DIAMOND DESIGN
A STUDY OF THE REFLECTION
AND REFRACTION OF LIGHT IN
A DIAMOND
=by
MARCEL TOLKOWSKY
B.Sc.~, A.C.G.I.~=
_WITH 37 ILLUSTRATIONS_
*London:*
=E.~ & F.~ N.~ SPON, Ltd.~, 57 HAYMARKET, S.W.~ 1=
*New York:*
=spon & chamberlain, 120 liberty street=
1919
CONTENTS
INTRODUCTION 5
PART I.  HISTORICAL 8
,, II.  OPTICAL 26
,, III.  MATHEMATICAL 53
=The Rose= 59
=The Brilliant= 64
=_A. Back_= 64
=_B. Front_= 80
=Faceting= 94
=Best Proportions of a Brilliant= 97
DIAMOND DESIGN
INTRODUCTION
=This= book is written principally for students
of precious stones and jewellers, and more
particularly for diamond manufacturers
and diamond cutters and polishers. The
author will follow the evolution of the
shape given to a cut diamond, and discuss
the values of the various shapes and the
reason for the discarding of the old shapes
and the practically universal adoption of
the brilliant.
It is a 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 con
temporary minds have been directed upon
the diamond, and the list of books written
on that subject increases rapidly, yet
nowhere can one find any mathematical
work determining the best shape for that
gem. The present volume's chief aim is
the calculation of that shape.
The calculations have been made as
simple as possible, so as not to be beyond
the range of readers with a knowledge of
elementary geometry, algebra, and trigo
nometry. Where, however, it 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 been
explained as an alternative.
The results of the calculations for the form
of brilliant now in use were verified by
actual mensuration from wellcut brilliants.
The measures of these brilliants are given
at the 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 used in the present work
will be found very useful for the design
of other transparent precious and semi
precious stones, although it will be found
advisable in the case of stones of an agree
able colour to cut the gem somewhat
thicker than the calculations warrant, so
as to take full advantage of the colour.
The same remark applies to diamonds of
some exceptional and beautiful colour, like
blue or pink, where the beauty or the value
of the stone increases with the depth of
its colour.
Part 1
HISTORICAL
=It= is to Indian manuscripts and early Indian
literature we turn when we want to find
the origin of diamond cutting, for India
has always been regarded as the natural
and ancient home of the diamond. It is
there that they were first found: up to
1728, the date of the discovery of the
Brazilian deposits, practically the whole
world's supply was derived from Indian
sources. They are found there in the
valleys and beds of streams, and also,
separated from the matrix in which they
were formed, in strata of detrital matter
that have since been covered by twelve
to sixteen feet of earth by the accumula
tion of later centuries. Diamonds have
existed in these deposits within the reach
of man for many ages, but the knowledge
of the diamond as a gem or as a crystal
with exceptional qualities does not go
back in India to the unfathomable antiquity to
which books on diamonds generally refer.
It was wholly unknown in the Vedic
period, from which no specific names for
precious stones are handed down at all.^"1"^
The earliest systematic reference appears
to be in the Arthaastra of Kautilya (about
third century =b.c.~=), 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 diamonds
should be large, regular, heavy, capable of
bearing blows,^"2"^ able to scratch metal,

 ^"1"^ Berthold Laufer, _The Diamond: a Study in
 Chinese and Hellenistic Folklore_ (Chicago, 1915).


 ^"2"^ This legend of the indestructibility 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. The
 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
 still current in Europe as late as the middle of the
 thirteenth 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.

refractive and brilliant. In the Milinda
paha (Questions of 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.
It is therefore permissible to estimate
with a sufficient degree of accuracy that
the diamond became known in India during
the Buddhist period, about the fourth
century =b.c.~=, and that its use as a gem
dates from that period.^"1"^
It is not known with certainty when
and where the art of grinding or polishing
diamonds originated. There is as yet 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 men

 ^"1"^ Laufer, _loc.~ cit.~_

tioned. It is, however, likely that, where
the polishing of other precious stones was
accomplished in that manner, that of
diamonds themselves cannot have been
entirely unknown. What polishing there
was must at first have been limited to
the smoothing of the faces of the crystals
as they were found. The first description
of cut diamonds is given by Tavernier,^"1"^
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 of
his visit (1665) the Indians were polishing
over the natural faces of the crystal, and
preferred, therefore, regularly crystallised
gems. They also used the knowledge they
had of grinding diamonds to remove faulty
places like spots, grains, or glesses. If the
fault 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 that

 ^"1"^ Tavernier, _Voyage en Turquie, en Perse et aux
 Indes_ (1679).

there were also European polishers in
India at that time, and that it was to them
the larger stones were given for cutting.
Whether they had learnt the art inde
pendently or from Indians and attained
greater proficiency than they, or whether
they were acting as instructors and teaching
the Indians a new or a forgotten art, is
uncertain. Both views are equally likely
in the present state of research upon that
subject: at the time of Tavernier's visit,
diamond cutting had been known in Europe
for more than two centuries.
Among the several remarkable gems that
Tavernier describes,
the most noteworthy
is the one known as
the Great Mogul.
This diamond was of
a weight of 280 cts.~
and was cut as
sketched in fig.~ 1.
The polishing was
the work of a Venetian, Hortensio Borgis,
to whom it was given for that purpose by
its owner, the Great Mogul Aurung Zeb, of
Delhi. This kind of cut is characteristic
of most of the
large *Indian*
stones, such as
the Orlow (fig.~ 2),
which is now the
largest diamond
of the Russian
crown jewels and weighs 193 3/4 cts. The Kohi
Noor (fig.~ 3), now among the British crown
jewels, was of a somewhat similar shape
before recutting.
It weighed then
186 cts.
Tavernier also
mentions several
other types of
cut which he
met in India.
The Great Table (fig.~ 4), which he saw in
1642, weighed 242 cts. Both the Great Table
and the Great Mogul seem to have dis
appeared: it is not known what has become
of them since the seventeenth century.
Various other shapes are described, such
as point stones, thick stones, table stones
(fig.~ 5), etc. But the
chief characteristic
remains: all these
diamonds have been
cut with one aim con
stantly in view  how
to polish the stone with the smallest
possible loss of weight. As a consequence
the polishing was generally accomplished
by covering the surface of the
stone with a large number of
facets, and the original shape
of the rough gem was, as far as
possible, left unaltered.
It was mentioned before that
the art of diamond polishing
had already been known in
Europe for several centuries when Tavernier
left for India. We have as yet no cer
tain source of information about diamond
cutting in Europe before the fourteenth
century. The first reference thereto men
tions that diamond polishers were work
ing in 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.^"1"^
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
more definite, more characteristic. And it
is from that time that both on its technical
and artistic sides progress is made at a
rate, slow at first, but increasing rapidly
later.
It is not difficult to find the chief reason
for that change.
Up to that time, diamonds had almost
exclusively been used by princes or priests.
To princes they were an emblem of 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

 ^"1"^ Jacobson's _Technologisches Worterbuch_ (1781).

for a very small weight, and could easily
be carried in case of flight. Priests used
them in the ornaments of temples or
churches; they have not infrequently been
set as eyes in the heads of statues of
Buddha.
In the fifteenth century it became the
fashion for women to wear diamonds as
jewels. This fashion was started by Agnes
Sorel (about 1450) at the Court of Charles
VII of France, and gradually spread from
there to all 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 increas
ing attractiveness of the gem increased
the demand and gave a new impulse to
the art.
At the beginning of the fifteenth century
a clever diamond cutter named Hermann
established a factory in Paris, where his
work met with success, and where the
industry started developing.
In or about 1476 Lodewyk (Louis) van
Berquem, a Flemish polisher of Bruges,
introduced absolute symmetry in the dis
position 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:
`After having ground off redundant
material from a stone by rubbing it against
another one (the process known in modern
practice as `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.'

