trying this, the Chamber was as dark as I could make
it. For, if these Colours be diluted and weakned by the Mixture of any
adventitious Light, the distance between the Places of the Paper will
not be so great. This distance in the second Experiment, where the
Colours of natural Bodies were made use of, was but an Inch and an half,
by reason of the Imperfection of those Colours. Here in the Colours of
the Prism, which are manifestly more full, intense, and lively than
those of natural Bodies, the distance is two Inches and three quarters.
And were the Colours still more full, I question not but that the
distance would be considerably greater. For the coloured Light of the
Prism, by the interfering of the Circles described in the second Figure
of the fifth Experiment, and also by the Light of the very bright Clouds
next the Sun's Body intermixing with these Colours, and by the Light
scattered by the Inequalities in the Polish of the Prism, was so very
much compounded, that the Species which those faint and dark Colours,
the indigo and violet, cast upon the Paper were not distinct enough to
be well observed.

_Exper._ 9. A Prism, whose two Angles at its Base were equal to one
another, and half right ones, and the third a right one, I placed in a
Beam of the Sun's Light let into a dark Chamber through a Hole in the
Window-shut, as in the third Experiment. And turning the Prism slowly
about its Axis, until all the Light which went through one of its
Angles, and was refracted by it began to be reflected by its Base, at
which till then it went out of the Glass, I observed that those Rays
which had suffered the greatest Refraction were sooner reflected than
the rest. I conceived therefore, that those Rays of the reflected Light,
which were most refrangible, did first of all by a total Reflexion
become more copious in that Light than the rest, and that afterwards the
rest also, by a total Reflexion, became as copious as these. To try
this, I made the reflected Light pass through another Prism, and being
refracted by it to fall afterwards upon a Sheet of white Paper placed
at some distance behind it, and there by that Refraction to paint the
usual Colours of the Prism. And then causing the first Prism to be
turned about its Axis as above, I observed that when those Rays, which
in this Prism had suffered the greatest Refraction, and appeared of a
blue and violet Colour began to be totally reflected, the blue and
violet Light on the Paper, which was most refracted in the second Prism,
received a sensible Increase above that of the red and yellow, which was
least refracted; and afterwards, when the rest of the Light which was
green, yellow, and red, began to be totally reflected in the first
Prism, the Light of those Colours on the Paper received as great an
Increase as the violet and blue had done before. Whence 'tis manifest,
that the Beam of Light reflected by the Base of the Prism, being
augmented first by the more refrangible Rays, and afterwards by the less
refrangible ones, is compounded of Rays differently refrangible. And
that all such reflected Light is of the same Nature with the Sun's Light
before its Incidence on the Base of the Prism, no Man ever doubted; it
being generally allowed, that Light by such Reflexions suffers no
Alteration in its Modifications and Properties. I do not here take
Notice of any Refractions made in the sides of the first Prism, because
the Light enters it perpendicularly at the first side, and goes out
perpendicularly at the second side, and therefore suffers none. So then,
the Sun's incident Light being of the same Temper and Constitution with
his emergent Light, and the last being compounded of Rays differently
refrangible, the first must be in like manner compounded.

[Illustration: FIG. 21.]

_Illustration._ In the twenty-first Figure, ABC is the first Prism, BC
its Base, B and C its equal Angles at the Base, each of 45 Degrees, A
its rectangular Vertex, FM a beam of the Sun's Light let into a dark
Room through a hole F one third part of an Inch broad, M its Incidence
on the Base of the Prism, MG a less refracted Ray, MH a more refracted
Ray, MN the beam of Light reflected from the Base, VXY the second Prism
by which this beam in passing through it is refracted, N_t_ the less
refracted Light of this beam, and N_p_ the more refracted part thereof.
When the first Prism ABC is turned about its Axis according to the order
of the Letters ABC, the Rays MH emerge more and more obliquely out of
that Prism, and at length after their most oblique Emergence are
reflected towards N, and going on to _p_ do increase the Number of the
Rays N_p_. Afterwards by continuing the Motion of the first Prism, the
Rays MG are also reflected to N and increase the number of the Rays
N_t_. And therefore the Light MN admits into its Composition, first the
more refrangible Rays, and then the less refrangible Rays, and yet after
this Composition is of the same Nature with the Sun's immediate Light
FM, the Reflexion of the specular Base BC causing no Alteration therein.

_Exper._ 10. Two Prisms, which were alike in Shape, I tied so together,
that their Axis and opposite Sides being parallel, they composed a
Parallelopiped. And, the Sun shining into my dark Chamber through a
little hole in the Window-shut, I placed that Parallelopiped in his beam
at some distance from the hole, in such a Posture, that the Axes of the
Prisms might be perpendicular to the incident Rays, and that those Rays
being incident upon the first Side of one Prism, might go on through the
two contiguous Sides of both Prisms, and emerge out of the last Side of
the second Prism. This Side being parallel to the first Side of the
first Prism, caused the emerging Light to be parallel to the incident.
Then, beyond these two Prisms I placed a third, which might refract that
emergent Light, and by that Refraction cast the usual Colours of the
Prism upon the opposite Wall, or upon a sheet of white Paper held at a
convenient Distance behind the Prism for that refracted Light to fall
upon it. After this I turned the Parallelopiped about its Axis, and
found that when the contiguous Sides of the two Prisms became so oblique
to the incident Rays, that those Rays began all of them to be
reflected, those Rays which in the third Prism had suffered the greatest
Refraction, and painted the Paper with violet and blue, were first of
all by a total Reflexion taken out of the transmitted Light, the rest
remaining and on the Paper painting their Colours of green, yellow,
orange and red, as before; and afterwards by continuing the Motion of
the two Prisms, the rest of the Rays also by a total Reflexion vanished
in order, according to their degrees of Refrangibility. The Light
therefore which emerged out of the two Prisms is compounded of Rays
differently refrangible, seeing the more refrangible Rays may be taken
out of it, while the less refrangible remain. But this Light being
trajected only through the parallel Superficies of the two Prisms, if it
suffer'd any change by the Refraction of one Superficies it lost that
Impression by the contrary Refraction of the other Superficies, and so
being restor'd to its pristine Constitution, became of the same Nature
and Condition as at first before its Incidence on those Prisms; and
therefore, before its Incidence, was as much compounded of Rays
differently refrangible, as afterwards.

[Illustration: FIG. 22.]

_Illustration._ In the twenty second Figure ABC and BCD are the two
Prisms tied together in the form of a Parallelopiped, their Sides BC and
CB being contiguous, and their Sides AB and CD parallel. And HJK is the
third Prism, by which the Sun's Light propagated through the hole F into
the dark Chamber, and there passing through those sides of the Prisms
AB, BC, CB and CD, is refracted at O to the white Paper PT, falling
there partly upon P by a greater Refraction, partly upon T by a less
Refraction, and partly upon R and other intermediate places by
intermediate Refractions. By turning the Parallelopiped ACBD about its
Axis, according to the order of the Letters A, C, D, B, at length when
the contiguous Planes BC and CB become sufficiently oblique to the Rays
FM, which are incident upon them at M, there will vanish totally out of
the refracted Light OPT, first of all the most refracted Rays OP, (the
rest OR and OT remaining as before) then the Rays OR and other
intermediate ones, and lastly, the least refracted Rays OT. For when
the Plane BC becomes sufficiently oblique to the Rays incident upon it,
those Rays will begin to be totally reflected by it towards N; and first
the most refrangible Rays will be totally reflected (as was explained in
the preceding Experiment) and by Consequence must first disappear at P,
and afterwards the rest as they are in order totally reflected to N,
they must disappear in the same order at R and T. So then the Rays which
at O suffer the greatest Refraction, may be taken out of the Light MO
whilst the rest of the Rays remain in it, and therefore that Light MO is
compounded of Rays differently refrangible. And because the Planes AB
and CD are parallel, and therefore by equal and contrary Refractions
destroy one anothers Effects, the incident Light FM must be of the same
Kind and Nature with the emergent Light MO, and therefore doth also
consist of Rays differently refrangible. These two Lights FM and MO,
before the most refrangible Rays are separated out of the emergent Light
MO, agree in Colour, and in all other Properties so far as my
Observation reaches, and therefore are deservedly reputed of the same
Nature and Constitution, and by Consequence the one is compounded as
well as the other. But after the most refrangible Rays begin to be
totally reflected, and thereby separated out of the emergent Light MO,
that Light changes its Colour from white to a dilute and faint yellow, a
pretty good orange, a very full red successively, and then totally
vanishes. For after the most refrangible Rays which paint the Paper at
P with a purple Colour, are by a total Reflexion taken out of the beam
of Light MO, the rest of the Colours which appear on the Paper at R and
T being mix'd in the Light MO compound there a faint yellow, and after
the blue and part of the green which appear on the Paper between P and R
are taken away, the rest which appear between R and T (that is the
yellow, orange, red and a little green) being mixed in the beam MO
compound there an orange; and when all the Rays are by Reflexion taken
out of the beam MO, except the least refrangible, which at T appear of a
full red, their Colour is the same in that beam MO as afterwards at T,
the Refraction of the Prism HJK serving only to separate the differently
refrangible Rays, without making any Alteration in their Colours, as
shall be more fully proved hereafter. All which confirms as well the
first Proposition as the second.

_Scholium._ If this Experiment and the former be conjoined and made one
by applying a fourth Prism VXY [in _Fig._ 22.] to refract the reflected
beam MN towards _tp_, the Conclusion will be clearer. For then the Light
N_p_ which in the fourth Prism is more refracted, will become fuller and
stronger when the Light OP, which in the third Prism HJK is more
refracted, vanishes at P; and afterwards when the less refracted Light
OT vanishes at T, the less refracted Light N_t_ will become increased
whilst the more refracted Light at _p_ receives no farther increase. And
as the trajected beam MO in vanishing is always of such a Colour as
ought to result from the mixture of the Colours which fall upon the
Paper PT, so is the reflected beam MN always of such a Colour as ought
to result from the mixture of the Colours which fall upon the Paper
_pt_. For when the most refrangible Rays are by a total Reflexion taken
out of the beam MO, and leave that beam of an orange Colour, the Excess
of those Rays in the reflected Light, does not only make the violet,
indigo and blue at _p_ more full, but also makes the beam MN change from
the yellowish Colour of the Sun's Light, to a pale white inclining to
blue, and afterward recover its yellowish Colour again, so soon as all
the rest of the transmitted Light MOT is reflected.

Now seeing that in all this variety of Experiments, whether the Trial be
made in Light reflected, and that either from natural Bodies, as in the
first and second Experiment, or specular, as in the ninth; or in Light
refracted, and that either before the unequally refracted Rays are by
diverging separated from one another, and losing their whiteness which
they have altogether, appear severally of several Colours, as in the
fifth Experiment; or after they are separated from one another, and
appear colour'd as in the sixth, seventh, and eighth Experiments; or in
Light trajected through parallel Superficies, destroying each others
Effects, as in the tenth Experiment; there are always found Rays, which
at equal Incidences on the same Medium suffer unequal Refractions, and
that without any splitting or dilating of single Rays, or contingence in
the inequality of the Refractions, as is proved in the fifth and sixth
Experiments. And seeing the Rays which differ in Refrangibility may be
parted and sorted from one another, and that either by Refraction as in
the third Experiment, or by Reflexion as in the tenth, and then the
several sorts apart at equal Incidences suffer unequal Refractions, and
those sorts are more refracted than others after Separation, which were
more refracted before it, as in the sixth and following Experiments, and
if the Sun's Light be trajected through three or more cross Prisms
successively, those Rays which in the first Prism are refracted more
than others, are in all the following Prisms refracted more than others
in the same Rate and Proportion, as appears by the fifth Experiment;
it's manifest that the Sun's Light is an heterogeneous Mixture of Rays,
some of which are constantly more refrangible than others, as was
proposed.


_PROP._ III. THEOR. III.

_The Sun's Light consists of Rays differing in Reflexibility, and those
Rays are more reflexible than others which are more refrangible._

This is manifest by the ninth and tenth Experiments: For in the ninth
Experiment, by turning the Prism about its Axis, until the Rays within
it which in going out into the Air were refracted by its Base, became so
oblique to that Base, as to begin to be totally reflected thereby; those
Rays became first of all totally reflected, which before at equal
Incidences with the rest had suffered the greatest Refraction. And the
same thing happens in the Reflexion made by the common Base of the two
Prisms in the tenth Experiment.


_PROP._ IV. PROB. I.

_To separate from one another the heterogeneous Rays of compound Light._

[Illustration: FIG. 23.]

