Full text of "Opticks: or, A treatise of the reflexions, refractions, inflexions and colours of light. Also two treatises of the species and magnitude of curvilinear figures"

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bPTICKS-

O R , A

TREATISE

O F T H E

REFLEXIONS, REFRACTIONS,
INFLEXIONS and COLOURS

L 1 Gr H T»

ALSO

Two TREATISES

O F T H E

SPECIES and MAGNITUDE

O F

Curvilinear Figures.

LONDON^ p

Printed for S.^m Smith, and Ben J, WaLf^r?;

Printers to the Royal S ociety , at the Vr'mci's Arms in
St. ?tiul\ Church-yard. MDCCIV* 4? ij /

i^/t/-^'

ADVERTISEMENT.

Art of the enfuing Difcourfe about Li^t wa^ written
at the defire ofjome Gentlemen of the Royal Society,

in the Tear i6'j^. and then fent to their Secretary y and
read at their Meetings ^ and the reft was added about
Twelve Tears after to complete the Theory ; except the
Third Book J and the laji Propofition of the Second^ which
were fine e put together out of f cotter ed Papers. To avoid
being engaged in Difputes about thefe Matters^ I have
hitherto delayed the Printings and fhould fliJl have dc"
layed 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 fet down^
and fuUy fatisfied my f elf about the Laws of Refractions
and Compqfition of Colours. I have here Publifhed what
I think proper to come abroad^ wifhing that it may not be
Tranflated into another Language 'without my Confent.

The Crowns of Colours^ which fometimes appear about
the Sun and Moon^ I have endeavoured to give an Ac--
count qf; but for want offufficient Obfervations leave that
Matter to be further examined. The Subject- of the Third
Book I have alfo left imperfect ^ not having tried all the

Expe-

iLxperiments which I intended when I was ahout theje
Matters, nor repeated fome of thofe which^I did try^ until
Ihadfatisfied my f elf ahutMJltheir CiriMmJiances. To
communicate what. I -have tried^ and leave the resi to
others for further Enquiry ^ ts all my Dejiga in puhlifhing
tkefe Papers,

In a Letter written to-MrJ-jdhnit'Lin the Tear i6j6,
and puhhfhed hy Dr. Waliis^ / mentioned a Method hy
wJjtch I had found fome- general Theorems ahout fquaring
■Curvilmear .Figures j^^orcomp^ them with the Conic
Sections J or other the JimpleB Figures with which they may
I^e compared. And fome Tears ago I lent out a Manufcript
containing fuchThcorems\ and having fine e met with fome
Things copied out of it^ I have -m this: Occafion made it
puhlickj prefixing to tt an Introdudlion and fuhjoyning a
Scholium concerning that Method. And I have joined
with it another fmall Traci concerning the Curvilinear
Figures of the Second Kind^ which was alfo written
many Tears ago^ and made Imown to fome Friends^ who
have folictted the making it puUich

CO
The FIRST BOOK

O F

O P T I C K S

PART I.

MY Defign in this Book is not to explain the Pro-
perties of Light by Hypothefes, but to propofc
and prove them by Reafon and Experiments :
In order to which , I fliall premife the following Defini-
tions and Axioms.

DEFINITIONS.

D E F I N. L

T lihe ^ys of Light I underfland its leajl Tarts , and thofe

as weliSucceJ/l'Ve in the fame Liries as Contemporary in fe^

l^eral Lines, For it is manifeft that Light confifts of parts
both Succeflive and Contemporary 3 becaufe in the fame
place you may flop that which comes one moment, and
let pafs that which comes prefently after3 and in the fame
time you may flop it in any one place, and let it pafs in
any other. For that part of Light which is ftopt cannot
be the fame with that which is let pafs. The leaft Light
or part of Light , which may be ftopt alone without the
reft of the Light, or propagated alone, or do or fufter any

A thing

thing alone, which the reft of thf Light doth not or fut
ers not, I all a Ray of jLi^.

D E F I N. II.

^frangtbilky of the ^ys of Light^ is their Difpojition to he
refrahed or turned out of their Way in pajjing out of one tranf^
parent ^ody or Medium into another. And a greater or lefs (?^e-
frangihility of^ys^ is their Difpojition to be turmd more or lefs
out of their Way in like Incidences on the fame Medium. Mathe-
maticians ufually confider the Rays of Light to be Lines
Teaching from the luminous Body to the body illumina-
ted, and the refradlion of thofe Rays to be the bending
or breaking of thofe Lines in their pafling out of one Me-
dium into another. And thus may Rays and Refradions
be confidered, if Light be propagated in an inftant. But
by an Argument taken from the Equations of the times
of the Eclipfes o? Jupiter's Satellites it feems that Light is
propagated in time, fpending in its paffage from the Sun
to us about Seven Minutes of time : And therefore I have
chofen to define Rays and Refradions in fuch general
terms as may agree to Light in both cafes.

DBF I N. IIL

'^flexibility of ^ys^ is their Difpojition to be turned baci ^^^^
■the fame Medium from any other Medium upon whofe Surface they
fall. And ^ys are 7?iore or le[^ reflexible , which are returned
hack ^ore oiSefs eaftly. As if Light pafs out of Glafs into
Air and by being incHned more and more to the com-
mon Surface of the Glafs and Air, begins at length to be
totally refleded by that Surface 3 thofe forts of Rays which,
at like Incidences are refleded moft copioufly , or by in-
clining the Rays begin fooneft to be totally refleded, are
moft reflexible. 53 E-

t 33

DEFiN. nr,

T%e Angle of Incidence^ is that Angle which the Line defcribei
hy the incident ^y contains with the Perpendicular to the reflc"
Bing or refraBing Surface at the ^oint of Incideme,

DEFIN. V.

The Angle of ^flexion or ^fraBion^ is the Angle which the
Line defcrihed hy the refleBed or refraBed ^y containeth with
the Perpendicular to the refleBing or refraBing Surface at the
Point of Incidence.

DEFIN. VI.

T%e Sines of Incidence^ ^flexion^ and ^fraBion^ are the
Sines of the Angles of Incidence^ ^flexion^ and ^efraBion,

DEFIN. VII.

The Light whofe (^ys are all alike ^frangihle^ I call Simm
pky Homogoieal and Similar-^ and that whofe ^ys are feme
more ^frangible than others^ I call Compound^ Heteroo-eneal and
DiJJlmilar. The former Light I call Homogeneal , not
becaiife I would affirm it fo in all relped:s 3 but becaufe
the Rays which agree in Refrangibility , agree at leaft ia
all thofe their other Properties. Which I confider in the
following Difcourfe.

DEFIN. VIII.

The Colours of Homogeneal Lights , I call Primary^ HomO"
geneal and Si?nple 5 and thofe of Heterogeneal Lights^ Heteroge"
neal and Compound, For thefe are always compounded of
the colours of Homogeneal Lights 3 as will appear in the
following Difcourfe. A 2 AX I-

c 4 J : : :

A X I M S.

A X, L

TH E Angles of Incidence^ (^flexion, and ^fraBioUy lye
in one and the fame ^latie.

A X. IL

The Angle of ^flexion is equal to the Angle of Incidence.

A X. III.

If the refraSied ^y he returned direBly hack^ to the ^oint
of Incidence , it fhall be ref rafted into the Line before defcri'
M hy the incident <S^y.

AX. IV.

^fraBion out of the rarer Medium into the denfer , is made:
towards the perpendicular 3 that is^ fo that the Angle of (^^fra-,
Bion he kf? than the Angle of Incidence.

A X. V.

T^he Sine of Incidence ^ is either accurately or Very nearly in a
given <^tio to the Sine of^efraBion..

Whence if that Proportion be known in any one Ihcli'^
nation of the incident Ray, 'tis known in all the Inclina-
tions and thereby the Refrad:ion in all cafes of Incidence
on the fame refrading Body may he determined. Thus
if the Refradion be made out of Air into Water, the Sine
of Incidence of the red Light is to the Sine of its Refra-
dicn as 4 to ?» If out of Air into Glafs, the Sines are.

as 17 to 1 1. In Light of other Colours the Sines have
other Proportions : but the difference is fo little that it
need feldom be confidered.

Suppofe therefore, that R S reprefents the Surface of F/?. s
ftagnating Water, and^C is the point of Incidence in.
which any Ray coming in the Air from A in the Line
A C is refleded or refraded, and I would know whether
this Ray fliall go after Reflexion or Refradion : I eredt
upon the Surface of the Water from the point of Inci*
dence the Perpendicular C P and produce it downwards,
to Q, and conclude by the firfl; Axiom, that the Ray af-
ter Reflexion and Refradion, fliall be found fomewhere in.
the Plane of the Angle of Incidence A C P produced. I
let fall therefore upon the Perpendicular CP the Sine of
Incidence A D, and if the reflected Ray be defired , I pro-
duce A D to B fo that D B be equal to A D, and draw:
C B. For this Line C B fliall be the reflected Ray 5 the
Angle of Reflexion B C P and its Sine B D being equal
to the Angle and Sine of Incidence, as they ought to be
by the fecond Axiom. But if the refradled Ray be de^
fired, I produce A D to H, fo that D H may be to A D
as the Sine of Refraction to the Sine of Incidence that is
as 3 to 4 5 and about the Center C and in the Plane A C P
with the Radius C A defcribing a Circle A B E I draw
Parallel to the Perpendicular C P Q, the Line H E cutting
the circumference in E, and joyning C E, this Line CE
fliall be the Line of the refraded Ray. For if E F be let
fall perpendicularly on the Line P Q , this Line E F fliall
be the Sine of Refradion of the Ray C E, the Angle of
Refradion being E C Q.3 and this Sine E F is equal ta
D H, and confequently in Proportion to the Sine of Inci-^-
dence AD as 3 to 4.

, )

In like maft^ner, if there be a Prifm of Glafs (that is a
Glafs bounded with two Equal and Parallel Triangular
ends, and three plane and well poliflied 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 paffing crofs this Prifm be defi-
red : Let ACB reprefent a Plane cutting this Prifm tranf-
verily to its three Parallel lines or edges there vvhere the
Light paffeth through it, and let BE be the Ray inci-
dent upon the firft fide of the Prifm A C where the Light
goes into the Glafs 3 And by putting the Proportion of
the Sine of Incidence to the Sine of Refraction as 1 7 to
1 1 find E F the firft refraCled Ray. Then taking this Ray
for the Incident Ray upon the fecond fide of the Glafs B C
where the Light goes out, find the next refracted Ray F G
by putting the Proportion of the Sine of Incidence to the
Sine of RefraClion as 11 to 17. For if the Sine of Inci-
dence out of Air into Glafs be to the Sine of Refra6tion
as 1 7 to II, the Sine of Incidence out of Glafs into Air
muft on the contrary be to the Sine of RefraCtion as 1 1
to 17, by the third Axiom.

Much after the fame manner , if A C B D reprefent a
Glafs fpherically Convex on both fides (ufually called a
Leyis^ fuch as is a Burning- glafs, or Spectacle-glafs, or an
Objed' glafs of a Telefcope) and it be required to know
how Light falling upon it from any lucid point Q. fliall
be refracted, let Q.M reprefent a Ray falling upon any
point M of its firft fpherical Surface ACB, and by erecting
a Perpendicular to the Glafs at the point M , find the firft
refracted Ray M N by the Proportion of the Sines 1 7
to 1 1 . Let that Ray in going out of the Glafs be inci-
dent upon N, and then find the fecond refracted Ray N ^
by the Proportion of the Sines 11 to 17. And after the

fame

[7]

fame ma^nner may the Refiadion he found when die
Lens is Convex on one fide and Plane or Concave on
the other, or Concave on both Sides.

AX. VI.

Hoinogenecd ^ys which flow from feVerdl joints of any Oh^
jeEi^ md fall almoft Perpendicularly on any refleHing or refra^
Bing <Plane or Spherical Surface y fhall afterwards diverge from
fo niafiy other Toints, or he Parallel to Jo many other Lines ^ or
C07iVerge to fo many other Toints, either accurately or without any
fenfihle Error. Jjid the fame thing will happen^ if the ^ys he
reflected or refraBed fuccejjlyely hy two or three or more ^lane
or jj^herical 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 refledled or re«
firadled ones may be found by finding the Refradlionof
any two Rays, as above 5 or more readily thus, ^

Caf 1. Let ACBbe a reflecting or refracting Plane, F^. 4^
and Q, the Focus of the incident Rays, and Q^ C a per-
pendicular to that Plane. And if this perpendicular be
produced to ^, fo that q C be equal to Q.C, the point q^
fhall be the Focus of the reflected Rays. Or if ^ C be
taken on the fame fide of the Plane with Q_C and in Pro-
portion to (iC as the Sine of Incidence to the Sine of
Refiradion, the point q fhall be the Focus of the refrac-
ted Rays.

Caf. 2. Let A C B be the reflecting Surface of any Fig^ 5. i

Sphere whofe Center is E. Bifect any Radius thereof (fup- C\/^fl/^&^ \
pofe E C) in T, and if in that Radius on the fame fide the p^<clU^^ctir.f>. ^
point T you take the Points Q. and ^, fo that T Q, T E, ^'"^1

and T^ be continual Proportionals, and the point Qbe

the.

C 8 ]

the Focus of the incident Rays , the point q ihall be the
Focus of the refledled ones.
SF^, 6. Caf. ^ . Let A C B be the refracting Surface of any

^^-(i)^//: tK^/' Sphere whofe Center is E. In any Radius thereof EC

g//c^l4. ^- n-a^^^ produced both ways take E T and C t lMM^ in fuch
Proportion ^to"^ that Radius as the lefler of the Sines of
Incidence and Refraction hath to the difference of thofe
Sines. And then if in the fame Line you find any two
Points Q. and q , fo that T Q. be to E T as E t to f ^,
taking t q the contrary way from t which T Q. lieth from
T, and if the Point Q.be the Focus of any incident Rays,
the Point q fhall be the Focus of the refradled ones.

And "by the fame means the Focus of the Rays after
two or more Reflexions or RefiraCtions may be found.
jF^'7' €af, /^, Let A C B D be any refracting Lens , fpheri-

Q^^-W?/i7at^>yfeally Convex or Concaye or Plane on either lide, and let
f, ^^. C D be its Axis (that is the Line which cuts boih its Sur-

faces perpendicularly, and paffes through the Centers of
the Spheres,) and in this Axis let F and/be the Foci of the
refracted Rays found as above , when the incident Rays
on both fides the Lens are Parallel to the fame Axis 3 and
upon the Diameter F/ bifected in E, defcribe a Circle.
Suppofe now that any Point Q be the Focus of any inci-
dent Rays. Draw Q.E cutting the faid Circle in T and f,
and therein take t q in fuch Proportion to ^ E as f E or T E
hath to T Q. Let t q lye the contrary way from t which
T Q, doth from T, and q fhall be the Focus of the refrac-
ted Rays without any feniible Error , provided the Point
Q be not fo remote from the Axis, nor the Lens fo broad
as to make any of the Rays fall too obliquely on the
refracting Surfaces.

And by the like Operations may the reflecting or re-
fracting Surfaces be found when the two Foci are given,

\ and

[p]

and thereby a Lens be formed, which fliall make the Rays
flow towards or from what place you pleafe.

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 Ql,
they fhall after Reflexion or Refraction flow from or to-
wards the Point q found by the foregoing Rules. And if
the incident Rays flow from or towards feveral points Q.,
the reflected or refracted Rays fhall flow from or towards
fo many other Points q found by the fame Rules. Whe-
ther the reflected and refracted Rays flow from or towards
the Point q is eafily known by the fituation of that Point.
For if that Point be on the fame fide of the reflecting or
refracting Surface or Lens with the Point Q, and the in-
cident Rays flow from the Point Q, , the refle(5led flow to-
wards the Point q and the refracted from it 5 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 fide of that Surface.

AX. VII.

Wherever the ^ys which come from all the joints of any Ob"
jeB meet again info many Joints after they ha'Ve been tnade to
converge hy ^flexion or ^fraBion^ there they ivill make a Tic"
ture of the Object upon any ivhite !Body on which they fall.

So if PR reprefent any Object without Doors, and AB Fig,
be a Lens placed at a hole in the Window-fliut 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 5 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

B and

[lOj

and Colours. For as the Light which comes from the
Point Qgoes to the Point qy fo the Light which comes
from other Points P and R of the Object, will go to fo
many other correlpondent Points p and r (as is ma:tiife{fc
by the fixth Axiom 3 ) fo that every Point of the Object
fliall illuminate a correfpondent Point of the Picture, and
thereby make a Picture like the Object in Shape and Co-
lour, this only excepted that the Picture fliall be inverted,
•And this is the reafon of that Vulgar Experiment of caft-
ingthe 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 P Q.R5
the Light which comes from the feveral Points olthe Ob-
ject is fo refracted by the tranfparent skins and humours
of the Eye, (that is by the outward coat EFG called the
Tunica Cornea, and by the cryftalline humour AB which is
beyond the Pupil m k,) as to converge and meet again at
fo many Points in the bottom of the Eye, and there to paint
the Picture of the Object upon that skin (called the 7«-
nica ^tina) with which the bottom of the Eye is covered.
For Anatomifts when they have taken off from the bot-
tom of the Eye that outward and moft thick Coat called
the Dura Mater, can then fee through the thinner Coats
the Pictures of Objects lively painted thereon. And thefe
Pictures propagated by Motion along the Fibres of the Op-
tick Nerves into the Brain, are the caufe of Vifion. For
accordingly as thefe Pictures are perfect or imperfect, the
Object is feen perfectly or imperfectly. If the Eye be tin-
ged with any colour fas in the Difeafe of the Jaundife) fo
as to tinge the Pictures in the bottom of the Eye with that
Co4our, then all Objects appear tinged with the fame Co-
lour. If the humours of the Eye by old Age decay, fo
as by flirinking to make the Cornea and Coat of the Cry-'

Jlalline

In]

fliiSkehumaur. grow &2cmxth^nh^ the Light will not be
refracted .enough, aad for want of a fufficient Reftadion
vvijlnot cQavergejto the bottom of the Eye but tofome
place beyond it , and by confequence paint in the bottom
of the Eye aconfufedPidure^and according to the indiftind-
jiefs of this Pidlure the Gbje(5l will appear confufed. This
is the reafon of the decay of Sight in old Men, and fliews
why their Sight is mended by Spe(5tacles. For thofe Con-
vex- glafles fupply the defe(5l ofplumpnefs in the Eye, and
by encreafingtheRefradlion make theRays converge fooner
:fo^s to convene diftindtly at the bottom of the Eye if the
Gkfs have a ducidegree of convexity. And the contrary
happens in fliort-fighted Men whofe 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 3
and therefore the Pidure made in the bottom and theVifion
caufed thereby will not be diftinft, unlefs the Objed be
brought fo near the Eye as that the place where the con-
verging Rays convene may be removed to the bottom, or
that the plumpnefs of the Eye be taken off and the Refra-
dlions diminifhed by a Concave-glafs of a due degree of
Concavity, or laftly that by Age the Eye grow flatter till it
come to a due Figure : For (hort-fighted Men fee remote
Objeds beft in Old Age, and therefore they are accounted
to have the mofl lafting Eyes.

A X. VIII.

Jn ObjeFt feen hy ^flexion or ^efraEiion^ appears in that place
from whence the ^ys after their laft ^flexion or ^fraSiion di'
"Verge in falling on the SpeBators Eye.

If the Objed A be feen by Reflexion of a Looking- Fig* p»
glafs m Hy it fliall appear, not in it's proper place A, but

B 2 behind

[12]

behind the Glafs at a^ from whence any Rays AB, AC,
AD, which flow from one and the fame Point of the Ol>
je<^5 do after their Reflexion made in the Points B,C, D,
diverge in going from the Glafs to E, F, G, where they
are incident on the Spectator's Eyes. For thefe Rays do
make the fame Picture in the bottom of the Eyes as if
they had come from the Object really placed at a without
the interpofition ©f the Looking- glafs 5 and all Vifion is
made according to the place and fhape of that Picture.

'k. 2. In like manner the Object D feen through a Prifm ap-
pears not in its proper place D, but is thence tranflated to
lome other place d fituated in the laft refracted Ray F G
drawn backward from F to d.

fig. \o> And fo the Object Q_ feen through the Lens A B, appears
at the place q from whence the Rays diverge in paffing
from the Lens to the Eye. Now it is to be noted, that the
Image of the Object at ^ is fo much bigger or leffer than
the Object it felf at Q., as the diftanee of the Image at
q from the Lens AB is bigger or lefs than the diftanee of
the Object at Q. from the fame Lens; And if the Object
be feen through two or more fuch Convex or Concave-
glafles, every Glafs fhall make a new Image, and the Ob-
jedfhal! appear in the place and of the bignefs of the laft
Image. Which confideration unfolds the Theory of Mi-
crofcopes and Telefcopes. For that Theory confifts in al-
moft nothing elfe than the defcribing fuch Glafles as fliall
make the laft Image of any Objed" as diftind and large:
and luminous as it can conveniently be made.

I have now given in Axioms and their Explications the
ftimm of what hath hitherto been treated of in Opticks»
For, what hath been generally agreed on I content my
felf to afllime under the notion of Principles, in order to
what I have further to write. And this may fuffice for an

Intro«

[13 1

TntroduAion to Readers of quick Wit and good Under^
ftanding not yet verfed in Opticks : Although thofe who
are already acquainted with this Science , and have
handled Glafles, will more readily apprehend what fol-
loweth.

PROPOSITIONS.

^%^0T, I. Theor. I.

I G H T S which differ in Colour, differ alfo in De-
grees of Refrangibility,

The Proof hy Experiments:

Exper. \ . I took a black oblong fliff Paper terminated
by Parallel Sides, and with a Perpendicular right Line
drawn crofs from one Side to the other , diflinguiflhed it
into two equal Parts. One of thefe Parts I painted with
a red Colour and the other with a blew. The Paper was
very black, and the Colours intenfe and thickly laid on,
that the Phaenomenon might be more confpicuous. This
Paper I viewed through a Prifm offolid Glafs, whofe two
Sides through which the Light pafled to the Eye were
plane and well poliflhed, and contained an Angle of about
Sixty Degrees : which Angle I call the refrading Angle of
the Prifm. And whilft I viewed it, I held it before a c j^"^*"^
Window in fuch manner that the Sides of the Paper were
parallel to the Prifm, and both thofe Sides and the Prifm
parallel to the Horizon, and the crofs Line perpendicular
to it 3 and that. the Light which fell from the- Window

-;pf on the Paper made an 4l^le with the Paper^ :equal tp
that Angle which was made with the fame Paper by the
,Light renefted from it to the Eye. Beyond the Prifm was
-the Wall of the Chamber under the Window covered, over
with black Cloth, and the Cloth was involved in Daifk-
nefs that no Light might be refleded from thence, which
in paffing by the edges of the Paper to the Eye , might
mipgle it felf with the Light of the Paper aad obfcure the
Phsenomenon thereof Thefc things being thus ordered,
I found that if the refradling Angle of the Prifm be turned
upwards, fo that the Paper may feem to be lifted upwards
by the Refradion, its blew half will be lifted higher by
#he fRefra<5lion than its red half But if the refracting
Angle of the Prifm be turned downward, fo that the Pa-
per may feem to be carried lower by the Refrad:ion, its
tlew half will be carried fomething lower thereby than
its red half Wherefore in both cafes the Light which
comes from the blew half of the Paper through the Prifm
^to the Eye, does in like Circumftances fuffer a greater R^-
fradlion than the Light which comes from the red half,
and by confequence is more refrangible.
Fig. 1 1. Ilkflratioru In the Eleventh Figure, M N reprefents the
Window,and D E the Paper terminated with parallel Sides
D J and HE, and by the tranfverfe Line F G diflinguiflhed
into two halfs, the one D G of an intenfely blew Colorur,
the other FEof an intenfely red. AndBACc^^ repr,e-
fents the Prifm whofe refracting Planes K^b a and KQca
meet in the edge of the refracting Angle A a. This edge
Act being upw^ard, is parallel Ippth jto the Horizon and -to
the parallel edges or tnePapd' DJ and H E.a And a^re-
prefents the Image of the Paper feen by RefraCtion up-
wards in fuch manner that the blew half D G is carried
higher to dg than the red half F E is to /e, and therefore

fufFers

'<i7ij f*r,/c

fuffers^ X gfcatet Refra€Bon. If: the edge of the refrafting
Angle be turned downward, the Image of the Paper will
be rcfrad:ed downward fuppofe to ^6, and the blew half
will be refrad^ed lower to ^ y- than the red half is to ?>f.

Exper. 1. About the aforefaid Paper, whofe two halfe
were painted over with red and blew/, and which was ftiff
hke thin Paftboard, I lapped feveral times a flender thred
of very black Silk, in fuch manner that the feveral parts
of the thred might appear upon the Colours like fo many
black Lines drawn over them , or like long and flender
dark Shadows cafl: upon them. I might have drawn black
Lines with a Pen, but the threds we^ fmalkr and better

►er thus coloured aaadiaea I let ;

defined. This Paper thus coloured arffiaeS I fet againft
a Wall perpendicularly to the Horizon, fo that one of the
Colours might fliand to the right hand and the other to
the left. Clofe before the Paper at the confine of the Co-
lours below I placed a Candle to illuminate the Paper
ftrongly : 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 difl:ance of
Silc Feet and one or two Inches from the Paper upon the
Floor I ere^led a glafs Lens four Inches and a quarter
broad, which might coUedl the Rays coming from the
feveral Points of the Paper, and make them converge to-
wards fo many other Points at the fame difl:ance of fix
Feet and one or two Inches on the other fide of the Lens^
and fo form the Image of the coloured Paper upon a white
Paper placed there 5 after the fame manner that a Lens at
a hole in a Window cafl:s the Images of Objeds abroad
upon a Sheet of white Paper in a dark Room. The afore-
faid white Paper, ereded perpendicular to the Horizon
and to the Rays which fell upon it fi'om the Lens, I moved
fometimes towards the Lens, fometimes firom it, to finct

the

12.

tlie places where the Images of the tlew and red parts of
the coloured Paper appeared moll: diftind. Thofe places
I eafily knew by the Images of the black Lines which I
had made by winding the Silk about the Paper. For the
Images of thofe fine and flender Lines (which by reafon of
their blacknefs were like Shadows on the Colours) were
Gonfiifed and fcarce vifible, unlefs when the Colours on ei-
ther fide of each Line were terminated moft diftindlly.
Noting therefore, as diligently as I could, the places where
the Images of the red and blew halfs of the coloured Pa-
per appeared moft diftin6l , I found that where the red
half of the Paper appeared diftinifl, the blew half appeared
eonfufed, fo that the black Lines drawn upon it could
fcarce be feen 3 and on the contrary where the blew half
appeared moft diftincS^ the red half appeared confufed, fo
that the black Lines upon it were fcarce vifible. And be-
tween the two places where thefe Images appeared diftinfl
there was the diftance of an Inch and a half : the diftance
of the white Paper from the Lens, when the Image of the
red half of the coloured Paper appeared moft diftin(5t, be-
ing greater by an Inch and an half than the diftance of the
fame white Paper from the Lens when the Image of the
blew half appeared moft diftin(5l. In like Incidences there-
fore of the blew and red upon the Lens, the blew was re-
fradted more by the Lens than the red^ fo as to converge
fooner by an Inch and an half, and therefore is more refran-
gible.

Illuftratioru In the Twelfth Figure, D E fignifies the co-
loured Paper, D G the blew half, F E the red half, M N
the Lens, H J the white Paper in that place where the red
half with its black Lines appeared diftindl, and hi the fame
Paper in that place where the blew half appeared diftindl.
The place hi was nearer to the Lens M N than the place
H J by an Inch and an half. Scholium.

[17]

Scholium, The fame things fucceed notwithftanding that
fome of the Circumftances be varied : as in the firft Ex-
periment when the Prifm and Paper are any ways inclined
to the Horizon , and in both when coloured Lines are
drawn upon very black Paper. But in the Defcription of
thefe Experiments , I have fet down fuch Circumftances
by which either the Phsenomenon might be rendred more
conlpicuous, or a Novice might more eafily try them, or
by which I did try them only. The fame thing I have
often done in the following Experiments : Concerning all
which this one Admonition may fuffice. Now from thefe
Experiments it follows not that all the Light of the blew
is more Refrangible than all the Light of the red 5 For
both Lights are mixed of Rays differently Refrangible,
So that in the red there are fome Rays not lefs Refrangible
than thofe of the blew , and in the blew there are fome
Rays not more Refrangible than thofe of the red 5 But
thefe Rays in Proportion to the whole Light are but fcWy
and ferve to diminifh the Event of the Experiment , but
are not able to deflroy it. For if the red and blew Co-
lours were more dilute and weak, the diftance of the Ima-
ges would be lefs than an Inch and an half 5 and if they
were more intenfe and full, that diftance would be greater,
as will appear hereafter. Thefe Experiments may fuffice
for the Colours of Natural Bodies. For in the Colours
made by the Refraction of Prifms this Propofition will
appear by the Experiments which are now to follow in the
next Propofition.

^(^Of.

[i8]
P ROF. II. Theor. 11.

The Light of the Sun confijls of ^ays differently (^frangible.

The Proof by Experiments.

Exper. 3. XN a Ycry dark Chamber at a round hole about
^ one third part of an Inch broad made in the
Shut of a Windovv/ I placed a Glafs Prifm, whereby the
beam of the Sun's Light which came in at that hole might
be refrad:ed upwards tovv^ard the oppofite Wall of the
Chamber , and there form a coloured Image of the
Sun. The Axis of the Prifm (that is the Line pafling
through the middle of the Prifm from one end of it to
the o:her end Parallel to the edge of the Refracflino; Angle)
was in this and the following Experiments perpendicular
to the incident Rays. About this Axis I turned tbe Prifm
flowly , and faw the refradied Light on the Wall or cO"
loured Image of the Sun firft to defcend and then to af^
etnd. Between the Defcent and Afcent when the Image
feemed Stationary , I ftopt the Prifm, and fixt it in that
Pofttire, that it (hould be moved no more. For in that
pofture the Refractions of the Light at the two fides of
the Refrading Angle, that is at the entrance of the Rays
into the Prifm and at their going out of it, were equal to
one another. So alfo in other Experiments as often as I
would have the Refradlions on both fides the Prifm to be
equal to one another, I noted the place where the Image
of the Sun formed by the refracted Light fl;ood ftill be-
tween its two contrary Motions, in the common Period
' of its progrefs and egrefs 3 and when the Image fell upon
tJiat fplace, I made faft the Prifm. A_nd in this pofture, as

the

[19]

die moft convenient,it is to be underftood that all the Prifms
are placed in the following Experiments, unlefs where fome
other pofture is defcribed. The Prifm therefore being pla-
ced in this pofture, I let the refraded Light fall perpendi-
cularly upon a Sheet of white Paper at the oppofite Wall
of the Chamber, and obfervcd the Figure and Dimeniions
of the Solar Image formed on the Paper by that Light.
This Image was Oblong and not Oval, but terminated
with two Recftilinear and Parallel Sides , and two Semi-
circular Ends. On its Sides it was bounded pretty diftindly,
but on its Ends very confufedly and indiftin6lly, the Light
there decaying and vanifliing by degrees. The breadth of
this Image anfwered 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
diftant from the Prifm, and at this diftance that breadth if
diminiflied by the Diameter of the hole in theWindow-flhut,
that is by a quarter of an Inch, fubtended an Angle at the
Prifm or about half a Degree, which is the Suns apparent
Diameter. But the length of the Image was about ten Inches
and a quarter, and the length of the Recftilinear Sides about
eight Inches 5 And the refradring Angle of the Prifm where-
by fo great a length was made, was 64. degr. With a kfs
Angle the length of the Image was lefs , the breadth re-*
maining the fame. If the Prifm was turned about its Axis
that way which made the Rays emerge more obliquely out
of the fecond refradinp; Surface of the Prifm, the Imao^e foon
became an Inch or two longer, or more 5 and if the Prifm
was turned about the contrary way, fo as to make the Rays
fall more obliquely on the firft refrading Surface, the Image
foon became an Inch or two fliorter. And therefore in try-
ing this Experiment, I was as curious as I could be in pla-
cing the Prifm by the above-mentionsd Rule exadly in

C 2 fuch

[20]

filch a pofture that the Refradions of the Rays at their emer-
gence out of the Prifm might be equal to that at their inci-
dence on it. This Prifm had fome Veins running along
within the Glafs from one end to the other , which feat-
tered fome of th€ Sun's Light irregularly,, but had no fen-
fible efFe6l in encreafing the length of the coloured Spec-
trum. For I tried the fame Experiment with other Prifms
with the fame Succefs. And particularly with a Prifm
which feemed free from fuch Veins, and whofe refracting
Angle was 61^ Degrees, I found the length of the Image 92

01^ 10 Inches at the diftance of i 8- Feet from the Prifm,
the breadth of the hole in the Window-fhut being i of an

Inch as before. And becaufe it is eafie to commit a mi-
ftake in placing the Prifm in its due pofture, I repeated
the Experiment four or five times, and always found the
length of the Image that which is fet down above. With
another Prifm of clearer Glafs and better PoUifh, which
feemed free from Veins and whofe refracting Angle was
63 - Degrees, the length of this Image at the &me diftance

of 1 8 { Feet was alfo about 1 o Inches, or 10^. Beyond

thefe Meafures for about • or - of an Inch at either end of

4 3

the Spedtrum the Light of the Clouds feemed to be a little
tin-ged with red and violet, but fo very faintly that I fufpe-
Cted that tinCture might either wholly or in great meafure
arife from fome Rays of the SpeCtrum fcattered irre-
gularly by fome inequalities in the Subftance and Polifh
of the Glafs , and therefore I did not include it in thefe
Meafures. Now the different Magnitude of the hole in
theWindow-fhut, and different thicknefs of the Prifm where
the Rays paffed through it, and different inclinations of the
Prifm to the Horizon, made no fenfible changes in the
length of the Image. Neither did the different matter of

the

[ 21 ]

the Prifms make any : for in a Veflel made of poliflied
Plates of Glafs cemented together in the fliape of a Prifm?
and filled with Water, there is the like Succefs of the Ex-
periment according to the quantity of the Refrad:ion. It
is further to be obferved, that the Rays went on in right
Lines from the Prifm to the Image, and therefore at their
very going out of the Prifm had all that Inclination to
one another from which the length of the Image pro-
ceeded, 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 poffibly be fo much in*
clined to one another. For let EG reprefent the Window- f/p-. j:9;
fhut, F the hole made therein through which a beam of the
Sun's Light was tranfmitted into the darkned Chamber, and
A B C a Triangular Imaginary Plane whereby the Prifm is
feigned to be cut tranfverfly through the middle of the
Light. Or if you pleafe, let A B G reprefent the Prifm it
felf, looking direcftly towards the Spe^fiator s Eye with its
nearer end : And let X Y be the Sun, M N the Paper upon
which the Solar Image or Spectrum is caft, and P T the
Image it felf whofe fides towards V and W are Redili-
near and Parallel, and ends towards P and T Semicir--
Gular. Y K HP and X L J T are -the two Rays, the firft
of which comes firom the lower part of the Sun to the>
higher part of the Image, and is refracted in the Prifm 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 Refradions on both fides the Prifm
are equal to one another, that is the Refradion at K equal
to the Refradion at J, and the Refradion at L. equal to
the Refradion at H, fo that the Refi:adions of the inci-
dent Rays at K and L taken together are equal to the
Refractions of the emergent Rays at H and J tai:ea toge^

ther ;

[22]

Aer : it follows by adding equal things to equal things,
that the Refradions at K and H taken together, are equal
to the Refractions at J and L taken together , and there-
fore the two Rays being equally refradred have the fame
Inclination to one another after Refradion which they had
before, that is the Inclination of half a Degree anfwering
to the Sun s Diameter. For fo great was the Inclination
of the Rays to one another before Refradlion. So then,
the length of the Image P T would by the Rules of VuU
gar Opticks fubtend an Angle of half a Degree at the
Prifm, and by confequence be equal to the breadth Vm -^
and therefore the Image would be round. Thus it would
be were the two Rays X L J T and Y K H P and all the
reft which form the Image P «» T >, alike Refrangible*
And therefore feeing 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 fuffer the greateft Refraction, muft be more Refran-
gible than thofe which go to the lower end T , unlefs the
inequality of Refraction be cafual.

This Image or SpeCtrum P T was coloured, being red
at its leaft rerraCted end T, and violet at its moft refraCted
end P, and yellow green and blew in the intermediate
fpaces. Which agrees with the firft Propofition, that Lights
which differ in Colour do alfo differ in RefrangibiHty,
The length of the Image in the foregoing Experiments I
meafured from the fainteft and outmoft red at one end, to
the fainteft and outmoft bjew at the other end. A/:^,:,^^^ ,

Exper, 4. In the Sun s beam which was propagated in-
to the Room through the hole in the Window-fhut, at
the diftance of fome Feet from the hole, I held the Prifm
in fuch a pofture that its Axis might be perpendicular to
diat beam. Then I looked through the Prifm upon the

hole,

[23]

hole, and turning the Prifm to and fro about its Axis to
make the Image of the hole afcend and defcend, when be*-
tween its two contrary Motions it feejned ftationary, I
ftopt the Prifm that the Refractions on both fides of the
refradling Angle might be equal to each other as in the
former Experiment. In this Situation of the Prifm view-
ing through it the faid hole, I obferved the length of its
refra6ted image to be many times greater than its breadth,,,
and that the moft refra<5led part thereof appeared violet,
the leaft refra6led red, the middle parts blew green and
yellow in order. The fame thing happened when I re*-
moved the Prifm out of the Sun s Light , and looked
through it upon the hole fliining by the Light of the
Clouds beyond it. And yet if the Refra6lion were done
regularly according to one certain Proportion of the Sines
of Incidence and Refra(5lion as is vulgarly fuppofed, the
r€fra<3:ed Image ought to have appeared round.

So then, by thefe two Experiments it appears that in
equal Incidences there is a confiderable inequality of Re-
fradions : But vv^hence this inequality arifes, whether it be
that feme of the incident Rays are refradte.d more and
others lefs, conftantly or by chance, or that one and the
fame Ray is by Refra6tion diftuAed, fhatiered, dilated,
and as it were fplit and fpread into many diverging Rays,
as Grimaldo fuppofes, does not yet appear by thefe Experi^
nients, but will appear by thofe that follow.

Exper. 5 . Confidering therefore, that if in the third Ex-
periment the image of the Sun fhould be drawn out into
an oblong form, erther by a Dilatation of every Ray, or
by any other cafual inequality of the Refractions, the fame
cblong Image ^vouid by a (econd Refra6lion made Side-
mzjs be drawn out as nmch in breadth by -the like Dila-
itatioa 'of the Ray^-or othe»r ■cafeai i-n-e-Qiiaiky oftheRe^

frad;ionS:

fraftions Sideways, I tried what would be the EfFeds of
fuch a fecond Refraftion. For this end I ordered all things
as in the third Experiment, and then placed a fecond Prilm
immediately after the firft in a crofs Pofition to it, that it
might again refradl the beam of the Sun's Light which
came to it through the firft Prifm. In the firft Prilm this
?beam was refracted upwards, and in the fecond Sideways.
And I found that by the Refradtion of the fecond Prifm
the breadth of the Image was not increafed, but its fupe-
>rior part which in the firft Prifm fuffered the greater Re-
fradion and appeared violet and blew, did again in the
.fecond Prilm fuffer a greater Refraction than its inferior
part, which appeared red and yellow , and this without
any Dilation of the Image in breadth.
#g. 14. Illujlration, Let S reprefent the Sun, F the hole in the
Window, A B C the firft Prifm, D H the fecond Prifm, Y
the round Image of the Sun made by a dired: beam of
Lic^ht when the Prifms are taken away, P T the oblong
Image of the Sun made by that beam paffing through the
firft Prifm alone when the fecond Prifm is taken away, and
ft the Image made by the crofs Refradlions of both
Prifms together. Now if the Rays which tend towards
the feveral Points of the round Image Y were dilated and
ipread by the Refradion of the firft Prifm, fo that they
fliould not any longer go in fingle Lines to fingle Points,
but that every Ray being fpHt, fliattered, and changed
from a Linear Ray to a Superficies of Rays diverging
from the Point of Reftadion, and lying in the Plane of
the Angles of Incidence and Refradion, they fhould
go in thofe Planes to fo many Lines reaching almoft
from one end of the Image P T to the other, and if
that Image fliould thence become oblong : thofe Rays
and their feveral parts tending towards the feveral Points of

the

[•25]

the Image P T ought to be again dilated and fpread Side-
ways by the tranfverfe Refradlion of the fecond Prifm , fo
as to compofe a fourfcjuare Image, fuch as is reprefented
at ^^7. For the better underftanding of which, let the hnage
PT be diftinguiflied into five equal Parts PQK, KQ^RL,
LRSM, MSVN, NVT. And by the fame irregularity
that the Orbicular Light Y is by the Refra6lion of the firft
Prifm dilated and drawn out into a long Image P T, the
the Light P Q.K which takes up a Ipace of the fame length
and breadth with the Light Y ought to be by the Refra-
ction of the fecond Prifm dilated and drawn out into the
long Image '^qkp-> ^nd the Light K Q_R L into the long
Image kqrl, and the Lights LRSM, MSVN, NVT
into fo many other long Images I r s m^ m s y 71^ 7iy tl ^ and
all thefe long Images would compofe the fourfquare Image
o-l. Thus it ought to be were every Ray dilated by Re-
fradion, and fpread into a triangular Superficies of Rays
diverging from the Point of Refradlion. For the fecond
Refradion would fpread the Rays one way as much as the
firfl: doth another , and fo dilate the Image in breadth as
much as the firft doth in length. And the fame thing
ought to happen, v/ere fome Rays cafually refradied more
than others. But the Event is otherwife. The Image P T
was not made broader by the Refradion of the fecond
Prifm, but only became oblique, as 'tis reprefented itpt^
its upper end P being by the Refradion tranflated to a
greater diftance than its lower end T. So then the Light
which went towards the upper end P of the Imac^e, \^as,
(at equal Incidences) more refraded in the feconcl Prifm
than the Light which tended towards the lower end T,
that is the blew and violet, than the red and yellow 5 and
therefore was more Refrangible. The fame Light was by
the Refradion of the firft Prifm tranflated further from the

D place

[2<5]

place Y to which it tended before Refradlion 5 and there*
Fore fufFered as well in the firft Prifm as in the fecond a
greater Refradion than the reft of the Light, and by con-
fequenee was more Refrangible than the reft^ even before
its incidence on the firft Prifm.
^' Sometimes I placed a third Prifm after the fecond, and

fometimes alfo a fourth after the third , by all which the
Image might be often refrad:ed fideways : but the Rays
which were more refradlcd than the reft in the firft Prifm.
were alfo more r€fra6led in all the reft, and that without
any Dilatation of the Image fideways : and therefore thofe
Rays for their conftancy of a greater Refraction are de-
fer vedly reputed more Refrangible.
Fig. 15. But that the meaning of this Experiment may more
clearly appear, it is to be confidered that the Rays whick
are equally Refrangible do fall upon a circle anfwering to
the Sun s Difque. For this was proved in the third Experi-
ment. By a circle I underftand not here a perfect Geo-
metrical Circle, but any Orbicular Figure whofe length is
equal to its breadth, and which, as to fenfe, may feem
circular. Let therefore A G reprefent the circle which all
the moft Refrangible Rays propagated from the whole
Difque of the Sun, would illuminate and paint upon the
oppofite Wall if they were alone 5 E L the circle which all
the leaft Refrangible Rays would in like manner illuminate
and paint if they were alone 3 B H, C J, D K, the circles
which fo many intermediate forts of Rays would fuccef-
fively paint upon the Wall, if they were fingly propagated
firom the Sun in fucceiEve Order, the reft being always in-
tercepted ^ And conceive that there are other intermediate
Circles without number which innumerable other inter-
mediate forts of Rays would fucceflively paint upon the
Wall if the Sun fliould fucceflively emit every fort apart.

[2?]

And feeing the Sun emits all thefe forts at once, they muft
all together illuminate and paint innumerable equal cir-
cles, of all which, being according to their degrees of Re-
frangibility placed in order in a continual fcries, that ob-
long Spedrum P T is compofed which I defcribed in the
third Experiment. Now if the Sun^s circular Image Y ficf. ]^h( ]^.
which is made by an unrefra^fted beam of Light was by
any dilatation of the fingle Rays, or by any other irregu-
larity in the Refra6lion of the firft Prifm, converted into
the Oblong Spedrum, P T : then ought every circle A G,
B H, C J, <src. in that Spedrum, by the crofs Refra-
(flion of the fecond Prifm again dilating or otherwife
fcattering the Rays as before, to be in like manner drawn
out and transformed into an Oblong Figure, and thereby
the breadth of the Image P T would be now as much aug-
mented as the length of the Image Y was before by the Re-
fradion of the firft Prifm 5 and thus by the Refractions of
both Prifms together would be form^ed a fourfquare Figure
f'^t'] as I defcribed above. Wherefore fince the breadth of
the Spedrum P T is not increafed by the Refradion fide-
ways, it is certain that the Rays are not fplit or dilated, or
otherways irregularly fcattered by that Refradrion, but
that every circle is by a regular and uniform R.efra6l:ion
tranflated entire into another place, as the circle A G by
the greateft Refradion into the place ag^ the circle B H by
a lefs Refradion into the place bh^ the circle C J by a Re-
fradion ftill lefs into the place ci^ and fo of the reft 3 by
which means a new Spedrum p t inclined to the former
P T is in like manner compofed of circles lying in a
right Line 5 and thefe circles muft be of the fame bignefs
with the former, becaufe the breadths of all the Spe-
dirums Y, P T and pt zt equal diftances from the PriiiTis
are equal.

D 2 I con-

[28]

I confidered further 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 Spedlrum
Y, and that Penumbra remains in the re6lilinear Sides of
the Spectrums P T and pt, I placed therefore at that hole
a Lens or Object- glafs of a Telefcope which miglit caft
the Image of the Sun diftin6lly on Y without any Penum-
bra at all, and found that the Penumbra of the Redili-
near Sides of the oblong Spedrums P T and pt was alfo
thereby taken away, fo that thofe Sides appeared as di-
ftindly defined as did the Circumference of the firfl Image
Y. Thus it happens if the Glafs of the Prifms be free
from veins, atd their Sides be accurately plane and well
polifiied without thofe numberlefs waves or curies which
ufually arife from Sand-holes a little fmoothed in polifli-
ing with Putty. If the Glafs be only well polifbed and
free from veins and the Sides not accurately plane but a
little Convex or Concave, as it frequently happens 5 yet
may the three Spe6trums Y, P T and pt want Penumbras,
but not in equal diftances from the Prifms. Now from
this waht of Penumbras, I knew more certainly that every
one of the circles was refra6ted according to feme moil
regular, uniform, and conftant law. For if there were
any irregularity in theRefradion, the right Lines A E and
G L which all the circles in the Spedrum P T do touch,
could not by that Refradlion be tranflated into the Lines
a e and g I as difl:in(5t and ftraight as they were before, but
there would arife in thofe tranflated Lines fome Penumbra
or crookednefs or undulation, or other fenfible Perturba-
tion contrary to what is found by Experience. Whatfo-
cver Penumbra or Perturbation flhould be made in the
circles by the crofs Refraction of the fecond Prifm , all
that Penumbra or Perturbation would be confpicuous in

the

[29]
the right Lines a e and g I which touch thofe circles. And
therefore fince there is no fuch Penumbra or Perturbation
in thofe right Lines there muft be none in the circles.
Since the diftance between thofe Tangents or breadth of
the Spedrum is not increafed by the Refrad;ions, the Dia-
meters of the circles are not increafed thereby. Since thofe
Tangents continue to be right Lines , every circle which
in the firft Prifm is more or lefs refracted , is exadlly in
the iame Proportion more or lefs refracted in the fecond.
And feeing all thefe things continue to fucceed after the
fame manner when the Rays are again in a third Prifm,
and again in a fourth refradted Sideways, it is evident that
the Rays of one and the fame circle as to their degree of
Refrangibiiity continue always Uniform and Homogeneal
to one another, and that thofe of feveral circles do differ
in degree of Refrangibiiity, and that in fome certain and
conftant Proportion. Which is the thing I was to prove.

There is yet another Circumftance or two of this Ex-Fg. i6,
periment by which it becomes ftill more plain and con-
vincing. Let the fecond Prifm D H be placed not imme-
ately after ^^ the firft, but at fome diftance from it 5
Suppofe in the mid-way between it and the Wall on which
the oblong Spedrum P T is caft, fo that the Light from
the firft Prifm may fall upon it in the form of an oblong
Spcdrum, ^7 Parallel to this fecond Prifm, and berefraded
Sideways to form the oblong Spedrum p t upon the Wall.
And you will find as before, that this Spedrum ft \s in-
clined to that Spedrum P T, which the firft Prifm forms
alone without the fecond 3 the blew ends P and f being fur-
ther diftant from one another than the red ones T and tj
and by confequence that the Rays which go to the blew
end '^ of the Image ^^1 and which therefore fuffer the greateft
Refraction in the firft Prifm, are again in the fecond Prifm
more reiraded than the reft. The

[30]

'Fig, 17. The fame thing I try'd alfo by letting the Suns Light
into a dark Room through two little round holes F and p
made in the Window, and with two Parallel Prifms ABC
and a0y placed at thofe holes ( one at each ) refradling
thofe two beams of Light to the oppofite Wall of the
Chamber, in fuch manner that the two coloured Images
P T and MH which they there painted were joyned end to
end and lay in one ftraight Line, the red end T of the
one touching the blew end JB of the other. For if thefe
two refradled beams were again by a third Prifm D H pla-
ced croft to the two firft, refradied Sideways, and the Spe-
d:rums thereby tranflated to fome other part of the Wall
of the Chamber , fuppofe the Spedirum P T to p t and

"^ulf jj'. the Spedlrum M N to in w, thefe tranflated Spe(ftrums p t
and m n would not lie in one ftraight Line with their ends
contiguous as before, but be broken off from one another
and become Parallel, the blew end of the Image m n being
by a greater Refra(5lion tranflated farther from its former
place M T, than the red end t of the other Image p t from
the fame place MT which puts the Propofition paft di-
Ipute. And this happens v/hether the third Prifm D H be
placed immediately after the two firft or at a great diftance
from them , fo that the Light refracbed in the two firft
Prifms be either white and circular, or coloured and ob-
long when it falls on the third,

Exper. 6. In the middle of tw^o thin Boards I made
round holes a third part of an Inch in Diameter, and in
the Window-iliut a much broader hole, being made to let
into my darkned Chamber a large beam of the Sun's
Light 5 I placed a Prifm behind the Shut in that beam to
refradt it towards the oppofite Wall, and clofe behind the
Prifm I fixed one of the Boards, m fuch manner that the
middle of the refraded Light might pafs through the hole

made

made in it, and the reft be intercepted by the Board.
Then at the diftance of about twelve Feet from the firft
Board I fixed the other Board, in fuch manner that the
middle of the refraded Light which came through the hole
in the firft Board and fell upon the oppofite Wall might
pafs through the hole in this other Board, and the reft be-
ing intercepted by the Board might paint upon it the co-
loured Spedrum of the Sun. And clofe behind this Board
I fixed another Prifm to refrad: the Light which came
through the hole. Then I returned fpeedily to the firft
Prifm, and by turning it flowly to and fro about its Axis,
I caufed the Image which fell upon the fecond Board to
move up and dov/n upon that Board, that all its parts
might fucceffively pafs through the hole in that Board and
fall upon the Prifm behind it. And in the mean time, I
noted the places on the oppofite Wall to which that Li8;ht
after its Refrad:ion in the fecond Prifm did pafs 5 and by
the difference of the places I found that the Light which
being moft refradled in the firft Prifm did go to the blew
end of the Image, was again more refrad:ed in the fecond
Prifm than the Light which went to the red end of that
Image, which proves as well the firft Propofition as the
fecond. And this happened whether the Axis of the two
Prifms were parallel, or inclined to one another and to the
H©rizon in any given Angles.

Illuftration, Let F be the wide hole in the Window-fliut, p^^ i g^
through which the Sun fliines upon the firft Prifm ABC,
and let the refraded Light fall upon the middle of the
Board D E, and the middle part of that Light upon the
hole G made in the middle of that Board. ~ Let this tra-
jededpart of the Light fall again upon the middle of the
fecond Board d e and there paint fuch an oblong coloured
Image of the Sun as was defcribed in the third Experiment.

[32]

By turning the Prifm ABC flowly to and fro about its
Axis this Image will be made to move up and down the
Board d e, and by this means all its parts from one end to
the other may be made to pafs fucceflively through the
hole g which is made in the middle of that Board. In the
mean while another Prifm a b c is to be fixed next after
that hole^ to refradl the trajed:ed Light a fecond time.
And thefe things being thus ordered, I marked the places
M and N of the oppofite Wall upon which the refracfled
Light fell,and found that whilfl: the two Boards and fecond
Prifm remained unmoved, thofe places by turning the firft
Prifm about its Axis were changed perpetually. For when
the lower part of the Light which fell upon the fecond
Board d e was caft through the hole g it went to a lower
place M on the Wall , and when the higher part of that
Light was call: through the fame hole^, it went to a higher
place N on the Wall, and when any intermediate part of
the Light was cafl; through that hole it went to fome place
on the Wall between M and N. The unchanged Pofition
of the holes in the Boards, made the Incidence of the Rays
upon the fecond Prifm to be the fame in all cafes. And
yet in that common Incidence fome of the Rays were more
refracted and others lefs. And thofe were more refradled
in this Prifm which by a greater Refraction in the firft
Prifm were more turned out of the way, and therefore for
their conftancy of being more refiraCled are defervedly cal-
led more Refrangible.

Exper. 7. At two holes made near one another in my
Window-flhut I placed two Prifms , one at each, which
might caft upon thi oppofite Wall ( after the manner of
the third Experiment ) two oblong coloured Images of the
Sun. And at a little diftance from the Wall I placed a
long flender Paper with flraight and parallel edges, and

ordered

C33 3

ordered tlie Prifms and Paper fo, that the red Colour of
one Image might fall diredly upon one half of the Paper,
and the violet colour of the other Image upon the other
half of the fame Papery fo that the Paper appeared of two
Colours , red and violet , much after the manner of the
painted Paper in the firft and fecond Experiments. Then
with a black Cloth I covered the Wall behind the Paper,
that no Light might be refleded from it to difturb the
Experiment, and viewing the Paper through a third Prifm <
held parallel to it, I faw that half of it which was illumi-
nated by the Violet-light to be divided from the other
half by a greater Refradion, efpecially 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 feemed contiguous at one
of their Angles like the painted Paper in the firft Expe-
riment. Which alfo happened when the Paper was too
broad.

Sometimes inftead of the Paper I ufed a white Thred,
and this appeared through the Prifm divided into two Pa-
rallel Threds as is reprefented in the 19th Figure, where Fig-. 19.
D G denotes the Thred illuminated with violet Light
from D to E and with red Light from F to G, and d e fg
are the parts of the Thred feen by RefracSiion. li one hali
of the Thred be conPcantly illuminated with red, and the
other half be illuminated with all the Colours fucceffively,
(which may be done by caufing one of the Prifms to be
turned about its Axis whilft the other remains unmoved)
this other half in viewing the Thred through the Prifm,
will appear in a continued right Line with the firft half
when illuminated with red , and begin to be a little divi-
ded from it v/hen illuminated with Orange, and remove
further from it when illuminated with Yellow, and ft-ill

E further

[34]

further when with Green, and further when with Blew, and
20 yet further off when illuminated with Indigo, and fur*
^ theft when with deep Violet. Which plainly flhews, that
the Lights of feveral Colours are more and more Refran-
gible one than another, in this order of their Colours, Red,
Orange, Yellow, Green, Blew, Indigo, deep Violet 3 and
fo proves as well the firft Propofition as the fecond.
F^^. 17. I caufed alfo the coloured Spedrums PT and MN
made in a dark Chamber by the Refradions of two Prifms
to lye in a right Line end to end, as was defcribed above
in the fifth Experiment, and viewing them through a third
Prifm held Parallel to their length, they appeared no longer
in a right Line, but became broken from one another, as
they are reprefented at pt and mri, the violet end m of the
Spedlrum m n being by a greater Refradlion tranflated
further from its former place M T than the red end^ of the
other Spedrum p t.
^i(r. 20. I further caufed thofe two Spedlrums P T and M N 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 reprefented in the oblong Figure P T M N 5:
and then viewing them through a Prifm D H held Paral-
lel to their length, they appeared not co-incident as when
viewed with the naked Eye , but in the form of two di-
ftind: Spedlrums p t and m n crofling one another in the
middle after the manner of the letter X. Which flhews
that the red of the one Sped:rum and violet of the other,
which were co-incident at PN and MT, being parted
from one another by a greater Refradion of the violet to
p and m than of the red to n and f, do differ in degrees of
Refrangibility.

I illuminated alfo a little circular piece of white Paper
all over with the Lights of both Prifms intermixed, and

when.

35]

when it was illuminated with the red of one Spedlrum and
deep violet of the other , fo as by the mixture of thofc
Colours to appear all over purple , I viewed the Paper,
firft ai a lefs diftance , and then at a greater , through a
third Prifm 5 and as I went from the Paper, the refracted
Image thereof became more and more divided by the un-
equal Refra6tion of the two mixed Colours, and at length
parted into two diftind: Images, a red one and a violet one,
whereof the violet was fijrtheft from the Paper, and there-
fore fuftered the greateft Pvefrad:ion. And when that Prifm
at the Window which caft the violet on the Paper was ta-
ken away,iiie violet Image difappeared^ but when the other
Prilm was taken away the red vanifhed : which fliews that
thefe two Images were nothing elfe than the Lights of the
two Prifms which had been intermixed on the purple Pa-
per, but were parted again by their unequal Refractions
made in the third Prifm through which the Paper was
viewed. This alfo was obfervable that if one of the
Prifms at the Window, fuppofe that which caft the violet
on the Paper, was turned about its Axis to make all the
Colours in this order, Violet, Indigo, Blew, Green, Yel-
low, Orange, Red, fall fucceffively on the Paper from that
Prifm, the violet Image changed Colour accordingly, and
in changing Colour came nearer to the red one, until when
it was alfo red they both became fully co-incident.

I placed alfo two paper circles very near one another,
the one in the red Light of one Prifm, and the other in
the violet Light of the other. The circles were each of
them an hich in Diameter, and behind them the Wall was
dark that the Experiment might not be difturbed by any
Light coming from thence. Thefe circles thus illuminated,
I viewed through a Prifm fo held that the Refradtion might
be made towards the red circle , and as I went from them

E 2 they

they came nearer and nearer together, and at length be-^
came co-incident 3 and afterwards when I went ftill further
off, they parted again in a cotitrary order, the violet by a
greater Refra<flion being carried beyond the red.

Exper. 8. In Summer when the Sun's Light ufes to
be ftrongeft, I placed a Prifm at the hole of the Window-
lliut, as in the third Experiment, yet fo that its Axis might
be Parallel to the Axis of the World, and at the oppofite
Wall in the Sun's refraded Light, I placed an open Book.
Then going Six Feet and two Inches from the Book, I
placed there the abovementioned Lens,by which the Light
reflected from the Book might be made to converge and
meet again at the diftance of fix Feet and two Inches be-
hind the Lens , and there paint the Species of the Book
upon a Hieet of white Paper much after the manner of the
fecond Experiment. The Book and Lens being made faft,
I noted the place where the Paper was, when the Letters
of the Book, illuminated by the fulleft red Light of the
Solar Image falling upon it, did caft their Species on that
Paper moft diftindly 3 And then I ftay'd till by the Mo*
tion of the Sun and confequent Motion of his Image on
the Book, all the Colours from that red to the middle of
the blew pafs'd over thofe Letters 5 and when thofe Letters
were illuminated by that blew, I noted again the place of
the Paper when they caft their Species moft diftindly upon
it : And I found that this laft place of the Paper was nearer
to the Lens than its former place by about two Inches and
an half, or tv/o and three quarters. So much fooner there-
fore did the Light in the violet end of the Image by a grea-
ter Pvefrad:ion 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 thefe Colours be diluted and weak-
ned by the mixture of any adventitious Light, the diftance

between

[ 37 ]

between the places of the Paper will not be fo great. This
diftance in the fecond Experiment where the Colours of
natural Bodies were made ufe of, was but an Inch and a
half, by reafon of the imperfedion of thofe Colours. Here
in the Colours of the Prifm , which are manifeftly more
full, intenfe, and lively than thofe of natural Bodies, the
diftance is two Inches and three quarters. And were the
Colours ftill more full , I queftion not but that the di»
ftance would be confiderably greater. For the coloured
Light of the Prifm, by the interfering of the Circles de-
fcribed in the i ith Figure of the fifth Experiment, and alfo
by the Light of the very bright Clouds next the Sun's
Body intermixing with thefe Colours, and by the Light
fcattered by the inequalities in the polifli of the Prifm, was
fo very much compounded, that the Species which thofe
faint and dark Colours, the Indigo and Violet, caft upon
the Paoer vvxre not diftindl enough to be well obferved.

Exper. p. A Prifm, whofe two Angles at its Bafe 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 in-
to a dark Chamber through a hole in the Window-fhut
as in the third Experiment. And turning the Prifm flowly
about ics Axis until all the Light which went through one
of its Angles and was refracted by it began to be refleded
by its Bafe , at which till then it went out of the Glafs,
I obferved that thofe Rays which had fuffered the greateft
Refraction were fooner reflected than the reft. I conceived
therefore that thofe Rays of the refledled Light, which
were moft Refrangible, did firft of all by a total Reflexion
become more copious in that Light than the reft , and
that afterwards the reft alfo, by a total Reflexion, be-
came as copious as thefe. To try this , I made the re-
fleded Light pafs through another Prifm, and being refra-

ded

[38]

ded by it to fall afterwards upon a fbeet of white Paper
placed at fome diftance behind it, and there by that Re-
fradion to paint the ufual Colours of the Prifm. And
tlien caufing the firft Prifm to be turned about its A:>cis as
above, I obferved that when thofeRays which in this Prifm
had fuffered the greateft Refradion and appeared of a blew
and violet Colour began to be totally refleded , the blew
and violet Light on the Paper which was moft refraded
in the fecond Prifm received a fenfible increafe above that
of the red and yellow, which was leaft refraded 5 and
afterwards when the reft of the Light which was green,
yellow and red began to be totally refleded in the firft
Prifm, the hghtofthofe Colours on the Paper received as
great an increafe as the violet and blew had done before.
Whence 'tis manifeft, that the beam of Light refleded by
the Bafe of the Prifm, being augmented firft by the more
Refrangible Rays and afterwards by the lefs Refrangible
ones, is compounded of Rays differently Refrangible.
And that all fuch refledled Light is of the fame Nature
with the Sun's Light, before its Incidence on the Bafe of
the Prifm, no Man ever doubted : it being generally al-
lowed, that Light by fuch Reflexions fuffers no Alteration
in its Modifications and Properties. I do not here take
notice of any Refradions made in the Sides of the firft
Prifm, becaufe the Light enters it perpendicularly at the
firft Side, and goes out perpendicularly at the fecond Side,
and therefore fuffers none. So then, the Sun's incident
Light being of the fame temper and conftitution with his
emergent Light, and the bft being compounded of Rays
differently Refrangible , the firft muft be in like manner
compounded.
Fi^. 1 1 . Illuftration. In the 1 1 th Figure, A B C is the firft Prifm,
B C its Bafe, B and C its equal Angles at the Bafe, each

[39]

of 45 degrees, A its Redangular Vertex, F M a beam of
the Sun's Light let into a dark Room through a hole B
one third part of an Inch broad, M its Incidence on theBafe
of the Prifm^M G a lefs refraded Ray, M H a more refra6t-
ed Ray, M N the beam of Light refle6led from the Bafe ,
V X Y the fecond Prifm by which this beam in paffing
through it is refrad:ed, N t the lefs refracted Light of this
beam, and N p the more refracted part thereof. When the
firft Prifm A B C is turned about its Axis according to the
order of the Letters ABC, the Rays M H emerge more
and more obliquely out of that Prifm, and at length after
their moft oblique Emergence are refle6led towards N,
and going on to p do increafe the number of the Rays N^^
Afterwards by continuing the motion of the firft Prifm, the
Rays M G are alfo reflected to N and increafe the number of
the Rays N t. And therefore the Light M N admits into
its Compofition, firft the more Refrangible Rays, and then
the lefs Refrangible Rays, and yet after this Compofition
is of the fame Nature with the Sun's immediate Light F M,
the Reflexion of the fpecular Bafe B C caufing no Altera-
tion therein.

Exper. I o. Two Prifms, which were alike in fliape, I
tied fo together, that their Axes and oppofite Sides being
Parallel, they compofed a Parallelopiped,. And, the Sun
fliining into my dark Chamber through a little hole in the
Window-fhut, I placed that Parallelopiped in his beam at
fome diftance from the hole, in fuch a pofture that the Axes
of the Prifms might be perpendicular to the incident Rays,
and that thofe Rays being incident upon the firft Side of
one Prifm, might go on through the two contiguous Sides
of both Prifms, and emerge out of the laft Side of the fe-
cond Prifm. This Side being Parallel to the firft Side of
the firft Prifm 5 caufed the emerging Light to be Parallel

[ 4° ]

to the Incident. Then, beyond thefe two Prifms I placed
a third, which might refrad that emergent Light, and by
that Refradion caft the ufual Colours of the Prifm upon
the oppofite Wall, or upon a (Tieet of white Paper held at
a convenient diftance behind the Prifm for that refradled
Light to fall upon it. After this I turned the Parallelopiped
about its Axis, and found that when the contiguous Sides
of the two Prifms became fo obHque to the incident Rays
that thofe Rays began all of them to be refie6ted , thofe
Rays which in the third Prifm had fuffered the greatefi; Re-
fraction and painted the Paper with violet and blew, were
firfl: of all by a total Reflexion taken out of the tranfmitted
Light, the reft remaining and on the Paper painting their
Colours of Green, Yellow, Orange, and Red as before 3
and afterwards by continuing the motion of the two Prifms,
the reft of the Rays alfo by a total Reflexion vaniflied in
order, according to their degrees of Refrangibility. The
Licrht therefore which emerged out of the two Prifms is
compounded of Rays differently Refrangible , feeing the
more Refrangible Rays may be taken out of it v/hile the
lefs Refrangible remain. But this Light being trajedted
only through the Parallel Superficies of the two Prifms, if
it fuffered any change by the Refradion of one Superficies
it loft that impreffion by the contrary Refradion of the
other Superficies, and fo being reftored to its priftine con-
ftitution became of the fame nature and condition as at firft
before its Incidence on thofe Prifms 3 and therefore, before
its Incidence, was as much compounded of Rays differently
Refrangible as afterwards.
Fig. 11. Iliuflration. In the nth Figure ABC and BCD are the
the two Prifms tied together in the form of a Parallelo-
piped, their Sides BC and CB being contiguous, and
their Sides A B and C D Parallel. And H J K is the third

Prifm,

C4-0

Prifm, by which the Sun's Light propagated through tlie
hole F into the dark Chamber, and there paffing through
thofe fides of the Prifms AB, BC, CB and CD, is refra-
died at O to the white Paper PT, falling there partly upon
P by a greater Refradiion, partly upon T by a lefs Refra-
dlion, 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 Let-
ters A,C,D,B, at length when the contiguous Planes BC
and CB become fufficiently oblique to the Rays F M,
which are incident upon them at M, there will vaniflh to-
tally out of the refraded Light OPT, firft of all the moft
refraded Rays O P, (the reft OR and O T remaining as
before) then the Rays O R and other intermediate ones,
and laftly, the leaft refracted Rays O T. For when the
Plane B C becomes fufficiently oblique to the Rays inci-
dent upon it, thofe Rays will begin to be totally reflect-
ed by it towards N 5 and firft the moft Refrangible Rays
will be totally reflected (as was explained in the preceding
experiment) and by confequence muft firft difappear at P,
and afi:erwards the reft as they are in order totally reflect-
ed to N, they muft difappear in the fame order at R and
T. So then the Rays which at O fuffer the greateft Re-
fraction, may be taken out of the Light M O whilft the reft
of the Rays remain in it, and therefore that Light MO is
Compounded of Rays diiferently Refrangible. And be-
caufe the Planes A B and C D are parallel, and therefore
by equal and contrary RefraCtions deftroy one anothers
Effects, the incident Light F M muft be of the fame kind
and nature with the emergent Light M O, and therefore
doth alfo confift of Rays differently Refrangible. Thefe
two Lights FM and ?vIO, before the moft reirangible Rays
are feparated out of the emergent Light M O agree in Co-

F lour.

