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Diamond Design Marcel Tolkowsky
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TS 753.T6
^Opy
1
DIAMOND DESIGN
DIAMOND DESIGNA STUDY OF THE REFLECTION AND REFRACTION OF LIGHT IN A DIAMOND
BY
MARCEL TOLKOWSKYB.Sc, 'A.C.G.I.
WITH
S7
ILLUSTRATIONS
E.
&
F. N.
SPON,
Xon^on: Ltd., 57 HAYMARKET,IWewll)orft:120
S.W.
i
SPON & CHAMBERLAIN,
LIBERTY STREET
I919
-X?)15.\
s
4di(.^'^^'J.I
3^4.
CONTENTSINTRODUCTION. . . .
DIAMOND DESIGNINTRODUCTIONThis bookis
written principally for students
and jewellers, and more particularly for diamond manufacturers and diamond cutters and polishers. Theof precious stones
author
will
follow
the
evolution
of
the
shape given to a cut diamond, and discussthe values of the various shapes and the
reason for the discarding of the old shapes
and theIt is a
practically^ universal
adoption of
the brilliant.
remarkable fact that, although the
art of cutting a
diamond has been known
for
more than two thousand years, it is entirely empirical, and that, though many keen contemporary minds have been directed upon the diamond, and the list of books written on that subject increases rapidly, yet
6
DIAMOND DESIGN
nowhere] can one find any mathematical
work determining the best shape for that gem. The present volume's chief aim isthe calculation of that shape.
The
calculations
have been
made
as
simple as possible, so as not to be beyondthe range of readers with a knowledge of
elementary geometry, algebra, and trigo-
was found that the accuracy of the results would be impaired without the introduction of more advanced mathematics, these have been used, and graphical methods have beennometry.
Where, however,
it
explained as an alternative.
Theof
results of the calculations for the
form
brilliant
now
in
use were verified by
actual mensuration from well-cut brilliants.
The measuresat the
of these brilliants are given
end
of the
volume both
in a tabulated
and in a graphical form.
It will
be seen
how
strikingly near the actual measures are
to the calculated ones.
The method usedwill
in the present
worksemi-
be found very useful for the designother transparent precious and
of
INTRODUCTIONprecious stones, althoughit
7
will
be found
advisable in the case of stones of an agreeable
colour
to
cut
the
gem somewhat
thicker than the calculations warrant, soas to take full advantage of the colour.
The same remark applies to diamonds of some exceptional and beautiful colour, likeblue or pink, where the beauty or the valueof the stone increases with the depth ofits colour.
Part
I
HISTORICALand early Indian literature we turn when we want to find the origin of diamond cutting, for IndiaItis
to Indian manuscripts
has always been regarded as the natural
and ancient home1728,
of thefirst
diamond.found:
It is
there that they were
upof
to
the
date of the discoverydeposits,
the
Brazilian
practically
the
whole
world'ssources.
supply was derived from Indian
They
are
and beds of separated from the matrixvalleys
found there in the streams, and also,
were formed, in strata of that have since been covered by twelveto sixteen feet of earth tionof
which they detrital matterin
later
centuries.
by the accumulaDiamonds haveknowledge
existed in these deposits within the reachof
man
for
many
ages, but the
HISTORICALof the
9
diamond
as a
gem
or as a crystal-
with
exceptional
qualities
does
not
go
back in India to the unfathomable antiquity fo which books on diamonds generally refer. It was wholly unknown in the Vedic Ji^f: period, from which no specific names for precious stones are handed down at all.^
Thethird
earliest
systematic reference appears
to be in the Arthagastra of Kautilya (about
where the author mentions six kinds of diamonds classified according to their mines, and describes them as differing in lustre and hardness. He also writes that the best diamondscenturyB.C.),
should be large, regular, heavy, capable ofbearing^
blows, ^Laufer,
able
to
scratcha
metal,Study in
Berthold
The Diamond
:
Chinese and Hellenistic Folklore (Chicago, 1915). ^ This legend of the indestructibihty of the diamond,
which reappears in many other places, and to which the test of the diamond's capacity of bearing the strongest blows was due, has caused the destruction of perhaps a very large number of fine stones, -^he legend was further embroidered by the remark that if the diamond had previously been placed in the fresh and still warm blood of a ram, it could then be broken, but with great difficulty. This legend was
10
DIAMOND DESIGNandbrilliant.
'
refractive
In the Milinda-
panha (Questions
King Milinda) (about first century B.C.) we read that the diamond ought to be pure throughout, and that it is mounted together with the most costly gems. This is the first manuscript in which the diamond is classed as a gem.ofItis
therefore permissible
to
estimate
with a sufficient degree of accuracy thatthe diamond becamethe
knownits
in India during
BuddhistB.C.,
period,
about
the
fourth
century
and that
use as a
gemwhen
dates from that period.Itis
not
known with
certainty
and where the art of grinding or polishing diamonds originated. There is as 3^et no source of ancient Indian literature in which the polishing of diamonds is distinctly set forth, although the fact that diamond is used for grinding gems generally is menstill
thirteenth
current in Europe as late as the middle of the century. The actual fact is that the
diamond, although exceedingly hard (it is the hardest substance known), can easily be split by a light blow along a plane of crystallisation.^
Laufer,
loc. cit.
HISTORICALtioned.Itis,
II'
however, likely that, where
the pohshing of other precious stones was accomplished in that manner, that of
diamonds themselves cannot have been What pohshing there entirel}^ unknown. was must at first have been limited tothe smoothing of the faces of the crystals The first description as they were found.of
diamonds is given by Tavernier,i a French jeweller who travelled through India, and to whom we owe most of our knowledge of diamond cutting in India in the seventeenth century. At the time ofcuthis visit (1665) the Indians
were pohshing
over the natural faces of the crystal, and preferred, therefore, regularly crystaUised
gems.
had
of
knowledge they grinding diamonds to remove faulty
They
also used the
places hke spots, grains, or glesses.fault
If
the
was too deep, they attempted to hide it by covering the surface under which it lay with a great number of small facets. It appears from Tavernier's writings that1
Tavernier,
Voyage en
Turqiiie,
en Perse
et
mix
Indes (1679).