 ^"1"^ Robert de Berquem, _Les merveilles des Indes:
 Trait des pierres prcieuses_ (Paris, in4"o", 1669),
 p.~ 12: `Louis de Berquem l'un de mes ayeuls a
 trouv le premier l'invention en mil quatre cent
 soixanteseize de les tailler avec la poudre de diamant
 mme. Auparavant on fut contraint de les mettre
 en "oe"uvre tels qu'on les rencontrait aux Indes, c'est
 dire tout fait bruts, sans ordre et sans grace,
 sinon quelques faces au hasard, irregulieres et mal
 polies, tels enfin que la nature les produit. Il mit
 deux diamants sur le ciment et aprs les avoir griss
 l'un contre l'autre, il vit manifestement que par le
 moyen de la poudre qui en tombait et l'aide du
 moulin et certaines roues de fer qu'il avait inventes,
 il pourrait venir bout de les polir parfaitement,
 mme de les tailler en telle manire qu'il voudrait.
 Charles devenu duc de Bourgogne lui mit trois grands
 diamants pour les tailler avantageusement selon
 son adresse. 11 les tailla aussitt, l'un pais, l'autre
 faible et le troisime en triangle et il y russit si bien
 que le duc, ravi d'une invention si surprenante, lui
 donna 3000 ducats de rcompense.'

As has already been shown, we know now
that 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 have
invented that part of the process. He may
perhaps have introduced some important
improvement like the use of castiron
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 therefore
correspondingly speedier.
What Van Berquem probably did ori
ginate is, as already stated, rigid symmetry
in the design of the cut stone. The intro
duction of the shape known as _pendeloque_
or _briolette_ is generally ascribed to him.
The Sancy and the Florentine, which are
both cut in this shape, have been said
by some to have been polished by him.
The Sancy (53 3/4 cts.~) belongs now to the
Maharaja of Guttiola, and the Florentine
(fig.~ 6), which is much larger (133 1/5 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 why
it is not at all certain that they were
the work of Van Berquem. At any rate,
they are typical of the kind of cut he
introduced.
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 shape
of the rough stone is especially suitable.
This unpopularity was largely due to the
fact that, although the loss of weight in
cutting was fairly high, the play of light
within the stone did not produce sufficient
fire 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 found
very advantageous in the polishing of fiat
pieces of rough or split diamond. Such
materia1 is even now frequently cut into
roses, chiefly in the smaller sizes.
In the chapter upon the design of
diamonds it will be shown that roses have
to be made thick (somewhat thicker than
in fig.~ 7) for the loss of light to be small,
and that the flatter the
rose the bigger the loss
of light. It will also be
seen there that the fire
of a rose cannot be
very high. These faults
caused the rose to be
superseded by the
brilliant.
We owe the introduction of the brilliant
in the middle of the seventeenth century
to Cardinal Mazarin  or at any rate to
his influence. As a matter of fact, the first
brilliants were known as Mazarins, and were
of the design of fig.~ 9. They had sixteen
facets, excluding the table, on the upper side.
They;are called doublecut brilliants. Vincent
Peruzzi, a Venetian polisher, increased the
number of facets from sixteen to thirty
two (fig.~ 10) (triplecut brilliants), thereby
increasing very much the fire and brilliancy
of the cut gem, which were already in the
doublecut brilliant incomparably better
than in the rose. Yet diamonds of that
cut, when seen nowadays, seem exceedingly
dull compared to moderncut 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 the
side facets, so that even if the first were
polished to the correct angle (which they
were not) the second would be cut too
steeply and give an effect of thickness.
Oldcut brilliants, as the triplecut brilliants
are generally called, were at first modified
by making the size and angle of the facets
more uniform (fig.~ 11), this bringing about
a somewhat rounder stone. With the in
troduction of mechanical bruting or cutting
(an operation distinct from polishing; see
p.~ 17) diamonds were made absolutely
circular in plan (fig.~ 37). The gradual
shrinkingin of the corners of an oldcut
brilliant necessitated a less thickly cut
stone with a consequent increasing fire and
life, until a point of maximum brilliancy
was reached. This is the presentday
brilliant.^"1"^
Other designs for the brilliant have been
tried, mostly attempts to decrease the loss
of weight in cutting without impairing the
brilliancy of the diamond, but they have
not met with success.
We may note here that the general
trend of European diamond polishing as
opposed to Indian is the constant search
for greater brilliancy, more life, a more

 ^"1"^ 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 have
 invented 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 first
 introduced.