C4^]

lour, and in all other properties fo far as my obfervation
reaches, and therefore are defervedly reputed of the fame
Nature and Conftitution, and by confequence the one is
compounded as well as the other. But after the moft Re*
frangible Rays begin to be totally refleded, and thereby
feparated out of the emergentLightMO,that Light changes
its Colour from white to a dilute and faint yellow, a pretty
good orange, a very full red fucceffively and then totally
vaniflbes. For after the moft Refrangible Rays which paint
the Paper at P Vvdth a Purple Colour, are by a total re-
flexion taken out of the Beam of hght M O, the reft of
the Colours which appear on the Paper at R and T being
mixed 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 reft which .-
appear between R and T (that is the Yellow, Orange, Red
and a little Green) being mixed in the Beam M O com-
pound there an Orange 5 and when all the Rays are by re-
flexion taken out of the Beam MO, except the leaft Refran-
gible, which at T appear of a full Red, their Colour is
the fame in that Beam MO as afterwards at T, the Re-
fraction of the Prifm HJK ferving only to feparate the
differently Refrangible Rays, without making any alteration
in their Colours, as (hall be more fully proved hereafter.
All which confirms as well the firft Propofition as the fe-
cond.

Scholium. If this Experiment and the former be conjoyned
w. 11* and made one, by applying a fourth Prifm VXY to re-
fradl the reflected Beam M N towards tp^ the conclufion
will be clearer. For then the light N^ which in the 4th
Prifm is more refracted, will become fuller and ftronger
when the Light O P, which in the third Prifm HJK is
more refraded, vanifhes at P 3 and afterwards when the lefs

refrs^ded

I43]

refra(fled Light O T vaniflies at T^the lefs refraded Light
Nf will become encreafed whilft the more refraded Light
at p receives no further encreafe. And as the trajeded
Beam M O in vanifliing is always of fuch a Colour as
ought to refult from the mixture of the Colours which
fall upon the Paper PT, fo is the refleded Beam MN al-
ways of fuch a Colour as ought to refult from the mix-
ture of the Colours which fall upon the Paper p t. For
when the moft refrangible Rays are by a total Reflexion
taken out of the Beam M O, and leave that Beam of an
Orange Colour, the excefs of thofe Rays in the refleded
Light, does not only make the Violet, Indigo and Blue at
p more full, but alfo makes the Beam M N change from
the yellowifli Colour of the Sun's Light, to a pale white in-
clining to blue, and afterward recover its yellowifli Co-
lour again, fo foon as all the reft of the tranfmitted light
MOT is refleded.

Now feeing that in all this variety of Experiments,
whether the trial be made in Light refleded, and that either
from natural Bodies, as in the firft and fecond Experiment,
or Specular, as in the Ninth 3 or in Light refiaded, and
that either before the unequally refraded Rays are by di-
verging feparated from one another, and loflng their white-
nefs which they have altogether, appear feverally of feve-
ral Colours, as in the fifth Experiment 5 or after they are
feparated from one another, and appear Coloured as in the
fixth, feventh, and eighth Experiments 3 or in Light tra-
jeded through Parallel fuperficies, deftroying each others
EfFeds as in the i oth Experiment 3 there are always found
Rays, which at equal Incidences on the fame Medium fuf-
fer unequal Refradions, and that without any Iplitting or
dilating of fingle Rays, or contingence in the inequality
of the Refradions, as is proved in the iifh and fixth Ex-

F 2 -^^^'iments ;

[44l

periments 3 and feeing the Rays which differ in Refrangibi^
lity may be parted and forted from one another, and that
cither by ReFra6lion as in the third Experiment, or by Re-
flexion as in the tenth^ and then the feveral forts apart at
equal Incidences fuffer unequal Refra6lions, and thofe forts^
are more refraded than others after feparation, which werer
more refraded before it, as in the fixth and following Ex-
periments, and if the Sun's Light be traje6led through three
or more crofs Prifms fucceffively, thofe Rays which in the
firft Prifm are refracfted more than others are in all the fol-
lowing Prifms, refradred more then others in the fame rate
and proportion, as appears by the fifth Experiment 3 it's
manifeft that the Sun's Light is an Heterogeneous mixture of
Rays, fome of which are conftantly more Refrangible then
others, as was to-be propofed.

T ROT. III. Theor. III.

T^be Suns Light conftfts of ^ys differing in ^flexibility., and
thofe ^ys are more ^flexible than others which are more (%-
frangible.

^ I ^HIS is manifeft by the ninth and tenth Experi-
I ments : For in the ninth Experiment, by turning,
the Prifm about its Axis, until the Rays within it which in
going out into the Air were refraded by its Bafe, became
lb oblique to that Bafe, as to begin to be totally reflected
thereby 5 thofe Rays became firft of all totally refleded,
which before at equal Incidences with the reft had fuffered
the greateft Refradion. And the fame thing happens in
the Reflexion made by the common Bafe of the two Prifms
in the. tenth Experimentc

[45]
V ROV. IV. Prob. I.

To feparate from one another the Heterogeneous ^ys of

Compound Light,

"^HE Heterogeneous Rays are in fome meafure fepa-

rated from one another by the Refraction of the

Prifm in the third Experiment, and in the fifth Experiment
by taking away the Penumbra from the Re6tiHnear fides of
the Coloured Image, that feparation in thofe very Rectili-
near fides or ftraight edges of the Image becomes perfect.
But in all places between thofe rectilinear edges, thofe in-
numerable Circles there defcribed, which are feverally illu-
minated by Homogeneral Rays, by interfering with one
another, and being every where commixt, do render the
Light fufficiently Compound. But if thefe Circles, whilft
their Centers keep their diftances and pofitions, could be
made lels in Diameter, their interfering one with another
and by confequence the mixture of the Heterogeneous
Rays would be proportionally diminiflied. In the 2 3th'Fg. 23.
Figure let AG, B H, CJ, D K, EL, F M be the Circles
which fo many forts of Rays flowing from the fameDifque
of the Sun, do in the third Experiment illuminate 3 of all
which and innumerable other intermediate ones lying in a
continual Series between the two Redilinear and Parallel-
edges of the Sun's oblong Image P T, that Image is com-
pofed as was explained in the fifth Experiment. And let'
agj bhy cij dk^j el ^ fm be fo many lefs Circles lying in
a like continual Series between two Parallel right Lines af
and ^ m with the fame diftances between their Centers,
and illuminated by the fame forts of Rays, that is the
Circle ag with the fame fort by which the correfponding

Circle

Circle A G was illuminated, and the Circle bh with the fame M
fort by which the correfponding Circle BHwas illuminated, )
and the reft of the Circles c i, dk^ el, f)n refpedtively,
with the fame forts of Rays by which the feveral corre-
fponding Circles CJ, D K, EL, FM v/ere illuminated.
In the Figure P T compofed of the greater Circles, three
of thofe Circles AG, B H, C J, are (b expanded into one
another, that the three forts of Pvays by which thofe Cir-
cles are illaminated, together with other imiamerable forts
of intermediate Rays, are mixed at Q. R in the middle of
the Circle B H. And the like mixture happens through-
out almoft the whole length of the Figure P T. But in
the Figure p t compofed of the lefs Circles, the three lefs
Circles ag^ hh^ c i, which anfwer to thofe three greater, do
not extend into one another 5 nor are there any where
mingled fo much as any two of the three forts of Rays
by which thofe Circles are illuminated, r.nd which in the
Figure P T are all of them intermingled at B H.

Now he that fliall thus confider it, will eafily underftand
that the mixture is diminiflhed in the fame Proportion
with the Diameters of the Circles. If the Diameters of
the Circles whilft their Centers remain the fame, be made
three times lefs than before, the mixture will be alfo three
times lefs 5 if ten times lefs, the mixture will be ten times
lefs, and fo of other Proportions. That is, the mixture
of the Rays in the greater Figure P T will be to their mix-
ture in the lefs p /•, as the Latitude of the greater Figure is
to the Latitude of the lefs. For the Latitudes of thefe Fi-
gures are equal to the Diameters of their Circles. And
hence it eafily follows, that the mixture of the Rays in the
refracted Spedrum ^ f is to the mixture of the Rays in the
dire6t and immediate Light of the Sun, as the breadth of
that Spedrum is to the difference between the length and
breadth of the fame Spedrum. <]ft^Ad7J:J^^:07iJ£<^^i So

[47]

So then, if we would diminifh the mixture of the Rays,
w€ are to diminifh the Diameters of the Circles. Now
thefe would be diminifh ed if the Sun's Diameter to which
they anfwer could be made lefs than it is^ or (which comes
to the fame purpofe) if without Doors, at a great diftance
from the Prifm towards the Sun, fome opake body were
placed, with a round hole in the middle of it, to intercept
all the Sun s Light, excepting fo much as coming from
the middle of his Body could pafs through that hole to
the Prifm. For fo the Circles AG, BH and the reftj
would not any longer anfwer to the whole Difque of the
Sun , but only to that part of it which could be feen
from the Prifm through that hole, that is to the apparent
magnitude of that hole viewed from the Prifm. But that
thefe Circles may anfwer more diftindlly to that hole a
Lens is to be placed by the Prifin to caft the Image of the
hole, (that is, every one of the Circles A G, B H, ^c.) di-
ftindly upon the Paper at P T, after fuch a manner as by
a Lens placed at a Window the Species of Objeds abroad
are caft diftindlly upon a Paper within the Room, and the
Redilinear Sides of the oblong folar Image in the fifth
Experiment became difl;in(5i: without any Penumbra. If
this be done it will not be neceffary to place that hole
very far off, no not beyond the Window. And therefore
inflead of that hole, I ufed the hole in the Window-fhut
as follows.

Exper. 1 1 . In the Sun's Light let into my darkned
Chamber through a fmall round hole in my Window^
fhut, at about i o or 12 Feet from the Window, I placed
a Lens , by which the Image of the hole might be di-
ftindly caft upon a fheet of white Paper, placed at the
diftance of fix, eight, ten or twelve Feet from the Lens.
For according to the difference of the Lenfes I ufed various

diftances,

[48

diftances , which I think not worth the while to defcrrbe.
Then immediately after the Lens I placed a Prifm, by
which the traje6ted Light might be refradled either up-
wards or £deways, and thereby the round Image which
the Lens alone did caftupon 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 fame diftance from the Prifm as be-
fore, moving the Paper either towards the Prifm or from
it, until I found the jufl: diftance where the Redilinear
Sides of the Image became moft diftind:. For in this cafe
the circular Images of the hole which compofe that Image
after the fame manner that the Circles ag^ bh^ ci^ 8cc. do
ft- 2 7. the Figure p f , were terminated moft diftindily without any
Penumbra, and therefore extended into one another the
leaft that they could, and by confequence the mixture of
the Heterogeneous Rays was now the leaft of all. By this
F/o-. 2^5 means I ufed to form an oblong Image (fuch as is pt) of
and 24. circular Images of the hole (fuch as are ag, bh^ ci^ &c.)
and by ufing a greater or lefs hole in the Window-ftiut, I
made the circular Images ag^ bh^ c i, &c. of which it was
formed, to become greater or lefs at pleafure, and thereby
the mixture of the Rays in the Image pt to be as much
or as little as I defired.
Fk. 24. Illujiration. In the 24th Figure, F reprefents the circular
hole in the Window-fliut, MN the Lens whereby the
Image or Species of that hole is caft diftin6lly upon a
Paper at J, ABC the Prifm whereby the Rays are at their
emerging out of the Lens refracted from J towards ano-
ther Paper at p f , and the round Image at J is turned into
an oblong Image p t falling on that other Paper. This
Image p t confifts of Circles placed one after another in a
Reftilinear order^ as was fufficiently explained in the fifth

Experiment 5

[49]

Experiment 3 and thefe Circles are equal to the Circle I,
and confequently anfwer in Magnitude to the hole F 5 and
therefore by diminifliing that hole they may be at pleafurc
diminiflied , whirft their Centers remain in their places.
By this means I made the breadth of the Image ^ t to be
forty times, and fometimes fixty or feventy times lefs than
its length. As for inftance, if the breadth of the hole F
be ^ of an Inch, and MF the diftance of the Lens from
the hole be 1 1 Feet 5 and if p B or pM the diftance of
the Image pt from the Prifm or Lens be 10 Feet, and the
refrading Angle of the Prifm be 62 degrees, the breadth
of the Image p t will be ~ of an Inch and the length about
fix Inches, and therefore the length to the breadth as 72
to I, and by confequence the Light of this Image 71 times
lefs compound than the Sun's dired: Light. And Light
thus far Simple and Homogeneal , is fufficient for trying
all the Experiments in this Book about fimple Light. For
the compofition of Heterogeneal Rays is in this Light fo
little that it is fcarce to be difcovered and perceived by
fenfe, except perhaps in the Indigo and Violet 3 for thefe
being dark Colours, do eafily fuffer a fenfible allay by that
little fcatterincr Light which ufes to be refraded irrep^ularlv
by the inequaiiteis of the Prijfin.

Yet inftead of the circular hole F, 'tis better to fubfti-
tute an oblong hole fliaped like a long Parallelogram
with its length Parallel to the Prifm 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-Imao^e
p t will be as Simple as before or fimpler, and the Image
will become much broader, and therefore more fit to have
Experiments tried in its Light than before.

Inftead of this Parallelogram-hole may be fubftituted a
Triangular one of equal Sides, whofe Bafe for inftance is

G about

about the tenth part of an Inch, and its height an Inch or
more. For by this means , if the Axis of the Prifm be
Parallel to the Perpendicular of the Triangle , the Image
Fig. ly pt will now be formed of Equicrural Triangles ag^ bh, ciy
dk^j el^ f m^ Sec, and innumerable other intermediate ones
anfwering to the Triangular hole in fhape and bignefs, and
lying one after another in a continual Series between two
Parallel Lines af zndgrn. Thefe Triangles are a little
intermingled at their Bafes but not at their Vertices, and
therefore the Light on the brighter fide af of the Image
where the Bafes of the Triangles are is a little compounded,
but on the darker fide ^ w is altogether uncompounded,
and in all places between the fides the Compofition is
Proportional to the diftances of the places from that ob-
fcurer fide g m. And having a Spe6trum p t of fuch a
Compofition, we may try Experiments either in its ftronger
and lefs fimple Light near the fide af or in its weaker
and fimpler Light near the other fide / m, as it fliall fe^n
moft convenient.

But in making Experiments of this kind the Chamber
ought to be made as dark as can be, leaft any forreign
Light mingle it felf with the Light of the Spedlrum p f,
and render it compound 3 efpecially if we would try Ex-
periments in the more fimple Light next the fide g ni of
the Spedlrumj which being fainter, will have a lefs Pro-
portion to the forreign Light, and fo by the mixture of
that Light be more troubled and made more compound.
The Lens alfo ought to be good, fuch as may ferve for
Optical Ufes, and the Prifm ought to have a large Angle,
fuppofe of^/o degrees, and to be well wrought, being
made of Glafs free from Bubbles and Veins, with its fides
not a little Convex or Concave as ufually happens but
truly Plane,and its poUifh elaborate, as in working Optick-

glafles

[51]

gkfles y and not fuch as is ufually wrought with Putty,
whereby the edges of the Sand-holes being worn away,
there are left all over the Glafs a numberlefs company of
very little Convex polite rifings like Waves. The edges
alfo of the Prifm and Lens fo far as they may make any
irregular Refra6tion, muft be covered with a black Paper
glewed on. And all the Light of the Sun's beam let into
the Chamber which is ufelefs and unprofitable to the Ex-
periment, ought to be intercepted with black Paper or other
black Obftacles. For otherwife the ufelefs Light being
refleded every way in the Chamber , will mix with the
oblong Spedrum and help to difturb it. In trying thefe
things fo much Diligence is not altogether neceflary, but
it will promote the fuccefs of the Experiments, and by a
very fcrupulous Examiner of things deferves to be applied.
It's difficult to get glafs Prifms fit for this purpofe, and
and therefore I ufed fometimes Prifmatick Vends made
with pieces of broken Looking-glafles, and filled with rain
Water. And to increafe the Refradion, I fometimes im-
pregnated the Water ftrongly with Saccharum Saturni,

PROP. v. Theor. IV.

Homogeneal Light is refraBed regularly without any Dilatation

f putting or Jhattering of the ^^ys , and the confufed Vtjton

of OhjeBs feen through ^fraEling 'Bodies hy Heterogejieal

Light arifes from the dtjferent ^efrangihtlity of federal forts

of %y. J,

TH E firft Part of this Propoficion has been already
fufficiently proved in the fifth Experiment, and will
further appear by the Experiments which follow.

G 2 Exper. i 2.

[52]

Exper, 12. In the middle of a black Paper I made a
round hole about a fifth or fixth part of an Inch in Dia-
meter. Upon this Paper I caufed the Spectrum of Homo-
geneal Light defcribed in the former Propofition , fo to
fall, that fome part of the Light might pafs through the
hole of the Paper. This tranfmitted part of the Light I
refra^led with a Prifm placed behind the Paper, and let-
ting this refraded Light fall perpendicularly upon a white
Paper two or three Feet diftant from the Prifm, I found
that the Spedrum 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 (fo far
as I could judge by my Eye) perfedly circular, the length
being no greater than the breadth. Which fliews that this
Light is refi:ad;ed regularly without any Dilatation of the

Exper. 1 ;. In the Homogeneal Light I placed a^Circle
of -^ of an Inch in Diameter, and in the Sun's unrefradted
Heterogeneal white Light I placed another Paper Circle of
the fame bignefs. And going from the Papers to thediftance
of fome Feet, I viewed both Circles through a Prifm. The
Circle illuminated by the Sun's Heterogeneal Light appear-
ed 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 Cir-
cular and diftindly defined as when 'tis viewed with the
naked Eye. Which proves the whole Propofition.

Exper. 14. In the Homogeneal Light I placed Flies and
fuch like Minute Objeds, and viewing them through a
Prifm , I faw their Parts as diftindlly defined as if I had
viewed them with the naked Eye. The fame Objeds pla-
ced in the Sun's unrefraded Heterogeneal Light which was
white I viewed alfo through a Prifm, and faw them moft

confufedly

[$3 3

Gonfufedly defined, fo thatlcould not diftinguiflh their fmaU
ler Parts from one another. I placed alfo the Letters of a
fmall Print one while in the Homogeneal Light and then
in the Heterogeneal, and viewing them through a Prifm,
they appeared in the latter cafe fo confufed and indiftind:
that I could not read them 5 but in the former they ap-
peared fo diftindl that I could read readily, and thought
I fav/ them: as diftind: as when I viewed them with my
naked Eye. In both cafes I viewed the fame Objedls
through the fame Prifm at the fame diftance from me and
in the fame Situation. There was no difference but in the
Light by which the Objects were illuminated , and which
in one cafe was Simple and in the other Compound, and
therefore the diftindl Vifion in the former cafe and confu-
fed in the latter could arife from nothing elfe than from
that difference of the Lights. Which proves the whole
Propoficion.

And in thefe three Experiments it is further very remar-
kable, that the Colour of Homogeneal Light was never:
changed by the Refradion..

PROP. Vr. Theor. V:

The Sine of Incidence of eVery ^ay confidered apart ^ is to its SifWi
of ^fraBion in a giyen ^l{atio.

THAT every Ray confidered apart is conftant to
it felf in fome certain degree of Refrangibility, is
fufficiently manifeft out of what has been faid. Thofe
Rays which in the firfl Refradion are at equal Incidences
moll refraded, are alfo in the following Refractions at
equal Incidences mofl refraded 5 and fo of the leafl Re-
frangible , and the reft which have any mean degree of

Refran-f

[$4]

Refran*gibility, as is manifeft by the 5th, 6th, 7th, 8th,
and 9th Experiments. And thofe which the firft time at
like Incidences are equally refraded, are again at like In-
cidences equally and uniformly refraded, and that whe-
ther they be refraded before they be feparated from one
another as in the 5 th Experiment, or whether they be re-
fraded apart, as in the i 2th, i ^th and 14th Experiments.
The Refradion therefore of every Ray apart is regular,
and what Rule that Refradion obferves we are now
to fhew.

The late Writers in Opticks teach, that the Sines of In-
cidence are in a given Proportion to the Sines of Refra-
dion, as was explained in the 5 th Axiom 5 and fome by
Inftruments fitted for meafuring Refradions, or otherwife
experimentally examining this Proportion, do acquaint us
that they have found it accurate. But whilft they, not
underftanding the different Refrangibility of feveral Rays,
conceived them all to be refraded according to one and
the fame Proportion, 'tis to be prefumed that they adapted
their Meafures only to the middle of the refraded Light 5
fo that from their Meafures we may conclude only that
the Rays which have a mean degree of Refrangibility ,
that is thofe which when feparated from the reft appear
green, are refraded according to a given Proportion of
their Sines. And therefore we are now to fliew that the
like given Proportions obtain in all the reft. That it
fhould be fo is very reafonable. Nature being ever confor-
mable to her felf : but an experimental Proof is defired.
And fuch a Proof will be had if we can flhew that the
Sines of Refradion of Rays differently Refrangible are
one to another in a given Proportion when their Sines of
Incidence are equal. For if the Sines of Refiradion of all
the Rays are in given Proportions to the Sine of Refradion

M5]

©f a Ray which has a mean degree of Refrangibility, and
this Sine is in a given Proportion to the equal Sines of
Incidence, thofe other Sines of Refradion will alfo 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 Refradion are
in a given Proportion to one another.

Exper. 15. The Sun flhining into a dark Chamber
through a little round hole in the Window-fliut, let S re- ^5?* ^^'
prefent his round white Image painted on the oppofite
Wall by his dired Light, P T his oblong coloured Image
made by refrading that Light with a Prifm placed at the
Windowj and pt^ or ip it^ or ^p 3 f, hisoblong coloured
Image made by refra6ling again the fame Light fideways
with a fecond Prifm placed immediately after the firft in
a crofs Pofition to it, as was explained in the fifth Experi-
ment : that is to fay, pt when the Refraction of the fecond
Prifm is fmall, ip it when its Refracftion is greater, and
3^ 3/- when it is greateft. For fuch will be the diverfity J(-
of the Refra6lions if the refra6ling Angle of the fecond
Prifm be of various Magnitudes 3 fuppofe of fifteen or
twenty degrees to make the Image p ?, of thirty or
forty to make the Image ip 2 f, and of fixty to make
the Image ip ^t. But for want of folid Glafs Prifms with
Angles of convenient bignefles, there may be Veflels
made of poliflhed Plates of Glafs cemented together in the
form of Prifms and filled with Water. Thefc things being
thus ordered, I obferved that all the folar Images or co-
loured Spedrums P T, ^^, ip it^ 3^ 3^ did very nearly
converge to the place S on which the dired: Light of the
Sun fell and painted his white round Image when the
Prifms were taken away. The Axis of the Spedrum PT,
that is the Line drawn through the middle of it Parallel to

its

its Redilincar Sides, did when produced pafs exadly through
the middle of that white round Image S. And when the
Refradion of the fecond Prifm was equal to the Refradlion
of the firft, the refradting Angles of them both being about
^o degrees, the Axis ofthe Spedrum ip ^t made by that
:Refra(5lion, did when produced pafs alfo through the mid-
dle of the fame white round Image S. But when the Re-
fraction of the fecond Prifm was lefs than that of the firft,
the produced Axes of the Spedirums tp ox it zp made
by that Refradion did cut the produced Axis of the Spe-
<5lrum TP in the Points m and w, a little beyond the Cen-
ter of that white round Image S. Whence the Proportion
ofthe Line ^ f T to the Line ipV was a little greater than
the Proportion of 2 f T to 2 /^P, and this Proportion a little
greater than that of tT to j?P. Now when the Light of
the Spc6lrumP T falls perpendicularly upon the Wall, thofe
Lines 3 tT, 3-^ P, and 2 1 T, if P and ^T, ^P,are the Tan-
gents of the Refrad:ions 3 and therefore by this Experiment
the Proportions of the Tangents of the Refradiions are ob-
tained, from whence the Proportions of the Sines being deriv-
ed, they come out equal, fo far as by viewing the Sped:rums
and ufing fome Mathematical reafoning I could Eftimate.
For I did not make an Accurate Computation. So then
the Propofition holds true in every Ray apart, fo far as ap-
pears by Experiment. And that it is accurately true may
j^-- be demonftrated upon this Suppofition, That bodies refraEi
Light by aBing upon its (^ys in Lines Perpendicular to their
Surfaces. But in order to this Demonftration , I muft di-
ftinguifh the Motion of every Ray into two Motions, the
one Perpendicular to the refra6ting Surface, the other Pa-
rallel to it, and concerning the Perpendicular Motion lay
down the following Propofition.

If any Motion or moving thing whatfoever be incident ^,'5 ii^V
with any velocity on any broad and thin Space termina- O^.-^^^/^
ted on both fides by two Parallel Planes, and in its paflage ^-^^'^
through that fpace be urged perpendicularly towards the
further Plane by any force which at given diftances from
the Plane is of given quantities 3 the perpendicular Velo-
city of that Motion or Thing, at its emerging out of that
fpace, fliall be always equal to the Square Root of the
Summ of the Square of the perpendicular Velocity of
that Motion or Thing at its Incidence on that fpace 5
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 fame Propofition holds true of any Motion or
Thing perpendicularly retarded in its paflage through that
fpace, if inftead of the Summ of the two Squares you take
their difference. The Demonflration Mathematicians will ^h.Qi(iJ---P^7,.9nctk
eafily find out, and therefore I fhall not trouble the Rea- ^^^--i9--'^(^7-^'r2
der with it.

Suppofe now that a Ray coming mofl obliquely in the/w-. u
Line MC be refraded 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 fliall be refraded 5 let MC, AD, be
the Sines of incidence of the two Rays, and NG, EF, their
Sines of Rcfradion, and let the equal Motions of the In-
cident Rays be reprefented by the equal Lines M C and
AC, and the Motion MC being confidered as parallel to
the refrading Plane, let the other Motion AC be diftin-
guifhed into two Motions AD and DC, one of which
AD is parallel, and the other DC perpendicular to the re-
fracting Surface. In like manner, let the Motions of the
emering Rays be diftinguifh'd into two, whereof the per-

H pendicular

perpendicular ones are -^ CG and ^p CF. And if the

force of the refrading Plane begins to ad upon the Rays
either in that Plane or at a certain diftance from it on the
one fide, and ends at a- certain diftance from it^ on the
other fide, and in all places between thofe two Limits ads
upon the Rays in Lines perpendicular to that rafrading
Plane, and the Adions upon the Rays at equal diftances
from the refrading Plane Se equal, and at unequal ones ei-
ther equal or unequal according to any rate whatever 3-
that motion of the Ray which is Parallel to the refrading
Plane will fufifer no alteration by that force 3 and that mo-
tion which is perpendicular to it will be altered according
to the rule of the foregoing Propofition. If therefore for
the perpendicular Velocity of the emerging Ray CN you

write 5^ CG as a-bove, then the perpendicular Velocity

of any other emerging Ray CE which was ^ CF, will be

A/fC n

equal to the fquare Root of CD^ + -^^^ CGq, And

by fquaring thefe equals, and adding to them the Equals
AD^ and MC^ — CD^, and dividing the Summ.s by the
Equals CVq+ EVq and CGq ^- NG^, you will have

11^ equal to f|A Whence AD, the Sine of Incidence,

is to EF the Sine of Refradion, as MC to NG, that isy
in a given ratio. And this Demonftration being general,
without determining what Light is, or by what kind of
force it is refraded, or afluming any thing further than
that the refrading Body ads upon the Rays in Lines per-
pendicular to its Surface 5 I take it to be a very convincing
Argument of the full Truth of this Propofition.

[$9]

So th^n, if the ratio of the Sines of Incidence and Re-
fraftion of any fort of Rays be found in any one Cafe, 'tis
given in all Cafes 5 and this may be readily found by the
Method in the following Propoiidon.

PROP. VII. Theor. VI.

The TerfeBmi of Tele/copes is impeded by the dijferent ^fraU"
^ihiitty of the ^ys of Light,

^ %^ H E imperfedion of Telefcopes is vulgarly attri-
J^ buted to the fpherical Figures of the Glafles, and
therefore Mathematicians have propounded to Figure them
by the Conical Sediions. To ftiew that they are mifta-
ken, I have inferted this Propofition^ the truth of which
w^ill appear by the meafures of the Refra6lions of the feve-
ral forts of Rays 3 and thefe meafures I thus determine.

In the third experiment of the firft Book, where the re^
frat^ing Angle of the Prifm was 62' degrees, the half of
that Angle 3 i deg. \ j min. is the Angle of Incidence of
the Rays at their going out of the Glafs into the Air 3 and
the Sine of this Angle is Jj^, the Radius being loooo. ^'^^7- r^
When the Axis of this Prifm was parallel to the tlorizon,
and the Refraction of the Rays at their Incidence on this
Prifm equal to that at their Emergence out of it, I obfervcd
with a Quadrantthe Angle which the mean refrangible Rays
(thatis, thofe which wentto the middle oftheSuns colour-
ed Image ) made with the Horizon and by this Aagle and
the Sun's altitude obferved at the fame 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 /Ingle ad-
ded to the Angle of Incidence 3 i deg. 1 5 min. makes the

H z Angle

[do]

Angle of Refradion,which is therefore 5 ; dcg. ^ 5 min. and
its Sine 8047. Thefe are the Sines of Incidence and Rc-r
fraaion of the mean refrangible Rays, and their proportion
in round numbers is 20 to 3 1 . This Gl?Js was of a colouring-
chnincT to green. The laft of the Prifms mentioned in the
third Experiment was of clear white Glafs. Its refrading
Angle 63 1 degrees. The Angle v/hich the emergent Rays
contained, with the incident 45 deg. 50 min. The Sine of
half the firft Angle 5262. The Sine of half the Summ.
of the Angles 8157. And their proportion in round num-
bers 20 to 31 as before.

From the Length of the Image, which was about 9I or
10 Inches, fubdud its Breadth, which was 2^ Inches, and
the Remainder 7^ Inches would be the length of the Image
were the Sun but a point, and therefore fubtends the An-
gle which the moft and leaft refrangible Rays, when inci-
dent on the Prifm in the fame Lines, do contain with one
another after their Emergence. Whence this Angle is
2 des. o.' 7' For the diftance between the Image and the
Prifm where this Angle is made, was 1 8^ Feet, and at that
diftance the Chord 7^ Inches fubtends an Angle of 2 deg.-
o.' 7." Now half this Angle is the Angle which thefe e-
mergent Rays contain with the emergent mean refrangible
Rays, and a quarter thereof, that is 30. 2." m^^y be ac-
counted the Angle which they would contain^'whidk the
fame emergent mean refrangible Rays, were they co-inci-
dent to them within the Glafs and fufFered no other Re-
ff action then that at their Emergence. For if two equal
Refra6tions the one at the incidence of the Rays on the
Prifm the other at their Emergence, make half the Angle
2 deg. 0.7. then one of thofe Refradions will make
about a quarter of that Angle, and this quarter added to

and

ami fubduded from the Angle of Reftadion of the mean
refrangible Rays, which was 5; deg. ^5', gives the An-
gles or Refradion of the moft and leaft refrangible Rays
54 deg. 5' 2", and 5^ deg. 4' 58", whofe Sines are 8099
and 7995, the common Angle of Incidence being ^i deg.
15' and its Sine 51885 and thefe Sines in the leaft round
numbers are in proportion to one another as 78 and 77
to 50.

No\v if you fubdudl the common Sine of Incidence 50 ^ t^
from the Sines of Refraction 77 and 78, the remainders
27 and 28 fliew that in fmall Refradlions the Refradion
of the leaft refrangible Rays is to the Refradion of the moft
refrangible ones as 27 to 28 very nearly, and that the dif-
ference of the Refractions of the leaft refrangible and moft
refrangible Rays is about the 27^th part of the whole Re-
fraction of the mean refrangible Rays.

Whence they that are skilled in Opticks will eafily un- ^
derftand, that the breadth of the leaft circular fpace into
which Objecc-Glafles of Telefcopes can collect all forts of
Parallel Rays, is about the 27jth part of half the aperture -
of the Glafs, or 55 th part of the whole aperture 3 and
that the Focus of the moft refrangible Rays is nearer to the
Object-Glafs thanthe Focus of the leaft refrangible ones, by
about the 27^'th part of the diftance between the Object-
Glafs and the Focus of the mean refrangible ones.

And if Rays of all forts,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 moft refrangible Rays fliall be
nearer to the Lens than the Focus of the leaft refrangible
ones, by a diftance which is to the 27-~th part of the di-
ftance of the Focus of the mean refrangible Rays from the
Lens as the diftance between that Focus and the lucid

point

point from whence the Rays flow is to the diftance be-
tween that lucid point and the Lens very neaily.

"Now to examine whether the difference between the Re-
fradlions which the moft refrangible and the leaft refran-
gible Rays flowing from the fame point fiiffer in the Ob-
jed-Glaffes of Telefcopes and fuch like Glaffes, be fo great
as is here defcribed, I contrived the following Experi-
ment.

Exper. 1 6. The Lens which I ufed in the fecond and
eighth Experiments, being placed fix Feet and an Inch dif-
tant from any Objecfb, coUeded the Species of that Object
by the mean refrangible Rays at the diftance of fix Feet
and an Inch from the Lens on the other fide. And there-
fore by the foregoing Rule it ought to colled: the Species of
that Obje6l by the leaft refrangible Rays at the diftance of
fix Feet and 3 - Inches from the Lens, and by the moft re-
frangible ones at the diftance of five Feet and lof Inches
from it : So that between the t\^ o Places where thefe leaft
and moft refrangible Rays colledl the Species, there may
be the diftance of about ^\ Inches. For by that Rule, as
fix Feet and an Inch ( the diftance of the Lens from the
lucid Object ) is to twelve Feet and two Inches ( the di-
ftance of the lucid Object from the Focus of the mean re-
frangible Rays) that is, as one is to two, fo is the 27j.th
part of fix Feet and an Inch (the diftance between the Lens
and the fame Focus ) to the diftance between the Focus of
the moft refrangible Rays and the Focus of the leaft re-
frangible ones, which is therefore 5 ^ Inches, that is very
nearly 5 : Inches. Now to know whether this meafure
was true, I repeated the fecond and eighth Experiment of
this Book with coloured Light, which Vv^as lefs compound-
ed than that I there made ufe of : For I now feparat:ed the

hetero-

h eterogeneous Rays from one another by Ae Method I de-
fc ribed in the i ith Experiment, fo as to make a coloured
Spedrum about twelve or fifteen times longer than broad.
This Spedrum I caft on a printed book, and placing the
above-mentioned Lens at the diftance of fix Feet and an
Inch from this Spedrum to colled the Species of the illu-
minated Letters at the fame diftance on the other fide, I
found that the Species of the Letters illuminated with Blue
were nearer to the Lens than thofe 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 fo confufed and indiftind, that I could not read
then-^ : Whereupon viewing the Prifm, I found it was full
of Veins running from one end of the Glafs to the other ^
fo that the Refradian could not be regular. I took ano-
ther Prifm therefore which was free from Veins, and in-
ftead of the Letters I ufed two or three Parallel black Lines
a little broader than the ftroakes of the Letters, and caft-
ing the Colours upon thefe Lines in fuch manner that the
Lines ran along the Colours from one end of the Spedum
to the other, I found that the Focus where the Indigo, or
confine of this colour and Violet caft the Species of the
black Lines moft diftindly,to be about 4 Inches or 4^ nea^-^ sl^ //?/v*/ 1
er to the Lens than the Focus where the deqt^ Red^caff^^^*-^!*^^^. j
the Species of the fame black Lines moft diftindly. ^ I

The violet was fo faint and dark, that I could not
difcern the Species of the Lines diftinctly by that Co-
lour 3 and therefore confidering that the Prifm was made
©f a dark coloured Glafs inchning to Green, I took another
Pifm of clear white Glafs 3 but the Spedlrum of Colours
which this Prifm made had long white Streams of faint
Light fliooting out from both ends of the Colours, which
made me conclude that fomething was amifs 3 and view-

iiig the Prifm, 1 found two or three little Bubbles in the
Glafs which refraded the Light irregularly. Wherefore I
covered that part of the Glafs with black Paper, and let-
ting the Light pafs through another part of it which was
free from fuch Rubles, the Spedlrum of Colours became
free from thofe irregular Streams of Light, and was now
fuch as I defired. But ftill I found the Violet fo dark and
faint, that I could fcarce fee thrfp^cies of the Lines by the
Violet, and not at all by the d^^St part of it, which was
next the end of the Spedrum. I fufpeded therefore that
this faint and dark Colour might be allayed by that fcat-
tering Light which was refracted, and reflected irregularly
partly by fome very fmall Bubbles in the Glafles and
partly by the inequalities of their PoliiGb: which Light,
tho' it was but little, yet it being of a White Colour,
might fuffice to affedl the Senfe fo ftrongly as to difturb
the Phenomena of that weak and dark Colour the Violet,
and therefore I tried, as in the nth, i5th5Ki4th Experi-
ments, whether the Light of this Colour did not confift of
a fenfible mixture of heterogeneous Rays, but found it did
not. Nor did the Refractions caufe any other fenfible
Colour than Violet to emerge out of this Light, as they
would have done out of White Light, and by con-
fequence out of this Violet Light had it been fenfi-
bly compounded with White Light. And therefore I con-
cluded, that the reafon why I could not fee the Species of
the Lines diflinClly by this Colour, was only the darknefs
of this Colour and Thinnefs of its Light, and its dif-
tance from the Axis of the Lens 5 I divided therefore thofe
Parallel Black Lines into equal Parts, by which I might
readily know the diftances of the Colours in the Spedrum
from one another, and noted the diftances of the Lens
from the Foci of fuch Colours as caft the Species of the

Lines

Lines diftin^lly, and then confidered whether the diffe-
rence of thofe diftances bear fuch proportion to 5 '-Inches,
the greateft difference of the diftances which the Foci of
the deepeft Red and Violet ought to have from the Lens,
as the diftance of the obferved Colours from one another
in the Spedrum bear to the like diftance of the deepeft Red
and Violet meafured in the redlilinear fides of the Speifl-
rum, that is, to the length of thofe fides or excefs oF the
length of the Spectrum above its breadth. And my Ob-
fervations were as follows. n^jy ^^.^r^u^i-Al^^'ifsa^.

When I obferved and compared the deepeft lenfibleRed,
and thc^Colour in the confine of Green and Blue, which
at that^''i:cd:ilinear fides of the Spectrum was diftant from it
half the length of thofe fides, the Focus where the confine
of Green and Blue caft the Species of the Lines diftindtly
on the Paper, was nearer to the Lens then the Focus where
the Red caft thofe Lines di6lin6lly on it by about 2 - or
2 ^ Inches. For fontetimes the Meafures were a little grea-
ter, fomctimes a little lefs, but feldom varied from one
another above \ of an Inch. For it was very difficult to
define the Places of the Foci, without fome little Errors.
Now if the Colours diftant half the length of the Image,
( meafured at its redtilinear fides ) give i ^ or 2 | difference
of the diftances of their Foci from the Lens, then the Co-
lours diftant the whole length ought to give 5 or 5-' Inches
difference of thofe diftances.

But here it's to be noted, that I could not fee the Red
to the full End of the Spedrum, but only to the Center
of the Semicircle which bounded that End, or a little far-
ther 5. and therefore I compared this Red not with that Co-
lour which was exadlly in the middle of the Specftrum, or
confine of Green and Blue, but with that which verged a
little more to the Blue than to the Green ; And as I reck-

oned the whole length of the Colours not to be the whole
length of the Spectrum, but the length of its redlilinear
fideSj fo completing theSemicirlar Ends into Circles, when
cither of the obfcrved Colours fell within thofe Circles, I
meafured the diftance of that Colour from the End of the
Spedrum, and fubduding half the diftance from the mea-
fured diftance of the Colours, I took the remainder for
their correded diftance 3 and in thefe Obfervations fee
down this correded diftance for the difference of their di«
ftances from the Lens. For as the length of the redlilinear
fides of the Spedrum would be the whole length of all the
Colours, were the Circles of which ( as we (hewed ) that
Spedrum confifts contra6ted and reduced to Phyfical
Points, fo in that Cafe this corredlcd diftance would be the
real diftance of the obferved Colours.

When therefore I further obferved the deepeftfenfible Red^
and that Blue whofe correded diftance from it was ^ parts
of the length of the redilinear fides of the SpecStrum, the
difference of the diftances of their Foci from the Lens was
about ^- Inches, and as 7 to 1 2 fo is 3 -^ to 5 ^.

When I obferved the deepeft fenfible Red, and that Indi-
go whofe correded diftance was -^ or ^ of the length of the
redilinear fides of the Spedlrum, the difference of the di-
ftances of their Foci from the Lens, was about 3 "^ Inches,
and as 2 to 3 fo is 3 ^to 5I;.

When I obferved the deepeft fenfible Red, and that deep
Indigo whofe correded diftance from one another was ^ or
'^- of the length of the redilinear fides of the Spedum, the
difference of the diftances of their Foci from the Lens was
about 4 Inches 5 and as 3 to 4 fo is 4 to 5 \,

When I obferved the deepeft fenfible Red, and that part
of the Violet next the Indigo whofe correded diftance from
the Red was ^^ or ^ of the length of the redilmear fides of

the

the SpeArum, the difference of the diftances of their Foci
from the Lens was about 4^ Inches 3 and as 5 to 6, fo is
47 to 57. For fometimes when the Lens was advantagi-
oufly placed, fo that its Axis relpeded the Blue, and ail
things elfe were well ordered, and the Sun flhone clear, and
I held my Eye very near to the Paper on which the Lens
caft the Species of the Lines, I could fee pretty diftinctly
the Species of thofe Lines by that part of the Violet which
was next the Indigo 3 and fometimes I could fee them by
above half the Violet. For in making thefe Experiments
I had obferved, that the Species of thofe Colours only ap-
peared diffinct which were in or near the Axis of the Lens :
So that if the Blue or Indigo were in the Axis, I could fee
their Species diftinctly 5 and then the Red appeared much
lefs diftinct than before. Wherefore I contrived to make
the Spectrum of Colours fliorter than before, fo that both
its Ends might be nearer to the Axis of the Lens. And
now its length was about li Inches and breadth about -or
i of an Inch. Alfo inftead of the black Lines on which the
Spectrum was caft, I made one black Line broader than
thofe, that I might fee its Species more eafily 3 and this
Line I divided by fliort crofs Lines into equal Parts, for
meafuringthe diftances of the obferved Colours. And now
I could fometimes fee the Species of this Line with its divi-
lions almoft as far as the Center^ of the Semicircular Violet
End of the Spectrum, and made thefe further Obfervations.
When I obferved the deepeft feofible Red, and that part
of the Violet whofe correded diftance frcni it v/as about
^ Parts of the re6tilinear fides of the Spedrum the difference
of the diftances of the Foci of thofe Colours from the Lens,
was one time 4-% another time 4^, anothertime 41^ Inches,
and as 8 to 9, fo are 4^, 4-;, 4I, to 5 -;, 5^^ 5IJ refpedively.

I 2 When

[68]

When I obferved the deepeft fenfible Red, and deeped
fenfible Violet, (the corrected diftance of which Colours
when all things were orderedto the bed advantage, andthe
Sun flione very clear, was about ^ or ^ parts of the length
of the rectilinear fides of the coloured Spectrum, ) I found
the difference of the diftances of their Foci from the Lens
fometimes 4-' fometimes 5^, and for the moft part 5 Inches
or thereabouts : and as 11 to 1 2 or 15 to |6, fo is five
Inches to 5 ^- or 5 i Inches.

And by this progreffion of Experiments I fatisfied my
felf, that had the light at the very Ends of the Spectrum been
ftrong enough to make the Species of the black Lines ap-
pear plainly on the Paper, the Focus of the dcepeft Vio-
let would have been found nearer to the Lens, than the Fo-
cus of the deepeft Red, by about y- Inches at leaft. And
this is a further Evidence, that the Sines of Incidence and
Refraction of the feveral forts of Rays, hold the fame pro-
portion to one another in the fmalleft Refractions which
they do in the greatefl:.

My progrefs in making this nice and troublefome Expe-
riment I have fet down more at large, that they that fhall
try it after me may be aware of the Circumfpe<5tion re-
quifite to make it fucceed well. And if they cannot make
it fucceed fo well as I did, they may notwithftanding col-
lect by the Proportion of the diftance of the Colours in the
Spedtrum, to the difference of the diftances of their Foci
from the Lens, what would be the fucccfs in the more di-
ftant Colours by a better Trial. And yet if they ufe a
broader Lens than I did, and fix it to a long ftreight Staff
by means of which it may be readily and truly directed to
the Colour whofe Focus is defired, I queftion not but the
Experiment will fucceed better with th^m than it did with
me. Fox I directed the Axis as nearly as I could to the

middle

L^9l

middle of the Colours, and then the faint Ends of the
Spedrum being remote from the Axis, caft their Species lefs
diftin(5lly on the Paper than they would have done had the
Axis been fucceffively diredred to them.

Now by what has been faid its certain, that the Rays
which differ in refrangibility do not converge to the fame
Focus, but if they flow from a lucid point, as far from
the Lens on one fide as their Foci are one the other, the
Focus of the moft refrangible Rays fliall be nearer to the
Lens than that of the leaft refrangible, by above the four-
teenth part of the whole diftance: and if they flow from a lu-
cid point, fo very remote from the Lens that before their
Incidence they may be accounted Parallel, the Focus of the
moil refrangible Rays fhall be nearer to the Lens than the
Focus of the leafl: refrangible, by about the 27th or 28th part
of their whole diflance from it. And the Diameter of the
Circle in the middle fpace between thofe two Foci which
they illuminate when they fall there on any Plane, perpen-
dicular to the Axis (which Circle is the leafl: into which
they can all be gathered) is about the 55th part of the Dia-
meter of the aperture of the Glafs. So that 'tis a wonder
that Telefcopes reprefent Objects fo diflrindt as they do. But
were all the Rays of Light equally refrangible, the Error
arifing only from the fphericalnefs of the Figures of Glafles
would be many hundred times lefs. For if-the Objed-
Glafs of a Telefcope be Plano-convex, and the Plane fide
be turned towards the Objed, and the Diameter of the
Sphere whereof this Glafs is a fegment,be called D, and the
Semidiameter of the aperture of the Glafs be called S, and
the Sine of Incidence out of Glafs into Air, be to the Sine of
Refradion as I to R: the Rays which come Parallel to the
Axis of the Glafs, fliall in the Place where the Image of the
Object is mofl: diftindly made, be fcattered all over a little

Circle

[7°]

Circle whofe Diameter is j ^ ^ — ^^ very nearly, as I ga-
ther by computing the Errors of the Rays by the method
of infinite Series, and rejeding the Terms whofe quanti-
ticies are inconfiderable. As for inftance, if the Sine of In-
cidence I, be to the Sine of Refradion R, as 20 to ^ i, and
if D the Diameter of the Sphere to which the Convex fide
of the Glafs is ground, be 100 Feet or 1200 Inches, and
S the Semidiameter of the aperture be two Inches, the

Diameter of the little Cirde ( that is j^D^Ii/. ) ^^'^ ^^

21 •><

20 >^ 1200 ■?< 120Q ^ 3

Diameter of the little Circle through which thefe Rays are
Scattered by unequal refrangibility, will be about the 55th
part of the aperture of the Objed-Glafs which here is four
Inches. And therefore the Error arifing from the fpherical
Figure of the Glafs, is to the Error anfing from the diffe-
rent Refrangibility of the Rays, as j^^ to ^ that is as i
to 8151 : and therefore being in Comparifon fo very little,
deferves not to be confidered.
# But you will fay, if the Errors caufed by the different re-
frangibility be fo very great, how comes it to pafs that Ob-
jeds appear through Telefcopes fo diflind as they do ? I an-
fwer, 'tis becaufe the erring Rays are not fcattered uniform-
ly over all that circular fpace, but collected infinitely more
denfely 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, fo as at the Circumference to
become infinitely rare 3 and by reafon of their rarity are
-. not fl:rong enough to bevifible, unlefs in the Center and ve-

^S* '^'^' j.y near it. Let AD E reprefent one of thofe Circles de-
fcribed vv^ith the Center C and Semidiameter AC, and let
BFG be a fmaller Circle concentric to the former, cutting

with

[71]

With its Circumference the Diameter AC in B^ and b^fect
AC in N, and by my reckoning the denfity of the Light
in any place B will be to its denfity inN, as AB to BC3
and the whole Light within the leffer Circle BFG, will be
to the whole Light within the greater AED, as the Excefs of
the Square of AC above the Square of AB, is tojthe Square
of AC. As if BC be the fifth part of AC, the Light will be
four times denfer in Bthan in N, and the whole Lightwith-
in the lefs Circle,will be to the whole Light within the grea-
ter, as nine to twenty five. Whence it's evident that the
Light within the lefs Circle, muflftrike the fenfe much more
ftrongly, than that faint and dilated light round about be-
tween it and the Circumference of the greater.

But its further to be noted, that the moft luminous of , e , ^^^
the prifmatick Colours are the Yellow and Orange. ^\\t{zW. JLr^-,u2^(SZe^y^H
affea the Senfes more ftrongly than all the reft" together, and -^-^^-f^"^.^"'^^^
next to thefe in ftrength are the Red and Green. The Blue
compared with thefe is a faint and dark Colour, and the In-
digo and Violet are much darker and fainter, fo that thefe
compared with the ftronger Colours are little to be regard-
ed. The Images of Objedls 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 thofe Rays
which are in the middle of the Orange and Yellow 3 there
where the Colour is moft luminous and fulgent, that is in
the brighteft Yellow, that Yellow which inclines more to
Orange than to Green. And by the Refradlion of thefe
Rays ( whofe Sines of Incidence and Refradlion in Glafs
are as 17 and 11) the Refradion of Glafs and Cryftal for
optical ufes is to be meafured. Let us therefore place the
Image of the Objed in the Focus of thefe Rays, and all the
Yellov/ and Orange will fall within a Circle, whofe Dia-
meter is about the zjoth part of the Diameter of the aper-
ture

[72]

ture of the Giafs. 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, ) a-
bout three fifth parts of the Light of thefe two Colours will
fall within the fame Circle,and two fifth parts will fall with-
out it round about 5 and that which falls without will be
ipread through almoft as much more fpace as that which
falls within, and fo in the grofs be almoft three times ra-
rer. 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 fpread through about four
or five times more fpace than that which fall$within5 and fo
in the grofs be rarer, and if compared with the whole Light
within it, will be about 25 times rarer than all that taken in
the grofs 3 or rather more than ^ o or 40 times rarer, be-
caufe the deep red in the end of the Spe6trum of Colours
made by a Prifm is very thin and rare, and the Willow Green
is fomething rarer than the Orange and Yellow. The Light
of thefe Colours therefore b£ing fo very much rarer than that
within the Circle, will fcarce affedt the Senfe efpecially fince
the deep Red and Willow Green of this Light, are much
darker Colours then the reft. And for the fame reafon the
Blue and Violet being much darker Colours than thefe, and
much more rarified, may be neglected. For the denfe and
bright Light of the Circle, will obfcure the rare and weak
Light of thefe dark Colours round about it, and render them
almoft infenfible. The fenfible Image of a lucid point is
therefore fcarce broader than a Circle Vv^hofe Diameter is
the 250th part of the diameter of the aperture of the Object
Glafs of a good Telefcope, or not much broader, if you
except a faint and dark mifty light round about it, which
a Spedator will fcarce regard. And therefore in a Telefcope

whofc

[73]

whofe aperture is four Inches, and length an hundrec! Feety
it exceeds not 2 '45", or 3". And in a Telcfcope whofe
aperture is two Inches, and length 20 or 30 Feet, it may
be 5 ' or 6" and fcarce above. And this Anfwers well to
Experience : For fome Aflronomers have found the Dia-
meters of the fixt Stars, in Telefcopes of between twenty
and fixty Feet in length, to be about 4" or 5" or at mofl:
^ in Diameter. But if the Eye-Glafs be tindled faintly
with the fmoke of a Lamp or Torch, to obfcure the Light
of the Star, the fiinter Light in the circumference of the
Scar ceafes to be vifible, and the Star (if the Glafs be fuffici-
ently foiled with fmoke) appears fomething more like a Ma-
thematical Point. And for the fame reafon, the enormous
part of the Light in the Circumference of every lucid Point
ought to be lefs difcernable in fliorter Telefcopes than in
longer, becaufe the fliorter tranfmit lefs Light to the Eye. ^^S^V^^w ^^*H''
Now if we fuppofe the fenfible Image of a lucid poinr/^;/* p^/^r^ tT^^!^
to be even 250 times narrower than the aperture of thei^^^^'l^f^'J
Glafs: yet were it not for the different refrangibility of ^z^^f/^^ PJih'^\
Rays, its breadth in an 100 Foot Telefcope whofe aperture.r^'^/.^^'^^v£|,
is 4 Inches would be but ,-^;^ parts of an Inch, as is ma-^^f^t fq^^j
nifcfl by the foregoing Computation. And therefore '^^^^xJ£l'^C^'C^

this Cafe the greatefl: Errors arifing from the fpherical Figure •^"^*||^ <^i^^^v^^-mU)

of the Glafs, would be to the greatefl fenfible Errors ari-«''^i^^'^^/2/JJ^i

fing from the different refrangibility of the Rays as ■^^^^'^f^'^

to ^^ at mofl, that is only as i to 1826. And this fufli-

ciently fliews that it is not the fpherical Figures of GlafTes

but the different refrangibility of the Rays which hinders the

perfection of Telefcopes.

There is another Argument by which it may appear that
the different refrangibility of Rays, is the true Caufe of the
impe^fedion of Telefcopes. For the Errors of the Rays
arifing from the fpherical Figures of Objed- GlafTes, are as

K the

972 /^im.

[74- 1

the Cubes of the apertures of the Objeft-Glafles^and thence
to make Telefcopes of various lengths, magnify with equal
diftindnefs, the apertures of the Objed-GlafTes, and the
Charges or magnifying Powers, ought to be as the Cubes of
the fquare Roots of their lengths 5 which doth not anfwer
to Experience. But the errors of the Rays arifing from,
the different refrangibiHty, are as the apertures of the Ob-
jed-»Glaffes, and thence to make Telefcopes of various
lengths, magnify with equal diftindnefs, their apertures and
charges ought to be as the fquare Roots of their lengths 5
and this anfwcrs to experience as is well known. For in-
ftance, a Telefcope of 64 Feet in length, with an aperture
of 1' Inches, magnifies about i 20 times, with as much dif-
tindnefs as one of a Foot in length, with ^ of an Inch aper-
ture, magnifies 1 5 times.

Now were it not for this different refrangibility of Rays,
Telefcopes might be brought to a greater Perfedion than
we have yet defcribed, by compofing the Objed^Glafs of I
two Glafles with Water between them. Let ADFC repre- I
fig, iS.f^nt the Gbjed-Gla(s compofed of two Glaffes ABED and |
and BEFC, alike convex on the outfides AGD and CHF,
and alike concave on the infides BME, BNE, with Water
in the concavity BMEN. Let the Sine of Incidence out of
Glafs into Air be as I to R and out of Water into Air as K
to R, and by confequence out of Glafs into Water, as I to
K : and let the Diameter of the Sphere to which the convex
fides AGD and CHF are ground be D, and the Diameter
of the Sphere to which the concave fides BME and BNE
are ground be to D, as the Cube Root of KK— KI to the
Cube Root of RK— RI: and the Refradions on the con-
cave fides of the Glaffes, will very much corred the Errors
of the Refradions on the convex fides, fo far as they arife
from the fphericalnefs of the Figure. And by this means

might

C 75 ]

might Telefcopes be brought to fufficientperfeaion, wereit
not forthediflferentreftangibiUty offeveralforsofRays. But
by reafon of this difFerent refrangibility, I do not yet fee any
other means of improving Telefcopes by Refractions alone
than that of increafing their lengths, for which end the late y
contrivance of Hugmms feems well accommodated. For
very long Tubes are cumberfome, and fcarce to be readily
managed, and by reafon of their length are very apt to
bend, and fhake by bending fo as to caufe a continual
trembling in the Objedls, whereby it becomes difficult to
fee them diftindlly : whereas by his contrivance the Glafles
are readily manageable, and the Obje6t-Glafs being fixt up-
on a ftrong upright Pole becomes more fteddy.

Seeing therefore the improvement of Telefcopes of given
lengths by Refra<5lions is delperate ^ I contrived heretofore a
Perlpeiflive by reflexion, ufing inftead of an Objed: Glafs
a concave Metal. The diameter of the Sphere to which
the Metal was ground concave was about 2 5 Englifh Inches,
and by confequence the length of the Inftrument about fix
Inches and a quarter. The Eye-Glafs was plano-convex,
and the Diameter of the Sphere to which the convex fide was
ground was about i of an Inch, or a little lefs, and by con-
fequence it magmfied between \ o and 40 times. By ano-
ther way of meafuring I found that it magnified about
\ 5 times. The Concave Metal bore an aperture of an Inch
and a third part 3 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 Eye-Glafs and the
Eye, and perforated in the middle with a little round hole
for the Rays to pafs through to the Eye. For this Circle
by being placed here, ftopt much of the erroneous Light,
which otherwife would have difturbed the Vifion. By com-
paring it with a pretty good Perfpedive of four Feet iti

K 2 length.

length, made with a concave Eye-Glafs, I could read at a
greater diftance with my own Inftrument than with the
Glafs. Yet Objeds appeared much darker in it than in the
Glafs, and that partly becaufe more Light was loft by re-
flexion in the Metal, then by refradion in the Glafs, and
partly becaufe my Inftrument was overcharged. Had it
magnified but ^oor 2 5 times it would have made the Object
appear more brisk and pleafant. Two of thefelmade about
1 6 Years ago, and have one of them ftill by me by which
I can prove the truth of what I write. Yet it is not fo good
as at thefirft. For the concave has been divers times tar-
niflied and cleared again, by rubbing it with very foft Lea-
ther. When I made thefe, an Artift in honion undertook
to imitate it 5 but ufing another way of poliflhing them
than I did, he fell much fliort of what I had attained to
as r afterwards underftood by difcourfing the under- Work-
man he had imployed. The Polifli I ufed was on this man-
ner. I had two round Copper Plates each fix Inches in
Diameter, the one convex the other concave, ground ve-
ry true to one another. On the convex I ground the Ob-
3e(5l-Metal or concave which v/as to be polifh'd, till it had
taken the Figure of the convex and was ready for a Polidi.
Then I pitched over the convex very thinly, by dropping
melted pitch upon it and warming it to keep the pitch
foft, whilft I ground it with the concave Copper wetted to
make it fpread evenly all over the convex. Thus by work-
ing it well I made it as thin as a Groat, and after the con-
vex 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 waflhing it from all its grofler Particles, and laying a lit-
tle of this upon the pitch, I ground it upon the Pitch with
the concave Copper till it had done making a noife^ and
then upon the Pitch I ground the Objeft-Metal with a brisk

Motion

[77]

Motion, for about two or three Minutes of time, leaning
hard upon it. Then I put frefli Putty upon the Pitch and
ground it again till it had done making a noife, and after-
wards ground the Objed Metal upon it as before. And
this Work I repeated till the Metal was polifhed, grinding
it the laft time with all my ftrength for a good while toge-
ther, and frequently breathing upon the Pitch to keep it
moift without laying on any more freflh Putty. The Ob-
jedi'Metal was two Inches broad and about one third part
of an Inch thick, to keep it from bending. I had two of
thefe Metals, and when I had poliflied them both I tried
which was beft, and ground the other again to fee if I could
make it better than that which I kept. And thus by many
Trials I learnt the way of polifliing, till I made thofe two
refleding Pelpe6lives I fpake of above. For this Art of
polifliing will be better learnt by repeated Practice than by
my defcription. Before I ground the ObjeA Metal on the
Pitch, I always ground the Putty on it with the concave
Copper till it had done making a noife, becaufe if the Par-
ticles of the Putty were not by this means made to ftick
faft in the Pitch, they would by rolling up and down grate
and fret the Objed Metal and fill it full of little holes.

But becaufe Metal is more difficult to polifli than Glafs
and is afterwards very apt to be fpoiled by tarnifliing, and
reflects not fo much Light as Glafs quick^filvered over does:
I would propound toufeinfteadof theMetal, a Glafs ground
concave on the forefide, and as much convex on the back-
fide, and quick-filvered over on the convex fide. The Glafs
muft be every where of the fame thicknefs exadly. Other-
wife it will make Objeds look coloured and indiftind. By
fuch a Glafs I tried about five or fix Years ago to make
a refleding Telefcope of four Feet in length to magnify a-
bout 150 times, and I fatisfied my felf that there wants no-
thing

[78]

thing but a good Attift to bring the defign to Perfe£i:iOft.
For the Glafs being wrought by one of our London Artifts
after fuch a manner as they grind Glaffes for Telefcopes,
tho it feemed as well wrought as the Objedl Glafles ufe to
be, yet when it was quick- filvered, the reflexion difcovered
innumerable Inequalities all over the Glafs. And by reafon
of thefe Inequalities, Obje6ls appeared indiftindl in this In-
ftrument. For the Errors of refleded Rays caufed by any
Inequality of the Glafs, are about fix times greater than the
Errors of refraded Rays caufed by the like InequaUties. Yet
by this Experiment I fatisfied my felf that the reflexion on
the concave fide of the Glafs, which I feared would difliurb
the vifion,didno fenfible prejudice to it, and by confequencc
that nothing is wanting to perfe6t thefe Telefcopes, but
good Workmen who can grind and polifli Glafles truly (phe- 1
rical. An Obje^t-Glafs or a fourteen Foot Telefcope, made
by one of our London Artificers, I once mended confidera-
bly, by grinding it on Pitch with Putty, and leaning ve-
ry eafily on it in the grinding, lefl: the Putty fliould fcratch
it. Whether this way may not do well enough for polifli-
ing thefe refleding Glafles, I have not yet tried. But he
that fliall try either this or any other way of polifliing which
he may think better, may do well to make his Glafles rea-
dy for polifliing by grinding them without that violence,
wherewith our London Workmen prefs their Glafles in grind-
ing. For by fuch violent preflure, Glafles are apt to bend
a little in the grinding, and fuch bending will certainly fpoil
their Figure. To recommend therefore the confideration
of thefe refleding Glafles, to fuch Artifts as are curious in
figuring <jlafles, I fliall defcribe this Optical Inftrument in
the following Propofition.

PROP.

[79]
VROV. VII. Prob. II.

To p?orlen Tele/copes.

TET ABDC reprefent a Glafs fpherically concave on p;^^ ^^
^ the forefide AB, and as much convex on the back-
fide CD, fo that it be every where of an equal thicknefs. Let
It not be thicker on one fide than on the other, lefl: it make
Objeds appear coloured and indiftind:, and let it be very
truly wrought and cjuick-filveredoveron the backfide ^ and
fet in the Tube VXYZ which muft be very black within.
Let EFG reprefent a Prifm of Glafs or Cryftal placed near
the other end of the Tube, in the middle of it, by means of
a handle of Brals or Iron FGK, to the end of which made
flat it is cemented. Let this Prifm be redangular at E, and
let the other two Angles at F and G be accurately equal to
each other, and by confequence equal to half right ones, and
let the plane fides FE and GE be fquare, and by confe-
quence the third fideFG a re(5tangular parallelogram, whofe
length is to its breath in a fubduplicate proportion of two
to one. Let it be fo placed in the Tube, that the Axis of
the Speculum may pafs through the middle of the fquare
fide EF perpendicularly, and by confequence through the
middle of the fide F G at an Angle of 45 degrees, and let the
fide EF be turned towards the Speculum, and the diftance
of this Prifm from the Speculum be fuch that the Raysof the
light PQ, RS, dec. which are incident upon the Speculum in
Lines Parallel to the Axis thereof, may enter the Prifm at
the fide EF, and be reiecSed by the fide F G, and thence
go out of it through the fide GE, to the point T which
muft be the common Focus of the Speculnm ABDC, and of
a Plano-convex Eye-Glafs H, through which thofe Rays
muft pafs to the Eye. And let the Rays at their coming

out

[8o]

out of the Glafs pafs through a fmall round hole, or aper-
ture made in a little Plate of Lead, Brafs, or Silver, where-
with the Glafs is to be covered, which hole muft be no
bigger than is neceflary for light enough to pafs through.
For fo it will render the Object diftind, 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 Inftrument well made if it be 6 Foot long, ( reckoning
the length from the Speculum to the Prifm, and thence to
the Focus T ) will bear an aperture of 6 Inches at the Spe-
culum, and magnify between two and three hundred times.
But the hole H here limits the aperture with more advan-
tage, then if the aperture was placed at the Speculum. If
che Inftrument be made longer or fliorter, the aperture muft
be in proportion as the Cube of the fquare Root of the
length, and the magnifying as the aperture. But its con-
venient that the Speculum be an Inch or two broader than
the aperture at the leaft, and that the Glafs of the Speculum
be thick, that it bend not in the working. The Prifm EFG
muft be no bigger than is neceflary, and its back fide FG
muft not be quick-filvered over. For without quick-filver
it will refled all the Light incident on it from the Speculum.
In this Inftrument the Objedl will be inverted, but may
be eredted by making the fquare fides EF and EG of the
Prifm EFG not plane but fpherically convex, that the Rays
may crofs as well before they come at it as afterwards be-
tween it and the Eye- Glafs. If it be defired that the Inftru-
ment bear a larger aperture, that may be alfo done by com-
pofing the Speculum of two Glaffes with Water between
•them. ^^'^ V " ^ /^^?^ «/ ^y/^ ^v/f./^;. W ^p^ -^-z ^c^j'-^^ j^ 9^r ^c

■\&l^cW ^^^^•t<-^n^ ca^ (^mo cfn^Sf^j ^ 9^^C 'Jay ij ^^'i^ Jt^J^ru^C^y />«-^^^ Ay

Book I 1'l.u.I.Parfl.