The heterogeneous Rays are in some measure separated from one another by
the Refraction of the Prism in the third Experiment, and in the fifth
Experiment, by taking away the Penumbra from the rectilinear sides of
the coloured Image, that Separation in those very rectilinear sides or
straight edges of the Image becomes perfect. But in all places between
those rectilinear edges, those innumerable Circles there described,
which are severally illuminated by homogeneal Rays, by interfering with
one another, and being every where commix'd, do render the Light
sufficiently compound. But if these Circles, whilst their Centers keep
their Distances and Positions, could be made less in Diameter, their
interfering one with another, and by Consequence the Mixture of the
heterogeneous Rays would be proportionally diminish'd. In the twenty
third Figure let AG, BH, CJ, DK, EL, FM be the Circles which so many
sorts of Rays flowing from the same disque of the Sun, do in the third
Experiment illuminate; of all which and innumerable other intermediate
ones lying in a continual Series between the two rectilinear and
parallel edges of the Sun's oblong Image PT, that Image is compos'd, as
was explained in the fifth Experiment. And let _ag_, _bh_, _ci_, _dk_,
_el_, _fm_ be so many less Circles lying in a like continual Series
between two parallel right Lines _af_ and _gm_ with the same distances
between their Centers, and illuminated by the same sorts of Rays, that
is the Circle _ag_ with the same sort by which the corresponding Circle
AG was illuminated, and the Circle _bh_ with the same sort by which the
corresponding Circle BH was illuminated, and the rest of the Circles
_ci_, _dk_, _el_, _fm_ respectively, with the same sorts of Rays by
which the several corresponding Circles CJ, DK, EL, FM were illuminated.
In the Figure PT composed of the greater Circles, three of those Circles
AG, BH, CJ, are so expanded into one another, that the three sorts of
Rays by which those Circles are illuminated, together with other
innumerable sorts of intermediate Rays, are mixed at QR in the middle
of the Circle BH. And the like Mixture happens throughout almost the
whole length of the Figure PT. But in the Figure _pt_ composed of the
less Circles, the three less Circles _ag_, _bh_, _ci_, which answer to
those three greater, do not extend into one another; nor are there any
where mingled so much as any two of the three sorts of Rays by which
those Circles are illuminated, and which in the Figure PT are all of
them intermingled at BH.

Now he that shall thus consider it, will easily understand that the
Mixture is diminished in the same Proportion with the Diameters of the
Circles. If the Diameters of the Circles whilst their Centers remain the
same, be made three times less than before, the Mixture will be also
three times less; if ten times less, the Mixture will be ten times less,
and so of other Proportions. That is, the Mixture of the Rays in the
greater Figure PT will be to their Mixture in the less _pt_, as the
Latitude of the greater Figure is to the Latitude of the less. For the
Latitudes of these Figures are equal to the Diameters of their Circles.
And hence it easily follows, that the Mixture of the Rays in the
refracted Spectrum _pt_ is to the Mixture of the Rays in the direct and
immediate Light of the Sun, as the breadth of that Spectrum is to the
difference between the length and breadth of the same Spectrum.

So then, if we would diminish the Mixture of the Rays, we are to
diminish the Diameters of the Circles. Now these would be diminished if
the Sun's Diameter to which they answer could be made less than it is,
or (which comes to the same Purpose) if without Doors, at a great
distance from the Prism towards the Sun, some opake Body were placed,
with a round hole in the middle of it, to intercept all the Sun's Light,
excepting so much as coming from the middle of his Body could pass
through that Hole to the Prism. For so the Circles AG, BH, and the rest,
would not any longer answer to the whole Disque of the Sun, but only to
that Part of it which could be seen from the Prism through that Hole,
that it is to the apparent Magnitude of that Hole view'd from the Prism.
But that these Circles may answer more distinctly to that Hole, a Lens
is to be placed by the Prism to cast the Image of the Hole, (that is,
every one of the Circles AG, BH, &c.) distinctly upon the Paper at PT,
after such a manner, as by a Lens placed at a Window, the Species of
Objects abroad are cast distinctly upon a Paper within the Room, and the
rectilinear Sides of the oblong Solar Image in the fifth Experiment
became distinct without any Penumbra. If this be done, it will not be
necessary to place that Hole very far off, no not beyond the Window. And
therefore instead of that Hole, I used the Hole in the Window-shut, as
follows.

_Exper._ 11. In the Sun's Light let into my darken'd Chamber through a
small round Hole in my Window-shut, at about ten or twelve Feet from the
Window, I placed a Lens, by which the Image of the Hole might be
distinctly cast upon a Sheet of white Paper, placed at the distance of
six, eight, ten, or twelve Feet from the Lens. For, according to the
difference of the Lenses I used various distances, which I think not
worth the while to describe. Then immediately after the Lens I placed a
Prism, by which the trajected Light might be refracted either upwards or
sideways, and thereby the round Image, which the Lens alone did cast
upon the Paper might be drawn out into a long one with Parallel Sides,
as in the third Experiment. This oblong Image I let fall upon another
Paper at about the same distance from the Prism as before, moving the
Paper either towards the Prism or from it, until I found the just
distance where the Rectilinear Sides of the Image became most distinct.
For in this Case, the Circular Images of the Hole, which compose that
Image after the same manner that the Circles _ag_, _bh_, _ci_, &c. do
the Figure _pt_ [in _Fig._ 23.] were terminated most distinctly without
any Penumbra, and therefore extended into one another the least that
they could, and by consequence the Mixture of the heterogeneous Rays was
now the least of all. By this means I used to form an oblong Image (such
as is _pt_) [in _Fig._ 23, and 24.] of Circular Images of the Hole,
(such as are _ag_, _bh_, _ci_, &c.) and by using a greater or less Hole
in the Window-shut, I made the Circular Images _ag_, _bh_, _ci_, &c. of
which it was formed, to become greater or less at pleasure, and thereby
the Mixture of the Rays in the Image _pt_ to be as much, or as little as
I desired.

[Illustration: FIG. 24.]

_Illustration._ In the twenty-fourth Figure, F represents the Circular
Hole in the Window-shut, MN the Lens, whereby the Image or Species of
that Hole is cast distinctly upon a Paper at J, ABC the Prism, whereby
the Rays are at their emerging out of the Lens refracted from J towards
another Paper at _pt_, and the round Image at J is turned into an oblong
Image _pt_ falling on that other Paper. This Image _pt_ consists of
Circles placed one after another in a Rectilinear Order, as was
sufficiently explained in the fifth Experiment; and these Circles are
equal to the Circle J, and consequently answer in magnitude to the Hole
F; and therefore by diminishing that Hole they may be at pleasure
diminished, whilst their Centers remain in their Places. By this means I
made the Breadth of the Image _pt_ to be forty times, and sometimes
sixty or seventy times less than its Length. As for instance, if the
Breadth of the Hole F be one tenth of an Inch, and MF the distance of
the Lens from the Hole be 12 Feet; and if _p_B or _p_M the distance of
the Image _pt_ from the Prism or Lens be 10 Feet, and the refracting
Angle of the Prism be 62 Degrees, the Breadth of the Image _pt_ will be
one twelfth of an Inch, and the Length about six Inches, and therefore
the Length to the Breadth as 72 to 1, and by consequence the Light of
this Image 71 times less compound than the Sun's direct Light. And Light
thus far simple and homogeneal, is sufficient for trying all the
Experiments in this Book about simple Light. For the Composition of
heterogeneal Rays is in this Light so little, that it is scarce to be
discovered and perceiv'd by Sense, except perhaps in the indigo and
violet. For these being dark Colours do easily suffer a sensible Allay
by that little scattering Light which uses to be refracted irregularly
by the Inequalities of the Prism.

Yet instead of the Circular Hole F, 'tis better to substitute an oblong
Hole shaped like a long Parallelogram with its Length parallel to the
Prism ABC. For if this Hole be an Inch or two long, and but a tenth or
twentieth Part of an Inch broad, or narrower; the Light of the Image
_pt_ will be as simple as before, or simpler, and the Image will become
much broader, and therefore more fit to have Experiments try'd in its
Light than before.

Instead of this Parallelogram Hole may be substituted a triangular one
of equal Sides, whose Base, for instance, is about the tenth Part of an
Inch, and its Height an Inch or more. For by this means, if the Axis of
the Prism be parallel to the Perpendicular of the Triangle, the Image
_pt_ [in _Fig._ 25.] will now be form'd of equicrural Triangles _ag_,
_bh_, _ci_, _dk_, _el_, _fm_, &c. and innumerable other intermediate
ones answering to the triangular Hole in Shape and Bigness, and lying
one after another in a continual Series between two Parallel Lines _af_
and _gm_. These Triangles are a little intermingled at their Bases, but
not at their Vertices; and therefore the Light on the brighter Side _af_
of the Image, where the Bases of the Triangles are, is a little
compounded, but on the darker Side _gm_ is altogether uncompounded, and
in all Places between the Sides the Composition is proportional to the
distances of the Places from that obscurer Side _gm_. And having a
Spectrum _pt_ of such a Composition, we may try Experiments either in
its stronger and less simple Light near the Side _af_, or in its weaker
and simpler Light near the other Side _gm_, as it shall seem most
convenient.

[Illustration: FIG. 25.]

But in making Experiments of this kind, the Chamber ought to be made as
dark as can be, lest any Foreign Light mingle it self with the Light of
the Spectrum _pt_, and render it compound; especially if we would try
Experiments in the more simple Light next the Side _gm_ of the Spectrum;
which being fainter, will have a less proportion to the Foreign Light;
and so by the mixture of that Light be more troubled, and made more
compound. The Lens also ought to be good, such as may serve for optical
Uses, and the Prism ought to have a large Angle, suppose of 65 or 70
Degrees, and to be well wrought, being made of Glass free from Bubbles
and Veins, with its Sides not a little convex or concave, as usually
happens, but truly plane, and its Polish elaborate, as in working
Optick-glasses, and not such as is usually wrought with Putty, whereby
the edges of the Sand-holes being worn away, there are left all over the
Glass a numberless Company of very little convex polite Risings like
Waves. The edges also of the Prism and Lens, so far as they may make any
irregular Refraction, must be covered with a black Paper glewed on. And
all the Light of the Sun's Beam let into the Chamber, which is useless
and unprofitable to the Experiment, ought to be intercepted with black
Paper, or other black Obstacles. For otherwise the useless Light being
reflected every way in the Chamber, will mix with the oblong Spectrum,
and help to disturb it. In trying these Things, so much diligence is not
altogether necessary, but it will promote the Success of the
Experiments, and by a very scrupulous Examiner of Things deserves to be
apply'd. It's difficult to get Glass Prisms fit for this Purpose, and
therefore I used sometimes prismatick Vessels made with pieces of broken
Looking-glasses, and filled with Rain Water. And to increase the
Refraction, I sometimes impregnated the Water strongly with _Saccharum
Saturni_.


_PROP._ V. THEOR. IV.

_Homogeneal Light is refracted regularly without any Dilatation
splitting or shattering of the Rays, and the confused Vision of Objects
seen through refracting Bodies by heterogeneal Light arises from the
different Refrangibility of several sorts of Rays._

The first Part of this Proposition has been already sufficiently proved
in the fifth Experiment, and will farther appear by the Experiments
which follow.

_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.

[Illustration: FIG. 26.]

_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 × 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 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 Phænomena 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 Circumspection 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 Space 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 Glass. 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 Glasses would be many hundred times
less. For, if the Object-glass of a Telescope be Plano-convex, and the
Plane side be turned towards the Object, and the Diameter of the
Sphere, whereof this Glass is a Segment, be called D, and the
Semi-diameter of the Aperture of the Glass be called S, and the Sine of
Incidence out of Glass into Air, be to the Sine of Refraction as I to R;
the Rays which come parallel to the Axis of the Glass, 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 Quantities 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 Glass 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-glass, which here
is four Inches. And therefore, the Error arising from the Spherical
Figure of the Glass, 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.

[Illustration: FIG. 27.]

But you will say, if the Errors caused by the different Refrangibility
be so very great, how comes it to pass, 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 Space, 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 Circumference, grow
continually rarer and rarer, so as at the Circumference 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 Circumference 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 Circumference 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 Glass are as 17 and 11) the Refraction of Glass 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 Glass. 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
space 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 space 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-glass 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-Glass be tincted faintly with the Smoak of a
Lamp or Torch, to obscure the Light of the Star, the fainter Light in
the Circumference of the Star ceases to be visible, and the Star (if the
Glass 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 Circumference 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 passes 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 Glass; yet this Image would be
still much greater than if it were only from the spherical Figure of the
Glass. 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 Glass, 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 Glasses, 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-glasses, are as the Cubes of the Apertures of the
Object Glasses; and thence to make Telescopes of various Lengths magnify
with equal distinctness, the Apertures of the Object-glasses, 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-glasses; 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.