12
DIAMOND DESIGNwere
there
India at
European polishers in that time, and that it was to themalso
the larger stones were given for cutting.
Whether they had
learnt
the
art
inde-
pendently or from Indians and attainedgreater proficiency than they, or whether
they were acting as instructors and teachingthe Indians auncertain.in
new
or a forgotten art, is
Both views are equally likely the present state of research upon that:
subject
at the time of Tavernier's visit,in
diamond cutting had been known for more than two centuries.
Europe
Among
the several remarkable gems that
Tavernier describes,the most noteworthyis
the one
known
as
the
Great
Mogul.
This diamond was ofa weight of 280cts.
andFig.
was
cutf^g.
asi.
sketched in
Theto
polishing
was
the work of a Venetian, Hortensio Borgis,
whom
it
was given
for that purpose
by
HISTORICALits
13
owner, the Great Mogul Aurung Zeb, of
Delhi.of
This kind of cutof
is
characteristic
most
the
largestones,
Tn d i a nsuch as
theOrlow(fig.2),
which is now thelargestof
diamondRussianFig.2.
the
crown jewels and weighs 193! cts. The Koh-iNoor (fig. 3), now among the British crown jewels, was of a somewhat similar shapebefore recutting.It
weighed thencts.
186
Ta vernier alsomentions severalothercutFig.3.
types
of
whichin
he
met(fig.
India.
The Great Table
1642, weighed 242 cts.
which he saw in Both the Great Table4),
and the Great Mogul seem to have disappeared it is not known what has become of them since the seventeenth century.:
14
DIAMOND DESIGNVarious other shapes are described, such
as point stones, thick stones, table stones(fig. 5),
etc.
But theall
chief characteristic
remains
:
these
Fig.
4.
diamonds have been cut with one aim constantly in view ^how
to
pohsh
the
stone
with
the
smallest
possible loss of weight.
As a consequence
the polishing was generally accomplished
by covering thefacets,
surface of the
stone with a large
number
\
/
of
of
and the original shape the rough gem was, as far as
possible, left unaltered.
was mentioned before that the art of diamond polishing had already been known in Europe for several centuries when Tavernier We have as yet no cerleft for India. tain source of information about diamond cutting in Europe before the fourteenth century. The first reference thereto mentions that diamond polishers were workIt
HISTORICALingill
15
Nurnberg (Germany) in 1375, where they formed a guild of free artisans, to which admission was only granted after an apprenticeship of five to six years. We do not know, however, in what shape and by what method the stones were cut.
^
It is in the fifteenth
century that Euro-
pean diamond cutting begins to become
moreis
definite,
more
characteristic.
And
it
from that time that both on its technical and artistic sides progress is made at a rate, slow at first, but increasing rapidlylater.
It is
not
difficult to find
the chief reason
for that change.
UpTo
to that time, diamonds had almost
exclusively been used
by
princes or priests.
power and wealth ^in those days diamonds were credited with extraordinary powers they were supposed to protect the wearer and to bring him luck. Princes also found them convenient, as they have great value
princes they were an
emblem
of
:
^
Jacobson's Technologisches Worterbuch (1781).
i6
DIAMOND DESIGNand couldof
for a very small weight,
easily
be carried in case of
flight.
Priests used
themset
in;
the
ornamentsthe heads
templesstatues
or
churchesas
they have not infrequently beenof
eyes in
of
Buddha.
became the fashion for women to wear diamonds as jewels. This fashion was started by AgnesIn the fifteenth centuryit
Sorel (about 1450) at the Court of Charles
VII
of France,all
and gradually spread from
there to
the Courts of Europe.
This resulted in a very greatly increased
demand, and gave a strong impulse to the development of diamond polishing. The production increased, more men applied their brains to the problems that arose, and, as they solved them and the result of their work grew better, the increasingattractivenessof
thea
thethe
demand and gaveart.
gem increased new impulse to
At the beginning of the fifteenth century a clever diamond cutter named Hermannestablished a factory in Paris, where his
HISTORICALwork met withsuccess,
'
17
and where the
industry started developing.
In or about 1476 Lodewyk (Louis) van Berquem, a Flemish polisher of Bruges,
introduced absolute symmetry in the disposition of the facets,
and probably
also
improved the process of polishing. Early authors gave credence to the statement of one of his descendants, Robert van Berquem, 1 who claims that his ancestor had invented the process of polishing the diamond by its own powder. He adds '' having ground off redundant After material from a stone by rubbing it against:
another one (the processpractice as'
known
in
modern
bruting
'
or cutting), he col-
lected the
powder produced, by means of which he polished the diamond on a mill and certain iron wheels of his invention.''^
Robert de Berquem,:
Les merveilles des Indes(Paris,in-40,
Traite des " Louis de p. 12
pierres precieuses
1669),
Berquem Tun de mes ayeuls a
premier Tinvention en mil quatre cent soixante-seize de les tailler avec la poudre de diamant meme. Auparavant on fut contraint de les mettre en oeuvre tels qu'on les rencontrait aux Indes, c'est2trouvele
i8
DIAMOND DESIGNAs has already been shown, we know now
diamonds were polished at least a century before Lodewyk van Berquem lived. And as diamond is the hardest substance known, it can only be polished by its own powder. Van Berquem cannot thus havethat
invented that part of the process.
He maycast-iron
perhaps have introduced some important
improvement
like
the
use
of
polishing wheels, or possibly have discovered
a more porous kind of cast iron
one
on
which the diamond powder finds a better hold, and on which polishing is thereforecorrespondingly speedier.a-dire tout a fait bruts, sans ordre et sans^ grace, sinon quelques faces au hasard, irregulieres et mal II mit polies, tels enfin que la nature les produit. deux diamants sur le ciment et apres les avoir egrises Tun contre I'autre, il vit manifestement que par Ic moyen de la poudre qui en tombait et a Faide du moulin et certaines roues de fer qu'il avait inventees, il pourrait venir a bout de les polir parfaitement, meme de les tailler en telle maniere qu'il voudrait.