Fig. 3.

Fig.4.

y Q: ^7

Fi^.5.

(I Eyx

t y

Fig. 7

Fie: 8.

Book I. Plate!. Part I.

FicT.p.

'€

lO. A

-%^

Bookl.Platein. Pai-tl.

Fig 17.

BoOK,l. Plate, IV. Part, I.
F.

G H I K L M

./ / .- J c f

p o.::::::n:::z:::n:::::::::n::::.::::D:::zD t
e k i k. I 771

Fl^. :26

DEF

■■\m.

Fie;. 19-

C80 •

THE

FIRST BOO

O P T I C K S

•

PART 11.

PROP. I. THEOR. L

The Thanomena of Colours in refraSed or refleBed Lioht
are not caujed i>j new modifications of the Light variouf^
ly imfreftj according to the variom terminations of the
LtQ-ht and Shadow,

-""C

The Proof i^y Experimenu.

EX PER. L

FOR if the Sun fliine into a very dark Chamber jr/o-
through an oblong Hole F, whofe breadth is the "^
fixth or eighth part of an Inch, or fomething lefs; and
his Beam FH do afterwards pafs firft through a very
large Prifm ABC, diftant about 20 Feet from the

L Hole,

Hole, and parallel to it, and then (with its white part)
through an oblong HoleH^ whofe breadth is about
the fortieth or fixtieth part of an Inch.^ and which is
made in a black opake Body G I, and placed at the
diftance of two or three Feet from the Prifm, in a pa-
rallel iituation both to the Prifm and to the former
Hole, and if this white Light thus tranfmitted through
the Hole H, fall afterwards upon a white Paper pt^
placed after that Hole H^ at the diftance of three or
four Feet from it^ and there paint the ufual Colours of
the Prifm, fuppofe red at t^ yellow at s, green at r,
blue at q, and violet at p ; you may with an iron Wire,
or any fuch like (lender opake Body, whofe breadth is
about the tenth part of an Inch, by intercepting the rays
at k, 1, m, n or o, take away any one of the Colours
at t, s, r, q or p, whilft the other Colours remain up-
on the Paper as before ; or with an obftacle fomething
bi^er you may take away any two, or three, or four Co-
lours together, the reft remaining: So that any one of
the Colours as well as violet may become outmoft in
the confine of the fhadow towards p, and any one of
them, as well as red may become outmoft in the confine
©f the fhadow towards t, and any one of them may alfo
border upon the fhadow made within the Colours by
the obftacle R intercepting fome intermediate part of;
the Light ; and, laftly, any one of them by 4 being
left alone may border upon the ftiadow on either hando -
AH the Colours have themfelves indifferently to any,
confines of fhadow, and therefore the differences of theie
Colours from one another, do not arife from the diffe-
rent confines of fhadow, whereby Liglit is varioufly
modified as has hitherto been the Opinion of Philofo-

pherSo

[83].

phers. In trying thefe things 'tis to be obferved, that
by how much the Holes F and H are narrower^ and the
intervals between them, and the Prilm greater, and the
Chamber darker, by fo much the better doth the Ex-
periment lucceed ; provided the Light be not fo far
diminifhed, but^ that the Colours at pt be fufficiently
viiible. To procure a Prifm of folid Glafs large enough
for this Experiment will be difficult, and therefore a
prifmatick Veflel muft be made of polifhed Glafs-plates
cemented together, and filled with TV'ater* ^ c^^^ ^.l.

EX PER. 11.

The Sun's Light let into a dark Chamber through Vig^ ^^
the round Hole F, half an Inch wide, paffed firft through
the Prifm ABC placed at the Hole, and then through
a Lens P T fomething more than four Inches broad, and
about eight Feet diftant from the Prifm,and thence coa*
verged to O the Focus of the Lens diftant from it about
three Feet, and there fell upon a white Paper DE. If
that Paper was perpendicular to that Light incident up--
on it, as 'tis reprelented in the pofture D E, all the Co-
lours -upon it at O appeared w^^iite. But if the Paper
being turned about an Axis parallel to the Prifm, be-
came very much inclined to the Light as 'tis reprefen-
ted in the pofitions d.e and ^^ _; the fame Light in the
one cafe appeared yellow and red, in the other blue.
Here one and the fame part of the Light in one and the
•fame place, according to the various inclinations of the
Paper, appeared in one cafe white, in another yellox^
or red, in a third blue, whilft the confine of Light and

L 2 Shadow-

[B4]

Shadow, and the refraftions of the Prifm in all thefe
cafes remained the fame.

EX PER. III.

Such another Experiment may be more eafily tried^
as follows. Let a broad beam of the Sun's Light coming
into a dark Chamber through a Hole in the Window
Khut be refrafted by a large Prifm ABC, whofe re^
ffadincf Angle C is more than 60 degrees, and fo. foon
as it comes out of the Prifm let it fall upon the white
Paper DE glewed upon a ftitf plane, and this Light^
when the Paper is perpendicular to it, as 'tis reprefen--
ted in DE, will appear perfectly vyhiteupon the Paper,
but when the Paper is very much inclined to it in fuch
a manner as to keep always parallel to the Axis of the
Prifm, the whitenefs 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 pofture de^ or into blue and violet, as
in the pofture ^^ And if the Light before it fall upon
the Paper be twice refrafted the fame way by two pa-
rallel Prifms, thefe Colours will become the more con>
fpicuous. Here all the middle parts of the broad beam
of white Light which fell upon the Paper, did without
any confine of fhadow to modify it, become coloured
all over with one uniform Colour, the Colour being al-
ways the famie in the middle of the Paper as at the
edges, and this Colour changed according the various
obliquity of the retleding Paper, without any change
in the refradions or fhadow, or in the Light which
fell upon the Paper, And therefore thefe Colours are

to be derived from forrte other caufe than the new mo-
difications of Light by refraftions and Ihadows.

If it be asked, What then is their caufe? I anfwer, ^
That the Paper in the pofture de , being more ob-
lique to the more refrangible rays than to the lefs re-
frangible ones, is more ftrongly illuminated by the lat-
ter than by the former, and therefore the lefs refran-
gible rays are predominant in the refleded Light. And
wherever they are predominant in any Light they tinge
it with red or yellow, as may in fome meafure appear by
the firft Propofition of the firft Book,and will more fully
appear hereafter. And the contrary happens in the
polture of the Paper ^^ , the more refrangible rays be-
ing then predominant which always tinge Light with
blues and violets.-

EX PER. iv;

The Colours of Bubbles with which Children play^^'
are various, and change their iituation variouily, with'
out any refpeft to any confine of Ihadow. If fuch a
Bubble be covered with a concave Glafs, to keep it from
being agitated by any wind or motion of the Air, the
Colours will flowly and regularly change their iitua-
tion, even whilft the Eye, and the Bubble, and all Bo-
dies which emit any Light, or caft any fliadow, re-
main unmoved. And therefore their Colours arife from
fome regular caufe which depends not on any confine of
ihadow. What this caufe is will be ftiewedin the next
Book.

To thefe Experiments may be added the tenth Ex-
periment of the firft Book, where the Sun's Light in a
dark Room being traje£ted through the parallel fuperfi-
cies of two Prifms tied together in the form of a Paral-
lelopide, became totally of one uniform yellow or red
Colour, at its emerging out of the Prifms. Here, in
the produdion of thefe Colours, the confine of fhadow
can have nothing to do. For the Light changes from
white to y ellow,orange and red fucceffively,withoutany
alteration of the confine of fhadow: And at both edges of
the emerging Light where the contrary confines of fha-
dow ought to produce different etfeds, the Colour is
one and the fame, whether it be white, yellow, orange
or red : And in the middle of the emerging Light,
where there is no confine of fhadow at all, the Colour
is the very fame as at the edges, the whole Light at its
very firfl emergence being of one uniform Colour, whe-
ther white, yellow, orange or red, and going on thence
perpetually without any change of Colour, Inch as the
confine of fhadow is vulgarly fuppofed to work in tc-
fraited Light after its emergence. Neither can thefe
Colours arife from any new modifications of the Light
by refrafldons, becaufe they change fucceffively from
wMte to yellow, orange and red, while the refradions
remain the faine, aiad alfo becaufe the refraftions are
made contrary ways by parallel fuperficies which de-
flroy on€ anothers effefts. They arife not therefore
from any modifications of Light made by rcfradions
itnd fhadow^, but have fome other caufe. What that
caufe is we fhewed above in this tenth Experiment,
and need not here repeat it.

There

^7l

There is yet another material circuinftance of this
Experiment* For this emerging Light being by a third Fig. 22.-
Prifm H I K refraded towards the Paper PT-> and there Tart i .
painting the ufual Colours of the Prifm^ red^ yellow,
green, blue, violet : If thefe Colours arofe from the
refradions of that Prifm modifying the Light, they
woald not be in the Light before its incidence on that
Prifm. And yet in that Experiment we found that
when by turning the two firft Prifms about their com-
mon Axis all the Colours were made to vanifli but the
red ; the Light which makes that red being left alone,
appeared of the very fame red Colour before its inci«
dence on the third Prifm. And in general we find by
other Experiments that when the rays which differ in
refrangibility are feparated from one another, and any
one fort of them is confidered apart, the Colour of the
Light which they compofe cannot be changed by any
refraftion or reflexion whatever, as it ought to be were
Colours nothing elfethan modifications of Light caufed
by refraftions, and reflexions, and lliadows. This un»
changeablenefs of Colour I am now to defcribe in the
following Propofition,

PROP. IL THEOR. IL

j^ll honiogeneal Light has its frofer Colour anfwering to^
its degree of refrangiiility^ and that Colour cannot he
changed by rejlexions and reffa<^ionSs

In the Experiments of the 4.th Propofition of the firft
Book, when I had feparated the heterogeneous rays
from one another, the Spectrum pt formed by the fepa-
rated

[ 88 ]

rated rays^ did in the progrefs from its end p, on which
the moft refrangible rays fell, unto its other end t, on
which the leaft refrangible rays fell, appear tinged with
this Series of Colours, violet, indico, blue, green, yel-
low, orange, red, together with all their intermediate
degrees in a continual fucceffion perpetually varying :
So that there appeared as many ii^gi-^ees ^ Colours, as M
there were forts of rays differing in refrangibility. /?r /-^y/^

EXP E R. V.

Now that thefe Colours could not be changed by re-
fradion, I knew by refrading with a Prifm fometimes
one very little part of this Light, fometimes another
very little part, as is defcribed in the 1 2 th Experiment
of the fir ft Book. For by this refradion the Colour of
the Light was never changed in the leaft. If any part
of the red Light was refraded, it remained totally of
the fame red Colour as before. No orange, no yel-
lov/, no green, or blue, no other new Colour was pro-
duced by that refraflion. Neither did the Colour any
ways change by repeated refradlions, but continued al-
ways the fame red entirely as at firft. The like con-
ftancy and immutability 1 found alfo in the blue, green,
and other Colours. So alfo if I looked through a Prifm
upon any body illuminated with any part of this homo-
geneal Light, as in the i^th Experiment of the firft
Book is defcribed j I could not perceive any new Co-
lour generated this way. All Bodies illuminated with
compound Light appear through Prifms confufed ( as
was faid above ) and tinged with various new Colours^
but thofe illuminated with homogeneal Light appeared

throuiih

through Priinis neither lets dlftinfl:., nor otherwife co-
loured, than when viewed with the naked Eyes. Their
Colours were not in the leaft changed by the refradion
of the interpofed Priim. I fpeak here of a fenfible
change of Colour : For the Light which I liere call ho»
mogeneal, being not abfolutely homogeneal, there ought
to arife fome little change of Colour from its heteroge-
neity. But if that heterogeneity was fo little as it might
be made, by the faid Experiments of the fourth Propo-
fition, that change was not fenfible, and therefore, • in
Experiments where fenfe is judge, ought to be accoun-
ted none at all.

EXPERo VL

And as thefe Colours were not changeable by refra- ^
(ftions, fo neither were they by reflexions. For all
white, grey, red.» yellow, green, blue, violet Bodies, as
Paper, Allies, red, Lead, Orpiment, Indico, Bile, Gold,
Silver, Copper, Grafs, blue Flowers, Violets, Bubbles
of Water tinged with various Colours, Peacock's Fea-
thers, the tinfture of Lignum Mefhriticum^ and fuch
like, in red homogeneal Light appeared totally red, in
blue Light totally blue, in green Light totally green,
and fo of other Colours. In the homogeneal Light of
of any Colour they all appeared totally of that fame
Colour, with this^nly difference, that fome of them
tefleded that Light more ftrongly, others more faintly.
1 never yet found any Body which by reflefting homo-
peneal Light could fenfibly change its Colour.

M From

From all which it is manifeft, that if the Sun's Light
confifted of but one fort of rays, there would be but
one Colour in the whole World, nor would it be pof-
iible to produce any new Colour by reflexions and re-
fractions, and by confequence that the variety of Co-
lours depends upon the compoiition of Light,

"DEFINIT ION.

The homogeneal light and rays which appear red,,
or rather make Objefts appear fo, I call rubrific •
or red-makng ; thofe which make ObjeSs appear
yellow, green, blue and violet, I call yellow-ma-
king, green-makingi, blue-making, violet-making,
and fo of the reft. And if at any time I fpeak of:
light and rays as coloured or endued with Co-
lours, I would be underftood to fpeak not philo-
fophically and properly, but groily, and accor-
ding to fuch conceptions as vulgar People in fee-
ing all thefe Experiments would be apt to frame. -
For the rays to fpeak properly are not coloured^-
In them there is nothing elfe than a certain power
and difpofition to ftir up a fenfation of this or that
Colour. For as found in a Bell ori mufical Stringy,
or other founding Body, is nothing but a trem-
bling Motion, and in the Air nothing but that
Motion propagated from the Object, and imthe
Senforium 'tis a fenfe of that Motion under the
form of found; fo Colours in theObjeft are no-
thing but a difpotition to reflet this or that fort
of rays more copioully than the reft ; in the rays
they are nothing but their difpofitions to propa-
gate

gate this or that Motion into the Senibrium, and
in the Senfbrium they are fenfations of thofe Mo-
tions under the forms of Colours. ^

PROP. III. PROB. I.

To define the refrangiiility of the feveral joints of homo^
geneal Light anjis>jertng to the feveral Colours.

For determining this Problem I made the following
Experiment.

EXPER. Vll.

' When I had caufed the redilinear line iides A F, G M, Kg-, 4..
of the Spedlrum of Colours made by the Prifm to be
diftindly denned, as in the fifth Experiment of the
firft Book is defcribed, there were found in it all the
homogeneal Colours in the fame order and fituation
one among another as in the Speftrum of fimple Light^
defcribed in the fourth Experiment of that Book. For
the Circles of which the Speftrum of compound Light
PT is com.pofed, and which in the middle parts of
the Spectrum interfere and are intermixt with one ano-
ther, are not intermixt in their outmoft parts where
they touch thofe reftilinear fides A F and G M. And
therefore in thofe redilinear fides when diftinftly defi-
ned, there is no new Colour generated by refraction. I
obferved alfo, that if any where between the two out-
moft Circles TMF and PGA a right line, as y^v, was
crofs to the Spedrum, lb as at both ends to fall per-
pendicularly upon its redtilinear fides, there appeared ,

M a one

'■■2

one and the iliiiie Colour and degree of Colour from one
end of this line to the other. I delineated therefore in
a Paper die perimeter of the Speitrum F APGMT,
and in trying the third Experiment of the firft Book, I
held the Paper fo that the Spearum might foil upon
this delineated Figure, and agree VN^ith it exaftly, whilft
an AiTiftant whofe Eyes for diftinguifliing Colours were
more critical than mine, did by right Vims ^cpo, yc\ ,'0-c.
drawn crofs the Spedrum, note the confines of the Co-
lours that isof the red M^.^F of theorange<x7^^^, of
the yellow y^?^, of the green =- ^ H' , of the blue ^^xS^,
of the indico ^x^ux, and of the violet xQAm. And
this operation being divers times repeated both in the
lame and in feveral Papers , I found that the Ob-
fervations agreed well enough with one another, and
that the redilinear fides M G and FA were by the faid
crofs lines divided after the manner of a mufical Chord.
Let GM be produced to X, that MX may be equal
toGM, and conceive GX, xX, 'X, ''X, ^X, yX, «X^
MX,, to be in proportion to one another, as the num-
bers I, I,. ^5 -^ p I? ?6' i' ^^i fo ^^ reprefent the
Chords of the Key, and of a Tone, a third Minor, a
fourth, a fifth, afixthMajor, aTeventh,, and aa eighth
above that Key : And the intervals M ^ ? ^ 7 , / ^ j^/- « , 71 ' ,.
^>^, andxG, will be the fpaces which the feveral Co-
lours ( red, orange,„yellow, green, blue, indico, violet ),
take up.

Now thefe intervals or fpaces- fubtending the diffe-
rences of the refraftions of the rays going to the limits,
of thofe Colours, tliat isi> to the points M, a, 7, c^ /;,/, x, G5.
may without any fenfi.ble Error be accounted propor-
tional to the differences of the fines of refraftion of thofe

rays

rays having one common fine of incidence, and there-
fore fince the common fine of incidence of the mod and
leafi: refrangible rays out of Glafs into Air was, (by a
method defcribed above ) found in proportion to their
fines of refradion, as 50 to 77 and 78, divide the dif-
ference between the fines of refradion 77 and 78, as the
line G M is divided by thofe intervals, you will have

11-f 11\^ 11\' llh^ 77i. 773' 779,78, the fi_nes of
refraction of thole nlys out of Glais into Air, their
common fine of incidence being 50. So then the fines
of the incidences of all the red-making rays out of
Glafs into Air, were to the fines of their refraftions,
not greater than 50 to 77, nor lefs than 50 to 77J-, but
varied from one another according to all interme-
diate Proportions. And the fines of the incidences
of the green-making rays were to the fines of
their refraftions in all proportions from that of 50
to 77^, unto that of 50 to 77^. And by the like limits
above-mentioned were the refradions of the rays be-
longing to the reft of the Colours defined, .the fines of
the red-making rays extending from 77 to 773- , thofe
of the orange-making from 77^ to 77^ , thofe of the yel-
low-making from 77J to 77 J, thofe of the green-making
from 77f to 77^5 thofe of the blue-making from 77^ to
77]^, thofe of the indico-making from 77-^ to 77^-, and
thofe of the violet from 77^ to 78, .

Thefe are the Laws of the refradions made out of
Glafs into Air, and. thence by the three Axioms of tiie
iirft Book the Laws of the refradions made out of Air

into Glafs are eafilv derived.

J..

E X P E R,

94]

EXPER. VIII.

I found moreover that when Light goes out of Air
through feveral contiguous refracting Mediums as
through Water and Glafs, and thence goes out again
into Air, whether the-refrafting fuperficies be parallel
or inclined to one another, that Light as often as by-
contrary refractions 'tis fo corrected, that it emergeth
in lines parallel to thofe in which it was incident,
continues ever after to be white. But if the emer-
gent rays be inclined to the incident, the whitenefs of
the emerging Light will by degrees in paffing on from
the place of emergence, become tinged in its edges with
Colours. This I tryed by refra£ting Light with Prifms
of Glafs within a prifmatick Veffel of Water. Now thofe
Colours argue a diverging and feparation of the hetero-
geneous rays from one another by means of their un-
equal refractions, as in what follows will more fully
appear.* And, on the contrary, the permanent white-
nefs argues, that in like incidences of the rays there is
no fuch feparation of the emerging rays, and by confe-
quence no inequality of their whole refraCtions. Whence
i leem to gether the two following Theorems.

^ I . The Exceffes of the fines of refraCtion of feveral
forts of rays above their common fine of incidence when
the refractions are made our of divers denfer^^edjums
immediately into one and the fame rarer medium,^'^afe
to one another in a given Proportion. ^f^m^<;^i- ^^^i- f'

^^y' f'^^ ji^^ 'yvo-^ eac-^iat^^'

*»e *^

[9S]

a. The Proportion of the fine of incidenceto the fine
of refraftion of one and the fame fort of rays out of one
medium into another, is compofed of the Proportion of
the fine of incidence to the fine of refraSion out of the
firft medium into any third medium, and of the Pro-
portion of the fine of incidence to the fine of refraftion
out of that third medium into the fecond medium.

By the firft Theorem the refraftions of the rays of
every fort made out of any medium into Air are known
by having the refraction of the rays of any one fort. As
for inftance, if the refraftions of the rays of every Ibrt
out of Rain-water into Air be defired, let the common
fine of incidence out of Glafs into Air be fubduded
from the fines of refradion, and the Exceffes will be

^7i ^7«-' ^7?' .^7^^7i' ^7h ^79% ^^^ Suppofenow
that the fine of incidence of the leaft refrangible rays be
to their fine of refradion out of Rain- water into Air as
three to four, and fay as i the difference of thofe fines
is to 3, the fine of incidence, fo is 27 the leaft of the
Exceffes above-mentioned to a fourth number 81 ; and
81 will be the common fign of incidence out of Rain-
water into Air, • to which fine if you add all the above-
mentioned Exceffes you will have, the defired fines of
the refradions 108^^. loSs ?. loSf, loSf, io8f, io8f^,
io8i, 109.

By the latter Theorem the refradion out of one me-
dium into another is gathered as often as you have
the refraftions out of them both into any third medium^
As if the fine of incidence of any ray out of Glafs into
Air be to its fine of refraction as ao to 3 1, and the fine
of incidence of the fame ray out of Air into Water, be

[96]

'to its fine of refradion as four to three; the fine of
incidence of that ray out of Glafs into Water will be to
its fine of refraction as a o to 31 and 4 to ^ joyntly, that
is, as the FaCtum of 20 and 4. to the Faftum of 3 1 and
5, or as 80 to 95.

And thefe Theorems being admitted into Opticks,
there would be fcope enough of handling that Science
voluminoufly after a new manner ; not only by teaching
thofe things which tend to the perfeftion of vifion, but
alfo by determining mathematically all kinds of Phaeno-
mena of Colours which could be produced by refra-
fbions. For to do this, there is nothing elfe requifite
than to find out the reparations of heterogeneous rays,
and their various mixtures and proportions in every
mixture. By this way of arguing I invented almoft
all the Pha^nomena defcribed in thefe Books, befide fome
others lefs necefifary to the Argument ; and by the
fucceffes I met with in the tryals, I dare promife, that
to him who fhali argue truly, and then try all things
with good Glaffes and fufficient circumfpeftion, the
expefted event will not be wanting. But he is firft to
know what Colours will arife from any others mixt in
.any affigned Proportion.

PROP. IV. THEOR. III.

^Colours ma^ he produced hy comfofitiGn "which JhaU he like
to the Colours of homogeneal Light as to the affear'ance
of Colour^ hut not as to the immutahtlity of Colony^ and
c'onjlttution of Light, ^nd thofe Colows h'j ho'w ?nuch
they are more compounded hy fo much are they lejs fuU
..and inteufe.^ and hy too much comfofition they may he

diluted and -z^eakened till the^ ceafe.^ ^!Z be?'e may be
aljo Colours froduced by comfofition^ ^juhich arc not fully
like any of the Colours of homogeneal Light, y

For a mixture of homogeneal red and yellow com- ^
pounds an orange^ like in appearance of Colour to that
orange which in the feries of unmixed prifmatick Co-
lours lies between them ; but the Light of one orange
is homogeneal as to refrangibility^ that of the other is
heterogeneal, and the Colour of the one , if viewed
through a Prifm, remains unchanged, that of the other
is changed and refoived into its component Colours red
and yellow. And after the fame manner other neigh-
bouring homogeneal Colours may compound new Co-
lours, like the intermediate homogeneal ones, as yel-
low and green, the Colour between them both, and af-
terwards, if blue be added, there will be made a green
the middle Colour of the three which enter the compo-
fition. For the yellow and blue on either hand,if they are
equal in quantity they draw the intermediate green equal-
ly towards themfeives in compofition, and fo keep it as
it were in equilibrio, that it verge not more to the
yellow on the one hand, than to the blue on the other,
but by their mixt actions remain ftill a middle Colour.
To this mixed green there may be further added
fome red and violet^ and yet the green will not prefent-
ly ceafe but only grow lefs full and vivid, and by in-
creafing 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 whitenefs, or fome other
Colourc So if to the Colour of any homogeneal Light,
the Sun's white Light compofed of all forts of rays be

N added.

[98]

added^ that Colour will not vaniih or change its fpe-
cies but be diluted, and by adding more and more white
it will be diluted more and more perpetually. Laft-
ly, if red and violet be mingled, there will be generated
according to their various Proportions various Purples,
fuch as are not like in appearance to the Colour of any
homogeneal Light, and of thefe Purples mixt with yel--
low and blue may be made other nev^ Colours..

PROP. V. THE OR. IT.

Whitenejs and all grey Colours bet'ween ^white and ifach^..
may be compounded of Colours^ and the isohttenejs of the
Suns Light is compunded of all the frimary Colours
mixt in a due p^o fort ion. ■-,

The Troofhy Experiments.

EXP E R, IX.

Fig' 5. The Sun fhining into a dark Chamber through a
little round Hole in the Window Ihut, and his Light
being there refracted by a Prifm to caft his coloured-
Image P T upon the oppofite Wall : I held a white Pa-
per V to that Image in fuch manner that it might be
illuminated by the coloured Light retieSed from thence,
and yet not intercept any part of that Light in its paf-
fage from the Prifm to the Speftrum. And I found that
when the Paper was held nearer to any Colour than to
the reft, it appeared of that Colour to which it ap-
proached nearcft ; but when it was equally or almoft

equally

[99]

equally diftant from all the Colours, Co that it might
be equally illuminated by them all it appeared white.
And in this lafl: lituation of the Paper, if ibme Colours
were intercepted, the Paper loft its white Colour, and
appeared of the Colour of the reft 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 re-
tained its proper Colour, imtil it w^as incident on the
Paper, and became reflected thence to the Eye ; fo that
if it had been either alone (the reft of the Light being
intercepted) or if it had abounded moft and been pre-
dominant in the Light reflected from thePaper,it would
have tinged the Paper with its own Colour; and yet be-
ing mixed with the reft of the Colours in a due propor*
tion, it made the Paper look white, and therefore by a
compofition with the reft produced that Colour. The
feveral parts of the coloured Light reflefted from the
Spe(3:rum,whilft they are propagated from thence thro'
the Air, do perpetually retain their proper Colours^
becauie wherever they fall upon the Eyes of any Specta-
tor, they make the feveral parts of the Spectrum to
appear under their proper Colours. They retain there-
fore their proper Colours when they fall upon the Pa-
per V, and lb by the confufion and perfed mixture of
thofe Colours compound the whitenefs of the Light
reflected from thence.

E X P E R. X*

Let that Speftrum or folar Image P T fall now upon Fig» 6,
the Lens M N above four Inches broad, and about fix

N 2 Feet

[ lOO ]

Feet diftant from the Piifm ABC, and fo figured that
it may caufe the coloured Light which divergeth from
the Prifin to converge and meet again at its Focus G,
about fix or eight Feet diftant from the Lens, and
thereto fall perpendicularly upon a vv^iite Paper DE.
And if you move this Paper to and fro, you will per-
ceive that near the Lens, as at de^ the whole folar Image
(fuppofe at pt) will appear upon it intenfly coloured
after the m.anner above-explained, and that by receding
from the Lens thofe Colours will perpetually come to-
wards one another, and by mixing more and more di-
lute one another continually, until at length the Paper
come to the Focus G, where by a perfed: mixture they
will wholly vanifh and be converted into whitenefs, the
whole Light appearing now upon the Paper like a little
white Circle. And afterwards by receding further from
the LenSj the rays which before converged will now
crofs one another in the Focus G, and diverge from
thence, and thereby make the Colours to appear again,
but yet in a contrary order ; fuppofe at o\ , where the
red t is now above which before was below, and the
violet p is below which before was above.

Let us now flop the Paper at the Focus G where
the Light appears totally white and circular, and let us
confider its whitenefs. I fay, that this is compofed of
ihe converging Colours. For if any of thofe Colours
be intercepted at the Lens, the whitenefs will ceafe and
degenerate into that Colour which arifeth from the
compofition of the other Colours which are not inter-
cepted. And then if the intercepted Colours be let
pafs and fall upon that compound Colour, they mix
with it, and by their mixture reftore the whitenefs.

[ lOI ]

So if the violet, blue and green be intercepted^ the re-
maining yellow, orange and red will compound upon
the Paper an orange, and then if the intercepted Co-
lours be let pafs they will fall upon this compounded
orange, and together with it decompound a white. So
alfo 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 pafs
will fall upon this green, and together with it decom-
pound a w^hite. And that in this compofition of white
the feveral rays do not fuifer any change in their colori-
fic qualities by acting upon one another, but are only
mixed, and by a mixture of their Colours produce
white, may further appear by thefe Arguments.

If the Paper be placed beyond the Focus G, fuppofe
at ^-'?, and then the red Colour at the Lens be alternate^
ly intercepted, and let pafs again, the violet Colour on
the Paper vvdll not fuffer any change thereby, as it ought
to do if the feveral forts of rays aded upon one another
in the Focus G, where they crofs. Neither will the
red upon the Paper be changed by any alternate flop-
ping, and letting pafs the violet which croffeth it.

And if the Paper be placed at the Focus G, and the
wdiite round Ima2,e at G be viewed through the Prifm
HIK, and by the refradion of that Prifm be tranflated
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 Colour at the Lens be
often ftopt and let pafs by turns, the red at r will ac-
cordingly difappear and return as often, but the violet
at v will not thereby fuifer any change. And lb by
flopping and letting pafs alternately the blue at the

Lens^

[102]

Lens^ the blue at r will accordingly dlfappear and re-
turn, v/ithout any change made in the red at r. The
red therefore depends on one fort of rays, and the qlue
on another fort, which in the Focus G where they are
commixt do nota£b on one another. And there is the
fame reafon of the other Colours.

I coniidered further, that when the moft refrangible
raysPp, and the leaft refrangible ones Tt, are by con-
verging inclined to one another, the Paper, if held very
oblique to thofe rays in the Focus G, might refled one
fort of them more copioufly than the other fort, and by
that means the refledted Light would be tinged in that
Focus with the Colour of the predominant rays, pro-
vided thofe rays feverally retained their Colours or co-
lorific qualities in the compofition of white made by
them in that Focus. But if they did not retain them
in that white, but became all of them feverally endued
there with a difpofition to ftrike the fenfe with the per-
ception of white, then they could never lofe their v/hite-
nefs by fuch reflexions. I inclined therefore the Paper
to the rays very obliquely, as in the fecond Experiment
of this Book, that the moft refrangible rays might be
more copioully refleded than the reft, and the white-
nefs at length changed fucceflively into blue, indico
and violet. Then I inclined it the contrary way, that
the moft refrangible rays might be more copious in the
reflected Light than the reft, and the whitenefs turned
fucceflively to yellow, orange and red.

Laftly, I made an Inftrument X Y in fafliion of a
Comb, whofe Teeth being in num.ber fixtecn were
about an Inch and an half broad, and the intervals of the
Teeth about two Inches wide. Then by interpofing

fuc-

fecceflively the Teeth of this Inftrument near tlie Lens
I intercepted part of the Colours by the interpofed
Tooth, whilft the reft of them went on through the in-
terval of the Teeth to the Paper D E, and there pain-
ted a round folar Image. But the Paper I had firft pla-
ced fo, that the Image might appear white as often
as the Comb was taken away; and then the Comb be-
ing as was faid interpofed, that whitenefs by reafon of
tlie intercepted part of the Colours at the Lens did al-
ways change into the Colour compounded of thofe
Colours which were not intercepted, and that Colour
was by the motion of the Comb perpetually varied fo
that in the palling of every Tooth over the Lens all .
thefe Colours red, yellow, green, blue and purple, did
always fucceed one another. I caufed therefore all the
Teeth to pafs fucceffively over the Lens, and when the
motion was How, there appeared a perpetual fucceflion
of the Colours upon the Paper : But if I fo much acce-
lerated the motion, that the Colours by reafon of their
quick fucceflion could not be diftinguillied from one
another,, the appearance of the fingle Colours ceafed»-
There was no red, no yellow, no green, no blue, nor
purple to be feen any longer, but from a confufion of
them all there arofe one uniform white Colour. Of the
Light which now by the mixture of all the Colours ap-
peared 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 ftrike the Senibrium. If the impreflions
follow one another f lowly, fo that they may be feve-
rally perceived, there is made a diftinft fenfation of all
the Colours one after another in a continual fucceflion.

But

[104]

But if the impreffions follow one another fo quickly
that they cannot be feverally perceived^ there arifetk
out of them all one common feniation, which is nei-
ther of this Colour alone nor -of that alone, but hath it
felf indifferently to 'em all, and this is a feniation of
whitenefs. By the quicknefs of the fucceffions the im-
preffions of the feveral Colours are confounded in the
Senforium, and out of that confuiion arileth a mixt fen-
_)(. fation. If a burning Coal be nimbly moved round in a
Circle with Gyrations continually repeated, the whole
Circle will appear Uke fire 3 the reafon of which is, that
the fenfation of the Coal in the feveral places of that
Circle remains impreft on the Senforium, until the
Coal return again to the &me place. And fo in a
quick confecution of the Colours the impreffion of every
Colour remains in the Senforium, until a revolution of
all the Colours be compleated, and that firft Colour re-
turn a^ain. The impreffions therefore of all the fucceffive
Colours are at once in theSenforium,and joyntly ftir up
a fenfation of them all ; and fo it is manifeil by this Ex-
periment, that the commixt im-preffions of all the Co-
lours do ftir up and beget a feniation of white, that is,
that whitenefs is compounded of all the Colours.

And if the Comb be now taken away, that all the
Colours may at once pafs from the Lens to the Paper,
and be there intermixed, and together retie£ted thence
to the Speftators Eyes ; their impreffions on the Senfo-
rium being now more fubtily^ and perfedly commixed
there, ought much m.ore to ftir up a fenfation of white-

You

105 ]

You may inftead of the Lens uie two Prifms HIK
andLMN, which by refrafting the coloured Light
the contrary way to that of the firft refraflion, may
make the diverging rays converge and meet again in G,
as you fee it reprefented in tlie feventh Figure. For Fig- 7.
where they meet and mix they will compofe a white
Light as when a Lens is ufed.

EX PER. XL

Let the Sun's coloured Image PT fall upon the Wall F%. 8,
of a dark Chamber, as in the third Experiment of the
firft Book, and let the fame be viewed through a Prifm
a b c, held parallel to the Prifm ABC, by whofe refra-
Sion that Image was made, and let it now appear lower
than before, fuppofe in the place S over againft the red-
colour T. And if you go near to the Image PT, the
Spedrum S will appear oblong and coloured like the
Image P T ; but if you recede from it, the Colours of
the Speftrum S will be contracted m.ore and more, and
at length vanifh, that Speftrum S becoming perfedly
round and white ; and if you recede yet further, the
Colours will emerge again, but in a contrary order.
Now that Spedrum S appears white in that cafe when
the rays of feveral forts which converge from the feve-
ral parts of the Image PT, to the Prifm a be, are fo
refracted unequally by it, that in their paffage from the
Prifm to the Eye they may diverge from one and the
faine point of the Spedrum S, and fo fall afterwards
upon one and the fame point in the bottom of the Eye^
and there be mingled.

O And

And further, if the Comb be here made ufe of, by-
whofe Teeth the Colours at the Image PT may be fuc-
ceffiveiy intercepted ; the Speftrum S when the Comb
is moved flowly will be perpetually tinged with fuc-
ceffive Colours : But when by accelerating the motion
of the Comb, thefucceffion of the Colours is fo quick
that they cannot be feverally feen, that Spe£trum S, by
a confufedand mixt fenfationof them all, will appear
white.

EXPER. XIL

Fio', Q. The Sun fliining through a large Prifm ABC upon
aCombXY, placed immediately behind the Prifm., his
Light which paffed through the interftices of the Teeth
fell upon a white Paper D E. The breadths of the
Teeth were equal to their interftices, and j^tven Teeth
together with their interftices took up an Inch in
breadth. Now when the Paper was about two or
three Inches diftant from the Comb, the Light which
paffed through its feveral interftices painted fo many
ranges of Colours kl, mn, op, qr, l5)'r. which were
parallel to one another and contiguous, and without any
mixture of white. And thefe ranges of Colours, if the
Comb was moved continually up and down with a re-
ciprocal motion, afcettded and defcended in the Paper^,
and when the motion of the Comb was fo quick, that
the Colours could not be diftinguiftied from one another^
the v/hole Paper by their confufion and mixture in the
Senforium appeared white.

Let

[ro;]

Let the Gomb now reft, and let the Paper be remo-
ved further from the Prifm, and the feveral ranges of
Colours will be dilated and expanded into one another
more and more, and by mixing their Colours will di-
lute one another, and at length, when the diftance
of the Paper from the Comb is about a Foot , or a
little more ( fuppofe in the place i D 2 E ) they will
fo far dilute one another as to become white.

With any Obftacle let all the Light be now ftopt
which pafles through any one interval of the Teeth, fo
that the range of Colours which comes from thence may
be taken away, and you will fee the Light of the reft of
the ranges to be expanded into the place of the range
taken away, and there to be coloured. Let the inter-
cepted range pafs on as before, and its Colours falling
upon the Colours of the other ranges, and mixing with
them, will reftore the whitenefs.

Let the Paper a D a E be now very much inclined to
the rays, fo that the moft refrangible rays may be more
copiouily refleded than the reft, and the white Colour
of the Paper through the excefs of thofe rays will be
changed into blue and violet. Let the Paper be as
much inclined the contrary way, that the leaft refran-
gible rays may be now morq copioufly reflefted than
the reft, and by their excefs the whitenefs will be
changed into yellow and red. The feveral rays there-
fore in that white Light do retain their colorific qua-
lities, by which thofe of any fort, when-ever they be-
come more copious than the reft, do by their excefs
-and predominance caule their proper Colour to ap-
pear.

O a ^ And

io8

And by the flime way of arguing, applied to the third
Experiment of this Book, it may be concluded, that
the white Colour of all refrafted Light at its very firft
emergence, where it appears as white as before its inci-
dence, is compounded of various Colours.

EXPER. XIII.

In the foregoing Experiment the feveral intervals of

the Teeth of the Comb do the office of fo many Prifms^

every interval producing the Phgenomenon of one Prifm,

Whence inftead of thofe intervals ufing feveral Prifms, I

try 'd to compound whitenefs by mixing their Colours,and

did it by ufing only three Prifms, as alfo by ufing only

Fio\ 10. two as follows. Let two Prifms ABC and a be, whole

refrafting Angles B and b are equal,be fo placed parallel

to one another, that the refradting Angle B of the one

may touch the Angle c at the bafe of the other, and

their planes C B aiid cb, at which the rays emerge, may

lye in diredum. Then let the Light trajeded through

: them fall upon the Paper M N, diftant about 8 or 12

Inches from the Prifms. And the Colours generated

by the interior limits B and c of the two Prifms, will

be mingled at PT.^ and there compound white. For if

either Prifm be taken away, the Colours made by the

other vs^ill appear in that place PT, and when the Prifm

is reftored to its place again, fo that its Colours may

there fall upon the Colours of the other, the mixture

of them both will reftore the whiteneise

This

[ lop]

This Experiment fucceeds alio, as I have tryed^whai
the Angle b of the lower Prifm, is a little greater than
the An^le B of the upper , and between the interior
Angles B and c, there intercedes fome fpace B c, as is
reprefented in the Figure, and the refrafting planes
BC and be, are neither in direftum, nor parallel to
one another. For there is nothing more requifite to
the fuccefs of this Experiment^ than that the rays of all
forts may be uniformly mixed upon the Paper in the
place FT. If the moft refrangible rays coming from
the fuperiorPrifm take up all thefpace from M to P, the
rays of the fame fort which come from the inferier
Prifm ought to begin at P, and take up all the reft of the
fpace from thence towards N. If tlie lea ft refrangible
rays coming from the fuperior Prifm take up the fpace
MT,. the rays of the fame kind which come from the
other Prifm ought to begin at T, and take up the remain^
ing fpace T N. If one ibrt of the rays which have in-
termediate degrees of refrangibility, and come from the
fuperior Prifm be extended through the fpace MQ, and
another fort of thofe rays through the fpace MR, and
a third ibrt of them through the fpace MS, the fame
forts of rays coming from the lower Prifm, ought to il^
luminate the remaining fpaces QN, RN, SN refpe-
ftively. And the fame is to be underftood of all the
other forts of rays. For thus the rays of every fort will
be fcattered uniformly and evenly through the whole
fpace MN, and ih being every where mixt in the fame
proportion, they mufl every where produce the lame
Colour. And therefore fince by this mixture they pro-
duce white in the exterior fpaces MP and TN, they
lEuft alfo produce white in the interior ipace P T» This

Is the reaTon of the compofition by v/hich whitenefs
was produced in this Experiment, and by what other .
way foever I made the like compofition the refult was J
whitenefs. '^

Laftly, If with the Teeth of a Comb of a due fize,
the coloured Lights of the two Prifms which fall upon
the fpace PT be alternately intercepted , that fpace
PT, when the motion of the Comb is flow, will always
appear coloured, but by accelerating the motion of
the Comb fo much, that the fucceflive Colours can-
not be diftinguiflied from one another, it will appear
white.

EXPER. XIV.

^^-: Hitherto I have produced whitenefs by mixing the
Colours of Prifms. If now the Colours of natural Bo-
dies are to be mingled, let Water a little thickned with
Soap be agitated to raife a froth, and after that froth
has flood a little, there will appear to one that fliall
view it intently various Colours every v/here in the
furfaces of the feveral Bubbles ; but to one that fhall
go fo far off that he cannot diftinguifli the Colours from
one another, the whole froth will grow white with a
perfect whitenefs.

EX PER. XV..

Laftly, in attempting to compound a white by mixing
the coloured Powders which Painters ufe, I confidered
that all coloured Powders do fupprefs and flop in
them a very confiderable part of the Light by which

they

[Ill]

they are illuminated. For they become coloured by
refledling the Light of their own Colours more copioufly^
and that of all other Colours morefparingly, and yet
they do not refleft the Light of their own Colours fo
copioufly as white Bodies do. If red Lead/or inftance
and a white Paper, be placed in the red Light of the
coloured Spettrum made in a dark Chamber by the re-
fraction of a Prifm, as is defcribed in the third Eperi-^
ment of the firft Book ; the Paper will appear more lu-^
cid than the red Lead, and therefore reilefts the red-
making rays more copioufly 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 re-
fieded by the red Lead in a much greater proportion.
And the like happens in Powders of other ColourSo
And therefore by mixing fuch Powders we are not to
exped: a ftrong and full white, fuch as is that of Paper,
but fome dusky obfcure one, fuch as might arife from a
mixture of light and darknefs, or from white and black,
that is, a grey, ordun, orruffetbrown, fuch as are the
Colours of a Man's Nail, ofaMoufe, of Aflies, of or-
dinary Stones, of Mortar, of Duft and Dirt in High-
ways, and the like. And fuch a dark white I have
often produced by mixing coloured Powders, For thus
one part of red Lead,and five parts of F^inde ^nsjCom-
pofed a dun Colour like that of a Moufe. For thefe
two Colours were feverally fo compounded of others,
that in both together were a mixture of all Colours j and
there was lefs red Lead ufed than Viride ^r?^, becaufe
of the fulnefs of its Colour. Again, one part of red
Lead, and four parts of blue Bife, compofed a dun Co-
lour verging a little to purple, and by adding to this a

certain

[112]

certain mixture of Orpiment and Viridi ^ris in a due
proportion, the mixture loft its purple tindure, and be-
came perfectly dun. But the Experiment fucceeded beft
without Minium thus. To Orpiment I added by little
and little a certain full bright purple, which Painters
Life until the Orpiment ceafed to be yellow, and became
ofa pale red. Then 1 diluted that red by adding a
little Firide Mns.^ and a little more blue Bife than 7A-
ridl jEris^ until it became of fuch a grey or pale white^
as verged to no one of the Colours more than to ano-
ther. For thus it became of a Colour equal in white-
neis to that of Afhes or of Wood newly cut, or of a
Man's Skin. The Orpiment relieved more Light than
did any other of the Powders, and therefore conduced
more to the whitenefs of the compounded Colour than .
they. To affign the proportions accurately may be
difficult, by reafon of the different goodnefs of Pow-
ders of the fame kind. Accordingly as the Colour of
any Powder is m.ore or lefs full and luminous, it ought
to be ufed in a lefs or greater proportion.

Now coniidering that thefe grey and dun Colours
may be alfo produced by mixing whites' and blacks, and
by confequence ditFer from perfeft whites not in Species
of Colours but only in degree of luminoufnefs, it is ma-
nifeft that there is nothing more requilite to make
them perfeftly white than to increafe their Light fuffi-
ciently ; and, on the contrary, if by increaiing their
Liglit they can be brought to perfeft whitenefs, it will
thence alio follow, that they are of the lame Species of
-Colour with the beft whites, and differ from them only
in the quantity of Light. And this I tryed as follows.
J took the third of the above-mentioned grey mixtures

(that

[113]

f*that which was compounded of Orpiment, Purple,
Bile and V'lr'de Mrk) and rubbed it thickly upon the
tloor of my Chamhev, where the Sun fhone upon it
through the opened Cafement ; and by it, in the flia-
dow, I laid a piece of white Paper of the fame bignefs.
Then going from them to the diftance of 11 or 1 8 Feet,
fo that I could not difcern the unevennefs of the furface
of the Powder, nor the little (hadows let fall from the
gritty particles thereof ; the Powder appeared intenfly
white, fo ^s to tranfcend even the Paper it felf in white-
nefs, efpecially if the Paper w^re a little Ihaded from
the Light of the Clouds, and then the Paper compared
with the Powder appeared of fuch a grey Colour as the
Powder had done before. But by laying the Paper
where the Sun ihines through the Glafs of the Window,
or by fhutting the Window that the Sun might fhine
through the Glafs upon the Powder, and by fuch other
fit means of increafing or decreafing the Lights where-
with the Powder and Paper were illuminated , the
Light wherewith the Powder is illuminated may be
made ftronger in fuch a due proportion than the Light
w^herewith the Paper is illuminated, that they fhall both
appear exactly alike in whitenefs. For w4ien I was
trying this, a Friend coming to vifit me, 1 ftopt him
at the door, and before 1 told him w^hat the Colours
were, or what I was doing \ I askt him. Which of the
two whites were the beft, and wherein they diflfered ?
And after he had at that diftance viewed them well, he
anfwered. That they were both good whites, and that
he could not fay which was beft, nor wherein their Co-
lours differed. Now if you confider, that this white
of the Powder in the Sun-ftiine was compounded of the

P Colours

["41

Colours which the component Powders ( Orpiment,
Purple, Bife, and Viride ^ru) have in the fame Sun>
fliine, you muft acknowledge by this Experiment, as
well as by the former, that per fed: whitenefs may be
compounded of Colours.

From what has been faid it is alfo evident, that the
whitenefs of the Sun's Light is compounded of all the
Colours wherewith the feveral forts of rays whereof
that Light confifts, when by their feveral refrangibili-
ties they are feparated from one another, do tinge Paper
©r any other white Body whereon they fall. For thofe
Colours by Prop. a. are unchangeable, and whenever
all thofe rays with thofe their Colours are mixt again^ .
they reproduce the fame white Light as before. .

PROP. VL PR OB. Ii:

Jn a mixture of frimary Colours^ the quantity and qualuy
of each being given j to-. know .the Colony^ of the com---
found,

II , With the Center O and Radius OD defcribe a Circle-
ADF, anddiftinguiiQiits-circumference into feven parts-
DE, EF, EG, GA, AB, BC, CD, proportional to
the feven muiicalTones or Intervals of the eight Sounds^
Sol^ la^ fa^ foly /^, mi, fa^ Jol^ contained in an- Eighty
that is, proportional to the -numbers-^, ;-, 7^? \y 7,, 7^,
:'. Let the hrft part DE reprefent a red Colour, the
fecond EF orange, the third EG. yellow,- the fourth
GH green, the fifth AB blue, the fixth BC indico^
and the feventh CD violet. Anch conceive, that thefe
are all the Colours of uncompoimded. Light gradually

paffing

[115 3

pa fling into one another, as they do when made by
Frifms ; the circumference DE FG A BCD, reprelen-
ting the whole leries of Colours from one end of the
Sun's coloured Image to the other, fo 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 de-
grees of yellow, and fo on. Let p be the center of
gravity of the Arch DE, and q, r, s, t, v, x, the centers
of gravity of the Arches EF, FG, GA, AB, BC
and CD refpeftively, and about thofe centers of gra-
vity let Circles proportional to the number of rays of
each Colour in the given mixture be defcribed; that is,
the circle p proportional to the number of the red-ma-
king rays in the mixture, the Circle q proportional to
the number of the orange-making rays in the mixture,
and fo of the reft. Find the common center of gravity
of all thofe Circles p, q, r, s, t, v, x. Let that center
be Z ; and from the center of the Circle A D F, through
Z to the circumference, drawing the right line O Y,
the place of the point Y in the circumference fhall Ihew
the Colour arifing from the compofition of all the Co-
lours in the given mixture, and the line OZ fhall be
proportional to the fulnefs or intenfenefs of the Colour,
that is, to its diftance from whitenefs. As if Y fall in
the middle between F and G, the compounded Colour
Ihall be the beft yellow ; if Y verge from the middle to=
wards F or G, the compounded Colour fhall according-
ly be a yellow, verging towards orange or green. If Z
fall upon the circumference the Colour fliall be intenfe
and florid inthehigheft degree; if it fall in , the mid
way between the circumferenceand center, it (hall be

P 2 ^ but

[ r 16 ]

but half fo Intenfe^ that is-, it Ihall be fiich a Colour as
would be made by diluting the intenfeft yellow with an
equal quantity of whitenefs ; and if it fall upon the
center O, the Colour fliall have loft all its intenfenefs^
and become a v/hite. But it is to be noted, That if the
point Z fall in or near the line O D-, the main ingredients
being the red and violet, the Colour compounded fhall
not beany of the priimatic Colours, but a purple, in-
clining to red or violet, accordingly as the point Z
lieth on the fide of the line D O towards E or towards C^
and in general the compounded violet is more bright and
more fiery than the uncompounded. Alio if only two
of the primary Colours which in the Circle are oppofite
to one another be mixed in an equal proportion, the
point Z (hall fall upon the center O, and yet the Co-
lour compounded of thofe two ihall not be perfectly
white, but Ibme 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 difliances in the circum-
ference I do not know, but of four or five 1 do not much
cjueftion but it may. But thefe are curiofities of little
or no moment to the underftanding the Phenomena of
nature. For in all whites produced by nature, there
nfes to be a mixture of all forts of rays, and by confe-
quence a compofition of aU Colours.

To give an initance of this Rule ; fuppofe a Colour is
compounded of thefe homogeneal Colours, of violet
1 part, of indico i part, of blue 2 parts, of green 3 parts^
of yellow 5 parts, of orange 6 parts, and of red iq parts.
Proportional to thefe parts I defcribe the Circles x, v, t,
h r, q, p refpeftively, that is^ fo that if the Circle x

[117]

be I, the Circle v may be i, the Circle t 2, the Circle
s ^, and the Circles r, qandp, 5, 6 and 10. Then I
find Z the common center of gravity of thefe Circles,
and through Z drawing the line O Y^ the point Y falls
upon the circumference between E and F, fome thino-
nearer to E than to F, and thence I conclude^ that the
Colour compounded of thefe ingredients will be an
orange^ verging a little more to red than to yellow.
Alfo I find that O Z is a little lefs than one half of
OY, and thence I conclude, that this orange hath a
little lefs than half the fulnefs or intenfenefs of an un-
compounded orange ; tte,t is to fay, that it is fuch an^
orange as may be made by mixing an homogeneal orange
Vvdth a good white in the proportion of the line OZ to
the line Z Y, this proportion being not of the quantities
of mixed orange and white powders, but of the quan-
tities of the lights reiieSed from them.

This Rule I conceive accurate enough for pradife^.
though not mathematically accurate ; and the truth of
it may be fufficiently proved to fenfe, by flopping any
of the Colours at the Lens in the tenth Experiment of
this Book. For the reft of the Colours which are riot
flopped, but pafs on to the Focus of the Lens, will'
there compound either accurately or very nearly fuch
a Colour as by this Rule ought to rcfult from, their
mixture^

PRO P^

[liS]

PROP. VIL THEOR. V.

^^4U the Colours in the Univerje 'Ui)htch are made l^ L'ght^^
and defend not on the jpower of imagination^ are
either the Colours of homogeneal Lights^ or com founded
of thefe and that either accurately or ver'j nearl'j^ ac*
wording to the Bjule of the foregoingTroMem. ^

For it has been proved ( in Prop. i. Lii.^.) that the
changes of Colours made by refractions do not arife
from any new modifications of the rays impreft bj/ thofe
refractions^ and by the various terminations of Hght
andfhadow, as has been the conftant and general opi-
nion of Philofophers. It has alfo been proved that the
leveral Colours of the homogeneal rays do conftantly
anfwerto theu' degrees ofrefrangibility, (Prop, i . L^z^. i .
and Prop.::. L^'/^.'^.J and that their degrees of refrangi^
bility cannot be changed by refractions and reflexions^
(Prop.i.X^/^.i.j and by confequence that thofe their
Colours are likewdfe immiutable. It has alfo been pro-
ved direjftly by refraCiing and reflecting homogeneal
Lights apart, that their Colours .cannot be changed^
(Prop.!. Lib,''^.) It ^has been proved alfo, that wdien
the feveral forts of rays are mixed, and in croffing pafs
through the lame fpace, they do not aCt on one another
ib as to change each others colorifick qualities, (Exper.
io.Lii.'2.) but by mixing their aCtions in the Senfo-
rium beget a fenfation differing from what either would
do apart, that is a fenfation of a mean Colour between
their proper Colours ; and particularly when by the
jQoncourfe and mixtures of all forts of rays, a white

Colour

[up]

Colour is produced, the white is a mixture of all the
Colours which the rays would have apart, ( Prop, 5.
Ltk a. J The rays in that mixture do not lofe or alter
their feveral colorifick qualities, but by all their various
kinds of adions mixt in the Senforium, beget a fenia-
tion of a middling Colour between all their Colours
which is whitenefs. For whitenefs is a mean between
all Colours, having it felf indifferently to themall^ fo
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 prefently lofe its Colour, but a
white Powder mixed with any Colour is prefently tin-
ged with that Colour, and is equally capable of beinc?
tinged with any Colour what-ever. It has been Ihewed
aifo, that as the Sun's Light is mixed of all forts of rays,
fo its whitenefs is a mixture of the Colours of all forts
of rays ; thofe rays having from the beginning their fe-
veraL colorific qualities as wallas their ieveral refrangi-'
bilities, and retaining them perpetually unchanged not^
withftanding any refractions or reflexions they m.ay at
any time fuffer, and that when-ever any fort of the
Sun's rays is by any means (as by reflexion in Exper. 9-
and io. LtL i. or by refraftion as happens in all re-'
fraftions ) fepa rated from the reft, they then- manifeit
theirproper Colours, Thefe things have been proved,'
and thefum. of all this amounts to the Prapoiition here
to be proved>:. Eor if the Sun's Light is mixed of fe-
veral forts of rays, each of which have originally their .
feveral refrangibilities and colorifick quilities, and not-
withftanding their refraftions and refleftions, and their
various feparations or mixtures, keep thofe their ori-
ginal properties perpetually. the fame without altera-
tion :

■*^'^' "^

[ 120]
^tion ) fiien all the Colours in the World muft be fuch as
conftantly ought to arife from the original colorific qua-
lities of the rays whereof the Lights confift by which
thofe Colours are feen. And therefore if thereafon of
any Colour what-ever be required, we have nothing elfe
to do then to confider how the rays in the Sun's Light
have by reflexions or refraftions, or other caufes been par-
ted from one another^or mixed together j or otherwiie to
find out what forts of rays are in the Light by which
that Colour is made, and in what proportion; and
then by the laft Problem to learn the Colour which
ought to arife by mixing thofe rays (or their Colours)
in that proportion. I fpeak here of Colours fo far as
they arife from Light. For they appear fometimes by
other caufes, as when by the power of phantafy we
fee Colours in a Dream, or a mad Man fees things before
him which are not there i or when we fee Fire by ftriking
the Eye, or fee Colours like the Eye of a. Peacock's
Feather, by prefiing our Eyes in either corner whilft
we look the other way. Where thefe and fuch like
caufes interpofe not, the Colour always anfwers to
the fort or forts of the rays whereof the Light confifts,
as I have conftantly found in what^ever Phenomena of
Colours I have hitherto been able to examin, I fliall in
the following Propofitions give inftances of this in the
.Jh^nomena of chiefeft note.

PROP.

PROP. VIIL PROB. III.

Bi the dijcovered Troperties of Light to explain the
Colours ynade b^ Prijms.

Let ABC reprefent a Prifm refracting the Light of p^o-. xi.
the Sun, which comes into a dark Chamber through a
Hole F <? almoft as broad as the Prifm, and let M N
reprefent a white Paper on which the refracted Light is
caft, and fuppofe the moft refrangible^or deepeft violet
making rays fall upon the fpace P-tt , the leaft refran-
gible or deepeft red-making rays upon the fpace Ti,
the middle fort between the Indico-making aud blue-
making rays upon the fpace d^ , the middle fort of the
green-making rays upon the fpaceRe, the middle fort
between the yellow-making and orange-making rays
upon the fpace S<T? and other intermediate forts upon
intermediate fpaces. For fo the fpaces upon which the
feveral forts adequately fall will by reafon of the diffe-
rent refrangibility of thofe forts be one lower than ano-
ther. Now if the Paper MN be fo near the Prifm that the
fpaces P T and ttT do not interfere with one another, the
diftance between them T TT will be illuminated by all
the forts of rays in that proportion to one another which
they have at their very firft coming out of the Prifm^
and confequently be white. But the fpaces PT and ^^
on either hand, will not be illuminated by them all,
and therefore will appear coloured. And particularly
at P, where the outmoft violet-making rays fall alone.,
the Colour muft be the deepeft violet. At Q. where the
violet-makino; and indico-makinp; rays are mixed ^ it

Q muft

[ 122]

nnift be a violet inclining much to indico. At R where
the violet-making , indico-making, blue-making, and
one half of the green-making rays are mixed, their Co-
lours muft ( by the conftruftion of the fecond Problem)
compound a middle Colour between indico and blue.
At S where all the rays are mixed except the red-ma-
king and orange-making,their Colours ought by the fame
Rule to compound a faint blue, verging more to green
than indie. And in the progrefs from S to T, this blue
will grow more and more faint and dilute, till atT^
where all the Colours begin to be mixed , it end iji
whitenefs.

So again, t)n the other fide of the white at T, w^here
the leaft refrangible or utmxoft red-making ^ays are alone
the Colour muft be the deepeft red. At cr the mixture
of red and orange will compound a red inclining ta
orange. At e the mixture of red, orange, yellow, and
one half of the green muft compound a middle Colour
between orange and yellow. At x the mixture of all
Colours but violet and indico will compound a faint
yellow, verging more to green than to orange. And
this yellow will grow more faint and dilute continually
in its progrefs from y^ to tt, where by a mixture of ail
forts of rays it will become white.

Thefe Colours ought to appear were the Sun's Light
perfectly white: But becaufe it inclines to yellow,theex-
cefs 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 fo. the
Colours in order from P to T ought to be violet, indico,
blue, very faint green, white, faint yellow, orange, red.
Thus it is by the computation : And they that pleafe to,.

view

[123]

view the Colours made by a Prifm will find it fo in
Nature.

Thele are the Colours on both fides the white when
the Paper is held between the Prifm, and the point X
where the Colours meet, and the interjacent white va-
nifhes. For if the Paper be held ftill farther off from the
Prifm, the moft refrangible and leaft refrangible rays
will be wanting in the middle of the Light, and the reft
ofthe rays which are found there, will by mixture pro-
duce a fuller green than before. Alfo the yellow and
blue will now become lefs compounded, and by con-
fequence more intenfe than before. And this alfo
agrees with experience.

And if one look through a Prifm upon a white Objed
encompafled with blacknefs or darknefs, the reafon of
the Colours arifing on the edges is much the fame, as
will appear to one that (hall a little confider it. If a
black Objeft be encompaffed with a white one, the Co-
lours which appear through the Prifm are to be derived
from the Light of the white one, fpreading into the Re*
gions of the black, and therefore they appear in a con*
trary order to that, in which they appear when a white
Objeft is furrounded with black. And the fame is to
be underftood when an Objed is viewed, whofe parts
are fome of them lefs luminous than others. For in the
Borders of the more and lefs luminous parts, Colours
ought always by the fame Principles to arife from the
excefs ofthe Light ofthe more luminous, and to be of
the fame kind as if the darker parts were blacky but yet
to be more faint and dilute.

'0.2 What

[124]

What is faid of Colours made by Prifms may be eafily
applied to Colours made by the Glafles of TelefeopeSj^
or Microfcopes, or by the humours of the Eye. For if
the Objedl'glafs of a Telefcope be thicker on one fide
than on the other^ or if one half of the Glafs, or one
half of the Pupil of the Eye be covered with any opake
fubftance : the Objed-glafs, or that part of it or of the
Eye which is not covered, may be confidered as a Wedge
with crooked fides, and every Wedge of Glafs^ or other-
pellucid fubftaace, has the effed of a Prifm in refrading
the Light which paflTes through it.

How the Colours in the 9th and loth Experiments
of the firft Part arife from the different reflexibility of
Lightjis evident by what was there faid. But it is obfer-
vable in the 9tii Experiment, that whilft the Sun's di-
red Light is yellow, the excefs of the blue-making
rays in the refleded Beam of Light MN, fuffices only;
to bring that yellow to a pale white inclining to blue^.
and not to tinge it with a manifefl:ly blue Colour. To
obtain therefore a better blue, I ufed inftead of the yel-
low Light of the Sun the white Light of the Clouds, by.,
varying a little the Experiment as follows^.

EX PER. XVL

15. Let H F G reprefent a Prifm in the open Air, and S
the Eye of the Spedator, viewing the Clouds by their.
Light coming into the Prifm at the plane fide F IG K^
and refleded in it by its bafe H E I G, and thence going
out through its plain fide HEFK to the Eye. And
when the Prifm and Eye are conveniently placed, fo
that the Angles of incidence and reflexion at the bafe

may

[125]

may be about 40 degrees^ the Spectator will fee a Bow^
MN of a blue Colour, running irom one end of the.
bafe to the other, with the concave fide towards him,,
and the part of the bafe IMNG beyond this Bow will
be brighter than the other part EMNH on the other
fide ol it. This blue Colour MN being made by no-
thing elfe than by reflexion of a fpecular iuperficies^
leems fo odd a Ph^nomienon, and fo unaccountable for •
by the vulgar Hypothefis of Philofophers, that I could
not but think it deferved to be taken notice of. Now
for underftanding the reafon of it, fuppofe the plane/
ABC to cut the plane fides and bafe of the Prifm per- ^
pendicularly. From the Eye to the line BC, wherein that
plane cuts the bafe, draw the lines S p and S t^ in the -
Angles Spc 50 degr. ;, andStc49 degr.-^g, and the.
point p will be the limit beyond which none of the moft./
refrangible rays can pafs through the bafe of the Prifm,
and be refradted, whofe incidence is fuch that they may
be refleded to the Eye ; and the point t wdll be the like
limit for the ieaft refrangible rays, that is, beyond
which none of them can pafs through the bafe, whofe :
incidence is fuch that by reflexion they may come to the?
Eye. And the point r taken in the middle vv^ay between
p and t, will be the like limit for the meanly refrangible
rays. And therefore all the refrangible rays which fall
upon the bafe beyond t, that i;s, between t and B, and.
can come from thence to the Eye will be reflefted thi-
ther : But on this fide t, that is, between t and c, many,
of thefe rays will be tranfmitted through the bafe. ,
And all the moil: refrangible rays which fall upon the.
bafe beyond p, that is , between p and B, and can by
r^eiiexion come from thence to the Eye, will be reflected/

thither^.

[ I^^ ]

thither, but every where between t and c, many of
thefe'rays will get through the bafe and be refrafted;
and the fame is to be underftood of the m.eanly refran-
gible rays on either fide of the point r. Whence it fol-
lows, that the bafe of the Prifm muft every vv^here be-
tween t and B, by a total reflexion of all forts of rays to
the Ey^, look white and bright. And every where
between/ andC, byreafon ofthe tranfmiffion ofmany
Tays of every fort, look more pale, obfcure and dark.
But at r, and in other places between p and t, where
all the more refrangible rays are reflefted to the Eye,
and many of the lefs refrangible are tranfmitted, the
-excefs ofthe mod refrangible in the refleded Light will
tinge that Light with their Colour, which is violet and
blue. And this happens by taking the line CprtB any
where between the ends ofthe Prifm H G and EI.

PROP. IX. PROB. IV.

■'By the dijcovered Troferties of Light to e,x^lain the
Colours of the Rjim^h'w.