vivid fire in the diamond, regardless of
the loss of weight. The weight of diamond
removed by bruting and by polishing
amounts even in the most favourable cases
to 52 per cent. of the original rough weight
for a perfectly cut brilliant. In the next
chapters the best proportions for a brilliant
will be calculated without reference to the
shape of a rough diamond, and it will be
seen how startlingly near the calculated
values the modern wellcut brilliant is
polished.
Part II
OPTICAL
=It= is to light, the play of light, its reflection
and its refraction, that a gem owes its
brilliancy, its fire, its colour. We have
therefore to study these optical properties
in order to be able to apply them to the
problem we have now before us: the cal
culation of the shape and proportions of
a perfectly cut diamond.
Of the total amount of light that falls
upon a material, part is returned or re
flected; the remainder penetrates into it,
and crosses it or is absorbed by it. The
first part of the light produces what is
termed the `lustre' of the material.
The second part is completely absorbed
if the material is black. If it is partly
absorbed the material will appear coloured,
and if transmitted unaltered it will appear
colourless.
The diamonds used as gems are generally
colourless or only faintly coloured; it can
be taken that all the light that passes into
the stones passes out again. The lustre
of the diamond is peculiar to that gem,
and is called adamantine for that reason.
It is not found in any other gem, although
zircon and demantoid or olivine have a
lustre approaching somewhat to the ada
mantine.
In gem stones of the same kind and of
the same grade of polish, we may take it
that the lustre only varies with the area
of the gem stone exposed to the light, and
that it is independent of the type of cut
or of the proportions given to the gem (in
so far as they do not affect the area); this
is why gems where the amount of light
that is reflected upon striking the surface
is great, or where much of the light that
penetrates into the stone is absorbed and
does not pass out again, are frequently cut
in such shapes as the cabochon (fig.~ 12),
so as to get as large an area as possible,
and in that way take full advantage of
the lustre.
In a diamond, the amount of light re
flected from the surface is much smaller
that that penetrating into the stone; more
over, a diamond is practically perfectly
transparent, so that all the light that passes.
into the stone has to pass out again. This
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 be
considered as negligible in the calculations.
The brilliancy or, as it is sometimes
termed, the `fire' or the `life' of a gem
thus depends entirely upon the play of
light in the gem, upon the path of rays
of light in the gem. If a gem is so cut or
designed that every ray. of light passing
into it 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 so
as to ensure this result.
If we want to design a gem or to calculate
its best shape and proportions, it is clear
that we must have sufficient 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 of
are the three fundamental ones of reflection,
refraction, and dispersion.
=Reflection=
Reflection occurs at the surface which
separates two different substances or media;
a portion or the whole of the light striking
that surface is thrown back, and does not
cross over from one medium into another.
This is the reflected light. There are dif
ferent kinds of reflected light according to
the nature of the surface of reflection. If
that surface is highly polished, as in the
case of mirrors, or polished metals or gems,
the reflection is perfect and an image is
formed. The surface may also be dull
or matt to a greater or smaller extent (as
in the case of, say, cloth, paper, or pearls).
The reflected light is then more or less
scattered and diffused.
It is the first kind of reflection that is of
importance to us here, as diamond, owing
to its extreme hardness, takes a very high
grade of polish and keeps it practically
for ever.
The laws of reflection can be studfed very
simply with a few pins and a mirror placed
at right angles upon a fiat sheet of paper.
A plan of the arrangement is shown in
fig.~ 13. The experiment is as follows: 
I. A straight line AB is drawn upon
the paper, and the mirror is stood on the
paper so that the plane of total reflection
(_i.e.~_ the silvered surface) is vertically over
that line. Two pins P and Q are stuck
anyhow on the paper, one as near the
mirror and the other as far away as possible.
Then the eye is placed in line with PQ at
1, so that Q is hidden by P. 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 that
their images appear in the mirror to lie
along PQ continued.
If the eye is now sighted from position
2 along SR, Q and P will appear in the
mirror to lie on SR continued.
The mirror is now removed, PQ and SR
are joined and will be found to intersect
on AB at M. If 'a perpendicular MN be
erected on AB at that point, the angles
NMP and NMS will be found equal.
The above experiment may be repeated
along other directions, but keeping the pin
S at the same point. The line of sight
will now lie on P"'"Q"'", and the angles
between P"'"Q"'", SR"'" and the normal will
again be found equal.
In the first experiment S appeared to
lie on the continuation of PQ, 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 equality
of 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 heads
are all at the same height above the plane
sheet of 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 be
hidden by the head of pin P.
The angle NMP (position I) is called
angle of incidence, and the angle NMS
angle of reflection.
The laws of reflection (verified by the
above tests) can now be formulated as
follows: 
I. The angle of reflection is equal to the
angle of incidence,
II. The paths of the incident and of the
reflected ray lie in the same plane.
From I it follows, as shown, that
III. The image formed in a plane re
flecting surface is at the same distance
from that surface as the object reflected,
and is of the same size as the object.
=Refraction=
When light passes from one substance
into another it suffers changes which are
somewhat more complicated than in the
case of reflection. Thus if we place a coin
at the bottom of a tumbler which we fill
with water, the coin appears to be higher
than when the tumbler was empty; also,
if we plunge a pencil into the water, it
will seem to be bent or broken at the surface,
except in the particular case when the
pencil is perfectly vertical.
We can study the laws of refraction in
a manner somewhat similar to that adopted
for the reflection tests. Upon a flat sheet
of paper (fig.~ 14) we place a fairly thick
rectangular glass plate with one of its
edges (which should be polished perpendi
cularly to the plane of the paper) along a
previously drawn line AB. 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 AB,
we 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 line of sight we stick a third
pin R, which therefore covers pin P. The
glass plate is now removed. PQ and PR
are joined, a perpendicular to AB, MM"'",
is 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"'", LM and
KM"'" are measured and the ratio LM / KM"'"
found.
The experiment is repeated for different
positions of P and Q and the corresponding
ratio LM / KM"'" calculated. It will be found that
for a given substance (as in this case glass)
this ratio is constant. It is called the
index of refraction, and generally repre
sented by the letter _n_.
Referring to fig.~ 14, we note that as
PK "=" PL "=" radius of the circle,
we can write
LM / KM"'" "=" LM/PL / KM"'"/PK "=" sin RPM / sin QPM"'".
Writing the angle of incidence RPM as _i_,
and the angle of refraction QPM"'" as _r_, this
equation becomes
_n_ "=" sin _i_ / sin _r_ (1)
or
_n_ sin _r_ "=" sin _i_ (2)
In this case the incident ray is in air,
the index of refraction of which is very
nearly unity. With another substance it
can be shown that equation (2) becomes
_n_ sin _r_ "=" _n"'"_ sin _i_ (3)
where _n"'"_ is the index of refraction of that
substance.
It can be seen easily, and in a way similar
to that used with reflection (_i.e.~_ sighting
along the heads of the pins), that, in re
fraction also:
The paths of the incident and of the
refracted ray lie in the same plane.
Of two substances with different index
of refraction, that which has the greater
index of refraction is called optically denser.
In the experiment the light passed from air
to glass, which is of greater optical density.
Let us now consider the reverse case, _i.e.~_
when light passes from one medium to
another less dense optically. Suppose a
beam of light AO (fig.~ 15) with a small
angle of incidence passes from water into
air. At the surface of separation a small
proportion of it is reflected to A"'""'" (as we
have seen under reflection). The remainder
is refracted in a direction OA"'" which is
more divergent from the normal NON"'"
than AO.
Suppose now that the angle AON gradu
ally increases. The proportion of reflected
light also increases, and the angle of re
fraction N"'"OA"'" increases steadily and at
a more rapid rate than NOA, until for a
certain value of the angle of incidence
BON the refracted angle will graze the
surface of separation. It is clear that under
these conditions the amount of light which
is refracted and passes in to the air is
zero. If the angle of incidence is still
greater, as at CON, there is no re
fracted ray, and the whole of the light is
reflected into the optically denser medium,
or, as it is termed, total reflection then
occurs. The angle BON is called critical
angle, and can easily be calculated by (3)
when the refractive indices _n_ and _n"'"_ are
known. It will be noted that when the
angle of incidence attains its critical
value _i"'"_, the angle of refraction becomes
a right angle, _i.e.~_ its sine becomes equal
to unity.
Substituting in (3)
_n_ sin _r_ "=" _n"'"_ sin _i"'"_
sin _r_ "=" 1
sin _i"'"_ "=" _n_ / _n"'"_ (4)
Or, if the less dense medium be air,
_n_ "=" 1
sin _i"'"_ "=" 1 / _n"'"_.^"*"^ (5)
This formula (5) is very important in
the design of gems, for by its means the
critical angle can be accurately calculated.
A precious stone, especially a colourless
and transparent one like the diamond, is
cut to the best advantage and with the
best possible effect when it sends to the
spectator as strong and as dazzling a beam
of light as possible. Now a gem, not being
in itself a source of light, cannot shine with
other than reflected light, The maximum
amount of light will be given off by the gem

 ^"*"^ No mention is made here of double refraction,
 as the diamond is a singly refractive substance, and
 it was considered unnecessary to introduce irrelevant
 matter.

if the whole of the light that strikes it 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 so
disposed that no light that enters it is let
out through its back, but that it is wholly
reflected. This result is obtained by having
the facets inclined in such a way that all
the light that strikes them does so at an
angle of incidence greater than the critical
angle. This point will be further dealt with
in a later chapter.
The following are a few indices of refrac
tion which may be useful or of interest: 
Water 1.33
Crown glass 1.5 approx.~
Quartz 1.541.55
Flint glass 1.576
Colourless strass 1.58
Spinel 1.72
Almandine 1.79
Lead borate 1.83
Demantoid 1.88
Lead silicate 2.12
Diamond 2.417 (6)
These indices have, of course, been found
by methods more accurate than the tests
described. One of these methods, one
particularly suitable for the accurate de
termination of the indices of refraction in
gems, will be explained later.
With this value for the index of re
fraction of diamond, the critical angle
works out at
sin _i_ "=" 1 / _n_
"=" 1 / 2.417 "=" .4136
_i_ "=" sin1 .4136
_i_ "=" 24"o" 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
decomposition of white light into its com
ponents that are due a variety of beautiful
phenomena like the rainbow or the colours
of the soap bubble  and, it may be added,
the `fire' of a diamond.
The index of refraction is found to be
different for light of different colours, red
being generally refracted least and violet
most, the order for the index of the various
colours being as follows: 
Red, orange, yellow, green, blue, indigo,
violet.
_Note._  In the list given above the
index of refraction is that of the
yellow light obtained by the incan
descence of a sodium salt. This colour
is used as a standard, as it is very
bright, very definite, and easily pro
duced.
If white light strikes a glass plate with
parallel surfaces (fig.~ 20) the different colours
are refracted as shown when passing into
the glass. Now for every colour the angle
of refraction is given by (equation (2))
_n_ sin _r_ "=" sin _i_.
When passing out of the glass, the angle
of refraction is given by
_n_ sin _i"'"_ "=" sin _r"'"_.
As the faces of the glass are parallel, _i"'"_ "=" _r_.
Therefore, _r"'"_ "=" _i_, and the ray when leaving
the glass is parallel to its original direction.
The various colours will thus follow parallel
paths as. shown in fig.~ 16, and as they are
very near together (the dispersion is very
much exaggerated), they will strike the eye
together and appear white. This is why
in the pin experiments on refraction, dis
persion was not apparent to any extent.
If, instead of using parallel surfaces as in
a glass plate, we place them at an angle,
as in a prism, light falling upon a face of
the prism will be dispersed as shown in
fig.~ 17; and, 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
colours, from red to violet. Such a ribbon
is called a spectrum. The colours of a
spectrum cannot be further decomposed by
the introduction of another prism.
The difference between the index of
refraction of extreme violet light and that
of extreme red is called dispersion.^"1"^ Dis

 ^"1"^ Generally two definite points on the spectrum
 are chosen; the values given here for gems are those
 between the B and G lines of the solar spectrum.