Charles devenu due de Bourgogne lui mit trois grands diamants pour les tailler avantageusement selon son adresse. II les tailla aussitot, I'un epais, Tautre faible et le troisieme en triangle et il y reussit si bien que le due, ravi d'une invention si surprenante, lui donna 3000 ducats de recompense."
HISTORICAL
19
What Van Berquem probablyginateis,
did
ori-
as already stated, rigid
in the design of the cut stone.
symmetry The intro-
duction of the shapeorbrioletteis
known
as pendeloque
generally ascribed to him.thfe
The Sancy and
Florentine, which are
both cut in this shape, have been said
Fig.
6.
by some to have been polished by him. The Sancy (53 J cts.) belongs now to the Maharaja of Guttiola, and the Florentine (fig. 6), which is much larger (133 1 cts.), is at present among the Austrian crown jewels. The history of both these gems is, however, very involved, and they may have been confused at some period or other with similar stones. That is whyit
is
not at
all
certain
that
the}^
were
20
DIAMOND DESIGNVan Berquem.At anyrate,
the work of
they are typical ofintroduced.
the kind of
cut he
The pendeloque shape did not meet with any very wide success. It was adopted in the case of a few large stones, but was gradually abandoned, and is not used to any large extent nowadays, and then in a modified form, and only when the shapeof the
rough stone
is
especialh^ suitable.
This unpopularity was largely due to thefact that,
although the loss of weight in
cutting was fairly high, the play of light
within the stone did not produce sufficientfire
or brilliancy.
About the middle of the sixteenth century a new form of cut diamond was introduced. It is known as the rose or rosette, and was made in various designs and proportions (figs. 7 and 8). The rose spread rapidly and was in high vogue for about a century, as it gave a more pleasant effect than the pendeloque, and could be cut with a much smaller loss of weight. It was also foundvery advantageous in the polishing offlat
HISTORICALpieces of rough
21
or split diamond.'
Such
material
is
even
now
frequently cut into
roses, chiefly in the smaller sizes.
Fig.
7.
In
the
chapter
diamonds it will to be made thick (somewhat thicker thaninfig.
upon the design of be shown that roses have
7) for
the loss of light to be small,flatter
and that theof light.
the
rose the bigger the lossIt will also
befire
seen there that theof
a
rose
cannot
be
very high.
These faults
caused the rose to be
superseded by thebrilliant.f^^
Weto
owe the introduction
of the brilliant
in the middle of the seventeenth centurv
Cardinal Mazarin
or
at
any rate to
his influence.
As a matter of fact, the first brilliants were known as Mazarins, and were
22
DIAMOND DESIGNfig.
of the design offacets,
9.
They had
sixteen
excluding the table, on the upper side.
Fig.
9,
They are called double-cut brilliants. VincentPeruzzi, a Venetian polisher, increased the
number
of facets
from sixteen to thirty-
FiG. 10.
two
(fig.
10) (triple-cut brilliants), thereby
increasing veryof the cut
much
the
fire
and
brilliancy
gem, which were already in the double-cut brilliant incomparably better
HISTORICALthan in therose.
23
Yet diamonds of that cut, when seen nowadays, seem exceedingly dull compared to modern-cut ones. This dullness is due to their too great thickness, and to a great extent also to the difference in angle between the corner facets and theside facets, so that evenif
the
first
were
Fig. II
polished to the correct angle (which they
were not) the second would be cut too steeply and give an effect of thickness.Old-cut brilliants, as the triple-cut brilliantsare generally called, were atfirst
modified
by making the size and angle of the facets more uniform (fig. 11), this bringing about a somewhat rounder stone. With the introduction of mechanical bruting or cutting(an operation distinct
from polishing
;
see
24p.
DIAMOND DESIGN17)
diamonds werein
circular
plan
(fig.
made absolutely The gradual 37).less
shrinking-in of the corners of an old-cutbrilliant
necessitated
a
thicklyfire
cut
stone with a consequent increasinglife,
and
until a point of
maximumis
brilliancy
was reached.brilliant.^
This
the
present-day
Other designs for thetried,
brilliant
have beenloss
mostly attempts to decrease the
of weight in cutting without impairing the
brilliancy of the
diamond, but they haveherethat
not met with success.
We may
note
the
general
trend of European diamond polishing as
opposed to Indianfor^
is
the constant search
greater brilliancy,
more
life,
a more
Some American
writers claim that this change
from the thick cut to that of maximum brilliancy was made by an American cutter, Henry D. Morse. It was, however, as explained, necessitated by the absolute roundness of the new cut. Mr Morse may haveinvented it independently in America. But it is highly probable that it originated where practically all the world's diamonds were polished, in Amsterdam or Antwerp, where also mechanical bruting was firstintroduced.
HISTORICALvividfire
25
in
the diamond,
regardless
of
The weight of diamond removed by bruting and by polishing amounts even in the most favourable casesthe loss of weight.to 52 per cent, of the original rough weightfor a perfectly cut brilliant.
In the next
chapters the best proportions for a brilliantwill
be calculated without reference to theit
shape of a rough diamond, andseen
will
be
howthe
startlingly
near the calculatedwell-cutbrilliantis
values
modern
polished.
Part II
OPTICALItis
to light, the play of Ught, its reflectionits refraction,itsfire,
and
thatits
a
gem owes
its
brilliancy,
colour.
We
have
therefore to study these optical properties in order to be able to apply them to the
the calproblem we have now before us culation of the shape and proportions of a perfectly cut diamond.:
Of the total amount
of light that fallsis
upon aflected;
material,
part
returned or reit,
the remainder penetrates intoit
andfirst
crosses
part of
The the Ught produces what isor is absorbed
by
it.
termedif
the
''lustre"is
of
the
material.
The second partthe materialis
completely absorbedIfit
black.will
is
partly
absorbed the material26
appear coloured,
OPTICALandif
27it
transmitted unaltered
will
appear
colourless.
