TS 753.T6

^Opy

1

DIAMOND DESIGN

DIAMOND DESIGNA STUDY OF THE REFLECTION AND REFRACTION OF LIGHT IN A DIAMOND

BY

MARCEL TOLKOWSKYB.Sc, 'A.C.G.I.

WITH

S7

ILLUSTRATIONS

E.

&

F. N.

SPON,

Xon^on: Ltd., 57 HAYMARKET,IWewll)orft:120

S.W.

i

SPON & CHAMBERLAIN,

LIBERTY STREET

I919

-X?)15.\

s

4di(.^'^^'J.I

3^4.

CONTENTSINTRODUCTION. . . .

DIAMOND DESIGNINTRODUCTIONThis bookis

written principally for students

and jewellers, and more particularly for diamond manufacturers and diamond cutters and polishers. Theof precious stones

author

will

follow

the

evolution

of

the

shape given to a cut diamond, and discussthe values of the various shapes and the

reason for the discarding of the old shapes

and theIt is a

practically^ universal

adoption of

the brilliant.

remarkable fact that, although the

art of cutting a

diamond has been known

for

more than two thousand years, it is entirely empirical, and that, though many keen contemporary minds have been directed upon the diamond, and the list of books written on that subject increases rapidly, yet

6

DIAMOND DESIGN

nowhere] can one find any mathematical

work determining the best shape for that gem. The present volume's chief aim isthe calculation of that shape.

The

calculations

have been

made

as

simple as possible, so as not to be beyondthe range of readers with a knowledge of

elementary geometry, algebra, and trigo-

was found that the accuracy of the results would be impaired without the introduction of more advanced mathematics, these have been used, and graphical methods have beennometry.

Where, however,

it

explained as an alternative.

Theof

results of the calculations for the

form

brilliant

now

in

use were verified by

actual mensuration from well-cut brilliants.

The measuresat the

of these brilliants are given

end

of the

volume both

in a tabulated

and in a graphical form.

It will

be seen

how

strikingly near the actual measures are

to the calculated ones.

The method usedwill

in the present

worksemi-

be found very useful for the designother transparent precious and

of

INTRODUCTIONprecious stones, althoughit

7

will

be found

advisable in the case of stones of an agreeable

colour

to

cut

the

gem somewhat

thicker than the calculations warrant, soas to take full advantage of the colour.

The same remark applies to diamonds of some exceptional and beautiful colour, likeblue or pink, where the beauty or the valueof the stone increases with the depth ofits colour.

Part

I

HISTORICALand early Indian literature we turn when we want to find the origin of diamond cutting, for IndiaItis

to Indian manuscripts

has always been regarded as the natural

and ancient home1728,

of thefirst

diamond.found:

It is

there that they were

upof

to

the

date of the discoverydeposits,

the

Brazilian

practically

the

whole

world'ssources.

supply was derived from Indian

They

are

and beds of separated from the matrixvalleys

found there in the streams, and also,

were formed, in strata of that have since been covered by twelveto sixteen feet of earth tionof

which they detrital matterin

later

centuries.

by the accumulaDiamonds haveknowledge

existed in these deposits within the reachof

man

for

many

ages, but the

HISTORICALof the

9

diamond

as a

gem

or as a crystal-

with

exceptional

qualities

does

not

go

back in India to the unfathomable antiquity fo which books on diamonds generally refer. It was wholly unknown in the Vedic Ji^f: period, from which no specific names for precious stones are handed down at all.^

Thethird

earliest

systematic reference appears

to be in the Arthagastra of Kautilya (about

where the author mentions six kinds of diamonds classified according to their mines, and describes them as differing in lustre and hardness. He also writes that the best diamondscenturyB.C.),

should be large, regular, heavy, capable ofbearing^

blows, ^Laufer,

able

to

scratcha

metal,Study in

Berthold

The Diamond

:

Chinese and Hellenistic Folklore (Chicago, 1915). ^ This legend of the indestructibihty of the diamond,

which reappears in many other places, and to which the test of the diamond's capacity of bearing the strongest blows was due, has caused the destruction of perhaps a very large number of fine stones, -^he legend was further embroidered by the remark that if the diamond had previously been placed in the fresh and still warm blood of a ram, it could then be broken, but with great difficulty. This legend was

10

DIAMOND DESIGNandbrilliant.