persion, on the whole, increases with the
refractive index, although with exception.
The dispersion of a number of gems and
glasses is given below: 
Quartz .013
Sapphire .018
Crown glass .019
Spinel .020
Almandine .024
Flint glass .036
Diamond .044
Demantoid .057
The greater the dispersion of a medium,
other things being equal, the greater the
difference between the angles of refraction
of the various colours, and the further
separated do they become. It is to its
very high dispersion (the greatest of all
colourless gemstones) that the diamond
owes its extraordinary `fire.' For when
a ray of light passes through a wellcut
diamond, it is refracted through a large
angle, and consequently the colours of the
spectrum, becoming widely separated, strike
a spectator's eye separately, so that at
one moment he sees a ray of vivid blue,
at another one of flaming scarlet or one of
shining green, while perhaps at the next
instant a beam of purest white may be
reflected in his direction. And all 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 in
diameter in its centre is placed in the
direct rays of the sun or another source
of light. The stone is held behind the
paper and facing it in the ray of light which
passes through the hole. A great number
of spots of the 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 and
their 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
chief reasons of the diamond's attraction,
and one of the main factors of its beauty.
=Measurement of Refraction=
In the study of refraction it was 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 gemstones.
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 ABC be the section of a prism of the
substance the refraction index of which
we want to calculate (fig.~ 18). A source
of light of the desired colour is placed at
R, and sends a beam RI upon the face AB
of the prism. The beam RI is broken,
crosses the prisln in the direction II"'", is
again broken, and leaves it along I"'"R"'".
Supposing now that we rotate prism ABC
about its edge A. The direction of I"'"R"'"
changes at the same tiIne; we note that as
we gradually turn the prism, I"'"R"'" turns
in a certain direction. But if we go on
turning the prism, I"'"R"'" will at a certain
moment stop and then begin to turn in
the reverse direction, although the rotation
of the prism was not reversed. We also
note that at the moment when the ray is
stationary the deviation has attained its
smallest value. It is not difficult to prove
that this is the case when the ray of light
passes through the prism symmetrically,
_i.e.~_ when angles _i_ and _i"'"_ (fig.~ 18) are equal.
Let AM be a line bisecting the angle A.
Then II"'" is perpendicular to AM. Let RI
be produced to Q and R"'"I"'" to O. They meet
on AM and the angle QOR"'" is the deviation _d_
(_i.e.~_ the angle between the original and
the final direction of the light passing
through the prism).
Therefore a OII"'" "=" 1/2 _d_.
Draw the normal at I, NN"'".
Then
MIN"'" "=" IAM "=" 1/2 _a_
if _a_ be the angle BAC of the prism.
Now by equation (1)
_n_ "=" sin _i_ / sin _r_
_i_ "=" NIR "=" OIN"'" "=" OIM + MIN"'"
"=" 1/2 _d_ + 1/2 _a_ "=" 1/2 (_d_ + _a_)
_r_ "=" MIN"'" "=" 1/2 _a_
therefore
_n_ "=" sin 1/2 (_d_ + _a_) / sin 1/2 _a_ (8)
The index of refraction can thus be cal
culated if the angles _d_ and _a_ are known.
These are found by means of a spectroscope.
This instrument consists of three parts: the
collimator, the table, and the telescope. The
light enters by the collimator (a long brass
tube fitted with a slit and a lens) passes
through the prism which is placed on the
table, and leaves by the telescope. The
collimator is usually mounted rigidly upon
the stand of the instrument. Its function
is to determine the direction of entry of
the light and to ensure its 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 de
gree, and by means of which the angles
are found.
Now two facets of a stone are selected,
and the stone is placed upon the table so
that these facets are perpendicular to the
table. The angle _a_ of the prism, _i.e.~_ the
angle between these facets, can be found
by direct measurement with a goniometer
or also by the spectroscope. The angle _d_
is found as follows:  The position of the
stone is arranged so that the light after
passing through the collimator enters it
from one selected facet and leaves it by the
other. The telescope is moved until the
spectral image of the source of light is
found. The table and the stone are now
rotated in the direction of minimum devia
tion, and at the same time the telescope
is moved so that the image is kept in view.
We know that at the point of minimum
deviation the direction of motion of the
telescope changes. When this exact point
is reached the movements of the stone
and of the telescope are stopped, and the
reading of the angle of deviation _d_ is taken
of the graduated scale.
The values of _a_ and _d_ are now introduced
in equation (8):
_n_ "=" sin 1/2 (_a_ + _d_) / sin 1/2 _a_ (8)
and the value of _n_ calculated with the help
of sine tables or logarithms.
The values for diamond are
_n_ "=" 2.417 for sodium light
Dispersion "=" _n_ red  _n_ violet "=" .044.
Part III
MATHEMATICAL
=In= the survey of the history of diamond
cutting, perhaps the 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 intro
duction of ameliorations. We have traced
the history of the art as far back as 1375,
when the earliest recorded diamond manu
factory 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 polish
ing 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 of
any purposeful planning of the design of
the diamond based upon fundamental optics.
Even the most bulky and thorough con
temporary 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 pro
perties of the gem explain its extraordinary
brilliancy; nowhere has the author seen
calculations determining its best shape and
proportions. It is the purpose of the
present chapter to establish this shape and
these proportions. The diamond will be
treated essentially as if it were a worth
less crystal in which the desired results
are to be obtained, _i.e.~_ without regard
to the great value which the relation
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 which will be used in
the calculations.
_Reflection_
1. 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.
_Refraction_
1. When a ray of light passes from one
medium into a second of different density,
it is refracted as by the following equation:
_n_ sin _r_ "=" _n"'"_ sin _i_ (3)
where
_r_ "=" angle of refraction.
_i_ "=" angle of incidence.
_n_ "=" index of refraction of the second
medium.
_n"'"_ "=" index of refraction of the first
medium.
If the first medium is air, _n"'"_ "=" 1, and
equation becomes
_n_ sin _r_ "=" sin _i_ (2)
2. When a ray of light passes from one
medium into another optically less dense,
total reflection occurs for all values of the
angle of incidence above a certain critical
value, This critical angle is given by
equation
sin _i"'"_ "=" _n_ / _n"'"_ (4)
Or, if the less dense medium be air,
sin _i"'"_ "=" 1 / _n"'"_ (5)
3. The paths of the incident and of the
refracted ray lie in the same plane.
_Dispersion_
When a ray of light is refracted, dis
persion occurs, _i.e.~_ the ray is split up into
a band or spectrum of various colours,
owing to the fact that each colour has a
different index of refraction. The disper
sion is the difference between these indices
for extreme rays on the spectrum.
_Data_
In a diamond :
Index of refraction : _n_ "=" 2.417 (for a
sodium light)
dispersion: _d_ "=" .044
critical angle : _i"'"_ "=" 24"o" 26"'" (7)
DETERMINATION OF THE BEST ANGLES
AND THE BEST PROPORTIONS
_Postulate._  The design of a diamond or
of any gemstone must be symmetrical
about an axis, for symmetry and regularity
in the disposition of the facets are essential
for a pleasing result.
Let us now consider a block of diamond
bounded by polished surfaces, and let us
consider the effect on the path of light of
a gradual change in shape; we will also
observe the postulate and keep the block
symmetrical about its axis.
Let us take as first section one having
parallel faces (fig.~ 20), and let MM"'" be its
axis of symmetry. Let us for convenience
place the axis of symmetry vertically in
all future work, so that surfaces crossing it
are horizontal.