This -Bow never appears but where it Rains in the
Sun-fhine, and may be made artificially by fpouting up
Water which may break aloft, and fcatter into Drops,
and faH<lown like Rain. For the Sun Ihining upon thefe
Drops certainly caufes the Bow to appear to a Speda-
Cor itanding in a due pofition to the Ram and Sun. And
hence it is now^ agreed upon, that this Bow is made by
refradion of the Sun's Light in Drops of falling Rain.
This was underftood by fome of the Ancients, and of
late more folly difcovered and explained by the Famous

j4ntonit^

C 127 ]

\Ayitonim de ^miinis Archbilhop of SfilatOy in his Boole.
"De Radiis Vift^ if)' L?4as^ publifhed by his Friend Bar^
tolm at I^enice^ in the Year 161 1, and written above
twenty Years before. For he teaches there how the
interior Bow is made in round Drops of Rain by two
refradlions of the Sun's Light, and one reflexion be-
tween them, and the exterior by two refraftions and
two forts of reflexions between them in each Drop of
Water 5 and proves his Explications by Experiments
made with a Phial full ofWater^and with Globes of Glais
filled with W^er, and placed in the Sun to make the
Colours of the two Bows appear in them. The fame
Explication ^es^Cartes hath purfued in his Meteors,
and mended that of the exterior Bow. But whilft they
underftood not the true origin of Colours, it's neceffary
to purfue it here a little further. For underfl:anding
therefore how the Bow is made, let a Drop of Rain or
any other fpherical tranfparent Body be reprefented by
the Sphere B N F G, defcribed with the Center C, and F%. i
Semi-diameter CN. And let AN be one of the Sun's
rays incident upon it at N, and thence refraft^ed to F, ,
wliere let it either go out of the Sphere by refraSion to-
wards 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 refradtion towards S, cutting the inci-
dent ray in Y j,; produce AN and RG, till they meet in
X, and upon A X and N F let fall the perpendiculars
CD and CE, and produce CD till it fall upon the cir-
cumference at L. Parallel to the incident ray AN draw,
the Diameter B Q, and let the fine of incidence out of
Air into Water be to the fine of refraftion as- 1 to.
R. Now if you fuppofe the point of incidence N to-

move

.[128]

move from the point B, continually till It come to L,
the Arch QF will firft increafe and then decrcafe, and
lb will the Angle AXR which the rays AN and GR
contain ; and the Arch Q F and Angle AXR will be
biggeit when ND is to CN as a^iiIrr to //^ RR,
in which cafe N E will be to N D as a R to I. Alfo the
Angle AYS which the rays A N and HS contain will
firft decreafe, and then increafe and grow leaft when
ND is to CN as /^II-RR to//8 RR, in which cafe
N E will be to N D as 3 R to I. And fo the Angle which
the qext emergent ray ( that is, the emergent ray after
three reflexions ) contains with the incident ray AN
will come to its limit whenND is to CN as i/ \\.-^^ to
7/15 RR, in which cafe N E will be to ND as 4.R to I,
and the Angle which the ray next after that emergent,
that is, the ray emergent after four reflexions, con-
tains with the incident will come to its limit^ when
ND is to C N/ as /^ii-RR to /ij^ RR , in which cafe
N E will be to N D as 5 R to Y\ and fo on infinitely,
the numbers 5, 8, 15^ ^4., )Bc\ being gathered by conti-
nual addition of the terms of the arithmetical progreflion
3, 5,7,9,}5)'<r. The truth of all this Mathematicians will
eafily examine. >TtW.i»>^/A i^. ^^z^^"?^/. /^ -^^z

Now it is to be obferved^ that as w^hen the Sun comes
to his Tropicks, days increafe and decreafe but a very
little for a great v/hile together ; fo when by increaf ng
the diftance C D, thefe Angles come to their limits,
they vary their quantity but very little for fome time
together, and therefore a far greater number of the rays
which fall upon all the points N in the Quadrant
B L , fhall emerge in the limiits of thefe Angles ,
tJien in any other inclinations. And further it is

[129]

to be obierved, that the rays which differ in refrangi-
bility will have different limits of their Angles of emer-
gence, and by confequence according tcftheir different
degrees of refrangibility emerge, moff copioufly in dif-
ferent Angles, and being feparated from one another
appear each in their proper Colours. And what thofe
Angles are may be eafily gathered from the foregoing
Theorem by computation.

For in the leaft refrangible rays the fines I and R (as
was found above) are io8 and 8i, and thence by
computation the greateft Angle A X R will be found
42 degrees and a minutes, and the leaft Angle AYS,
50 degr. and 57 minutes. And in the moft refrangible
rays the fines I andR are 109 and 81, and thence by
computation the greateft Angle AX R will be found
4.0 degrees and 17 minutes, and the leaft Angle AYS
54. degrees and 7 minutes.

Suppofe now that O is the Spectator's Eye, and OP a line pig, 1 5 .
drawn parallel to the Sun's rays, and let P O E, P O F,
POG, POH, be Angles of 40 degr. i7min. 41 degn
0, min. 5odegi. 57 min. and 54 degr. 7 min. refped:ively,
and thefe Angles turned about their common fide O P5
(hall with their other fides OE, OF; OG, OH de-
defcribe the verges of two* Rain-bows A F B E and
CHDG. For if E, F, G, H, be Drops placed any
where in the conical fuperficies defcribed by O E, O F,
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. ftiali be the greateft
Angle in which the moft refrangible rays can after one
reflexion be refraded to the Eye, and therefore all the
Drops in the line O E fhall fend the moft refrangible

R rays

[ ISO ]

rays moft copioudy to the Eye, and thereby ftrike the
lenfes with tli£ deepefl: violet Colour in that region.
And in like manner the Angle SFO being equal to
the Angle P OF, or 4.1 deg. 2 min. fhall be the greatefi:
in which the leaft refrangible rays after one reflexion
can emerge out of the Drops, and therefore thofe rays
fhall come moft copioufly to the Eye from the Drops in
the line O F, and ftrike the fenfes with the deepeft red
Colour in that region. And by the fame argument,,
the rays which have intermediate degrees of refrangibi-
lity fhall come moft copioufly from Drops between
E and F, and ftrike the fenfes with the intermedi;ite
Colours in the order which their degrees of refrangibi-
lity require, that is, in the progrefs from E to F, or
from the iniide of the Bow to the outhde in this order^
violet, indico, 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.

Again, the Angle S G O being equal to Angle P O G,
or 50 gr. 51 min. fhall be the leaft Angle in which the
leaft refrangible rays can after two reflexions emerge out
oftheDrops,and therefore the leaft refrangible rays fhall
come moft copioufly to the Eye from the Drops in the
line O G, and ftrike the fenfe with the deepeft red in
that region. And the Angle S HO being equal to the
Angle POH or 54. gr. 7 min. fhall be the leaft Angle in
which the moft refrangible rays after two refledions can
emerge out of the Drops, and therefore thofe rays fliall
come moft copioufly to the Eye from the Drops in the
line O H, and ftrike the fenfes with the deepeft violet in
that region. And by the fame argument, the Drops in
the regions between G and H fliall ftrike the fenfe with

the

the Intermediate Colours in the order which their de>
grees of refrangibility require, that is, in the prooTefs
from G to H, or from the infide of the Bow to the out-
iide in this order^ red, orange, yellow, green, blue, in-
dico, violet. And fmce theie four lines O E, O F, O G.
O H, may be fituated any where in the above-mentioned
conical fuperficies, what is faid of the Drops and Co-
lours in thefe lines is to be underftood of the Drops
and Colours every where in thofe fuperficies.

Thus fhall there be made two Bows of Colours, an
interior and ftronger, by one reflexion in the Drops,
and an exterior and fainter by two j for the Light be-
comes fainter by every reflexion. And their Colours
iliall ly in a contrary order to one another, the red of
both Bows bordering upon the fpace G F which is be-
tween the Bows. The breadth of the interior Bow
EOF meafured crofs the Colours fliall be i degr. 45 min.
and the breadth of the exterior GOH fliall be 3
degr. 10 min. and the diftance between them GOF
(hall be 8gr. 55 min. the greatefl: Semi-diameter of the
innermufl:, that is, the Angle POF being 4a gr. 1 min.
and the leaft Semi-diameter of the outermofl P O G, be-
ing 50 gr. 57 min. Thefe are the meafures 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 in-
ereafed and their diftance decreafed by half a degreej
and fo the breadth of the interior Iris will be 1 degn
1 5 mm. that of the exterior 3 degr, 40 min. their di-
ftance 8 degr, a 5 min. the greateft Semi-diameter of the
interior Bow 4adggr. 17 min. and the leaft of the ex»
terior 50 degn 4^ min. And fiich .are, the dimenfions
of the Bows in the Heavens found to be very nearly,

R a when

[132]

when their Colours appear ftrong and perfed:. For
once, by inch means as I then had, I meafured the
greateft Semi-diameter of the interior Iris about 4.2 de-
grees, the breadth of the red, yellow and green in that
Iris 63 or 64. minutes, befides the outmoft faint red ob-
fcured by brightnefs of the Clouds, for which we
may allow 3 or 4. minutes more. The breadth of the
blue was about 40 minutes more befides the violet,
which was fo much obfcured by the brightnefs of the
Clouds, that I could not meafure its breadth. But
luppofing the breadth of the blue and violet together
to equal that of the red, yellow and green together, the
whole breadth of this Iris will be about 1^ degrees as
above. The leaft diftance between this Iris and the ex-
terior Iris was about 8 degrees and go minutes. The ex-
terior Iris was broader than the interior, but fo faint^
efpecially on the blue fide, that I could not meafure its
breadth diftinftly. At another time when both Bows
appeared more diftindl:, I meafured the breadth of the
interior Iris a gr. ic, and the breadth of the red, yel-
low and green in the exterior Iris, was to the breadth
of the fame Colours in the interior as g to a.

This Explication of the Rain-bow is yet further con-
firmed by the known Experiment ( m.ade by y^ntoni'us
de T)ominis and "Des^Cartes) of hanging^ up any where
in the Sun-lliine a Glafs-Globe filled with Water, and
viewing it in fuch a pofl:ure that the rays which come
from the Globe to the Eye may contain with the Sun's
rays an Angle of either 4^ or 50 degrees. For if the
Angle be about 42 or 43 degrees, the Spectator ( fup-
pofe at O) fliall fee a full red Colour in that fide of the
Globe oppofed to the Sun as 'tis reprefented at F, and

[133]

if that Angle become lefs ( luppole by depreffing the
Globe to E ) there will appear other Colours^ yellow,
green and blue fucceffively in the lame fide of the Globe,
But if the Angle be made about 50 degrees (luppole by
lifting up the Globe to G)there will appear a red Colour
in that fide of the Globe towards the Sun, and if the
Angle be made greater (fuppole by lifting up the Globe
to H) the red will turn fucceffively to the other Colours
yellow, green and blue. The fame thing I have tried by
letting a Globe reft, and railing or depreffing the Eye,
or otherwife moving it to make the Angle of a juft
magnitude.

I have heard it reprefented,. that if the Light of a
Candle be refraSed by a Prifm to the Eye ; when the
blue Colour falls upon the Eye the Spefilator lliall fee
red in the Prifm, and when the red falls upon the Eye
he Ihallfee blue ; and if this were certain, the Colours
of the Globe and Rain-bow ought to appear in a con-
trary order to w^hat we find. But the Colours of the
Candle being very faint, the miftake feems to arife from
the difficulty of difcerning what Colours fall on the
Eye. For, on the contrary, I have fometimes had oc-
cafion to obferve in the Sun's Light refraded by a Prifm,
that the Spectator always fees that Colour in the Prifm
which falls upon his Eye. And the fame I have found
true alfo in Candle-Light. For when the Prifm is mo-
ved llowly from the line which is drawn direftly from the
Candle to the Eye,the red appears firft in the Prifm, and
then the blue, and therefore each of them is feen when^
it falls upon the Eye. For the red pafTes over the Eye
firft, and then the blue.

The

[134-3

The Light which, comes through Drops of Rain by
two refradions without any reflexion, ought to appear
ftrongeft at the diftance of about a 6 degrees from the
Sun, and to decay gradually both ways as the diftance
from him increafes and decreafes. And the fame is to
be underftood of Light tranfmitted through fpherical
Hail-ftones. And if the Hail be a little flatted, as it
often is, the Light tranfmitted may grow fo ftrong at
a little lefs diftance than that of a 6 degrees, as to form
a Halo about the Sun or Moon ; w4iich Halo, as often
as the Hail-ftones are duly figured may be coloured,
and then it muft be red within by the leaft refrangible
rays,and blue without by the moft refrangible ones,efpe-
daily if the Hail-ftones have opake Globules of Snow in
their center to intercept the Light within the Halo ( as
Htigenins has obferved) and make the infide thereof more
diftinflly defined than it would otherwife be. For
fuch Hail-ftones, though fpherical, by terminating the
Light by the Snow, may make a Halo red within and
colourlefs without, and darker in the red than with-
out, as Halos ufe to be. For of thofe rays which pais
clofe by the Snow the rubriform will be leaft refradted^^
and fo come to the Eye in the diredeft lines.

The Light which paffes through a Drop of rain after
two refractions, and three or- more reflexions, is fcarce
ftrons enough to^ caule a feniible Bow^ ; but in thofe Cy-
Hnders of Ice by w^iich Hug^nm explains the TarkeUa^
it may perhaps be fenfible.

PROP.

[135 J

P R O p. X. P R O B. V.

Bi the difcovered "properties of Light to explain the per-^
manent Colours of natural Bodies,

Thefe Colours arife from hence, that fome natural
Bodies relied fome forts of rays, others other forts more
copiouily than the reft^ Minium reflets the leafl: re-
frangible or red-making rays moft copioufly, and thence
appears red. Violets reflect the moft refrangible, moft
copioufly, and thence have their Colour, and fo of other
Bodies. Every Body reflects the rays of its own Colour
more copiouily than the reft, and from their excefs and
predominance in the reflected Light has its Colour.

EX PER. XVII.

For if the homogeneal Lights obtained by the folu-
tion of the Problem propofed in the 4.th Propofition of
the lirftBook you place Bodies, of feveral Colours, you
will find, as I have done, that every Body looks moft
fplendid and luminous in the Light of its ov^n Colour.
Cinnaber in the homogeneal red Light is moft refplen-
dent, in the green Light it is manifeftly lefs refplen-
dent, and in the blue Light ftill lefs. Indico in the
violet blue Light is mioft refplendent, and its fplendor
is gradually diminiftied as it is removed thence by de-
grees through the green and yellow Light to the red.
By a Leek the green Light, and next that the blue and
yellow which compound green, are more ftrongly re-

fleded

fleded than the other Colours red and violet^and fo of the
reft. But to make thefe Experiments the more manifeft,
fuch Bodies ought to be chofen as have the fulleft and
moft vivid Colours, and two of thofe Bodies are to be
compared together. Thus, for inftance, if Cinnaber
Ca^Y^urrr Cyp^'u^ and ultrci marine blue , or fome other full blue be
^d .^Z...^^— -j^^j^ t^S^etherln^e homogeneal Light, they will both
appear red, but the Cinnaber will appear of a ftrongly
luminous and refplendent red, and the ultra marine
blue of a faint obfcure and dark red^; afid if they be
held together in the blue homogeneal Light they will
both appear blue, but the ultra marine will appear of
a ftrongly luminous and refpleij^^ent blue, and the
Cinnaber of a faint and dark blue^ "'Which puts it out
of difpute , that the Cinnaber reflefts the red Light
much more copioully than the ultra marine doth, and
the ultra marine retlefts the blue Light much more co-
pioufly than the Cinnaber doth. The fame Experiment
may be tryed fuccesfuUy with red Lead and Indico, or
with any other two coloured Bodies, if due allowance
be made for the different ftrength or weaknefs of their
Colour and Light.

And as the reafon of the Colours of natural Bodies is
evident by thefe Experimenrs, fo it is further confirmed
and put paft difpute by the two firft Experiments of the
firft Book, whereby 'twas proved in fuch Bodies that
the reflefted Light which differ in Colours do differ alfo
in degrees of refrangibility. For thence it's certain,
that fome Bodies retied the more refrangible, others
the lefs refrangible rays more copioufly.

And

[i37]

And that this is not only a true reafon of thefe Co-
lours, but even the only reafon may appear further
iroiii this confideration, that the Colour of homogeneal
Light cannot be changed by the reflexion of natural
Bodies.

For if Bodies by reflexion cannot in the leaft change
the Colour of any one fort of rays, they cannot appear
coloured by any other means than by refleding thofe
which either are of their own Colour, or which by
mixture mufl: produce it.

But in trying Experiments of this kind care muft be
had that the Light be fufficlently homogeneal. For if
Bodies be illuminated by the ordinary prifmatick Co-
lours, they will appear neither of their own day-light
Colours, nor of the Colour of the Light cafl: on them,
but of fome middle Colour between both, as I have
found by Experience. Thus red Lead ( for inftance )
illuminated with the ordinary prifmatick green will
not appear either red or green, but orange or yellow,
or between yellow and green accordingly, as the green
Light by which "'tis' illuminated is more or lefs com-
pounded. For becaufe red Lead appears red when il-
luminated with white Light, wherein all forts of rays
are equally mixed, and in the green Light all forts of
rays are not equally mixed, the excefs of the yellow-
making, green-making and blue-making rays in the
incident green Light, will caufe thofe rays to abound
fo m.uch in the reflefted Light as to draw the Colour
from red towards their Colour. And becaufe the red
Lead refleds the red-making rays moft copioufly in
proportion to their number, and next after them the
orange-making and yellow-making rays ; thefe rays in

S the

the reflefted Light will be more in proportion to the
Light than they were in the incident green Light, and
thereby will draw the refleded Light from green to-
wards their Colour. And therefore the red Lead will ap-
pear neither red nor green ^but of a Colour between both.
In tranfparently coloured Liquors 'tis oblervable,
that their Colour ufes to vary with their thicknefs.
Thus, for inftance, a red Liquor in a conical Glafs
held between the Light and the Eye, looks of a pale
and dilute yellow at the bottom where 'tis thin, and a
little higher where 'tis thicker grows orange,and where
'tis ftill thicker becomes red, and where 'tis thickeft
the red is deepeft and darkeft. For it is to be conceived
that fuch a Liquor ftops the indico-making and violet-
making rays moft eafily, the blue- making rays more
difficultly, the green^making rays ftill more difficultly,
and the red-making moft difficultly : And that if the
thicknefs of the Liquor be only fo much as fuffices to
ftop a competent number of the violet-making and in-
dico-making rays, without diminilhing much the num-
ber of the reft, the reft muft ( by Prop. 6. TJib, i.) com-
pound a pale yellow. But if the Liquor be fo much
thicker as to ftop alfo a great number of the blue-making
rays, and Ibme of the green-making, the reft muft com-
pound an orange ; and w^here it is fo thick as to ftop
alfo a great number of the green-making and a confi-
derable number of the yellow-making, the reft muft
begin to compound a red, and this red muft grow deeper
and darker as the yellow making and orange-making
rays are more and more ftopt by increaiing the thick-
nefs of the Liquor, fo that few rays beiides the red-
making can get through. . ^

^ Of

[ 1 3P ]

Of this kind Is an Experiment lately related to me by
- Mr. Halle^j^ who, in diving deep into the Sea, found
in a clear Sun-(hine day, that when he was funk many
Fathoms deep into the Water, the upper part of his
Hand in which the Sun fhone diredly through the
Water looked of a red Colour, and the under part of
his Hand illuminated by Light reHe£ted from the Water
below looked green. For thence it, may be gathered,
that the Sea -water reflects back the violet and blue-
making rays moft ealily, and lets the red-making rays
pafs moft freely and copioully to great depths. For
thereby the Sun's dire£t Light at all great depths, by
reafon of the predominating red-making rays, muft
appear red; and the greater the depth is, the fuller
and intenfer muft that red be. And at fuch depths as
the violet-making rays fcarce penetrate unto, the blue-
making, greei)-m.aking and yellow-making rays being
reflected from below iti ore copioully than the red-making
ones, muft compound a green.

Now if there be two Liquors of full Colours, fup-
pofe a red and a blue, and both of them fo thick as
iuffices to make their Colours fufficiently full ; though
either Liquor be fufficiently tranfparent apart, yet
will you not be able to fee through both together. For
if only the red-making rays pafs through one Liquor^
and only the blue-making through the other, no rays
can pafs through both« This Mr. Hook tried cafually
with Glafs-wedges filled with red and blue Liquors^
and was furprized at the unexpected event, the reafon
of it being then unknown ; which makes me truft the
more to his Experiment, though I have not tryed it
my felf But he that would repeat it, muft take care
the Liquors be of very good and full Colours.

S !• Now

ho]

Now whilft Bodies become coloured by rciletfting or
tranfmitting this or that fort of rays more copioufly than
the reft, it is to be conceived that they ftop and ftifle in
themfelves the rays which they do not reflector tranfmit.
For if Gold be foliated and held between your Eye and
the Light, the Light looks blue, and therefore maffy Gold
lets into its Body the blue-making rays to be relied ed
to and fro within it till they be ftopt and ftifled, whilft
it rellefts the yellow-making outvv^ards, and thereby
looks yellow. And miuch after the fame manner that
Leaf-gold is yellow by refledted, and blue by tranfmit-
ted Light, and maffy Gold is yellow in allpoiitions of
the Eye ; there are Ibme Liquors as the tinfture of
luignum Nefhriticum^ and fom.e forts ofGlafs which
tranfmit one fort of Light moft copioufty^ and refiedt
another fort, and thereby look of feveral Colours, ac-
cording to the pofition of the Eye to the Light. But if -
thefe Liquors or Glaffes were fo thick and mafly that
no Light could get through them, I queftion not but
that they would like all other opake Bodies appear of
one and, the fame Colour in all poiitions of the Eye,,
though this I cannot yet affirm by experience. For all
coloured Bodies, fo far as my Qbiervation reaches, may
be feen through if made fufficiently thin, and therefore
are in fome m.eafure tranfparent, and differ only in de-
grees of tranfparency from tinged tranfparent Liquors;
thefe Liquors, as well as thofe Bodies, by a fufiicient
thicknefs becoming opake. A tranfparent Body which
Iboks of any Colour by tranfmitred Light, may alfo..
look of the fame Colour by refle&ed Light, the Light
of that Colour being refteiied by the further furface of
the Body, or by the Air beyond it. And then the re-
flected Colour will be dimiiiifhed^ and perhaps ceafe, by

making

[ H^- ]

making the Body very thick, and pitching it on the
back-fide to diminifh the reflexion of its further furface,
lb that the Light refleded from the tinging particles
may predominate. In fuch cafes, the Colour of the re-
flefted Light will be apt to vary from that of the Light
tranfmitted. But v^hence it is that tinged Bodies and
Liquors refleft fome fort of rays, and intromit or tranf-
mit other forts, Ihall belaid in the next Book. In this
Propofition I content my felf to have put it paft difpute,^
that Bodies have fuch Properties, and thence appear
coloured.

PROP. XL PROB. VL

5y mixing coloured Lights to comfound a Beam of Liohp
of thejame Colour and NaXvjre 'with a Beam of the Suns'^
dtreiH L'^ght^ and therein to experience the truth of the.
foregoing Trofofitions,

Let A B C a b c reprefent a Prifm by which the Sun's Fig^ r&
Light let into a dark Chamber through the Hole F, may.
be refracted towards the Lens M N, and paint upon it
at p, q, r, s and t, the uliial Colours violet, blue, green^,
yellow and red, and let the diverging rays by the re-
fraftion of this Lens converge again towards X, and.
there,by the mixture of all thofe their Colours,compound.
a white according to what was (liewn above. Then let
another Prifm DEGdeg, parallel to the former, be
placed at X, to refrad: that white Light upwards to-
v/ards Y. Let the refrading Angles of the Prifms^,,
aad their diftances from the Lens be equal, fo that the.
rays which converged from the Lens towards X, and.
without refradion, would there have croffed and diver-
ged again, may by the refradion of the fecond Priim be

redaced.

reduced into Parallelifm and diverge no more. For
then thofe rays will recompofe a Beam of white Light
XY. If the refracting Angle of either Prifm be the
bigger, that Prifm mult be fb much the nearer to the
Lens. You will know when the Prifms and the Lens
are well fet together by obferving if the Beam of Light
XY which comes out of the fecond Prifm be perfedly
white to the very edges of the Light, and at all diftan-
•ces from the Prifm continue perfedly and totally white
like a Beam of the Sun's Light. For till this happens,
the pofition of the Prifms and Lens to one another mult
be correded, and then if by the help of a long Beam of
Wood, as is reprefented in the Figure, or by a Tube,
or fome other fuch inftrument made for that purpofe,
they be made faft in that fituation, you may try all the
lame Experiments in this compounded Beam of Light^
XY, which in the foregoing Experiments have been
nrade in the Sun's direft Light. For this compounded
Beam of Light has the fame appearance, and is endowed
with all the fame Properties with a direCt Beam of the
Sun's Light, fo far as my Obfervation reaches. And in
trying Experiments in this Beam you may by flopping
any of the Colours p, q, r, s and t, at the Lens, fee how
the Colours produced in the Experiments are no other
than thofe which the rays had at the Lens before they
entered the compolition of this Beam : And by confe-
quence tiiat they arife not from any new modifications
of the Light by refraftions and reliexions, but from the
various feparations and mixtures of the rays originally
endowed with their colour-making qualities.

So, for inftance, having with a Lens 4; Inches broad,
and two Prifms on either Hand 6\ Feet diftant from the
Lens, made fuch a Beam of compounded Light : to

examin

CH3]

€xamm the reafon of the Colours made by Priiliis^ 1
refraded this compounded Beam of Light X Y with
another Prifm HIK kh, and thereby call the ufual prit
matick Colours P QR S T upon the Paper LV placed be-
hind. And then by ftopping any of the Colours p, q^
r, s, t^ at the Lens^ I found that the fame Colour would
vanifh at the Paper. So if the purple P was flopped at
the Lens, the purple P upon the Paper would vanifh,
and the reft of the Colours would remain unaltered,
unlefs perhaps the blue, fo far as fome purple latent in
it at the Lens might be feparated from it by the fol-
lowing refra£tions. And fo by intercepting the green
upon the Lens, the green R upon the Paper would va-
nilh, and fo of the reft ; which plainly Ihev/s, that as
the white Beam of Light X Y was compounded of fe-
ve Lights varioufly coloured at the Lens, fo the Co-
lours w^hich afterwards emerge out of it by new refra-
ftions are no other than thofe of which its whitenefs
was compounded. The refradlion of the Prifm HIK
kh generates the Colours PQR ST upon the Paper,
not by changing the colorific qualities of the rays, but
by feparating the rays which had the very fame colorific
qualities before they entered the compofition of the re-
fraded Beam white of Light X Y. For otherwife the rays
which were of one Colour at the Lens might be of ano-
ther upon the Paper, contrary to w^hat we find.

So again, to examin the reafon of the Colours of na-
tural Bodies, I placed fuch Bodies in the Beam of Light
X Y, and found that they all appeared there of thofe
their own Colours which they have in Day-light, and
that thofe Colours depend upon the rays which had the
fame Colours at the Lens before they enti^ed the compo-

[H4-]

fition of that Beam. Thus, for infcancc^Cinnaber illumi-
nated by this Beam appears of the fame red Colour as in
Day-light ; and if at the Lens you intercept the green-
making and blue-making rays^ its rednefs will become
more full and lively : But if you there intercept the red-
making rays, it will not any longer appear red, but be-
come yellow or green, or of fom.e other Colour, accor-
ding to the forts of rays which you do not intercept.
So Gold in this Light XY appears of the fame yellow
Colour as in Day-light, but by intercepting at the Lens a
due quantity of the yellow-making rays it will appear
white like Silver (as 1 have tryed) which fiiews that its
yellownefs arifes from the excefs of the intercepted rays
tinging that whitenefs with their Colour when they are
let pals. So the infufion oi Lignum Nefhritlcum ( as I
have alfo tryed ) when held in this Beam of Light XY,
looks blue by the reflefted part of the Light, and yellow^
by the tranimitted part of it, as when 'tis viewed in Day-
light, but if you intercept the blue at the Lens the infu-
fion will lofe its reflefted blue Colour, whilft its tranf-
mitted red remains perfeft and by the iofs of fome blue-
making rays vv^herewith it was allayed becomes morein-
tenie and full. And, on the contrary, if the red and orange-
making rays be intercepted at Lens, the infuiion will
iole its tranimitted red, whilfl its blue will remain and
become more full and perfedl. Which ihews, that the in-
fufion does not tinge the rays with blue and yellow, but
- only tranlmit thofe moft copioufly which were red-ma-
king before, and refleds thofe molt copioufly w^hich were
blue-making before. And after the fame manner may the
reafons..of other Phaenomena be examined, by trying
themx in this artificial Beam of Light X Y.

THE

Book I. Part H. Plate I.

J^i 2a-

6 ^ ^ ^ F

Fl^^l,. B,/^

i G....-- ^

L 5 I

^7 4

&_^

<Zl -■■■:.. M-

3 ^

IS, ±

3 \S 2

^^5-

Book I. Part n. Plate E.

sa:

X Bookl.PartH. Plate m.

Book 1. Part I.Place R".

THE

SECOND BOOK

O F

O P T I C K S.

PARTI.

Oifervations concerning the Reflexions^ RefraBions^ and
Colours of thin tranjfarent Bodies,

IT has been obferved by others that tranfparent
Subftances, as Glafs, Water, Air, Jf^r. when made
very thin by being blown into Bubbles, or otherwife
formed into Plates, do exhibit various Colours accor-
ding to their various thinnefs, although at a greater
thicknefs they appear very clear and colourlefs. In
the former Book I forbore to treat of thefe Colours^
becaufe they feemed of a more difficult confideration,
and were not neceffary for eftablifhing the Properties
of Light there difcourfed of. But becaufe they may
conduce to further difcoveries for completing the
Theory of Light, efpecially as to the conftitution of
the parts of natural Bodies, on which their Colours or
Tranfparency depend ; I have here fet down an ac-
count of them. To render this Difcourfe fliort and
diftinCt, I have firft defcribed the principal of my

[2]

Obfervations, and then confidered and made ufe of
them. The Obfervations are thefe.

O B S. I.

Compreffing two Prifms hard together that their
Sides (which by chance were a very little convex)might
fomewhere touch one another : I found the place in
which they touched to become abfolutely tranfparent^
as if they had there been one continued piece of Glafs.
For when the Light fell fo obliquely on the Air, which
in other places was between them^as to be all refleded ;
it feemed in that place of contad to be wholly tranf-
mitted, infomuch that v/hen looked upon, it appeared
like a black or dark Spot, by reafon that little or no
fenfible Light was refleded from thence, as from other
places; and when looked through it feemed (as it were)
a hole in that Air w^iich was formed into a thin Plate^
by being compreffed between the Glaffes. And through
this hole Objects that were beyond might be feen di-
ftindly, which could not at all be feen through other
parts of the Glaffes where the Air was interjacent. Al-
though the Glaffes were a little convex, yet this tranf-
parent Spot was of a confiderable breadth,which breadth
feemed principally to proceed from the yielding inwards
of the parts of the Glaffes, by reafon of their mutual
preffure. For by preffing them very hard together it
would become much broader than otherwile.

OBS.

O B S. 11.

When the Plate of Air, by turning thePrifms about
their common Axis., became fo little inclined to the in-
cident RaySj that fome of them began to be tranfmit-
ted, there arofe in it many flender Arcs of Colours
which at fir ft were fhaped almoft like the Conchoid,
as you fee them delineated in the firft Figure. AndF^^^ ^
by continuing the motion of the Prifms, thefe Arcs in-
creafed and bended more and more about the faid tranf-
--parent Spot, till they were completed into Circles or
Rings incompaffing it, and afterwards continually grew
more and m.ore contrafted.

Thefe Arcs at their firft appearance were of a violet
and blue Colour, and between them were white Arcs
of Circles, which prefently by continuing the motion of
the Prifms became a little tinged in their inward Limbs
with red and yellow, and to their outward Limbs the
blue was adjacent. So that the order of thefe Colours
from the central dark Spot, was at that time white,
blue, violet ; black ; red, orange, yellow, white, blue,
violet, Js'r. But the yellow and red were much fainter
than the blue and violet.

The motion of the Prifms about their Axis being con-
tinued, thefe Colours contracted more and more,ftirink=
ing towards the whitenefs on either fide of it, until they
totally vanifhed into it. And then the Circles in thofe
•parts appeared black and white, without any other Co-
lours intermixed. But by further moving the Prifms
about, the Colours again emerged out of the whitenefs,
the violet and blue as its inward Limb, and at its out-

A a 2-

[4]

ward Limb the red and yellow. So that now their order
from the central Spot was white, yellow, red ; black ;
violet, blue, white, yellow, red, Oc, contrary to what
it was before.

O B S. Ill

When the Rings or fome parts of them appeared only
black and white, they were very diftin£t and v/ell de-
fined, and the backnefs feemed as intenfe as that of
the central Spot. Alfo in the borders of the Rings,,
where the Colours began to emerge out of the white-
nefs, they were pretty diftinCt, which made them vi-
iible to a very great Multitude. I have fom.etimes
numbred above thirty Succeffions ( reckoning every
black and white Ring for one Succeffion ) and feen
more of them., which by reafon of their fmalnefs I could
not number. But in other Pofitions of the Prifms, at
which the Rings appeared of many Colours, I could not
diftinguifh above eight or nine of them, and the exte-
rior of thofe were very confufed and dilute.

In thefe two Obfervations to fee the Rings diftinft,^
and without any other Colour than black and white,!
found it neceflary to hold my Eye at a good diftance
from them. For by approaching nearer, although in the
fame inclination of my Eye to the plane of the Rings,
there emerged a blueifh Colour out of the white,
which by dilating it felf more and more into the black
tendred the Circles lefs diftind, and left the white a
little tinged with red and yellow. I found alfo by
looking through a flit or oblong hole , which was
flarrower than, the Pupil of my Eye, and held clofe to

[5]

it parallel to the Prlfms^ I could fee the Circles much
diftinder and vifible to a far greater number than
otherwife.

O B S. IV.

To obferve more nicely by the order of the Colours-
which arofe out of the white Circles as the Rays be-
came lefs and lefs inclined to the plate of Air; 1 took
two Object Glaffes, the one a Plano-convex for a four-
teen-foot Telefcope, and the other a large double con-
vex for one of about fifty-foot; and upon this^laying the
other with its its plane-fide downwards, I prefled them
llowly together^to make the Colours fucceffively emerge
in the middle of the Circles, and then flowly lifted
the upper Glafs from the lower to make them fuccef-
fively vanilh again in the fame place. The Colour,
which by preffing the Glaffes together emerged laft in
the middle of the other Colours, would upon its firft
appearance look like a Circle of a Colour ahnoft uni-
form from the circumference to the center , and by
comprefling the Glaffes ftill more, grow continually
broader until a new Colour emerged in its center, and
thereby it became a Ring encompafling that new Co^
lour. And by comprefling the Glaffes ftill more, the
Diameter of this Ring would encreafe, and the breadth
of its Orbit or Perimeter decreafe until another new
Colour emerged in the center of the laft : And fo on
until a third, a fourth, a fifth, and other following
new Colours fucceflively emerged there, and became
Rings encompaffmg the innermoft Colour, the laft of
which was the black Spot. And, on the contrary, by

lifting up the upper Glafs from the lower, the diameter
of the Rings would decreafe, and the breadth of their
Orbit encreafe, until their Colours reached fucceffively
to the center ; and then they being of a confiderable
breadth, I could more eafily difcern and diltinguifh
their Species than before. And by this means 1 ob-
ferved their Succeffion and Quantity to be as fol-
loweth.

Next, to the pellucid central Spot made by the con-
taft of the Glaffes fucceeded blue, white, yellow, and
red, the blue was fo little in quantity that I could not
difcern it in the circles made by the Prifms, nor could
I well diftinguifh any violet in it, but the yellow and
red were pretty copious, and feemed about as much
in extent as the white , and four or five times more
than the blue. The next Circuit in order of Colours
immediately encompaffing thefe were violet, blue,
green, yellow, and red, and thefe were all of them co-
pious and vivid, excepting the green, which was very
little in quantity, and feemed much more faint and
dilute than the other Colours. Of the other four, the
violet was the leaft in extent , and the blue lefs than
the yellow or red. The third Circuit or Order was
purple, blue, green, yellow, and red ; in which the
purple feemed more reddifli than the violet in the
former Circuit, and the green was much more confpi-
cuous, being as brifque and copious as any of the other
Colours, except the yellow ; but the red began to be
a little faded, inclining very much to purple. After
this fucceeded tlie fourth Circuit of green and red. The
green was very copious and lively, inclining on the one
lide to blue, and on the other fide to yellow. But in

this

C7],

this fourth Circuit there was neither violet, blue, nor
yellow, and the red was very imperfed and dirty.
Alfo the fucceeding Colours became more and more im-
perfed and dilute, till after three or four Revolutions
they ended in perfect whitenefs. Their Form, when the
Glaffes were moft compreffed fo as to make the black
Spot appear in the Center, is delineated in the Second
Figure ; where a^ b^ r, J, e ;/, o, /?, i, ^ : /, m^ n^ o^ f : q^ ?- ; F^g'. ^
J", t : v^x'.y denote the Colours reck'ned in order from
the center, black, blue, white, yellow, red : violet,
blue, green, yellow, red : purple, blue, green, yellow,
red : green, red : greenifh blue, red : greenilTi blue^
pale red : greenifli blue, reddiih white,

O B S. V.

To determine the interval of the Glaffes, or thick-
nefs of the interjacent Air, by which each Colour was
produced, I meafured the Diameters of the firft fix
Rings at the moft lucid part of their Orbits, and fqua-
ring them, I found their Squares to be in the Arith-
metical Progreffion of the odd Numbers, 1,3.5.7,9.11.
And fince one of thefe Glaffes was Plain, and the other
Spherical, their Intervals at thofe Rings muft be in the
fame Progreffion. I meafured alfo the Diameters of
the dark or faint Rings between the more lucid Co-
lours, and found their Squares to be in the Arithme-
tical Progreffion of the even Numbers, 1. 4.. 6c 8. 10= i cj.
And it being very nice and difficult to take thefe mea-
fures exaftly ; I repeated them at divers times at divers
parts of the Glaffes, that by their Agreement I might
be confirmed in them.. And the fame Method I ufed in

determining fome others of the following Obferva*
tions.

O B S. VI.

The Diameter of the fixth Ring at the moft lucid
part of its Orbit was ^^ parts of an Inch, and the Dia-
meter of the Sphere on which the double convex Ob-
jeft'Glafs was ground was about loa Feet, and hence
I gathered the thicknefs of the Air or Aereal Interval
of the Glafles at that Ring. But fome time after, fuf-
ped:ing that in making this Obfervation I had not de-
termined the Diameter of the Sphere with fufficient ac-
curatenefs, and being uncertain whether the Plano-
convex Glafs was truly plain, and not fomething con-
cave or convex on that fide which I accounted plain ;
and whether I had not preffed the Glafles together, as

I often did, to make them touch, (for by prefling fuch
Glafles together their parts eafily yield inwards, and
the Rings thereby become fenfibly broader than they
would be, did the Glafles keep their Figures.) I re-
peated the Experiment, and tound the Diameter of
the fixth lucid Ring about 7^ parts of an Inch. I re-
peated the Experiment alfo with fuch an Objedt-Glafs
of another Telefcope as I had at hand. This was a double
convex ground on both fides to one and the fame
Sphere, and its Focus was diftant from it 8^j Inches.
And thence, if the Sines of incidence and refra£tion of
the bright yellow Light be aflumed in proportion as

II to 17, the Diameter of the Sphere to which the
Glafs was figured will by computation be found 1 82 In-
ches. This Glafs I laid upon a flat one, fo that the

black

black Spot appeared in the middle of the Rings of Colours
without any other preffure than that of the weight of
the Glafs. And now meafuring the Diameter of the
fifth dark Circle as accurately as 1 could, I found it the
fifth part of an Inch precifely. This meafure was taken
with the points of a pair of Compafles on the upper fur-
face on the upper Glafs, and my Eye was about eight
or nine Inches diftance from the Glafs, almoft perpen-
dicularly aver it, and the Glafs w^as I of an Inch thick,
and thence it is eafy to colleft that the true Diameter
of the Ring between the Glaflfes was greater than its
mcafured Diameter above the Glaflfes in the proportion
of 80 to 79 or thereabouts, and by confequence equal
to ~ paTts of an Inch, and its true Semi-diameter equal
to ^ parts. Now as the Diameter of the Sphere ( 1 81 In-
ches) is to the Semi-diameter of this fifth dark Ring
( ~ parts of an Inch ) fo is this Semi-diameter to the
.thicknefs of the Air at this fifth dark Ring ; which is
therefore ^, or j^^ parts of an Inch, and the fifth
part thereof; viz. the sgy^^th part of an Inch, is the
thicknefs of the Air at the firft of thefe dark Rings.

The fame Experiment I repeated with another dou-
ble convex Objeft-glafs ground on both fides to one and
the fame Sphere. Its Focus was diftant from it i68[
Inches, and therefore the Diameter of that Sphere was
184 Inches. This Glafs being laid upon the fame
plain Glafs, the Diameter of the fifth of the dark
Rings, when the black Spot in their center appeared
plainly without preffing the Glaflfes , was by the mea»
fure of the Compaflfes upon the upper Glafs --^ parts
of an Inch, and by confequence between the Glafles it
waS|^. For the upper Glafs was ^ of an Inch thick^

Bb and

[lo]

and my Eye was diftant from it 8 Inches. And a third
proportional to half this from the Diameter of the
Sphere is gglfo parts of an Inch. This is therefore the
thicknefs of the Air at this Ring^ and a fifth part there-
of^ vit. the ggg^o^h part of an Inch is the thicknefs there-
of at the firft of the Rings as above.

I tryed the fame thing by laying thefe Objedt-Glaffes
upon liar pieces of a broken Looking-gkifs, and found
the fame meafures of the Rings : Which makes me
rely upon them till they can be determined more ac-
curately by GlalTes ground to larger Spheres^ though
in fuch Glaffes greater care muft be taken of a true
plain.

Thefe Dimenfions were taken when my Eye was
placed almoft perpendicularly over tlie Glaffes^ being
about an Inch ^ or an Inch and a quarter^ diftant from
the incident rays^ and eight Inches diftant from the
Glafs ; fo that the rays were inclined to the Glafs in an
Angle of about 4. degrees. Whence by the following
Obfervation you will underftand ^ that had the rays
been perpendicular to the Glaffes., the thicknefs of the
Air at thefe Rings would have been lefs in the propor-
tion of the Radius to the fecant of 4 degrees, that is of
loooo. Let the thickneffes found be therefore dimi-
nillied in this proportion, and they will becomie — ^ and
&9^5 or ( to ufe the neareft round number ) the g^th
part of an Inch. This is the thicknefs of the Air at the
darkeft part of the firft dark Ring made by perpendi-
cular rays, and half this thicknefs multiplied by the
progreffion,i,3,5,7,9, i \^c. gives the thickneffes of the
Air at the moft luminous parts of all the brighteft
Rings, viz. :^^r.T^,,,,T^^,Tjt;,,^c, their arithmetical

means

[II]

, b'/T. being its thickneffes at the

iliedllb i^acoo? 17600CJ i7«oo . _

darkeft parts of all the dark ones

O B S. VII.

The Rings were leaf!: when my Eye was placed per-
pendicularly over the Glaffes in the Axis of the Rings :
And when I viewed them obliquely they became big-
ger, continually fwelling as I removed my Eye further
from the Axis. And partly by meafuring the Diameter
of the fam.e Circle at feveral obliquities of my Eye,
partly by other means, as alfo by making ufe of the
two Frifms for very great obliquities. I found its Dia-
meter, and confequently the thicknefs of the Air at its
perimeter in all thofe obliquities to be very nearly in the
proportions exprefled in this Table.

Angle

of In-

Angle of Re-

Diameter of

Thicknefs of

cidence

on the

fra^ion into

the King,

the Air.

Air,

the Air,

deg.

min.

00 00

10 00

lorj

1°^ .

10 00

1°^

lOf

50 00

I of

II-:

40 00

"-:

50 00

i2i

15^

60 00

65 00

157

^3^

70 00

i6|

i8f

.38

75 00

195

37,

80 00

aaf

52J

85 00

1 1

90 00

iaa|

i5b 2

[12]

In the two firft Columns are expreffed the QbHq;Uities
of the incident and emergent rays to the plate of the
Air, that is^ their angles of incidence and refradion. In
the third Column the Diameter of any coloured Ring
at thofe obliquities is expreffed in parts, of which ten
conftitute that Diameter when the rays are perpendicu-
lar. And in the fourth Column the thickneis of the Air
at the circumference of that Ring is expreffed in parts
of which alio ten conftitute that thicknefs when the rays«
are perpendicular.

And from thefe meafures I feem to gather this Rule r
That the thicknefs of the Air is proportional to the fe-*
cant of an angle, whofe Sine is a certain mean propor-
tional between the Sines of incidence and retraftion.,-
And that mean proportional, fo far as by thefe meafures^
I can determine it, is the firft of an hundred and fix.
arithmietical mean proportionals between thofe Sines
counted from the Sine of refraction when the refra-
filion is made out of the Glafs into the plate of Air, or
from the Sine of incidence when the refraction is
made out of the plate of Air into the Glafs o

O B S. VIIL

The dark Spot in the middle of the Rings increafed
alfo by the obliquation of the Eye, although almoft in-
fenfibly. But if infteadoftheObjeft-Glaffes tbePrifms
were made ufe of, its increafe was more manifeft when
viewed fo obliquely that no Colours appeared about it.
It was leaft when the rays were incident moft obliquely
on the interjacent Air, and as the obliquity decreafed
it increafed more and more imtil the coloured Rings ap-
peared.

C 13 ].

peared^ and then d£creafed agaiia^ but rtot fo much as
it increafed before. And hence it is evident, that tlie
tranfparency was not only at the abiblute contaft of the
Glafles, but alfo where they, had fome Httle interval.
1 have ibmetimes obferved the Diameter of that Spot to
be between half and two fifth parts of the Di-ameter of
the exterior circumference of the red in the firft cir-
cuit or revolution of Colours when viewed almoft per-
pendicularly ; whereas when viewed obliquely it hath
wholly vanilhed and become opake a*nd white like the
other parts of the Glafs ; whence it may be colledted
that the GlalTes did then fcarcely, or not at all, touch
one another, and that their interval at the perimeter
of that Spot when viewed perpendicularly was about a
fifth or fixth part of their interval at the circumference
of the faid red.

O B S. IX.

By looking through the two contiguous Objeff-
Glaffes, I found that the interjacent Air exhibited Rings
of Colours, as well by tranfmitting Light as by reflect-
ing it. The central Spot was now white, and from it
the order of the Colours were yellowifh red ; black ;
violet, blue, white, yellow, red; violet, blue, green^
yellow, red, '^c\ But thefe Colours were very faint
and dilute unlefs when the Light was trajefted very
obliquely through the Glafles : For by that means they
became pretty vivid. Only the firft yellowifh red, like
the blue in the fourth Qbfervation, was fo little and
faint as fcarcely to be difcerned. Comparing the co-
loured Riiigs made by reflexion, with thefe made by

tranl"

tranfmiffion of the Light ; I found that white was op-
pofite to black, red to blue, yellow to violet, and green
to a compound of red and violet. That is, thofe parts
oftheGlafs were black when looked through, which
when looked upon appeared white, and on the con-
trary. And fo thofe which in one cafe exhibited blue,
did in the other cafe exhibit red. And the like of the
K^". 3- other Colours. The manner you have reprefen ted in
the third Figure, where A B, C D, are the furfaces of
the Glafles contiguous at E, and the black lines be-
tween them are their diftances in arithmetical progref-
fion, and the Colours written above are feen by re-
flected Light, and thofe below by Light tranfmitted.

O B S. X.

Wetting the Objed-Glafles a little at their edges,
the water crept in flowly between them, and the Cir-
cles thereby bec-ame lefs and the Colours more faint :
Infomuch that as the water crept along one half of
them at w^hicli it firft arrived would appear broken oif
from the other half, and contracted into a lefs room.
By meafuring them I found the proportions of their
Diameters to the Diameters of the like Circles made by
Air to be about feven to eight, and confequently the in-
tervals of the Glafles at like Circles, caufed by thofe
two mediums Water and Air,are as about three to four.
Perhaps it may be a general Rule, That if any other
medium more or lefs denfe than water be comprefled
between the Glafles, their intervals at the Rings caufed
thereby will be to their intervals caufed by interjacent

/Vir*

J ^5 1

Air, as the Sines are which meafure the refradtion made
out of that medium into Air.

OB S. XL

When the water was between the Glafles, if I pref-
fed the upper Glafs varioufly at its edges to make the
Rings move nimbly from one place to another, a little
white Spot would immediately follow the center of
them, which upon creeping in of the ambient water
into that place w^ould prefently vanifh. Its appearance
was fuch as interjacent Air would have caufed, and it-
exhibited the fame Colours. But it was not Air, for
where any bubbles of Air were in the water they would
not vanifh.. The reflexion muft have rather been caufed
by^a^ fuEtiler medium, which could recede through the,,
GlalTes at the creeping in of the water,

OB S. XII.

Thefe Obfervations were made in the open Air. Bute
further to examin the efFeds of coloured Light falling
on the Glafles, I darkened the Room, and viewed them
by reflexion of the Colours of a Prifm caft on a Sheet
of white Paper, my Eye being fo placed that I could
fee the coloured Paper by reflexion in the Glafles, as.
in a Looking-glafs. And by this means the Rings be-
came diftin£ter and vifible to a far greater number than-
in the open Air. I have fometimes feen more than
twenty of them, whereas in the open Air I could not
difcern above eight or nine.

Id]

O B S. XII I.

Appointing an affiftant to move the Prifm to and
fro about its Axis, that all the Colours might fuccef-
fively fall on that part of the Paper which I faw by
reflexion from that part of the Glaffes, v/here the Cir-
cles appeared, fo that all the Colours might be fuccef-
lively refledled from the Circles to my Eye whilfl: I held
it immovable, I found the Circles which the red Light
made to be manifeftly bigger than thofe which were
made by tlie blue and violet. And it was very plea-
fant to fee them gradually fwell or contrad: according
as the Colour of the Light was changed. The inter-
val of the Glafles at any of the Rings when they were
made by the utmoft red Light, was to their interval at
the fame Ring when made bythe utmoft violet, greater
than as 3 to a,and lefs than as 1 9 to 8,by the moft of my
Obfervations it was as 14. to 9. And this proportion
feemed very nearly the fame in all obliquities of my
Eye ; unlets when two Prifms were made ufe of inftead
of the Objeft-Gtaffes. For then at a certain great
obliquity of my Eye, the Rings made by the fet^ral
Colours feemed equal, and at a greater obliquity thofe'
made by the violet would be greater than the fame
Rings made by the red. The refraftion of the Prifm
in this cafe caufing the moft refrangible rays to fall
more obliquely on that plate of the Air than the leaft
refrangible ones. Thus the Experiment fucceeded in
the^ coloured Light, which was fufficiently ftrong and
copious to make the Rings fenfible. And thence it
may be gathered, that if the moft refrangible and leaft

refran-

[I?]

refrangible rays had been copious enough to make the
Rings feniible without the mixture of other rays, the
proportion which here was 14 to 9 would have been a
little greater^ fuppofe 14. » or 14. Uo 9.

O B S. XIV.

Whilil the Prifm was turn'd about its Axis with an
uniform motion, to make all the feveral Colours fall
fucceffively upon the Objeft-Glafles, and thereby to
make the Rings contraft and dilate : The contraftion
or dilation of each Ring thus made by the variation of
its Colour was fwiftcft in the red, and floweft in the
violet, and in the intermediate Colours it had inter-
mediate degrees of celerity. Comparing the quantity
o{ contraftion and dilation made by all the degrees of
each Colour, I found that it was greateft in the red ;
lefs in the yellow, ftill lefs in the blue, and leaft in the
violet. And to make as juft an eftimation as I could of the
proportions of their contractions or dilations, I obferved
that the w^hole contraftion or dilation of the Diameter
of any Ring made by all the degrees of red, was to that
of the Diameter of the fame Ring made by all the de-
grees of violet, as about four to three, or five to four, and
that when the Light was of the middle Colour between
yellow and green, the Diameter of the Ring was very
nearly an arithmetical mean between the greateft Dia-
meter of the fame Ring made by die outmoft red, and
the leaft Diameter thereof made by the outmoft violet :
Contrary to what happens in the Colours of the oblong
Spedrum made by the refradion of a Prifm, where the
red ismoft contracted, the violet moft expanded, and

D d * in

[i8]

ill the midft of all the Colours is the confine of green
and blue. And hence 1 feem to colleft that the thick-
neffes of the Air between the Glaffes there, where the
Ring is fucceffively made by the limits of the five prin-
cipal Colours (red, yellow, green, blue, violet) in order
( that is, by the extreme red, by the limit of red and
yellow in the middle of the orange, by the limit of
yellow^ and green, by the limit of green and blue, by
the limit of blue and violet in the middle of the in-
digo, and by the extreme violet ) are to one another
very nearly as the fix lengths of a Chord which found
the notes in a fixth Major, fol^ la^ ml^ fa^ fol^ la. But
it agrees fomething better with the Obfervation to fay,
that the thicknefles of the Air between the Glaffes there,
where the Rings are fucceffively made by the limits of
the feven Colours, red, orange, yellow, green, blue, in-
digo, violet in order, are to one another as the Cube-
roots of the Squares of the eight lengths of a Chord^
which found the notes in an eighth , fol^ la^ fa^ fol^ la^
mij fii^ fol ; that is, as the Cube-roots of the Sq[uares
of the Numbers, i,|;a, i i, f, tI,i--

OB a XV.

Thefe Rings were not of various Colours like thofe
made in the open Air^ but appeared all over of that
prifmatique Colour only with which they were illu-
minated. And by projefting the prifmatique Colours
immediately upon the Glaffes, 1 found that the Light
which fell on the dark Spaces which were between
the coloured Rings , was tranfmitted through the
Glaffes without any variation of Colour. For on a

white

J9]

white Paper placed behind, it would paint Rings of
the fame Colour with thofe which were reflefted, and
of the bignefs of their immediate Spaces. And from
thence the origin of thefe Rings is manifeft ; namely,
That the Air between the GlajQTes, according to its va-
rious thicknefs, is difpofed in fome places to reflied:,
and in others to tranfmit the Light of any one Co-
lour (as you may fee reprefented in the fourth Figure ) pig
and in the fame place to refled that of one Colour
where it tranfmits that of another.

O B S. XVL^

The Squares of the Diameters of thefe Rings made
any prifmatique Colour were in arithmetical pro-
greffion as in the fifth Obfervation. And the Diameter
of the fixth Circle, when made by the citrine yellow^
and viewed almoit perpendicularly, was about f|;7 parts
of an Inch, or a little lefs, agreeable to the iixth Ob-
fervation.

. The precedent Obfervations were made with a rarer
thin medium, terminated by a denfer, fuchas was Air
or Water compreffed between two Glaffes. In thofe
that follow are fet down the appearances of a denfer
medium thin^'d within a rarer, fuch as are plates of
Mufcovy-glafs, Bubbles of Water, and fome other thin
fubftances terminated on all fides with Air»

U uJii^

d 2 OfiS.

O B S. XVII.

If a Bubble be blown with Water firft made tenacious
by diflblving a little Soap in it, 'tis a common Obfer-
varion, that after a while it will appear tinged with a
great variety of Colours. To defend theie Bubbles
from being agitated by the external Air (w- hereby their
Colours are irregularly moved one among another, fo
that no accurate Obfervation can be made of them,) as
foon as 1 had blown any of them I covered it with a
clear Glafs, and by that means its Colours emerged in
a very regular order, like fo many concentrick Rings
incompaffing the top of the Bubble. And as the
Bubble grew thinner by the continual fubfiding of the
Water, thefe Rings dilated ilow^ly and over-fpread the
ivhole Bubble, defcending in order to the bottom of it,
where they vaniihted fucceffively. In the mean while,
after all the Colours were emerged at the top, there
grew in the Center of the Rings a fmall round black
Spot, like that in the firft Obfervation, which conti-
nually dilated it felf till it became fometimes more than
I or I of an Inch in breadth before the Bubble broke.
At firft I thought there had been no Light reflefted from
the Water in that place, but obferving it more cu-
rioufly, I faw within it feveral fmaller round Spots,
which appeared much blacker and darker than the reft,
whereby I knew that there was fome reflexion at the
other places which were not fo dark as thofe Spots.
And by further tryal I found that I could fee the Images
of fokiue things ( as of a Candle or the Sun ) very faint-
ly reflected, not only from the great black Spot, but

alio

[21]

alio from the little darker Spots whicii were with-
in it.

Befides the aforefaid coloured Rings there would
often appear fmall Spots of Colours, afcending and de-
fcending up and down the fides of the Bubble, by reafon
of fome inequalities in the fubfiding of the Water.
And fometimes fmall black Spots generated at the fides
would afcend up to the larger black Spot at the top of
the Bubble, and unite with it.

O B S. XVIIL

Becaufe the Colours of thefe Bubbles were more ex-
tended and lively than thofe of the Air thin'd between
two GlafiTes, and fo more eafy to to diftinguilhed , I
fhail here give you a further delcriptior^ of their order,,
as they were obferved in viewing them by reflexion of
the Skies when of a white Colour, whilft a black Sub-
Itance was placed behind the Bubble. And they were
thefe, red, blue; red, blue; red, blue; red, green ;.
red, yellow, green, blue, purple y red, yellow, green,_
blue, violet ; red, yellow, white, blue, black..

The three firft Succeffions of red and blue w^ere very
dilute and dirty, efpecially the firft, where the red
feemed in a manner to be white. Among thefe there,
was fcarce any other Colour fenfible befides red and
blue, only the blues ( and principally the fecond blue )
inclined a little to green.

The fourth red was alfo dilute and dirty, but not
fo much as the former three ; after that fucceeded little
or no yellow, but a copious green, which at firft incli-
ned a little to yellow, and then became a pretty brifque

and:

and good willow green, and afterwards changed to a
bluiih Colour; but there fucceeded neither blue nor
violet.

The fifth red at firft inclined very much to purple,
and afterwards became more bright and brifque, but
yet not very pure. This- was fucceeded with a very
bright and intenfe yellow, which w^as but little in
quantity, and foon changed to green : But that green
was copious and fomething more pure, deep and lively,
than the former green. After that followed an excel-
lent blue of a bright sky-colour, and then a purple,
which was lefs in quantity than the blue, and much
inclined to red.

The fixth Red was at firft of a very fair and lively
Scarlet, and foon after of a brighter Colour , being
very pure and brifque , and the beft of all the
reds. Then after a lively orange followed an intenle
bright and copious yellow, which was alfo the beft
of all tl;ie yellows, and this changed firft to a greenifli
yellow, and then to a greenilh blue ; but the green
between the yellow and the blue, was very little and
dilute, feeming rather a greenilh white than a green.
The blue w^hich fucceeded became very good, and of a
very fair bright sky-colour, but yet fomething inferior
to the former blue ; ^and the violet was intenfe and
deep with little or no rednefs in it. And lefs in quan-
tity than the blue.

In the laft red appeared a tinfture of fcarlet next
to violet, which foon changed to a brighter Colour,
inclining to an orange ; and the yellow which followed
was at firft pretty good and lively , but afterwards it
grew more dilute, until by degrees it ended in perfed

wliite-

[23]

whitenefs. And this whitenefs, if the Water was very
tenacious and well-tempered, would flowly fpread and
dilate it felf over the greater part of the Bubble ; con-
tinually growing paler at the top, w^here at length it
would crack in many places, and thofe cracks, as they
dilated, would appear of a pretty good, but yet obfcure
and dark sky-colour; the white betvs^een the blue Spots
diminiihing, until it refembled the threds of an irre-
gular Net- work, and foon after vanifhed and left all
the upper part of the Bubble of the faid dark blue
Colour. And this Colour, after the aforeiaid manner,
dilated it felf downwards , until fometimes it hath
overfpread the whole Bubble. In the mean while at
the top, which was of a darker blue than the bottom,
and appeared alfo full ofm.any round blue Spots, fome-
thing; darker than the reft . there would emerge one

D 7

or more very black Spots, and v/ithin thofe other Spots
of an intenfer blacknefs, which I mentioned in the
former Obfervation ; and thefe continually dilated
themfelves until the Bubble broke»

If the Water was not very tenacious the black Spots
would break forth in the white, without any fenlible
intervention of the blue. And fometimes they would
break forth within the precedent yellow , or red, or
perhaps within the blue of the fecond order, before
the intermediate Colours had time to difplay them-
felves.

By this defcriptioB you may perceive how great an
affinity thefe Colours have with thofe of Air defcri-
bed in the fourth Obfervation, although fet down in
a contrary order, by reafon that they begin to appear
when the Bubble is thickeft , and are moll conve-
niently

Ch]

niently leckoned from the loweft and thickeft part of
the Bubble upwards.

O B S. XIX.

Viewing in feveral oblique pofitions of my Eye
the Rings of Colours emerging on the top of the Bubble,
I found that they were fenfibly dilated by increafing
the obliquity , but yet not fo much by far as thofe
made by thin'd Air in the feventh Obfervation. For
there they w^ere dilated fo much as, when viewed
moft obliquely, to arrive at a part of the plate more
than twelve times thicker than that where they ap-
peared when viewed perpendicularly ; whereas in this
cafe the thicknefs of the Water, at which they arrived
when viewed moft obliquely, was to that thicknefs
which exhibited them by perpendicular rays, fome-
thing lefs than as 8 to 5. By the beft of my Obfervations
it was between 15 and 15I to 10, an increafe about
14. times lefs than in the other cafe.

Sometimes the Bubble would become of an uniform
thicknefs all over, except at the top of it near the black
Spot, as I knew, becaufe it would exhibit the fame
appearance of Colours in all pofitions of the Eye. And
then the Colours which were feen at its apparent cir-
cumference by the obliqueft rays, would be different
from thofe that were feen in other places, by rays lefs
oblique to it. And divers Spectators might fee the
fame part of it of differing Colours, by viewing it at
very differing obliquities. Now obferving how much
the Colours at the fame places of the Bubble, or at di-
vers places of equal thicknefs , were varied by the

feveral

fevcral obliquities of the rays ;.. by the affiftance of the
4th, i+th, 1 6th and 18th Obfervations, as they are
hereafter explained, I colleft the thicknefs of the Water
requifite to exhibit any one and the fame Colour, at fe«
veral obliquities , to be very nearly in the proportion
exprefled in this Table*

Ificidence on
the Water,

Refraction in- I Thicknefs of
totheWaterA the Water »

In the two firft Columns are exprefled the obliqui-
ties of the rays to the fuperficies of the Water, that
is, their Angles of incidence and refraction. Where
I fuppofe that the Sines which meafure them are in
round numbers as 3 to 4, though probably the diflb-
lution of Soap in the Water ^ may a little alter its
refradive Vertue. In the third Column the thicknefs
of the Bubble, at which any one Colour is exhibited
in thofe feveral obliquities, is expreft in parts,of which
ten conftitute that thicknefs when the rays are perpen=
dicular, x

I have fometimes obferved, that the Colours which
arife on polifhed Steel by heating it, or on Bell-metal^
and fome other metalline fubftances. when melted and

2($]

poured on the ground , where they may cool in the
open Air, have, like the Colours of Water-bubbles,
been a little changed by viewing them at divers ob-
liquities, and particularly that a deep blue^ or violet,
when viewed very obliquely, hath been changed to a
deep red. But the changes of thefe Colours are not fo
great and fenfible as of thofe made by Water. For the
Scoria or vitrified part of the Metal, which moft Me-
tals wlien heated or melted do continually protrude,
and fend out to their furface, and which by covering
the Metals in form of a thin glaffy skin, caufes thefe
Colours, is much denfer than Water ; and I find that
the change made by the obliquation of the Eye is leaft.
In Colours of the denfeft thin fubftances.

OB S. XX.

As in the ninth Obfervation, fo here, the Bubble, by
Iranfmitted Light, appeared of a contrary Colour to
that which it exhibited by reflexion. Thus when the
Bubble being looked on by the Light of the Clouds re-
fledJed from it, fecmed red at its apparent circumfe-
rence, if the Clouds at the fame time, or immediately
after, were viewed through it, the Colour at its cir-
cumference would be blue. And, on the contrary^
when by reflefted Light it appeared blue, it would ap-
pear red by tranfmitted Light..

O B S. XXL

By wetting very thin plates of Mufcovy-glafs, wbofe
thinnefs made the like ^ Colours appear, the Colours

became

[27]

became more faint and languid j efpeeiaUy by wetting
the plates on that fide oppofite to the Eye : But I could
not perceive any variation of their fpecies. So then
the thicknefs of a plate requifite to produce any Co^
lour, depends only on the denfity of the plate, and
not on that of the ambient medium: And hence, by the
loth and i6th Obfervations, may be known the thick-
nefs which Bubbles of Water, or Plates of Mufcovy«
glafs, or other fubftances, have at any Colour pro-
duced by them.

O B S. XXIL

A thin tranfparent Body, which is denfer than its
ambient medium, exhibits more brifque and vivid Co^
lours than that which is fo much rarer; as I have
particularly obferved in the Air and Glafs, For blow-
ing Glafs very thin at a Lamp-furnace, thofe plates \f^'^^'fr^
incompaffed with Air did exhibit Colours much
more vivid than thofe of Air made thin between tW0
Glaffes.'

O B S. XXIII.

Comparing the quantity of Light refkifted from the
feveral Rings, I found that it was moft copious from
the firft or inmoft, and in the exterior Rings be-^
came gradually lefs and lefs. Alfo the whitenefs of
the firft Ring was ftronger than that refleded from
thofe parts of the thinner medium which were with-^
out the Rings; as I could manifeftly perceive by view-
ing at a diftance the Rings made by the two Obje£l:»

E e 1 Glaffes^

^iC^

[28]

Glaffesi or by comparing two Bubbles of Water blown
at diftant times, in the firft of which the whitenefs
appeared, which fucceeded all the Colours, and in:
the other, the whitenefs which preceded them all.

OB S. XXIV.

When the two Objed-Glafles were lay'd upon one
another, fo as to make the Rings of the Colours ap-
pear, though with my naked Eye I could not difcern
above 8 or 9 of thofe Rings, yet by viewing them
through a Prifm I have feen a far greater multitude^
infomuch that I could number more than forty, befides
many others, that were fo very fmall and clofe toge-
ther, that I could not keep my Eye fteddy gr^jheiij^
feverally fo as to number them, but by their extend Hiav&
fometimes eftimated them to be more than a hundred.
And 1 believe the Experiment may be improved to the
difcovery of far greater numbers. For they feem to
be really unlimited, though vifible only fo far as they
can be feparated by the refraftion, as I fhall hereafter
explain.

But It was but one fide of thefe Rings, namely, that
towards which the refraftion was made, which by that
refraSion was rendered diftinfl:, and the other fide be-
came more confufed than when viewed by the naked
Eye, infomuch that there I could not difcern above
one or two, and fometimes none of thofe Rings, of
which I could difcern eight or nine with my naked
Eye. And their Segments or Arcs, which on the
other fide appeared fo numerous, for the moft part

exceeded

[29

exceeded not the third part of a Circko If the Re»
fradion was very great, or the Prifm very diftant from
the Objeft-Glaffes, the middle part of thofe Arcs be-
came alfo confufed^ fo as to difappear and conftitute an
even whitenefs, whilft on either fide their ends, as alfo
the v^hole Arcs furtheft from the center, became di-
ftinfter than before, appearing In the form as you fee.
them defigned in the fifth Figure. Fig.