'

refractive

In the Milinda-

panha (Questions

King Milinda) (about first century B.C.) we read that the diamond ought to be pure throughout, and that it is mounted together with the most costly gems. This is the first manuscript in which the diamond is classed as a gem.ofItis

therefore permissible

to

estimate

with a sufficient degree of accuracy thatthe diamond becamethe

knownits

in India during

BuddhistB.C.,

period,

about

the

fourth

century

and that

use as a

gemwhen

dates from that period.Itis

not

known with

certainty

and where the art of grinding or polishing diamonds originated. There is as 3^et no source of ancient Indian literature in which the polishing of diamonds is distinctly set forth, although the fact that diamond is used for grinding gems generally is menstill

thirteenth

current in Europe as late as the middle of the century. The actual fact is that the

diamond, although exceedingly hard (it is the hardest substance known), can easily be split by a light blow along a plane of crystallisation.^

Laufer,

loc. cit.

HISTORICALtioned.Itis,

II'

however, likely that, where

the pohshing of other precious stones was accomplished in that manner, that of

diamonds themselves cannot have been What pohshing there entirel}^ unknown. was must at first have been limited tothe smoothing of the faces of the crystals The first description as they were found.of

diamonds is given by Tavernier,i a French jeweller who travelled through India, and to whom we owe most of our knowledge of diamond cutting in India in the seventeenth century. At the time ofcuthis visit (1665) the Indians

were pohshing

over the natural faces of the crystal, and preferred, therefore, regularly crystaUised

gems.

had

of

knowledge they grinding diamonds to remove faulty

They

also used the

places hke spots, grains, or glesses.fault

If

the

was too deep, they attempted to hide it by covering the surface under which it lay with a great number of small facets. It appears from Tavernier's writings that1

Tavernier,

Voyage en

Turqiiie,

en Perse

et

mix

Indes (1679).

12

DIAMOND DESIGNwere

there

India at

European polishers in that time, and that it was to themalso

the larger stones were given for cutting.

Whether they had

learnt

the

art

inde-

pendently or from Indians and attainedgreater proficiency than they, or whether

they were acting as instructors and teachingthe Indians auncertain.in

new

or a forgotten art, is

Both views are equally likely the present state of research upon that:

subject

at the time of Tavernier's visit,in

diamond cutting had been known for more than two centuries.

Europe

Among

the several remarkable gems that

Tavernier describes,the most noteworthyis

the one

known

as

the

Great

Mogul.

This diamond was ofa weight of 280cts.

andFig.

was

cutf^g.

asi.

sketched in

Theto

polishing

was

the work of a Venetian, Hortensio Borgis,

whom

it

was given

for that purpose

by

HISTORICALits

13

owner, the Great Mogul Aurung Zeb, of

Delhi.of

This kind of cutof

is

characteristic

most

the

largestones,

Tn d i a nsuch as

theOrlow(fig.2),

which is now thelargestof

diamondRussianFig.2.

the

crown jewels and weighs 193! cts. The Koh-iNoor (fig. 3), now among the British crown jewels, was of a somewhat similar shapebefore recutting.It

weighed thencts.

186

Ta vernier alsomentions severalothercutFig.3.

types

of

whichin

he

met(fig.

India.

The Great Table

1642, weighed 242 cts.

which he saw in Both the Great Table4),

and the Great Mogul seem to have disappeared it is not known what has become of them since the seventeenth century.:

14

DIAMOND DESIGNVarious other shapes are described, such

as point stones, thick stones, table stones(fig. 5),

etc.

But theall

chief characteristic

remains

:

these

Fig.

4.

diamonds have been cut with one aim constantly in view ^how

to

pohsh

the

stone

with

the

smallest

possible loss of weight.

As a consequence

the polishing was generally accomplished

by covering thefacets,

surface of the

stone with a large

number

\

/

of

of

and the original shape the rough gem was, as far as

possible, left unaltered.

was mentioned before that the art of diamond polishing had already been known in Europe for several centuries when Tavernier We have as yet no cerleft for India. tain source of information about diamond cutting in Europe before the fourteenth century. The first reference thereto mentions that diamond polishers were workIt

HISTORICALingill

15

Nurnberg (Germany) in 1375, where they formed a guild of free artisans, to which admission was only granted after an apprenticeship of five to six years. We do not know, however, in what shape and by what method the stones were cut.