Consider a ray of light SP striking face
AB. It will be refracted along PQ and
leave by QR, parallel to SP (as we have
seen in studying dispersion). We also know
that if _NN"'"_ is the normal at Q, angles
NQP and QPM"'" are equal. Therefore, for
total reflection,
QPM"'" "=" 24"o" 26"'",
but at that angle of refraction the angle of
incidence SPM 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 light
passing in by the front of a gem passes out
again by the back without any reflection.
We can avoid parallelism by inclining
either the top or the bottom faces at an
angle with the direction AB. In the first
case we obtain the shape of a rosecut
diamond and in the second case that of a
brilliant cut. We will examine the rose
cut in the first instance.
=The Rose=
Consider (fig.~ 21) a section having the
bottom surface horizontal, and let us incline
the top surface AB at 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 reflection
occurs at AC. Now for this to be the case,
the minimum angle of incidence upon AC
must be 24~ 26"'". Let us draw such an
incident ray PQ. To ensure that no light
is incident at a smaller angle, we must make
the angle of refraction at entry 24"o" 26"'" and
arrange the surface of entry as shown, AB,
for we know that then no light will enter
at an angle more oblique to AB or more
vertical to AC. This gives to a a value of
twice the critical angle, _i.e.~_ 48"o" 52"'". Such
a section is very satisfactory indeed as
regards reflection, as, owing to its deriva
tion, all the light entering it leaves by the
front part. Is it also satisfactory as regards
refraction?
Let us follow the path of a ray of light of
any single colour of the spectrum, SPQRT
(fig.~ 22). Let _i_ and _r_ be the angles of
incidence upon and of refraction out of the
diamond.
At Q, PQN "=" RQN, and therefore in
triangles APQ and RCQ
angle AQP "=" angle CQR.
Also by symmetry A "=" C,
therefore
angle APQ "=" angle CRQ;
it follows that _i_ "=" _r_.
As the angle _i_ is the same for all colours
of a white ray of light, the various colours
will emerge parallel out of the diamond
and give white light. This is the funda
mental reason of the unpopularity of the
rose; there is no fire.
This effect may be remedied to a small
extent by breaking the inclined facet (figs.~
23 and 24), so that the angle be not the same
at entry as at exit. This breaking is harm
ful to the amount of light reflected which
ever way we arrange it; if we steepen the
facet near the edge, there is a large propor
tion of light projected backwards and being
lost, for we may take it that the spectator
will not look at the rose from the side of
the mounting (fig.~ 23). If, on the other
hand, we flatten the apex of the rose (fig.~ 24)
(which is the usual method), a leakage will
occur through its base. There is, of course,
no amelioration in the refraction if the
light passes from one facet to another
similarly placed (as shown in fig.~ 23, path
S"'"P"'"Q"'"R"'"T"'"). Taking the effect as a whole,
the least unsatisfactory shape is as shown
in fig.~ 24, with the angles _a_ about 49"o" and
30"o" for the base and the apex respectively.
The rose cut, however, is fundamentally
wrong, as we have seen above, and should
be abolished altogether. It is the high
cost of the material that is the cause of its
still being used in cases where the rough
shape is especially suitable, and then only
in small sizes. In actual practice the
proportions of the cut rose depend largely
upon those of the rough diamond, the
stone being cut with as small a loss of
material as possible. Generally the
values of _a_ are much below those given
above, _i.e.~_ 49"o" and 30"o", as where the
material is thick enough to allow such
steep angles it is much better to cut it
into a brilliant.
=The Brilliant=
_A. Back of the Brilliant_
Let us now pass to the consideration of
the other alternative, _i.e.~_ where the top
surface is a horizontal plane AB and
where the bottom surface AC is inclined
at an angle _a_ to the horizontal (fig.~
25). As before, we have to introduce
a third plane BC to have a symmetrical
section.
_First Reflection_
Let a vertical ray PQ strike AB. As the
angle of incidence 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"o" 26"'".
But
angle NRQ "=" angle QAR "=" _a_,
as AQ and QR, AR and RN are per
pendicular.
Therefore, for total reflection of a vertical
ray,
_a_ "=" 24"o" 26"'".
Let us now incline the ray PQ so that it
gradually changes from a vertical to a
horizontal direction, and let P"'"Q"'" be such
a ray. Upon passing into the diamond it
is refracted, and strikes AC at an angle
Q"'"R"'"N"'" where R"'"N"'" is the normal to AC.
When P"'"Q"'" becomes horizontal, the angle
of refraction T"'"Q"'"R"'" becomes equal to
24"o" 26"'". This is the extreme value attain
able by that angle; also, for total reflection,
angle Q"'"R"'"N"'" must not be less than 24"o" 26"'".
If we draw R"'"V, vertical angle VR"'"Q"'" "="
R"'"Q"'"T"'" "=" 24"o" 26"'", and
angle VR"'"N"'" "=" VR"'"Q"'" + Q"'"R"'"N"'"
"=" 24"o" 26"'" + 24"o" 26"'"
"=" 48"o" 52"'".
as before,
_a_ "=" angle VR"'"N"'",
and therefore
_a_ "=" 48"o" 52"'" (9)
_For absolute total reflection to occur at the
first facet, the inclined facets must make an
angle of not less than _48"o" 52"'"_ with the horizontal._
_Second Reflection_
When the ray of light is reflected from
the first inclined facet AC (fig.~ 26), it
strikes the opposite one BC. Here too the
light must be totally reflected, for other
wise there would be a leakage of light
through the back of the gemstone. Let us
consider, in the first instance, a ray of light
vertically incident upon the stone. The
path of the ray will be PQRST. If RN
and SN"'" are the normals at R and S respec
tively, then for total reflection,
angle N"'"SR "=" 24"o" 26"'".
Let us find the value of _a_ to fulfil that
condition:
angle QRN "=" angle QAR "=" _a_
as having perpendicular sides.
angle SRN "=" angle QRN
as angles of incidence and reflection.
Therefore
angle NRS "=" _a_.
Now let
angle N"'"SR "=" _x_
Then, in triangle RSC,
angle SRC "=" 90"o"  _a_
angle RSC "=" 90"o"  _x_
angle RCS "=" 2 "x" angle RCM
"=" 2 "x" ARQ "=" 2 (90"o"  _a_).
The sum of these three angles equals
two right angles,
90"o"  _a_ + 90"o"  _x_ + 180"o"  2_a_ "=" 180"o",
or
3_a_ + _x_ "=" 180"o"
3_a_ "=" 180"o"  _x_.
Now, _x_ is not less than 24"o" 26"'", therefore
_a_ is not greater than
a "=" 180  24"o" 26"'" / 3 "=" 51"o" 51"'".
Let us again incline PQ from the vertical
until it becomes horizontal, but in this case
in the other direction, to obtain the inferior
limit.
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,
TQR will be 24"o" 26"'". Draw RV vertical
at R.
Then
angle QRV "=" angle TQR "=" 24"o" 26"'"
angle VRN "=" _a_.
As before, in triang le RSC,
angle SRC "=" 90"o"  NRS "=" 90"o"  _a_  24"o" 26"'"
angle RCS "=" 2 (90"o"  _a_)
angle RSC "=" 90"o"  _x_
Then
90  _a_  24"o" 26"'" + 180"o"  2_a_ + 90"o"  _x_ "=" 180"o"
3_a_ + _x_ "=" 180"o"  24"o" 26"'" "=" 155"o" 34"'"
3_a_ "=" 155"o" 34"'"  _x_.
In the case now considered,
_x_ "=" 24"o" 26"'".
Then
3_a_ "=" 155"o" 34"'"  24"o" 26"'" "=" 131"o" 8"'"
_a_ "=" 43"o" 43"'" (10)
_For absolute total reflection at the second
facet, the inclined facets must make an angle
of not more than _43"o" 43"'"_ with the horizontal._
We will note here that this condition
and the one arrived at on page 66 are in
opposition. We will discuss this later, and
will pass now to considerations of refraction.
_Refraction_
_First case:_ _a_ is less than 45"o"
In the discussion of refraction in a
diamond, we have to consider two cases,
_i.e.~_ _a_ is less than 45"o" or it is more than 45"o".
Let us take the former case first and let
PQRST (fig.~ 28) be path of the ray. Then,
if SN is the normal at S, we know that for
total reflection at S angle RSN "=" 24"o" 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
ray will be broken too frequently and the
result will be disagreeable.
Therefore,
angle RSN "<" 24"o" 26"'" (11)
Suppose this condition is fulfilled and the
light leaves the stone along ST. It is re
fracted, and its colours are dispersed into
a spectrum. It is desirable to have this
spectrum as long as possible, so as to disperse
the various colours far away from each other.
As we know, this will give us the best
possible `fire.'
This result will be obtained when the