The Arcs, where they feemed diftinftefl:, were only
white and black fucceflively, without any other Co-
lours intermixed. But in other places there appeared.
Colours, whofe order was inverted' by the refraftion
in fuch manner, that if I firfl: held the Prifm very near
the Objeft-Glaffes , and then gradually removed^ it-
further off towards my Eye, the Colours of the ad,
5d, 4.th, and following Rings llirunk towards the. white
that ejnerged between them , until they wholly va-
nilhed into it at the middle of the Arcs, and after-
wards emerged again in a contrary order. But at
the ends of the Arcs they retained their order un-
changed,

I have fometimes fo lay'd one Obje6l:-Glafs upon
the other, that to the naked Eye they have all over
feemed uniformly white, without the leaft appearance
of any of the coloured Rings ; and yet by viewing
them through a Prifm, great multitudes of thofe Rings
have difcovered themfelves. And in like manner plates
of Mufcovy-glafs, and Bubbles of Glafs blown at a
Lamp-furnace, which were not fo thin as to exhibit
any Colours to the naked Eye, have through the Prifm
exhibited a great variety of them ranged irregu-*
larly up and down in the form of waves. And fo

Bubbles

Bubbles of Water, before they began to exhibit their
Colours to the naked Eye or a By-ftander, have ap*
peared through a Prifm, girded about with many pa-
rallel and horizontal Rings ; to produce which efFedt,
it was neceffary to hold the Prifm parallel, or very
nearly parallel to the Horizon, and to difpofe it fo
that the rays might be refrafted upwards.

THE

C 31 ]

It ii

SECOND BOOK

O P T I

i A xv jy X As

Remarks u^onthe foregoing O^Jervations.
of the

'Aving given my Obfervations of thefe Colours^

before I make ufe of them to unfold the Caufes

of the Colours of natural Bodies, it is convenient that
by the fimpleft of tliem, fuch as are the id^ ^d, 4th5
9th) lath, 1 8th,. aoth, and 14th , I firft explain the
more expounded. And firft tofhewhowthe Colours
in the fourth and eighteenth Obfervations are produ^
ced, let there be taken m any ri^ht line from the point
Y, the lengths ¥ A, Y B, Y (^ YD, YE, YF, YG.Bg^
YHl^ in proportion to one another, as the Cube^roots
©f the Squares of the numbers, {, y|, ^ j, ^) 1, 1, 1 5 w^here-
by the lengths of a mufical Chord to found all the Notes
is an Eighth are reprefented; tliat is, in the propor^
tion of the numbers 6500, 6814., 71 14, 7631, 8155,,
8855, 9H3>. 1000.Q. And at the points A, B, C, D,

E, F^

[32]

E,F, G, H, let perpendiculars Aa^ B ^^ 15^6% be erefted,
by whole intervals the extent of the feveral Colours
fet underneath againft them, is to be reprefented. Then
divide the line A a in fuch proportion as the numbers
I, a, 5, 5, 6, 7, 9, lo, 1 1, ISJ'^r. fet at the points of divi-
fion denote. And through thofe divifions from Y
draw lines i I, i K, 3 L, 5 M, 6 N, 7 0^}^c,

Now if A ci be luppofed to reprefent the thicknefs
of any thin tranfparent Body , at which the outmoft
violet is moft copioufly refleded in the firft Ring, or
Series of Colours, then by the i gth Obfervation HK,
will reprefent its thicknefs, at which the utmoft red
is moft copioufly refleded in the fame Series. Alfo
by the 5 th and 1 6th Obfervations, A 6 and HNwill
denote the thickneffes at which thofe extreme Colours
are moft copioufly refleded in the fecond Series, and
A I o and H Q the thickneffes , at which they are
moft copioufly reflefted in the tliird Series, and fo on.
And the thicknefs at which any of the intermediate
Colours are reflefted moft copioufly, wifl, according to
the i4.th Obfervation, be defined by the diftance of the
line A H from the intermediate parts of the lines 1 K,
6N, loQ, }^c, againft which the names of thofe Co-
lours are written below.

But further, to define the latitude of thefe Colours in
each Ring or Series, let A i defign the leaft thicknefs,
and A 3 the greateft thicknefs, at which the extreme
violet in the firft Series is refleded, and let H I, and
H L, defign the like limits for the extreme red, and
let the intermediate Colours be limited by the inter-
mediate parts of the lines i I, and 3 L, againft which
the oames of thofe Colours are written, and fo on : But

C333

yet with this caution, that the refle<9:ions be fuppofed
ttrongeft at the intermediate Spaces, aK, 6N, loQ^JfTr,
and from tiience to decreafe gradually towards thefe li-
mits, 1 1, ? L, 5 M, 7 O, )3c. on either fide ; where
you Jnuft not conceive them to be precifeiy limited,
but to decay indefinitely. And whereas I have affigned
the fame latitude to every Series, 1 did it, becaufe al-
tiiough the Colours in the firfl: Series feem to be a little
broader than the reft, by reafon of a ftronger reflexion
there, yet that inequality is fo infenfible as fcarcely to
be determined by Obfervation.

Now according to this defcrlption, conceiving that
the rays originally of feveral Colours are by turns re-
fleded at the Spaces 1 1 L 5, 5 M O 7, 9 r R 1 1, l?r»
atidtranfmitted at the Spaces AHIi,5LM5,70P9,
b'r. it is eafy to know what Colour muft in the open Air
be exhibited at any thicknefs of a tranfparent thin body.
For if a Ruler be applied parallel to A H, at that di»
ftance from it by which the thicknefs of the body is
reprefented, the alternate Spaces 1 1 L 5, 5 M O 7,l5?'r,
which it crofleth wall denote the reflefted original Co-
lours, of which the Colour exhibited in the open Air
is compounded. Thus if the conftitution of the green
in the third Series of Colours be defired, apply the
Ruler as you fee at ^ e^<P, and by its paffing through
fome of the blue at -^ and yellow at o-, as well as through
the green at ^, you may conclude that the green exhi^
bited at that thicknefs of the body is principally con»
ftituted of original green, but not without a mixture
of fome blue and yellow,

■ Ff B¥

By this means you may know how the Colours from
the center of the Rings outward ought to fucceed in
order as they were defcribed in the 4th and 18th Ob-
fervations. For if you move the Ruler gradually from
AH through all diftances, having paft over the firft
fpace which denotes little or no reflexion to be made
by thinned fubftances, it will firft arrive at i the violet,
and then very quickly at the blue and green, which
together with that violet compound blue, and then at
the yellow and red , by whofe further addition that
blue is converted into whitenefs, which whitenefs con*
tinues during the tranfit of the edge of the Ruler from
I to 5, and after that by the fucceffive deficience of
its component Colours, turns firft to compound yellow^
and then to red, and laft of all the red ceafeth at L.
Then begin the Colours of the fecond Series, which
fucceed in order during the tranfit of the edge of the
Ruler from 5 toO, and are more lively than before,,
becaufe more expanded and fevered. And for the
fame reafon, inftead of the former white there inter-
cedes between the blue and yellow a mixture of orange,
yellow, green, blue and indico, all which together ought
to exhibit a dilute and imperfeft green. So the Co-
lours of the third Series all fucceed in order ; firft, the
violet, which a little interferes with the red of the fe-
cond order, and is thereby inclined to a reddifti purple;
then the blue and green , which are lefs mixed with
other Colours, and confequently more lively than be-
fore, efpecially the green : Then follows the yellow,
fomeof which towards the green is diftind and good, but
that part of it towards the fucceeding red, as alfo that
led is mixed with the violet and blue of the fourth Se-
ries,

lies, whereby various degrees of red very much incli-
ning to purple are compounded. This violet and blue,
which fhould fucceed this red, being mixed with, and
hidden in it, there fucceeds a green. And this at firft
is much inclined to blue, but foon becomes a good
green , the only unmixed and lively Colour in this
fourth Series. For as it verges towards the yellow, it
begins to interfere with the Colours of the fifth Series,
by whofe mixture the fucceeding yellow and red are
very much diluted and made dirty, efpecially the yel«
low, which being the weaker Colour is fcarce able to
Ihew it felf. After this the feveral Series interfere more
and more, and their Colours become more and more
intermixed, till after three or four more revolutions
( in which the red and blue predominate by turns )
all forts of Colours are in all places pretty equally ben«
ded, and compound an even whitenefs.

And fince by the 15th Obfervation the rays indued
with one Colour are tranfmitted, where thofe of ano-
ther Colour are reiieded, the reafon of the Colours
made by the tranfmitted Light in the 9th and 10th Ob-
fervations is from hence evident.

If not only the order and fpecies of thefe ColourSj
but alio the precife thicknefs of the plate, or thin body
at which they are exhibited, be defired in parts of an
Inch, that may be. alfo obtained by affiftance of the 6tli
or 1 6th Obfervations. For according to thofe Obferva^
tions the thicknefs of the thinned Air, which between
two Glafles exhibited the moft luminous parts of the
tirit lix Kmgs were 1-^3^5 iTSoSS"? TtHooSj i^aH^? itSSoo? vt^qSo psrts 01
an Inch. Suppofe the Light reflefted moft copioufly
at thefe thickneffes be the bright citrine yellow^ or con-

Ff 1 fine

C3^]

fine of yellow and orange, and thefe thicknefles will
be<JMj Gv, G^, Gp, G^. And this being known, it is
eafy to determine what thicknefs of Air is reprefented'
by G^, or by any other diftance of the ruler from
AH.

But further, fince by the i oth Obfervation the thick-
nefs of Air was to the thicknefs of Water, which be-
tween the fame GlajGTes exhibited the fame Colour, as
4 to 5, and by the aith Obfervation the Colours of
thin bodies are not varied by varying the ambient me-
dium ; the thicknefs of a Bubble of Water, exhibiting
any Colour, will be ^ of the thicknefs of Air producing
the fame Colour. And fo according to the fame loth
and ^ith Obfervations the thicknefs of a plate of
Glafs, v^hofe refraftion of the mean refrangible i*ay, is
meafured by the proportion of the Sines 31 to qo,
may be f^ of the thicknefs of Air producing the fame
Colours ; and the like of other mediums. 1 do not
affirm, that this proportion of ao to 31, holds in all
the rays ; for th^ Sines of other forts of rays have other
proportions. But the differences of thofe proportions
are fo little that 1 do not here confider them. Oil
thefe Grounds 1 have compofed the following Table,
wherein the thicknefs of Air, Water, and Glafs, at
which each Colour is moft intenfe and fpecifick, is ex-
preffed in parts of an Inch divided into Ten hundred
thoufand equal parts.

[37 J

The thichnefs of coloured Tlates and T articles of

r Colours of the'
firll Order,

0f the.fecond Order,

fVery Black
Black
Beginning of

Black-
Blue
White
Yellow-
Orange

^Redi

^Violet
Indico
Blue
Green
i Yellow
Orange
Bright. Red
LScarlet

rpurple.
Indico
Blue
Of the third'.Order, ^ Green

Yellow.
Red
i^Bluifh Red

Bluifh Green
iGreen

lYellowifh Green
Red

5Greenifli Blue
iRed

^Greenifh Blue
^Red

OfthefeventhOrderJGreeniih Blue

'jRuddy White

Of the fourth Order,

Ofthe fifth Order

Of the fixth Order,

Jir,

•y

Water, GUfs,

i 1 1. 1

li^

lit

I2|

8f.

loi

iif

i6f

I2t

lot

i7t

III

i8f

i3i

ii|

191

I2f .

i5i

l^is

22Ti

i^f

i4t

2?t

i7ii

1515

^51

i8fo

i^i

277

20|

'Z^

2Ii

i8'f

20f

25t

35f

26i

22i

2|l

4o|

30*

H»

29t

521

19^^

34 _

48i

42.

534

1 45?

?7„

57i

1 49t .

Now

Can

Now if this Table be compared with the 6th Scheme,
you will there fee the conftitution of each Colour, as
to its Ingredients, or the original Colours of which it
is compounded, and thence be enabled to judge of its
intenfenefs or imperfedion ; which may fuffice in ex-
plication of the 4th and 1 8th Obfervations, unlefs it
be further defired to delineate the manner how the Co-
lours appear, when the two Objed-Glafles are lay'd
upon one another. To do which, let there be de-
fcribed a large Arc of a Circle, and a ft r eight Line
which may touch that Arc, and parallel to that Tan-
gent feveral occult Lines, at fuch diftances from it, as
the numbers fet againft the feveral Colours in the Table
denote. For the Arc, and its Tangent, will reprefent
the fuperficies of the Glaffes terminating the interjacent
Air j and the places where the occult Lines cut the
Arc will lliow at what diftances from the Center, or
Point of contaft, each Colour is reflefted.

There are alfo other ufes of this Table : For by its
affiftance the thicknefs of the Bubble in the 1 9th Ob-
fervatiDn was determined by the Colours which it ex-
hibited. And fo the bignefs of the parts of natural
Bodies may be conjedured by their Colours, as Oiall be
hereafter Ihewn. Alfo, if two or more very thin plates
be lay'd one upon another, fo as to compofe one plate
equalling them all in thicknefs, the refulting Colour
may be hereby determined e For inftance, Mr. Hook in
^ his Miftrografbia obferves, that a faint yellow plate of
^p;A^ ^p^^/Mufcov y-g lafs lay'd upon a blue one, conftituted a very
Jeep purple. The yellow of the firft Order is a faint
one^ and the thicknefs of the plate exhibiting it, ac-
cortog to the Table is 4f , to which add 9, the thick-
nefs

C39]

nels exhibiting blue of the fecond Order, and the fum
will be 13', which is the thicknefs exhibiting the
purple of the third Order.

To explain, in the next place, the Circumftances of
the ad and ^d Obfervations ; that is, how the Rings of
the Colours may ( by turning the Prifms about their
common Axis the contrary way to that expreffed in
thofe Obfervations) be converted into white and black
Rings,and afterwards into Rings of Colours again, the
Colours of each Ring lying now in an inverted order; it
muft be remembred, that thofe Rings of Colours are di-
lated by the obliquation of the rays to the Air which
intercedes the Glafles, and that according to the Table
in the 7th Obfervation, their dilatation or increafe of
their Diameter is moft manifeft and fpeedy when they
are obliqueft. Now the rays of yellow being more re-
fracted by the firft fuperficies of thefaid Air than thofe
of red, are thereby made more oblique to the fecond fu^
perficies, at which they are reflefted to produce the co--
loured Rings, and confequently the yellow Circle in each
Ring will be more dilated than the red; and the excefs of
its dilatation will be fo much the greater, by how much
the greater is the obliquity of the rays, until at laft it be-
come of equal extent with the red of the fame Ring. And
for the fame reafon the green, blue and violet, will be alio
fo much dilated by the ftill greater obliquity of their
rays, as to become all very nearly of equal extent with
the red, that is, equally diftant from the center of the
Rings. And then all the Colours of the fame Ring
muft be coincident, and by their mixture exhibit a.
white Ring. And thefe white Rings muft have black
and dark Rings betw^een them ^ becaufe they do not

fpread

fpread and interfere with one another as before. And
for that reafon alfo they muft become diftinfter and vi-
fible to far greater Numbers. But yet the violet being
obliqueft wilbbe fomething more dilated in proportion
to its extent then the other Colours, and fo very apt to
appear at the exterior verges of the white.

Afterwards, by a greater obliquity of the rays, the
violet and blue become more fenfibly dilated than the
red and yellow, and fo being further removed from the
center or the Rings, the Colours muft emerge out of the
white in an order contrary to that which they had be-
fore, the violet and blue at the exterior limos of each
Ring,and the red and ydlow at the interior. And the vio-
let, by reafon of the greateft obliquity of its rays, being
in proportion moft of all expanded, will fooneft appear
at the exterior limb of each white Ring, and become
more confpicuous than the reft. And the feveral Series
of Colours belonging to the feveral Rings , will, by
their unfolding and fpreading, begin again to interfere,
and thereby render the Rings lefs diftinft, and not vifi-
ble to fo great numbers.

If inftead of the Prifms the Objeil-glafles be made
ufe of, the Rings which they exhibit become not white
and diftindt by the obliquity of the Eye, by reafon that
the rays in their paflage through that Air which inter-
cedes the Glaffes are very nearly parallel to thofe Lines
in which they were firft incident on the Glaffes, and con-
fequently the rays indued with feveral Colours are not
inclined one more than another to that Air, as it hap-
pens in the Prifms.

There is yet another circumftance of thefe Experiments
to be confidered, and that is why the black and white

Rings

Rings which when viewed at a diftance appear diftind^
fliould not only become confufed by viewing them near
at hand , but alfo yield a violet Colour at both the
edges of every white Ring. And the reafon is, that the
rays which enter, the Eye at feveral parts of the Pupil,
have feveral obliquities to the Glaffes, and thofe which
are moft oblique^ if confidered apart, would reprefent
the Rings bigger tlian thofe which are the leaft oblique.
Whence the breadth of the perimeter of every white
Ring is expanded outwards by the obliqueft rays^
and inwards by the leaft oblique. And this expanfion
is fo much the greater by how much the greater is the
difference of the obliquity ; that is, by how much the
Pupil is wider, or the Eye nearer to the Glaffes. And
the breadth of the violet muft be moft expanded, be-
caufe the rays apt to excite a fenfation of that Colour
are moft oblique to a fecond, or further fuperficies of
the thin'd Air at which they are reflefted, and have
alfo the greateft variation of obliquity , which makes
that Colour fooneft €merge out pf the edges of the
white. And as the breadth of every Ring is thus aoig-.
mented, the dark intervals muft be.diminiftied, until
the neighbouring Rings become continuous, and are
blended, the exterior firft, and then thofe nearer the
Center, fo that they can no longer be diftinguifli-d
apart , but feem to conftitute an even and uniform
whitenefs.

^ Among all the Obfervations there is none accompa^
nied witn fo odd circumftances as the ^^th. Of thofe
the principal are, that in thin plates^ which to the
naked Eye feem of an even and uniform tranfparent

[42].

whitenefs, without any terminations of (hadows^ the
refraftion of a Prifm ftiouM make Rings af Colours ap-
pear, whereas it ufually makes Objedts appear coloured
only there where they are terminated with (hadows, or
haveparts unequally luminous; and that it fhouldinake
thofe Rings exceedingly diftinft and white, although
it ufually renders Objeds confufed and coloured. The
caufe of thefe things you will underftand by confidering^
that all the Rings of Colours are really in the plate^
when viewed with the naked Eye, although by reafon
of the great breadth of their circumferences they fo
much interfere and are blended together^that they {eem
to conftitute an even whitenefs. But when the rays
pafs through the Prifm to the Eye, the orbits of the
ieveral Colours in every Ring are refrafted, fome more
than others, according to their degrees of refi-angibility r
By which means the Colours on one fide of the Ring
f that is on one fide of its Center) become more unfolded
md dilated, and thofe on the other fide more compli-
cated and contrafted. And where by a due refradioB
they are fo much contraded, that the fevral Rings be-
come narrower than to interfere with one another, they
muft appear diftinft, and alfo white^ if the conftituent
Colours be fo much eontradedas to be wholly coincident.
But^ oii the other fide, where the orbit of every Ring
is made broader by the further unfolding of its Co-
lours, it muft interfere more with other Rings than
before, and fo become lefs diftinft.

To explain this a little further, fuppofe the concen-

•pl„ 7. ^^ Circles A V, and BX, reprefent the red and violet

' of any order^ which, together with the intermediate

Colours,

[43]

Colours, conftitute any one of thefe Rings. Now thefe
being viewed through a Prifm, the violet Circle B X,
will by a greater refraftion be further tranflated from
its place than the red A V, and fo approach nearer to
it on that ftde, towards which the refraftions are made.
For inftance, if the red be tranflated to av^ the violet
may be tranllated to b x, fo as to approach nearer to it
at X than before, and if the red be further tranflated
to a V, the violet may be fo much further tranflated to
b X as to convene with it at x, and if the red be yet
further tranflated to * ^\ the violet may be ftill fo much
further tranflated to /3| as to pafs beyond it at I, and
convene with it at e and/. And this being underftood
not only of the red and violet, but of all the other in-
termediate Colours, and alfo of every revolution of
thofe Colours, you will eafily perceive how thofe of the
fame revolution or order, by their nearnefs at xi/and
'^ ^, and their coincidence at xv, ^ ^n<l/) ought to con-.
flitute pretty diflinft Arcs of Circles, efpecially at xv^
or at ^and/, and that they will appear feverally at
X z;, and at x v exhibit whitenefs by their coincidencey
and again appear feveral at ^ I, but yet in a contrary
order to that which they had before, and flill ittmh.
beyond^ and/. But, on the other fide, ^t ai^ ab^
or a ^, thefe Colours muft become much more confu*
fed by being dilated and fpread fo, as to interfere with
thofe of other Orders. And the fame confufion will
happen at ^^ i between e and/, if the refraction be very
great, or the Prifm very diftant from theObjed-Glaffes :
In which cafe no parts of the Rings will be feen, fave
only two little Arcs at e and/, whofe diftance from one

Gg a another^

44-

another will be augmented by removing the Prlfm'
ftill further from the Objeft-Glaffes : And thefe little
Arcs muft be diftinftefl: and whiteft at their middle, and
at their ends, where they begin to grow confuied they
muft be coloured. And the Colours at one end of
every Arc muft be in a contrary order to thofe at the
other end, by reafon that they crofs in the interme^
diate white; namely their ends, which verge towards
'^^, will be red and yellow on that fide next the Cen-
ter, and blue and violet on the other fide. But their
other ends which verge from '^ ^ will on the contrary
be blue and violet on that fide towards the Center, and:
on the other fide red and yellow.

Now as all thefe things follow from the Properties^
©f Light by a mathematical way of reafoning, fo the
truth of them may be manifefted by Experiments* For
in a dark room, by viewing thefe Rings through a
Prifm, by reflexion of the feveral prifmatique Colours,
which an affiftant caufes to move to and fro upon ^
Wall or Paper from whence they are reflected, whilft^
the Speftator's Eye, thePrifm, and the Objeftr-Glaffes
(as in the 13th Obfervation) are placed fteddy : the
pofition of the Circles made fucceffively by the feveral
Colours, will be found fuch, in refped: of one another^,
as I have defcribed in the Figures aixv^ or abxvy;
or aJgT, And by the fame method the truth of
the Explications of other Obfervations may be exa-
mined.

By what hath been faid the like PhKnomlna of
Water, and thin plates of Glafs may be underftood.
But in fmall fragments of thofe plates, there is this^

further

C45]

further obfervable, that where they lye flat upon a
Table and are turned about their Centers whilft they are
viewed through a Prifm , they will in fome poftures
exhibit waves of various Colours, and fome of them ex-
hibit thefe waves in one or two pofitions only, but the
Hioft of them do in all politions exhibit them, and make
them for the moft part appear almoft all over the plates.
The reafon is, that the fuperficies of fuch plates are not
even, but have many cavities and fwellings, which how
Ihallow foever do a little vary the thicknefs of the
plate. For at the feveral fides of thofe cavities, for^^
the reafons newly defcribed, there ought to be produ^
ced waves in feveral poftures of the Prifm. Nowthough^
it be but fome very fmall, and narrower parts of the
Glafs, by which thefe waves for the moft part are cau-
fed, yet they may feem to extend themfelves over the
whole Glafs, becaufefrom the narroweft of thofe parts
there are Colours of feveral Orders that is of feveral
Rings, confufedly refleded, which by refradion of the-
Prifm are unfolded, feparated, and according to their
degrees of refraftion, difperfed to feveral places, fo as to
Gonftitute fo many feveral waves, as there were divers
orders of Colours promifcuoufly refle£led from that
part of the Glafs.

Thefe are the principal Phaenomena of thin Plates^
or Bubbles, whofe explications depend on the pro-
perties of Light, which I have heretofore delivered.
And thefe you fee do neceffarily follow from them, and
agree with them, even to their very leaft circumftances;
and not only fo, but do very much tend to their proof.
Thus, by the a^th Obfervation, it appears^ that the

rays-..-.

C4«]

rays of feveral Colours made as well by thin Plates or
Bubbles, as by refradtions of a Prifm, have feveral de-
grees of refrangibility, v^hereby thofe of each order,
which at their reflexion from the Plate or Bubble are
intermixed with thofe of other orders, are feparated
from them by refra£i:ion,and affociated together lb as to
become vifible by themfelves like Arcs ot Circles. For
if the rays were all alike refrangible, 'tis impoffible that
the whitenefs, which to the naked fence appears uni-
.form, fhould by refraction have its parts tranfpofed and
ranged into thofe black and white Arcs.

It appears alfo that the unequal refractions of dif-
form rays proceed not from any contingent irregulari-
ties ; fuch as are veins, an uneven polifh, or fortuitous
poiition of the pores of Glafs , unequal and cafual mo-
tions in the Air or ^ther ; the fpreading, breaking, or
dividing the fame ray into many diverging parts, or
the like. For, admitting any fuch irregularities, it would
be impoflible for refractions to render thofe Rings fo
very diftinCt , and well defined , as they do in the
a^th Obfervation. It is neceflary therefore that eve-
ry ray have its proper and conftant degree of refran-
gibility connate with it,according to which its refraftion
is ever juftly and regularly performed, and that feve-
ral rays have feveral of thofe degrees.

And what is faid of their refrangibility may be alfo
underftood of their reflexibility, that is of their difpo-
(itions to be reflected fome at a greater, and others at a
lefs thicknefs, of thin Plates or Bubbles, namely, that
thofe dlfpofitions are alfo connate with the rays, and
immutable; as may appear by the 13th, 14th, and

15th

C47I

15th Obfervations compared with the fourth and
eighth.

By the precedent Obfervations it appears alfo, that
whitenefs is a diffimilar mixture of all Colours, and that
Light is a mixture of rays indued with all thofe Co«
lours. For confidering the multitude of the Rings of
Colours, in the gd, nth and ^4.th Obfervations, it is
manifeft that although in the 4th and 18th Obferva-
tions there appear no more than eight or nine of thofe
Rings, yet there are really a far greater number, which
fo much interfere and mingle with one another, as after
thofe eight or nine revolutions to dilute one another
wholly, and conftitute an even and fenfibly uniform
whitenefe. And confequently that whitenefs muft be
allowed a mixture of all Colours, and the Light which
conveys it to the Eye muft be a mixture of rays indued .
with all thofe Colours.

But further, by the a^th Obfervation , it appears^.
that there is a conftant relation between Colours and
Refrangibility, the moft refrangible rays being violet^
the leaft refrangible red, and thofe of intermediate Co-
lours having proportionably intermediate degrees of re^
frangibility. And by the 1 3th, 14th and 1 5th Obfer-
vations, compared with the 4.th or i8th, there appears
to be the fame conftant relation between Colour and
Reflexibility, the violet being in like circumftances re-
flected at leaft thickneffes of any thin Plate or Bubble,
the red at greateft thickneffes , and the intermediate
Colours at intermediate thickneffes. Whence it fol-
lows, that the colorifique difpofitions of rays are alfo
connate with them and immutable, and by confequence

that

48]

that all the produftions and appearances of Colours
in the World are derived not from any phyfical change
caufed in Light by refraction or reflexion, but only
from the various mixtures or feparations of rays, by
virtue of their different Refrangibility or Reflexibility,
And in this refpeft the Science of Colours becomes a
Speculation as truly mathematical as any other part of
Optiques. I mean fo far as they depend on the nature
of Light, and are not produced or altered by the power
of imagination, or by (triking or prefling the Eyes.

Fig. 2.

BooK,n. Plate,I.

*!••

Zy ^ I' t s r ./pcnmlkih e^ f : d c

Aj fg h ik I mmp ,jr s tir xf^i

Book.n

riarr.n.

ex, $ y- 6

FliTn-

-y o

Fig. 6.

B C

F G

■*i-i

^^g-- 7-

SECOND BOOK

O P T I

if A Jtv L JLJLx.

Of the permanent Colours of natural Bodies j and the
Analogy iet%<ueen them and the Colours of thin tranf^
parent Tlates.

I Am now come to another part of this Defign, which
is to confider how the Phasnomena of thin tranfpa-
rent Plates ftand related to thofe of all other natural
Bodies. Of thefe Bodies I have already told you that
they appear of divers Colours, accordingly as they are
difpofed to refled moft copioufly the rays originally
indued with thofe Colours. But their Conftitutions^
whereby they refieft fome rays more copioufly than
others, remains to be difcovered, and thefe I (hi '^
deavour to manifeft in the following Propofitions

PROP. L

T'hofefuperficieroftranf parent Bodies reflet thegreatefi
quantity of Light ^ "which have the great eft ref racing poisuer;
that is^ "which intercede mediums that differ moft m their
refractive denfities. And in the confines of equally re-
framing mediums there is no reflexion. ^
• The Analogy between reflexion and refraftlon will
appear by confidering, that w^hen Light paffeth ob-
liquely out of one medium into another which refrafts
from the perpendicular, the greater is difference of
their refractive denhty, the lels obliquity^ is requmte
to caufe a ^tal reflexion. For as the Sines are which
meafure the refradion, fo is the Sine of incidence at
which the total reflexion begins, to the radius of the
Circle, and confequently that incidence is leaft where
there is the greateit difference of the Sines. Thus in the
paffing of Light out of Water into Air, where the
refraftion is meafured by the Ratio of the Sines 5 to 4^
the total reflexion begins when the Angle of incidence
i% about 48 degrees 35 minutes. In paffing out ofGlafs
iiito Air, where the refradion is meafured by the Ratio
of the Sines qo to 51, the total reflexion begins when
the Angle of incidence is 40 deg. 10 min. and fo in
paffing out of cryftal, or more fl:rongly refra^Sing me-
diums into Air, there is fl:ill a lefs obliquity requifite
to caufe a total reflexion. Superficies therefore which
refraft mofl: do foonefl: refleft all the Light which is in*
cident on them, and fo muff be allowed mofl: ftrongly
reflexive.

Bur

But the truth of this Propofition will further appear
by obferving , that in the fuperficies interceding two
tranfparent mediums, fuch as are (Air, Water ,Oyl, Com-
mon-Glafs, Cryftal, Metalline-Glafles , Ifland-Glaffes,
white tranlparent Arfnick, Diamonds, If^c. ) the re-
flexion is ftronger or weaker accordingly, as the fuper-
ficies hath a greater or lefs refracting power. For in
the confine of Air and Sal-gemm 'tis ftronger than in
the confine of Air and Water, and ftill ftronger in the
confine of Air andCommon'GlafsorCryftal,and ftronger
in the confine of Air and a Diamond. If any of thefe,and
fuch like tranfparent Solids, be immerged in Water, its
reflexion becomes much weaker than before, and ftill
weaker if they be immerged in the more ftrongly re--
framing Liquors of well-rectified oyl of Vitriol or fpirit
of Turpentine. If Water be diftinguiflied into two parts,
by any imaginary furface, the reflexion in the coejfine
of thofe two parts is none at all. In the confine of Wa-
ter and Ice 'tis very little, in that of Water and Oyl 'tis
fomething greater, in that of Water and Sal-gemm ftill
greater, and in that of Water and Glafs, or Cryftal, >dr
other denfer fubftances ftill greater, accordingly = as thofe
mediums diifer more or lefs in their refraSing powersi
Hence in the confine of Common-Glafs and Gryftal,
there ought to be a weak reflexion, and a ftronger re-
flexion in the confine of Common and Metalline-Glafs,
though I have not yet tried this. But, in the confine of
two Glafles of equal denfity, there is not any fenfible re-
flexion, as was (hewn in the firft Obfervation. And
the fame may be underftood of the fuperficies interce-
ding two Cryftals^ or two Liquors, or any other Sub**
ftances in which no refraftion is caufed. So then the

Hh 2 reafon

[ 52 3

reafon why uniform pellucid mediums, (fuch as Water,
Glafs, or Cryftal) have no fenfible reflexion but ia
their external fuperficies, where they are adjacent to
other mediums of a different denfity , is becaufe ;^11
their contiguous parts have one and the fame degree
of denfity.

PROP. II.

The ieafl farts of almojl all natural Bodies are in fome
meafure tranffarent : jind the ofacity of thofe Bodies
arijeth from the multitude of reflexions caufed in their in^
ternal "Parts. ^

That this is fo has been obferved by others, and
will eafily be granted by them that have been conver-
fant with Mifcrofcopes. And it may be alfo tryed by
applying any fubftance to a Hole through which fome
Light is immitted into a dark room. For how opake
foever that fubftance may feem in the open Air, it will
by that means appear very manifeftly tranfparent, if
it be of a fuflficient thinnefs. Only white metalline Bo-
dies muft be excepted, which by reafon of their excef-
five denfity feem to refleft almoft all the Light inci-
dent on their firft fuperficies , unlefs by folution in
menftruums they be reduced into very fmall particles^
and then they become tranfparent.

PROP. IlL

*'' Betisoeen the farts of ofake and coloured Bodies are
many f faces ^ either emfty or reflenijhed^ ^with mediums
of other denfities ; ^ Water ietmeen the tinging corfufcles
'wherewith any Liquor k imfregnatedy Air between the

' 7 a(j^ueo'Wi

aqueom glohules that conftitute Clouds or Mifts | and for
the mojl fart [faces void of ioth Air and Watery hut yet
perhafs not ^wholly void of all f ub fiance^ het'wten the farts
of hard Bodies. "^

The truth of this Is evinced by the two precedent
Propofitions : For by the fecond Propofition there are
many reflexions made by the internal parts of Bodies^
which, by the firft Propofition, would not happen if
the parts of thofe Bodies were continued without any
fuch inter ft ices between them, becaufe reflexions are
caufed only in fuperficies, which intercede mediums of
a differing denfity by Prop, i .

But further, that this difcontinuity of parts is the
principal caufe of the opacity of Bodies^ will appear by
confidering, that opake fubftances become tranfparent
by filling their pores with any fubftance of equal or al-
moft equal denfity with their parts. Thus Paper dip-
ped in Water or Oyl, the Oculm mundi Stone fteep'd in
Water, Linnen-cloth oyled or varniftied,and many other
fubftances foaked in fuch Liquors as will intimately
pervade their little pores, become by that means more
tranfparent than otherwife ; fo, on the contrary, the
moft tranfparent fubftances may by evacuating their
pores, or feparating their parts, be rendred fufficiently
opake, as Salts or wet Paper, or the Oculm mundi Stone
by being dried, Horn by being fcraped, Glafs by being
reduced^ ,|qjDowder, or otherwife flawed, Turpen^
tinetBy !)eirfg ftirred about witfiT^Vater till they mix
imperfeftly , and Water by being formed into many
fmall Bubbles, either alone in the form of froth, or
by ftiaking it together with Oyl of Turpentine , or
with fome other convenient Liquor, with which it will

not

5+ ]

not perfedly incorporate. And to the increafe of the
opacity of thefe Bodies it conduces fomething, that by
the agth Obfervation the reflexions of very thin trans-
parent fubftances are confiderably {Ironger than thofe
inade by the fame fubftances of a greater thicknefs.

PROP. IV.

T'he farts . of Bodies and their Inter flic es mufl not he
lefs than offome definite hignejs^ to render them opake and
coloured.

For the opakeft Bodies, if their parts be fubtily
divided, ( as Metals by being diflblved in acid men-
ftruums, %c.) become perfedly tranfparent. And you
may alfo remember, that in the eighth Obfervation
there was no fenfible reflexion at the fuperficies of
the Objeft-Glafles where they were very near one
another, though they did not abfolutely touch. And
in the i yth Obfervation the reflexion of the Water-bubble
where it became thinneft was almoft in fenfible, fo as
to caiife very black Spots to appear on the top of the
Bubble by the want of refleded Light.
■ On thefe grounds I perceive it is that Water, Salt,
Glafs, Stones, and fuch like fubftances, are tranfparent.
For, upon divers conliderations, they feem to be as full
of pores or interftices between their parts as other Bo-
dies are, but yet their parts and interftices to be too
fmall to caufe reflexions in their common furfaces.

PROP.

1 55 ]

PROP, V.

T^he tranffarent farts of Bodies according ta their fe^
veral fizes rauji rejleB ra'js of one Colour^ and tranfmit
thofe of another^ on the fame grounds tha.t thin plates or
Bubbles do refieH or tranfmit thofe rays. j4nd this I take
to be the ground of all their Colours.

For if a thin'd or plated Body, which being of an
even thicknefs, appears all over of one uniform Co-
lour, fhould be llit into threds, or broken into frag-
ments, of the fame thicknefs with the plate ; I fee no
reafon why every thred or fragment fhould not keep its
Colour, and by confequence why a heap of thofe threds
or fragments ftiould not conftitute a mafs or powder of
the lame Colour, which the plate exhibited before it
was broken. And the parts of all natural Bodies being
like fo many fragments of a Plate, muft on the fame
grounds exhibit the fame Colours.

Now that they do fo, will appear by the afftnity of
their properties. The finely coloured Feathers of fome
Birds, and particularly thofe of Peacocks Tails, do in
the very fame part of the Feather appear of feveral Co»
lours in feveral pofitions of the Eye, after the very fame
manner that thin Plates were found to do in the 7th
and 19th Obfervations, and therefore arife from the
thinnefs of the tranfparent parts of the Feathers | that
is, from the flendernefs of the very fine Hairs, or CafiUa^
mentaj, which grow out of the fides of the groffer late-
ral branches or fibres of thofe FeatherSo And to the
lame purpofe it is, that the Webs of fome Spiders by

beins

being fpun very fine have appeared coloured, as fome
have obferved^ and that the coloured fibres of fome filks
by varying the pofition of the Eye do vary their Co-
lour. Alfo the Colours of filks, cloths, and other fub-
ftances, which Water or Oyl can intimately penetrate,
become more faint and obfcure by being immerged in
thofe liquors, and recover their vigor again by being
dried, much after the manner declared of thin Bodies
in the loth and nth Obfervations. Leaf-gold, fome
forts of painted Glafs, the infufion of Lignum Mefhru
t'tcum^ and fome other fubftances reflect one Colour,
and tranfmit another, like thin Bodies in the 9th and
aoth Obfervations. And fome of thofe coloured pow-
ders which Painters ufe, may have their Colours a little
changed, by being very elaborately and finely ground.
Where I fee not what can be juftly pretended for thofe
changes, befides the breaking of their parts into lefs
parts by that contrition after the fame manner that the
Colour of a thin Plate is changed by varying its thick-
nefs. For which reafon alfo it is that the coloured flowers
of Plants and Vegitables by being bruifed ufually be-
come more tranfparent than before, or at leaft in fome
degree or other change their Colours. Nor is it much
lefs to my purpofe, that by mixing divers liquors very
odd and remarquable productions and changes of Co«
lours may be effefted, of which no caufe can be more
obvious and rational than that the faline corpufcles of
one liquor do varioufly aft upon or unite with the
tinging corpufcles of another, fo as to make them fwell,
or Ihrink (whereby not only their bulk but their den-
fity alfo may be changed ) or to divide them into
fmaller corpufcles, (whereby a coloured liquor may be-
come

C57]

come tranfparent) or to make many of them aflbciate
into one clufter, whereby two tranfparent liquors may
compofe a coloured one. For we fee how apt thofe
faline menftruums are to penetrate and diffolve fub-
ftances to which they are applied, and fome of them
to precipitate what others diffolveo In like manner, if
we confider the various Phaenomena of the Atmofphaere,
we may obferve, that when Vapors are firft raifed, they
hinder not the tranfparency of the Air, being divided
into parts too fmall to caufe any reflexion in their fuper-
ficies. But when in order to compofe drops of rain they
begin to coalefce and conflitute globules of all inter-
mediate fizes, thofe globules when they become of a
convenient fize to refled fome Colours and tranfmit
others, may conftitute Clouds of various Colours accor-
ding to their fizes. And I fee not what can be ratio-
nally conceived in fo tranfparent a fubftance as Water for
the produftion of thefe Colours, befides the various
fizes of its fluid and globuler parcels^

P R O R VL

The farts of Bodies on "which their Colours defend^
are denjer than the medium , "which fervades their in^
terjiices, /

This will appear by confidering, that the Colour of
a Body depends not only on the rays which are inci-
dent perpendicularly on its parts, but on thofe alf©
which are incident at all other Angles^ And that ac-
cording to the 7th Obfervation, a very little variation
of obliquity will change the refleded Colour where the
thin body or fmall particle is rarer than the ambient

I i medium*

inedium,. infomuch that fuch a fmall particle will at di-
verily oblique incidences refled all forts of Colours, "in
fo great a variety that the Colour relulting from them
all, confufedly reflected from a heap of fuch particles,
muft rather be a white or grey than any other Colour,
or at beft it muft be but a very imperfeft and dirty Co-
lour. Whereas if the thin body or flnall particle be
much denfer than the ambient medium, the Colours
according to the 1 9th Obfervation are fo little changed
by the variation of obliquity, that the rays which are
reflefted leaft obliquely may predominate over the reft
fo much as to caufe a heap of fuch particles to appear
very intenfly of their Colour.

It conduces alfo fomething to the confirmation of this
Propolition, that, according to the aith Obfervation,
the Colours exhibited by the denfer thin body within
the rarer, are more brifque than thofe exhibited by the
rarer within the denfer.

P R OP. VI I.

The hignejs of the component farts of natural Bodies
mWj he conjeBured i>j their Colours.

For fince the parts of thefe Bodies by Prop. 5. do
moft probably exhibit the fame Colours with a Plate of
equal thicknefs, provided they have the fame refractive
denfity ; and fince their parts feem for the moft part to
have much the fame denfity with Water or Glafs, as
by many circumftances is obvious to colled: ; to deter-
mine the fizes of thole parts you need only have recourfe
to the precedent Tables, in which the thicknefs of Wa-
ter ojj Glals exhibiting any Colour is expreffed. Thus

if it be defired to know the Diameter of a corpufcle,
which being of equal denfity with Glafs fhall refleft
green of the third order ; the number 1 6\ lliews it to
be ^^^ parts of an Inch.

lOCOOO

The greateft difficulty is here to know of what order
the Colour of any Body is. And for this end we muft
have recourfe to the 4.th and 1 8th Obfervations, from
whence may be collected thefe particulars.

Scarlets J and other reds^ oranges and 'jeUo'ws^ if they
be pure and intenfe are moft probably of the fecond or>
der. Thofe of the firft and third order alfo may be
pretty^ good, only the yellow of the firft order is faint,
and the orange and red of the third order have a great
mixture of violet and blue.

There may be good greens of the fourth or4er^ but
the pureft are of the third. And of this ordex the green
of all vegitables feem to be, partly by reafon of the in-
tenfenefs of their Colours , and partly becaufe when
they wither forne of them turn to a greenlfli yellow,
and others to a more perfect yellow or orange, or per^
haps to red, paffing firft through all the aforefaid in=>
termediate Colours. Which changes feem to be efffe£ted
by the exhaling of the moifture which may leave the
tinging corpuCcles more denfe, and fomething augmen-
ted by the accretion of the oyly and earthy part of
that moifture. Now the green without doubt is of the
lame order with thofe Colours into which it changeth,
becaufe the changes are gradual, and thofe Colours,
though ufually not very full, yet are often too full and
lively to be of the fourth order.

. I i 2 Blues

[do]

Blues and fur fies maybe either of the fecond or third
order, but the beft are of the third. Thus the Colour
of violets feems to be of that order, becaufe their Syrup
by acid Liquors turns red, and by urinous and alcali-
zale turns green. For fince it is of the nature of Acids
to diffolve or attenuate, and of Alcalies to precipitate
or incraflate, if the purple Colour of the Syrup was of
the fecond order, an acid Liquor by attenuating its ting-
ing corpufcles would change it to a red of the firft
order, and an Alcaly by incralTating them would change
it to a green of the fecond order ; which red and green,
efpecially the green, feem too imperfe£t to be the Co-
lours produced by thefe changes. But if the laid purple
be fuppofed of the third order, its change to red of the
fecond, and green of the third, may without any in-
convenience be allowed.

If there be found any Body of a deeper and lefs red-
difli purple than that of the violets, its Colour moft'
probably is of the fecond order. But yet their being
no Body commonly known whofe Colour is conftantly
more deep than theirs, I have made ufe of their name to
denote the deepeft and leaft reddifh purples, fuch as
manifeftly tranfcend their Colour in purity.

The Hue of the firft order , though very faint and
little, may poffibly be the Colour of fome fubftances ;
and particularly the azure Colour of the Skys feems to
be of this order. For all vapours when they begin to
condenfe and coalefce into fmall parcels, become firft of
that bignefs whereby fuch an Azure muft be reflected
before they can conftitute Clouds of other Colours. And
fo this being the firft Colour which vapors begin to
reiieft, it ought to be the Colour of the tineft and moft

tranf-

tranfparent Skys in which vapors are not arrived to that
groihefs requifite to refled other Colours, as we find it
is by experience.

TVbitenefs^ if moft intenfe and luminous, is that of the
firft order, if lefs ftrong and luminous a mixtui'e of the
Colours of feveral orders. Of this laft kind is the
whitenefs of Froth, Paper, Linnen, and moft white fub-
ftances ; of the former I reckon that of white metals to
be. For whilft the denfeft of metals. Gold, if foliated
is tranfparent, and all metals become tranfparent if
diffolved in menftruums or vitrified, the opacity of
white metals arifeth not from their denfity alone. They
being lefs denfe than Gold would be more tranfparent
than it, did not fome other caufe concur with their den-
fity to make them opake. And this caufe I take to be
fuch a bignefs of their particles as fits them to refle£t
the white of the firft order. For if they be of other
thicknelTes they may refleft other Colours,, as is mani-
feft by the Colours which appear upon hot Steel in tem^
pcring it, and fometimes upon the furface of melted
metals in the Skin or Scoria which arifes upon them in
their cooling. And as the white of the firft order is
the ftrongeft which can be made by Plates of tranfparent
fubftances, fo it ought to be ftronger in the denfer fub-
ftances of metals than in the rarer of Air, Water and
Glafs.. Nor do I fee but that metallic fubftances -of fuch.
a thicknefs as may fit them to refled the white of the
firft order, may, by reafon of their great denfity (accor-
ding to the tenour of the firft of thefe Propofitions) res-
iled: all the Light incident upon them, and fo be as
opake and fplendent as its poinble for any Body to be.
Gold, or Copper mixed with lefs than h^lf their weight

q£

I 62 J

of Silver, or Tin, or Regulus of Antimony, in fufion
or amalgamed with a very little Mercury become white;
which Ihews both that the particles of white metals
have much more fuperficies, and fo are fmaller, than
thofe of Gold and Copper, and alfo that they are lb
opake as not to fuifer the particles of Gold or Copper to
fhine through them. Now it is fcarce to be doubted,
but that the Colours of Gold and Copper are of the fe-
eond or third order, and therefore the particles of white
metals cannot be much bigger than is requifite to make
them refled the white of the firft order. The volati-
lity of Mercury argues that they are not much bigger,
nor may they be much lefs, leaft they lofe their opacity,
and become either tranfparent as they do when attenua-
ted by vitrification, or by folution in menftruums, or
black as they do when ground fmaller, by rubbing Sil-
ver,or Tin, or Lead, upon other fubftances to draw black
Lines. The firit and only Colour which white metals
take by grinding their particles fmaller is black, and
therefore their white ought to be that which borders
upon the black Spot in the center of the Rings of Co-
lours, that is, the white of the firft order. But if you
would hence gather the bignefs of metallic particles,
you muft allow for their denfity. For were Mercury
tranfparent, its denfity is fuch that the Sine of inci-
dence upon it (by my computation) w^ould be to the
fine of its refraftion, as 71 to ao, or 7 to o>. And
therefore the thickneis of its particles, that they may
exhibit the fiime Colours with thofe of Bubbles of War
ter, ought to be lefs than the thicknefs of the Skin of
thofe Bubbles in the proportion of 2 to 7. Whence
it;s poffible that the particles of Mercury may be as little

C ^3 3

as the particles of fome tranfparent and volatile fluids,
and yet retled the white of the firft order.

Laftly, for the production of hlack^ the corpufcles
muft be lefs than any of thofe which exhibit Colours.
For at all greater fizes there is too much Light refle-'
ded to conftitute this Colour. But if they be fuppo-
fed a little lefs than is requifite to refle£t the white and
very faint blue of the hrft order, they will, according
to the 4th, 8th, 17th and i8th Obfervations, refleft
fo very little as to appear intenfly black, and yet may
perhaps variouily refraft it to and fro within them*
felves fo long, until it happen to be ftifled and loft,
by which means they will appear black in all pofitions
of the Eye without any tranfparency. And from hence
may be underftood why Fire , and the more fubtile
diffolver Putrefadion, by dividing the particles of fub-
ftances, turn them to black, why fmall quantities of
black fubftances impart their Colour very freely and in-
tenfly to other fubflances to which they are applied ;
the minute particles of thefe, by reafon of their very
great number, eafily overfpreading the grofs particles
of others ; vdiy Glafs ground very elaborately with
Sand on a copper Plate, 'till it be well poliflied, makes
the Sand, together with what is worn off from the Glafs
and Copper^ become very black : why black fubftances
do fooneft of all others become hot in the Sun's Light
and burn, (which effed: may proceed partly from the
multitude of refractions in a little room, and partly
from the eafy commotiofii of fo very fmall corpufcles;)
and why blacks are ufually a little inclined to a bluifh
Colour. For that they are fo may be feen by illumina-
ting white Paper by Light retledted from black fub*

ftances,-

d4]

ftances. For the Paper will ufually appear of a bluifli
white ; and the reafon is, that black borders on the
obfcure blue of the firft order defcribed in the i8th
Obfervation, and therefore reflefts more rays of that
Colour than of any other.

In thefe Defcriptions I have been the more particu-
lar, becaufe it is not impoffible but that Mifcrofcopes
may at length be improved to the difcovery of the
particles of Bodies on which their Colours depend, if
they are not already in fome meafure arrived to that de-
gree of perfection. For if thofe Inftruments are or can
be fo far improved as with fufficient diftindnefs^ to re-
prefent Objeds five or fix hundred times biggerttlian
at a Foot diftance they appear to our naked Eyes, I
fhould hope that we might be able to difcover fome of
the greateft of thofe corpufcles. And by one that would
magnify three or four thoufand times perhaps they
might all be difcovered, but thofe which produce black-
nefs. In the mean while I fee nothing material in this
Difcourfe that may rationally be doubted of excepting
liiis Pofition, That tranfparent corpufcles of the fame
thicknefs and denfity with a Plate, do exhibit the fa#e
Colour. And this I would have underftood not with-
out fome latitude, as well becaufe thofe corpufcles may
be of irregular Figures, and many rays muft be oblique-
ly incident on them, and fo have a fhorter way through
them than the length of their Diameters, as becaufe the
ftraitnefs of the medium pent in on all fides within fuch
corpufcles may a little alter its motions or other qua-
lities on which the reflexion depends. But yet I can-
not much fufpeft the laft, becaufe I have obferved of
fome fmall Plates of Mufcovy-Glafs which were of an

even

even thicknefs, that through a Mifcrofcope they have
appeared of the fame Colour at their edges and cor-
ners where the included medium w^as terminated, which
they appeared of in other places. However it will add
much to our fatisfaftion, if thofe corpufcles could be dif-
covered with Mifcrofcopes ; which if we fhall at length
attain to, I fear it will be the utmoft improvement of
this fenfe. For it feems impoffible to fee the more fe-
cret and noble works of nature within the corpufcles
by reafon of their tranfparency.

P R O R VIIL

T^he caufe of Reflexion is not the imfinging of Light on
the folid or im^ervtom farts of Bodies ^ as is commonly ie*
lieved.

This will appear by the following Confiderations»
Firft, That in the paffage of Light out of Glafs into
Air there is a reflexion as flrong as in its paffage out of
Air into Glafs, or rather a little ftronger, and by many
degrees ftronger than in its paffage out of Glafs into
Water. And it feems not probable that Air fhould have
more reflefting parts than Water or Glafs. But if that
Ihould poffibly be fuppofed, yet it will avail nothing |
for the reflexion is as ftrong or fl:ronger when the Air is
drawn away from the Glafs, (fuppofe in the Air-pump
invented by Mr. Boyle ) as when it is adjacent to it.
Secondly, If Light in its paffage out of Glafs into Air
be incident more obliquely than at an Angle of 40 or
41 degrees it is wholly reflected, if lefs obliquely it is
in great meafure tranfmitted. Now it is not to be ima-
gined that Light at one degree of obliquity fliould meet

K k with

with pores enough in the Air to tranfmit the greater
part of it, and at another degree of obUquity fhould
meet with nothing but parts to refleft it wholly, efpe^
daily conlidering that in its paffage out of Air into
Glafs , how oblique foever be its incidence , it finds
pores enough in the Glafs to tranfmit the greateft part
of it. If any Man fuppofe that it is not refleded by the
Air,- but by the outmoft fuperficial parts of the Glafs^
there is ftill the fame difficulty : Beiides, that fuch a
Suppofition is unintelligible, and will alfo appear to be
falfe by applying Water behind fome part of the Glafs
inftead of Air. For fo in a convenient obliquity of the
rays fuppofe of 45 or 46 degrees, at which they are all
refieded where the Air is adjacent to the Glafs, they
fliall be in great meafure tranfmitted where the Water
is adjacent "to it; which argues, that their refkxion
or. tranfmiffion depends on the conftitution of the Air
and Water behind the Glafs, and not on the ftriking
off the rays upon the parts of the Glafs. Thirdly, If
the Colours made by a Prifm placed at the entrance of
a beam of Light into a darkened room be fucceffively
caft on a fecond Prifm placed, at a greater diftance from
the former, in fach manner that they are all alike inci-
dent upon it, the fecond Prifm may be fo inclined to
the incident rays, that thofe which are of a blue Colour
fhall be all reflefted by it, and yet thofe of a red Colour
pretty copioufly tranfmitted. Now if the reflexion be
caufed by the parts of Air or Glafs, I would ask, why
at the fame obliquity of incidence the blue fhould whol-
ly impinge on thofe parts fo as to be all refleded, and
yet the red find pores enough to be in great meafure
tranfmitted. Fourthly, where two Glalles touch one

another^

another, there is no fenfible reflexion as was declared
in the firft Obfervation; and yet I fee no reafon why
the rays fhould not impinge on the parts of Glafs as
much when contiguous to other Glafs as when con-
tiguous to Air. Fifthly, When the top of a Water-
bubble (in the i yth Obfervation) by the continual fub-
fiding and exhaling of the Water grew very thin, there
was fuch a little and almoft infenfible quantity of Light
refle£led from it, that it appeared intently black ; where-
as round about that black Spot, where the Water was
thicker, the reflexion was lb ftrong as to make the
Water feem very white. Nor is it only at the leaft
thicknefs of thin Plates or Bubbles, that there is no
manifefl: reflexion, but at many other thicknelTes con-
tinually greater and greater. For in the 1 5 th Obfer-
vation the rays of the fame Colour were by turns tranf-
mitted at one thicknefs^ and reflected at another thick-
nefs, for an indeterminate number of fucceflions. And
yet in the fuperficies of the thinned Body, w^here it is
of any one thicknefs, there are as many parts for the
rays to impinge on, as where it is of any other thick-
nefs. Sixthly, If reflexion were caufed by the parts of
reflecting Bodies, it would be impoflible for thin Plates
or Bubbles at the fame place to reflect the rays of one
Colour and tranfmit thofe of another, as they do accor^
ding to the 13th and 15th Obfervations. For it is
not to be imagined that at one place the rays which
for infl:ance exhibit a blue Colour, ftiould have the for-
tune to dafli upon the parts, and thofe which exhibit
a red to hit upon the pores of the Body ; and then at
another place, where the Body is either a little thicker,
or a Uttle thinner, that on the contrary the blue fliould

K k a * hit

d8]

hit upon its pores, and the red upon its parts, Laftly^
were the rays of Light reflefted by impinging on the
folid parts of Bodies, their reflexions from poliflied Bo-
dies could not be fo regular as they are. For in po-
lilTiing Glafs with Sand, Putty or Tripoly, it is not to
be imagined that thofe fubftances can by grating and
fretting the Glafs bring all its leaft particles to an ac-
curate polifh ; fo that all their furfaces fhall be truly
plain or truly fpherical^ and look all the fame way, fo
as together to compofe one even furface. The fmaller
the particles of thole fubftances are, the fmaller will
be the fcratches by which they continually fret and wear
away the Glafs until it be polifhed, but be they never
fo fmall they can wear away the Glafs no otherwife
than by grating and fcratching it , and breaking the
proturberances , and therefore polifh it no otherwife
than by bringing its roughnefs to a very fine Grain, fo
that the fcratches and frettings of the furface become
too fmall to be vifible* And therefore if Light were
refleded by impinging upon the folid parts of the Glafs^
it would be Icattered as much by the moft poliflied
Glafs as by the rougheft. So then it remains a Pro-
bkm, how Glafs poliflied by fretting fubftances can re-
fled Light fo regularly as it does. And this Problem
is fcarce otherwife to be folved than by faying, that
the reflexion of a ray is efFeded, not by a Angle point of
the refleding Body, but by fome power of the Body
which is evenly diffufed aU over its furface, and. by
which it ads upon the ray without immediate contad.
For that the parts of Bodies do ad upon Light at a di-
ftance fliall be fliewn hereafter^

Now

Now if Light be reflefted not by impinging on the
folid parts of Bodies, but by fome other principle ; its
probable that as many of its rays as impinge on the
folid parts of Bodies are not reflefted but ftifled and
loft in the Bodies. For otherwife we muft allow two
forts of reflexions. Should all the rays be reflected which
impinge on the internal parts of clear Water or Cryftal,
thofe fubftances would rather have a cloudy Colour
than a clear tranfparency. To make Bodies look black,
its neceflary that many rays be ftopt, retained and loft
in them, and it feems not probable that any rays can
be ftopt and ftifled in them which do not impinge on
their parts.

And hence we may underftand that Bodies are much
more rare and porous than is commonly believed. Wa-
ter is 1 9^ times lighter, and by confequence 1 9 times
rarer than Gold , and Gold is fo rare as very readily
and without the leaft oppofition to tranfmit the mag-
netick. EfHuvia, and eafily to admit Quick-filver into
its pores, and to let Water pafs through it. For a con-
cave Sphere of Gold filled, with Water, and fodered up,'
has upon prefling^ the Sphere with great force, let the
Water fqueeze througli it, and ftand. all over its out-
fide in multitudes of fmall Drops, like dew, without
burfting or cracking the Body of the Gold as I have
been informed by an Eye^witnefs. From all which we
may conclude, that Gold has more pores than folid
parts, and by confequence that Water has above forty-
times more pores than parts. And he that fhali find out
anHypothefis, by which Water may be fo rare, and yet
not be capable of compreflion by force, may doubtlefs
by the fame Hypothefis make Gold and Water, and all

other

C7o]

Other Bodies as much rarer as he pleafes, fo that Light
may find a ready paflage through tranfparent fub- ,,
ftances. ^d-. aJirU^^ ^ ^n^-pslkj <^^^ki>i4^l^ti,>6die.^ "^h^x^S

P R O P. IX,

^Bodies refieB and refraB Light iy one and the fame
fo^wer varioujly exercijed in variom circumflances, v

This appears by feveral Confiderations. Firft, Be^
caufe when Light goes out of Glafs into Air, as ob-
liquely as it can poffibly do, if its incidence be made
ftill more oblique , it becomes totally refleded. For
the power of the Glafs after it has refracted the Light
as obliquely as is poffible if the incidence be ftill made
more oblique, becomes too ftrong to let any of its rays
go through, and by confequence caufes total reflexions.
Secondly , Becaufe Light is alternately reflefted and
tranfmitted by thin Plates of Glafs for many fucceflions
accordingly , as the thicknefs of the Plate increafes
in an arithmetical Progreflion. For here the thicknefs
of the Glafs determines whether that power by which
Glafs afts upon Light fhall caufe it to be reflefted, or
fuifer it to be tranfmitted. And, Thirdly, becaufe thofe
furfaces of tranfparent Bodies which have the greateft
refrafting power, refleft the greateft quantity of Light,
as was fhewed in the firft Propofition.

PROP. X.

s If Light he fvoifter in Bodies than in Vacuo in the
ffofcrtion of the Sines 'which meafure the refraction of the
B^dies^ the forces of the Bodies to rejleB and refraB Light ^

are

[71]

are ver^ nearly i^r of onioned to the denfities of the jame
Bodies J excepting that undiuom and fulfhureom Bodies re^
fra^ more than others of this fame denfit'j . ^ '

Let A B reprefent the refrafting plane furfacc of any '4/rA9^//i?4>f ^J^
Body/ and IC a ray incident very obliquely upon the '?^- ^^^

Body in C, fo that the Angle A CI may be infinitely
little^ and let C R be the refracted ray. From a given
point B perpendicular to the refracting furface ere£t
B R meeting w^ith the refradied ray C R in R, and if
CR reprefent the motion of the refracted ray, and this^
motion be diftinguilhed into two motions CB and BR^
whereof CB is a parallel to the refracting plane, and
BR perpendicular to it : CB fhall reprefent the motion
of the incident ray, and B R the motion generated by
the refraction, as Opticians have of late explained.

Now if any body or thing in moving through any
fpace of a giving breadth terminated on both fides by
two parallel plains, be urged forward in all parts of
that fpace by forces tending direCtly forwards towards
the lafl: plain , and before its incidence on the firil
plane, had no motion towards it, or but an infinitiy
little one ; and if the forces in all parts of that fpace^
between the planes be at equal difl:ances from the planes
equal to one another, but at feveral difl:ances be bigger
or lefs in any given proportion, the motion generated
by the forces in the whole paflage of the body or thing

tlirough.

[72]

through that fpace (hall be in a fubduplicate proportion
of the forces, as Mathematicians will eafily underftand.
And therefore if the fpace of aftivity of the refradling
fuperficies of the Body be confidered as fuch a fpace,
the motion of the ray generated by the refrading force
of the Body , during its paflage through that fpace
that is the motion BR muft be in a fubduplicate
proportion of that refra61:ing force : I fay therefore that
the fquare of the Line BR, and by confequence the
refracting force of the Body is very nearly as the den-
fity of the fame Body. For this will appear by the fol-
lowingTable, wherein the proportion of the Sines which
meafure the refraxions of feveral Bodies, the fquare
of BR fuppofing CB an unite, the denfities of the
Bodies eftimated by their fpecifick gravities, and their
refraftive power in refpeft of their denfities are fet
down in feveral Columns.

The

[73]

The refrading Bodies.

The Proportion
of the Sines of
incidence and
refraction of
yellow Light,

A Pfeudo-Topazius, be-
ing a natural,pellucid,
brittle, hairy Stone, of
a yellow Colour

Air

Glafs of Antimony

A Selenitis

Glafs vulgar

Cryftal of the Rock

Ifland Cryflal

Sal Gemmae

Alume

Borax

Niter

Dantzick Vitriol

Oyl of Vitriol

Rain Water

Gumm Arabic

Spirit of Wine well redi
fied

Camphire

Oyl Olive

Lintfeed Oyl

Spirit of Turpentine

Ambar

A Diamond

23 to 14

3851

3850

£0

?o?

200

529

396

100 to 73

3 to
22 to
40 to
25 to
14 to
100 to

The Square of
BR, to which
the refra^ing
force oftheBo.
dy is propor-
tionat-e.

The denfitjCThe refra-
and fpeci- £tive power

fc gravity
bf the Bo-

4r-

i'699

o'ooo52
2'568

l'2l3

i'4025
i'445

i'388

i'i267

i'i5ii

i'345
i'295

I '041

o'7845

i'i79

o'8765

I I5II

i'i948
i'i626

l'42

4^949

427

o'ooi2 5

5'28

2*252

2'58

2^65

2'72

2'i43
i'7i4
i'7i4
I '9
i'7i5

i'7
I.

i'375

o'866

©'996
0^913
o'932
o'874
I '04

?'4

of the Body
in refpeCi
of its den-

S979

4160
4864
5386
543^

5450
6536

6477

6570

6716
7079

7551
6124

7S45
8574

10121

12551

12607
12819

13222
13654

14556

The refra£tion of the Air in this Table is determined
by that of the Atmofphere obferved by Aftronomers.
For if Light pafs through many refracting fubftances or
mediums gradually denier and denier, and terminated

L 1 with

[74]

with parallel furfaces, the fumm of all the refraftions
will be equal to the fingle refraStion which it would
have fufFered in palling immediately out of the firft
medium ifito the laft. And this holds true, though the
number of the refrafting fubftances be increafed to infi-
nity, and the diftances from one another as much de-
creafed, fo that the Light may be rcfrafted in every
point of its paffage, and by continual refraftions bent
into a curve Line. And therefore the whole refradion
of Light in paffing through the Atmofphere from the
higheft and rareft part thereof down to the loweft and
denfeft part, muft be equal to the refraction which it
would futfer in paffing at like obliquity out of a Va-
cuum immediately into Air of equal denfity with that
in the loweft part of the Atmofphere,

Nov/, by this Table, the refraftions of a Pfeudo-To-
paz, a Selenitis, Rock Cryftal, liland Cryftal, Vulgar
Glafs ( that is. Sand melted together ) and Glafs of
Antimony, which are terreftrial ftony alcalizate con-
cretes,and Air which probably arifes from fuch fubftances
by fermentation, though thefebe fubftances very differing
from one another in denfity, yet they have their refra-
ctive powers almoft in the fame proportion to one ano-
ther as their denfities are, excepting that the refraction of
that ftrange fubftance Illand-Cryftal is a little bigger
than the reft. And particularly Air, which is 5490 times
rarer than the Pleudo-Topaz, and 4100 times ra^'er than
Glafs of Antimony, has notwithftanding its rarity the
fame refraCtivc power in refpeCl of its denfity which
thofe two very denfe fubftances have in refpeCt of theirs,
excepting fo tar as thofe two differ from one another.

Again,

Again, the refraftion of Camphire, Oyl-Olive, Lint-
feed Oyl, Spirit of Turpentine and Amber, which are
fat fulphureous unftuous Bodies, and a Diamond, which
probably is an unftuous fubftance coagulated, have their
refraftive powers in proportion to one another as their
denfities without any confiderable variation. But the
refradive power of thefe unftuous fubftances is two
or three times greater in refpeft of their denfities than
the refractive powers of the former fubftances in refped
of theirs.

Water has a refra<flive power in a middle degree be-
tween thofe two forts of fubftances, and probably is of
a middle nature. For out of it grow all vegetable and
animal fubftances, which confift as well of fulphureous
fat and inflamable parts, as of earthy lean and alcali-
zate ones.

Salts and Vitriols have refradive powers in a middle
degree between thofe of earthy fubftances and Water,
and accordingly are compofed of thofe two forts of fub-
ftances. For by diftillation and rectification of their
Spirits a great part of them goes into Water, and a great
part remains behind in the form of a dry fixt earth ca«
pable of vitrification.