[Illustration: 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-glass of two Glasses with Water between them. Let
ADFC [in _Fig._ 28.] represent the Object-glass composed of two Glasses
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 Glass into Air be as I to R, and out of
Water into Air, as K to R, and by consequence out of Glass 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 Glasses, 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 cumbersome, 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 Glasses
are readily manageable, and the Object-glass 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-glass 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-glass 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 Eyeglass and the Eye, and perforated in the middle
with a little round hole for the Rays to pass 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-glass, I could read at a greater distance with my own Instrument
than with the Glass. Yet Objects appeared much darker in it than in the
Glass, and that partly because more Light was lost by Reflexion in the
Metal, than by Refraction in the Glass, 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 polishing 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 washing 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 polishing, till I made those two
reflecting Perspectives I spake of above. For this Art of polishing 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 Glass, and is
afterwards very apt to be spoiled by tarnishing, and reflects not so
much Light as Glass quick-silver'd over does: I would propound to use
instead of the Metal, a Glass ground concave on the foreside, and as
much convex on the backside, and quick-silver'd over on the convex side.
The Glass must be every where of the same thickness exactly. Otherwise
it will make Objects look colour'd and indistinct. By such a Glass 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 Glass being wrought by one of our _London_ Artists
after such a manner as they grind Glasses for Telescopes, though it
seemed as well wrought as the Object-glasses use to be, yet when it was
quick-silver'd, the Reflexion discovered innumerable Inequalities all
over the Glass. And by reason of these Inequalities, Objects appeared
indistinct in this Instrument. For the Errors of reflected Rays caused
by any Inequality of the Glass, 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 Glass, 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 Glasses
truly spherical. An Object-glass 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
polishing these reflecting Glasses, I have not yet tried. But he that
shall try either this or any other way of polishing which he may think
better, may do well to make his Glasses ready for polishing, by grinding
them without that Violence, wherewith our _London_ Workmen press their
Glasses in grinding. For by such violent pressure, Glasses 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 Glasses to such Artists as are curious in figuring Glasses, I
shall describe this optical Instrument in the following Proposition.


_PROP._ VIII. PROB. II.

_To shorten Telescopes._

Let ABCD [in _Fig._ 29.] represent a Glass 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 Glass or Crystal placed near the other end of the Tube, in
the middle of it, by means of a handle of Brass 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 pass 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-glass H, through which
those Rays must pass to the Eye. And let the Rays at their coming out of
the Glass pass through a small round hole, or aperture made in a little
plate of Lead, Brass, or Silver, wherewith the Glass is to be covered,
which hole must be no bigger than is necessary for Light enough to pass
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 Glass 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.

[Illustration: 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-glass. If it be desired that the
Instrument bear a larger aperture, that may be also done by composing
the Speculum of two Glasses 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 pass 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 constitute 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 Opticæ § 10. _Sect. II. § 29. and Sect.
III. Prop. 25._

[D] See our Author's _Lectiones Opticæ_, 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.
_Sect._ IV. _Prop._ 31.




THE FIRST BOOK OF OPTICKS




_PART II._


_PROP._ I. THEOR. I.

_The Phænomena of Colours in refracted or reflected Light are not caused
by new Modifications of the Light variously impress'd, according to the
various Terminations of the Light and Shadow_.

The PROOF by Experiments.

_Exper._ 1. For if the Sun shine into a very dark Chamber through an
oblong hole F, [in _Fig._ 1.] whose breadth is the sixth or eighth part
of an Inch, or something less; and his beam FH do afterwards pass first
through a very large Prism ABC, distant about 20 Feet from the hole, and
parallel to it, and then (with its white part) through an oblong hole H,
whose breadth is about the fortieth or sixtieth part of an Inch, and
which is made in a black opake Body GI, and placed at the distance of
two or three Feet from the Prism, in a parallel Situation both to the
Prism and to the former hole, and if this white Light thus transmitted
through the hole H, fall afterwards upon a white Paper _pt_, placed
after that hole H, at the distance of three or four Feet from it, and
there paint the usual Colours of the Prism, suppose red at _t_, yellow
at _s_, green at _r_, blue at _q_, and violet at _p_; you may with an
Iron Wire, or any such like slender opake Body, whose breadth is about
the tenth part of an Inch, by intercepting the Rays at _k_, _l_, _m_,
_n_ or _o_, take away any one of the Colours at _t_, _s_, _r_, _q_ or
_p_, whilst the other Colours remain upon the Paper as before; or with
an Obstacle something bigger you may take away any two, or three, or
four Colours together, the rest remaining: So that any one of the
Colours as well as violet may become outmost in the Confine of the
Shadow towards _p_, and any one of them as well as red may become
outmost in the Confine of the Shadow towards _t_, and any one of them
may also border upon the Shadow made within the Colours by the Obstacle
R intercepting some intermediate part of the Light; and, lastly, any one
of them by being left alone, may border upon the Shadow on either hand.
All the Colours have themselves indifferently to any Confines of Shadow,
and therefore the differences of these Colours from one another, do not
arise from the different Confines of Shadow, whereby Light is variously
modified, as has hitherto been the Opinion of Philosophers. In trying
these things 'tis to be observed, that by how much the holes F and H are
narrower, and the Intervals between them and the Prism greater, and the
Chamber darker, by so much the better doth the Experiment succeed;
provided the Light be not so far diminished, but that the Colours at
_pt_ be sufficiently visible. To procure a Prism of solid Glass large
enough for this Experiment will be difficult, and therefore a prismatick
Vessel must be made of polish'd Glass Plates cemented together, and
filled with salt Water or clear Oil.

[Illustration: FIG. 1.]

_Exper._ 2. The Sun's Light let into a dark Chamber through the round
hole F, [in _Fig._ 2.] half an Inch wide, passed first through the Prism
ABC placed at the hole, and then through a Lens PT something more than
four Inches broad, and about eight Feet distant from the Prism, and
thence converged to O the Focus of the Lens distant from it about three
Feet, and there fell upon a white Paper DE. If that Paper was
perpendicular to that Light incident upon it, as 'tis represented in the
posture DE, all the Colours upon it at O appeared white. But if the
Paper being turned about an Axis parallel to the Prism, became very much
inclined to the Light, as 'tis represented in the Positions _de_ and
_[Greek: de]_; the same Light in the one case appeared yellow and red,
in the other blue. Here one and the same part of the Light in one and
the same place, according to the various Inclinations of the Paper,
appeared in one case white, in another yellow or red, in a third blue,
whilst the Confine of Light and shadow, and the Refractions of the Prism
in all these cases remained the same.

[Illustration: FIG. 2.]

[Illustration: FIG. 3.]

_Exper._ 3. Such another Experiment may be more easily tried as follows.
Let a broad beam of the Sun's Light coming into a dark Chamber through a
hole in the Window-shut be refracted by a large Prism ABC, [in _Fig._
3.] whose refracting Angle C is more than 60 Degrees, and so soon as it
comes out of the Prism, let it fall upon the white Paper DE glewed upon
a stiff Plane; and this Light, when the Paper is perpendicular to it, as
'tis represented in DE, will appear perfectly white upon the Paper; but
when the Paper is very much inclin'd to it in such a manner as to keep
always parallel to the Axis of the Prism, the whiteness of the whole
Light upon the Paper will according to the inclination of the Paper this
way or that way, change either into yellow and red, as in the posture
_de_, or into blue and violet, as in the posture [Greek: de]. And if the
Light before it fall upon the Paper be twice refracted the same way by
two parallel Prisms, these Colours will become the more conspicuous.
Here all the middle parts of the broad beam of white Light which fell
upon the Paper, did without any Confine of Shadow to modify it, become
colour'd all over with one uniform Colour, the Colour being always the
same in the middle of the Paper as at the edges, and this Colour changed
according to the various Obliquity of the reflecting Paper, without any
change in the Refractions or Shadow, or in the Light which fell upon the
Paper. And therefore these Colours are to be derived from some other
Cause than the new Modifications of Light by Refractions and Shadows.

If it be asked, what then is their Cause? I answer, That the Paper in
the posture _de_, being more oblique to the more refrangible Rays than
to the less refrangible ones, is more strongly illuminated by the latter
than by the former, and therefore the less refrangible Rays are
predominant in the reflected Light. And where-ever they are predominant
in any Light, they tinge it with red or yellow, as may in some measure
appear by the first Proposition of the first Part of this Book, and will
more fully appear hereafter. And the contrary happens in the posture of
the Paper [Greek: de], the more refrangible Rays being then predominant
which always tinge Light with blues and violets.

_Exper._ 4. The Colours of Bubbles with which Children play are various,
and change their Situation variously, without any respect to any Confine
or Shadow. If such a Bubble be cover'd with a concave Glass, to keep it
from being agitated by any Wind or Motion of the Air, the Colours will
slowly and regularly change their situation, even whilst the Eye and the
Bubble, and all Bodies which emit any Light, or cast any Shadow, remain
unmoved. And therefore their Colours arise from some regular Cause which
depends not on any Confine of Shadow. What this Cause is will be shewed
in the next Book.

To these Experiments may be added the tenth Experiment of the first Part
of this first Book, where the Sun's Light in a dark Room being
trajected through the parallel Superficies of two Prisms tied together
in the form of a Parallelopipede, became totally of one uniform yellow
or red Colour, at its emerging out of the Prisms. Here, in the
production of these Colours, the Confine of Shadow can have nothing to
do. For the Light changes from white to yellow, orange and red
successively, without any alteration of the Confine of Shadow: And at
both edges of the emerging Light where the contrary Confines of Shadow
ought to produce different Effects, the Colour is one and the same,
whether it be white, yellow, orange or red: And in the middle of the
emerging Light, where there is no Confine of Shadow at all, the Colour
is the very same as at the edges, the whole Light at its very first
Emergence being of one uniform Colour, whether white, yellow, orange or
red, and going on thence perpetually without any change of Colour, such
as the Confine of Shadow is vulgarly supposed to work in refracted Light
after its Emergence. Neither can these Colours arise from any new
Modifications of the Light by Refractions, because they change
successively from white to yellow, orange and red, while the Refractions
remain the same, and also because the Refractions are made contrary ways
by parallel Superficies which destroy one another's Effects. They arise
not therefore from any Modifications of Light made by Refractions and
Shadows, but have some other Cause. What that Cause is we shewed above
in this tenth Experiment, and need not here repeat it.

There is yet another material Circumstance of this Experiment. For this
emerging Light being by a third Prism HIK [in _Fig._ 22. _Part_ I.][I]
refracted towards the Paper PT, and there painting the usual Colours of
the Prism, red, yellow, green, blue, violet: If these Colours arose from
the Refractions of that Prism modifying the Light, they would not be in
the Light before its Incidence on that Prism. And yet in that Experiment
we found, that when by turning the two first Prisms about their common
Axis all the Colours were made to vanish but the red; the Light which
makes that red being left alone, appeared of the very same red Colour
before its Incidence on the third Prism. And in general we find by other
Experiments, that when the Rays which differ in Refrangibility are
separated from one another, and any one Sort of them is considered
apart, the Colour of the Light which they compose cannot be changed by
any Refraction or Reflexion whatever, as it ought to be were Colours
nothing else than Modifications of Light caused by Refractions, and
Reflexions, and Shadows. This Unchangeableness of Colour I am now to
describe in the following Proposition.


_PROP._ II. THEOR. II.

_All homogeneal Light has its proper Colour answering to its Degree of
Refrangibility, and that Colour cannot be changed by Reflexions and
Refractions._

In the Experiments of the fourth Proposition of the first Part of this
first Book, when I had separated the heterogeneous Rays from one
another, the Spectrum _pt_ formed by the separated Rays, did in the
Progress from its End _p_, on which the most refrangible Rays fell, unto
its other End _t_, on which the least refrangible Rays fell, appear
tinged with this Series of Colours, violet, indigo, blue, green, yellow,
orange, red, together with all their intermediate Degrees in a continual
Succession perpetually varying. So that there appeared as many Degrees
of Colours, as there were sorts of Rays differing in Refrangibility.