^

It is in the fifteenth

century that Euro-

pean diamond cutting begins to become

moreis

definite,

more

characteristic.

And

it

from that time that both on its technical and artistic sides progress is made at a rate, slow at first, but increasing rapidlylater.

It is

not

difficult to find

the chief reason

for that change.

UpTo

to that time, diamonds had almost

exclusively been used

by

princes or priests.

power and wealth ^in those days diamonds were credited with extraordinary powers they were supposed to protect the wearer and to bring him luck. Princes also found them convenient, as they have great value

princes they were an

emblem

of

:

^

Jacobson's Technologisches Worterbuch (1781).

i6

DIAMOND DESIGNand couldof

for a very small weight,

easily

be carried in case of

flight.

Priests used

themset

in;

the

ornamentsthe heads

templesstatues

or

churchesas

they have not infrequently beenof

eyes in

of

Buddha.

became the fashion for women to wear diamonds as jewels. This fashion was started by AgnesIn the fifteenth centuryit

Sorel (about 1450) at the Court of Charles

VII

of France,all

and gradually spread from

there to

the Courts of Europe.

This resulted in a very greatly increased

demand, and gave a strong impulse to the development of diamond polishing. The production increased, more men applied their brains to the problems that arose, and, as they solved them and the result of their work grew better, the increasingattractivenessof

thea

thethe

demand and gaveart.

gem increased new impulse to

At the beginning of the fifteenth century a clever diamond cutter named Hermannestablished a factory in Paris, where his

HISTORICALwork met withsuccess,

'

17

and where the

industry started developing.

In or about 1476 Lodewyk (Louis) van Berquem, a Flemish polisher of Bruges,

introduced absolute symmetry in the disposition of the facets,

and probably

also

improved the process of polishing. Early authors gave credence to the statement of one of his descendants, Robert van Berquem, 1 who claims that his ancestor had invented the process of polishing the diamond by its own powder. He adds '' having ground off redundant After material from a stone by rubbing it against:

another one (the processpractice as'

known

in

modern

bruting

'

or cutting), he col-

lected the

powder produced, by means of which he polished the diamond on a mill and certain iron wheels of his invention.''^

Robert de Berquem,:

Les merveilles des Indes(Paris,in-40,

Traite des " Louis de p. 12

pierres precieuses

1669),

Berquem Tun de mes ayeuls a

premier Tinvention en mil quatre cent soixante-seize de les tailler avec la poudre de diamant meme. Auparavant on fut contraint de les mettre en oeuvre tels qu'on les rencontrait aux Indes, c'est2trouvele

i8

DIAMOND DESIGNAs has already been shown, we know now

diamonds were polished at least a century before Lodewyk van Berquem lived. And as diamond is the hardest substance known, it can only be polished by its own powder. Van Berquem cannot thus havethat

invented that part of the process.

He maycast-iron

perhaps have introduced some important

improvement

like

the

use

of

polishing wheels, or possibly have discovered

a more porous kind of cast iron

one

on

which the diamond powder finds a better hold, and on which polishing is thereforecorrespondingly speedier.a-dire tout a fait bruts, sans ordre et sans^ grace, sinon quelques faces au hasard, irregulieres et mal II mit polies, tels enfin que la nature les produit. deux diamants sur le ciment et apres les avoir egrises Tun contre I'autre, il vit manifestement que par Ic moyen de la poudre qui en tombait et a Faide du moulin et certaines roues de fer qu'il avait inventees, il pourrait venir a bout de les polir parfaitement, meme de les tailler en telle maniere qu'il voudrait.

Charles devenu due de Bourgogne lui mit trois grands diamants pour les tailler avantageusement selon son adresse. II les tailla aussitot, I'un epais, Tautre faible et le troisieme en triangle et il y reussit si bien que le due, ravi d'une invention si surprenante, lui donna 3000 ducats de recompense."

HISTORICAL

19

What Van Berquem probablyginateis,

did

ori-

as already stated, rigid

in the design of the cut stone.

symmetry The intro-

duction of the shapeorbrioletteis

known

as pendeloque

generally ascribed to him.thfe

The Sancy and

Florentine, which are

both cut in this shape, have been said

Fig.