The diamonds used as gems are generallycolourless or only faintly coloured;
it
can
be taken thatof the
all
the light that passes into
the stones passes out again.
The
lustre
diamondcalled
is
peculiar to that gem,
adamantine for that reason. It is not found in any other gem, although zircon and demantoid or olivine have ais
and
lustre approaching
somewhat
to the ada-
mantine.In
gem
stones of the
same kind and
ofit
the same grade of polish,of the
we may take
that the lustre only varies with the area
gem
stone exposed to the light, and
that
it is
independent of the type of cut
or of the proportions given to theso far as theyis
gem;
(in
do not affect the area) why gems where the amount ofis
thislight
thatis
reflected
upon
striking the surfaceof the light thatis
great, or
where much
penetrates into the stone
absorbed and
does not pass out again, are frequently cut in such shapes as the cabochon (fig. 12),
28
DIAMOND DESIGNin that
so as to get as large an area as possible,
and
way
take
full
advantage of
the lustre.
In a diamond, the amount of light reflected
from the surfacea
is
much;
smaller
that that penetrating into the stoneover,
more-
diamond
is
practically
perfectly
transparent, so that
all
the light that passes
into the stone has to pass out again.
This
Fig. 12.
is
why
lustre
may
be ignored in the working
out of the' correct shape for a diamond,
and why any variation in the amount of light reflected from the exposed surface due to a change in that surface may beconsidered as negligible in the calculations.
The
brilliancy
or,
as
it
is
sometimesa
termed, thelight in the
''fire''
or the
''life'' of
gem
thus depends entirely upon the play of
gem, upon the path of raysgem.If
of light in the
a
gem
is
so cut or
designed that every ray. of light passing
OPTICALintoit
29
follows the best path possible for
producing pleasing effects upon the eye,then the
gem
is
perfectly cut.
The whole
art of the lapidary consists in proportion-
ing his stone and disposing his facets soas to ensure this result.Ifits
we want
to design a
gem
or to calculateit is
best shape and proportions,
clear
knowledge to be able to work out the path of any ray of light passing through it. This knowledge comprises the essential part of optics, and the laws which have to be made use ofthatsufficient
we must have
are the three fundamental ones of reflection,refraction,
and
dispersion.
ReflectionReflection occurs at the surface which
separates two different substances or media
a portion or the whole of the light striking
thrown back, and does not cross over from one medium into another. This is the reflected light. There are difthat surfaceis
ferent kinds of reflected light according to
the nature of the surface of reflection.
If
30
DIAMOND DESIGNis
that surface
highly polished, as in the
case of mirrors, or polished metals or gems,
the reflection
is
perfect
and an imagealso
is
formed.or
The
surface
may
be dull(as
matt to a greater or smaller extentreflected lightis
in the case of, say, cloth, paper, or pearls).
The
then more or
less
scatteredIt is
and
diffused.
kind of reflection that is of importance to us here, as diamond, owing to its extreme hardness, takes a very highthefirst
grade of polish and keepsfor ever.
it
practically^
The laws
of reflection
can be studied verysheet of paper.is
simply with a few pins and a mirror placedat right angles
upon a
flat
Afig.I.
plan of the arrangement13.
shown:
in
The experimentstraight line
is
as followsis
A
AB
drawn uponstood on the
the paper, and the mirror
is
paper so that the plane of total reflection (i.e. the silvered surface) is vertically overthatline.
Two
pins
P and Qaway
are stuck
anyhow on the
paper,
one as near theas possible.
mirror and the other as far
OPTICALThen the eyeI,
^.i
is
placed in line with
FO
at
Without moving the eye, two more pins R and S are inserted, one near to and the other far from the mirror, in such positions thatso
that
Q
is
hidden by P.
Fig. 13.
their
images appear in the mirror to
lie
alongIf
PQ
continued.is
the eye
now
sighted from positionwill
2 along S R,
Q and Pon Sis
appear in the
mirror to
lie
R
continued.
The mirroron
now removed, PIf
and S
Rbe
are joined and will be found to intersect
AB
at M.
a perpendicular
MN
32
DIAMOND DESIGNand The above experiment may be repeated
erected on
NMP
AB at that point, the angles N M S will be found equal.
along other directions, but keeping the pin
S at the same point.will
now
lie
between P' Q', again be found equal.In thelie
The line of sight on P' Q', and the angles S R' and the normal willexperiment S appeared to
first
on the continuation of P Q, in the second it appears to be situated on P' Q' produced. Its image is thus at the intersection of these two lines, at L. It can easily be proved
by elementary geometry (from the equalityof angles) that the
image L of the pin is at the same distance from the mirror as the pin S itself, and is of the same size.II.
If
the
pins
P, Q, R, S
in
the
first
experiment be placed so that their headsareall
at the
sheet of
same height above the plane paper, and the eye be placed in a
line of sight
with the heads P, Q, the images of the heads R, S in the mirror will beb}/
hidden
the head of pin P.
The angle
NMP
(position
I)
is
called
OPTICALangle of incidence, angle of reflection.
33
and the angle(verified
NMSby theas
The lawsabovefollowsI.
of reflection
tests):
can
now be formulated
The angle
of reflection is equal to the
angle of incidence.,
IL The paths
of the incident
and
of the
reflected ray lie in the
same
plane.
Fromflecting
I it follows,
as shown, thatre-
in. The image formed in a planesurfaceis
at
the
same distance
from that surface as the object reflected, and is of the same size as the object..
JRefractionpasses from one substancesuffers
When
lightit
into another
changes which are
somewhat more complicated than in the Thus if we place a coin case of reflection. at the bottom of a tumbler which we fillwith water, the coin appears to be higher
than when the tumbler was empty;if
also,it
we plungeseem
a pencil into the water,3
will
to be bent or broken at the surface,
34
DIAMOND DESIGNtheparticular
except inpencilis
case
when
the
perfectly vertical.of refraction in
We
can study the laws
a manner somewhat similar to that adoptedfor the reflection tests.of
Upon
a
fiat
sheet
paper
(fig.
