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DIAMOND DESIGN



DIAMOND DESIGN

A STUDY OF THE REFLECTION

AND REFRACTION OF LIGHT IN

A DIAMOND



BY



MARCEL TOLKOWSKY

B.Sc, 'A.C.G.I.



WITH S7 ILLUSTRATIONS




Xon^on:

E. & F. N. SPON, Ltd., 57 HAYMARKET, S.W. i
IWew ll)orft:

SPON & CHAMBERLAIN, 120 LIBERTY STREET
I919



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CONTENTS



INTRODUCTION . . . .


PAGE

5


PART L-


-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



6 DIAMOND DESIGN

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 well-cut 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-



INTRODUCTION 7

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 I
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



HISTORICAL 9

of the 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.
Ji^f: It was wholly unknown in the Vedic
period, from which no specific names for
precious stones are handed down at all.^
The earliest systematic reference appears
to be in the Arthagastra 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, ^ able to scratch metal,

^ Berthold Laufer, The Diamond : a Study in
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 DESIGN '

refractive and brilliant. In the Milinda-
panha (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. ^

It is not known with certainty when
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 men-
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.

^ Laufer, loc. cit.



HISTORICAL II

tioned. It is, however, likely that, where '
the pohshing of other precious stones was
accomplished in that manner, that of
diamonds themselves cannot have been
entirel}^ unknown. What pohshing 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,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 of
his visit (1665) the Indians were pohshing
over the natural faces of the crystal, and
preferred, therefore, regularly crystaUised
gems. They also used the knowledge they
had of grinding diamonds to remove faulty
places hke 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 Turqiiie, en Perse et mix
Indes (1679).



12 DIAMOND DESIGN

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 f^g. i.
The polishing was
the work of a Venetian, Hortensio Borgis,
to whom it was given for that purpose by




Fig.



HISTORICAL



13




its owner, the Great Mogul Aurung Zeb, of

Delhi. This kind of cut is characteristic

of most of the

large Tn d i a n

stones, such as

theOrlow(fig.2),

which is now the

largest diamond

of the Russian

crown jewels and weighs 193! cts. The Koh-i-

Noor (fig. 3), now among the British crown

jewels, was of a somewhat similar shape

before recutting.
It weighed then
186 cts.

Ta vernier also



Fig. 2.




Fig. 3.



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.



14



DIAMOND DESIGN




Fig. 4.



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 pohsh 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-



\ /


/








\




Fig.



HISTORICAL 15

ing ill 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
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
^ Jacobson's Technologisches Worterbuch (1781).



i6 DIAMOND DESIGN

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



HISTORICAL ' 17

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.''

^ Robert de Berquem, Les merveilles des Indes :
Traite des pierres precieuses (Paris, in-40, 1669),
p. 12 : " Louis de Berquem Tun de mes ayeuls a
trouve le 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'est-

2



i8 DIAMOND DESIGN

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 cast-iron
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.

a-dire tout a fait bruts, sans ordre et sans^ grace,
sinon quelques faces au hasard, irregulieres et mal
polies, tels enfin que la nature les produit. II mit
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 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 thfe 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 why
it is not at all certain that the}^ were



20 DIAMOND DESIGN

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 especialh^ 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 flat



HISTORICAL 21

pieces of rough or split diamond. Such
material is even ' now frequently cut into
roses, chiefly in the smaller sizes.




Fig. 7.

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. f^°- ^•

We owe the introduction of the brilliant
in the middle of the seventeenth centurv
to 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 DESIGN



of the design of fig. 9. They had sixteen
facets, excluding the table, on the upper side.




Fig. 9,



They are called double-cut brilliants. Vincent
Peruzzi, a Venetian polisher, increased the
number of facets from sixteen to thirty-




FiG. 10.



two (fig. 10) (triple-cut brilliants), thereby
increasing very much the fire and brilliancy
of the cut gem, which were already in the
double-cut brilliant incomparably better



HISTORICAL 23

than in the rose. 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 the
side facets, so that even if 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 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



24 DIAMOND DESIGN

p. 17) diamonds were made absolutely
circular in plan (fig. 37). The gradual
shrinking-in of the corners of an old-cut
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 present-day
brilliant.^

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

^ 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.



HISTORICAL



25



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 well-cut brilliant is
polished.



Part II

OPTICAL

It is to light, the play of Ught, 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 Ught 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,

26



OPTICAL 27

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),



28 DIAMOND DESIGN

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




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 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



OPTICAL 29

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



30 DIAMOND DESIGN

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 studied very
simply with a few pins and a mirror placed
at right angles upon a flat sheet of paper.

A plan of the arrangement is shown in
fig. 13. The experiment is as follows : —

I. A straight line A B 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.



OPTICAL



^.i



Then the eye is placed in line with FO at
I, 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




Fig. 13.

their images appear in the mirror to lie
along PQ continued.

If the eye is now sighted from position
2 along S R, Q and P will appear in the
mirror to lie on S R continued.

The mirror is now removed, P and S R
are joined and will be found to intersect
on A B at M. If a perpendicular M N be



32 DIAMOND DESIGN

erected on AB at that point, the angles
N M P and N M S will be found equal.


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