6.

by some to have been polished by him. The Sancy (53 J cts.) belongs now to the Maharaja of Guttiola, and the Florentine (fig. 6), which is much larger (133 1 cts.), is at present among the Austrian crown jewels. The history of both these gems is, however, very involved, and they may have been confused at some period or other with similar stones. That is whyit

is

not at

all

certain

that

the}^

were

20

DIAMOND DESIGNVan Berquem.At anyrate,

the work of

they are typical ofintroduced.

the kind of

cut he

The pendeloque shape did not meet with any very wide success. It was adopted in the case of a few large stones, but was gradually abandoned, and is not used to any large extent nowadays, and then in a modified form, and only when the shapeof the

rough stone

is

especialh^ suitable.

This unpopularity was largely due to thefact that,

although the loss of weight in

cutting was fairly high, the play of light

within the stone did not produce sufficientfire

or brilliancy.

About the middle of the sixteenth century a new form of cut diamond was introduced. It is known as the rose or rosette, and was made in various designs and proportions (figs. 7 and 8). The rose spread rapidly and was in high vogue for about a century, as it gave a more pleasant effect than the pendeloque, and could be cut with a much smaller loss of weight. It was also foundvery advantageous in the polishing offlat

HISTORICALpieces of rough

21

or split diamond.'

Such

material

is

even

now

frequently cut into

roses, chiefly in the smaller sizes.

Fig.

7.

In

the

chapter

diamonds it will to be made thick (somewhat thicker thaninfig.

upon the design of be shown that roses have

7) for

the loss of light to be small,flatter

and that theof light.

the

rose the bigger the lossIt will also

befire

seen there that theof

a

rose

cannot

be

very high.

These faults

caused the rose to be

superseded by thebrilliant.f^^

Weto

owe the introduction

of the brilliant

in the middle of the seventeenth centurv

Cardinal Mazarin

or

at

any rate to

his influence.

As a matter of fact, the first brilliants were known as Mazarins, and were

22

DIAMOND DESIGNfig.

of the design offacets,

9.

They had

sixteen

excluding the table, on the upper side.

Fig.

9,

They are called double-cut brilliants. VincentPeruzzi, a Venetian polisher, increased the

number

of facets

from sixteen to thirty-

FiG. 10.

two

(fig.

10) (triple-cut brilliants), thereby

increasing veryof the cut

much

the

fire

and

brilliancy

gem, which were already in the double-cut brilliant incomparably better

HISTORICALthan in therose.

23

Yet diamonds of that cut, when seen nowadays, seem exceedingly dull compared to modern-cut ones. This dullness is due to their too great thickness, and to a great extent also to the difference in angle between the corner facets and theside facets, so that evenif

the

first

were

Fig. II

polished to the correct angle (which they

were not) the second would be cut too steeply and give an effect of thickness.Old-cut brilliants, as the triple-cut brilliantsare generally called, were atfirst

modified

by making the size and angle of the facets more uniform (fig. 11), this bringing about a somewhat rounder stone. With the introduction of mechanical bruting or cutting(an operation distinct

from polishing

;

see

24p.

DIAMOND DESIGN17)

diamonds werein

circular

plan

(fig.

made absolutely The gradual 37).less

shrinking-in of the corners of an old-cutbrilliant

necessitated

a

thicklyfire

cut

stone with a consequent increasinglife,

and

until a point of

maximumis

brilliancy

was reached.brilliant.^

This

the

present-day

Other designs for thetried,

brilliant

have beenloss

mostly attempts to decrease the

of weight in cutting without impairing the

brilliancy of the

diamond, but they haveherethat

not met with success.

We may

note

the

general

trend of European diamond polishing as

opposed to Indianfor^

is

the constant search

greater brilliancy,

more

life,

a more

Some American

writers claim that this change

from the thick cut to that of maximum brilliancy was made by an American cutter, Henry D. Morse. It was, however, as explained, necessitated by the absolute roundness of the new cut. Mr Morse may haveinvented it independently in America. But it is highly probable that it originated where practically all the world's diamonds were polished, in Amsterdam or Antwerp, where also mechanical bruting was firstintroduced.