14)
we
place a fairly thick
Fig. 14.
rectangular
glass
plate
with
one
of
its
edges (which should be poHshed perpendicularly to the plane of the paper) along apreviously drawn line
A B.
We
place a
pin, P, close to the edge
AB
of the glass
plate and another, Q, close to the further edge. Looking through the surface A B,
OPTICAL we
35
place our eye in such a position that the pin Q as viewed through the glass is
covered by pin P.
Near to the eye and
on the same hne of sight we stick a third pin R, which therefore covers pin P. Theglass plate is
now removed.
PQ
and
PR
are joined, a perpendicular tois
AB, MM',
erected at P, and a circle of any radius
drawn with P as centre. This circle cuts PQ at K and RP at L. LM and KM' are drawn perpendicular to MM', L M and
K M'found.
are measured
and the
ratio
tfitt?
The experiment positions of P and..
is
repeated for differentthe corresponding
Q and
LMratio
,
ratio
j^^, calculated.
,
It will
be found that
for a given substance (as in this case glass)thisis
constant.
It
is
called
the
index j4 refraction, and generally represented by the letter n.Referring tofig.
14,
we note
that as
PK
=
PL
=
radius of the
circle.
36
DIAMOND DESIGNwrite
we can
LMsin
RPM
OPTICAL
37
Of two substances with different index' qf refraction, that which has the greaterindex of refractionto glass, whichis
called optically denser.
In the experiment the light passed from airis
of greater optical density.i.e.
Let us
now
consider the reverse case,N'A'
when
lightless
passes
another
from one medium to dense optically. Suppose a
beamair.
of light
AOit is
(fig.
15)
with a small
angle of incidence passes from water into
At the surface
of separation a small(as
proportion of
reflected to A''.
weis
have seen underis
reflection)
The remainder
refracted in a direction
O A'
which
38
DIAMOND DESIGN
more divergent from the normalthan
N O N'gradu-
AO.
Suppose now that the angleally increases.light also increases,
AON
The proportion
of reflected
and the angle
of re-
fraction
N'OA'valueof
increases steadily
and
at
a more rapid rate thancertain
N O A,angle
until for a
the
of
incidence
BONis
the refracted angle will graze theIt is clear that
surface of separation.
under
these conditions therefractedIf
amount
of light
whichairisstill
and passes into theangleat
zero.
theas
of incidence is
greater,
CON,
there
is
no
reis
fracted ray, and the whole of the lightreflected into the optically denseror,
medium,
as
it
is
termed, total reflection thenis
occurs.angle,
The angle B O Nand caneasily
called critical(3)
be calculated by
when theknown.angleofi' ,
refractive indicesIt will
n and n' are be noted that when theattainsits
incidence
critical
value
the angle of refraction becomesi.e.
a right angle,to unity.
its sine
becomes equal
OPTICALSubstituting in(3)^'
39
n
sin r
= n^ sin sin r I^
smOr,if
=- -7
n
.
(4)
the less dense
medium be1
air,
nsini'
=
=^^ nis
.
.
(5)
This formula
(5)
very important in
the design of gems, forcritical
by
its
means the
angle can be accuratety calculated.especiallylike the
A
precious stone,
a colourless
and transparent onebest possible effect
diamond,
is
cut to the best advantage and with the
when
it
sends to the
spectator as strong and as dazzling aof light as possible.
beam
Now
a gem, not being
in itself a source of light, cannot shine with
other than reflected light.
amount*
of light willis
The maximum be given off by the gem
here of double refraction, as the diamond is a singly refractive substance, and it was considered unnecessary to introduce irrelevantmatter.
No mention
made
40if
DIAMOND DESIGNthe whole of the light that strikesit is
reflected
by the back of the gem, i.e. by that part hidden by the setting, and sent out into the air by its front part. The facets of the stone must therefore be soit is let
disposed that no light that enters
out throughreflected.
its
back, but thatis
it is
wholly
This result
obtained by having
the facets inclined in such a the light that strikes
way
that
all
them does
so at an
angle of incidence greater than the criticalangle.
This point will be further dealt withare a few indices of refrac-
in a later chapter.
The followingtion
which
may.
be useful or of interest
:
Water
OPTICALLeadsilicate.
41.
.
2-12
Diamond
.
.
2*417
.
(6)
These indices have, of course, been found by methods more accurate than the testsdescribed.
One
of
these
methods,
one
particularly suitable for the accurate de-
termination of the indices of refraction in gems, will be explained later.
Withfraction
this value
for
the the^
indexcritical
of
re-
of
diamond,
angle
works out at
sm^
I
nI
2-417i i
=.
-4136
= sin~^
'4136.
=- 24 26'
.
(7)
This angle will be found very important.
Dispersion
What we
call
white light
is
made up
of
a variety of different colours which produce white by their superposition. It is to the
42
DIAMOND DESIGNits
decomposition of white light into
com-
ponents that are due a variety of beautiful
phenomenaof the soap
like the
rainbow or the coloursit
bubble
and,
mayis
be added,
the
''
fire
" of a diamond.of refraction
The index
found to be
different for light of different colours, red
being generally refracted least and violetmost, the order for the index of the variouscolours being as follows:
Red, orange, yellow, green, blue, indigo,violet.
Note.
Inof
the
list
given above theis
indexyellow
refraction
that
of
the
light
obtained
by theit
incan-
descence of a sodiumis
salt.
This colouris
used as a standard, asdefinite,
verypro-
bright, very
and
easily
duced.If
white light strikes a glass plate with(fig.
parallel surfaces
20) the different colours
are refracted as
shown when passing
into
the glass.
Now#
for every colour the angle
of refraction is givensin r
by (equation
(2))
= sin
i.
OPTICAL
43
When
passing out of the glass, the angle
of refraction is given
by
n sm
I
= sm ri'
As the facesTherefore,
of the glass are parallel,
r.
/
=
i,
and the ray when leaving
the glass
is
parallel to its original direction.
The various
colours will thus follow parallel
Fig.