_Exper._ 5. Now, that these Colours could not be changed by Refraction,
I knew by refracting with a Prism sometimes one very little Part of this
Light, sometimes another very little Part, as is described in the
twelfth Experiment of the first Part of this Book. For by this
Refraction the Colour of the Light was never changed in the least. If
any Part of the red Light was refracted, it remained totally of the same
red Colour as before. No orange, no yellow, no green or blue, no other
new Colour was produced by that Refraction. Neither did the Colour any
ways change by repeated Refractions, but continued always the same red
entirely as at first. The like Constancy and Immutability I found also
in the blue, green, and other Colours. So also, if I looked through a
Prism upon any Body illuminated with any part of this homogeneal Light,
as in the fourteenth Experiment of the first Part of this Book is
described; I could not perceive any new Colour generated this way. All
Bodies illuminated with compound Light appear through Prisms confused,
(as was said above) and tinged with various new Colours, but those
illuminated with homogeneal Light appeared through Prisms neither less
distinct, nor otherwise colour'd, than when viewed with the naked Eyes.
Their Colours were not in the least changed by the Refraction of the
interposed Prism. I speak here of a sensible Change of Colour: For the
Light which I here call homogeneal, being not absolutely homogeneal,
there ought to arise some little Change of Colour from its
Heterogeneity. But, if that Heterogeneity was so little as it might be
made by the said Experiments of the fourth Proposition, that Change was
not sensible, and therefore in Experiments, where Sense is Judge, ought
to be accounted none at all.

_Exper._ 6. And as these Colours were not changeable by Refractions, so
neither were they by Reflexions. For all white, grey, red, yellow,
green, blue, violet Bodies, as Paper, Ashes, red Lead, Orpiment, Indico
Bise, Gold, Silver, Copper, Grass, blue Flowers, Violets, Bubbles of
Water tinged with various Colours, Peacock's Feathers, the Tincture of
_Lignum Nephriticum_, and such-like, in red homogeneal Light appeared
totally red, in blue Light totally blue, in green Light totally green,
and so of other Colours. In the homogeneal Light of any Colour they all
appeared totally of that same Colour, with this only Difference, that
some of them reflected that Light more strongly, others more faintly. I
never yet found any Body, which by reflecting homogeneal Light could
sensibly change its Colour.

From all which it is manifest, that if the Sun's Light consisted of but
one sort of Rays, there would be but one Colour in the whole World, nor
would it be possible to produce any new Colour by Reflexions and
Refractions, and by consequence that the variety of Colours depends upon
the Composition of Light.


_DEFINITION._

The homogeneal Light and Rays which appear red, or rather make Objects
appear so, I call Rubrifick or Red-making; those which make Objects
appear yellow, green, blue, and violet, I call Yellow-making,
Green-making, Blue-making, Violet-making, and so of the rest. And if at
any time I speak of Light and Rays as coloured or endued with Colours, I
would be understood to speak not philosophically and properly, but
grossly, and accordingly to such Conceptions as vulgar People in seeing
all these Experiments would be apt to frame. For the Rays to speak
properly are not coloured. In them there is nothing else than a certain
Power and Disposition to stir up a Sensation of this or that Colour.
For as Sound in a Bell or musical String, or other sounding Body, is
nothing but a trembling Motion, and in the Air nothing but that Motion
propagated from the Object, and in the Sensorium 'tis a Sense of that
Motion under the Form of Sound; so Colours in the Object are nothing but
a Disposition to reflect this or that sort of Rays more copiously than
the rest; in the Rays they are nothing but their Dispositions to
propagate this or that Motion into the Sensorium, and in the Sensorium
they are Sensations of those Motions under the Forms of Colours.


_PROP._ III. PROB. I.

_To define the Refrangibility of the several sorts of homogeneal Light
answering to the several Colours._

For determining this Problem I made the following Experiment.[J]

_Exper._ 7. When I had caused the Rectilinear Sides AF, GM, [in _Fig._
4.] of the Spectrum of Colours made by the Prism to be distinctly
defined, as in the fifth Experiment of the first Part of this Book is
described, there were found in it all the homogeneal Colours in the same
Order and Situation one among another as in the Spectrum of simple
Light, described in the fourth Proposition of that Part. For the Circles
of which the Spectrum of compound Light PT is composed, and which in
the middle Parts of the Spectrum interfere, and are intermix'd with one
another, are not intermix'd in their outmost Parts where they touch
those Rectilinear Sides AF and GM. And therefore, in those Rectilinear
Sides when distinctly defined, there is no new Colour generated by
Refraction. I observed also, that if any where between the two outmost
Circles TMF and PGA a Right Line, as [Greek: gd], was cross to the
Spectrum, so as both Ends to fall perpendicularly upon its Rectilinear
Sides, there appeared one and the same Colour, and degree of Colour from
one End of this Line to the other. I delineated therefore in a Paper the
Perimeter of the Spectrum FAP GMT, and in trying the third Experiment of
the first Part of this Book, I held the Paper so that the Spectrum might
fall upon this delineated Figure, and agree with it exactly, whilst an
Assistant, whose Eyes for distinguishing Colours were more critical than
mine, did by Right Lines [Greek: ab, gd, ez,] &c. drawn cross the
Spectrum, note the Confines of the Colours, that is of the red M[Greek:
ab]F, of the orange [Greek: agdb], of the yellow [Greek: gezd], of the
green [Greek: eêthz], of the blue [Greek: êikth], of the indico [Greek:
ilmk], and of the violet [Greek: l]GA[Greek: m]. And this Operation
being divers times repeated both in the same, and in several Papers, I
found that the Observations agreed well enough with one another, and
that the Rectilinear Sides MG and FA were by the said cross Lines
divided after the manner of a Musical Chord. Let GM be produced to X,
that MX may be equal to GM, and conceive GX, [Greek: l]X, [Greek: i]X,
[Greek: ê]X, [Greek: e]X, [Greek: g]X, [Greek: a]X, MX, to be in
proportion to one another, as the Numbers, 1, 8/9, 5/6, 3/4, 2/3, 3/5,
9/16, 1/2, and so to represent the Chords of the Key, and of a Tone, a
third Minor, a fourth, a fifth, a sixth Major, a seventh and an eighth
above that Key: And the Intervals M[Greek: a], [Greek: ag], [Greek: ge],
[Greek: eê], [Greek: êi], [Greek: il], and [Greek: l]G, will be the
Spaces which the several Colours (red, orange, yellow, green, blue,
indigo, violet) take up.

[Illustration: FIG. 4.]

[Illustration: FIG. 5.]

Now these Intervals or Spaces subtending the Differences of the
Refractions of the Rays going to the Limits of those Colours, that is,
to the Points M, [Greek: a], [Greek: g], [Greek: e], [Greek: ê], [Greek:
i], [Greek: l], G, may without any sensible Error be accounted
proportional to the Differences of the Sines of Refraction of those Rays
having one common Sine of Incidence, and therefore since the common Sine
of Incidence of the most and least refrangible Rays out of Glass into
Air was (by a Method described above) found in proportion to their Sines
of Refraction, as 50 to 77 and 78, divide the Difference between the
Sines of Refraction 77 and 78, as the Line GM is divided by those
Intervals, and you will have 77, 77-1/8, 77-1/5, 77-1/3, 77-1/2, 77-2/3,
77-7/9, 78, the Sines of Refraction of those Rays out of Glass into Air,
their common Sine of Incidence being 50. So then the Sines of the
Incidences of all the red-making Rays out of Glass into Air, were to the
Sines of their Refractions, not greater than 50 to 77, nor less than 50
to 77-1/8, but they varied from one another according to all
intermediate Proportions. And the Sines of the Incidences of the
green-making Rays were to the Sines of their Refractions in all
Proportions from that of 50 to 77-1/3, unto that of 50 to 77-1/2. And
by the like Limits above-mentioned were the Refractions of the Rays
belonging to the rest of the Colours defined, the Sines of the
red-making Rays extending from 77 to 77-1/8, those of the orange-making
from 77-1/8 to 77-1/5, those of the yellow-making from 77-1/5 to 77-1/3,
those of the green-making from 77-1/3 to 77-1/2, those of the
blue-making from 77-1/2 to 77-2/3, those of the indigo-making from
77-2/3 to 77-7/9, and those of the violet from 77-7/9, to 78.

These are the Laws of the Refractions made out of Glass into Air, and
thence by the third Axiom of the first Part of this Book, the Laws of
the Refractions made out of Air into Glass are easily derived.

_Exper._ 8. I found moreover, that when Light goes out of Air through
several contiguous refracting Mediums as through Water and Glass, and
thence goes out again into Air, whether the refracting Superficies be
parallel or inclin'd to one another, that Light as often as by contrary
Refractions 'tis so corrected, that it emergeth in Lines parallel to
those in which it was incident, continues ever after to be white. But if
the emergent Rays be inclined to the incident, the Whiteness of the
emerging Light will by degrees in passing on from the Place of
Emergence, become tinged in its Edges with Colours. This I try'd by
refracting Light with Prisms of Glass placed within a Prismatick Vessel
of Water. Now those Colours argue a diverging and separation of the
heterogeneous Rays from one another by means of their unequal
Refractions, as in what follows will more fully appear. And, on the
contrary, the permanent whiteness argues, that in like Incidences of the
Rays there is no such separation of the emerging Rays, and by
consequence no inequality of their whole Refractions. Whence I seem to
gather the two following Theorems.

1. The Excesses of the Sines of Refraction of several sorts of Rays
above their common Sine of Incidence when the Refractions are made out
of divers denser Mediums immediately into one and the same rarer Medium,
suppose of Air, are to one another in a given Proportion.

2. The Proportion of the Sine of Incidence to the Sine of Refraction of
one and the same sort of Rays out of one Medium into another, is
composed of the Proportion of the Sine of Incidence to the Sine of
Refraction out of the first Medium into any third Medium, and of the
Proportion of the Sine of Incidence to the Sine of Refraction out of
that third Medium into the second Medium.

By the first Theorem the Refractions of the Rays of every sort made out
of any Medium into Air are known by having the Refraction of the Rays of
any one sort. As for instance, if the Refractions of the Rays of every
sort out of Rain-water into Air be desired, let the common Sine of
Incidence out of Glass into Air be subducted from the Sines of
Refraction, and the Excesses will be 27, 27-1/8, 27-1/5, 27-1/3, 27-1/2,
27-2/3, 27-7/9, 28. Suppose now that the Sine of Incidence of the least
refrangible Rays be to their Sine of Refraction out of Rain-water into
Air as 3 to 4, and say as 1 the difference of those Sines is to 3 the
Sine of Incidence, so is 27 the least of the Excesses above-mentioned to
a fourth Number 81; and 81 will be the common Sine of Incidence out of
Rain-water into Air, to which Sine if you add all the above-mentioned
Excesses, you will have the desired Sines of the Refractions 108,
108-1/8, 108-1/5, 108-1/3, 108-1/2, 108-2/3, 108-7/9, 109.

By the latter Theorem the Refraction out of one Medium into another is
gathered as often as you have the Refractions out of them both into any
third Medium. As if the Sine of Incidence of any Ray out of Glass into
Air be to its Sine of Refraction, as 20 to 31, and the Sine of Incidence
of the same Ray out of Air into Water, be to its Sine of Refraction as 4
to 3; the Sine of Incidence of that Ray out of Glass into Water will be
to its Sine of Refraction as 20 to 31 and 4 to 3 jointly, that is, as
the Factum of 20 and 4 to the Factum of 31 and 3, or as 80 to 93.

And these Theorems being admitted into Opticks, there would be scope
enough of handling that Science voluminously after a new manner,[K] not
only by teaching those things which tend to the perfection of Vision,
but also by determining mathematically all kinds of Phænomena of Colours
which could be produced by Refractions. For to do this, there is nothing
else requisite than to find out the Separations of heterogeneous Rays,
and their various Mixtures and Proportions in every Mixture. By this
way of arguing I invented almost all the Phænomena described in these
Books, beside some others less necessary to the Argument; and by the
successes I met with in the Trials, I dare promise, that to him who
shall argue truly, and then try all things with good Glasses and
sufficient Circumspection, the expected Event will not be wanting. But
he is first to know what Colours will arise from any others mix'd in any
assigned Proportion.


_PROP._ IV. THEOR. III.