HISTORICALvividfire

25

in

the diamond,

regardless

of

The weight of diamond removed by bruting and by polishing amounts even in the most favourable casesthe loss of weight.to 52 per cent, of the original rough weightfor a perfectly cut brilliant.

In the next

chapters the best proportions for a brilliantwill

be calculated without reference to theit

shape of a rough diamond, andseen

will

be

howthe

startlingly

near the calculatedwell-cutbrilliantis

values

modern

polished.

Part II

OPTICALItis

to light, the play of Ught, its reflectionits refraction,itsfire,

and

thatits

a

gem owes

its

brilliancy,

colour.

We

have

therefore to study these optical properties in order to be able to apply them to the

the calproblem we have now before us culation of the shape and proportions of a perfectly cut diamond.:

Of the total amount

of light that fallsis

upon aflected;

material,

part

returned or reit,

the remainder penetrates intoit

andfirst

crosses

part of

The the Ught produces what isor is absorbed

by

it.

termedif

the

''lustre"is

of

the

material.

The second partthe materialis

completely absorbedIfit

black.will

is

partly

absorbed the material26

appear coloured,

OPTICALandif

27it

transmitted unaltered

will

appear

colourless.

The diamonds used as gems are generallycolourless or only faintly coloured;

it

can

be taken thatof the

all

the light that passes into

the stones passes out again.

The

lustre

diamondcalled

is

peculiar to that gem,

adamantine for that reason. It is not found in any other gem, although zircon and demantoid or olivine have ais

and

lustre approaching

somewhat

to the ada-

mantine.In

gem

stones of the

same kind and

ofit

the same grade of polish,of the

we may take

that the lustre only varies with the area

gem

stone exposed to the light, and

that

it is

independent of the type of cut

or of the proportions given to theso far as theyis

gem;

(in

do not affect the area) why gems where the amount ofis

thislight

thatis

reflected

upon

striking the surfaceof the light thatis

great, or

where much

penetrates into the stone

absorbed and

does not pass out again, are frequently cut in such shapes as the cabochon (fig. 12),

28

DIAMOND DESIGNin that

so as to get as large an area as possible,

and

way

take

full

advantage of

the lustre.

In a diamond, the amount of light reflected

from the surfacea

is

much;

smaller

that that penetrating into the stoneover,

more-

diamond

is

practically

perfectly

transparent, so that

all

the light that passes

into the stone has to pass out again.

This

Fig. 12.

is

why

lustre

may

be ignored in the working

out of the' correct shape for a diamond,

and why any variation in the amount of light reflected from the exposed surface due to a change in that surface may beconsidered as negligible in the calculations.

The

brilliancy

or,

as

it

is

sometimesa

termed, thelight in the

''fire''

or the

''life'' of

gem

thus depends entirely upon the play of

gem, upon the path of raysgem.If

of light in the

a

gem

is

so cut or

designed that every ray. of light passing

OPTICALintoit

29

follows the best path possible for

producing pleasing effects upon the eye,then the

gem

is

perfectly cut.

The whole

art of the lapidary consists in proportion-

ing his stone and disposing his facets soas to ensure this result.Ifits

we want

to design a

gem

or to calculateit is

best shape and proportions,

clear

knowledge to be able to work out the path of any ray of light passing through it. This knowledge comprises the essential part of optics, and the laws which have to be made use ofthatsufficient

we must have

are the three fundamental ones of reflection,refraction,

and

dispersion.

ReflectionReflection occurs at the surface which

separates two different substances or media

a portion or the whole of the light striking

thrown back, and does not cross over from one medium into another. This is the reflected light. There are difthat surfaceis

ferent kinds of reflected light according to

the nature of the surface of reflection.

If

30

DIAMOND DESIGNis

that surface

highly polished, as in the

case of mirrors, or polished metals or gems,

the reflection

is

perfect

and an imagealso

is

formed.or

The

surface

may

be dull(as

matt to a greater or smaller extentreflected lightis

in the case of, say, cloth, paper, or pearls).

The

then more or

less

scatteredIt is

and

diffused.

kind of reflection that is of importance to us here, as diamond, owing to its extreme hardness, takes a very highthefirst

grade of polish and keepsfor ever.

it

practically^

The laws

of reflection

can be studied verysheet of paper.is

simply with a few pins and a mirror placedat right angles

upon a

flat

Afig.I.

plan of the arrangement13.

shown:

in

The experimentstraight line

is

as followsis

A

AB

drawn uponstood on the

the paper, and the mirror

is

paper so that the plane of total reflection (i.e. the silvered surface) is vertically overthatline.