I 6.
paths as shown in
fig.
i6,
and as they areis
very near together (the dispersion
very
much
exaggerated), they will strike the eye
together and appear white.in the pin experiments
This
is
why
on refraction, dispersion was not apparent to any extent.If,
instead of using parallel surfaces as in
a glass plate,
we
place
as in a prism, light
them at an angle, falling upon a face of
44
DIAMOND DESIGN17and,
the prism will be dispersed as shown infig.;
when,
leaving
by another
face, the light, instead of
combining to form
white
(as in a plate) is still further dispersed
and forms a ribbon
of lights of the different
Fig. 17.
colours,is
from red to
violet.
Such a ribboncoloursof
called
a spectrum.
The
a
spectrum cannot be further decomposed by the introduction of another prism. The difference between the index ofrefraction of extreme violet lightof1
and thatDis-
extreme red
is
called dispersion.^
Generally two definite points on the spectrum the values given here for gems are those are chosen between the B and G lines of the solar spectrum.;
OPTICALpersion,
45
on the whole, increases with theof a
refractive index, although with exception.
The dispersionglasses is given
number:
of
gems and
below
Quartz
46
DIAMOND DESIGNso
a spectator's eye separately,
that
at
one moment he sees a ray of vivid blue, at another one of flaming scarlet or one ofshining green, while perhaps at the nextinstant a
beam
of purest
white
mayall
be
reflected in his direction.
And
these
colours change incessantly with the slightest
motion
of the
diamond.:
The effect of refraction in a diamond can be shown very interestingly as follows
A
piece of white cardboard or fairly
stiff
paper with a hole about half an inch indiameter inoflight.its
centre
is
placed in the
direct rays of the sun or another source
The stoneit
is
held behind the
paper and facingof spots of the
in the ray of light
which
passes through the hole.
A
great
number
most diverse colours appear then upon the paper, and with the slightest motion of the stone some vanish, others appear, and all change their position andtheir colour.If
the stone
is
held with the
hand,
its
slight
unsteadiness will give a
startling appearance of life to the
image
upon the paper.
This
life
is
one of the
OPTICALchief reasons of the
47
diamond's attraction,factors of its beauty.
and one
of the
main
Measurement of Refractionwas pointed out that the manner by which the index of refraction was calculated there, although the simplest, was both not sufficiently accurate and unsuitable for gem-stones. One of the best methods, and perhaps the one giving the most correct results, is that known as method of minimum deviation. Owing to the higher index of refraction of diamond it is especially suitable in its case, where others might not be convenient. The theory of that method is as follows Let A B C be the section of a prism of theit:
In the study of refraction
substance the refraction index of which
we want
to calculate
(fig.
i8).
A
source
of light of the desired colour is placed at
R, and sends aof
the prism.
beam R I upon the face A B The beam RI is broken,and leavesit
crosses the prism in the direction IF, is
again broken,
along
F
R'.
Supposing now that we rotate prism
ABC
48
DIAMOND DESIGNits
about
edge A.
The
direction of;
F
R''
changes at the same time
we noteif
that as
we
gradually turn the prism,
V R'
turns
in a certain direction.
But
we go ona certain
turning the prism,
F R'
will at
moment
stop and then begin to turn in
Fig. iS.
the reverse direction, although the rotationof the prism
was not
reversed.
We
alsois
note that at thesmallest value.
moment whenIt is
the ray
stationary the deviation has attained
its
not
difficult to
prove
that thispassesi.e.
is
the case
when the rayi'
of light
through the prism symmetrically,anglesi
when
and
(fig.
i8) are equal.
OPTICALLet
49
AMI
be a
line bisecting the angle A.
Then
T
is
perpendicular to
A M.
Let
R
be produced to
Q and R' I' to O. They meet on A M and the angle Q O R' is the deviation d[i.e.
the angle betweenfinal
the original andthelight
the
direction
of
passing
through the prism).Therefore
OIF-:
\d.I,
DrawThen
the normal at
N N'.
MIN' = IAM=-i^BAC(i)
if
a be the angle
of the prism.
Now by
equation
n
=
smsin -
i
r
i::::.NIR-:OIN'-OIM+MIN'
^ Id+la r - M I N' = 1^therefore
=^{d+a)
n
=
sin Md-\-a)
^A_^ sm fa
L
.
.
,n\ (8)
The indexculatedif
of refraction
can thus be
cal-
the angles d and a are known.of a spectroscope.
These are found by means
50
DIAMOND DESIGN:
This instrument consists of three partscollimator, the table,light enters
the
and the telescope.slit
The
by the
collimator (a long brass
and a lens) passes through the prism which is placed on the The table, and leaves by the telescope. colUmator is usually mounted rigidly upontube fitted with athe stand of the instrument.is
Its function
to determine the direction of entry ofits
the light and to ensure
being parallel.
Both the table and the telescope are movable about the centre of the table, and are fitted with circular scales which are graduated in degrees and parts of a degree, and by means of which the anglesare found.
Now
two
facets of a stone are selected,is
and the stonetable.
placed upon the table soof the prism,
that these facets are perpendicular to the
The angle a
i.e.
the
angle between these facets, can be found
measurement with a goniometer The angle d or also by the spectroscope. The position of the is found as followsdirect:
by
stone
is
arranged so that the light after
OPTICALpassing
51it
through the coUimator entersit
from one selected facet and leavesother.
by theis
TheThe
telescopeof
is
moved
mitil the
spectral
image
the source of light
found.
table
and the stone are now
rotated in the direction of
minimum
devia-
F1G/19.
tion,is
same time the telescope moved so that the image is kept in view.at the
and
We know
that at the point of
minimum
deviation the direction of motion of thetelescope changes.is
When
this exact point
reached the movements of the stoneof the telescope are stopped,
and
and the
52
DIAMOND DESIGNis
reading of the angle of deviation d
taken
on the graduated
scale.
The values
of a
and d are now introducedsin|(a+^) sin \a
in equation (8)^^
_
and the value
of
n calculated with the help
of sine tables or logarithms.