Spirit of Wine has a refractive power in a middle
degree between thofe of Water and oyly fubftances, and
accordingly feems to be compofed of both, united by
fermentation ; the Water, by means of fome faline Spi-
rits with which 'tis impregnated, diffolving the Oyl,
and volatizing it by the aCtion. For Spirit of Wine is
inflamable by means of its oyly parts, and being diftil-
led often from Salt of Tartar, grows by every diftilla-
tion more and more aqueous and flegmatick. And

LI 2 Chymifts

Ghymifts obferve, that Vegitables (as Lavender, Rue,
Marjoram, )^c\) diftilled fer fe , before fermentation
yield Oyls without any burning Spirits, but after fer-
mentation yield ardent Spirits without Oyls : Which
fliews, that their Oyl is by fermentation converted into
Spirit. They find alfo, that if Oyls be poured in fmall
quantity upon fermentating Vegetables, they diftil over
after fermentation in the form of Spirits.

So then, by the foregoing Table, all Bodies feem to
have their refraSive powers proportional to their
denfities, ( or very nearly ; ) excepting fo far as they
partake more or lefs of fulphurous oyly particles, and
thereby have their refractive power made greater or
lefs. Whence it feems rational to attribute the refra-
ctive power of all Bodies chiefly, if not wholly, to the
fulphurous parts with which they abound. For it's
probable that all Bodies abound more or lefs with Sul-
phurs. And as Light congregated by a Burning-glafs
a£ts moft upon fulphurous Bodies, to turn them in-
to fire and flame ; fo, fince all a£tion is mutual. Sul-
phurs ought to a£t mofl: upon Light. For that the
action between Light and Bodies is mutual, may appear
from this Confideration, That the denfefl: Bodies which
refrad: and reflect Light mofl: ftrongly grow hotteft in
the Summer-Sun, by the a£tion of the refracted or re-
flefted Light.

I have hitherto explained the power of Bodies to re>
fleft and refra£t, and (hewed, that thin tranfparent
plates, fibres and particles do, according to their leveral
thicknefles and denfities, reflect feveral forts of rays,
and thereby appear of feveral Colours, and by conle-
quence that nothing more is requifite for producing all

the.

[77]

the Colours of natural Bodies than the ieveral fizes and
denfities of their tranfparent particles. But whence it
is that thefe plates, fibres and particles do, according
to their feveral thicknefles and denfities, refled: fev^eral
forts of rays, I have not yet explained. To give fome
infight into this matter, and make way for underflian-
ding the next Part of this Book, I fliall conclude this-
Part with a few more Propofitions. Thofe which pre-
ceded refpedl the nature of Bodies, thefe the nature of
Light : For both mufl: be underfliood before the reafom
of their adlions upon one another can be known. And^
becaufe the lafl: Propofition depended upon the velo-
city of Light, 1 will begin with a Propofition of that
kind.

PROP. XL

^' Light is frofagated from luminom Bodies in time^ and
fiends about [even or eight minutes of an hour in 2^ffi^ig-
from the Sun to the Earth,

This was obferved firft by Romer^ and then by others3.
by means of the Eclipfes of the Satellites oi'^upter,-
For thefe Eclipfes, when the Earth is between the Sun
^nAjufiter^ happen about feven or eight minutes fooner
than they ought to do by the Tables, and when the Earth
is beyond the Sun they happen about feven or eight mi-
nutes later than they ought to do; the reafon being, that
the Light of the Satellites has farther to go in the latter
cafe than in the former by the Diameter of the Earth's
Orbit. Some inequalities of time may arife from the
excentricities of the Orbs of the Satellites ; but thofe
eannot anfwer in all the Satellites , and at all times

to the pofition and diftance of the Earth from the Sun.
The mean motions of ^ufiter's Satellites is alfo fwifter
in his defcent from his Aphelium to^ his Perihelium,
than in his afcent in the other half of his Orb : But this
inequality has no refped to the pofition of the Earth,
and in the three interior Satellites is infenfible, as I find
by computation from the Theory of their gravity.

PROP. XIL

iv Every r^y^i>f Light in its fajfage through any refra^
Bin^ furface is fut into a certain tranfient conflitution
or hate , "which in the frogrejs of the ray returns at
equal intervals^ and dijfojes the ray at every return
to he eafdy tranjmitted through the next refra(Bing fur^
facej and het'ween the returns to be eafdy refiedied by

it' \{

This is manifeft by the 5th, 9th5 i ith and 1 5th Ob-
fervations. For by thofe Obfervations it appears, that
, one and the fame fort of rays at equal Angles of inci-
dence on any thin tranfparent plate, is alternately refle-
cted and tranfmitted for many fucceflions accordingly,
as the thicknefs of the plate increafes in arithmetical
progreffion of the numbers o, 1,1, 3, 4, 5, 6, 7, 8, Iffc,
fo that if the firft reflexion (that which makes the firfl:
or innermofl; of the Rings of Colours there defcribed )
be made at the thicknefs i,the rays fliallbe tranfmitted at
the thicknefles o, a, 4, 6, 8, 10, la, Jf^r. and thereby
make the central Spot and Rings of Light, which ap-
pear by tranfmiflion, and be reflected at the thicknefs
I, 3, 5, 7, 9,ai,}5)'^\ and thereby make the Rings which

appear

[79 3

appear by reflexion. And this alternate reflexion and
tranfmiflion, as 1 gather by the 14th Obiervation, con-
tinues for above an hundred viciflitudes, and by the
the Obfervations in the next part of this Book^ for many
thoufands, being propagated from one furface of a Glafs-
plate to the other, though the thicknefs of the plate
be a quarter of an Inch or above : So that this alter-
nation feems to be propagated from every refracting
furface to all diftances without end or limitation.

This alternate reflexion and refradion depends on
both the furfaces of every thin plate, becaufe it de-
pends on their diltance. By the aith Obfervation, if
either furface of a thin plate of Mufcovy-Glafs be wet-
ted, the Colours caufed by the alternate reflexion
and refraction grow faint, and therefore it depends on
them both.

It is therefore performed at the fecond furface, for
if it were performed at the firfl:, before the rays ar-
rive at the fecond, it would not depend on the fe-
cond «

It is alfo influenced by fome aftion or difpofition,
propagated from the firfl: to the fecond, becaufe other-
wife at the fecond it would not depend on the firfl:. And
this action or difpofition, in its propagation, intermits
and returns by equal intervals, becaufe in all its pro-
grefs it inclines the ray at one diftance from the firfl:
furface to be reflected by the fecond, at another to be
tranfmitted by it, and that by equal intervals for innu-
merable viciflitudes. And becaufe the ray is difpofed
to reflexion at the diftances i, 3:, 5> 7-, 9>J55'^- and to
tranfmiflion at the diftances o, a, 4, 6, 8, lo^'^c^ ( for
its traafmiflion through the firft furface, is at the di-
ftance

[8o]

ftance o, and it is tranfmitted through both toge-
ther, if their diftance be infinitely Uttle or much lefs
than I ) the ditpofition to be tranfmitted at the diftances
1^ 4., 6, 8, 10, ]5fc. is to be accounted a return of the
fame difpofition which the ray firft had at the diftance o,
that is at its tranfmiffion through the firft refrafting fur-
face. AH which is the thing I would prove.

What kind of aftion or difpofition this is ? Whether
it confift in ' a circulating or a vibrating motion of the
ray, or of the medium, or fomething elfe ? I do not
here enquire. Thpfe that are averfe from affenting to
any new difcoveries, but fuch as they can explain by an
Hypothefis, may for the prefent fuppofe, that as Stones
by falling upon Water put the Water into an undula-
ting motion, and all Bodies by percuflion excite vibra-
tions in the Air; fo the rays of Light, by impinging on
any refrafting or reflecting furface, excite vibrations in
the refrafting or refleding medium or fubftance, and
by exciting them agitate the folid parts of the refrafiting
or refleding Body, and by agitating them caufe the Body
to grow warm or hot 3 that the vibrations thus excited
are propagated in the refrafting or reflefting medium
or fubftance, much after the manner that vibrations are
propagated in the Air for caufing found, and move
fafter than the rays fo as to overtake them ; and that
when any ray is in that part of the vibration which con-
fpires with its motion, it eafily breaks through a re-
frading furface, but when it is in the contrary part of
the vibration which impedes its motion, it is eafily
reflected ; and, by confequence, that every ray is fuc-
ceflively difpofed to be eafily reflected, or eafily tranf-
mitted, by every' vibration which overtakes it. But

whether

[8ij

whether thisHypothefis be true or falfe I do not here
confider. I content my felf with the bare dilcovery,
that the rays of Light are by fome caufe or other alter-
nately difpofed to be refleded or refrafted for many vi-
ciffitudes.

"DEFINITION.

The returns of the diffofition of any ray to be refle^ed
I zmll call its Fits of eafy reflexion, and thofe of
its diffofition to be tranfmttted its Fits of eafy tranf-
miflion, and the fj^ace it fajfes between every re-
turn and the next return^ the Interval of its
Fits.

PROP. XIII.

' The reafon ^why the fur faces of all thick tranffarent
Bodies refieB fart of the Light incident on them^ and
refract the reft^ u^ that fome rays at their incidence are
in Fits of eafy reflexion^ and others in Fits of eafy tranf^
mi£lon, V

This may be gathered from the i^th Obfervation,-
where the Light refieSed by thin plates of Air andGIafs,
which to the naked Eye appeared evenly white all over
the plate, did through a Prifm appear waved with many
fucceflions of Light and Darknefs made by alternate fits
of eafy reflexion and eafy tranfmiflion , the Prifm
fevering and diftinguifliing the waves of which the
white refle<3;ed Light was compofed, as was explained
above.

M m And

And hence Light is in fits of eafy reflexion and eafjr
tranfmiffion, before its incidence on tranfparent Bodies,
And probably it is put into fuch fits at its firft emiffion
from luminous Bodies, and continues in them during
all its progrefs. For thefe fits are of a lafting Nature,
as will appear by the next part of this Book.

In this Propofition I fuppofe the tranfparent Bodies
to be thick, becaufe if the thicknefs of the Body be
much lefs than the interval of the fits of eafy reflexion
and tranfmiflion of the rays, the Body lofethits reflecting
power. For if the rays, which at their entering into
the Body are put into fits of eafy tranfmiflion, arrive at
the furthefl: furface of the Body before they be out of
thofe fits they mufl: be tranfmitted. And this is the
reafon why Bubbles of Water lofe their reflecting power
when they grow very thin, and why all opake Bo-
dies when reduced into very fmall parts become tranf-
parent.

PROP. XIV.

T'hoje Jur faces of tranfparent Bodies^ "which if the ra^
h in a fit ofrefradion do refraB it mop f^^ongl^y if the
f(Vj he in a fit of re^exion do refle3 it mofl eafdp

For we fliewed above in Prop. 8. that the caufe of
reflexion is not the impinging of Light on the folid
impervious parts of Bodies, but fome other power by
which thofe folid parts aft on Light at a diftance. We
fliewed alfo in Prop. 9. that Bodies refled and refraft
Light by one and the fame power varioufly exercifed in
various circumftances, and in Prop. i. that the mofl:
ftrongly refrafting furfaces refleft the moft Light : All

which

[83]

which compared together evince and ratify both this
and the laft Propolition.

PROP. XV.

In an^j one and the fame fort of rays emerging in any
j4ngle out of any refraSing furface into one and the fame
medium^ the interval of the foUo'wing jits of eafy reflexion
and tranfmijflon are either accurately or very nearly^ as
the Redangle of the fecant of the Angle of refra^ion^ and
^f ^^^ fecant of another jdngle^ "whofe fine is the firft of
1 06 arithmetical mean froprtionals ^ Set ween the fines
of incidence and refra^iion counted from the fine of re^
frailion, , (^ ij^

This is manifeft by the ythpbfervation.

PRO R XVL

In fever al forts of rays emerging in equal jingles out
of any refraSling furface into the fame medium^ the inter^
vals of the foUo'uuing fits of eafy reflexion and eafy tranp
mijfflon are either accurately j or very nearly j as the Cube^
roots of the Squares of the lengths of a Chord j which found
the notes in an Eighty foi, la, fa, fol, la, mi, fa, fol, with
all their intermediate degrees anfwering to the Colours of
thofe rays^ according to the Analogy defcribed in the fe^
venth Experiment of the fecond Booh

This is manifeft by the 13 th and i4thObfer¥ations.

Mm 2 PROP,

[84]

PROP. XVII.

Jfrap of any one fort fafs ferfendicularly into fever al
mediums^ the intervals of the fits of eafy reflexion and
tranfmijjlon in any one medium^ ts to thafe intervals in
any other a^ the fine of incidence to the fine of refraSion^
*when the rays fafs out of the firfi of thofe fwo mediums
into the fecond.

This is manifeft by the loth Obfervation.

P R O P. XVIIL

Jf the rays "which faint the Colour in the confine of
yellois) and orange fafs ferfendicularly out gf any medium
into jtir^ the intervals of their fits of eafy reflexion are
the ^Jih fart of an Inch, u4nd of the fame length are
the intervals of their fits of eafy tranfmijfion.

This is manifeft by the 6th Obfervation.

From thefe Propofitions it is eafy to colle£l: the in-
tervals of the fits of eafy reflexion' and eafy tranfmif-
fion of any fort of rays refrafted in any Angle into
any medium, and thence to know, whether the rays
fliall be refleded or tranfmltted at their fubfequent
incidence upon any other pellucid medium. Which
thing being ufeful for under ftanding, the next part of
this Book was here to be fet down. And for the fame
reafoa I add the two following Propol&tions*

' PROP.

PROP. XIX.

• If my fort of rays falling on the polite furface of am
fellucid medium he refleB>ed hach^ the fits of eafy re^
flexion "which they have at the point of reflexion , Jhall
fiiU continue to return^ and the returns fhall he at di-
fiances from the point of reflexion in the arithmetical
progrejfion of the numhers o,^ 4.^ 6, 8, 10, ii^&c. and be-
tween thefe fits the rays Jhall be in fits of eafy tranf-
mijfi^on.

For fince the fits of ealy reflexion and eafy tranf-
miflion are of a returning nature, there is no reafon
why thefe fits, which continued till the ray arrived at
the refleding medium, and there inclined the ray to
reflexion, fliould there ceafe. And if the ray at the
point of reflexion was in a fit of eafy reflexion, the
progrefiion of the difl:ances of thefe fits from that point
muft begin from o, and fo be of the numbers o, 2,4.5
6, 8, "^c. And therefore the progrefiion of the di-
ftances of the intermediate fits of eafy tranfmiflion rec-
koned from the fame point, mufl: be in the progrefiion
of the odd numbers i, 5, 5, 7, 9,^'^- contrary to what
happens when the fits are propagated from points of
refraftion,

PROP. XX

The intervals of the fits of eafy reflexion and eafy
tranfrrujfon^ propagated fram points of reflexion into any
medium;^ are equal to the intervals of the like fits -which
the fame rays would have^ if refradied into the fame

medium

[85]

medium in Angles of jrfra^ion equol to their Angles of
reflexion .

For when Light is refleded by the fecond furface of
thin plates, it goes out afterwards freely at the firft fur-
face to make the Rings of Colours which appear by
reflexion, and by the freedom of its egrefs, makes the
Colours of thefe Rings more vivid and ftrong than thofe
which appear on the other fide of the plates by the
tranfmitted Light. The refleded rays are therefore in
fits of eafy tranfmiffion at their egrefs ; which would
not always happen, if the intervals of the fits within
the plate after reflexion were not equal both in length
and number to their intervals before it. And this confirms
alio the proportions fet down in the former Propofition.
For if the rays both in going in and out at the firft furface
be in fits of eafy tranfmiflion,andthe intervals and num-
bers of thofe fits between the firft and fecond furface,
before and after reflexion, be equal | the diftances or
the fits of eafy tranfmiflion from either furface, muft be
in the fame progr-eflion after reflexion as before ; that
is, from the firft furface which tranfmitted them, in
the progreflion of the even numbers o, 2, 4, 6, S, }5^c.
and from the fecond which reflected them, in that of
the odd numbers i^ 3^ 5, 7j ^^* But thefe two Pro-
pofi.tions will become much more evident by the Obfer^
vations in the following part of this Book*

T H E^

]

THE

SECOND BOOK

O F

OPT

PARTI

Oifervations concerning the Reflexions and Colours of
thick tranf^arent folijhed Vlates,

THere is no Glafs or Speculum how well foever
poliflied, but, befides the Light which it refrads
or refieias regularly , fcatters every way irregularly a
faint Light, by means of which the polilhed furface,
when illuminated in a dark Room by a beam of the
Sun^s Light, may be eafily feen in all pofitions of the
Eye, There are certain Phgenomena or this fcattered
Light, which when I firft obferved them, feemed very
ftrange and furprifing to me. My Obfervations were
as follows.

OBS.

[88]

O B S. I.

The Sun fhinlng into my darkened Chamber through
a Hole I of an Inch wide, I let the intromitted beam
of Light fall perpendicularly upon a Glafs Speculum
ground concave on one fide and convex on the other,
to a Sphere of five Feet and eleven Inches Radius, and
quick'filvered over on the convex fide. And holding
a white opake Chart, or a Quire of Paper at the Center
of the Spheres to which the Speculum was ground, that
is, at the diftance of about five Feet and eleven Inches
from the Speculum, in fuch manner, that the beam of
Light might pafs through a little Hole made in the
middle of the Chart to the Speculum, and thence be
refleded back to the fame Hole : I obferved upon the
Chart foyr or five concentric Irifes or Rings of Colours,
like Rain-bows, encompaffing the Hole much after the
manner that thofe, which in the fourth and following
Obfervations of the fir ft part of this third Book appeared
between the Objeft-GlaffeSjencompaffed the black Spot,
but yet larger and fainter than thofe. Thele Rings as
they grew larger and larger became diluter and fainter,
ib that the fifth was fcarce vifible. Yet fometimes,
when the Sun fhone very clear, there appeared faint
Lineaments of a fixth and feventh. If the diftance of
the Chart from the Speculum was much greater or much
lefs than that of fix Feet, the Rings became dilute and
vanifhed. And if the diftance of the Speculum from
the Window was much greater than that of fix Feet,
the reflected beam of Light would be fo broad at the
diftance of fix Feet from the Speculum where the Rings

appeared,

appeared, as to obfcure one^r t^o of the innennoft
Rmgs.^ And therefore I ufually placed the Speculum
at about fix Feet from the Window ; fo that its Focus
might there fall in with the center of its concavity at the
Rings upon the Chart. And this pofture is always to
be underftood in the following Oblervations where no
other is expreft.

O B S. 11.

The Colours of thefe Rain-bows fucceeded one anO'-
ther from the center outwards, in the fame form and
order with thofe which were made in the ninth Obfer-
vation of the firft Part of this Book by Light not re-
fleded, but tranfmitted through the twoObjeft-Glaffes^
For, firft, there was in their common center a white
round Spot of faint Light, fomething broader than the
reflefted beam of Light ; which beam fometimes fell
upon the middle of the Spot, and fometimes by a little
inclination of the Speculum receded from the middle^
and left the Spot white to the center.

This white Spot was immediately encompaffed with
a dark grey or ruffet, and that darknefs with the Co-
lours of^e fii'f|^J^j^^which were on the infide next
the darkne'^little vSSlet and indico, and next to that
a blue, which on the outfide grew pale, and then fuc-
ceeded a little greenifh yellow, and after that a brighter
yellow, and then on the outward edge of the Iris a red
which on the outfide inclined to purple.

This Iris was immediately encompaffed With a fe>-
cond, whofe Colours were in order from the infide

Nn out-

[90]

outwards, purple, blue, green, yellow, light red, a red
mixed with purple.

Then immediately followed the Colours of the third
Iris, which were in order outwards a green inclining
to purple, a good green, and a red more bright than
that of the former Iris.

The fourth and fifth Iris feemed of a bluifh green
within, and red without, but fo faintly that it was dif-
ficult to difcern the Colours.

O B S. IIL

Meafuring the Diameters of thefe Rings upon the
Chart as accurately as I could, I found them alfo in
the fame proportion to one another with the Rings
made by Light tranfmitted through the two ObjeiS-
Glaffes. For the Diameters of the four firft of the
bright Rings meafured between the brighteft parts of
their orbits, at the diftance of fix Feet from the Specu-
lum were iJJ, a^^ ijj, jf Inches, whofe fquares are in
arithmetical progreffion of the numbers i, i, 3, 4. If
the white circular Spot in the middle be reckoned
amongft the Rings, and its central Light , where it
feems to be mofi: luminous, be put equipollent to an
infinitely little Ring ; the fquares of the Diameters of the
Rings will be in the progreffion o, i, a, 5, 4, '^c, I
meafured alfo the Diameters of the dark Circles be-
tween thefe luminous ones, and found their fquares
in the progreffion of the numbers \^ i', i{^ ^it^^f-'-^
the Diameters of the firft four at the diftance of fix Feet
from the Speculum, being ij6,2[g,2|j 3f, Inches. If
the diftance of the Chart from the Speculum was in-

crealed

t90

creafcd or diminifhed, the Diameters of the Circles were
iacreafed or diminiihed proportionally.

O B S. IV.

By the analogy between thefe Rings and thofe de-
fcribed in the Obfervations of the fir ft Part of this Book,
I fufpeded that there were many more of them which
fpread into one another, and by interfering mixed their
Colours, and diluted one another fo that they could
not^^e feen apart. I viewed them therefore through a
Prifm, as I did thofe in the 24th Obfervation pf the
firft Part of this Book. And when the Prifm was fo
placed as by refrading the Light of their mixed Co-
lours to feparate them, and diftinguiih the Rings from
one another, as it did thofe in that Obfervation, I could
then fee them diftinder than before, and eafily num-
ber eight or nine of them, and fometimes twelve or
thirteen. And had not their Light been fo very faint^
I queftion not but that I might have feen many more,

. O B S. V.

Placing a Prifm at the Window to refract the intro-
mitted beam of Light, and caft the oblong Spe(5lrum
of Colours on the Speculum : I covered the Speculum
with a black Paper which had in the middle of it a Hole
to let any one of the Colours pafs through to the Spe-
culum, whilft the reft were intercepted by the Paper.
And now I found Rings of that Colour only which fell
upon the Speculum. If the Speculum was illuminated
with red the Rings were totally red with dark inter-

Nn ^ vals,

[92]

vals, if with blue they were totally blue, and lb of the
other Colours. And when they were illuminated with
any one Colour, the Squares of their Diameters mea-
fured between their moft luminous parts, were in the
arithmetical progreffion of the numbers o, 1,^,9,4, and
the Squares of the Diameters of their dark intervals in
the progreffion of the intermediate numbers 7, i{, a{, :^\i
But if the Colour was varied they varied their magni^
tude. In the red they were largeft, in the indico and
violet leaft, and in the intermediate Colours yellov/,
green and blue; they were of ieveral intermediate big-
neffes anfwering to the Colour, that is, greater in yel-
low than in green^ and greater in green than in blue.
And hence I knew that when the Speculum was illumi-
nated with white Light, the red and yellow on the out-
fide of the Rings were produced by the leaft refrangible
rays, and the blue and violet by the moft refrangible^
and that the Colours of each Ring fpread into the Co-
lours of the neighbouring Rings on either fide, after
the manner explained in the firft and fecond Part of this
Book, and by mixing diluted one another fo that they
could not be diftinguiflied, unlefs near the center where
they were leaft mixed. For in this Obfervation I could
fee the Rings more diftinftly, and to a greater number
than before, being able in the yellow Light to number
eight or nine of them, beiides a faint fhadow of a tenth.
To fatisfy my felf how much the Colours of the feveral
Rings fpread into one another, I meafured the Diame-
ters of the fecond and third Rings , and found them
when made by the confine of the red and orange to be
the fame Diameters when made by the confine of blue
md indico;^ as 9 to 85 or thereabouts. For it was hard

[93]

to determine this proportion accurately. Alfo the Cir<-
cles made fucceffively by the red, yellow and green^
differed more from one another than thofe made fuccef-
fiv^ely by the green, blue and indico. For the Circle
made by the violet was too dark to be feen. To carry
on the computation, Let us therefore fuppofe that the
diiferences of the Diameters of the Circles made by the
outmoft red, the confine of red and orange, the confine
of orange and yellow, the confine of yellow and green,
the confine of green and blue, the confine of blue and
indico, the confine of indico and violet, and outmoft vio-
let, are in proportion as the diiferences of the lengths
of a Monochord which found the tones in an Eight ;
fol^la^fa^fol^la^mi^fa^fol^ that is, as the numbers ^^
h, t:? ?z, i, [-J, Is- And if the Diameter of the Circle made
by the confine of red and orange be 9 A, and that of
the Circle made by the confine of blue and indico be
8 A as above, their diiference 9 A — — 8 A will be to
the difference of the Diameters of the Circles made by
the outmoft red, and by the confine of red and orange^,
as f 8 + Ta + ^* + i7 to 9, that is as fr to t or 8 to 5, and to
the difference of the Circles made by the outmoft violet^
and by the confine of blue and indico, as ^s +72 -f t* + 1?
to i? + ^8, that is, as 17 to h, or as 1 6 to 5. And there-
fore thefe differences will be i A and U A. Add the
firft to 9 A and fubdud the laft from 8 A, and you
will have the Diameters of the Circles made by the
leaft and moft refrangible rays V A and pf A. Thefe
Diameters are therefore to one another as 75 to 61^ or
50, to 41, and their Squares as 1500 to 1681, that is^,
as 3 to 1 very nearly. Which proportion differs not
much from the proportion of the Diameters of the

[94]

^tl^ircles made by the outmoft red and outmoft violet in
the 1 5 th Obfervation of the firft part of this Book,

O B S. VL

Placing my Eye where thefe Rings appeared plaineflr,
I faw the Speculum tinged all over with waves of Co-
lours ( red, yellow, green, blue ; ) like thofe which in
the Obfervations of the firft Part of this Book appeared
between the Obje£t-Glafles and upon Bubbles of Water,
but much larger. And after the manner of thofe, they
w^ere of various magnitudes in various pofitions of the
Eye, fwelling and {hrinking as I moved my Eye this
way and that way. They w^re formed like Arcs of
xoncentrick Circles as thofe were, and when my Eye
was over againft the center of the concavity of the Spe-
culum (that is, 5 Feet and 10 Inches diftance from the
Speculum) their common center was in a right Line
with that center of concavity, and with the Hole in the
Window. But in*other poftures of my Eye their center
had other pofitions. They appeared by the Light of
the Clouds propagated to the Speculum through the
Hole in the Window, and when the Sun fhone through
that Hole upon the Speculum, his Light upon it was
of the Colour of the Ring whereon it fell, but by its
fplendor obfcured the Rings m.ade by the Light of the
Clouds, unlefs when the Speculum was removed to a
great diftance from the Window, fo that his Light upon
it might be broad and faint. By varying the pofition of
my Eye, and moving it nearer to or farther from the
direft beam of the Sun's Light, the Colour of the Sun's
i:efieded Light conftantly varied upon the Speculum,

as It did upon my Eye, the lame Colour always ap-
pearing to a By-ftander upon my Eye which to me ap-
peared upon the Speculum. And thence I knew that
the Rings of Colours upon the Chart were made by thefe
reflected Colours propagated thither from the Specu-
lum In feveral Angles, and that their production de-
pended not upon the termination of Light and Shad-
dow.

O B S. VIL

By the Analogy of all thefe Phaenomena with thofe of
the like Rings of Colours defcrlbed In the firft Part of
this Book, It feemed to me that thefe Colours were
produced by this thick plate of Glafs, much after the
manner that thofe were produced by very thia
plates. For, upon tryal, I found that if the Quick-
lilver were rubbed off from the back-fide of the Specu-
lum, the Glafs alone would caufe the fame Rings of
Colours, but much mor€ faint than before ; and there-
fore the Phaenomenon depends not upon the Quick-
filver, unlefs fo far as the Quick-filver by the increafing
the reflexion of the back- fide of the Glafs increafes the
Light of the Rln^s of Colours. I found alfo that a Spe-
culum of metal without Glafs made fome years fince
for optical ufes, and very well wrought, produced none
of thofe Rings ; and thence I underftood that thefe
Rings arlfe not from one fpecular furface alone , but
depend upon the two furfaces of the plate of Glafs where-
of the Speculum was made, ancl upon the thicknefs of
the Glafs between them. For as in the 7th and 19th
Obfervations of the firft Part of this Book a thin plate

'©f Air, Water, or Glafs of an even thicknefs appeared
of one Colour when the rays were perpendicular to it,
of another when they were a little oblique, of another
when more oblique, of another when ftill more oblique,
and fo on ; fo here, in the fixth Obfervation, the Light
which emerged out of the Glafs in feveral obliquities,
made the Glafs appear of feveral Colours, and being
propagated in thofe obliquities to the Chart, there pain-
ted Rings of thofe Colours. And as the reafon why a
thin plate appeared of feveral Colours in feveral obli-
quities of the rays,was,that the rays of one and the fame
fort are reflected by the thin plate at one obliquity and
tranfmitted at another, and thofe of other forts tranf-
mitted where thefe are retledted, and reflefted where
thefe are tranfmitted : So the reafon why the thick
plate of Glafs whereof the Speculum was made did ap-
pear of various Colours in various obliquities, and in
thofe obliquities propagated thofe Colours to the Chart,
was, that the rays of one and the fame fort did at one
obliquity emerge out of the Glafs, at another did not
emerge but were reflefted back towards the Quick-fil-
ver by the hither furface of the Glafs, and accordingly
as the obliquity became greater and greater emerged
and were retiefted alternately for many fucceffions, and
that in one and the fame obliquity the rays of one fort
were refleded, and thofe of another tranfmitted. This
is manifeft by the firft Obfervation of this Book : For
in that Obfervation, when the Speculum was illumi-
nated by any one of the prifmatick Colours, that Light
made many Rings of the fame Colour upon the Chart
with dark intervals, and therefore at its emergence out
of the Speculum was alternately tranfmitted, and not

tranf-

97]

tranfmitted from the Speculum to the Chart for many
fucceffions^ according to the various obliquities of its
emergence. And when the Colour caft on the Specu-
lum by the Prifm was varied, the Rings became o-f
the Colour caft on it, and varied their bignefs with their
Colour, and therefore the Light was now alternately
tranfmitted and not tranfmitted from the Speculum to
the Lens at other obliquities than before. It feemed to
me therefore that thefe Rings were of one and the fame
original with thofe of thin plates, but yet with this
difference that thofe of thin plates are made by the al-
ternate reflexions and tranfmiflions of the rays at the
fecond furface of the plate after one paffage through it :
But here the rays go twice through the plate before
they are alternately reflefted and tranfmitted ; firft,
they go through it from the firft furface to the Quick-
filver, and then return through it from the Quick-filver
to the firft furface, and there are either tranfmitted to
the Chart or refleded back to the Quick-filver, ac-
cordingly as they are in their fits of eafie reflexion or
tranfmilnon when they arrive at that furface. For the
intervals of the fits of the rays which fall perpendicu-
larly on the Speculum, and are reflected back in the
fame perpendicular Lines, by reafon of the equality of
thefe Angles and Lines,are of the fame length and num-
ber within the Glafs after reflexion as before by the
19th Propofition of the third Part of this Book. And
therefore fince all the rays that enter through tlie firft
furface are in their fits or eafy tranfmiffion at their en*
trance, and as many of thefe as are refleSed by the fe»
cond are in their fits of eafy reflexion there, all thefe
muft be again in their fits of, eafy tranfmiflion at their

O o return

«f

C 98 ]

return to the firft, and by confequence there go out of
the Glafs to the Chart, and form upon it the white
Spot of Light in the center of the Rings. For the rea-
fon holds good in all forts of rays , and therefore all
forts muft go out promifcuoully to that Spot, and by
their mixture cauie it to be white. But the intervals
of the fits of thofe rays which are reflefted more ob-
liquely than they enter, muft be greater after reflexion
than before by the 15 th and 10th Prop. And thence
it may happen that the rays at their return to the firft
furface, may in certain obliquities be in fits of eafy re-
flexion, and return back to the Quick-filver, and in
other intermediate obliquities be again in fits of eafy
tranfmiflion, and fo go out to the Chart, and paint on
it the Rings of Colours about the white Spot. And
becaufe the intervals of the fits at equal obliquities are
greater and fewer in the lefs refrangible rays, and lefs
and more numerous in the more refrangible, therefore
the leis refrangible at equal obliquities fhall make fewer
Rings than the more refrangible, and the Rings made
by thofe fhall be larger than the like number of Rings
made by thefe ; that is, the red Rings fhall be larger
than the yellow, the yellow than the green, the green
than the blue^ and the blue than the violet, as they
were really found to be in the 5th Obfervation. And
therefore the firft Ring of all Colours incompafiing the
white Spot of Light ihall be red without and violet
within, and yellow, and green, and blue in the middle,
as it was found in the fecond Obfervation; and thefe
Colours in the lecond Ring, and thofe that follow ihall
be more expanded till they fpread into one another,,
and blend one another by interfering.

Thefe

[99 3

Thefe feem to be the reafons of thefe Rings in ge-
neral, and this put me upon obferving the thicknefs of
the Glafs, and confidering whether the dimenfijons and
proportions of the Rings rnay be. truly derived from it
by computation.

O B S. VIIL

I meafured therefore the thicknefs of this concavo-
convex plate of Glafs, and found it every- where 4 of an
Inch precifely. Now, by the 6th Obfervation of the
firft Part of this Book, a thin plate of Air tranfmits the
brighteft Light of the firft Ring, that is the bright yel-
low, when its thicknefs is the ggoooth part of an Inch,
and by the i oth Obfervation of the fame part, a thin
plate of Glafs tranfmits the lame Light of the fame Ring
when its thicknefs is lefs in proportion of the fine of
refraction to the fine of incidence, that is, when its
thicknefs is the r^th or ,375^5th part of an Inch, fup-
pofing the fines areas 11 to 17. And if this thicknefs
be doubled it tranfmits the fame bright Light of the
fecond Ring, if tripled it trantmits that of the third,
and fo on, the bright yellow Light in all thefe cafes be-
ing in its fits of tranfmiflion. And the-refore if its thick-
nefs be multiplied ^4586 times fo as to become \ of an
Inch it tranfmits the fame bright Light of the 34386th
Ring. Suppofe this be the bright yellow Light tranf-
mitted perpendicularly from the reflefting convex fide
of the Glafs through the concave fide to the white Spot
in the center of the Rings of Colours on the Chart : And
by a rule in the feventh Obfervation in the firft Part of
the firft Book, and by the 15th and ^ oth Propofitions

O o 2 of

[ lOO ]

of the third Part of this Book, if the rays be madeob-
fique to the Glafs, the thicknefs of the Glafsrequi-
fite to tranfmit the fame bright Light of the fame Ring
in any obliquity. is to t^is thicj^ne^ of '- of an Inch, as
the fecant of ast^nsle ^fTSfeMeTs the firft of an hun-
dred and fix arithmetical means between the fines of
incidence and refraftion, counted from the fine of inci-
dence when the refraftion is made out of any plated Bo-
dy into any medium incompafling it, that is, in this cafe,
out of Glafs into Air. Now if the thicknefs of the Glafs
be increafed by degrees,fo as to bear to its firft thicknefs,
( viz. that of a quarter of an Inch ) the proportions
which 54-386 (the number of fits of the perpendicular
rays in going through the Glafs towards the white Spot
in the center of the Rings,) hath to 34-385, 34384,
54383 and 3438a (the numbers of thefits of the oblique
rays in going through the Glafs towards the firft, fe-
cond, third and fourth Rings of Colours,) and if the
firfl thicknefs be divided into 1 00000000 equal parts,
the increafed thicknefles will be 100002908, 100005816,
100008725 and 100011633^ and the Angles of which thefe
thickneffes are fecants will be a6' 13", 37' tj",^ 45' 6" and
5 a' a 6", the Radius being 1 00000000 ; and the fines of
thefe Angles are 762, 1079,, 1321 and 15^5, and the
proportional lines ofrefraftion 1172, 1659, 2031 and
2.34.5, the Radius being 1 00000. For fince the fines
of incidence out of Glafs into Air are to the fines,
of refraftion as 11 to 1 7, and to the above-mentioned
fecants as 11 to the firfl of 106 arithmetical means
between 11 and 17, that is as 11 to n,^, thofe fe-
cants will be to the fines of refraction as ii^-^ to 17,
and by this Analogy will give thefe fines. So then

[ lOI ]

if the obliquities of the rays to the concave furtacc of
the Glafs be luch that the fines of their refradion in
paffing out of the Glafs through that furface into the.
Air be 1172, ^^59^ ^031,^ ^i^S-) the bright Light of
the 34:586th Ring (hall emerge at the thickneffes of the
Glafs which are to \ of an Inch as 34.386 to 34385^
34.384, 34.383, 34.382, refpeftively. And therefore if
the thicknefs in all thefe cafes be \ of an Inch, (as it is in
the Glals of which the Speculum was made) the bright
Light of the 34.385th Ring (hall emerge where tlw. fine,
ofrefradtionis 1 171, and thatof the.34.384.th,384.383th
and 34381th Ring where the fine is i659,.^q3J, and
1345 refpedively. And in thefe Angles of refraction,
the Light of thefe Rings (hall be propagated from the.
Speculum to the Chart, and there paint Rings about the
white central round Spot of Light which we laid was
the Light of the 34386th Ring. And the Semidiame-
ters of thefe Rings (hall fubtend the Angles of refraftion
made at the concave iurface of the Speculum, and by
confequence their Diameters (hall be to the diftance of
the Chart from the Speculum as thofe fines of refradion
doubled are to the Radius that is as 11 7a, 1659, ^03 1^
and a 345,. doubled are to looooo. And therefore if.
the diftance of the. Chart from the concave furface of
the Speculum be fix Feet (as it was in the third of thefe.
Obfervations) the Diameters of the Rings of this bright
yellow Light upon the Chart fhall be i'688, 2^389^,.
2V5j 3'37 5 Inches : For thefe Diameters are to 6 Feet
as the above-mentioned fines doubled are to the Radius.
Now thefe Diameters of the bright yellow Rings, thus
found by computation are the very fame with thofe
found in the third of thefe Obfervations by meafuring^

them^

tl02]

them, (viz, with ifi' ^^ ^li')and 3'- Inches, and there-
fore the Theory of deriving thefe Rings from the thick-
nefs of the plate of Glafs of which the Speculum was
made, and from the obliquity of the emerging rays agrees
with the Obfervation. In this computation I have
equalled the Diameters of the bright Rings made by
Light of all Colours, to the Diameters of the Rings
made by the bright yellow. For this yellow makes the
brighteft part of the Rings of all Colours. If you defire
the Diameters of the Rings made by the Light of any
other unmixed Colour, you may find them readily by
putting them, to the Diameters of the bright yellow ones
in a fubduplicate proportion of the intervals of the fits
of the rays of thofe Colours when equally inclined to
the refrading or refleding furface which caufed thofe
fits, that is, by putting the Diameters of the Rings made
by the rays in the extremities and limits of the feven
Colours, red, orange, yellow, green, blue, indico, violet,
proportional the Cube-roots of the numbers, i, f , 6 ' 4 5
Mo ?6> i' which exprefs the lengths of a Monochard
founding the notes in an Eight : For by this means the
Diameter of the Rings of thefe Colours will be found
pretty nearly in the fame proportion to one another,
which they ought to have by the fifth of thefe Obfer-
vations.

And thus I fatisfied my felf that thefe Rings were of
the fame kind and original with thofe of thin plates,
and by confequence that the fits or alternate difpofi-
tions of the rays to be reflefted and tranfmitted are pro-
pagated to great diftances from every reflefting and re-
fracting furface. But yet to put the matter out of doubt
1 added the foUowincr Obfervation.

O B S.

C 103 ]

O B S- IX.

If thefe Rings thus depend on the thicknefs of the plate
of Glafs their Diameters at equal di fiances from feveral
Speculums made of fuch concavo-convex plates of Glafs
as are ground on tl\e fame Sphere, ought to be recipro-
cally in a fubduplicate proportion of the thicknefles of
the plates of Glafs. And if this proportion be found
true by experience it v^ill amount to a demonftration
that thefe Rings ( like thofe formed in thin plates ) do
depend on the thicknefs of the Glafs. I procured there-
fore another concavo-convex plate of Glafs ground on
both fides to the fame Sphere with the former plate :
Its thicknefs was |, parts of an Inch ; and the Diameters
of the three fir ft bright Rings meafured between the
brighteft parts of their orbits at the diftance of 6 Feet
from the Glafs were 3. 4^. 5g. Inches. Now the thick-
nefs of the other Glafs being \ of an Inch was to thick-
nefs of this Glafs as^to^j that is as ^i to 10, or
310000000 to loooooooo^ and the roots of thefe numbers
are 17607 and loooo, & in the proportion of the firft
of thefe roots to the fecond are the Diameters of the
bright Rings made in this Obfervation by the thinner
Glafs, 5. 4-1. 55 to the Diameters of the fame Rings made
in the third of thefe Obfervations by the thicker Glafs
i;^. a2 i;]j that is, the Diam.eters of the Rings are reci-
procally in a fubduplicate proportion of thickneflfes of
the plates of Glafs.

So then in plates of Glafs which are alike concave on
one fide, and alike convex on the other fide, and alike
quick-filvered on the convex fides, and differ in nothing

but

[io4l

but their thickueis, the Diameters of the Rings are re-
ciprocally in a fubduplic ate proportion of the thickneffes
of the plates. And this iliews lufficiently that the Rings
'depend on both the furfaces of the Glafs. They de-
pend on the convex furface becaufe they are more lu-
minous when that furface is quick-filvered over than
when it is without Quick-filver. They depend alfo
upon the concave furface, becaufe without that furface
a Speculum makes them not. They depend on both
furfaces and on the diftances between them , becaufe
their bignefs is varied by varying only that diftance.
And this dependance is of the fame kind with that
iwhich the Colours of thin plates have on the diftance
of the furfaces of thofe plates , becaufe the bignefs
of the Rings and their proportion to one another,
and the variation of their bignefs ariling from the varia-
tion of the thicknefs of the Glafs, and the orders of
their Colours, is fuch as ought to relult from the Propo-
litions in the end of the third Part of this Book, derived
from the the Phaenomena of the Colours of thin plates
fet down in the firft Part.

There are yet other Phaenomena of thefe Rings of
Colours but fuch as follow from the fame Propofitions,
and therefore confirm both the truth of thofe Propofi-
tions, and the Analogy between thefe Rings and the
Rings of Colours made by very thin plates. I fhall
fubjoyn fome of them.

O B S.

O B S. X.

When the beam of the Sun's Light was refleded back
from the Speculum not direCtly to the Hole in the Win-
dow, but to a place a little diftant from it, the common
center of that Spot, and of all the Rings of Colours fell
in the middle way between the beam of the incident
Light, and the beam of the refleded Light, and by
confequence in the center of the fpherical concavity of
the Speculujn, whenever the Chart on which the Rings
of Colours tell was placed at that center. And as the
beam of relieved Light by inclining the Speculum re-
ceded more and more from the beam of incident Light
and from the common center of the coloured Rings be-
tween them, thole Rings grew bigger and bigger, and
lb alfo did the white round Spot, and new Rings of Co-
lours emerged fucceffively out of their common center^
and the white Spot became a white Ring encompaffing
them ; and the incident and reiieded beams of Light
alw^ays fell upon the oppofite parts of this Ring, illumi-
nating its perimeter like two mock Suns in the oppofite
parts of an Iris. So then the Diameter of this Ring,
meafured from the middle of its Light on one fide to
the middle of its Liglii: on the other fide, was always
equal to the diftance between the middle of the incident
beam of Light, and the middle of the refleded beam
meafured at the Chart on which the Rings appeared :
And the rays which formed this Ring were refleSed by
the Speculum in Angles equal to their Angles of inci-
dence, and by coniequence to their Angles of refradion
at their entrance into the Glafs, but yet their Angles of

P p reflexion

[10^]

reflexion were not in the fame planes with their Angles
of incidence,

O B S. XL

The Colours of the new Rings were in a contrary-
order to thofe of the former, and arofe after this man-
ner. The white round Spot of Light in the middle of
the Rings continued white to the center till the diftance
of the incident and reflected beams at the chart was
about I parts of an Inch, and then it began to grow
dark in the middle. And when that difliance was about
if^of an Inch, the white Spot was become a Ring en-
compafling a dark round Spot which in the middle in-
clined to violet and indico. And the luminous Rings
incompafling it were grown equal to thofe dark ones
which in the four firfl: Obfervations encompafled them,
that is to fay, the white Spot was grown a white Ring
equal to the firfl: of thofe dark Rings, and the firfl: of
thofe luminous Rings was now grown equal to the fe-
cond of thofe dark ones, and the fecond of thofe lumi-
nous ones to the third of thofe dark ones, and fo on.
For the Diameters of the luminous Rings were now i J3,
^76 5 ^i? 3h,^^^ Inches.

When the diftance between the incident and refleded
beams of Light became a little bigger, there emerged
out of the middle of the dark Spot after the indico a
blue, and then out of that blue a pale green, and foon
after a yellow and red. And when the Colour at the
center was brighteft, being between yellow and red,
the bright Rings were grown equal to thofe Rings which
in the four firfl Obfervations next encompafled them;

tliat

[loy]

that Is to fay^ the white Spot in the middle of thofe
Rings was now become a white Ring equal to the firft
of thofe bright Rings, and the firft of thofe bright ones
was now becomie equal to the fecond of thofe, and fo
on. For the Diameters of the white Rings, and of the
other luminous Rings incompaffing it, were now lii ^
qI, lii, ^s^lf^c. or thereabouts.

When the diftance of the two beams of Light at the
Chart was a little more increafed, there emerged out
of the middle in order after the red, a purple, a blue,
a green, a yellow, and a red inclining much to purple,
and when the Colour was brighteft being between yel-
low and red, the former indico, blue, green, yellow and
red, were become an Iris or Ring of Colours equal
to the firft of thofe luminous Rings which appeared in
the four firft Obfervations, and the white Ring which
was now become the fecond of the luminous Rings was
grown equal to the fecond of thofe, and the firft of
thofe which was now become the third Ring was be-
come the third of thofe, and fo on. For their Diame-
ters were 1^6, ^8, afi, ^f Inches, the diftance of the
two beams of Light, and the Diameter of the white
Ring being 2^ Inches.

When thefe two beams became more diftant there
emerged out of the middle of the purplifli red, firft a
darker round Spot, and then out ot the middle of that
Spot a brighter. And now the former Colours (purple,
blue, green, yellow, and purplifh red ) were become a
Ring equal to the firft of the bright Rings mentioned in
the four firft Obfervations , and the Ring about this
Ring were grown ^qual to the Rings about that re-
fpeStively ; the diftance between the two beams of

P'p a Light

[io8] ;

Li^ht and the Diameter of the white Ring ( which
wa^ now become the third Ring) being about 3 In-
ches.

The Colours of the Rings in the middle began rrow
to grow very dilute^ and if the diftance between the
two beams was increafed half an Inch, or an Inch m.ore,
they vanifhed whilft the white Ring, with one or two
of the Rings next it on either fide, continued ftii! vi-
fible. But if the diftance of the two beams of Light
was ftill more increafed thefe alfo vanifhed : For the
Light which coming from feveral parts of the Hole in
the Window fell upon the Speculum in feveral Angles of
incidence made Rings of feveral bigneffes, which diluted
and blotted out one another, as I knew by intercepting
fome part of that Light. For if I intercepted that part
which was neareft to the Axis of the Speculum the
Rings would be lefs, if the other part which was re>
moteft from it they would be bigger.

O B S, XIL

When the Colours of the Prifm were eaft fucceffively
on the Speculum, that Ring which in the two laft Ob-
fervations was white, was of the fame bignefs in all the
Colours, but the Rings without it were greater in the
green than in the blue, and ftill greater in the yellow,
and greateft in the red. And, on the contrary, the
Rings within that white Circle were lefs in the green
than in the blue, and ftill lefs in the yellow, and leaft
in the red. For the Angles of reflexion ofthofe rays
which made this Ring being equal to their Angles of
incidence, the fits of every refieded ray within the Glafs

after

^ . [109]

after reflexion are equal in length and nuinber to tlie
i fits of the fame ray within the Glafs before its incidence
jt on the refleding furface; and therefore iince all the rays
of all forts at their entrance into the Glafs were in a fit
of tranfmiflion, they were alio in a fit of tranfmiffion at
their returning to the fame furface after reflexion ; and
by confequence were tranfmitted and went out to the
white Ring on the Chart. This is the reafon why that
Ring was of the fame bignefs in all the Colours,, and
why in a mixture of all it appears white. But in rays
which are reflefted in other Angles^ the intervals of the
fits of the leaft refrangible being greatefl:, make the
Rings of their Colour in their progrefs from this white
Ring, either outwards or inwards, increafe or decreafe
by the greateft fleps ; fo that the Rings of this Colour
without are greateft, and within leaft. And this is the
reafon why in the laft Obfervation, when the Specu-
lum was illuminated with white Light, the exterior
Rings made by all Colours appeared red without and
blue within, and the interior blue without and red
within.

Thefe are the Phaenomena of thick convexo-concave
plates of Glafs, which are every where of the fame
thicknefs. There are yet other Phenomena when thefe
plates are a little thicker on one fide than en the
other, and others when the plates are more or lefs con-
cave than convex, or plano-convex,, or double-conveXo
For in all thefe cafes the plates make Rings of Colours^
but after various manners ^ all which, fo far as I have
yet obferved, follow from the Propolitions in the end
of the third part of this Book, and fo confpire to con-
firm the truth of thofe Propofitions. But the Phaeno-
mena.

[no]

niena are too various, and the Calculations whereby
they follow from thofe Propofitions too intricate to be
here profecuted. I content my felf with having profe-
cuted this kind of Phsenomena fo far as to difcover their
caufe, and by difcovering it to ratify the Propofitions
in the third Part of this Book.

O B S. XIII.

As Liglit reflected by a Lens quick- filvered on the
back'fide makes the Rings of Colours above de-
fcribed, fo it ought to make the like Rings of Colours
in paffing through a drop of Water. At the firft re-
flexion of the rays within the drop, fome Colours ought
to be tranfmitted, as in the cafe of a Lens, and others
to be reflefted back to the Eye. For inftance, if the
Diameter of a fmall drop or globule of Water be about
the 5octh part of an Inch, fo that a red-making ray in
paffing through the middle of this globule has a 50 fits
of eafy tranfmiffion within the globule, and that all the
red -making rays whicli are at a certain diftance from
this middle ray round about it have ^4.9 fits within the
globule, and all the like rays at a certain further di-
ftance round about it have 148 fits, and all thofe at a
certain further diftance 24.7 fits, and fo on ; thefe con-
centrick Circles of rays after their tranfmiffion, falling
on a white Paper, will make concentrick rings of red
upon the Paper , fuppofing the Light which paffes
through one imgle globule ftrong enough to be fenfible.
And, in like manner, the rays of other Colours will
make Rings of other Colours. Suppofe now that in a
fair day the Sun lliines through a thin Cloud of fuch

globules

^\}l 1 1 ]

globules of Water or Hail^ and that the globules are all
of the fame bignefs,and the Sun feen through this Cloud
(hall appear incompafled with the hke concentrick Rings
of Colours, and the Diameter of the firft Ring of red
(hall be 7; degrees, that of the fecond i O; degrees, that
of the third 12 degrees 33 minutes. And accordingly
as the globules of Water are bigger or lefs, the Rings
fhall be lefs or bigger. This is the Theory, and expe-
rience anfwers it. For in June 1691. I faw by reflexion
in a Veflel of ftagnating Watier tliree Halos Crowns or
Rings of Colours about the Sun, like three little Rain-
bows, concentrick to his Body. The Colours of the
firft or innermoft Crown were blue next the Sun, red
without, and white in the middle between the blue
and red. Thofe of the fecond Crown were purple and
blue within, and pale red without, and green in the
middle. And thofe of the third were pale blue with-
in, and pale red without; theie Crowns inclofed one
another immediately, fo that their Colours proceeded
in this continual order from the Sun outward : blue,
white, red ; purple, blue, green, pale yellow and red ;
pale blue, pale red. The Diameter of the fecond Crown
meafured from the middle of the yellow and red on one
fide of the Sun, to the middle of the fame Colour on
the other fide was 9^ degrees, or thereabouts. The Dia-
meters of the firft and third I had not time to meafure,
but that of the firft feemed to be about five or fix de-
grees, and that of the third about twelve. The like
Crowns appear fometimes about the Moon 3 for in the
beginning of the year 1 664, Feh\ 1 9th at night, I faw
two fuch Crowns about her. The Diameter of the firft
or innermoft was about three degrees, and that of the

[II2]

lecond about five degrees and an half. Next about the
Moon was a Circle of white, and next about that the
inner Crown which was of a bluiih green within next the
white, and of a yellow and red without, and next about
thefe Colours were blue and green on the infide of the
outward Crown, and red on the outfide of it. At the
fame time there appeared a Halo about 12 degre-:^^ 35'
diftant from the center of the Moon. It was Elliptical,
and its long Diameter was perpendicular to the Horizon
verging below fartheft from the Moon. I am told thatS
the Moon has fometimes three or more concentrick
Crowns of Colours incompaffing one another next about
her Body. The more equal the globules of Water or
Ice are to one another, the more Crowns of Colours
will appear, and the Colours will be the more lively.
The Halo ar the diftance of cli'- degrees from the Moon
is of another fort. By its being oval and remoter from
the Moon below than above, I conclude, that it v/as
made by refraftion in fome fort of Hail or Snow floaling
in the Air in an horizontal Pofture, the refrafting Angle
being about 58 or 60 degrees.

T H

["3]

THE

THIRD BOOK

O F

O P T I C K

Oifervations concerning the Inflexions of the rays of Light j
and the Colours made thereby.

GRimaldo has informed us, that if a beam of the
Sun's Light be let into a dark Room through a
very fmall Hole, the fliadows of things in this Light
will be larger than they ought to be if the rays went
on by the Bodies in ftreight Lines, and that thefe fha-
dows have three parallel fringes, bands or ranks of co-
loured Light adjacent to them. But if the Hole be
enlarged the fringes grow broad and run into one ano-
ther, fo that they cannot be dirtinguiflied. Thefe broad
fhadows and fringes have been reckoned by fome to pro-
ceed from the ordinary refraftion of the Air, but with-
out due examination of the matter. For the circum«
ftances of the Phaenomenon, fo far as I have obferved
them, are as follows.

Qq OBS.

["4]

O B S. L

I made in a piece of Lead a fmall Hole with a Pin,
whofe breadth wis ihe^ath part of an Incli. For 21
of thofePins laid together took up the breadth of half
an Inch. Through this Hole I let into my darkened
Chamber a beam of the Sun's Light, and found that the
ihadows of Hairs^Thred^Pins^Straws, and fuch like llen-
der fubftances placed in this beam of Light, were confider-
ably broader than they ought to be, if the rays of Light
pafled on by thefe Bodies in right Lines. And particu-
larly a Hair of a Man's Head, whofe breadth was but
the 280th part of an Inch, being held in this Light, at
the diftance of about twelve Feet from the Hole, did
caft a ftiadow which at the diftance of four Inches from
the Hair was the fixtieth part of an Inch broad, that is,
above four times broader than the Hair, and at the di-
ftance of tw^o Feet from the Hair was about the eight
and twentieth part of an Inch broad, that is, ten times
broader than the Hair, and at the diftance often Feet
was the eighth part of an Inch broad, that is 35 times
broader.

Nor is it material W'hether the Hair be incompaffed
with Air, or with any other pellucid fubftance. For I
wetted a poliftied plate of Glafs, and laid the Hair in
the Water upon the Glafs, and then laying another po-
liftied plate of. Glafs upon it, fo that the Water might
fill up the fpace between the Giafles, I held tljiem in
the aforeiaid beam of Light, fo that the Lighl:4night
pafs through them perpendicularly, and the ftiadow
of the Hair was at the iame diftances as big as before.

The

[115]

The fhadows of fcratches made in polifhed plates of
Glafs were alio much broader than they ought to be,
and the Veins in polifhed plates of Glafs did alio caft the
like broad (liadovvs. And therefore the great breadth
of thefe ihadows proceeds from fome other caufe than
the refradion of the Air.

Let the Circle X reprefent the middle of the Hair ; Fig, i
ADG, BEH, CFI, three rays paffing by one fide of
the Hair at feveral diftances ; KNQ, LOR^ MPS,
three other rays pafling by the other fide of the Hair at
the like diftances; D, E, F and N, O, P, the places
where the rays are bent in their pafTage by the Hair ;
G, H, I and Q, R, S, the places where the rays fail on
a Paper G Q j I S the breadth of the fhadow of the Hair
caft on the Paper, and T I, V S, two rays paffing to the
points I and S without bending when the Hair is taken
away. And it's manifeft that all the Light between
thefe two rays A I and V S is bent in paffing by the
Hair, and turned afide from the fhadow IS, becauie if
any part of this Light were not bent it would fall on
the Paper within the fhadow, and there illuminate the
Paper contrary to experience. And becaufe when the
Paper is at a great diiliance from the Hair, the fhadow
is broad, and therefore the rays TI and VS are at a
great diitance from one another, it follows that the
Hair ads upon the rays of Light at a good diftance in
their paffing by it. But the a£i:ion is ftrongeil on the
rays which paf^ by at leaft difl:ances, and grows weaker
and weaker accordingly as the rays pafs ' by at diftances
greater and greater, as is reprefented in the Scheme :
For thence it comes to pais, that the fhadow of the
Hair is much broader in proportion to the diflance of

(Iq 2 the

the Paper from the Hair, when the Paper is nearer the
Hair than when it is at a great diftance from it,

O B S. IL

The flhadows of all Bodies ( Metals, Stones, Glafs^
Wood, Horn, Ice, tor. ) in this Light were bordered
with three parallel fringes or bands of coloured Light,
whereof that which was contiguous to the Ihadow was
broadeft and moft luminous, and that which was re-
moteft from it was narroweft, and fo faint, as not eafily
to be vifible. It was difficult to diftinguifh the Colours
unlefs when the Light fell very obliquely upon a fmooth
Paper, or fome other fmooth white Body, io as to. make
them appear much broader than they would otherwife
do. And then the Colours were plainly vifible in this
order : The firft or innermoft fringe was violet and deep
blue next the fhadavv, and then light blue, green and
yellow in the middle, and red without. The fecond
fringe was almoft contiguous to the firft, and the third
to the fecond, and both were blue within and yellow
and red without, but their Colours were very faint
efpecially thofe of the third. The Colours therefore
proceeded in this order from the fhadow, violet, indico,
pale blue, green, yellow, red ; blue, yellow, red ; pale
blue, pale yellow and red. The Ihadows made by
fcratches and bubbles in polifhed plates of Glafs were
bordered with the like fringes of coloured Light. And
if plates of Looking-glafs floop'd off near the edges with
a Diamond cut, be held in the fame beam of Light, the
Light which pafles through the parallel planes of the
Glafs will be be bordered with the like fringes of Cor

lours

["7]

lours where thofe Planes meet with the Diamond cut,
and by this means there will fometimes aj3pear four or
five fringes of Colours. Let AB, CD reprefent theKi>, 2
parallel planes of a Looking-glafs^ and BD the plane
of the Diamond-cut, making at B a very obtufe An^le
with the plane A B. And let all the Light between die
rays EN I and FBM piafs diredly through the parallel
planes of the Glafs, and fall upon the Paper between I
andM, and all the Light between the rays GO and
HD be refrafted by the oblique plane of the Diamond
cut B D,and fall upon the Paper between K and L ; and
the Light which pafles diredly through the parallel
planes of the Glafs, and falls upon the Paper between.
I and M, will be bordered with three or more fringes,
at M.

O B S. III.

When the Hair was twelve Feet diftant from the
Hole, and its fliadow fell pbliq^uely upon a flat white
fcale of Inches and parts of an Inch placed half a Foot -
beyond it, and alfo when the fliadow fell perpendicu-
larly upon the fame fcale placed nine Feet beyond it;
I meafured the breadth of the fliadow and fringes as
accurately as I could, and found them in parts of art-
Inch as follows.

The

[ii8]

The breadth of the Shadow

The breadth between the middles of the
brighteft Light of the innermoft fringes
;-.U^ on either fide the (hadow

The breadth between the middles of the
brishteft Light of the middlemoft frin-
g;es on either fide the fhadow

The breadth between the middles of the
brighteft Light of the outmoft fringes
y&Q) on either fide the fhadow

The diftance between the middles of the
brighteft Light of the firft and fecond
fringes

The diftance between the middles of the
brighteft Light of the fecond and third
fringes

The breadth of the luminous part (green,
white, yellow and red ) of the firft

fringe

The breadth of the darker fpace between
the firft and fecond fringes.

The breaddi of the luminous part of the
fecond fringe

The breadth of the darker fpace between
the fecond and third fringes.

half O'

nine

Foot,

Feet,

'i'i

;»or;,

7_
SO

•

23i

^ «^ T8i

lio

. 2X

■ T

J 70

~ J

173

[up]

Thefe meafures I took by letting the fhadow of the
Hair at half a Foot diftance fall lb obliquely on the
fcale as to appear twelve times broader than when it
fell perpendicularly on it at the fame diftance, and fht-
ring down in this Table the twelfth part of the mea-
fures I then took.

O B S. IV,

When the ftiadovv and fringes were caft obliquely
upon a fmooth white Body, and that Body was remo-
ved further and further from the Hair, the firft fringe
began to appear and look brighter than the reft of the
Light at the diftance of lefs than a quarter of an Inch
from the Hair, and the dark line or fhadow between
that and the fecond fringe began to appear at a lefs di-
ftance from the Hair than that of the third part of an
Inch. The fecond fringe began to appear at a diftance
from the Hair of lefs than half an Inch, and the fliadow
between that and the third fringe at a diftance lefs than,
an Inch, and the third fringe at a diftance lefs than three
Inches. At greater diftances they became much more
fenfible, but kept very nearly the fame proportion o£
their breadths and intervals which they had at their firft
appearing. For the diftance between the middle of the
firft and middle of the fecond fringe, was to the diftance
between the middle of the fecond and middle of the
third fringe, as three to two, or ten to feven. And
the laft of thefe two diftances was equal to the breadth
of the bright Light or luminous part of the firft fringe«
And this breadth was to the breadth of the bright Light
of the fecond fringe as feven to four, and to the dark

mtervai:

interval of the firft and fecond fringe as three to two,
and to the hke dark interval between the lecond and
third as two to one. For the breadths of the fringes
feemed to be in the progreffion of the numbers i, /

rX:

V \ and their intervals to be in the fame progreffion
with them ; that is, the fringes and their intervals to-
gether to be in the continual progreffion of the numbers
I , /^ -S /\-, /^ J- , l^]^ or thereabouts. And thefe pro-
portions held the fame very nearly at all diftances from
the Hair 3 the dark Intervals of the fringes being as
broad in proportion to the fringes at their firft appea-
rance as afterwards at great diftances from the Hair,
though not fo dark and diftinft.

O B S. V.

The Sun ffiining into my darkened Chamber through
a Hole a quarter of an Inch broad ; I placed at the di^
ftance of tw^o or three Feet from the Hole a Sheet of
Paft-board, which was black'd all over on both fides,
and in the middle of it had a Hole about three quarters
of an Inch fquare for the Light to pafs through. And
behind the Hole I faftened tothePaft-board with Pitch
the blade of a fharp Knife, to intercept fome part of
the Light which paffed through the Hole. The planes
of the Paft'board and blade of the Knife were parallel
to one another, and perpendicular to the rays. And
vvlien they were fo placed that none of the Sun's Light
fell on the Paft-board, but all of it paffed through the
Hole to the Knife, and there part of it fell upon the
blade of the Knife, and part of it- paffed by its edge :
I let this part of the Light which paffed by, fall on a

white

[ 121 ]

white Paper two or three Feet beyond the Knife, and
there faw two ftreams of faint Light (hoot out both
ways from the beam of Light into the fhadow Hke the
tails of Comets. But becaufe the Sun's direct Light by
its brightnefs upon the Paper obfcured thefe faint
ftreams, fo that I could fcarce fee them, I made a little
Hole in the midft of the Paper for that Light to pafs
through and fall on a black cloth behind it ; and then
I Hiw the two ftreams plainly. They were like one
another, and pretty nearly equal In length and breadth,
and quantity of Light. Their Light at that end next
the Sun's dire£t Light was pretty ftrong for the fpace of
about a quarter of an Inch, or half an Inch, and in all
its progrefs from that dire£t Light decreafed gradually
till it became infenfible. The whole length of either of
thefe ftreams meafured upon the Paper at the diftance
of three Feet from the Knife was about fix or eight
Inches ; fo that it fubtended an Angle at the edge of
the Knife of about lo or 12, or at moft 14 degrees.
Yet fometimes I thought I faw it ftioot three or four
degrees further, but with a Light fo very faint that I
could fcarce perceive it, and fufpeded it might ( in
fome meafure at leaft) arife from fome other caufe than
the two ftreams did. For placing my Eye in that Light
beyond the end of that ftream which was behind the
Knife, and looking towards the Knife, I could fee a
line of Light upon its edge, and that not only when
my Eye was in the line of the ftreams, but alfo when
it was without that line either towards the point of the
Knife, or towards the handle. This line of Light ap-
peared contiguous to the edge of the Knife, and was
narrower than the Light of the innermoft fringe, and

R r narroweft

[122]

mrro,weft when my Eye was furtheft from the dlrefl;
Light, and therefore feemed to pafs between the Light
of that fringe and the edge of the Knife , and that
which palled neareft the edge to be moft bent, though
not all of it.

O B S. VI.

I placed another Knife by this fo that their edges
might he parallel and look towards one another, and
that the beam of Light might fall upon both the Knives^
and fome part of it pafs betw^een their edges. And
when the diftance of their edges was about the 400th
part of an Inch tlie ftream parted in the middle, and
left a fhadow between the two parts. This ihadow
was fo black and dark that all the Light which paffed
between the Knives feemed to be bent, and turned afide
to the one hand or to the other. And as the Knives ftill
approached one another the fhadow grew broader, and
the ftreams fhorter at their inward ends which were
next the fhadow, until upon the contad of the Knives
the whole Light vanifhed leaving its place to the
fhadow.

And hence I gather that the Light which is leaft
bent, and goes to the inward ends of the ftreams, paf-
fesby the edges of the Knives at the greateft diftance,
and this diftance when the ftiadow begins to appear be-
tween the ftreams is about the eight- hundredth part of
an Inch. And the Light which pafles by the edges of
the Knives at diftances ftill lefs and lels is more and
more bent, and goes to thofe parts of the ftreams w^hich
are further and further from the, direct Light, becaufc

when

[ t23 ]

when the Knives approach one another till they toiich^
thofe parts of the ftreams vaniftl laft which ^re furtheft
from the dired Light.