ray is refracted through the maximum
angle. By (11) the value for that angle
is 24"o" 26"'", and (11) becomes
angle RSN "=" 24"o" 26"'" for maximum dis
persIon.
But then the light leaves AB tangentially,
and the amount of light passing is zero.
To increase that amount, the angle of
refraction has to be reduced: the angle of
dispersion decreases simultaneously, but
the amount of light dispersed increases
much more rapidly. Now we know that
the angle of dispersion is proportional to
the sine of the angle of refraction. It is,
moreover, proved in optics that the amount
of light passing through a surface as at
AB is proportional to the cosine of the
angle of refraction. The brilliancy pro
duced is proportional both to the amount
of light and to the angle of dispersion, and
therefore to their product, and by the
theory of maxima and minima) will be
maximum when they are equal, _i.e.~_ when
the sine and cosine of the angle of refraction
are equal. For maximum brilliancy, there
fore, the angle of refraction should be 45"o".
This gives for angle RSN
sin RSN "=" sin 45 / 2.417 "=" .707 1 / 2.417 "=" .2930,
therefore
angle RSN "=" 17"o" for optimum brilliancy (12)
Let (fig.~ 28) QX and RY be the normals
at Q and R respectively, and let ZZ"'" be
vertical through R.
We know that
angle RQX "=" angle PQX "=" _a_,
therefore
angle PQR "=" 2_a_.
Produce QR to Q"'".
Then, as PQ and ZZ"'" are parallel,
angle ZRQ"'" "=" angle PQR "=" 2_a_.
Now, let
angle RSN "=" _x_ ( "=" 17"o" for optimum
brilliancy).
Then, as ZZ"'" and SN are parallel,
angle ZRS "=" _x_.
As they are complements to angles of
incidence,
angle QRC "=" angle SRB "=" _i_ (say),
but
angle Q"'"RB "=" angle QRC,
therefore
angle SRQ"'" "=" 2_i_.
In angle ZRQ"'" we have
angle ZRQ"'" "=" angle ZRS + angle SRQ"'"
2_a_ "=" _x_ + 2_i_ (13)
In triangle QCR
angle RCQ "=" 90"o"  _a_
angle QRC "=" _i_
angle QCR "=" 2 (90"o"  _a_),
therefore
(90  _a_) + _i_ + 180"o"  2_a_ "=" 180"o",
or
_i_ "=" 3_a_  90"o".
Introduce this value of _i_ in (13),
2_a_ "=" _x_ + 6_a_  180"o"
4_a_ "=" 180"o"  _x_
and giving _x_ its value 17"o",
4_a_ "=" 180"o"  17"o" "=" 163"o"
_a_ "=" 40"o" 45"'" (14)
If we adopt this value for _a_, the paths of
oblique rays will be as shown in fig.~ 29,
PQRST when incident from the left of the
figure, and P"'"Q"'"R"'"S"'"T"'" when incident from
the right. Ray PQRST will leave the dia
m?nd after the second reflection, but with a
smaller refraction than that of a vertically
incident ray, and therefore with less `fire.'
Oblique rays incident from the left are,
however, small in number owing to the acute
angle QRA with which they strike AC; the
loss of fire may therefore be neglected.
Ray P"'"Q"'"R"'"S"'"T"'" will strike AB at a
greater angle of incidence than 24"o" 26"'", and
will be reflected back into the stone. This
is a fault that can be corrected by the
introduction of inclined facets DE, FG;
ray P"'"Q"'"R"'"S"'"T"'" will then strike FG at an
angle less than 24"o" 26"'", and this angle can
be arranged by suitably inclining FG to
the horizontal so as to give the best possible
refraction. The amelioration obtained by
thus taking full advantage of the refraction
is so great that the small loss of light caused
by that arrangement of the facts is in
significant: the leakage occurs through the
facet CB, near C, where the introduction
of the facet DE allows light to reach CB
at an angle less than the critical. In a
brilliant, where CB is the section of the
triangular side of an eightsided pyramid,
the area near the apex C is very small, and
the leakage may therefore be considered
negligible.
_Second case:_ _a_ is greater than 45"o"
In this case the path of a vertical ray
will be as shown by PQRST in fig.~ 30, and
the optimum value for _a_, which may be
calculated as before, will be
a "=" 49"o" I3"'" (15)
As regards the vertical rays, this value
gives a fire just as satisfactory as (14)
(_a_ "=" 40"o" 45"'"); let us consider what happens
to oblique rays.
Rays incident from the left as _pqrstu_
may strike BC at an angle of incidence less
than the critical, and will then leak out
backwards. Or they may be reflected along
_st_, and may then be reflected into the stone.
Both alternatives are undesirable, but they
do not greatly affect the brilliancy of the
gem, because, as we have seen, the amount
of light incident from the left is small.
That incident from the right is, on the
contrary, large.
Let us follow ray P"'"Q"'"R"'"S"'"T"'". It will
be reflected twice, and will leave the
diamond after the second reflection, like
the vertically incident ray, but with a
smaller refraction, and consequently less
fire; most of the light will be striking
the face AB nearly vertically when leaving
the stone, and the fire will be very small.
This time it is impossible to correct the
defect by introducing accessory facets, as
the paths S"'"T"'" of the various oblique rays
are not localised near the edge B, but are
spread over the whole of the face; we are
therefore forced to abandon this design.
_Summary of the Results obtained for a,_
We have found that 
For first reflection, _a_ must be greater
than 48"o" 52"'".
For second reflection, _a_ must be less than
43"o" 43"'"
For refraction, _a_ may be less or more
than 45"o". When more, the best value is
49"o" 15"'", but it is unsatisfactory. When less,
the best value is 40"o" 45"'", and is very satis
factory, as the light can be arranged to
leave with the best possible dispersion.
Upon consideration of the above results,
we conclude that the correct value for _a_ is
40"o" 45"'", and gives the most vivid fire and
the greatest brilliancy, and that although
a greater angle would give better reflection,
this would not compensate for the loss due
to the corresponding reduction in dispersion.
In all future work upon the modern brilliant
we will therefore take
_a_ "=" 40"o" 45"'".
_B. Front of the Brilliant_
When arriving at the value of a "=" 40"o" 45"'",
we have explained how the use of that
angle introduced
defects which
could be cor
rected. by the use
of extra facets.
The section will
therefore be
shaped somewhat
as in fig.~ 31. It will be convenient to give
to the different facets the names by which
they are known in the diamondcutting
industry. These are as follows: 
AC and BC are called pavilions or quoins,
(according to their position relative
to the axis of crystallisation of the
diamond).
AD and EB are similarly called bezels
or quoins.
DE is the table.
FG is the culet, which is made very
small and whose only purpose is to
avoid a sharp point.
Through A and B passes the girdle of
the stone.
We have to find the proportions and in
clination of the bezels and the table. These
are best found graphically. We know that
the introduction of the bezels is due to the
oblique rays; it is therefore necessary to
study the distribution of these rays about
the table, and to find what proportion of
them is incident in any particular direction.
Consider a surface AB (fig.~ 32) upon which
a beam of light falls at an angle _a_. Let
us rotate the beam so that the angle
becomes _b_ (for convenience, the figure
shows the surface AB rotated instead to
A"'"B, but the effect is the same). The light
falling upon AB can be stopped in the
first case by intercepting it with screen
BC, and in the second with a screen BC"'"
where BCC"'" is at right angles to the
direction of the beam.
And if the intensity
of the light is uni
form, the length of
BC and BC"'" will be
a measure of the
amount of light fall
ing upon AB and AB"'" respectively.
Now
BC "=" AB sin _a_
BC"'" "=" A"'"B sin _b_ "=" AB sin _b_.
Therefore, other things being equal, the
amount of light falling upon a surface is pro
portional to the sine of the angle between
the surface and the direction of the light.
We can put it as follows: 
If uniformly distributed light is falling
from various directions upon a surface AB,
the amount of light striking it from any
particular direction will be proportional
to the sine of the angle between the surface
and that direction.
If we draw a curve between the amount
of light striking a surface from any parti
cular direction, and the angle between the.
surface and that direction, the curve will
be a sine curve (fig.~ 33) if the light is equally
distributed and of equal intensity in all
directions.
For calculations we can assume this to
be the case, and we will take the distri