The values
for
diamond are
;^
Dispersion
= 2'4i7 for sodium light == 044= ^red^vioiet
Part III
MATHEMATICALIn the survey of the history of diamond
most remarkable fact is that so old an art should have progressed entirely by trial and error, by gradual correction and slow progress, by the almost accidental elimination of faults and introduction of ameliorations. We have traced the history of the art as far back as 1375, when the earliest recorded diamond manufactory existed, and when the polishers had already attained a high degree of guild organisation. We have every reason to believe that the process of diamond polishing was known centuries before. And yet all these centuries, when numerous keen minds were directed upon the fashioning of the gem, have left no single record ofcutting, perhaps the53
54
DIAMOND DESIGN
any purposeful planning of the design of the diamond based upon fundamental optics. Even the most bulky and thorough contemporary works upon the diamond or upon gems generally rest content with explaining the basic optical principles, and do no more than roughly indicate how these principles and the exceptional optical properties of thebrilliancy;
gem
explain
its
extraordinary
nowhere has the author seen calculations determining its best shape andItis
proportions.
the
purpose
of
the
present chapter to establish this shape andthese proportions.
The diamondif
treated essentially asless
it
be were a worthwill
crystal
in
which the desiredi.e.
results
areto
to
be obtained,great
without regardtherelation
the
value
which
between
a
great
demand
and
a
very
small supply gives to the least weight of
the material.It is useful to recall here the principles
and the properties whichthe calculations.
will
be used
in
MATHEMATICALReflection1.
55
The angles
of
incidence and
of
re-
flection are equal.2.
The paths
of the incident
and
of the
reflected ray lie in the
same
plane.
Refraction1.
When
a ray of light passes from one
mediumit is
into a second of different density,
refracted as:^
by the following equation
:
sin f
=
n' sin
i
.
.
(3)
where
= angle of refraction. i angle of incidence. n = index of refraction of the secondr
medium.n'
= indexfirst
of refraction of the first
medium.If
the
mediumnsinr
is
air,
n'
=
1,
and
equation becomes
=
sin
i
.
.
(2)
2.
When
medium
a ray of light passes from one into another optically less dense,
total reflection occurs for all values of the
angle of incidence above a certain critical
56value.
DIAMOND DESIGNThiscritical
angle
is
given
by
equation'
smt
=n
.
.
(4)
Or,
if
the less dense
medium be
air,
sin.-'
= ^,same
..
(5)
3.
The pathslie
of the incident
and
of the
refracted ra}^
in the
plane.
Dispersion
Whena band
a ray of lighti.e.
is
refracted,is split
dis-
persion occurs,or
the ray
up
into
spectrum of various colours, owing to the fact that each colour has a The disperdifferent index of refraction.sionforis
the difference between these indices
extreme rays on the spectrum.
Data
MATHEMATICAL
57
DETERMINATION OF THE BEST ANGLES AND THE BEST PROPORTIONS,
Postulate.
^The
design of a diamond or
any gem-stone must be symmetrical about an axis, for symmetry and regularityof
in the disposition of the facets are essentialfor a pleasing result.
Let us
now
consider a block of
bounded by polished surfaces, consider the effect on the patha gradual change in shape;
diamond and let usof light ofwill also
we
observe the postulate and keep the block
symmetrical aboutLet us take asparallel faces(fig.
its axis.first
section one having
20),
and
let
MM'
be
its
axis of
symmetry.
Let us for convenience
place the axis ofall
symmetry
vertically init
future work, so that surfaces crossing
are horizontal.
Consider a ray of light S
P P
striking face
A B.leave
It will
be refracted alongparallel to S.
P Q and
by Q R,if
(as
seen in studying dispersion)that
We
we have also know
NN'
is
the normal at Q, angles
58
DIAMOND DESIGNand
NQP
Q P M'
are equal.
Therefore, for
total reflection,
O P M'
== 24 26',
but at that angle of refraction the angle of incidence S P M becomes a right angle and
no
light penetrates into the stone.
It is
thus obvious that parallel faces in a
gem
are very unsatisfactory,
as
all
the lightpasses out
passing in by the front of a
gem
again by the back without any reflection.
can avoid parallelism by inclining either the top or the bottom faces at anangle with the directioncase
We
AB.of
In the
first
we obtain the shapein the
a rose-cut
diamond and
second case that of a
MATHEMATICALbrilliant
59rose'
cut.first
We
will
examine the
cut in the
instance.
The RoseConsider(fig.
21)
a section having thelet
bottom surface horizontal, and
us incline
an angle a with it. To maintain symmetry, another surface BC is introduced. We have now to find the value of a for which total reflectionthe top surfaceat
AB
occurs at
A C. Now26'.
for this to be the case,
the
minimum
angle of incidence upon
AC
must be 24incident ray
Let us draw such an
P Q.
To ensure
that no light
6ois
DIAMOND DESIGNincident at a smaller angle,
we must make
the angle of refraction at entry 24 26' and
arrange the surface of entry as shown,for
A B,moreSuch
we know
that then no light will enter
at an anglevertical to
more obliquei.e.
to
AB
or
A C.is
This gives to a a value of48 52'.
twice the critical angle,asection
very satisfactory indeed asas,
regards reflection,
owing to
its
deriva-
tion, all the light entering it leaves
by the
front part.refraction?
Is it also satisfactory as regards
Let us follow the path of a ray of light of any single colour of the spectrum, S P Q R T
and r be the angles of incidence upon and of refraction out of the(fig.
22).
Let
i
diamond.
R Q N, and therefore triangles A P Q and R C Q angle A Q P == angle COR. Also by symmetry A == C,At Q, P Q N--
in
therefore
angleit
APQ i
angle
C
RQ
;
follows that
=
r.
MATHEMATICALAs the anglewilli is
6iall
the same for
colours
of a white ray of light, the various colours
emerge
parallellight.
out of the diamond
and give whitemental reasonrose;
This
is
the funda-
of the unpopularity of the
there
is
no
fire.