O B S. VIL

In the fifth Obfervatioh the fringes did riot Appear,
but by reafon of the breadth of the Hole in the Win-
dow became fo broad as to run into one another, and
by joyning make one continued Light in the beginning
of the ftreams. But in the fixth^ as the Knives ap-
proached one another, a little before the fhadow ap-
peared between the two ftreams, the fringes began to'
appear on the inner ends of the ftreams on either fide
of the direct Light, three on one fide made by the edge
of one Knife, and three on the other fide made by the
edge of the other Knife. They were diftindeft when
the Knives were placed at the greateft diftance from the
Hole in the Window, and ftill became more diftind by
making the Hole lefs, infomuch that I could fometiriies
fee a faint lineament of a fourth fringe beydnd the three
above-mentioned. And as the Knives continually ap-
proached one another, the fringes grew diftinder and
larger until they vaniftied. The outmoft fringe va-
niftied firft, and the middlemoft next, and the inner-
moft laft. And after they were all vanifhed, and tJie
line of Light which was in the middle between them
was grown very broad, enlarging it felf on both fides
into the ftreams of Light defcribed in the fifth Obfer-
vation, the above-mentioned fliadow began to appear
in the middle of this line, and divide it along the middle
mto two lines of Light, and increafed until the whole

R r 2 Lisht

[124]

Light vanifhed. This inlargement of the fringes was
fo great that the rays which go to the innermoft fringe
feemed to be bent above twenty times more when this
fringe was ready to vanifh^ than when one of the Knives
was taken away.

And from this and the former Obfervation compared,
I gather, that the Light of the firft fringe paffed by the
edge of the Knife at a diftance greater than the eight-
hundredth part of an Inch, and the Light of the fecond
fringe paffed by the edge of the Knife at a greater di-
ftance than the Light of the firft fringe did, and that
of the third at a greater diftance than that of the fe-
cond, and that of the ftreams of Light defcribed in
the fifth and fixth Obfervations paffed by the edges
of the Knives at lefs diftances than that of any of the

fringes.

O B S. VIII.

I caufed the edges of two Knives to be ground truly
ftreight, and pricking their points into a board fo that
their edges might look towards one another, and meet-
ing near their points contain a reftihnear Angle, I faft-
ned their handles together with Pitch to make this.
Angle invariable. The diftance of the edges of the
Knives from one another at the diftance of four Inches
from the angular point, where the edges of the Knives
met, was the eighth part of an Inch, and therefore the
Angle contained by the edges was about i degr, 54,'.
The Knives thus fixed together I placed in a beam of
the Sun's Light, let into my darkened Chamber through
a Hole the ^ith part of an Inch wide, at the diftance.

[125]
of ten or fifteen F^et from the Hole, and let the Light
which paffed between their edges fell very obliquely
upon a fmooth white Ruler at the diftance of half an
Inch, or an Inch from the Knives, and there faw the
fringes made by the two edges of the Knives run along
the edges of the fhadows of the Knives in lines parallel
to thofe edges without growing fenfibly broader, till
they met in Angles equal to the Angle contained by the
edges of the Knives, and where they met and joyned
they ended without croffing one another. But if the
Ruler was held at a much greater diftance from the
Paper, the fringes became fomething broader and broader
as they approached one another, and after they met
they crofled one another, and then became much broader
than before.

Whence I gather that the diftances at which the
fringes pafs by the Knives are not increafed nor altered
by the approach of the Knives^ but the Angles in which
the rays are there bent are much increafed by that ap-
proach ; and that the Knife which is neareft any ray
determines which way the ray fliall be bent, and the
other Knife increafes the bent.

O B S. IX.

When the rays fell very obliquely upon the Ruler at
the diftance of the third part of an Inch from the Knives.^,
the dark line between the firft and fecond fringe of the
fliadow of one Knife, and the dark line between the:
firft and fecond fringe of the ftiadow of the other Knife
met with one another, at the diftance of the fifth part
of an Inch from the end of the Light which paffed be-

tween the Knives at the concourfe of their edges. And
therefore the diftance of the edges of the Knives at the
meeting of thefe dark lines was the i6oth part of an
Inch. For as four Inches to the eighth part of an Inch,
fo is any length of the edges of the Knives rneafured
from the point of their concourfe to the diftance of the
edges of the Knives at the end of that length, and fo is
the fifth part of an Inch to the 1 6oth part. So then the
dark lines above-mentioned meet in the middle of the
Light which pafles between the Knives where they are
diftant the 1 6oth part of an Inch, and the one half of
that Light pafles by the edge of one Knife at a diftance
not greater than the 3aoth part of an Inch, and falling
upon the Paper makes the fringes of the lliadow of that
Knife, and the other half paffes by the edge of the
other Knife, at a diftance not greater than the 320th
part of an Inch, and falling upon the Paper makes the
fringes of the (hadow of the other Knife. But if the
Paper be held at a diftance from the Knives greater than
the third part of an Inch, the dark lines above-men-
tioned meet at a greater diftance than the fifth part of
an Inch from the end of the Light which pafled be-
tween the Knives at the concourfe of their edges; and
therefore the Light which falls upon the Paper where
thofe dark lines meet pafles between the Knives
where their edges are diftant above the 1 6cth part of
an Inch.

For at another time when the two Knives were di-^
ftant eight Feet and five Inches from the little Hole in
the Window, made with a fmall Pin as above, the Light
which fell upon the Paper where the aforefaid dark
lines met. pafled between the Knives, where the di-
ftance

[127]

ftance between their edges was as in the following
Table, when the diftance of the Paper from the Knivel
was alfo as follows.

Dijiances of the Paper
from the Kjiives in
Inches,

Difimces between the edges
of the Kj^ives in milU"
fimal parts of an Inch.

li-

0'0I2.

3i-

0'020.

8;-

o'o34.

3^-

o'o57. .

96.

o'oSi.

151.

o'oS?. 1

And hence I gather that the Light which makes the
fringes upon the Paper is not the fame Light at all di-
ftances of the Paper from the Knives, but when the Pa»
per is held near the Knives, the fringes are made by
Light which paffes by the edges of the Knives at a lels
diftance, and is more bent than when the Paper is held
at a greater diftance from the Knives,

O B S. X.

When the fringes of the fiiadows of the Knives fell
perpendicularly upon a Paper at a great diftance from
the Knives, they w^ere in the form of Hyperbolas, and
their dimenfions were as follows. Let C A, CB repre^
fent lines drawn upon the Paper parallel to the edges of
the Knives, and between which all the Light would
fall, if it paffed between the edges of the Knives with-
out iniiexion; DE a right line drawn through C making

the

[128]

th e Angles A C D, B C E, equal to one another, and
terminating all the Light whith falls upon the Paper from
the point where the edges of the Knives meet j eis, fk t,
and glv, three hyperbolical lines reprefenting the ter-
minus of the fhadow of one of the Knives, the dark line
between the firft and fecond fringes of that fhadow, and
the dark line between the fecond and third fringes of
the fame fhadow ; x i p, y k q and z 1 r, three other Hy-
perbolical lines reprefenting the terminus of the fhadow
of the other Knife, the dark line between the firft and
fecond fringes of that fhadow, and the dark line be-
tween the fecond and third fringes of the fame fhadow.
And conceive that thefe three Hyperbolas are like and
equal to the former three, and crofs them in the points
i, k and 1, and that the fhadows of the Knives are termi-
nated and diftinguifhed from the firft luminous fringes
by the lines e is and xip, until the meeting andcrof-
iing of the fringes, and then thofe lines crofs the fringes
in the form of dark lines, terminating the firft luminous
fringes within fide, and diftinguiftiing them from ano-
ther Light which begins to appear at i, and illuminates
all the triangular fpace ipDEs comprehended by thefe
dark lines, and the right line DE. Of thefe Hy-
perbolas one Afymptote is the line DE, and their other
Afymptotes are parallel to the lines CA and CB. Let
rv reprefent a line drawn any where upon the Paper
parallel to the Afymptote D E, and let this line crofs
the right lines A C in m and B C in n, and the fix dark
hyperbolical lines in p, q, r ; s, t, v ; and by meafuring
the diftances ps, qt, rv, and thence cofledting the
the lengths of the ordinatesnp, nq, nr or ms, mt,
mv, and doing this at feveral diftances of the line rv^

irom

from the Afymptote DE you may find as many points
of thefe Hyperbolas as you pleafe, and thereby know
that thefe curve lines are Hyperbolas diifering little from
the conical Hyperbola. And by meafuring the lines
C i , C k , CI, you may find other points of thefe
Curves.

For inftance, when the Knives were diftant from the
Hole in the Window ten Feet, and the Paper from the
Knives 9 Feet, and the Angle contained by the edges of
the Knives to which the Angle ACB is equal, was fub-
tended by a chord which was to the Radius as i to 3 1,
and the diftance of the line r v from the Afymptote DE
was half an Inch: I meafured the lines ps, qt, rv,
and found them o'^ 5, 0*65, o'^S Inches refpedively,
and by adding to their halfs the line i mn (which here
was the 1:28th part of an Inch, or o'ooyS Inches ) the
funis np, nq, nr, were o'i8i8, o'^^iS, o\(^j^ In-
ches. 1 meafured alfo the diftances of the brighteft
parts of the fringes which run between pqand st, qr
and tv, and next beyond r and v, and found them o'*'^^
o'S, and I'ly Inches.

O B S. XL

The Sun Ihining into my daAened Room through a
fmall round Hole made in a plate of Lead with a llender
Pin as above ; I placed at the Hole a Prifm to refrad
the Light, and form on the oppofite Wall theSpeftrum
of Colours, defcribed in the third Experiment of the
firft Book. And then I found that the (badows of all
Bodies held in the coloured Light between the Prifm
and the Wall, were bordered with fringes of the Colour

S s of

[130]

of that Light in which they were held. In the full red
Light they were totally red without any fenfible blue
or violet^ and in the deep blue Light they were totally
blue without any fenfible red or yellow ; and fo in the
green Light they were totally green, excepting a little
yellow and blue, which were mixed in the green Light
of the Prifin. And comparing the fringes made in the
leveral coloured Lights, 1 found that thole made in the
red Light were largeft, thofe made in the violet were
leaft, and thofe made in the green were of a middle
bignefs. For the fringes with which the Ihadow of a
Man's Hair were bordered, being meafured crofs the
fhadow at the diftance of fix Inches from the Hair ; the
diftance between the middle and moll: luminous part of
the firft or innermoft fringe on one fide of the fhadow^,
and that of the like fringe on the other fide of the Iha-
dow, was in the full red Light /^^ of an Inch, and in
the full violet ^. And the like diftance between the
middle and moil: luminous parts of the fecond fringes on
either fide the (hadow was in the full red Light {^ , and
in the violet '- of an Inch. And thefe diftances of the
fringes held the fame proportion at all diftances from
the Hair without any fenfible variation.

So then the rays which made thefe fringes in the red
Light paffed by the Hair at a greater diftance than thofe
did which made the like fringes irl the violet ; and there-
fore the Hair in caufing thefe fringes adted alike upon,
the red Light or leaft refrangible rays at a greater di-
ftance, and upon the violet or moft refrangible rays at
a lefs diftance, and by thofe adions difpofed the red
Light into larger fringes, and the violet into fmaller,
-md the Lights of interm.ediate Colours into fringes of

inter-

[I3IJ

intermediate bigneffes without changing the Colour of
of any fort of Light*

When therefore the Hair in tiie firft and fecond of
thele Obfervations was held in the white beam of the
Sun's Light, and cafi: a fhadow which was bordered with
three fringes of coloured Light, thofe Colours arofe not
from any new modifications impreft upon the rays of
Light by the Hair, but only from the various intiedlions
whereby the feveral forts of rays were feparated from
one another, wliich before reparation by the mixture
of all their Colours, compofed the white beam of the
Sun's Light, but whenever fepajated compofe Lights
of the feveral Colours which they are originally dii'po-
fed to exhibit. In this 15th Obfervation, where the
Colours are feparated before the Light paffes by the
Hair, the leatl: refrangible rays, which when fepara-
ted from the reft make red, were infleded at a greater
diftance from the Hair, fo as to make three red fringes
at a greater diftance from the middle of the fhadow of
the Hair 3 and the moft refrangible rays which when
feparated make violet, w^re inflefted at a lefs diftance
from the Hair, fo as to make three violet fringes at a
lefs diftance from the middle of the ftiadow of the Hair.
And other rays of intermediate degrees of refrangibi-
lity were infledted at intermediate diftances from the
Hair, fo as to make friezes of intermediate Colours at
intermediate diftances from the middle of the fhadow
of the Hair. And in the fecond Obfervation, where
all the Colours are mixed in the v/hite Light which
paifes by the Hair, thefe Colours are feparated by the
various iniiexions of the rays, and the fringes which
they make appear all together , and the innermoft

S s a fringes

fringes being contiguous make one broad fringe compo-
fed of all the Colours in due order, the violet lying
on the infide of the fringe next the fhadow., the red on
the outfide furtheft from the fliadow, and the blue,
green and yellow, in the middle. And, in like man-
ner, the middlemoft fringes of all the Colours lying iii
order, and being contiguous, make another broad fringe
compofed of all the Colours ; and the outmoft fringes
of all the Colours lying in order, and being contiguous,,
make a third broad fringe compofed of all the Colours.
Thefe are the three fringes of coloured Light with
which the fhadows of all Bodies are bordered in the fe-
cond Obfervation.

When I m^ade the foregoing Obfervations, I defigned
to repeat moft of them with more care and exadtnefs^
and to make fome new ones for determining the man-
ner how the rays of Light are bent in their paflage by
Bodies for making the fringes of Colours with the-
dark lines between them. But I was then interrup-
ted, and cannot now think of taking thele things into
further confideration. And iince 1 have not .hnifhed:
this part of my Defign, I (hall conclude, with propo-
fing only fome Queries in order to a further fearch ta
be made by others.

^ery 1 . Do not Bodies a(3 upon Light at a diftance^v
and by their aftion bend its rays, and is not this a£tioa
(ceteris fartim) ftrongeft at the leaft diftance?

<%. a. Do not the rays which differ in refrangibility
differ alfo in flexibility, and are they not by their dif-
ferent inflexions feparated from one another , fo as
after feparation to make the Colours in the three fringes

above

[133]

above defcribed ? And after what manner are they in-
fleifled to make thofe fringes ?

^. 5. Are not the rays of Light in paffing by the
edges and fides of Bodies^ bent feveral times backwards
and forwards, with a motion like that of an Eel ? And
do not the three frin2;es of coloured Light above-men-
tioned, arife from three fuch bendings ?

^, 4. Do not the rays of Light which fill upon Bo-
dies, and are reflected or refraded, begin to bend be-
fore they arrive at the Bodies ; and are they not re-
flected, refracted and infleded by one and the fame
Principle, acting varioufly in various circumftances ?

^, 5. Do not Bodies and Light aft mutually upon
one another, that is to fay, Bodies upon Light in emit-
ting, reflecting, refraCting and inflefting it, and Light
upon Bodies for heating them, and putting their parts
into a vibrating motion wherein heatconfifts ?

^. 6. Do not black Bodies conceive heat more eafily
from Light than thofe of other Colours do, by reafon
that the Light falling on them is not refleded outwards^
but enters the Bodies, and is often reflected and re-
frafted within them, until it be ftifled and loft ?

*^. 7. Is not the ftrength and vigor of the action
between Light and fulphureous Bodies obferved above,,
one reafon why fulphureous Bodies take fire more
readily, and burn more vehemently, then other Bo-,
dies do ?

^. 8.. Do not all fixt Bodies when heated beyond a
certain degree, emit Light and fhine, and is not this
emifiion performed by the vibrating motions of their

parts? v5..^5^eO*,

[134]

^i. 9. Is not fire a Body heated fo hot as to emit
Light copioudy ? For what elfe is a red hot Iron than
fire ? And what elfe is a burning Coal than red hot
Wood ?

c%. 10. Is not flame a vapour, fume or exhalation
heated red hot, that is, fo hot as toihine? For Bodies
do not flame without emitting a copious fumie, and this
fume burns in the flame. The Ig^m Fatum is a vapour
fliining without heat, and is there not the fame diffe-
rence between this vapour and flame, as between rot-
ten Wood fliining v/ithout heat and burning Coals of
fire ? In diftilling hoj^ Spirits, if the head of the fl:iri be
taken off, the vapo^ir which afcends out of the Still will
take fire at the flame of a Candle, and turn into flame,
and the name wifl riin along the vapour from the Candle
to tlie Still. Some Bodies heated by motion or fermen-
tation, if the heat grow intenfe fume copioufly, and if
the heat be great enough the fumes will fliine and be-
com.e flame. Metals in fufion do not flame for want of
a copious fume, except Spelter which fumes copioufly,
and thereby flames. AU flaming Bodies, as Oyl, Tal-
low, Wax, Wood, foffil Coals, Pitch, Sulphur, by
flaming wafl:e and vanifli into burning fmoke, which
fmoke, af the flame be put out, is very thick and vifible,
and fomxtimes fmefls ftrongly, but in the flame lofes
its fmell by burning, and according to the nature of the
fmoke the flame is of feveral Colours, as that of Sul-
phur blue, that of Copper opened with Sublimate
green, that of Tallow yellow. Smoke paffing through
flame cannot but grow red hot, and red hot imoke can
have no other appearance than that of flame.

^, 1 1 .

^. 1 1. Do not great Bodies conferve their heat the
longeft, their parts heating one another, and may not
great denfe and fix'd Bodies, when heated beyond a
certain degree, emit Light fo copioufly, as by the emif-
fion and readion of its l-.ight, and the reflexions and re-
fraftions of its rays within its pores to grow ftill hot-
ter, till it comes to a certain period of heat, fuch as is
that of the Sun ? And are not the Sun and fix'd Stars
great Earths vehemently hot, whofe heat is conferved
by the greatnefs of the Bodies, and the mutual action
and readion between them, and the Light which they
emit, and whofe parts are kept from fuming away, not
only by their fixity, but alfo by the vaft weight and
denfity of the Atmofpheres incumbent upon them, and
very ftrongly comprefling them, and condenfing the va->
pours and exhalations which arifefrom them? ^U-. a))ei^^,

^. 12, Do not the rays of Light in falling upon the
bottom of the Eye excite vibrations in the Tunico. re^
tina ? Which vibrations, being propagated along the
folid fibres of the optick Nerves into the Brain, caufe
the fenfe of feeing. For becaufe denfe Bodies conferve
their heat a long time, and the denfeft Bodies conferve
their heat the longeft, the vibrations of their parts are
of a lafting nature, and therefore may be propagated^
along folid fibres of uniform denfe matter to a great di-
ftance, for conveying into the Brain the impreffions
made upon all the Organs of fenfe. For that motion; *
which can continue long in one and the fame part of a
Body, can be propagated a long way from one part to-
another, fuppofing the Body homogeneal, fo that the
motion may not be reiledted, refraded, interrupted or
difordered by any unevemiefs of the Bodyo

1 16 ]

c%. 13. Do not feveral fort of rays make vibrations
of feveral bigneflcs, which according to their bignefles
excite feniations of feveral Colours^ much after the
manner that the vibrations of the Air, according to their
feveral bigneffes excite fenfations of feveral founds ?
And particularly do not the moft refrangible rays ex-
cite the fhorteft vibrations for making a fenfation of
deep violet, the leaft refrangible the largeft for making
a fenfation of deep red, and the feveral intermediate
forts of rays, vibrations of feveral intermediate bignef-
fes to make fenfations of the feveral intermediate Co-
lours ?

^i. 1 4. May not the harmony and difcord of Co-
lours arife from the proportions of the vibrations propa-
gated through the fibres of the optick Nerves into the
Brain, as the harmony and difcord of founds arifes from
the proportions of the vibrations of the Air ? For fome
Colours are agreeable, as thofeofGold andlndico, and
others difagree.

^. 1 5. Are not the Species of Objefts feen with both
Eyes united where the optick Nerves meet before
tJiey come into the Brain, the fibres on the right fide
of both Nerves uniting there, and after union going
thence into the Brain in the Nerve which is on the
right fide of the Head, and the fibres on the left fide
of both Nerves uniting in the fame place, and after
union going into the Brain in the Nerve which is on
the left fide of the Head, and thefe two Nerves meet-
ing in the Brain in fuch a manner that their fibres
make but one entire Species or Pifture, half of which
on the right fide of the Senforium comes from the
right fide of both Eyes through the right fide of

both

[137]

both optick Nerves to the place where the Nerves
meet, and from thence on the right fide of the Head
into the Brain, and the other half on the left fide of the
Senforium comes in like manner from the left fide of
both Eyes. For the optick Nerves of fuch Animals as
look the fame way with both Eyes ( as of Men, Dogs,
Sheep, Oxen, }sfc. ) meet before they come into the
Brain, but the optick Nerves of fuch Animals as do
not look the fame way with both Eyes (as of Fillies and
of the Chameleon) do not meet, if I am rightly in-
formed.

^, 1 6. When a Man in the dark preffes either cor-
ner of his Eye with his Finger, and turns his Eye away
from his Finger, he will fee a Circle of Colours like
thofe in the Feather of a Peacock's Tail ? Do not thefe
Colours arife from fuch motions excited in the bottom
of the Eye by the prejQTure of the Finger, as at other
times are excited there by Light for caufing Vifion ? And
when a Man by a ftroke upon his Eye fees a Flafli of
Light, are not the like Motions excited in the Retina
by the ftroke ?

Book m. Plate I.

^^^S-

j^^'"!l „!^'(^;',i:;^ 1 1 [ i[ I m L m „..„.■ .i n m 1/

x ii i S iiin SS '"""""

C 1 38 3

-. ■ w» ^*. .» » ^ . ^j.

ENUMERATIO

LINEA

TERTII ORDINI&

— ^i^Wgl H I J. -I -

[ISP

ENUMERATIO

LIN

TERTII ORDINIS.

LIneae Geometries fecundum numerum dimen- r:
fionum aequationis qua relatio inter Ordinatas,.^"^^'^^^ ^^'°
& Abfeiflas definitur, vel (quod perinde eft) fecun-
dum numerum punftorum in quibus a linea reda
fecari poffunt, optime diftinguuntur in OrdineSs
Qua ratione linea primi Ordinis erit Refta fola, ex
fecundi five quadratici ordinis erunt feftiones Conicae
& Circulus, & ex tertii five cubici Ordinis Parabola
Cubica, Parabola Neiliana, Ciflbis veterum & reli-
qux quas hie enumerare iuicepimus. Curva autem
primi generis, (fiquidem refta inter Curvas non eft
numeranda) eadem eft cum. Linea fecundi Ordinis^
& Curva fecundi generis eadem cum Linea Ordinis
tertii. Et Linea Ordinis infinitefimi ea eft quam
refta in punftis infinitis fecare poteft, qualis eft Spi-
ralis, Cyclois, Quadratrix & linea omnis quae per
radii vel rotae revolutiones infinitas generatur.

T t 1 Sedionum

[ho]

"• Sedionum Conicarum proprietates praecipuae a

^^XX%?-' Geometris paffim traduntur. Et confimiles funt pro-
rum comfetunt prictatcs Curvaruiii fecundi generis & reliquarum, ut
cttrvtsfufenorum-^^ fequcnti proprietatum prascipuarum enumera-

tione conttabit.
III. Nam fi reflae plures parallelae & ad conicam fe-

Curvtwurn
€undi

dinatdi.

r^^)^««? A- ftionem utrinq; terminatae ducantur, reda duas ea-
AlmameS rUiii bifccans bifecabit alias omnes,ideoq; dicitur Ih'^-
tri^J^ertkesfen'r^QXer^^\x^^^ rcftse bifcdiae dicuntur Ordtnatim ap-
tra.Axej. pUcat^ ^d Diametrum, & concurfus omnium Dia-
metrorum eft Centrum figurae, & interleftio Curv^ &
diametri Vertex nominatur, & diameter iWiAxi^
eft cui ordinatim applicator infiftunt ad angulos re-
d:os. Et ad eundem modum in Curvis fecundi ge-
neris, fi re£i:se duae qusevis parallelae ducantur occur-
rentes Curvae in tribus punftis : reda quae ita fecat
has parallelas ut fumma duarum partium ex uno fe-
cantis latere ad curvam terminatarum, aequetur parti
tertiae ex altero latere ad curvam terminatce, eodem
modo fecabit omnes alias his parallelas curvaeq; in
tribus punftis occurrentes redas, hoc eft, ita ut llim-
ma partium duarum ex uno ipiius latere Temper
aequetur parti tertis ex altero latere. Has itaq; tres
partes quae hinc inde ^quantur, Ordinatim affli-
catas & rectam fecantem cui ordinatim applicantur.
T)iametrum & interfeftionem diametri & cuYVxVer"
ticem & concurfum duarum diametrorum Centrum
nominare licet. Diameter autem ad Ordinatas re-
dangula fi modo aliqua fit, etiam Axis dici poteft^,
& ubi omnes diametri in eodem pundo concurrunt
iftud erit Centrum generate. .

Hyper-«

[ HI ]

Hyperbola primi generis duas u4[fmj^totoSy ea fe- iv.
eundi tres^ea tertii quatuor & non plures habere. 2^-'earl!m'^lpietT
teft, & fie in reliquis. Et quemadmodum partes f^-^v-"
lineae cujufvis refta^ inter Hyperbolam Conicam &
duas ejus Afymptotos lunt hinc inde acquales : fie iii
Hyperbolis fecundi geaeris fi ducaturi|:efta quaevife
fecans tarn Curvam quam tres ejus Afymptotos in
tribus punftis, fumma duarum partium iftius redse
qua? a duobus quibufvis Alymptotis in eandem pla-
gam ad duo punfta Curvge extenduntur aequalis erit
parti tertiae quae a tertia Afymptoto iii plagam con-
trariam ad tertium Curva? pundum extenditur.

Et quemadmodum in Conicis feftionibus non Pa- v.
rabolicis quadra turn Ordinatim applicatae^ hoc eil, Laterareaa^^^^-
redtangulum Ordinatarum quae ad contrarias par-^"^ "^^
tes Diametri ducuntur, eft ad redangulum partium
Diametri qu^ ad Vertices Ellipfeos vel Hyperbolas
terminantur,ut data quaedam linea quae dicitur Latm
• re(Hum^ ad partem diametri qus inter Vertices jacet
& dicitur ia^m tranfverfum : fie in Curvis non Para-
bolicis fecundi generis Parallelepipedum fub tribus
Ordinatim applicatis eft ad Parallelepipedum fub par-
tibus Diametri ad Ordinatas & tres Vertices figurae ab-
fciffis, in ratione quadam data : in qua ratione fi fu-
mantur tres reftaead tres partes diametri inter ver--
tices figurae fitas fingulaead fingulas, tunc ilte tres- ^
redas dici poflunt Later a re<Ha Rguvx^ & illae partes <
Diametri inter Vertices Later a tranfverfa. Et ficut:
in Parabola Conica quae ad unam & eandem diame-
trum unicum tantum habet Verticem, reftangulum^
fub Ordinatis aequatur redangulo fub parte Diametri
quae ad Ordinatas & Verticem abfcinditur & refta^

quadam ;

14^

quadam data quae Latus reftum dicitur,fic in Curvis
lecundi generis quas non nil! duos habent Vertices ad
eandemDiametrum, ParallelepipedumfubOrdinatis
tribus aequatur Parallelepipedo iub duabus partibus
Diametri ad Ordinatas & Vertices illos duos abfciffis,
& refta quadam data quae proinde Latm return
dici poteft.

VI. Deniq; ficut in Conicis fe£tionibus ubi duae paral-
ruljulZr'Jie'^ Ick ad Curvam utrinq; terminata? fecantur a dua-
Urumfegmemis. bus parallelis ad Curvam utrinq; terminatis, prima

a tertia & lecunda a quarta, rectangulum partium
primiB eft ad redangulum partium tertiae ut reftan-
gulum partium fecundae ad reftangulum partium
quartan : fie ubi quatuor tales redas occurrunt Curvae
lecundi generis fingulae in tribus punftis, parallele-
pipedum partium primae redae erit ad parallelepide-
dum partium tertian, ut parallelepipedum partium
fecundcE ad parallelepipedum partium quartae.

VII. Curvarum fecundi & fuperiorum generum aeque
^./Se^p?r£^^tq;pi"™i crura omnia in infinitum progredientia
ika& eorum fa- yqI Hy^erMici fuut gcueris vcl Taf^dMict. Crus Hy^
^^' ferhoikum voco quod ad Afymptoton aliquam in in-
finitum appropinquate Para/'<?/2r^^w quod Afymptoto
deftituitur. Haec crura ex tangentibus optime dig-
nofcuntur. Nam fipunftum contaftus in infinitum
abeat tangens cruris Hyperbolici cum Afymptoto
coincidet & tangens cruris Parabolici in infinitum
recedet, evaneicet & nuUibi reperietur. Invenitur
igitur Aiymptotos cruris cujufvis qua^rendo tangen-
tem cruris illius ad pundum infinite diftans. Plaga
autem cruris infiniti invenitur quaerendo pofitionem
redas cujutVis quae tangenti parallela elt ubi pun-

d»um

[ H3 3

ftum contaftus in infinitum abit. Nalll h^c rect^
in eandem plagam cum crure infinito dirigitur.

Linear omnes Ordinis primi, tertii, quinti, fep- VIII.
timi & imparis cujufq; duo habent ad minimum ^^f^'f;'^^^;;;
ctura in infinitum veiius plagas oppofitas pYogYQ-s^-^ensfecwridiad
dientia. Et line^ omnes tertii Ordinis duo habent ^^^f^^^^^f^
ejufmodi crura m plagas oppofitas progredientia m primus.
quas nulla alia earum crura infinita (prasterquam
in Parabola Cartefiana ) tendunt. Si crura ilia
fint Hyperbolici generis , fit G A S eorum Afymp-
totos & huic parallela agatur reda qusevis CBc
ad Curvam utrinque ( fi fieri potefl: ) terminata
eademq; biiecetur in pundtoX, & locus pun£ti il-^'if- r-
lius X erit Hyperbola Conica ( puta X $ ) cujus
una Afymptotos eft AS. Sit ejus altera Afymp-
totos A B, &: a^quatio qua relatio inter Ordinatam
BC & Abfciffam AB definitur, fi AB dicatur x &
B C y^ femper induct hanc formam xyy-^-ey — ax^
-j^bxx-j-cx-j-d. Ubi termini e, a, b^ c, d, defig-
nant quantitates datas cum fignis fuis -J- &— < aife-
ftas, quarum quaellbet deeflfe poffunt modo ex earum
defeftu figura in fedtionem conicam non vertatur..
Poteft autem Hyberbola ilia Conica cum afymp to-
lls fuis coincidere^ id eft pundum X in reda AB
locari : & tunc terminus -|-ey deeft.

At fi reda ilia CBc non poteft utrinq; ad Cur\ram
terminari fed Curvas in unico tantum pun£to occur- ^ r ^?~' ,

r ' 1 n \ -n r Calm ecundm,

rit : age quamvis politione datamrectam A d aiymp-
toto AS occurrentem in A, ut & aliam quamvis BC
afymptoto illi parallelam CurviEque occurrentem in
pundoC, & cequatio qua relatio inter Ordinatam

X.
Cafus tertms.

XI.

-Cajm quartm.

144]

BC & Abfciffam AB definitur^ femper induct hanc
for mam x y == a x^ -|- b x x -y c x -J- d.

Quod fi crura ilia oppofita Parabolici lint generis,
refta CB cad Curvam utrinque, fi fieri poteft, ter^
minata in plagam crurum ducatur & bifecetur in B,
& locus punSti B erit linea reda. Sit ifta AB, ter-
minata ad datum quodvis pundum A, & oequatio
qua relatio inter Ordinatam BC ScAbfciflam AB
definitur, femper induct hanc formam, yy = ax^
^-bxx-^cx-j-d.

At vero fi redla ilia CB c in unico tantum pundo
occurrat Curvas, ideoq; ad Curvam utrinq; terminari
non poflit : fit punftum illud C, & incidat reCta ilk
ad punttum B in redam quamvis aliam pofitione
datam & ad datum quodvis pundum A terminatam
AB : & aequatio qua relatio inter Ordinatam BC &
Abfciffam AC definitur femper induct hanc formam,
y = ax^-^bxx-|-cx-i-d.

Enumerando curvas horum cafuum, Hyperbolam
Nominaforma- yocabimus infcrtftam quae tota jacet in Afymptot6n
a ngulo ad inftar Hyperbolae conicae, circumjcriftam
quae Afymptotos fecat & partes abfciffas in finu fuo
ampleftitur, ambigenam quae uno crure infinito in-
fcribitur & altero circumfcribitur , convergentem
cujus crura concavitate fua feinvicem refpiciunt &
in phg^m^^nd^mdinguntUT ^divergent em cujus crura
convexitate fua feinvicem recipiunt & in plagas con-
trarias diriguntur, crunbus contrariu fr^ditam cujus
crura in partes contrarias convexa funt & in plagas
contr^iims infiimtd.^ Conchoidalem quae vertice concavo
& cruribus divergentibus ad afymptoton applicatur,
anguineam quae flexibus contrariis afymptoton fecat

xn.

rum

[H5] .

& utrinq; in crura contraria producitur, cfuciformem
qua: conjugatam decuflat, nodatam qux feipfam de-
cuffat in orbem redeundo, cujpdatam ciijus partes
duae in angulo contadus concurrunt & ibi terminan-
tur, funSatam quse conjugatam habet Ovalem infi-
nite parvam id eft pun£tuni, & furam quae per im-
poffibilitatem duarum radicum Ovali, Nodo, Cuf-
pide & Pundo conjugate privatur. Eodem fenfu
rarabolam quoq; convey'gentem^ divergent em^ cruri-
hm contrarm frccditam^ cruciformeni^ nodatam^ cuj^
fid at am J funHatam Sl furam nominabimus.

In calu primo fi terminus a x? affirmativus eft Fi- ™-
gura erit Hyperbola triplex cum fex cruribus Hy- redundante'^& '^
perbolicis quae iuxta tres Alymptotos quarum nulte^-/^ ^^''^^ ^^
iunt parallelae in mfinitum progrediuntur,binae juxta^ ^ '
unamquamq; in plagas contrarias. Et hx Afymp-
toti li terminus bxx non deeft fe mutuo fecabunt
in tribus pundis triangulum (Dd^^j inter fe con-
tinentes, fin terminus bxx deeft convergent omnes
ad idem pundlum. In priori cafu capeAD=:
-~, & Ad=^Ac/ = j^, ac junge Dd, Dc/^, & erunt
AD, Ddj Dj^tres Afymptoti. In pofteriori due
ordinatam quamvis B C, & in ea utrinq; produfta
cape hine inde BF & Bffibi mutuo aequales &
in ea ratione ad A B quam habet /d. ad a, j'angeq;
AF, At% & erunt AB, AF, Af tres Alympoti.
Hanc autem Hyperbolam vocamus redundantem
quia nurnero crurum Hyperbolicorum Sediones Co-
nicas fuperat.

In Hyperbola omni redundante fi neq; terminus B^hlm Hy-
e y defit neq; fit b b - 4 a c aequale + a e ^a curva mA-^prboU diametris
lam habebit diametrum, fin eorum alterutrum ac- ^f^!" crumm

U u cidat

Gidat curva habebit unicam diametrum, & tres fi
utrumque. Diameter autem femper tranfit per In^
terfeftionem duaruin Afymptoton & bilecat redas
omnes quas ad Afymptotos illas utrinq; terminantnr
& parallelse funt & Aiymptoto tertiae. Eftq; abfcifla
AB diameter Figurae quoties terminus ey deeft.
Diametrum vcro abiblute diftam hie & in fequen-
tibus in vulgari fignificatu ufurpo, nempe pro ab-
fcifla quae paflim habet ordinatas binas a^quales ad
idem punftum hinc inde infiftentes.
XV. Si Hyperbola redundans nullam habet diametrum

^.^CJi'LTJ qu^rantur ^quationis hujus ax^-bx'+cxx-l-dx
^m diametro de- -|- ;J = o radices quatuor feu valores ipfius x. Eae
trrS/t;:::f"nto a P, a ^ , a , , a p. Engantur ordinate
tos trianguium PT, -nrr, ttT, p t, & hoe tangent Curvam in punftis
capemes,. totidcm T, r, "^ , t, & taugendo dabunt limites Cur-

vae per quos fpecies ejus innotefcet.
Hf. 1,2, Nam fi radices omnes AP, A*^, A^, Ap funt

reales^ ejufdem figni & in^quales, Curva conftat ex
tribus Hyperbolis , ( infcripta circumfcripta Sc am-
bigena ) cum Ovali. Hyperbolarum una jacet ver-
fus D, altera verfus d, tertia verfus ^, & Ovalis
femper jacet intra triangulamDd^, atq; etiam in-
ter medios limites ^ & r , in quibus utiq; tangitur
ab ordinatis -rf^ & '^'^^ Et haec eft fpecies prima.
^^' J, 4- Si e radicibus duae maximae Att, A f^ vel duae mi-

nimal AP, A^ aequantur inter fe, & ejufdem funt
figni cum alteris duobus, Ovalis & Hyperbola cir-
cumfcripta fibi inxicem junguntur coeuntibus earum
punftis conta£tus7 & t vel T & t, & crura Hyper-
bolae fefe decuflando in Ovalem continuantur, iigu-
ram nodatam efficientia* Quae fpecies eft fecunda.

CH7]

Si e radicibus tres maximae A/, At, A to-, vel tres fi^: 5, <?.
minimse A tt, a te-, A P iiequentur inter fe, Nodus in
€'ufpdem acutiffimum convertetur. Nam crura duo
Hyperbolae circumfcriptoe ibi in angulo contaftus
concurrent & non ultra producentur. Et haec eft
fpecies tertia.

Si e radicibus dua? mediae At«r ^ Att aequentur in- iFiv. 7.
ter fe, punda contaftus t &7 coincidunt, & propte-
rea Ovalis interjefita in punftum evanuit, & conftat
figura ex tribus Hyperbolis, infcripta, circumfcripta
& ambigena cum fun^o conjugato. Quae eft fpecies
quarta.

Si duae ex radicibus funt impoffibiles & reliquae%'7->S,s3,i4^
duae inaequales & ejufdem figni ( nam figna contraria
habere nequeunt^) fur^ habebuntur Hyperbolae tres
fine Ovali vel Nodo vel cufpide vel punfto conju-
gato, &hae Hyperbolae vel ad latera trianguli ab
Afymptotis comprehenfi vel ad angulos ejus jacebunt
& perinde fpeciem vel quintam vel fextam confti«
tuent.

Si e radicibus duae funt sequales & alterae du3e%. 9rio^iS?J<
vel impoffibiles funt vel reales cum fignis quae a fig-
nis aequalium radicum diverfa funt, figura cruciform
mis habebitur, nempe dua: ex Hyperbolis feinvicem
decuffabunt idq; vel ad verticem trianguli ab A-
fymptotis comprehenfi, vel ad ejus bafem. Quae
duae fpecies funt feptima & odava.

Si deniq; radices omnes funt impoffibiles vel fi^ir- 11512;
omnes funt reales & in^quales & ecirum duae funt
affirmativae & alterae duae negativae, tunc duse habe-
buntur Hyperbolae ad angulos .oppofitos duarum

Mu 1 -Afymp^

[148]

Afymptoton cum Hyperbola anguinea circa Afymp-
toton tertiam. Quas fpecies eft nona.

Et hi funt omnes radicum cafus poffibiles. Nam

fi duae radices funt aequales inter le, & aliae dua^ funt

etiam inter fe oequales, Figura evadet Se6tio Conica

cum linea redla.

xVL Si Hyperbola redundans habet unicam tantum

,^yp'''^'ff"Dmmetmm fit ejus Diameter Abfciffa AB, & aequa-

tescnrminkatan-tiomshujus ax^ -|- bx x-j- cX'l-d^ o qusere tres ra-

turn Diametro. ^^qq<^ fg^ valores X.

Fig. 17. Si radices illae funt omnes reales & ejufdem figni^

Figura conftabit ex Ovali intra triangulum D d o'^ ja-
cente & tribus Hyperbolis ad angulos ejus, nempe
circumfcripta ad angulum D Sc infcriptis duabus ad
angulos d & cT. Et haec eft fpecies decima.

Fi^' 1^' Si radices duae majores funt aequales & tertia ejuf-

dem figni, crura Hyperbolae jacentis verfus D fefe
deculTabunt in forma Modi propter conta£tum Ova-
lis. Quae fpecies eft undecima.

F^i- ip- Si tres radices funt aequales. Hyperbola ifta fit

cujpdata fine Ovali. Quae fpecies eft duodecima.

fpg, 20. Si radices duae minores funt aequales & tertia ejuf-

dem figni, Ovalis in funSium evanuit. Quae fpecies
eft decima tertia. In fpeciebus quatuor noviffimis
Hyperbola quae jacet verfus D Afymptotos in finu
fuo ampleditur, reliqua^ duae in finu Afymptoton
jacent.

Fig. 20, Si duae ex radicibus funt impoflibiles habebuntur tres

pfg. Ill Hyperbolae fur^ fine Ovali decuflatione vel cufpide.

i%. 23. Et hujus cafus fpecies funt quatuor, nempe decima

quarta fi Hyperbola circumi'cripta jacet verfus D &

decima

declma quinta fi Hyperbola infcripta jacetTerfus D,
decima lexta li Hyperbola circumfcripta jacet fub
bafid^ trianguli Dd^, & decima feptima fi Hyper-
bola infcripta jacet fub eadem bafi.

Si du32 radices funt oequales & tertia figni dWevfi^'^'^^-
figura mt cruciformts, Nempe diia? ex tribus Hy- "°^^"
perbolis leinvicem decuffabunt idq; vel ad verticem
trianguli ab Afymptotis comprehenfi vel ad ejus ba-
fem. Qu« duse fpecies funt decima octava Sc decima
nona.

Si duse radices funt inasquales & ejufdem figni &
tertia eft figni diverfi, duac habebuntur Hyperbolae
in oppofitis angulis duarum afymptoton cum Con^
choidali intermedia. Conchoidalis autem vel jace- ^^i"- ^>
bit ad eafdem partes afymptoti fuae cum triangulo '^' ^ *
ab afymptotis conftituto, vel ad partes contrarias j
& hi duo cafus conftituunt fpeciem vigefimam & vi-
gefimam primam.

Hyperbola redundans quas habet tres diametros rr^'^lh ,
conltat ex tribus Hyperbolis m linubus alymptoton redundames cum
jacentibus, idq; vel ad angulos trianguli ab afympto- trihmDUmetris.
tis comprehenfi vel ad ejus latera. Cafus prior dat pfg. ip]
fpeciem vigefimam fecundam^Sc pofterior ipeciem vi-
gefimam tertiam.

Si tres afymptoti in punfto communi fe mutuo xvni.'
decuffant, vertuntur fpecies quinta & fexta in ^^g^- ^.XuXZs
fimam quartam , feptima & odava in vigefimam cum Afym^mk
quintam. & nona in vigefimam fextam ubi Ansuinea ^'''^^^ ^^^°^'^^-

A f p o»«r-^^ punctum con*'

non tranht per concurium alymptoton^ oc m vigeh- vergemibm.
mam feptimam ubi tranfit per concurfum ilium, quo ^J^- 3o-
cafu termmi b ac d defunt, & concurfus afympto- b|. 32*:
ton eft centrum figuros ab omnibus ejus partibus^^-33'

fitis

oppofitis ^qualiter diftans. Et hse quatuor ipecies
Diametrum non habent.
F^' 34' Vertuntur etiam fpecies decima quarta ac decima

§f. \l\ fata in vigefimam oCtavam^ decima quinta ac de-

^^- 37- cima feptima in vigefimam nonam, decima odava

& decima nona in tricefimam^ 8c vigefima cum vige-
fima prima in tricefimam primam. Et hae fpecies
unicam iiabent diametrum.
F^. 3%. Ac deniq; fpecies vigefima fecunda & vigefima

tertia vertuntur in fpeciem tricefimam fecundam cu-
jus tres funt Diametri per concurfum alymptoton
tranfeuntes. Quae omnesj converfiones facillime in-
telliguntur faciendo ut triangulum ab afymptotis'
comprehenfum diminuatur donee in punftum eva-
nefcat.
^ix. Si in primo sequationum cafu terminus a x^ ne-

defr^^'^JZ g^^ivus eft, Figura erit Hyberbola defeftiva unicam
tfrum non hahen- habcus afymptotou & duo tautum crura Hyperbo-
^^"^° lica juxta afymptoton illam in plagas contrarias in-

finite progredientia. Et afymptotos ilia eft Ordi-
nata prima & principalis A G. Si terminus e y non
deeft figura nullam habebit Diametrum, fi deeft ha-
bebit unicam. In priori cafu fpecies fie enume-
rantur.
^'^l' W' Si sequationis hujus a x'* = b x^ 4- c x x -|r d x -\- ^ e e,

radices omnes Att, AP, A/, A-nr^ funt reales & in-
^quales, Figura erit Hyperbola anguinea afympto-
ton flexu contrario amplexa, cum Ovali conjugata.
Quae fpecies eft tricefima tertia.
v%.4c. Si radices duae mediae AP & Af cequentur inter

fe, Ovalis & Anguinea junguntur fefe decuflfantes
m iovvm Modu Quae eft fpecies tricefima quarta.

SI tres radices funt ^quales, Nodus vertetur m^'i^'4i»
cuff idem acutlffiinum In vertice anguines. Et haec
eft fpecles triceslma quinta.

Si e tribus radicibus ejufdem fignl duae maximae ^'^' 43-
A / Sc A -^ fibi mutuo aequantur, Ovalis in f unburn
evanuit. Qua? fpecies eft tricefima fexta.

Si radices duss qua^vis imaginariae funt^ fola ma-
nebit Anguinea fura fine Ovali, decuffatione, Guf-
pide vel puniSto conjugate. Si Anguinea ilia non%.42»
tranfit per pundum A fpecies eft tricefima feptlma,
fin tranfit per pundum illud A ( id quod contingit ^-^i"- 43*
ubi termini b ac d defunt,) pundium illud A erit
centrum figuras reftas omnes per ipfum du£tas &
ad Curvam utrinq; terminatas bilecans, Et haec
eft fpecies tricefima oftava.

In altero cafu ubi terminus ey deeft Sc propterea xx;
figiira Diametrum habet, fi aequatlonls hujus ^'^' tem^mivlli-

= bxX-|-CX-|-d radices omnes AT^ Atj At, {uut ametrum hdln^

realeSj inaequales & ejufdem fignl, figura erit Hyper- ^5"
bola Conchoidalis cum Ovali ad convexitatem. Quae ' "*" \
eft fpecies tricefima nona.

Si duae radices funt inaequales & ejufdem fignl &%-44'
tertia eft figni contrarii, Ovalis jacebit ad concavi-
tatem Conchoidalis. Eftq; fpecies quadragefima.

Si radices duae minores AT, At, funt aequales =%, 4^,
& tertia At eft ejufdem figni, Ovalis & Conchoi-
dalis jungentur fefe decuffando in modum Modi.
Quae fpecies eft quadragefima prima.

Si tres radices funt aequales, Nodus mutabitur inHf^47>
Cuffidem & figura erit CiJJ^ois P^eterum. Et haec eft
fpecies quadragefima fecunda.

C 1^2 ]

3%-. 49. Si radices duae majores funt aequales, Sc tcrtia eft

cjufdem figni^Conchoidalis habebit fun6lum conju-
gatuin ad convexitatem fuam^ eftq; fpecies quadra-
gefima tertla.
Fi^. 4p: Si radices duae funt aequales & tertia eft figni con-

trarii Conchoidalis habebit funSum conjugatum
ad concavitatem luam, eftq; fpecies quadragefima
quarta.
.:i7f..48,49. Si radices duae funt impoflibiles habebitur Con-

choidalis furct fine Ovali , Nodo , Cufpide vel
pun£to conjugate. Quae Ipecies eft quadragefima
quinta.
XXI. Siquando in primo aequationum cafu terminus a x^

tem^plMki' ^^^^ & terminus bxx non deeft, Figura erit Hy-
Diamemm non perbola ParaboHca duo habens crura Hyperbolica ad
hahenm. uuam Afymptoton SAG & duo Parabolica in pla-

gam unam & eandem convergentia. Si terminus
ey non deeft figura nuUam habebit diametrum, fin
deeft habebit unicam. In priori cafu fpecies funt
hse.
.■Tig.^oZ Si tres radices AP, kt^^ At aequationis hujus

bx^-j-cx ^-dx-|-^ ee==o funt inaequales & ejufdem
figni, figura conftabit cxOvali ScaUis duabus Curvis
quae partim HyperboHcae funt & partim ParaboHca.
Nempe crura ParaboHca continuo du£tu junguntur
eruribus HyperboHcis fibi proximis. Et haec eft
fpecies quadregefima lexta.
.^^.5=1. Si radices duae minores funt aequales Sc tertia eft

ejufdem figni, Ovahs & una Curvarum illarum
Hyperbolo-ParaboUcarum. junguntur &: fe deculTant
in formam Nodi, Qjax Ipecies eft quadragefima
feptima.

Si tres radices funt irquales, Nodus ille in Cuf-F/^. 51.
pidem veititur. Eftq;fpecies quadragefima oftava.

Si radices duae inajores iunt aequales & tertia eft ^^^i- S3-
ejufdem iigni-, Ovalis in. funclum conjugatum eva-
nuit. QviX fpecies eft quadragefima nona.

Si duce radices ilant impoiiibiles^ manebmit y?<r^%- S3,54«
ilte dus curva: Hyperbolo-parabolicge fine Ovali,
dectiflation e^cufpide vel punfto conjugato, & fpe-
ciem quill quagefimam conftituent.

Si radices duae funtsequales & tertia eft figni con- ^''^" ^^^
trarii, Curvoe ilte hyperboio-parabolicae junguntur
fefe decuffando in morem crucis. Eftq; fpecies quin-
quagefima prima.

Si radices duas funt inaequales & ejufdem figni & ^'^' '^^-
tertia eft figni contrarii, figura evadet Hyperbola
anguinea circa Afymptoton AG, cum Parabola con-
jugata. Et haec eft fpecies quinquagefima fecunda.

In altero cafu ubi terminus ey deeft & figura ^™-
Diametrum habet, fi duse radices aequationis hujus tmrTarlboikt
b X x-J- c x-j- d = o funt impoffibileSj duae habentur ^i^metrum ha-
figurse hyperbolo-parabolicae a Diametro A B hinc fZ%j.
inde aequaliter diftantes. Quae fpecies eft quinqua-
gefima tertia.

Si aequationis iliius radices duae funt impoffibiies, %' S§*
Figurae hyperbolo-parabolicae junguntur fefe de-
cuffantes in morem crucis, & fpeciem quinquagefi-
mam quartam conftituunt.

Si radices illae funt inaequales & ejufdem figni, ha- ^'^- !^^
betur Hyperbola Conchoidalis cum Parabola ex
eodem latere Afymptoti» Eftq; fpecies quinquage^-
fimaquinta«

[154]

ng.eol Si radices ilk funt figni contrarii, habetur Con-

choidalis cum Parabola ad alteras partes Afymptoti.

Quae fpecies eft quinquagefima fexta.

xxiii. Siquando in primo sequationum cafu terminus

Oumior Hy- ^terq ; ax' &bxx deeft, figura erit Hyperbolifmus

hu. ^^ feftionis ahcujus Conicae. Hyperbolifmum figure

voco cujus Ordinataproditapplicandocontentumfub

Ordinata figurse illius & reda data ad Abfciflam com<«^

munem. Hac ratione linea refta vertitur in hyper-

bolam Conicam, & feSio omnis Conica v^titur in

^ aliquam figurarum quas hie Hyperbolifmos leaio-

num Conicarum voco. Nam aequatio ad figuras

de quibus animus, nempe xyy -|-ey-cx-|-d^ feu

_ etA^ee'i"4dx -|- 4 cxx generatur appli-

cando contentum fub Ordinata fe£tionis Conicae
e4A^ee-l-4dx'|-4cxx & re£ta data m ad curvarum

Abfciflam communem x. Unde liquet quod figura
genita Hyperbolifmus erit Hyperbola?, Ellipfeos vel
Parabolae perinde ut terminus ex affirmativus eft
¥el negativus vel nullus.

Hyperbolifmus Hyperbolae tres habet afymptotos
quarum una eft Ordinata prima & principalis A d,
alterae duae funt parallelse Abfciffae A B «& ab eadem
hinc inde sequaliter diftant. In Ordinata principal!
Ad cape Ad, A^ hinc inde ^quales quantitati /^c
& per pun£ta d ac ^ age d g, ^7 Afymptotos Ab-
fciflae A B parallelas.

Ubi terminus ey non deeft figura nullam ha-
bet diametrum. In hoc cafu fi sequationis hujus
c x x -|- d X 'Y ;e e ==^o radices duce A P, Af funt r eales

c 155 ]

oc ina'quales ( nam aequales effe nequeunt miiBgm3.P^i*<su
lit Conica feftio ) figura conftabit ex tribus Hyper-
bolis fibi oppoiitis quarum una jacet inter afymp-
totos parallelas & alterae duae jacent extra. Et haec
eft fpecies quinquagefima feptima.

Si radices illoe duae lunt impoffibiles^habentur Hy-
perbolae duae oppofitae extra alymptotos paralleks &
Anguinea hyperbolica intra eafdern. Haec figura
duarum eft fpecierum. Nam centrum non habet%-<^2.j
ubi terminus d non deeft ; fed fi terminus ille deeft '^' ^^^
punftum A eft ejus centrum. Prior fpecies eft quin«
quagefima oCtava, pofterior quinquagefima nona.

Quod fi terminus e y deeft, figura conftabit ex Fig. 642
tribus hyperbolis oppofitis quarum una jacet inter
Efymptotos parallelas & alterae duse jacent extra ut
in fpecie quinquagefima quarta, & praeterea diame-
trum habet quae eft abfciffa AB. Et haec eft fpecies
fexagefima.

Hyperbolifmus Ellipfeos per hanc aequationem de- xxiv:
finitur X y y + e y = c X -|- d, & unicam habet afymp- ^^^ „-^^^^^^^'^^
toton qus eft Ordmata prmcipalis Ad. Si termmus Fig. 6%.
ey non deeft, figura eft Hyperbola anguinea fine dia-
metro atq; etiam fine centro fi terminus d non deeft »
Quae fpeeies eft fexagefima prima.

At fi terminus d deeft, figura habet centrum fine ^^i^^tf^r
diametro & centrum ejus eft punftum A. Species
vero eft fexagefima fecunda.

Et fi terminus ey deeft & terminus d non deeft^^^^'^T^^
figura eft Conchoidalis ad afymptoton A G, habetq j
diametrum fine centro, & diameter ejus eft Abfciffa
A B. Quae fpecies eft fexagefima tertia.

X x 2 Hyper-

[15^ J

XXV. Hyperboliimus Parabolae per hanc aeqiiationem^
lifm^'p^i^' definitur x y y -|- e y ^ d ; & duas habet aiymptotos^.

Abfciffam AB & Ordinatam primam & principalem
A G. Hyperbolae vero in hac figura funt duge, non
in aiymptoton angulis oppofitis fed in angulis qui

Bv.(58. funt deinceps jacentes, idq; ad utrumq; latus ab-

feiflk A B^ & vel fine diametro fi terminus e y ha-

fzv. 6p. betur, vel cum diametro fi terminus ilie deeft. Qua?

duae fpecies funt fexagefima quarta & fexagefima.
quinta.

XXVI. In. fecundo aequationum cafu habebatur asquatio-
Tndens. xy = ax^-|-bxx-|-cx-|-d. Et figura in hoc cafu.

habet quatuor crura infinita quorum duo funt hy-

perbolica circa afymptoton A G in contrarias partes

tendentia & duo Parabolica convergentia & cum

prioribus fpeciem Tridentis fere efformantia. Eftq;

%. 7<y. haec Figura Parabola ilia per quam Cartefius aequa-

tiones fex dimenfionum conftruxit* Haec eft igitur

fpecies fexagefima fexta.

xxvii. In tertio cafu aequatio eratyy = ax^-[-bxx-|-cx

TatahoUqmn- \.A & Parabolcim defio;nat cuius crura divergunt

que dtvergtntes. 1.^. o* • ' r •

ab invicem oc m contrarias partes mnnite progre-
diuntur. Abfcifla AB eft ejus diameter & fpecies ejus
funt quinq; fequentes.
f^^' 70,71. Si3equationisax^-l-bx^-l-cx^-d=o radices om-

nes At f AT5 At funt reales & in^equales^ figura eft
Parabola divergens campaniformis cum Ovali ad
verticem« Et fpecies eft fexagefima feptima.
Bg, 72. Si radices duae funt aequales. Parabola prodit vel

%• 73- nodata contingendo Ovalem, vel funBata ob Ovalem

infinite parvam. Quae duae fpecies funt fexagefima
o£tava & fexagefima nona*

Si tres radices llint a'quales Parabola erk cufp-J^'^^-j^
data in vevticc. Et haec eft Parabola Neiliana qiice
vulgQ iemlcubica dicitur.

Si radices duse funt impoffibiles^ habetur Parabola fi^. 73, 74..
pura campaniformis fpeciem feptuagefiinam primam
conftituens.

In quarto cafu aequato erat y — ax -l-bxx-j-cx xxviii.
'4-d, & hsc i]equatio Parabolam illam IValli/mnam ^^'""^'^f ''■^^''''
defignat quae crura habet contraria & cubic a di- '^'
ci iblet. Et fie fpecies omnino funt feptuaginta
duae,-

Si in planum infinitum a pundo lucido illumina- ^^}\'
tum umbr^ figurarum projiciantur, umbras fed:io-'^^^p^^'j,;^^^-^,v
numConicarum femper erunt fediones Conical, eas
Curvarum fecundi generis femper erunt Curvae fe-
cundi generis^, eae curvarum tertii generis femper
erunt Curvae tertii generis, & ficdeinceps in infini-
tum. Et quemadmodum Circulus umbram proji-
ciendo generat fe£tiones omnes conicas, fie Parabolse
quinq; divergentes umbris fuis generant & exhi-
bent alias omnes fecundi generis curvas ^ & fie
Curv^ quaedam fimpliciores aliorum generum inve-
niri poffunt quae alias omnes eorundem generum
curvas umbris fuis a punfto lucido in planum pro^
jeftis formabunt. .

Diximus Curvas fecundi generis a llnea refta in xxx.
punctis tribus iecari polie. riorum duo nonnun^ ^^ ^w/e-/^; ^
quam coincidunt lit cum re£ta per Ovalem infi-
nite parvam' tranfit vel per concurfum duarum par-
tium Curvae fe mutuo fecantium vel in cufpidem
coeuntium ducitur. Et fiquando reftae omnes in

plagam

plagam cruris aliciijus infiniti tendentes Curvam
in nnico tantiim punfto lecant ( ut fit in ordinatis
Parabolae Cartefianae & Parabolge cubical, nee non in
re£tis Abfcifife Hyperbolifraorum Hyberbolge & Para-
bolae pkrallelis ) concipiendum eft quod redtae illae
per alia duo Curvati pun£ta ad infinitam diftan-
tiam fita ( ut ita dicam ) tranfeunt. Hujufmodi
interfettiones duas coincidentes five ad firiitam
fint diftantiam five ad infinitam, vocabimus pun-
dum duplex. Curvse autem quae^ habent pun-
ftum duplex defcribi poffunt per fequ^ntia Theo-
remata»

I . Si anguliduo magnitudine dati PAD, PBD circa

nica.
Fig, 78.

XXXL
Hoeoremata de
CiiYvarum de- - j^, 1 ""a -n a '

fcriftiom orga- polos pofitione datos A, B rotentur, & eorum crura
A P, B P concurfu fuo P percurrant lineam re6tam ;
crura duo reliqua A D, B D concurfu fuo D defcri-
bent feftionem Conicam per polos A, B tranfeun-
tern : praeterquam ubi linea ilia reda tranfit per po-
lorum alterutrum A vel B, vel anguliBAD, ABD
fimul evanefcunt, quibus in cafibus pundum D de-
fcribet lineam reftam.

1. Si crura prima A P, B P concurfu fuo P
percurrant fe£t:ionem Conicam per polum alter-
utrum A tranfeuntem, crura duo reliqua AD, B D
concurfu fuo D defcribent Curvam fecundi gene-
ris per polum alterum B tranfeuntem & pun-
ctum duplex habentem in polo primo A per quem
feSio Conica tranfit : prasterquam ubi anguli
BAD, ABD fimul evanefcunt, quo cafu pun-

6tum

£tum D defcribet aliam fedtionem Conlcam per po-
lum A tranfeuntem.

3. At fi feftio Conica quam punftum P percur-
rit tranfeat per neutrum polorum A, B, pundura
D defcribet curvam fecund i vel tertii generis pun-
dum duplex habentem. Et punftum illud duplex
in concurfu crurum defcribentiuin, AD, BD in-
venietur ubi anguliBAP, ABP limul evanefcunt.
Curva autem defcripta fecundi erit generis fi an-
guli BAD, ABD iimul evanefcunt, alias erit ter-
tii generis & alia duo habebit punfta duplicia in
polls A & B,

Jam fedio Conica determinatur ex datis ejus xxxii.
punais quinq; & per cadem fic defcribi poteft. ^i^'T^^
Dentur ejus punda quinq; A, B, C, D, E. ]un- tio per data qmn^
gantur eoruni tria qucevis A, B, C & trianguli ABC ^'''^'''''^^^
rotentur anguli duo quivis CAB, CBA circa ver-
tices fuos A & B, & ubi crurum AC, BC interfedio
Cfucceffive applicatur ad pundaduo reliquaD,E5
incidat interfedio crurum reliquorumAB & BA
in punda P & Q. Agatur & infinite producatur
redaPQ, & anguli mobiles ita rotentur ut inter-
fedio crurum AB, BA percurrat redam PQ, &
crurum reliquorum interfedio C defcribet propofi-
tam fedionem Conicam per Theorema primum.

XXXIIL
/-^ r* 1 • * n If Curvariim /e-

Curvse omnes lecundi generis punctum dupltXctmdige^erisptm-
habentes determinantur ex datis earum pundis f ^'^. ^^^^^ ^^"

z'. _n a, 'Hill bentiiim defer Ip-^

leptem, quorum mium elt punctum illud dw^ltyi^ tio per data fJp^

[ i6o ]

& per eadem punfta lie deicribi poiTunt. Dentui

'Curv^ defcribenda^ pundla quselibet feptem A, B, C,

D, E, F, G quorum A eft punftum duplex. Jun-

gantur pundum A & alia duo qusevis e pundls puta

;B & C ; Sc trianguli ABC rotetur turn angulus

CAB circa verticem fuum A, turn angulorum reli-

quorum alteruter ABC circa verticem fuum B. Et

ubi crurum AC, BC concurfus C fucceffive appli-

catur ad punfta quatuor reliqua D, E, F, G incidat

concurfus crurum reliquorum A B & B A in punda

quatuor P, Q, R, S. Per punda ilia quatuor &

quintum A defcribatur fe61:io Conica, & anguli prae-

fati CAB, CBA ita rotentur ut crurum A B, B A

concurfus percurrat feftionem illam Conicam , &

concurfus reliquorum crurum A C, B C defcribet

Curvam propoiitam per Theorema fecundum.

Si vice pun£ti C datur pofitione reda B C quae
Curvam defcribendam tangit inB, lineis AD, AP
coincident, & vice anguli DAP habebitur linea reda
circa polum A rotanda.

Si pun£tum duplex A infinite diftat debebit Reda
ad plagam pundi illius perpetuo dirigi & motu pa-
rallelo ferri interea dum angulus ABC circa polum
;B rotatur.

Defcribi etiam poflunt hae curvae paulo aliter per
Theorema tertium, fed defcriptionem fimpliciorem
pofuiffe fufficit.

Eadem methodo Curvas tertii, quarti & fuperio-
rum generum defcribere licet, non omnes quidem
fed quotquot ratione aliqua commoda per motum
localem defcribi poflunt. Nam curvam aliquam

fecundi

[l^l]

iecundi vel fupeiiorls generis pundum duplex noii
habentem comnK)de defcribere Problema eft inter
difficiliora numerandum.

Curvarum ufus in Geometria eft ut per earum xxxiv.
interfediones Problemata folvantur. Proponatur ciuaHo!Z%lZ
iEquatio conftruenda dimenfionum novem y^^'^-\-h^'^ f^^iftionemCur-
-j- c x^ -\- d x^ \ e xH f x3 -)- g X X -]- h X -j- k = o. Ubi ''^''''^•

b, c, d, 15c. fignificant quantitates quafvis datas
lignis fuis 4" & -^ affectas. Aflumatur sequatio ad
Parabolam cubicam x^ = y, & aequatio prior, fcri-
bendo y pro x', evadet y''|"bxyy-|- cyy-j^ dxxy
--r e X y -j- m y \- fx^ -j-g x x -j- h x -j-k = o, oequatio ad
Curvam aliam fecundi generis. Ubi m vel f deefle
poteft vel pro lubitu aflumi. Et per harum Curva-
rum defcriptiones & interfediones dabuntur radices
aequationis conftruendae. Parabolam cubicam femel
defcribere fufficit.

Si aequatio conftruenda per defectum duorum ter-
minorum ultimorum hx & k reducatur ad feptem
dimenfiones, Curva altera delendo m, habebit pun-
dum duplex in principio abfciflae, & inde facile de-*
fcribi poteft ut fupra.

Si aequatio conftruenda per defedum termino-
rum trium ultimorum gxx-i-hx-Hk reducatur ad
fex dimenfiones , Curva altera delendo f evadet
fedio Conica.

Et ft per defedum fex ultimorum terminorum
Gequatio conftruenda reducatur ad tres dimenfiones^
incidetur in conftrudionem JVallifianam per Para-
bolam cubicam & lineam reftam.

Yy Coii^

[ t<^2 ]

Conftrui etiam pofiiint a^quatlones per Hyperbo-
Urmum Parabolce cum diainetro. lit fi conftruenda
fit hcEC asquatio dimeniionum novem termiiio penul-
timo carens, a'^-cxx-j-dx^'i-ex^-x-fx -j-gx^^-j-hx^

+ m
^-kx^-V-lx^ = o ; affumatur a^quatio ad Hyperbolif-^
mum ilium xxy= i, & fcribendo y pro ^, sequatio
conftruenda vertetur in banc ay^ i-c y y + d x yy -j- e y
-j- fxy-j-m XX y-\-g-|-hx'Hk xx-j- 1x3 = 0, quae cur-
vam fecundi generis defignat cujus defcriptione
Problema folvetur. Et quantitatum m ac g alter-
utra hie deeffe poteft, vel pro lubitu aflumi.