bution of the quantity of light at different
angles to follow a sine law.
It is convenient to divide all the light
entering a diamond into three groups, one
of vertical rays and two of oblique rays,
such that the amount of light entering
from each group is the same. Now in the
sine curve (fig.~ 33) the horizontal distances
are proportional to the angles between the
table of a diamond and the direction of the
entering rays; the vertical distances are
proportional to the amount of light entering
at these angles. The total amount of light
entering will be proportional to the area
shaded. That area must therefore be
divided into three equal parts; this may
be done by integrals, or by drawing the
curve on squared paper, counting the
squares, and drawing two vertical lines on
the paper so that onethird of the number
of the squares is on either side of each line.
By integrals,
area "=" "I" sin _x_ _dx_ "=" cos _x_.
The total area "=" [ cos _x_] 0 180 "=" 1 + 1 "=" 2,
therefore
1/3 area "=" 2/3.
The value of _a_ corresponding to the vertical
dividing lines on the curve is thus given
by
cos _x_ "=" 1  2/3 "=" 1/3
cos _x_ "=" 1  4/3 "=" 1/3,
therefore
_x_ "=" 70 1/2"o" approximately
and
_x_ "=" 109 1/2"o".
Taking the value _x_ "=" 90"o" as zero for
reckoning the angles of incidence,
_i_ "=" 90"o"  70 1/2"o" "=" 19 1/2
and
_i_ "=" 90"o"  109 1/2"o" "=" 19 1/2
The corresponding angles of refraction are
sin _r_ "=" sin _i_ / _n_ "=" sin 19 1/2"o" / 2.417 "=" .333 3 / 2.1417 "=" .137 7
_r_ "=" 7"o" 52"'".
The range of the different classes is thus
as follows: 
Angle of incidence:
vertical rays 19 1/2"o" to "+"19 1/2"o"
oblique rays 90"o" to 19 1/2"o"
and "+"19 1/2"o" to "+"90"o"
Angle of refraction:
vertical rays 7"o" 52"'" to "+"7"o" 52"'"
oblique rays 24"o" 26"'" to 7"o" 52"'"
and "+"7"o" 52"'" to "+"24"o" 26"'".
The average angle of each of these classes
may be obtained by dividing each of the
corresponding parts on the sine curve in two
equal parts. The results are as follows: 
Angle of incidence:
vertical rays 0"o"
oblique rays 42"o"
and "+"42"o".
Angle of refraction:
vertical rays 0"o"
oblique rays 16"o"
and "+"16"o".
For the design of the table and bezels,
we have to know the directions and positions
of the rays leaving the stone. The values
just obtained would enable us to do so if
all the rays entering the front of the gem
also left there. We have, however, adopted
a value for _a_ (_a_ "=" 40"o" 45"'") which we know
permits leakage, and we have to take that
leakage into consideration.
The angle where leakage begins is in
clined at 24"o" 26"'" to the pavilion (fig.~ 24).
We have thus
Q"'"R"'"N"'" "=" 24"o" 26"'",
therefore
Q"'"R"'"A"'" "=" 90"o"  24"o" 26"'" "=" 65"o" 34"'".
Now in triangle AQ"'"R"'",
Q"'"R"'"A "+" AQ"'"R"'" "+" R"'"AQ"'" "=" 180"o",
therefore
AQ"'"R"'" "=" 180"o"  65"o" 34"'"  40"o" 45"'"
"=" 73"o" 41"'".
The limiting angle of refraction R"'"Q"'"T
is thus
"=" 90"o"  73"o" 41"'" "=" 16"o" 19"'",
corresponding to an angle of incidence of
sin _i_ "=" _n_ sin _r_ "=" 2.417 sin 16"o" 19"'"
"=" 2.417 "x" .281 "=" .678.
_i_ "=" 42 1/2"o".
Upon referring to the sine curve, we find
that the area shaded (fig.~ 34), which repre
sents the amount of light lost by leakage,
although not so large as if the same number
of degrees leakage had occurred at the
middle part of the curve, is still very ap
preciable, forming as it does about one
sixth of the total area. Just under one
half (exactly .493) of the light incident
obliquely from the right (fig.~ 25) is effective,
the other half being lost by leakage. Still,
the sacrifice is worth while, as it produces
the best possible fire.
The oblique rays incident from the right
range therefore 19 1/2"o" to 42 1/2"o", with an average
(obtained as before) of 30"o" 15"'". The corre
sponding refracted rays are 7"o" 52"'", 16"o" 19"'",
and 12"o" 0"'".
We have now all the information necessary
for the design of the table and the bezels.
_Design of Table and Bezels_ (fig.~ 35)
Let us start with the fundamental section
ABC symmetrical about MM"'", making the
angles ACB and ABC 40"o" 45"'".
The bezels have been introduced into the
design to disperse the rays which were
originally incident from the right upon the
facet AB. To find the limits of the table,
we have therefore to consider the path of
limiting oblique ray. We know that this
ray has an angle of incidence of 42 1/2"o" and
an angle of refraction of 16"o" 19"'". Let us
draw such a ray PQ: it will be totally
reflected along QR, if we make PQN "="
NQR, where QN is the normal. Now QR
should meet a bezel.
If the ray PQR was drawn such that
MP "=" MR, then P and R will be the points
at which the bezels should meet the table.
For if PQ be drawn nearer to the centre of
the stone, QR will then meet the bezel,
and if PQ be drawn further away, it will
meet the opposite bezel upon its entry into
the stone and will be deflected.
The first point to strike us is that no
oblique rays incident from the left upon
the table strike the pavilion AB, owing to
the fact that the table stops at P. We
will, therefore, treat them as nonexistent,
and confine our attention to the vertical
rays and those incident from the right.
Let us draw the limiting average rays
of these two groups, _i.e.~_ the rays of the
average refractions 0"o" and 12"o" passing
through P, PS, and PT. The length of the
pavilion upon which the rays of these two
groups fall are thus respectively CS and CT.
The rays of the first group P"'"Q"'"R"'"S"'" are
all reflected twice before passing out of
the stone, and make, after the second
reflection, an angle of I7"o" with the vertical
(as by eq.~ (12)). Of the rays of the second
group, most are reflected once only (P1Q1R1)
and make then an angle of 69 1/2"o" with the
vertical (this angle may be found by
measurement or by calculation). Part
of the second group is reflected twice
(P3Q3R3S3), and strikes the bezel at 29"o"
to the vertical. This last part will be
considered later, and may be neglected
for the moment.
We have to determine the relation be
tween the amount of light of the first
group and of the first part of the second
group. Now we know that the amount
of oblique light reflected from a surface on
pavilion AC is .493 of the amount of verti
cal light reflected (cp.~ fig.~ 34 and context).
If we take as limit for the oncereflected
oblique ray the point E (as a trial) on
pavilion BC, _i.e.~_ if it is at E that the girdle
is situated, then the corresponding point
of reflection for that oblique ray will be
Q2 (fig.~ 35). The surface of pavilion upon
which the oblique rays then act will be
limited by S and Q2, and as in a brilliant
the face AC is triangular, the surface will
be proportional to
SC"2"  Q2C"2".
Similarly, the surface upon which the
vertical group falls will be proportional to
TC"2".
Thus we have as relative amounts of
light 
for vertical rays TC"2"
for oblique rays .493 (SC"2"  QC"2").
The first group strikes the bezel at 17"o"
to the vertical, and the second at 69 1/2"o" to
the vertical. The average inclination to
the vertical will thus be
17 "x" TC"2" "+" 69 1/2 "x" .493 (SC"2"  QC"2") /
TC"2" "+" .493 (SC"2"  QC"2")
Let us draw a line in that direction
(through R, say), and let us draw a perpen
dicular to it through R, RE; then that
perpendicular will be the best direction for
the bezel, as a facet in that direction takes
the best possible advantage of both groups
of rays.
If the point E originally selected was not
correct, then. the perpendicular through R
will not pass through E, and the position
of E has to be corrected and the corre
sponding value of CQ2 correspondingly
altered until the correct position of E is
obtained.
For that position of E (shown on fig.~ 35),
measures scaled off the drawing give
CS "=" 2.67 CS"2" "=" 7.12
CT "=" 2.13 CT"2" "=" 4.54 CS"2"  CQ"2" "=" 4.57.
CQ2 "=" 1.60 CQ2"2" "=" 2.56
Therefore the average resultant inclina
tion will be
17 "x" CT"2" + 69 1/2 "x" .493 (CS"2"  CQ"2") / (CT"2" "+" .493 (CS"2"  CQ"2")
"=" 17 "x" 4.54 "+" 69.5 "x" .493 "x" 4.57 / 4.54 "+" .493 "x" 4.57
"=" 77.2 + 156.2 / 4.54 + 2.24 "=" 233.4 / 6.78 "=" 34.45 "=" 34 1/2"o"
to the vertical.
By the construction, the angle _b_, _i.e.~_ the
angle between the bezel and the horizontal,
has the same value
_b_ "=" 34 1/2"o".
The small proportion of oblique rays
which are reflected twice meet the bezel