_Colours may be produced by Composition which shall be like to the
Colours of homogeneal Light as to the Appearance of Colour, but not as
to the Immutability of Colour and Constitution of Light. And those
Colours by how much they are more compounded by so much are they less
full and intense, and by too much Composition they maybe diluted and
weaken'd till they cease, and the Mixture becomes white or grey. There
may be also Colours produced by Composition, which are not fully like
any of the Colours of homogeneal Light._

For a Mixture of homogeneal red and yellow compounds an Orange, like in
appearance of Colour to that orange which in the series of unmixed
prismatick Colours lies between them; but the Light of one orange is
homogeneal as to Refrangibility, and that of the other is heterogeneal,
and the Colour of the one, if viewed through a Prism, remains unchanged,
that of the other is changed and resolved into its component Colours red
and yellow. And after the same manner other neighbouring homogeneal
Colours may compound new Colours, like the intermediate homogeneal ones,
as yellow and green, the Colour between them both, and afterwards, if
blue be added, there will be made a green the middle Colour of the three
which enter the Composition. For the yellow and blue on either hand, if
they are equal in quantity they draw the intermediate green equally
towards themselves in Composition, and so keep it as it were in
Æquilibrion, that it verge not more to the yellow on the one hand, and
to the blue on the other, but by their mix'd Actions remain still a
middle Colour. To this mix'd green there may be farther added some red
and violet, and yet the green will not presently cease, but only grow
less full and vivid, and by increasing the red and violet, it will grow
more and more dilute, until by the prevalence of the added Colours it be
overcome and turned into whiteness, or some other Colour. So if to the
Colour of any homogeneal Light, the Sun's white Light composed of all
sorts of Rays be added, that Colour will not vanish or change its
Species, but be diluted, and by adding more and more white it will be
diluted more and more perpetually. Lastly, If red and violet be mingled,
there will be generated according to their various Proportions various
Purples, such as are not like in appearance to the Colour of any
homogeneal Light, and of these Purples mix'd with yellow and blue may be
made other new Colours.


_PROP._ V. THEOR. IV.

_Whiteness and all grey Colours between white and black, may be
compounded of Colours, and the whiteness of the Sun's Light is
compounded of all the primary Colours mix'd in a due Proportion._

The PROOF by Experiments.

_Exper._ 9. The Sun shining into a dark Chamber through a little round
hole in the Window-shut, and his Light being there refracted by a Prism
to cast his coloured Image PT [in _Fig._ 5.] upon the opposite Wall: I
held a white Paper V to that image in such manner that it might be
illuminated by the colour'd Light reflected from thence, and yet not
intercept any part of that Light in its passage from the Prism to the
Spectrum. And I found that when the Paper was held nearer to any Colour
than to the rest, it appeared of that Colour to which it approached
nearest; but when it was equally or almost equally distant from all the
Colours, so that it might be equally illuminated by them all it appeared
white. And in this last situation of the Paper, if some Colours were
intercepted, the Paper lost its white Colour, and appeared of the Colour
of the rest of the Light which was not intercepted. So then the Paper
was illuminated with Lights of various Colours, namely, red, yellow,
green, blue and violet, and every part of the Light retained its proper
Colour, until it was incident on the Paper, and became reflected thence
to the Eye; so that if it had been either alone (the rest of the Light
being intercepted) or if it had abounded most, and been predominant in
the Light reflected from the Paper, it would have tinged the Paper with
its own Colour; and yet being mixed with the rest of the Colours in a
due proportion, it made the Paper look white, and therefore by a
Composition with the rest produced that Colour. The several parts of the
coloured Light reflected from the Spectrum, whilst they are propagated
from thence through the Air, do perpetually retain their proper Colours,
because wherever they fall upon the Eyes of any Spectator, they make the
several parts of the Spectrum to appear under their proper Colours. They
retain therefore their proper Colours when they fall upon the Paper V,
and so by the confusion and perfect mixture of those Colours compound
the whiteness of the Light reflected from thence.

_Exper._ 10. Let that Spectrum or solar Image PT [in _Fig._ 6.] fall now
upon the Lens MN above four Inches broad, and about six Feet distant
from the Prism ABC and so figured that it may cause the coloured Light
which divergeth from the Prism to converge and meet again at its Focus
G, about six or eight Feet distant from the Lens, and there to fall
perpendicularly upon a white Paper DE. And if you move this Paper to and
fro, you will perceive that near the Lens, as at _de_, the whole solar
Image (suppose at _pt_) will appear upon it intensely coloured after the
manner above-explained, and that by receding from the Lens those Colours
will perpetually come towards one another, and by mixing more and more
dilute one another continually, until at length the Paper come to the
Focus G, where by a perfect mixture they will wholly vanish and be
converted into whiteness, the whole Light appearing now upon the Paper
like a little white Circle. And afterwards by receding farther from the
Lens, the Rays which before converged will now cross one another in the
Focus G, and diverge from thence, and thereby make the Colours to appear
again, but yet in a contrary order; suppose at [Greek: de], where the
red _t_ is now above which before was below, and the violet _p_ is below
which before was above.

Let us now stop the Paper at the Focus G, where the Light appears
totally white and circular, and let us consider its whiteness. I say,
that this is composed of the converging Colours. For if any of those
Colours be intercepted at the Lens, the whiteness will cease and
degenerate into that Colour which ariseth from the composition of the
other Colours which are not intercepted. And then if the intercepted
Colours be let pass and fall upon that compound Colour, they mix with
it, and by their mixture restore the whiteness. So if the violet, blue
and green be intercepted, the remaining yellow, orange and red will
compound upon the Paper an orange, and then if the intercepted Colours
be let pass, they will fall upon this compounded orange, and together
with it decompound a white. So also if the red and violet be
intercepted, the remaining yellow, green and blue, will compound a green
upon the Paper, and then the red and violet being let pass will fall
upon this green, and together with it decompound a white. And that in
this Composition of white the several Rays do not suffer any Change in
their colorific Qualities by acting upon one another, but are only
mixed, and by a mixture of their Colours produce white, may farther
appear by these Arguments.

[Illustration: FIG. 6.]

If the Paper be placed beyond the Focus G, suppose at [Greek: de], and
then the red Colour at the Lens be alternately intercepted, and let pass
again, the violet Colour on the Paper will not suffer any Change
thereby, as it ought to do if the several sorts of Rays acted upon one
another in the Focus G, where they cross. Neither will the red upon the
Paper be changed by any alternate stopping, and letting pass the violet
which crosseth it.

And if the Paper be placed at the Focus G, and the white round Image at
G be viewed through the Prism HIK, and by the Refraction of that Prism
be translated to the place _rv_, and there appear tinged with various
Colours, namely, the violet at _v_ and red at _r_, and others between,
and then the red Colours at the Lens be often stopp'd and let pass by
turns, the red at _r_ will accordingly disappear, and return as often,
but the violet at _v_ will not thereby suffer any Change. And so by
stopping and letting pass alternately the blue at the Lens, the blue at
_v_ will accordingly disappear and return, without any Change made in
the red at _r_. The red therefore depends on one sort of Rays, and the
blue on another sort, which in the Focus G where they are commix'd, do
not act on one another. And there is the same Reason of the other
Colours.

I considered farther, that when the most refrangible Rays P_p_, and the
least refrangible ones T_t_, are by converging inclined to one another,
the Paper, if held very oblique to those Rays in the Focus G, might
reflect one sort of them more copiously than the other sort, and by that
Means the reflected Light would be tinged in that Focus with the Colour
of the predominant Rays, provided those Rays severally retained their
Colours, or colorific Qualities in the Composition of White made by them
in that Focus. But if they did not retain them in that White, but became
all of them severally endued there with a Disposition to strike the
Sense with the Perception of White, then they could never lose their
Whiteness by such Reflexions. I inclined therefore the Paper to the Rays
very obliquely, as in the second Experiment of this second Part of the
first Book, that the most refrangible Rays, might be more copiously
reflected than the rest, and the Whiteness at Length changed
successively into blue, indigo, and violet. Then I inclined it the
contrary Way, that the least refrangible Rays might be more copious in
the reflected Light than the rest, and the Whiteness turned successively
to yellow, orange, and red.

Lastly, I made an Instrument XY in fashion of a Comb, whose Teeth being
in number sixteen, were about an Inch and a half broad, and the
Intervals of the Teeth about two Inches wide. Then by interposing
successively the Teeth of this Instrument near the Lens, I intercepted
Part of the Colours by the interposed Tooth, whilst the rest of them
went on through the Interval of the Teeth to the Paper DE, and there
painted a round Solar Image. But the Paper I had first placed so, that
the Image might appear white as often as the Comb was taken away; and
then the Comb being as was said interposed, that Whiteness by reason of
the intercepted Part of the Colours at the Lens did always change into
the Colour compounded of those Colours which were not intercepted, and
that Colour was by the Motion of the Comb perpetually varied so, that in
the passing of every Tooth over the Lens all these Colours, red, yellow,
green, blue, and purple, did always succeed one another. I caused
therefore all the Teeth to pass successively over the Lens, and when the
Motion was slow, there appeared a perpetual Succession of the Colours
upon the Paper: But if I so much accelerated the Motion, that the
Colours by reason of their quick Succession could not be distinguished
from one another, the Appearance of the single Colours ceased. There was
no red, no yellow, no green, no blue, nor purple to be seen any longer,
but from a Confusion of them all there arose one uniform white Colour.
Of the Light which now by the Mixture of all the Colours appeared white,
there was no Part really white. One Part was red, another yellow, a
third green, a fourth blue, a fifth purple, and every Part retains its
proper Colour till it strike the Sensorium. If the Impressions follow
one another slowly, so that they may be severally perceived, there is
made a distinct Sensation of all the Colours one after another in a
continual Succession. But if the Impressions follow one another so
quickly, that they cannot be severally perceived, there ariseth out of
them all one common Sensation, which is neither of this Colour alone nor
of that alone, but hath it self indifferently to 'em all, and this is a
Sensation of Whiteness. By the Quickness of the Successions, the
Impressions of the several Colours are confounded in the Sensorium, and
out of that Confusion ariseth a mix'd Sensation. If a burning Coal be
nimbly moved round in a Circle with Gyrations continually repeated, the
whole Circle will appear like Fire; the reason of which is, that the
Sensation of the Coal in the several Places of that Circle remains
impress'd on the Sensorium, until the Coal return again to the same
Place. And so in a quick Consecution of the Colours the Impression of
every Colour remains in the Sensorium, until a Revolution of all the
Colours be compleated, and that first Colour return again. The
Impressions therefore of all the successive Colours are at once in the
Sensorium, and jointly stir up a Sensation of them all; and so it is
manifest by this Experiment, that the commix'd Impressions of all the
Colours do stir up and beget a Sensation of white, that is, that
Whiteness is compounded of all the Colours.

And if the Comb be now taken away, that all the Colours may at once pass
from the Lens to the Paper, and be there intermixed, and together
reflected thence to the Spectator's Eyes; their Impressions on the
Sensorium being now more subtilly and perfectly commixed there, ought
much more to stir up a Sensation of Whiteness.

You may instead of the Lens use two Prisms HIK and LMN, which by
refracting the coloured Light the contrary Way to that of the first
Refraction, may make the diverging Rays converge and meet again in G, as
you see represented in the seventh Figure. For where they meet and mix,
they will compose a white Light, as when a Lens is used.

_Exper._ 11. Let the Sun's coloured Image PT [in _Fig._ 8.] fall upon
the Wall of a dark Chamber, as in the third Experiment of the first
Book, and let the same be viewed through a Prism _abc_, held parallel to
the Prism ABC, by whose Refraction that Image was made, and let it now
appear lower than before, suppose in the Place S over-against the red
Colour T. And if you go near to the Image PT, the Spectrum S will appear
oblong and coloured like the Image PT; but if you recede from it, the
Colours of the spectrum S will be contracted more and more, and at
length vanish, that Spectrum S becoming perfectly round and white; and
if you recede yet farther, the Colours will emerge again, but in a
contrary Order. Now that Spectrum S appears white in that Case, when the
Rays of several sorts which converge from the several Parts of the Image
PT, to the Prism _abc_, are so refracted unequally by it, that in their
Passage from the Prism to the Eye they may diverge from one and the same
Point of the Spectrum S, and so fall afterwards upon one and the same
Point in the bottom of the Eye, and there be mingled.

[Illustration: FIG. 7.]

[Illustration: FIG. 8.]

And farther, if the Comb be here made use of, by whose Teeth the Colours
at the Image PT may be successively intercepted; the Spectrum S, when
the Comb is moved slowly, will be perpetually tinged with successive
Colours: But when by accelerating the Motion of the Comb, the Succession
of the Colours is so quick that they cannot be severally seen, that
Spectrum S, by a confused and mix'd Sensation of them all, will appear
white.