Per Parabolam cubicam & Curvas tertii generis
conftruuntur etiam a?quationes omnes dimenfionum
non plufquam duodecim, & per eandem Parabolam
& curvas quarti generis conftruuntur omnes dimen-
fionum non plufquam quindecim, Et fie deinceps in.
infinitum. Et curvae illas tertii quarti & fiiperiorum
generum defcribi femper poffunt inveniendo eorum>
pun£ta per Geometriam planam. Ut fi conftruenda
fit aequatiox" ^ +ax^°+bx9+cx^+dx7-j-ex^-l-fx^
4-gx^-|-hx5'^ixx-l- kx -|- 1 = , & defcripta
liabeatur Parabola Cubica ; fit aequatio ad Pa-
rabolam illarn cubicam x^ =: y ^ & fcribendo y
pro x^ aequatio conftruenda vertetur in hane
y4 '|-axy^ ^|-cxxyy -f^fxxy -^ixx^o , quae eft
Arh +dx +g^ -V^^

sequatio ad Curvam tertii generis cujus defcriptione
Problema folvetur. Defcribi autem poteft hsecCurva
inveniendo ejus punfta per Geometriam planam,prop-
terea quod indeterminata quantitas x non nifi ad
duas dimenfiones afcendit.

( (/nmru/ji TalrJ.

LuruarunvTab. H.

Cui^mnun Tahlll.

( iirndruni 7alin^

( itrvaruni Tab. T

( ({/iranim Talr VI.

y .

7;^

fl.

Xt/

/'/-

t(l

TRACTATUS

D E

Quadratura Ciirvarmn.

Yy 2

1 1^$ ]

wil l ' " > '-»->X f'^m^mm

INTRODUCTIO-

QUantitates Mathematicas non ut ex partibus
quam minimis conftantes, fed ut motu conti-
nuo defcriptas hie conlidero. Lineae delcri-
buntur ac defcribendo generantur non per appofi-
tionem partium fed per motum continuimi pun£to-
rmn, fuperficies per motum linearum, folida per
motum fuperficierum, anguli per rotationem late-
rum, tempora per fluxum continuum, & fie in cae-
teris. Hx Genefes in rerum natura locum vere ha-
bent & in motu corporum quotidie cernuntur. Et
ad hunc modum Veteres ducendo redtas mobiles in
longitudinem reftarum immobilium genefin docue-
runt redangulorum.

Confiderando igitur quod quantitates aequalibus
temporibus crefcentes & crefcendo genitae, pro velo»
citate majori vel minori qua crefcunt ac generantur,
evadunt majores vel minores j methodum quaerebam

determinandi quantitates ex velocitalibus motuum
vel incrementorum quibus generantur j & has mo-
tuum vel incrementorum velocitates nominando Flu-
xtones & quantitates genitas nominando Fluent es^ in-
cidi paulatim^wm i665&i666in Methodum Flu-
xionum qua hie ufus fum in Quadratura Curvarum.

Fluxiones funt quam proxime ut Fluentium aug-
menta sequalibus temporis particulis quam minimis
genita, & ut accurate loquar, funt in prima ratione
. augmentorum nafcentium ; exponi autem poffunt per
Jineas quafcunq; quae funt ipfis proportionales. Ut
-"^V. I, fiaregeABC, ABDG Ordinatis BC, BD fuper

ball A B uniformi cum motu progredientibus defcri-
bantur, harum arearum fluxiones erunt inter fe ut
Ordinate defcribentes BC & BD, & per Ordinatas
illas exponi poffunt, propterea quod Ordinatae illa^
funt ut arearum augmenta nafcentia. Progre-
diatur Ordinata BC de loco fuo BC in locum
quemvis novum b c. Compleatur parallelogram-
mum BCEb, ac ducatur refta VTH quae Cur-
vam tangat in C ipfifq; b c ScBAproduSis occur-
rat in T & V : & Abfciff^ AB, Ordinatse BC, &
Lineae Curva: A C c augmenta modo genita erunt
Bb, Ec & Cc; & in horum augmentorum nafcen-
tium ratione prima funt latera trianguli CET,ideoq;
fluxiones ipfarum AB, BC & AC funt ut trianguli
illius CET latera CE, ET & CT & per eadem
latera exponi poffunt, vel quod perinde eft per la-
tera trianguli confimilis VBC«

Eodem recidit li fumantur fluxiones in ultima
iiatione partium evanefcentium. Agatur refta Cc
-& producatur eadem ad K. Redeat Ordinata be

in locum fuum priorem B C, & coeuntibus punftis
C & c, refta CK coincidet cum tangente CH^ &
triangulum evanefcens CEc in ultima fua forma
evadet fimile triangulo GET, & ejus latera evanef-
centia CE^ Ec & Cc erunt ultimo inter feut funt
trianguli alterius GET latera GE, ET &GT, &
propterea in hac ratione funt fluxiones linearum A B,
BG «& AG. Si pun£ta G & c parvo quovis inter-
vallo ab invicem diftant reda G K parvo intervallo a
tangente GH diftabit. lit recta CK cum tangente
G H coincidat & rationes ultimse linearum G E, E c &
Gc inveniantur, debent pundta G & c coire & om-
nino coincidere. Errores quam minimi in rebus
mathematicis non funt contemnendi.

Simili argumento fi circulus centro B radio B G
defcriptus in longitudinemAbfciffse AB ad angulos
reftos uniformi cum motu ducatur^ fluxio folidi ge-
niti ABG erit ut circulus ille generans, & fluxio fu-
perficiei ejus erit ut perimeter Girculi illius &
fluxio lineae curvse A G conjunftim. Nam quo tem-
pore folidum ABG generatur ducendo circulum
ilium in longitudinem Abfcilfe A B, eodem fuper-
ficies ejus generatur ducendo perimetrum circuli il-
lius in longitudinem Gurvae A G.

R^^a TB circa plum datum T revolvens fecet aliam Tig
fofitione datam rei^amAB: qu^ritur froprtio fluxio^-
num reBarum iUarum j4B !5 P5» Progrediatur
refta PB de loco fuo PB in locum novum Pk In
P b capiatur P G ipii P B aequalis, & ad A B ducatur
P D fie, ut angulus b P D sequalis fit angulo b B G ^
& ob fimilitudmem triangulorum bBG, bPDerit
augmentum Bb ad augmentum Cb ut Pb ad Db,

Redeat

[ 1^8 ]

Redeat jam Pb in locum fuum piiorem PB ut aug-
menta ilia evanefcant, & evaneicentium ratio ulti-
ma, id eft ratio ultima Pb ad Db, ea erit quae eft
-PB ad D B, exiftente angulo PDB refto, & prop-
terea in hac ratione eft tluxlo ipfius A B ad fluxionem
ipfius P B.

Re(^a T B circa datum Tolum T revolvens fecet
aliojs dim fofitione datas redJas ABh' AE in B b*
E : quc^ntur p'^ofortio jluxionum re^ay-um iUarum
A B '^ AE. Progrediatur refta revolvens P B de
loco fuo P B in locum novum P b reflas A B, A E in
pundis b &e fecantem, & re£tce AE parallela BC
ducatur ipfi Pb occurrens in C, Sc erit Bb ad BC ut
Ab ad Ae, & BC ad Eeut P B ad P E, & conjunftis
rationibus Bb ad Ee ut AbxPB ad AexPE»
Redeat jam linea Pb in locum fuum priorem PB, &:
augmentum evanefcens Bb erit ad augmentum eva-
nefcens Ee ut ABxPB ad AExPE, ideoq; in
hac ratione eft fluxio redae A B ad fluxionem reftcs
A E.

Hinc C refta revolvens PB lineas qualvis Curvas
pofitione datas fecet in pun6tis B & E, & re^ta? jam
mobiles AB,AE Curvas illas tangant in Seftionum
punftis B & E : erit fluxio Curvas quam refta, A B
tangit ad fluxionem Curvae quam refta AE tangit
utABxPB ad AExPE. Id quod etiam eveniet
fl reda P B Curvam aliquam pofitione datam perpe-
tuo tangat in punfto mobili P.

Fluat quant'it/18 x uniformiter }^ invenienda/it fluxio
qMantitatts x^. Quo tempore quantitas x fluendo

evadit x-j-o, quantitas x^ evadet x-|-ol"^ id eft
jper metliodum ferierum infinitarum, x''-] nox""^

Lj-^^i^oox^^-Mlfy^:'. Et augmenta o & nox^-M^"oox*'^
^-to'^r.funt adinvicem ut i & nx"-'~|-Hi^ox"-^-j-. Jc;^.
Evanefcant jam augmenta ilia , & eorum ratio
ultima erit i ad nx"'^ : ideoq; fluxio quantitatis
X eft ad fluxionem quantitatis x" ut i ad nx'^"^

Similibus argumentis per methodum rationum
primarum & ultimarum colligi poffiint fluxiones li-
nearum feu redtarum feu curvarum in cafibus qui-
bufcunque, ut & fluxiones fuperficierum, anguio-
rum & aliarum quantitatum. In finitis autem quan-
titatibus Analyfin fie inftituere, & finitarum nafcen-
tium vel evanefcentium rationes primas vel ultimas
inveftigare, confonum eft Geometriae Veterum : &
volui oftendere quod in Methodo Fluxionum norl
opus fit figuras infinite parvas in Geometriam intro-
ducere. Peragi tamen poteft Analyfis in figuris qui-
bufcunq; feu finitis feu infinite parvis quae figuris
evanefcentibus finguntur fimiles, ut & in figuris quae
pro infinite parvis haberi folent^ modo caute pro-
cedas.

Ex Fluxionibus in venire Fluentes Problema dif-
ficilius eft, & folutionis primus gradus aequipollet
Quadraturae Curvarum : de qua fequentia olim

JLt Z xJ Ju<

[lyo]

JL xv A Cj 1 Ax LI w3

D E

Quadratura Curvarum.

QUantitates indetermihatas ut motu perpetuo j
. crefcentes vel decrefcentes, id eft ut fluen- ]
tes vel defluentes in fequentibus confidero^defignoq;
Uteris z, y, x, v, & earum fluxiones feu celeritates

crefcendi noto iifdem Uteris pun£tatis z, y, x^ v.
Sunt & harum fluxionum fluxiones feu mutationes
magis aut minus celeres quas ipfarum z, y, x^ v
fluxiones fecundas nominare licet & fie dignare

Zj y, X, V, & harum fluxiones primas feu ipfarum :

Zj y, X, V fluxiones tertias fie z^ y, x, v, & quartas fie

z, y, x, Ve / Et quemadmodum z, y, x^ v funt flu-

xiones quantitatum z, y, x^ v, & hae funt fluxiones

quantitatum z, y, Xj v & hae funt fluxiones quantita-
tum primarum z, y^x, V : fie hae quantitates confide-
rari poITurit ut fluxiones aliarum quas fie defignabo.

z, y, X, V, & hae ut fluxiones aliarum z, y, x, v, &

hx ut fluxiones aliarum z, y, x, v. Defignant igitur
II I ..«••♦• ...» ft .

z, z, z, z, z, Z-, z, z to'r. feriem quantitatum quarum
quGelibet pofterior eft fluxio prsecedentis & quaelibet
prior eft fluens quantitas fluxionem habens fubfe-

quentem. Similis eft feries /^az— zz, ^^az— zz,

f/?iz—zz , /^az— zz , >^az— zz ^ /^az— zz , ut &

^ ^. az-4-z^ az-\-z^ az-pz^ az-|-z^ az^-z^

a z ai~'^"°°z d z a z a z

a -l-^^ ^

.^ J...-__ . Et notandum eft quod quantitas quaelibet

a— z

prior in his feriebus eft ut area figurae curviliniae
cujus ordinatim applicata reftangula eft quantitas

pofterior & abfcifla eft z : uti /^az-==-zz area curvge

cujus ordinata eft /^az— zz & abfcifla z. Quo au^
tern fpeftant hsec omnia patebit in Propofitionibus
quae fequuntuia

P R O P,

C 172 3

PROP. I. PROB. L

^ata ^equatione quotcunq* fluent es quantitates invoU
vente^ invenire fiuxiones,

Solutio.

Multipllcetur omnis aequatlonis terminus per in-
dicem dignitatis quantitatis cujufq; fluentis quam
involvit, & in fingulis multiplicationibus mutetur
dignitatis latus in fluxionem fuam, & aggrega-
turn faftorum omnium fub propriis fignis erit
aequatio nova.

Expltcatio.

Sunto a, b^ c, d Is)'^. quantitates determinata? &
immutabiles, & proponatur aequatio quaevis quan-
titates fluentes z, y, x '^c, in volvens^ uti x^ — x y y
-|- a a z — b^ = o. Multiplicentur termini primo per
indices dignitatum x, & in fingulis multiplicationi-
bus pro dignitatis latere, feu x unius dimenfionis,

fcribatur X5& fumma fadorum erit 3 x x' — x y y .Idem
fiat in y & prodibit — ^xy y. Idem fiat in z & pro-
dibit a a z. Ponatur fumma faftorum aequalis ni-
bilo, & habebitur sequatio 3 x x^ — x y y" — 'x y y

'-j-a a z = o. Dico quod hac aequatione definitur re-
latio fluxionum.

[173]
Demonftratio.

Nam fit o quantitas admodum parva &: iunto

0Z5 oy, ox, quantitatum z, y, x momenta id eft In-
crementa momentanea fynchrona. Et fi quantita-
tes fluentes jam iunt z, y & x, hae poft momentum

temporis incrementis fuis oz, oy, ox auda?, evadent

• • »

z^-oz, y^-oy, x-j-ox, quae in aequatione prima pro

z, y & X fcriptae dant aequationem x^ -l^^xxox

^|- 3X00XX -j- 0^x3 — ^xyy — oxyy — ^xoyy — ^xooyy

— xooyy-— -xo^yy -j-aaz-l-aaoz— b^ = o, Subducatur

sequatio prior, & refiduum divifum per o erit ^xx^

-]- 3 xxox -j-^x^oo — xyy — ixyy -ixoyy — xoyy —xooyy
-j-aaz^ o. Minuatur quantitas o in infinitum^Sc neg»

le£tis terminis evanefcentibus reftabit gxx^-— xyy

— ^xyy -f-aaz = o. Q. E, D.

ExpUcatio plenion

Ad eundem modum fi asquatio eflet xs— xyy
4-aa f^ax — -yy—b^ =0, produceretur gx^x— xyy
— 2xyy -^^aar'lax-— -yy — o. Ubi fi fluxionem/^ax — -yy
tollere velis, pone //ax— yy = z, & erit ax— yy — z®

[174]

(per hanc Propofitionem ) ax-— -^yy^V.z feu

^J- — z 5 hoc eft ■ JJ.^ = /^ax— yy , ht

^z ^/^ax — -yy

. . . a^x— laayy

mae 3 x^x — xy y— 2xy y \ ^-— ' -- =0

Vax— yy

Et per operationem repetitam pergitur ad fluxio-

nes iecundas, tertias & lequentes. Sit aequatio

zy^"— z4-|-a4 = o, & fiet per operationem primam

^y^^"'3^yy^"^42^^~o 5 perfecundam zy3-|-^6zyy2
"\-3zyy2-|-6zy^y— 4.ZZ3— i^z2z^ = o , per tertiam

-2y^-i- 9zyy' + 9zyy2 ^-. iSzy^y 4-.3zyy^^-^i8zyyy

-|-^'6zy^"4.zz3— '5 6zzz2— 24.z^z ~ Oe

llbi vero fie pergitiirad fluxiones fecundas, ter-
tias & fequentes^ convenit quantitatem aliquam ut
uniformiter fluentem con{iderare5& pro ejusfluxione
prima unitatem fcribere^ pro fecunda vero & fe-
quentibus nihil. Sit aequatio zy ^ — z"^ -]-; a4 — o, ut
lupra; & fluat Ziiniformiter5fitq;.ejus fluxio unitas^

&fiet per operationem primam y^-]-^ 5 Zyy^—4.Z3 = 05

per fecundam 6yy^ \i S^yy^ -^l- 6zy^y — - 1 2z^ = o,

per tertiam 9yy'+i8y2y-]-^zyy^4.i8zyyy-1^6zy3
—242 = 0.

. In

In hujus autem generis ^quationlbus concipieii-
dum eft quod tluxiones in iingulis terminis fiat ejuf-

dem ordinis, id eft vel omnes primi ordinis y^ z,

vel omnes fecundi y, y^^ yz, z% vel omnes tertii

y^ yy^ y^^ yS Th y^^ ^^ &<^'- Et ubi res aliter fe
habet complendus eft ordo per fubintelleftas fluxio-
nes quantitatis unlformiter fluentis. Sic aequatio

Hoviffima complendo ordinem tertium fit 9Zyy^
+ ^^^Tl\V^lT\ ^ 8zyyy~^-.6zy3~i4.zz3 = o,

PROP. IL PROB. IL

Jnvenire Curvas qu^ quadrari fo^unt.

Sit ABC figura invenienda, BC Ordinatim z^-Fig> ^.
plica ta redangula^ & AB abfciffa. Producatur
CB ad E ut fit BE—i^ & compleatur parallelo-
grammum ABED: & arearum ABC, ABED
fluxiones erunt ut BC & BE. Affumatur igitur
sequatio quaevis qua relatio arearum definiatur^ &
inde dabitur relatio ordinatarum BC & BE per

Hujus rei exempla habentur in Propolitionibus
duabus fequentibus.

17^]

PROP. III. THEOR. L

Si pro abfclffa AB & area AE feu ABxi pro-
miicuefcribaturZj&fipro e -[^ft!' ]-gz^'^ -{-hzh-^-Sac,
fcribatur R : fit autem area Curvae zeR'^ erit,
ordinatim applicata BC =

^e t L fz' IJ gz'" IL hz3« -H &c. in z ^-^ R ^-^

Demonftratio.

Nam fi fit z^R^^v, erit per Prop, i, ^zz^-'R;^

« • «

'^-AZ^RR'^'^ = V. Pro Ra in primo sequationis ter-
mino & z® in fecundo fcribe RR^"^ & zz^"^, & fiet

szR -|- azR in z®'' R^^ = v, Erat autem R = e -|- f z»

-l-gz^«^[-hz3« &c. & inde per Prop. i. fit R =;

?3fzz«"V-[-2Hgzz^»'\4^iMhzz3«-^^j- &c. quibus fubftitu-

tis & fcripta BE feu i pro z, fiet
»e.mfzi|^^gz^»:|:«^hz3»4.&c. in z8-R^- = v=BC.

Cjt» E. De

PROP.

1^71 1

PROP. IV. THEOR, 11.

Si Curvae abfciffa A B fit z, & fi pro q\- f z" -)-gz'«
-]-&c. fcribatur R, & pro k-4"lz«-j~^mz^"-|- &c. fcri-
batur S ; fit autem area Curva? z^ R^ S^ : erit or-
dinatim applicata B C == ,

9ek~!^ f k z" i^ gkz^'* ' ^ * i

M-XM H-2AMD B

glz3» *

2A„ 5^^ ^ i^ z^-'R^-'S«*"'

-AM '1-2AM

Demonftratur ad modum Propofitionis fiiperioris.

PROP. V. THEOR. IIL

Si Curv^ abfcifla AB fit z, & pro e-j-fz^'-l-gz^"
■^-hz^" -Y &c. fcribatur R : fit autem ordinatim ap-
plicata z^-'R'^'' in a -|-bz" ^l-cz^" -l-dz3«-i- &c, & po-

natur J-r. r-j-^^s. s-j-'^^t. t-l-A==v<&c. erit area

re r-h I ,e M^T, e r^+lje

-^- -- -^^ ^^ z4« ~1- &c. Ubi A, B, C, D, &c.

r-t-4, e

Aaa denotant

[178]

denotant totas coefficieiites datas termlnorum (ingu-
lorum in ferie cum fignis fuls-|-& — ^nempe A primi

termini coefficientem ^ B fecundi coefficientem

i-e°)

^b~sfA r>. .- rr • .7cZ;fB— tgA ^

"._-____, C terhi ropfRriPnl-pm " ^ , &

r -H- I , e r — h 2, e

fic deinceps.

Demonjiratio.
Sunto juxta Propofitionem tertiam^

Curvarum Ordinatae & earundem areas.

I . eeA tlf Az" tLg Az^« '\l h Az^^'Sc c. 1 ' Az^ R^ ,

s etii, eBz" t®-i"«f B z^" t^-UgBzS" &c. I Bze-h" R\

3 ' - • ' +9T^,eCz'«:p-l-2«fCz3^ &c.

^ z^-iRvi,

Cz9-NhR\

+9-'h«5eDz3"&c.j Dz^-'^3« R^

Et fi fumma ordinatarum ponatur ^qualis ordi-
natae a^|-bz"'|-cz^^~dz3"-|- &c. in z^-'R^-', fumma
arearum z^R^ in A~j-Bz"'pCz^"-j-Dz3"-|- &c. sequa-
liseritare^ Curves cujus ifta eft ordinata. ^quen-
tur igitur Ordinatarum termini correipondentes, &
fiet a-eeA, b = .i_«„fA:l^eB, c= .y^^t^' fB

.^j- 0-l-2„5eC 6CC. & mde g^ =^ A.

C~[9-1-2A„. gA-;9-i-r!-A«,fB ^ T- r 1 •

-~- — — Q_|_2„, e ~— =: i^. jit lie demceps m mfi

3S i

nitum

C 179 }

iiitiim. Pone jamS-r. r-j-A^^s. s-J^x^t &c. &

in area z^R^x A-VBz"^J-Cz^"-|-"Dz3» &c. fcribe ip
forum A^ B, C, &c. valores inventos & prodibit
feries propofita. Q. E. D.

Et notandum eft quod Ordinata omnis duobus
modis iu feriem reiblvitur. Nam index "vel afEr-
mativus eft poteft vel negativus. Prpponatur Ordi-

nata ----^^ -p-^- — ^, Haec vel fie fcribi poteft

z-ix 5 k— Izzxk — IzzA-mz^j'^^ vel fie zx-l-j-^kF^
xm-lz""'-l-kz"~^, -i. In cafu priore eft a— ^Eb^o.

C=— 1. e==k. frrrO. g= J. h=m. ^^=—1. n=:I,
9-1=-^. ^=-l=l\ S=-I. t=--^. Vrr=0. In

pofteriore eft a = — 1. b=o. c^^k. e=m. f==— 1.

g=0. h^=I. ^=—1 ^i=—l. Q— 1:^1. 0=2. r=— 2,

s=— i^. t=— I. v=— '. Tentandus eft cafus uter-
que. Et fi ferierum aiterutra ob terminos tandem
deficientes abrumpitur ac terminatur, habebitur area
Curvae in terminis finitis. Sic in exempli hujus
priore cafu fcribendo in ferie valores ipforum a, b^
c, e, f, g, h, A, 9, r, s^ t, v, termini omnes poft pri-
mum evanefcunt in infinitum Sz. area Curra prodit

— ■2V^ '~z3^ '''^\ Et ha?c area ob fignum negativum
adjacet abfciffe ultra ordinatam produd^. Nam
area omnis affirmativa adjacet tarn abfciife quam
ordinate, negativa vero cadit ad contrarias par-
tes ordinatae & adjacet abfcilfe produdse^ manente
fcilicet figno Ordinatce. Hoc modo feries aiter-
utra & nonnunquam utraque fem^per termlnatur
& finita evadit fi Curva geometrice quadrari po-
teft. At fi Curva talem quadraturam non admit-
tit, feries utraq; continuabitur in infinitum^ & ea-

A a a 2

[i8o]

rum altera converget 8c areamdabit approximandoy
praeterquam ubi r ( propter aream infinitam ) vel
nihil eft vel Humerus integer & negativus, vel ubi ^
iiequalis eft unitati. Si '^ minor eft unitate, conver-
get feries in qua index „ affirmativus eft : fin I unita

te major eft, converget feries altera. In uno cafii
area adjacet abfcilTas ad ufq; ordinatam dufta?, in
altero adjacet ablciffas ultra ordinatam produilo?.

Nota infuper quod ii Ordinata contentum eft fub
feftore rationali Q & fadore furdo irreducibili R'^^
& faftoris furdi latus R non dividit factorem ratio-
nalem Qj erit a-i =t & R'^'^ - R'^. Sin factoris fur-
di latus R dividit faftorem rationalem femel, erit
A-»i=r=7r-|- I & R^'^ r=z=R''+^ \ ft divldlt bls , erit
A=-i=r=r7r^|-ci & R^^'^ =:R'^-1"^: fi ter, erit a-i^tt-J^^^,
& R^-^=:R''-»'3 : & ficdeinceps.

Si Ordinata eft fradio rationalis irreducibilis cum
Denominatore ex duobus vel pluribus terminis com-
pofito : refolvendus eft denominator in divifores
faos omnes primes. Et fi divifor fit aliquis cui
iiullus alius eft sequalis , Curva quadrari nequit :
Sin duo vel plures lint divifores sequales, rejicien-
dus eft eorum unus, & fi adhuc alii duo vel plures
fint fibi mutuo ^quales & prioribus ina^quales, re-
jiciendus eft etiam eorum unus, & fie in aliis omni-
bus aequalibus fi adhuc plures fint : deinde divifor
qui relinquitur vel contentum fub diviibribus omni-
bus qui relinquuntur, fi plures funt, ponendum eft
pro R, & ejus quadrati reciprocum R'^ pro R'^'^^prse-
terquam ubi contentum illud eft quadratum vel cu-
bus vel quadrato quadratum^&c. quo cafu ejus latus

ponen-

[i8i]

ponendum eft pro R & poteftatis index 2 vel 3 vel 4
negative fumptus pro a. & Ordinata ad deiio'iiina^
torem R^ vel R^ vel R^ vel R^ &c. reduceoda.

lit ft ordinata fit ^^l±^4=:i23 . anoni'^m li-r

fraftio irreducibilis eft & denominatoris divi fores
funt pares.^^ nempe z— i, z— i, z— i & z-]-a,
Z'j-i, rejicm magnitudinis utriufque diviforem
unum & reliquomm z— i, z — i , z-|-2 content
turn z^ — ^3^-1-^ pono pro R & ejus quadra ti re-
ciprccum "^^ feu R~2 p^.Q ^k-^i j^^^^ Ordina>

tarn ad denominatorem R^ feu'R'-^reduco, & fit

z^- 9z^-l-8z3 ______ ■

Et inde eft a = 8, b=-9. c = o. d^=-i^ &c
e=a. f=~^, g = o. h==i. ^-i=r-.2. ;,~_K

„— I. 9-1 — ^. 9=r4=:r. S— 3. t==:2. V=I. Et Ills

in ferie fcriptis prodit area -11-— ^ terminis om-
nibus in tota ferie poft primuin evanefcentibus.

Si deniq; Ordinata eft fraffio irreducibilis & ejus
denominator contentum eft fob fadore rationali Q
& faftore furdo irreducibili R'^, inveniendi funt la-
teris R diviforcs omnes primi«, & rejiciendus eft di-
vifor unus magnitudinis cujufq; & per divifores
qui reftant , fiqui fint , multiplicandus eft fa£tor
rationalis Q : & fi fadum a^quale eft lateri R vel
lateris illius poteftati alicui cujus index eft numerus
integer, efto index ille m^^ & erit a— i ^— ^— m, &.

R^--R— . UtfiOrdinatafit-^^-^^ "^-^^-^"-^^"-^^^^%

quonj'am.

.[182]

quoniamfaftoris fiirdi latusR feu q3-]-qqx-qxx-X5
diviibres habet q-1 X5 q^-x, q— xqui duarumfunt
magiiitudinuin, rejicio diviforem uiium magnitudi-
nis utriufq; Sc per diviforem q-j-x qui relinquitur
multiplico faCtorem rationalem qq — xx. Et quo-
niam faftum q^-j-qqx — qxx — x^ ^quale eft la-
teri R^pono m=i. & inde^ cum tt fit |, fit ^-^i =-.^.
Ordinatam igitur reduce ad denominatorem R:l

& fit Z°x 3qM-^q^x+8q^xx-l-8q^x^~7qqx'~6qx^
X q5 -j- qqx — qxx— xl^^^-.Unde eft a — 5 q^ b = iq^ &c.
e-qs. f-qq&c. 9 — 1=^0. 0=1— «. ;, = — ■. r= i.
tr='. v:=o. Et his in ferie fcriptis prodit

area . ^ , , — - ^ termmis ommbus m ferie tota

^cub. a3-l-aax — axx — x^ ^

poft tertium evanefcentibus.

PROP. VI. THE OR. IV.

Si Curvae abfciffa AB fit z, & fcribantur R pro

e^-fz« -|-gz^« -|-'hz3»-l-&c. & S pro k "l-lz« -l-mz^^
"j-nz^" &c. fit autem ordinatim applicata z^-'R^"^ S^-'
in a-|-bz« -j-cz^" --\-dz3" &c. 8c fi terminorum, e^ f,
gj \ &c. & k, I5 m^ n. &c. reftangula fint.

ek fk gk hk &c.

el fl gl hi &c.

em fm gm hm&c.

en fn gu hn &c.

Et fi reSanguloriim illorum coefficieiites nume-
rales fint refpeftive

«"9 = r. Y-\-y\^S. S-|-A-t. t-'p^=V. &C.'

. S-j-/,— t. t-|-^==V. V-\--"^W. W'j-/"=X. &c.

area Curvae erit h^c

'-— 1 2 k
- q -h — 5 ^^ A ' r —5^-17 f k D — t' f 1 A

6D>C • " «^_3,ei^ hC-.s'4i el J^-t"enA

z^R^S/^m - — ^ j_ — -^ ,rJ--^-~^ — — , 72«.

rek 1 r-|-i,ek ' fTT,ek ^

___ __v hkA ' -

___ -t-)-i,gk^_v'gl^

Lj ~s-1-2, fkp — t'-l-i,fl -D _v"fm
« <^ — s'-l-2, e l"^ -t'Vi-i,e m _v'"e n

,z5« M-, Sec.

r-i-3»ek

Ubi A denotat termini primi coefficientem datani

»L^ cum figno fuo ■\^ vel — , B coefficientem datam

fecundi, C coefficientem datam tertii^Sc fie deinceps.
Terminorum vero^ a^ b, c, &c. k, 1, m, &c. unus
vel plures deeffe poffunt. Demonftratur Propofitio
ad modum prsecedentis, & qu-^ ibi notantur hie ob-
tinent. Pergit autem feries talium Propcfitionum in
infinitum^ &Progreffio feriei.manifefta eft.

PROP.

liHl

PROP. VII. THEOR, V.

Si pro e 4-fz"-|-gz^«4- &c. fcribatur R ut fupra, &
in Curvs alicujus Ordinata z^inor'R_K±T maneant
quantitates datae 0, h, a^ e, £> g, &c. & pro ^ ac t fcri-
bantur fucceflive numeri quicuiiq; integri : & fi
detur area unius ex Curvis quos per Ordinatas in-
niimeras (ic prodeuntes defignantur fi Ordinatoe funt
duorum nominum in vinculo radicis^ vel fi dentur
areoe duarum ex Curvis fi Ordinatoe lunt trium no-
ininum in vinculo radicis, vel area: trium ex Curvis
fi Ordinatoe funt quatuor nominum in vinculo radi-
cis, & fie deinceps in infinitum : dico quod dabun-
tur areoe curvarum omnium. Pro nominibus hie
liabeo terminos gmnes in vinculo radicis tarn de-
ficientes quam plenos quorum indices dignitatum
funt in progreffione arithmetica. Sic ordinata
Va^ — ax3 -j- x^ ob terminos duos inter a*& — ax^
deficientes pro quinquinomio haberi debet. At

Va'^-i-X4 binomium eft & V^'^-J-^x^ — ^ trinonium^
cum progrelTio jam per majores differentias proce-
dat. Propofitio vero fie demonftratur,

Sunto Curvarum duarum Ordinatoe pz®'^ R'^-' &
qzM«-iR_A-i^ & areoe p A & qB, exiftente R quanti-
tate trium nominum e-j~fz«'j-gz'«. Et cum per

Prop,

Prop. III. fit z^R^ area curvae cujus Ordinata eft
fle:|:^/z«:jN ^gz^« in z«-R-^,fubduc Ordinatas & areas
priores de area & Ordinata pofteriori, & manebit
% 11/^V|l« g^'"^« ^"''^"' Ordinata nova Curv^,&

— qz"

z^R'^ — pA— ^qB ejufdem area. Pone se-p &
df-j-Awf rr=q & Ordinata evadet J 2z^» in z^-'R'^'L &
area z'R'^ — eeA — efB — A»fB. Divide utramq; per
9g-|-.^A«g^ & aream prodeuntem die C, & afliimpta
utcunq; r, erit r C area Cuvvx cujus Ordinata eft
i-zfl-Htf-iR^-i, Y.t qua ratione ex areis pA & qB
aream rC Ordinata? rz^'l"^«'^ R^"' congruentem inve-
nimus, licebit ex areis qB & rC aream quartam
puta sD, ordinatae sz^+^n |{^a-i congruentem invenire,
& fic deinceps in infinitum. Et par eft ratio pro-
greflionis ab areis B & A in partem contrariam
pergentis. Si terminorum 95 9-|-'^»,& 9-^-2a„ aliquis de-
ficit & feriem abrumpit^ allumatur area pA in prin-
cipio progreffionis unius & area qB in principio a.U
terius, & ex his duabus areis dabuntur areae omnes
in progreffione utraque. Et contra, ex aliis duabus
areis alTumptis fit regreffus per analyfin ad areas A
& B, adeo ut ex duabus datis caeterae omnes den-
tur. Q. E. O. Hie eft cafus Curvarum ubi ipfius z
index G augetur vel diminuitur perpetua additione vel
fubdudione quantitatis **. Cafus alter eft Curva*^
rum ubi index ^ augetur vel diminuitur unitatibuso

■L ji S»

[i8d3

C jl S. IL

Ordinatse pz^'^R^ & qz^+^-'R'^, quibus areae pA
& qB jam refpondeant, fi in R<feu e-j-fz^ + gz^" du-
cantur ac deinde ad R viciffim applicentur, eva-
dunt pe -\- pfz^ -\- pgz^" x z^-'R'^"^ & qez'' -j- qfz^*^
J-qgz3« X z^-^R^-'. Et per Prop. HI. eft az^R^^
area Curvae cujus Ordinata eft sae :j:J^afz\|^^^agz^*'
in z^-^R^"^ 5 & bz^'^'^R^ area Curvae cujus ordinata
eft .|-5bez« tjbfz^" t-J^gz^" in z®-'R'^-^ Et harum qua-

tuor arearum fumma eft pA-|-qB'|"' az^R'^-j-bz^'l'^R^
& fumma refpondentium ordinatarum

^ae t® afz« tl asz^« V bgz^" in z^-^R^^-^
. ^ f be f bf ^

+ pf

+ qe + qf

Si terminus primus tertius & quartus ponantur fe-
orfim aequales nitiilo, per primum fiet 0ae^i-pe = o
feu --«a = p5 per quartum — 9b — »b — 2 Awb = q ^ & per

tertium (eliminando p & q) t = b. Unde fecundus

fit ^^-7^^^^, adeoq; fumma quatuor Ordinatarum eft

^^^'z^'^'«'^R^-S& fumma totidem refpondentium

arearum eft azSR^^^ip^s+^R^^eaA— '-^^^^^

Divi-

C 187 ] ^^

Dividantur hse fummas per ^. — r^*'^ & fi Quotum
pofterius dicatur D, erit D area curvge cujus ordi-
nata eft Quotum prius z^+^'^R^'^ Et eadem ratione
ponendo omnes Ordinatae terminos praeter primum
aequales nihilo poteft area Curvas inveniri cujus Or-
dinata eft z^'^R^'^ Dicatur area ifta C, & qua ra-
tione ex areis A & B inventas funt areae C ac D, ex
his areis C ac D inveniri poffunt aliae duae E & F
ordinatis z^'^R^'^ & z^-V^^^R^-^ congruentes, & fie de-
inceps in infinitum. Et per analyfin contrariam
regredi f licet ab areis E&Fad areas CacD, &
inde ad areas A & B, aliafq; quae in progreflione fe-
quuntur. Igitur fi index ^ perpetua unitatum ad-
ditione vel liibduftione augeatur vel minuatur, &
ex areis quae Ordinatis fie prodeuntibus refpondent
duae fimpliciflimae habentur j dantur aliae om_nes in
infinitum. Q, E, O.

C ^ S. Ill

Et per cafus hofce duos conjunftos, fi tarn in-
dex e perpetua additione vel fubdudione ipfius^^
quam index x perpetua additione vel fubduCtione
unitatis, utcunq; augeatur vel minuatur, dabuntur
areae fingulis prodeuntibus Ordinatis refpondentes,
Q. E. O.

Bbbi CJtS.

[i88]

C A S^ IV.

Et fiinili augmento fi ordinata conftat ex qua-
tuor nominibus in vinculo radicali &: dantur tres
arearum, vel fi conftat ex quinq; nominibus &
dantur quatuor arearum, & fie deinceps : dabun-
tur areoe omnes c\nx addendo vel fubducendo nume-
rum n indici 6 vel unitatem indici x generari poffunt.
Et par eft ratio Curvarum ubi ordinatae ex binomiis
conflantur, & area una earum quos non font geome-
trice quadrabiles datur. Q. E. O.

PROP. VIII. THEOR. VL

Si pro e4-fz"-l-gz^«'|-&c. & k + Iz" -l-mz^-|-&e.
fcribantur R & S ut fopra,& in Curvoe alicujus Or-
dinata z®+"'' R^l^ S^^l ** maneant quantitates datae e,
M, A, F, e, f, g, k, 1, m, &c. & pro ^^^ t, & v^ fcri-
bantur fucceffive numeri quicunq; integri : & fi
dentur area? duarum ex curvis quae per ordinatas
fie prodeuntes defignantur fi quantitates R.& S font
binomia, vel fi dentur arese trium ex. curvis fi R
& S conjunftim ex quinq; nominibus conftant, vel
areae quatuor ex curvis fi R &S conjundim ex fex
nominibus conftant, & fie deinceps in infinitum :
dico quod dabuntur areae curvarum omnium.

Demonftratur ad modum Propofitionis fiiperioris.

PROP.

[i89]

PROP. IX. THE OR. VIL

^quantur Curvarum arese inter fe quarum Or-
dinate funt reciproce ut fluxiones Abfciflarum.^

Nam contenta fub Ordinatis & fluxionibus Ab-
fciflarum erunt asqualia^ & fluxiones arearum funt
ut hgec contenta.

COROL. L

Si aiTumatur relatio qusevis inter Abfciffas dua^
rum Curvarum, & inde per Prop. i. quseratur
relatio fluxionum Abfciflarum, & ponantur Ordi^
natae reciproce proportionates fluxionibus, inveniri
poflRmt innumerae Curvas quarum areas fibi mutuo
aequales erunt.

COROL. IL

Sic enim Curva omnis cujus haec eft Ordinata
z^' in e -y- fz^-^gz^" -[- &C.1'' affumendo quantitatem
quamvis pro r & ponendo ^--s Sc z^^x, migrat in
aliam fibi aequalem cujus ordinata eft =x— ^ in

^fx^_+gx^^4.&c;|^«

1 19^ ]
COR.OL^ III.

Et Curva omnis cu jiis Ordinata e ft z^'' in
a"^bz«~^Pci2^&c". X e-i-fz«'i-gz^« &c.Kaffumen-
do quantitatem quamvis pro »' & ponendo J.= s &
z^^x, migrat in aliam fibi oequalem cujus ordinata

eft V-^ in a+bx''+cx^''-t&c."xe4-fx''J-gx^^'i-&c.^

COROL. IV.

Et C urva omnis cu jus Ordinata eft z^" in
r+rbz""'!- c z"» + Sec . X e -(- fz« i- gz^" -i- Sec. j^
xk 4- Iz" -b mz^» -r Scc.^ affumendo quantitatem
quamvis pro v & ponendo l-=s & z^ = x, migrat in
aliam fibiaequalem cujus ordinata eft ^x'i=-" in a-p bxi^

COROL, V.

Et Curva omnis cujus Ordinata eft z^'^ in
eTuf^>ripp^irip&^j^ ponendo i= x migrat in

aliam fibi ^qualem cujus ordinata eft ^^ x^'V'^^''

»--»—,, . , n. —^ v/f J-.ex«h fiduo funt

'T"^-2« ^ ^c> id eft xH-^-l-«^^ .^ - ^1

nomina in vinculo radicis vel -^p^K x g+f^M'
fi tria funt nomina ; & fie deinceps.

ex^"l

CO-

[ 191 ]

COROL. VI.

Et Curva omnis cujus Ordinata eft z^"' in

e -\-{z»-\- gz^-P SccJ'^ X k -|- lz« -|- mz^"-)- &c.[f^

ponendo J — x migrat in aliam fibi aequalem cu^

jus ordinata eft ~^p, x e -|- f x-" ^|- gx"^" -|- *^c^|^

X '

xk+lx-« +mx--«| &c|^ id eft ^q,!.;.,-,,.;,,^ x f+exf
xl-i kx^i'^ fi bina funt nomina in vincuHs radicum,
vel x^T-i-F^iM^ii^ X g -j- fx" -|- ex^wp x l-|-kx«p fi tria

funt nomina in vinculo radicis prioris ac duo in
vinculo pofterioris : & fie in aliis. Et nota quod
areae duse aequales in noviffimis hifce duobus Co-
roUariis jacent ad contrarias partes ordinatarum.
Si area in alterutra curva adjacet abfcilTce , area
huic aequalis in altera curva adjacet abfciflb pro»
dud:a\

COROL. VIL

Si relatio inter Curvae alicujus Ordinatam y &
Abfciffam z definiatur per aequationem quamvis
feftam hujus formse^y « in e ~|- fyz^-j-gy^wz^^-f- hy3«z^^
+ &c. = z^ in k -i" ly"zJ^ -j- my ^^z"^ -|- &c. ha^c
figuraaffumendos-:2=i, x-^l^^' 8cx=^;^^^ migrat
m aliam fibi aequalem cujus AbfcifTa x/ex data

Ordinata

[^92]

Ordinata v, determinatur per a^quation em non

^i-mv'"-l-&c.r:=x.

COROL, VIII.

Si relatio inter Curvss alicujus Ordinatam y &
Abfciffam z definitur per aequationem quamvis
atfeitam hujiis format, y* in Q\--iyz^'-\-gy'^'^ z^^ -H&c
==z^ in k^^'\Yz^^my^''^'^\^. +z>' in p^-qy V
j.y2M22dr^j_ ^j^ggc figuj;^ afluniendo s- ^,x= '-z^^

M^^^'Sc ^ = ^— J, migrat in aliam fibi asqualem
cujus Abfciffa x ex data Ordinata v determinatur
per aequationem minus afFeftam v* in e -j- fyw-l-gv^"*

-\- &c. = s'^x'^ in k 4- lv« -\- mv^'' ^-^ &c. -Y s'x' in

CO ROL. IX.

Curva omnis cujus Ordinata^ eft tz^'' in

(a -1- b \qz' + fzH-H -1 - gz^t« ._|- &c.lf ^ fi fit 9 ^ 'x &

<r-t

affamantur x-ez" 4:^fz-l-« +gz''+^»^-&c.h ,
& -^ =^ in, migrat in aliam fibi sequalem cujus ordi-
nata eft x*^^ xa-j bx'^j *'. Et nota quod ordinata prior

[193].

in hoc CoroUario evadit fimplicior ponendo k
vel ponendo ^ = i & efficiendo ut radix dignitatis
extrahi poffit cujus index eft «, vel etiam ponendo
«»=— i&^ = i=T = '^—TTj ut alios cafus praete-
ream.

COR.OL. X.

Pro ez»' 4- fz'-i-^ -|- gz'-^-^» -|- &c. ^qz'-' :}:;;fz^^''*-'

T2 gz"'^'"^"'' -V ^^- '^ V ^^^ "i' ^^""^ '1'^ ^^- ^ "^^""^
,-^ inmz^"-' 4'^^' fcribantur R, r, S & s refpeftive, &
Curva omnis cujus ordinata eft TrSr -^ p Rs in R'^-' S<*"'
X aS^4-bR^ , ft fitiJiziii* = '^ = 5 i =r cr ^-'f = ^

& R*S9 = x, migrat in aliam fibi aequalem cujus or-
dinata eft x^ X a^pbPp. Et nota quod Ordinata
prior evadit fimplicior^ ponendo unitates pro t, v^
& ^ vel i*.^ & faciendo ut radix dignitatis extrahi
poffit cujus index eft «, vel ponendo « — r-i vel

PROP. X. PROB. IIL

Invenire figuras fimpliciffimas cum quibus Curva
qucEvis geometrice compari poteft, cujus ordinatim
applicata y per sequationem non affe6tam ex data ab-
fcifla z determinatur.

C c c

[194]

C A S. I.

Sit Ordinata az^-\ & area erit iaz\ ut ex Prop.V.
ponendob^o = c = d = f^g=h Sce^i, facile col-

ligitur.

C AS, 11.

Sit Ordinata az^-^ x^e^j-^fz^^- gz^f^'' -]- &c. & fi
curva cum figuris reftilineis geometrice comparari
poteft, quadrabitur per Prop. V. ponendo b = o-c
c=d. Sin minus convertetur in aliam curvam fibi

K'l

aqualem cujus Ordinata eft -„x^ x e-j-fx-j-gx^&c.
per Corol. a. Prop. IX. Deinde fi de dignitatum
indicibus ^ g^ ^-i per Prop. VIL rejiciantur uni-
tates donee dignitates ilk fiant quam minima, de-
venietur ad figuras fimpliciffimas qux hac ratione
colligi poffunt. Dein harum unaqua^q; per Corol. 5.
Prop. IX. dat aliam quae nonnunquam fimplicior
eft, Et ex his per Prop. III. & Corol. 9 & lo^
Prop. IX. inter le coUatis, figura: adhuc (impliciores
quandoq; prodeunt. Deniq; ex figuris fimplicif-
fimis ajGTumptis fafto regreffu computabitur area
qusefita.

CAS.

[195 3

C A S. III.

Sit Ordinata z^'' x a -^^ bz« -j- cz^« 4-- &c.

X e -\-' fz" -I" gz^" 4^ &c.l^"' , & haec figura fi quadrari
poteftj quadrabitur per Prop. V. Sin minus, di-
ftinguenda eft ordinata in partes z^-' x a x e -j- f z«

+ gz^" -1- 8cc.l^-^, z®-^ X bz« X e^- f z« +-g z^«-|-&c.H,
&c. & per Caf. 2. inveniendae funt figurae fimpli-
ciffimse cum quibus figurse partibus illis refpon-
dentes comparari poflunt. Nam areae figurarum
partibus illis refpondentium lub fignis fuis -j- & —
conjunfitae component aream totam qusefitam.

CAS. IV.

Sit Ordinata z^'^ x a -(- bz^J-Hcz^" +&c x
e-j-^ fz» -j- gz2« -|- &C.H X k -[- Iz» -^mz^«i-&c.|t^-f:
& fi Curva quadrari poreft^quadrabitur per Prop. Vt
Sin minus, convertetur in fimpliciorem per Corol.4.
Prop. IX. ac deinde comparabitur cum figuris fim-
pliciffimis per Prop. VIII. & Corol. 6, 9 & 10.
Prop. IX. ut tit in Cafu 2 & 5.

C A S, V,

Si Ordinata ex variis partibus conftat 5 partes
fingulse pro ordinatis curvarum totidem habendae
funt5& curvae illae quotquot quadrari poffunt^figilla-

Ccc a tim

tim quadrandae funt, earumq; ordinatae de ordinata
tota demendae. Dein Curva quam ordinatos pars
refidua defignat feorfim ( ut in Cafu i^ 3 & 4.5)
cum figuris iimpliciffimis comparanda eft cum qui-
bus comparari poteft. Et fumma arearum omnium
pro area Curvss propofite habenda eft<

COROL. I.

Mine etiam Curva .omnis cujus Ordinata eft ra-
dix quadratica affe£i:a oequationis fuoe, cum figuris
fimpliciffimis feu redilineis leu curvilineis com-
pari poteft. Nam radix ilia ex duabus partibus
lemper conftat quae feorfim fpectat^ non fant aequa-
num radices affeftas. Proponatur aequatio aayy
-]- zzyy — ^a'y -|-'z3y — z% & extra£la radix erit

_ a^ -\- z^^ aVa^-l-'^z^— z* cujus pars rationalis

aa - - zz

ar-Vz:? ^ . . -. aVa^ -1^ laz^^ - z''

aa-^-zz & pars irrationalis ^^r^ii ~ l^fit

ordinatas curvarum quae per banc Propofitionem
vel quadrari poffunt vel cum figuris fimpliciffimis
comparari cum quibus coUationem geometricam ad-
mittunt.

VOROL. IL

Et curva omnis cujus Ordinata per aequationem
quamvis affeftam definitur quae per Corol. 7. Prop.
IX. in aequationem non affeftam migrat, vel qua-

dratur

[ 197 ]

dratur per hanc Propofitionem li quadrari poteft vel
comparatur cum figuris limpliciffimis cum quibus
compari poteft. Et hac ratione Curva omnis quadra-
tur cujus aequatio eft trium terminorum. Nam squa-
tio ilia ft affedta fit tranfmutatur in non afFedam per
Corol.y. Prop.IX. ac delude per Corol. ^ & 5. Prop.
IX. in fimplicflimam migrando^ dat vel quadratu-
ram figurae fi quadrari poteft, vel curvam fimplicif-
fimam quacum comparatur.

COROL. III.

Et Curva omnis cujus Ordinata per isquationem
quamvis affedam definitur qucE per Corol. 8. Prop.
IX. in aequationem quadraticam aiFedam migrat;
vel quadratur per hanc Propofitionem & hujus Co-
rol. I . fi quadrari poteft, vel comparatur cum figu-
ris fimpliciflimis cum quibus coUationem geometric
camadmittit.

SCHOLIUM.

Uti quadrandas funt figurae; ad Regulas hafce
generales femper recurrere nimis moleftum effet :
prseftat Figuras quae fimpliciores funt & magis ufui
effe poflfunt femel quadrare & quadraturas in Ta-
bulam referre, deinde Tabulam confulere quoties
ejufmodi Curvam aliquam quadrare oportet. Hu-
jus autem generis funt Tabulae duae fequentes, in
quibus z denotat Abfciffam, y Ordinatam redan-

gulam

[198]

gulam & t Aream Curvse quadrandae, & d,, e, f, g,
g. h, " funt quantitates datae cum fignis fuis-|-'&— .

TABULA

Curvarumjimpliciorum qu^ quadrari pojfmt.

Curvarum formge. Curvarum areae.
Forma prima.
dz«-' = y, -z" = t.

Forma fecunda.
dz*^' dzn — d ^^

Forma tertia.

i.dz,iVe^|-'fz« = y. |;fR^^ = t, exiftente R^V^-j-^fz"

a. dz!?\/e-l-fz*'=^y. °~" i5«ff " dR^ —t.

, „ / — —-r — i6ee— .24efzM-|-3offz2M ,^

3 . dz?« Ve 'l-fz« = y . ^^^ — dR^ =. t.

1 ^« /"^Tr^ ^96e3-\-i44eefzn— i8oeffz2H-l-2iof323 ,),.^

4. dzljVe-i-tz*'==y. -^ ^:;^r- ^dR^^t.

Forma quarta.

dz»'i 2d

;^€^-fZtj

dz^«-

^. =y- :=l|^dR-t.

dz3«^

Ay^n-i J6ee-~8efz„-|-6ffz2„

(|2i4ri — 96e3-|-4Seefz>r-36effz2,rl-3of3Z3,)

TABULA

Curvarum Jhnpliciwum qua cum EUi^ S
Hyperbola compart pojfmt.

Sit jam aGD vel PGD vel GDS Seaio
Conica cujus area ad Quadraturam Curvse pro- ^^g* 5?<^s7>S.
pofitae requiritur, fitq; ejus centrum A, Axis K a,
Vertex a, Semiaxis conjugatus AP^ datum Abfciffe
principium A vel a vel a^ AbfcilTa AB vel a B vel
aB^^x^ Ordinata reftangula BD = v, & Area
A B DP vel aBDG vel aBDG = s, exiftente ^G Or-
dinata ad punftum «. Jungantur KD, AD, aD. Du-
catur Tangens DT occurrens Abfciflfe AB in T,
& compleatur parallelogrammum ABDO. Et
fiquando ad quadraturam Curvge propofitse requi-
runtur arese duarum Sedionem Conicarum, dica-
tur pofterioris AbfcilTa?, Ordinata T^ & Area o-.
Sit autem ^ differentia duarum quantitatum ubi in-
certum eft utrum pofterior de priori an prior de po-
fteriori fubduci debeat,

Curva-

[200]

4->

• »H

C/5

• ■— I

1 *

c
o

• •-»

S— (

-II

-a

C/i

1. L^

G
O

-1«

«J 1'+-.

C4 I

>
11

X
X

r "

•T3 ri

U l<4-l

^.^
>

v>|rr

€%

Q
H
-I-
pq
Q

H
Q
O

r-1

-1

• .-H

4-J

111

II I

I 11 I c<

"«

':i-|

s -^

II II

11 *^

>. u

4-i

'^ a->

T^ >

tJ |N

1— t

®4»

csl

Q
<!

Q
O

> -^[^

X
X

C '201 ]

0—4

r-a I

C/)

>►

"d cs 4— "^ I f^"i~

tSJ

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4->
11

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11 I cs

X
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Q
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^l^r

if)

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O
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II
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II II

111

r,X
|X

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w 1 »

Ddd

ISl

T*"*4

-r

«j

i>".

*4-

m
Q

<
•I-

ill

1+=

■^ l« '-'In'

isi
rn

a
N

I
1

" 1 "

[202]

f' 11

N 'I

lU ,N

I- bO

bJ3

• — (
Jd

• — (
»-(

I I,

bO'

Lr"^

bJD

JkXJ>

rv^s^^

11 11

3CVJ>

M ID

£S

_l_

-^ri

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^^^r

«*-j

bJ3

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X

JvVr

,'VX/

11 11
11

Q
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d c/) _4-.

-cl

(/i

fX
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WD

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)• I

IS]

> >

II
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bD

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pq
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bJ3

W3l

00 '

6J3

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X*
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:^ I

•tf bO

f X

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tS3
11

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bflu-ii
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bJ3

[204]

a
o

4~»

-T?

> IX

M( en "^
>

1^ 1

»,

^i^

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WD ^'T

-d

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11

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B
o

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rri

[205]

In Tabulis hifcc, feries Curvarum cujufq; formae,
utrinq; in infinitum continuari poteft. Scilicet
in Tabula prima, in numeratoribus arearum for-
mx tertise & quarts, numeri coefficientes initialium
terminorum (1, — +«,i6, — ^96, 868^&c.) generan-
tuT multiplicando numeros^ — 2, — ^^ — 6, —10 &c«
in fe continuo, & fubfequentium terminorum coef-
ficientes ex initialibus derivantur multiplicando
ipfos gradatim, in Forma quidem tertia , per — '-
—h—ly—h —To &<^'- in quarta vero per — I, _i''
—h — h — h &c. Et Denominatorum coefficientes
5, 15, 105, &c. prodeunt multiplicando numeros
I5 5, 5, 7, 9, &c. infe continuo.

In fecunda vero Tabula, feries Curvarum fornix
prim^, fecundae, quintae, fextae, nonae & decim^ ope
folius divifionis, & formae reliqus ope Propofitio-
nis tertiiB & quartae, utrinq; producuntur in in-
finitum.

Quinetiam has feries mutando fignum numeri ^^
variari folent. Sic enim, e. g. Curva |\/e+fz«~ y
evadit -r--- yf-V-ezw,

P R O P. IX. T H E O R. VIIL

fSit A Die Curva quavis Abfciflam habens
AB=z & Ordinatam BD=y, & fitAEKC Curva
alia cujus Ordinata BE squalls eft prions ares

ABC

?•

[20(J]

ADB ad unitatem applicat-3e^ &AFLC Curva
tertia cujus Ordinata BF aequalis eft fecundae areae
A E B ad unitatem applicatae, & A G M C Curva
quarta cujus Ordinata B G aequalis eft tertiae areae
A F B ad unitatem applicatae , & A H N C Curva
quinta cujus Ordinata BH aequalis eft quartae areae
AGB ad unitatem applicatoe, & fie deinceps in
infinitum. Et funto A, B, C, D, E, &c. Areae Cur-
varum Ordinatas habentium, y, zy, z'y, z^y, z^'y,
& Abfciffam communem z.

Detur AbfcilTa quaevis AC=t, fitq; BC=t— z
= x, & funto P, Q, R, S, T areae Curvarum Ordi-
natas habentium x, xy, xxy, x'y, x^y & Abfciffam
communem x.

Terminenter autem hae areae omnes ad Abfciffam
totam da tarn A C, nee non ad Ordinatam pofitione
datam & infinite produ£!:am C I : & erit arearum
fub initio pofitarum prima ADIC=A=P5 fecunda
AEKC=tA-B=Q.Tertia AFLC-^-^^=f5±c ^,j^^

Quarta AG MC :^ 5 . 3^-3 Wc-d _,^^ q^j^^^^

A H N C rrtrr l4A>-4t:!B4-6ttC--4tD ^E ___ I ^

24 24 1 ,

CO-

[ 207 J

COROL.

Unde fi Curvs3 quarum Ordinata- funt y, zy,
z*y, z'y, &c. vel y, xy, x'y, x^y, &c. quadrari
poffunt^ quadrabuntur etiam Curva^ ADIC^ AEKC,
AFLC, AGMC, &c. & habebuntur Ordinatae BE,
BF^ BG, BH areis Cur varum proportionales.

Quantitatum fluentium fluxiones efle primas ^
fecundas, tertias, quartas , aliafq; diximus fupra.
Hae tluxiones funt ut termini ferierum infinita=
rum convergentium. Uc fi z" fit quantitas fluens &
fluendo evadat z-l-o]", deinde refolvatur in feriem
convergentem z^'-j-MOZ^-^-i-^ooz «-'-f- "^ - ^'^^ "^ ' Vzr^

\^ &c. terminus primus hujus feriei z" erit quan-
titas ilia fiuens, fecundus «oz""' erit ejus increment
tum primum feu differentia prima cui nafcenti pro-
portionalis eft ejus fluxio prima ^ tertius —^ oz«'^
erit ejus incrementum fecundum feu differentia fe-
cunda cui nafcenti proportionalis eft ejus fluxio
fecunda, quartus "3-3wi-4- ^" qSzh-? erit ejus increment
tum tertium feu differentia tertia cui nafcenti
fluxio tertia proportionalis eft, & fie deinceps in
infinitum.

t T? - -

Jbxponi

[ 2o8 ]

Exponi autem poffuiit hoefluxiones per Curvarurn
Ordinatas BD, BE, BF, BG, BH, &c. Ut fi
Ordinata BE (=^) fit quantitas fluens, erit
ejus fluxio prima ut ordinata B D. Si B F (=M?^

fit quantitas fluens, erit ejus fluxio prima ut Or-
dinata BE & fluxio lecunda ut Ordinata BD. Si
BH ("=—2) fit quantitas fluens, erunt ejus fluxio-
nes, prima, fecunda, tertia & quarta, ut Ordinata^
BG, BF, BE, BDreipeaive.

Et hinc in aequationibus quae quantitates tantum
duas incognitas involvunt, quarum una eft quan-
titas uniformiter fluens & altera eft fluxio qugelibet
quantitatis alterius fluentis , inveniri poteft fluens
iila altera per quadraturam Curvarurn. Exponatur
enim fluxio ejus per Ordinatam B D, & fi hoec fit
fluxio prima, qua^atur area ADB=BExi, ii
fluxio lecunda, quaeratur area AEB:=BFxi^ fi
fluxio tertia, quaeratur area AFB^^BGx i, &c.=
& area inventa erit exponens fluentis quaefit^.

Sed & in aequationibus qu^ fluentem & ejus
fluxionem primam fine altera fluente , vel duas
qufdem fluentis fiuxiones, primam & fecundam,
vel fecundam & tertiam, vel tertiam & quartam,
&c, fine aiterutra fluente involvunt : inveniri pof-
funt fluentes per quadraturam Curvarum. Sit

tsquatio aav = av -U vv , exiftente v :== B E ,

v =BD, z=:==AB & Zr=i, & aequatio ilia com-

plendo dimenfiones fluxionum, evadet aav == avz

-|- vvz, feu ^fqr^ ~z. Jam fluat v uniformiter &

[209]

fit ejus fluxio v=i & erit £^=z, & quadrando
Curvam cuius Ordinata eft ^^^ & Abfclffa v, ha-
bebitur ttuens z. Adhaec fit aequatio aav=av^-vY
exiftente v=BF, v=:BE, 'v=BD & z=AB &
per relationem inter V & v feu BD & BE invenie-
tur relatio inter A B & B E ut in exemplo fuperiore.
Deinde per banc relationem invenietur relatio in-
ter AB & BF quadrando Cur\ram AEB.

^quationes quse tres incognitas quantitates invol-
vunt aliquando reduci polTunt ad sequationes quae
duas tantum involvunt, & in his cafibus fluentes
invenientur ex fluxionibus ut fupra. Sit sequatio
a — ^bx^=cxyMy -(-dy^^'yy. Ponatur y«y=v & erit
a — bx^"cxV']-dvv. Ha^c sequatio quadrando Cur-
vam cujus Abfciffa eft x & Ordinata v dat aream
v^ & oequatio altera y"y=^v regrediendo ad fluentes
dat 4-7*^^"^ =v. Uiide habetur fluens y .

Quinetiam in irquationibus quae tres incognitas
involvunt & ad aequationes quse duas tantum in-
volvunt reduci non poffunt, fluentes quandoq;
prodeunt per quadraturam Curvarum. Sit ^quatio
a x'^+ b x^p = r e x^-' y ' -]- s e x^ y y'-' — f y y^^ exiftente

♦ «

X = I . Et pars pofterior r e x^'^ y ^ -'l- s e x^ y y ^"^ — f y y ^5
regrediendo ad fluentes, fit exry' — ^JL.yM-r q^^^

proinde eft ut area Curvas cujus Abfciffa eft x &
Ordinata ax"' i-bx')^ & inde datur fluens y.

E e e Sit

[2I0]

Sit aquatio X X aJr + bx^P := JntL. Et fluens 1

cujus fluxio eft X X ax^^i-bx*'^ erit rut area Curvag
cujus Abfcifla eft x & Ordinata eft a x^^ .4- bx«l^.
Item fluens cuius fluxio eft -ip^ erit ut area Curv^ ^

cuius Abfcifla eft y & Ordinata -^J^^ id eft

(per Cafum i. Formae quartae Tab. I.) ut area
^^fy%^y«r Pone ergo ""^Ve-^fy" aequalem areae

Curvae cujus Abfcifla eft x & Ordinata ax^^^j^ bx^l^?
& habebitur fluens y.

Etnota quod fluens omnis quae exfluxione prim^
coUigitur augeri poteft vel minui quantitate quavis
son fluente. (lux ex fluxione fecunda colligitur
augeri poteft vel minui quantitate quavis cujus
fluxio fecunda nulla eft. Quae ex fluxione tertia
(^oUigitur augeri poteft vel minui quantitate quavis
cujus fluxio tertia nulla eft. Et fie deinceps in in-
finitum,

Poftquam vero fluentes ex fluxlonibus coUedt^.
font fi de veritate Conclufionis dubitatur, fluxio-
nes fluentium inventarum viciflim colligend^ funt
& cumfluxionibus fub initio propofitis comparandae.
Nam fi prodeunt aequales Conclufio rede fe Ha-

bet:

[211 J
bet: fin minus, corrigenda^ funt fluentes fie, ut
earum fluxiones fluxionibus fub Initio propofitis
aequentur. Nam & Fluens pro lubitu affumi po-
teft & affumptio corrlgi ponendo fluxionem flu-
entis alTumptae oequalem fluxioni propofitas, & ter-
minos homologos inter fe comparando.

Et his principiis via ad majora fternltur.

F I N I

ERR JT J.
BOOK I. OfOptlch.

PArt I. p. 3. 1.20. Tropertiesrohich, ib.p.5. 1.5. and that C, p. 6. I.9. DE^ p.2i. I.23.
arj twoJ{ajs, p.27. 1.6. f;? t/^e M^irgmput Fig.14 ^ i^y.-p.-^oA.j.MNyl.g. M, p.
44. 1.15. a/srvaspropofed, p.52. 1. 17. ^ i'-^i"?^ C/Vde, p.57. l.ult. emerging, t;).6o. 1.25.'
£-o«toWTO/t/» tky p.64. 1.18. ^wrf i4t^> p.65. i.13. ^rrk, p.66. 1.3.>S'e»x/c/?T«/^r, p.67.
1.25.Cew?e?-, 1.31. 4I /;?ck^, p.68. 1.8. to 16, I.9. or ^|, p.71.1.1. ^i/eff, p.72.1.13.
/^&, 1.20. i'ew^. Part II. p.S6. 1.5. klopipede, p.89. I.9. wi^^e ^j/, p.93. 1.18. to 77^^
i.28,29, by the third Jxiom of the firft Pan of this Book, the Lawsy p. 105. l.'^.fee repre-
fented, p. 144. 1. 24, i, j-^^, /„, f, A, f^g, ^. P- "S, "P- ^r X/^.i. if^.2. write
-n. ., n....^ i_ j„j.Vo -r. Tor^ 1 To +/i the /ivinifl f^ i'^ 2, 1.6. bj the bright'

BOOK. II.

P.5. 1. i^.vicely t/;e,p. 7. 1.9.^? ^ detjote, I.28. tkw divers,-p. 10. 1. 24. 1000 w 1024,
p.ii.Lii. oli^uitiesyj. p.17. 1.4^ Hf ^9, p. 25. 1. 11. i of^ p.31. 1. 12. wiore cor/i-
pounded, p.55. [.-^.fi^esrefleB:, I.24. (?wi therefore their Colours arife, p.65. 1.5. cor/?w_/:.
c/a- a?/, p.71. 1.17. give?t breadth, p.84. 1. 4. are to thofe, p. 96. 1. 24. Ohfervation of
thU Part of this Booh, p.103. 1.17. wa/s to the thicHefs, p. 105. 1. 19. of this r?hite E^/jg,
p. 1 07. 1.20. become equal to the thiM oj thofe.

£mimeratio Line arum.

•^.\\l,\.^Q. ditas fig;MsfiiiSy ^,\4\.\.2l.refpiciunt, p. 146. l.'j.funtJfymptoto, p,
154. 1.13. cx-\-d dat Ordinatmy =5 , i.i4» q^u^ generatur.

Ouadratura Cwvarum..
p.i68.1.24.re5.z^5,p.i76.1.ult. ^^ ^^^ fz.«,p.i83.1.i3.^,&,c,SJ(r. e,/;^,^^. fc>^ w»

■^c, p. 18^. 1.4. i;z z&-i» P'lS^- l-H- zB±m p. 190. 1. 19- 'y^^ ^9-ri- F^.
^. 192. 1. iS. a2,pH-2fj. p.193.1.11. aSy-l'bRr;*^'

•4|:

PH^ 2 It.

Qiiiuin Ta/\L

JTJ

A
G

K A T ^^ B N T A cr B N A

B C

;t- (0- V7