near its edge, striking it nearly normally:
they make an angle of 29"o" with the vertical.
Facets more steeply inclined to the hori
zontal than the bezel should therefore be
provided there. The best angle for re
fraction would be 29"o" "+" 17"o" "=" 46"o", but if
such an angle were adopted most of the
light would leave in a backward direction,
which is not desirable. It is therefore
advisable to adopt a somewhat smaller
value; all angle of about 42"o" is best.
=Faceting=
The faceting which is added to the
brilliant is shown in fig.~ 43. Near the
table, `star' facets are introduced, and
near the girdle, `cross' or `half' facets
are used both at the front and at the back
of the stone.
We have seen that it is desirable to intro
duce near the girdle facets somewhat steeper
than the bezel, at an angle of about 42"o",
by which facets the twicereflected oblique
rays might be suitably refracted. The
front `half' facets fulfil this purpose.
We have remarked that the angle (42"o")
had to be made smaller than the best angle
for refraction (46"o") to avoid light being
sent in a backward direction, where it is
unlikely to meet either a spectator or a
source of light.
To obviate this disadvantage, a facet two
degrees steeper than the pavilion should
be introduced near the girdle on the back
side of the stone; for then the second
reflection of the oblique rays will send them
at an angle of 25"o" to the vertical (instead
of 29"o"), and the best value for refraction
for the front half facets will be between
25"o" "+" 17"o" "=" 42"o".
These values are satisfactory also as
regards the distribution of light; for now
the greater part of the light is sent not in
a backward, but in a forward, direction.
The facet two or three degrees steeper
than the pavilion is obtained in the brilliant
by the introduction of the back `half'
facet, which is, as a matter of fact, generally
found to be about 2"o" steeper than the
pavilion in wellcut stones. Where the
cut is somewhat less fine and the girdle is
left somewhat thick (to save weight), that
facet is sometimes made 3"o" steeper, or
even more, than the pavilion.
The `star' facet was probably intro
duced to complete the design of the brilliant,
which without its use would be lacking in
harmony, but which its introduction makes
exceedingly pleasing from the point of view
of the balance of lines.
Let us examine the optical consequences
of the use of `star' facets.
On the one hand, their inclination 
about 15"o" to the horizontal  permits a
certain amount of light to leave the stone
without being sufficiently refracted. On
the other, they diminish the area of the
bezels and consequently decrease the leak
age of light which occurs through the bezel
and the opposite pavilion (owing to the
surfaces being nearly parallel). They also
cause a somewhat better distribution of
light, for they deflect part of the rays
which would otherwise have increased the
strength of the spectra refracted by the
bezels, and create therewith spectra along
other directions; it is true that, as seen
above, these spectra will be shorter. But
they will be more numerous; and though
the `fire'  as consequent from the great
dispersion of the rays of light  will be
slightly diminished, the `life'  if we may
term `life' the frequency with which a
single source of light will be reflected and
refracted to a single spectator upon a
rotation of the stone  will be increased to
a greater degree. And if we take into ac
count the decrease in the leakage of light,
we may conclude that the introduction of
the stars, on the whole, is decidedly ad
vantageous in the brilliant.
=Best Proportions of a Brilliant=
We have thus as best section of a brilliant
one as given in fig.~ 35, ABCDE, where
_a_ "=" 40"o" 45"'"
_b_ "=" 34"o" 30"'".
DE is obtained from PR in fig.~ 35.
If we make the diameter AB of the stone
100 units, then th~ main dimensions are
in the following proportions (fig.~ 35): 
Diameter AB 100
Table DE 53.0
Total thickness MC 59.3
Thickness above girdle MM"'" 16.2
,, below ,, M"'"C 43.1
Fig.~ 36 shows the outline of a brilliant
with these proportions.
These proportions can be approximated
as follows: 
In a wellcut brilliant the diameter of
the table is onehalf of the total diameter,
and the thickness is sixtenths of the total
diameter, rather more than onequarter
of the thickness being above the girdle and
rather less than threequarters below.
It is to be noted here that a different pro
portion is generally stated for the thickness
above the girdle (`onethird of the total
thickness'), both in works upon diamonds
and by diamond polishers themselves. It
is true that diamonds were cut thicker
above the girdle and with a smaller table
before the introduction of sawing, for
then the table was obtained by grinding
away a corner or an edge of the stone, and
the loss in weight was thus considerable,
and would have been very much greater
still if the calculated proportions had been
adopted. With the use of the saw, the loss
in weight was enormously reduced and the
manufacture of sawn stones became there
fore much finer and more in accordance
with the results given above. It is a
remarkable illustration of conservatism that
although diamonds have been cut for de
cades with 1/4 (approximately) of the thick
ness above the girdle, yet even now the
rule is generally stated as 1/3 of the
thickness.
Stones are still cut according to that
rule, but then they are not sawn stones as
a rule, and the thickness is left greater
to diminish the loss in weight. The
brilliancy is not greatly diminished by
making the stone slightly thicker over the
girdle.
=Comparison of the theoretically best
Value with those used in Practice=
In the course of his connection with the
diamondcutting industry the author has
controlled and assisted in the control of
the manufacture of some million pounds'~
worth of diamonds, which were all cut
regardless of loss of weight, the only aim
being to obtain the liveliest fire and the
greatest brilliancy. The most brilliant
larger stones were measured and their
measures noted. It is interesting to note
how remarkably close these measures, which
are based upon empirical amelioration and
ruleofthumb correction, come to the
calculated values.
As an instance the following measures,
chosen at random, are given (the dimensions
are in millimetres): 
=Table I=
_a_ 40 3/4"o" 40 3/4"o" 40"o" 41"o" 41"o"
_b_ 35"o" 35"o" 34 1/2"o" 33"o" 34"o"
AB 7.00 7.08 6.50 21.07 9.12
MC 4.12 4.35 3.61 12.34 5.47
MM"'" 1.08 1.32 0.85 3.31 1.61
These measures, worked out in percentage
of AB, give: 
=Table II=
_a_ 40 3/4"o" 40 3/4"o" 40"o" 41"o" 41"o" 40"o" 42"'" 40"o" 45"'"
_b_ 35"o" 35"o" 34 1/2"o" 33"o" 34"o" 34"o" 18"'" 34"o" 30"'"
AB 100 100 100 100 100 100 100
MC 58.7 61.4 55.4 58.5 60 58.9 59.3
MM"'" 15.7 18.6 13.3 15.7 17.8 16.2 16.2
M"'"C 43.0 42.8 42.1 42.8 42.2 42.6 43.1
In the seventh column the averages of
the measures are worked out, and the eighth
gives the calculated theoretical values. It
will be noted that the values of _a_, _b_, and
MM"'" correspond very closely indeed, but
that MC and M"'"C are very slightly less than
they should be theoretically.
The very slight difference between the
theoretical and the measured values is due
to the introduction of a tiny facet, the
collet, at the apex of the pavilions. This
facet is introduced to avoid a sharp point
which might cause a split or a breakage
of the diamond.
What makes the agreement of these
results even more remarkable is that in the
manufacture of the diamond the polishers
do not measure the angles, etc.~, by any
instrument, but judge of their values en
tirely by the eye. And such is the skill
they develop, that if the angles of two
pavilions of a brilliant be measured, the
difference between them will be in
appreciable.
We may thus say that in the presentday
wellcut brilliant, perfection is practically
reached: the highclass brilliant is cut as
near the theoretic value's as is possible in
practice, and gives a magnificent brilliancy
to the diamond.
That some new shape will be evolved
which will cause even greater fire and life
than the brilliant is, of course, always
possible, but it appears very doubtful; and
it seems likely that the brilliant will be
supreme for, at any rate, a long time yet.
=(pr.~ 1605.)
printed in great britain by neill and co.~, ltd.~, edinburgh.=