_Exper._ 12. The Sun shining through a large Prism ABC [in _Fig._ 9.]
upon a Comb XY, placed immediately behind the Prism, his Light which
passed through the Interstices of the Teeth fell upon a white Paper DE.
The Breadths of the Teeth were equal to their Interstices, and seven
Teeth together with their Interstices took up an Inch in Breadth. Now,
when the Paper was about two or three Inches distant from the Comb, the
Light which passed through its several Interstices painted so many
Ranges of Colours, _kl_, _mn_, _op_, _qr_, &c. which were parallel to
one another, and contiguous, and without any Mixture of white. And these
Ranges of Colours, if the Comb was moved continually up and down with a
reciprocal Motion, ascended and descended in the Paper, and when the
Motion of the Comb was so quick, that the Colours could not be
distinguished from one another, the whole Paper by their Confusion and
Mixture in the Sensorium appeared white.

[Illustration: FIG. 9.]

Let the Comb now rest, and let the Paper be removed farther from the
Prism, and the several Ranges of Colours will be dilated and expanded
into one another more and more, and by mixing their Colours will dilute
one another, and at length, when the distance of the Paper from the Comb
is about a Foot, or a little more (suppose in the Place 2D 2E) they will
so far dilute one another, as to become white.

With any Obstacle, let all the Light be now stopp'd which passes through
any one Interval of the Teeth, so that the Range of Colours which comes
from thence may be taken away, and you will see the Light of the rest of
the Ranges to be expanded into the Place of the Range taken away, and
there to be coloured. Let the intercepted Range pass on as before, and
its Colours falling upon the Colours of the other Ranges, and mixing
with them, will restore the Whiteness.

Let the Paper 2D 2E be now very much inclined to the Rays, so that the
most refrangible Rays may be more copiously reflected than the rest, and
the white Colour of the Paper through the Excess of those Rays will be
changed into blue and violet. Let the Paper be as much inclined the
contrary way, that the least refrangible Rays may be now more copiously
reflected than the rest, and by their Excess the Whiteness will be
changed into yellow and red. The several Rays therefore in that white
Light do retain their colorific Qualities, by which those of any sort,
whenever they become more copious than the rest, do by their Excess and
Predominance cause their proper Colour to appear.

And by the same way of arguing, applied to the third Experiment of this
second Part of the first Book, it may be concluded, that the white
Colour of all refracted Light at its very first Emergence, where it
appears as white as before its Incidence, is compounded of various
Colours.

[Illustration: FIG. 10.]

_Exper._ 13. In the foregoing Experiment the several Intervals of the
Teeth of the Comb do the Office of so many Prisms, every Interval
producing the Phænomenon of one Prism. Whence instead of those Intervals
using several Prisms, I try'd to compound Whiteness by mixing their
Colours, and did it by using only three Prisms, as also by using only
two as follows. Let two Prisms ABC and _abc_, [in _Fig._ 10.] whose
refracting Angles B and _b_ are equal, be so placed parallel to one
another, that the refracting Angle B of the one may touch the Angle _c_
at the Base of the other, and their Planes CB and _cb_, at which the
Rays emerge, may lie in Directum. Then let the Light trajected through
them fall upon the Paper MN, distant about 8 or 12 Inches from the
Prisms. And the Colours generated by the interior Limits B and _c_ of
the two Prisms, will be mingled at PT, and there compound white. For if
either Prism be taken away, the Colours made by the other will appear in
that Place PT, and when the Prism is restored to its Place again, so
that its Colours may there fall upon the Colours of the other, the
Mixture of them both will restore the Whiteness.

This Experiment succeeds also, as I have tried, when the Angle _b_ of
the lower Prism, is a little greater than the Angle B of the upper, and
between the interior Angles B and _c_, there intercedes some Space B_c_,
as is represented in the Figure, and the refracting Planes BC and _bc_,
are neither in Directum, nor parallel to one another. For there is
nothing more requisite to the Success of this Experiment, than that the
Rays of all sorts may be uniformly mixed upon the Paper in the Place PT.
If the most refrangible Rays coming from the superior Prism take up all
the Space from M to P, the Rays of the same sort which come from the
inferior Prism ought to begin at P, and take up all the rest of the
Space from thence towards N. If the least refrangible Rays coming from
the superior Prism take up the Space MT, the Rays of the same kind which
come from the other Prism ought to begin at T, and take up the
remaining Space TN. If one sort of the Rays which have intermediate
Degrees of Refrangibility, and come from the superior Prism be extended
through the Space MQ, and another sort of those Rays through the Space
MR, and a third sort of them through the Space MS, the same sorts of
Rays coming from the lower Prism, ought to illuminate the remaining
Spaces QN, RN, SN, respectively. And the same is to be understood of all
the other sorts of Rays. For thus the Rays of every sort will be
scattered uniformly and evenly through the whole Space MN, and so being
every where mix'd in the same Proportion, they must every where produce
the same Colour. And therefore, since by this Mixture they produce white
in the Exterior Spaces MP and TN, they must also produce white in the
Interior Space PT. This is the reason of the Composition by which
Whiteness was produced in this Experiment, and by what other way soever
I made the like Composition, the Result was Whiteness.

Lastly, If with the Teeth of a Comb of a due Size, the coloured Lights
of the two Prisms which fall upon the Space PT be alternately
intercepted, that Space PT, when the Motion of the Comb is slow, will
always appear coloured, but by accelerating the Motion of the Comb so
much that the successive Colours cannot be distinguished from one
another, it will appear white.

_Exper._ 14. Hitherto I have produced Whiteness by mixing the Colours of
Prisms. If now the Colours of natural Bodies are to be mingled, let
Water a little thicken'd with Soap be agitated to raise a Froth, and
after that Froth has stood a little, there will appear to one that shall
view it intently various Colours every where in the Surfaces of the
several Bubbles; but to one that shall go so far off, that he cannot
distinguish the Colours from one another, the whole Froth will grow
white with a perfect Whiteness.

_Exper._ 15. Lastly, In attempting to compound a white, by mixing the
coloured Powders which Painters use, I consider'd that all colour'd
Powders do suppress and stop in them a very considerable Part of the
Light by which they are illuminated. For they become colour'd by
reflecting the Light of their own Colours more copiously, and that of
all other Colours more sparingly, and yet they do not reflect the Light
of their own Colours so copiously as white Bodies do. If red Lead, for
instance, and a white Paper, be placed in the red Light of the colour'd
Spectrum made in a dark Chamber by the Refraction of a Prism, as is
described in the third Experiment of the first Part of this Book; the
Paper will appear more lucid than the red Lead, and therefore reflects
the red-making Rays more copiously than red Lead doth. And if they be
held in the Light of any other Colour, the Light reflected by the Paper
will exceed the Light reflected by the red Lead in a much greater
Proportion. And the like happens in Powders of other Colours. And
therefore by mixing such Powders, we are not to expect a strong and
full White, such as is that of Paper, but some dusky obscure one, such
as might arise from a Mixture of Light and Darkness, or from white and
black, that is, a grey, or dun, or russet brown, such as are the Colours
of a Man's Nail, of a Mouse, of Ashes, of ordinary Stones, of Mortar, of
Dust and Dirt in High-ways, and the like. And such a dark white I have
often produced by mixing colour'd Powders. For thus one Part of red
Lead, and five Parts of _Viride Æris_, composed a dun Colour like that
of a Mouse. For these two Colours were severally so compounded of
others, that in both together were a Mixture of all Colours; and there
was less red Lead used than _Viride Æris_, because of the Fulness of its
Colour. Again, one Part of red Lead, and four Parts of blue Bise,
composed a dun Colour verging a little to purple, and by adding to this
a certain Mixture of Orpiment and _Viride Æris_ in a due Proportion, the
Mixture lost its purple Tincture, and became perfectly dun. But the
Experiment succeeded best without Minium thus. To Orpiment I added by
little and little a certain full bright purple, which Painters use,
until the Orpiment ceased to be yellow, and became of a pale red. Then I
diluted that red by adding a little _Viride Æris_, and a little more
blue Bise than _Viride Æris_, until it became of such a grey or pale
white, as verged to no one of the Colours more than to another. For thus
it became of a Colour equal in Whiteness to that of Ashes, or of Wood
newly cut, or of a Man's Skin. The Orpiment reflected more Light than
did any other of the Powders, and therefore conduced more to the
Whiteness of the compounded Colour than they. To assign the Proportions
accurately may be difficult, by reason of the different Goodness of
Powders of the same kind. Accordingly, as the Colour of any Powder is
more or less full and luminous, it ought to be used in a less or greater
Proportion.

Now, considering that these grey and dun Colours may be also produced by
mixing Whites and Blacks, and by consequence differ from perfect Whites,
not in Species of Colours, but only in degree of Luminousness, it is
manifest that there is nothing more requisite to make them perfectly
white than to increase their Light sufficiently; and, on the contrary,
if by increasing their Light they can be brought to perfect Whiteness,
it will thence also follow, that they are of the same Species of Colour
with the best Whites, and differ from them only in the Quantity of
Light. And this I tried as follows. I took the third of the
above-mention'd grey Mixtures, (that which was compounded of Orpiment,
Purple, Bise, and _Viride Æris_) and rubbed it thickly upon the Floor of
my Chamber, where the Sun shone upon it through the opened Casement; and
by it, in the shadow, I laid a Piece of white Paper of the same Bigness.
Then going from them to the distance of 12 or 18 Feet, so that I could
not discern the Unevenness of the Surface of the Powder, nor the little
Shadows let fall from the gritty Particles thereof; the Powder appeared
intensely white, so as to transcend even the Paper it self in Whiteness,
especially if the Paper were a little shaded from the Light of the
Clouds, and then the Paper compared with the Powder appeared of such a
grey Colour as the Powder had done before. But by laying the Paper where
the Sun shines through the Glass of the Window, or by shutting the
Window that the Sun might shine through the Glass upon the Powder, and
by such other fit Means of increasing or decreasing the Lights wherewith
the Powder and Paper were illuminated, the Light wherewith the Powder is
illuminated may be made stronger in such a due Proportion than the Light
wherewith the Paper is illuminated, that they shall both appear exactly
alike in Whiteness. For when I was trying this, a Friend coming to visit
me, I stopp'd him at the Door, and before I told him what the Colours
were, or what I was doing; I asked him, Which of the two Whites were the
best, and wherein they differed? And after he had at that distance
viewed them well, he answer'd, that they were both good Whites, and that
he could not say which was best, nor wherein their Colours differed.
Now, if you consider, that this White of the Powder in the Sun-shine was
compounded of the Colours which the component Powders (Orpiment, Purple,
Bise, and _Viride Æris_) have in the same Sun-shine, you must
acknowledge by this Experiment, as well as by the former, that perfect
Whiteness may be compounded of Colours.

From what has been said it is also evident, that the Whiteness of the
Sun's Light is compounded of all the Colours wherewith the several sorts
of Rays whereof that Light consists, when by their several
Refrangibilities they are separated from one another, do tinge Paper or
any other white Body whereon they fall. For those Colours (by _Prop._
II. _Part_ 2.) are unchangeable, and whenever all those Rays with those
their Colours are mix'd again, they reproduce the same white Light as
before.


_PROP._ VI. PROB. II.

_In a mixture of Primary Colours, the Quantity and Quality of each being
given, to know the Colour of the Compound._

[Illustration: FIG. 11.]