This effect 23 and
may
be remedied to a small(figs.
extent by breaking the inclined facet24), so that
the angle be not the same
at entry as at exit.ful to the
This breaking
is
harm-
amount of hght reflected whichif we steepen the ever way we arrange it;
facet near the edge, there is a large propor-
tion of Hght projected
backwards and being
62lost, for
DIAMOND DESIGNwe may take(fig.
it
that the spectator
will
not look at the rose from the side of23).If,
the mounting
on the other(fig.
hand,
we flattenis
the apex of the rose
24)
(which
the usual method), a leakage willits base.
occur through
There
is,
of course.
no amelioration in the refraction if the light passes from one facet to anothersimilarly placed (asS' P' Q'
shown
in
fig.
23,
path
Taking the effect as a whole, the least unsatisfactory shape is as shown in fig. 24, with the angles a about 49 and 30 for the base and the apex respectively. The rose cut, however, is fundamentallyR'T').
MATHEMATICAL
63
wrong, as we have seen above, and should be abohshed altogether. It is the highcost of the material thatstill
is
the cause of
its
being used in cases where the rough shape is especially suitable, and then only in small sizes. In actual practice the
Fig. 24.
proportions of the cut rose depend largely upon those of the rough diamond, thestone being cut with as small a loss of material as possible. Generally the values of a are much below those given
above,material
i.e.
is
and 30, thick enough49it
asto
where theallow
suchcutit
steep angles
is
much
better to
into a brilhant.
64
DIAMOND DESIGNThe Brilliant
A Back.
of the Brilliant
Let us
now
pass to the consideration ofi.e.
the other alternative,
where the top
surface
is
a
horizontal
plane
AB
and
where the bottom surface A C is inclined (fig. at an angle a to the horizontal As before, we have to introduce 25). a third plane B C to have a symmetricalsection.
MATHEMATICAL'
65
First Reflection
Let a vertical rayangle of incidence
PQ
strike
A B.
As the
is zero, it
passes into the
stone without refraction and meets plane
AC
at R.
Let
RN
be the normal at that
point, then, for total reflection to occur,
angle
NRQ==
24 26'.
Butangleas
NRQ
:= angle
QAR
===
a,
AQ
and QR,
AR
and
RN
are per-
pendicular.
Therefore, for total reflection of a verticalray,
a
=
24 26'.
Let usgradually
now
incline the ray
PQ
so that
it
changes from a vertical to a horizontal direction, and let P' Q' be sucha ray.is
Upon
passing into the diamond
it
and strikes AC at an angle Q' R' N' where R' is the normal to A C. When P' Q' becomes horizontal, the angle of refraction T' Q' R' becomes equal torefracted,
W
24
26'.
This
is
the extreme value attain5
66 able
DIAMOND DESIGN;
by that angle also, for total reflection, angle Q' R' N' must not be less than 24 26'. If we draw R' V, vertical angle V R' Q' = R' Q' T' = 24 26', andangle
V R' N' - V R' Q' +-: 24 26'
Q' R'
N'
+ 24 26'
= 48 52'as before,a
= angle V R' N',.
and thereforea :^ 48 52'.
.
(9)
For
absolute total reflection to occur at thethe inclined facets
first facet,
must make an
angle of not less than 48 52' with the horizontal.
Second Reflection
Whenthefirst
the ray of lightinclinedfacet
is
reflected(fig.
fromit
ACB C.
26),
strikes the oppositelight
one
Here too the
must be totally reflected, for otherwise there would be a leakage of lightLet us
through the back of the gem-stone.vertically
consider, in the first instance, a ray of light
incident
upon the
stone.
The
MATHEMATICAL
67
path of the ray will be P Q R S T. If R N and S N' are the normals at R and S respectively, then for total reflection,angle N' S
R -=M
24 26'.
P
QNo(\oC
B
N'
Fig. 26.
Let us find the value of a toconditionangle
fulfil
that
QRN
=
angle
QAR
=a
as having perpendicular sides.
angle S
RN
=:=
angle
QRN
68
DIAMOND DESIGNandreflection.
as angles of incidence
Thereforeangle
NRS = a.R=a;
Now
let
angle N' S
Then, in triangleangle angle
R S C, S R C = 90 a R S C = 90 x
angle
RCS
== 2
X
angle
RCMa).
-=2 X
ARQ =- 2(90=
The sum of these three angles equals two right angles,
90 a +90 +180 2 a:\;
180,
or
3a -\-x 3a
= =
180
i8o
X.
Now, Xais
is
not less than 24
26', therefore
not greater than
3
Let us again inclineuntil it
P Q from the verticalin this case
becomes horizontal, but
in the other direction, to obtain the inferiorlimit.
MATHEMATICAL
69
Then (fig. 27) the path will be PQRS. Let QT, RN, SN' be the normals at Q, R, and S respectively. At the extreme case,
TQRat R.
will
be 24
26'.
Draw
RV
vertical
Fig. 27.
Thenangle angle
QRV =
anglea.
TQR
=
24 26'
VRN =
Asangle
before, in triangle
R S C,
SRC = go^-NRS = 9o-a-2426' angle RCS = 2(90- a)angle
R S C = 90 x.
70
DIAMOND DESIGN
Then
90 a 24 26'+i8o 2a+90 3a +% = 180^-24 26' - 155In the case
;t;
=34'
180
nowX
considered,
=
24 26'.
Then3a
=
155 34'-24' 26'.
=
131 8'..
== 43 43'
(10)
For
absolute total reflection at the second
facet, the inclined facets
must make an angle
of not more than 43 43' with the horizontal.
We
will
note here that this condition
and the one arrivedopposition.will
We
will
on page 66 are in discuss this later, andat
pass
now
to considerations of refraction.Refraction
First case
:
ql
is less
than 45refractionin
In
the
discussion
of
a
diamond, we have to consider two cases, i.e. a is less than 45 or it is more than 45. Let us take the former case first and let P Q R S T (fig. 28) be path of the ray. Then,
MATHEMATICALif
71
SN
is
the normal at
S,
we know
that for
total reflection at S angle
RSN
== 24 26'.
We
want to avoid total reflection, for if the light is thrown back into the stone, some of it may be lost, and in any case the
Fig. 28.
ray
be broken too frequently and the result will be disagreeable.will
Therefore,
angleRSN