With the Center O [in _Fig._ 11.] and Radius OD describe a Circle ADF,
and distinguish its Circumference into seven Parts DE, EF, FG, GA, AB,
BC, CD, proportional to the seven Musical Tones or Intervals of the
eight Sounds, _Sol_, _la_, _fa_, _sol_, _la_, _mi_, _fa_, _sol_,
contained in an eight, that is, proportional to the Number 1/9, 1/16,
1/10, 1/9, 1/16, 1/16, 1/9. Let the first Part DE represent a red
Colour, the second EF orange, the third FG yellow, the fourth CA green,
the fifth AB blue, the sixth BC indigo, and the seventh CD violet. And
conceive that these are all the Colours of uncompounded Light gradually
passing into one another, as they do when made by Prisms; the
Circumference DEFGABCD, representing the whole Series of Colours from
one end of the Sun's colour'd Image to the other, so that from D to E be
all degrees of red, at E the mean Colour between red and orange, from E
to F all degrees of orange, at F the mean between orange and yellow,
from F to G all degrees of yellow, and so on. Let _p_ be the Center of
Gravity of the Arch DE, and _q_, _r_, _s_, _t_, _u_, _x_, the Centers of
Gravity of the Arches EF, FG, GA, AB, BC, and CD respectively, and about
those Centers of Gravity let Circles proportional to the Number of Rays
of each Colour in the given Mixture be describ'd: that is, the Circle
_p_ proportional to the Number of the red-making Rays in the Mixture,
the Circle _q_ proportional to the Number of the orange-making Rays in
the Mixture, and so of the rest. Find the common Center of Gravity of
all those Circles, _p_, _q_, _r_, _s_, _t_, _u_, _x_. Let that Center be
Z; and from the Center of the Circle ADF, through Z to the
Circumference, drawing the Right Line OY, the Place of the Point Y in
the Circumference shall shew the Colour arising from the Composition of
all the Colours in the given Mixture, and the Line OZ shall be
proportional to the Fulness or Intenseness of the Colour, that is, to
its distance from Whiteness. As if Y fall in the middle between F and G,
the compounded Colour shall be the best yellow; if Y verge from the
middle towards F or G, the compound Colour shall accordingly be a
yellow, verging towards orange or green. If Z fall upon the
Circumference, the Colour shall be intense and florid in the highest
Degree; if it fall in the mid-way between the Circumference and Center,
it shall be but half so intense, that is, it shall be such a Colour as
would be made by diluting the intensest yellow with an equal quantity of
whiteness; and if it fall upon the center O, the Colour shall have lost
all its intenseness, and become a white. But it is to be noted, That if
the point Z fall in or near the line OD, the main ingredients being the
red and violet, the Colour compounded shall not be any of the prismatick
Colours, but a purple, inclining to red or violet, accordingly as the
point Z lieth on the side of the line DO towards E or towards C, and in
general the compounded violet is more bright and more fiery than the
uncompounded. Also if only two of the primary Colours which in the
circle are opposite to one another be mixed in an equal proportion, the
point Z shall fall upon the center O, and yet the Colour compounded of
those two shall not be perfectly white, but some faint anonymous Colour.
For I could never yet by mixing only two primary Colours produce a
perfect white. Whether it may be compounded of a mixture of three taken
at equal distances in the circumference I do not know, but of four or
five I do not much question but it may. But these are Curiosities of
little or no moment to the understanding the Phænomena of Nature. For in
all whites produced by Nature, there uses to be a mixture of all sorts
of Rays, and by consequence a composition of all Colours.

To give an instance of this Rule; suppose a Colour is compounded of
these homogeneal Colours, of violet one part, of indigo one part, of
blue two parts, of green three parts, of yellow five parts, of orange
six parts, and of red ten parts. Proportional to these parts describe
the Circles _x_, _v_, _t_, _s_, _r_, _q_, _p_, respectively, that is, so
that if the Circle _x_ be one, the Circle _v_ may be one, the Circle _t_
two, the Circle _s_ three, and the Circles _r_, _q_ and _p_, five, six
and ten. Then I find Z the common center of gravity of these Circles,
and through Z drawing the Line OY, the Point Y falls upon the
circumference between E and F, something nearer to E than to F, and
thence I conclude, that the Colour compounded of these Ingredients will
be an orange, verging a little more to red than to yellow. Also I find
that OZ is a little less than one half of OY, and thence I conclude,
that this orange hath a little less than half the fulness or intenseness
of an uncompounded orange; that is to say, that it is such an orange as
may be made by mixing an homogeneal orange with a good white in the
proportion of the Line OZ to the Line ZY, this Proportion being not of
the quantities of mixed orange and white Powders, but of the quantities
of the Lights reflected from them.

This Rule I conceive accurate enough for practice, though not
mathematically accurate; and the truth of it may be sufficiently proved
to Sense, by stopping any of the Colours at the Lens in the tenth
Experiment of this Book. For the rest of the Colours which are not
stopp'd, but pass on to the Focus of the Lens, will there compound
either accurately or very nearly such a Colour, as by this Rule ought to
result from their Mixture.


_PROP._ VII. THEOR. V.

_All the Colours in the Universe which are made by Light, and depend not
on the Power of Imagination, are either the Colours of homogeneal
Lights, or compounded of these, and that either accurately or very
nearly, according to the Rule of the foregoing Problem._

For it has been proved (in _Prop. 1. Part 2._) that the changes of
Colours made by Refractions do not arise from any new Modifications of
the Rays impress'd by those Refractions, and by the various Terminations
of Light and Shadow, as has been the constant and general Opinion of
Philosophers. It has also been proved that the several Colours of the
homogeneal Rays do constantly answer to their degrees of Refrangibility,
(_Prop._ 1. _Part_ 1. and _Prop._ 2. _Part_ 2.) and that their degrees
of Refrangibility cannot be changed by Refractions and Reflexions
(_Prop._ 2. _Part_ 1.) and by consequence that those their Colours are
likewise immutable. It has also been proved directly by refracting and
reflecting homogeneal Lights apart, that their Colours cannot be
changed, (_Prop._ 2. _Part_ 2.) It has been proved also, that when the
several sorts of Rays are mixed, and in crossing pass through the same
space, they do not act on one another so as to change each others
colorific qualities. (_Exper._ 10. _Part_ 2.) but by mixing their
Actions in the Sensorium beget a Sensation differing from what either
would do apart, that is a Sensation of a mean Colour between their
proper Colours; and particularly when by the concourse and mixtures of
all sorts of Rays, a white Colour is produced, the white is a mixture of
all the Colours which the Rays would have apart, (_Prop._ 5. _Part_ 2.)
The Rays in that mixture do not lose or alter their several colorific
qualities, but by all their various kinds of Actions mix'd in the
Sensorium, beget a Sensation of a middling Colour between all their
Colours, which is whiteness. For whiteness is a mean between all
Colours, having it self indifferently to them all, so as with equal
facility to be tinged with any of them. A red Powder mixed with a little
blue, or a blue with a little red, doth not presently lose its Colour,
but a white Powder mix'd with any Colour is presently tinged with that
Colour, and is equally capable of being tinged with any Colour whatever.
It has been shewed also, that as the Sun's Light is mix'd of all sorts
of Rays, so its whiteness is a mixture of the Colours of all sorts of
Rays; those Rays having from the beginning their several colorific
qualities as well as their several Refrangibilities, and retaining them
perpetually unchanged notwithstanding any Refractions or Reflexions they
may at any time suffer, and that whenever any sort of the Sun's Rays is
by any means (as by Reflexion in _Exper._ 9, and 10. _Part_ 1. or by
Refraction as happens in all Refractions) separated from the rest, they
then manifest their proper Colours. These things have been prov'd, and
the sum of all this amounts to the Proposition here to be proved. For if
the Sun's Light is mix'd of several sorts of Rays, each of which have
originally their several Refrangibilities and colorific Qualities, and
notwithstanding their Refractions and Reflexions, and their various
Separations or Mixtures, keep those their original Properties
perpetually the same without alteration; then all the Colours in the
World must be such as constantly ought to arise from the original
colorific qualities of the Rays whereof the Lights consist by which
those Colours are seen. And therefore if the reason of any Colour
whatever be required, we have nothing else to do than to consider how
the Rays in the Sun's Light have by Reflexions or Refractions, or other
causes, been parted from one another, or mixed together; or otherwise to
find out what sorts of Rays are in the Light by which that Colour is
made, and in what Proportion; and then by the last Problem to learn the
Colour which ought to arise by mixing those Rays (or their Colours) in
that proportion. I speak here of Colours so far as they arise from
Light. For they appear sometimes by other Causes, as when by the power
of Phantasy we see Colours in a Dream, or a Mad-man sees things before
him which are not there; or when we see Fire by striking the Eye, or see
Colours like the Eye of a Peacock's Feather, by pressing our Eyes in
either corner whilst we look the other way. Where these and such like
Causes interpose not, the Colour always answers to the sort or sorts of
the Rays whereof the Light consists, as I have constantly found in
whatever Phænomena of Colours I have hitherto been able to examine. I
shall in the following Propositions give instances of this in the
Phænomena of chiefest note.


_PROP._ VIII. PROB. III.

_By the discovered Properties of Light to explain the Colours made by
Prisms._

Let ABC [in _Fig._ 12.] represent a Prism refracting the Light of the
Sun, which comes into a dark Chamber through a hole F[Greek: ph] almost
as broad as the Prism, and let MN represent a white Paper on which the
refracted Light is cast, and suppose the most refrangible or deepest
violet-making Rays fall upon the Space P[Greek: p], the least
refrangible or deepest red-making Rays upon the Space T[Greek: t], the
middle sort between the indigo-making and blue-making Rays upon the
Space Q[Greek: ch], the middle sort of the green-making Rays upon the
Space R, the middle sort between the yellow-making and orange-making
Rays upon the Space S[Greek: s], and other intermediate sorts upon
intermediate Spaces. For so the Spaces upon which the several sorts
adequately fall will by reason of the different Refrangibility of those
sorts be one lower than another. Now if the Paper MN be so near the
Prism that the Spaces PT and [Greek: pt] do not interfere with one
another, the distance between them T[Greek: p] will be illuminated by
all the sorts of Rays in that proportion to one another which they have
at their very first coming out of the Prism, and consequently be white.
But the Spaces PT and [Greek: pt] on either hand, will not be
illuminated by them all, and therefore will appear coloured. And
particularly at P, where the outmost violet-making Rays fall alone, the
Colour must be the deepest violet. At Q where the violet-making and
indigo-making Rays are mixed, it must be a violet inclining much to
indigo. At R where the violet-making, indigo-making, blue-making, and
one half of the green-making Rays are mixed, their Colours must (by the
construction of the second Problem) compound a middle Colour between
indigo and blue. At S where all the Rays are mixed, except the
red-making and orange-making, their Colours ought by the same Rule to
compound a faint blue, verging more to green than indigo. And in the
progress from S to T, this blue will grow more and more faint and
dilute, till at T, where all the Colours begin to be mixed, it ends in
whiteness.

[Illustration: FIG. 12.]

So again, on the other side of the white at [Greek: t], where the least
refrangible or utmost red-making Rays are alone, the Colour must be the
deepest red. At [Greek: s] the mixture of red and orange will compound a
red inclining to orange. At [Greek: r] the mixture of red, orange,
yellow, and one half of the green must compound a middle Colour between
orange and yellow. At [Greek: ch] the mixture of all Colours but violet
and indigo will compound a faint yellow, verging more to green than to
orange. And this yellow will grow more faint and dilute continually in
its progress from [Greek: ch] to [Greek: p], where by a mixture of all
sorts of Rays it will become white.

These Colours ought to appear were the Sun's Light perfectly white: But
because it inclines to yellow, the Excess of the yellow-making Rays
whereby 'tis tinged with that Colour, being mixed with the faint blue
between S and T, will draw it to a faint green. And so the Colours in
order from P to [Greek: t] ought to be violet, indigo, blue, very faint
green, white, faint yellow, orange, red. Thus it is by the computation:
And they that please to view the Colours made by a Prism will find it so
in Nature.

These are the Colours on both sides the white when the Paper is held
between the Prism and the Point X where the Colours meet, and the
interjacent white vanishes. For if the Paper be held still farther off
from the Prism, the most refrangible and least refrangible Rays will be
wanting in the middle of the Light, and the rest of the Rays which are
found there, will by mixture produce a fuller green than before. Also
the yellow and blue will now become less compounded, and by consequence
more intense than before. And this also agrees with experience.

And if one look through a Prism upon a white Object encompassed with
blackness or darkness, the reason of the Colours arising on the edges is
much the same, as will appear to one that shall a little consider it. If
a black Object be encompassed with a white one, the Colours which appear
through the Prism are to be derived from the Light of the white one,
spreading into the Regions of the black, and therefore they appear in a
contrary order to that, when a white Object is surrounded with black.
And the same is to be understood when an Object is viewed, whose parts
are some of them less luminous than others. For in the borders of the
more and less luminous Parts, Colours ought always by the same
Principles to arise from the Excess of the Light of the more luminous,
and to be of the same kind as if the darker parts were black, but yet to
be more faint and dilute.

What is said of Colours made by Prisms may be easily applied to Colours
made by the Glasses of Telescopes or Microscopes, or by the Humours of
the Eye. For if the Object-glass of a Telescope be thicker on one side
than on the other, or if one half of the Glass, or one half of the Pupil
of the Eye be cover'd with any opake substance; the Object-glass, or
that part of it or of the Eye which is not cover'd, may be consider'd as
a Wedge with crooked Sides, and every Wedge of Glass or other pellucid
Substance has the effect of a Prism in refracting the Light which passes
through it.[L]

How the Colours in the ninth and tenth Experiments of the first Part
arise from the different Reflexibility of Light, is evident by what was
there said. But it is observable in the ninth Experiment, that whilst
the Sun's direct Light is yellow, the Excess of the blue-making Rays in
the reflected beam of Light MN, suffices only to bring that yellow to a
pale white inclining to blue, and not to tinge it with a manifestly blue
Colour. To obtain therefore a better blue, I used instead of the yellow
Light of the Sun the white Light of the Clouds, by varying a little the
Experiment, as follows.

[Illustration: FIG. 13.]

_Exper._ 16 Let HFG [in _Fig._ 13.] represent a Prism in the open Air,
and S the Eye of the Spectator, viewing the Clouds by their Light coming
into the Prism at the Plane Side FIGK, and reflected in it by its Base
HEIG, and thence going out through its Plane Side HEFK to the Eye. And
when the Prism and Eye are conveniently placed, so that the Angles of
Incidence and Reflexion at the Base may be about 40 Degrees, the
Spectator will see a Bow MN of a blue Colour, running from one End of
the Base to the other, with the Concave Side towards him, and the Part
of the Base IMNG beyond this Bow will be brighter than the other Part
EMNH on the other Side of it. This blue Colour MN being made by nothing
else than by Reflexion of a specular Superficies, seems so odd a
Phænomenon, and so difficult to be explained by the vulgar Hypothesis of
Philosophers, that I could not but think it deserved to be taken Notice
of. Now for understanding the Reason of it, suppose the Plane ABC to cut
the Plane Sides and Base of the Prism perpendicularly. From the Eye to
the Line BC, wherein that Plane cuts the Base, draw the Lines S_p_ and
S_t_, in the Angles S_pc_ 50 degr. 1/9, and S_tc_ 49 degr. 1/28, and the
Point _p_ will be the Limit beyond which none of the most refrangible
Rays can pass through the Base of the Prism, and be refracted, whose
Incidence is such that they may be reflected to the Eye; and the Point
_t_ will be the like Limit for the least refrangible Rays, that is,
beyond which none of them can pass through the Base, whose Incidence is
such that by Reflexion they may come to the Eye. And the Point _r_ taken
in the middle Way between _p_ and _t_, will be the like Limit for the
meanly refrangible Rays. And therefore all the least refrangible Rays
which fall upon the Base beyond _t_, that is, between _t_ and B, and can
come from thence to the Eye, will be reflected thither: But on this side
_t_, that is, between _t_ and _c_, many of these Rays will be
transmitted through the Base. And all the most refrangible Rays which
fall upon the Base beyond _p_, that is, between, _p_ and B, and can by
Reflexion come from thence to the Eye, will be reflected thither, but
every where between _p_ and _c_, many of these Rays will get through the
Base, and be refracted; and the same is to be understood of the meanly
refrangible Rays on either side of the Point _r_. Whence it follows,
that the Base of the Prism must every where between _t_ and B, by a
total Reflexion of all sorts of Rays to the Eye, look white and bright.
And every where between _p_ and C, by reason of the Transmission of many
Rays of every sort, look more pale, obscure, and dark. But at _r_, and
in other Places between _p_ and _t_, where all the more refrangible Rays
are reflected to the Eye, and many of the less refrangible are
transmitted, the Excess of the most refrangible in the reflected Light
will tinge that Light with their Colour, which is violet and blue. And
this happens by taking the Line C _prt_ B any where between the Ends of
the Prism HG and EI.


_PROP._ IX. PROB. IV.

_By the discovered Properties of Light to explain the Colours of the
Rain-bow._

[Illustration: FIG. 14.]

This Bow never appears, but where it rains in the Sun-shine, and may be
made artificially by spouting up Water which may break aloft, and
scatter into Drops, and fall down like Rain. For the Sun shining upon
these Drops certainly causes the Bow to appear to a Spectator standing
in a due Position to the Rain and Sun. And hence it is now agreed upon,
that this Bow is made by Refraction of the Sun's Light in drops of
falling Rain. This was understood by some of the Antients, and of late
more fully discover'd and explain'd by the famous _Antonius de Dominis_
Archbishop of _Spalato_, in his book _De Radiis Visûs & Lucis_,
published by his Friend _Bartolus_ at _Venice_, in the Year 1611, and
written above 20 Years before. For he teaches there how the interior Bow
is made in round Drops of Rain by two Refractions of the Sun's Light,
and one Reflexion between them, and the exterior by two Refractions, and
two sorts of Reflexions between them in each Drop of Water, and proves
his Explications by Experiments made with a Phial full of Water, and
with Globes of Glass filled with Water, and placed in the Sun to make
the Colours of the two Bows appear in them. The same Explication
_Des-Cartes_ hath pursued in his Meteors, and mended that of the
exterior Bow. But whilst they understood not the true Origin of Colours,
it's necessary to pursue it here a little farther. For understanding
therefore how the Bow is made, let a Drop of Rain, or any other
spherical transparent Body be represented by the Sphere BNFG, [in _Fig._
14.] described with the Center C, and Semi-diameter CN. And let AN be
one of the Sun's Rays incident upon it at N, and thence refracted to F,
where let it either go out of the Sphere by Refraction towards V, or be
reflected to G; and at G let it either go out by Refraction to R, or be
reflected to H; and at H let it go out by Refraction towards S, cutting
the incident Ray in Y. Produce AN and RG, till they meet in X, and upon
AX and NF, let fall the Perpendiculars CD and CE, and produce CD till it
fall upon the Circumference at L. Parallel to the incident Ray AN draw
the Diameter BQ, and let the Sine of Incidence out of Air into Water be
to the Sine of Refraction as I to R. Now, if you suppose the Point of
Incidence N to move from the Point B, continually till it come to L, the
Arch QF will first increase and then decrease, and so will the Angle AXR
which the Rays AN and GR contain; and the Arch QF and Angle AXR will be
biggest when ND is to CN as sqrt(II - RR) to sqrt(3)RR, in which
case NE will be to ND as 2R to I. Also the Angle AYS, which the Rays AN
and HS contain will first decrease, and then increase and grow least
when ND is to CN as sqrt(II - RR) to sqrt(8)RR, in which case NE
will be to ND, as 3R to I. And so the Angle which the next emergent Ray
(that is, the emergent Ray after three Reflexions) contains with the
incident Ray AN will come to its Limit when ND is to CN as sqrt(II -
RR) to sqrt(15)RR, in which case NE will be to ND as 4R to I. And the
Angle which the Ray next after that Emergent, that is, the Ray emergent
after four Reflexions, contains with the Incident, will come to its
Limit, when ND is to CN as sqrt(II - RR) to sqrt(24)RR, in which
case NE will be to ND as 5R to I; and so on infinitely, the Numbers 3,
8, 15, 24, &c. being gather'd by continual Addition of the Terms of the
arithmetical Progression 3, 5, 7, 9, &c. The Truth of all this
Mathematicians will easily examine.[M]

Now it is to be observed, that as when the Sun comes to his Tropicks,
Days increase and decrease but a very little for a great while together;
so when by increasing the distance CD, these Angles come to their
Limits, they vary their quantity but very little for some time together,
and therefore a far greater number of the Rays which fall upon all the
Points N in the Quadrant BL, shall emerge in the Limits of these Angles,
than in any other Inclinations. And farther it is to be observed, that
the Rays which differ in Refrangibility will have different Limits of
their Angles of Emergence, and by consequence according to their
different Degrees of Refrangibility emerge most copiously in different
Angles, and being separated from one another appear each in their proper
Colours. And what those Angles are may be easily gather'd from the
foregoing Theorem by Computation.

For in the least refrangible Rays the Sines I and R (as was found above)
are 108 and 81, and thence by Computation the greatest Angle AXR will be
found 42 Degrees and 2 Minutes, and the least Angle AYS, 50 Degrees and
57 Minutes. And in the most refrangible Rays the Sines I and R are 109
and 81, and thence by Computation the greatest Angle AXR will be found
40 Degrees and 17 Minutes, and the least Angle AYS 54 Degrees and 7
Minutes.

Suppose now that O [in _Fig._ 15.] is the Spectator's Eye, and OP a Line
drawn parallel to the Sun's Rays and let POE, POF, POG, POH, be Angles
of 40 Degr. 17 Min. 42 Degr. 2 Min. 50 Degr. 57 Min. and 54 Degr. 7 Min.
respectively, and these Angles turned about their common Side OP, shall
with their other Sides OE, OF; OG, OH, describe the Verges of two
Rain-bows AF, BE and CHDG. For if E, F, G, H, be drops placed any where
in the conical Superficies described by OE, OF, OG, OH, and be
illuminated by the Sun's Rays SE, SF, SG, SH; the Angle SEO being equal
to the Angle POE, or 40 Degr. 17 Min. shall be the greatest Angle in
which the most refrangible Rays can after one Reflexion be refracted to
the Eye, and therefore all the Drops in the Line OE shall send the most
refrangible Rays most copiously to the Eye, and thereby strike the
Senses with the deepest violet Colour in that Region. And in like
manner the Angle SFO being equal to the Angle POF, or 42 Degr. 2 Min.
shall be the greatest in which the least refrangible Rays after one
Reflexion can emerge out of the Drops, and therefore those Rays shall
come most copiously to the Eye from the Drops in the Line OF, and strike
the Senses with the deepest red Colour in that Region. And by the same
Argument, the Rays which have intermediate Degrees of Refrangibility
shall come most copiously from Drops between E and F, and strike the
Senses with the intermediate Colours, in the Order which their Degrees
of Refrangibility require, that is in the Progress from E to F, or from
the inside of the Bow to the outside in this order, violet, indigo,
blue, green, yellow, orange, red. But the violet, by the mixture of the
white Light of the Clouds, will appear faint and incline to purple.

[Illustration: FIG. 15.]

Again, the Angle SGO being equal to the Angle POG, or 50 Gr. 51 Min.
shall be the least Angle in which the least refrangible Rays can after
two Reflexions emerge out of the Drops, and therefore the least
refrangible Rays shall come most copiously to the Eye from the Drops in
the Line OG, and strike the Sense with the deepest red in that Region.
And the Angle SHO being equal to the Angle POH, or 54 Gr. 7 Min. shall
be the least Angle, in which the most refrangible Rays after two
Reflexions can emerge out of the Drops; and therefore those Rays shall
come most copiously to the Eye from the Drops in the Line OH, and strike
the Senses with the deepest violet in that Region. And by the same
Argument, the Drops in the Regions between G and H shall strike the
Sense with the intermediate Colours in the Order which their Degrees of
Refrangibility require, that is, in the Progress from G to H, or from
the inside of the Bow to the outside in this order, red, orange, yellow,
green, blue, indigo, violet. And since these four Lines OE, OF, OG, OH,
may be situated any where in the above-mention'd conical Superficies;
what is said of the Drops and Colours in these Lines is to be understood
of the Drops and Colours every where in those Superficies.

Thus shall there be made two Bows of Colours, an interior and stronger,
by one Reflexion in the Drops, and an exterior and fainter by two; for
the Light becomes fainter by every Reflexion. And their Colours shall
lie in a contrary Order to one another, the red of both Bows bordering
upon the Space GF, which is between the Bows. The Breadth of the
interior Bow EOF measured cross the Colours shall be 1 Degr. 45 Min. and
the Breadth of the exterior GOH shall be 3 Degr. 10 Min. and the
distance between them GOF shall be 8 Gr. 15 Min. the greatest
Semi-diameter of the innermost, that is, the Angle POF being 42 Gr. 2
Min. and the least Semi-diameter of the outermost POG, being 50 Gr. 57
Min. These are the Measures of the Bows, as they would be were the Sun
but a Point; for by the Breadth of his Body, the Breadth of the Bows
will be increased, and their Distance decreased by half a Degree, and so
the breadth of the interior Iris will be 2 Degr. 15 Min. that of the
exterior 3 Degr. 40 Min. their distance 8 Degr. 25 Min. the greatest
Semi-diameter of the interior Bow 42 Degr. 17 Min. and the least of the
exterior 50 Degr. 42 Min. And such are the Dimensions of the Bows in the
Heavens found to be very nearly, when their Colours appear strong and
perfect. For once, by such means as I then had, I measured the greatest
Semi-diameter of the interior Iris about 42 Degrees, and the breadth of
the red, yellow and green in that Iris 63 or 64 Minutes, besides the
outmost faint red obscured by the brightness of the Clouds, for which we
may allow 3 or 4 Minutes more. The breadth of the blue was about 40
Minutes more besides the violet, which was so much obscured by the