Two

pins

P and Qaway

are stuck

anyhow on the

paper,

one as near theas possible.

mirror and the other as far

OPTICALThen the eyeI,

^.i

is

placed in line with

FO

at

Without moving the eye, two more pins R and S are inserted, one near to and the other far from the mirror, in such positions thatso

that

Q

is

hidden by P.

Fig. 13.

their

images appear in the mirror to

lie

alongIf

PQ

continued.is

the eye

now

sighted from positionwill

2 along S R,

Q and Pon Sis

appear in the

mirror to

lie

R

continued.

The mirroron

now removed, PIf

and S

Rbe

are joined and will be found to intersect

AB

at M.

a perpendicular

MN

32

DIAMOND DESIGNand The above experiment may be repeated

erected on

NMP

AB at that point, the angles N M S will be found equal.

along other directions, but keeping the pin

S at the same point.will

now

lie

between P' Q', again be found equal.In thelie

The line of sight on P' Q', and the angles S R' and the normal willexperiment S appeared to

first

on the continuation of P Q, in the second it appears to be situated on P' Q' produced. Its image is thus at the intersection of these two lines, at L. It can easily be proved

by elementary geometry (from the equalityof angles) that the

image L of the pin is at the same distance from the mirror as the pin S itself, and is of the same size.II.

If

the

pins

P, Q, R, S

in

the

first

experiment be placed so that their headsareall

at the

sheet of

same height above the plane paper, and the eye be placed in a

line of sight

with the heads P, Q, the images of the heads R, S in the mirror will beb}/

hidden

the head of pin P.

The angle

NMP

(position

I)

is

called

OPTICALangle of incidence, angle of reflection.

33

and the angle(verified

NMSby theas

The lawsabovefollowsI.

of reflection

tests):

can

now be formulated

The angle

of reflection is equal to the

angle of incidence.,

IL The paths

of the incident

and

of the

reflected ray lie in the

same

plane.

Fromflecting

I it follows,

as shown, thatre-

in. The image formed in a planesurfaceis

at

the

same distance

from that surface as the object reflected, and is of the same size as the object..

JRefractionpasses from one substancesuffers

When

lightit

into another

changes which are

somewhat more complicated than in the Thus if we place a coin case of reflection. at the bottom of a tumbler which we fillwith water, the coin appears to be higher

than when the tumbler was empty;if

also,it

we plungeseem

a pencil into the water,3

will

to be bent or broken at the surface,

34

DIAMOND DESIGNtheparticular

except inpencilis

case

when

the

perfectly vertical.of refraction in

We

can study the laws

a manner somewhat similar to that adoptedfor the reflection tests.of

Upon

a

fiat

sheet

paper

(fig.

14)

we

place a fairly thick

Fig. 14.

rectangular

glass

plate

with

one

of

its

edges (which should be poHshed perpendicularly to the plane of the paper) along apreviously drawn line

A B.

We

place a

pin, P, close to the edge

AB

of the glass

plate and another, Q, close to the further edge. Looking through the surface A B,

OPTICAL we

35

place our eye in such a position that the pin Q as viewed through the glass is

covered by pin P.

Near to the eye and

on the same hne of sight we stick a third pin R, which therefore covers pin P. Theglass plate is

now removed.

PQ

and

PR

are joined, a perpendicular tois

AB, MM',

erected at P, and a circle of any radius

drawn with P as centre. This circle cuts PQ at K and RP at L. LM and KM' are drawn perpendicular to MM', L M and

K M'found.

are measured

and the

ratio

tfitt?

The experiment positions of P and..

is

repeated for differentthe corresponding

Q and

LMratio

,

ratio

j^^, calculated.

,

It will

be found that

for a given substance (as in this case glass)thisis

constant.

It

is

called

the

index j4 refraction, and generally represented by the letter n.Referring tofig.

14,

we note

that as

PK

=

PL

=

radius of the

circle.

36

DIAMOND DESIGNwrite

we can

LMsin

RPM

OPTICAL

37

Of two substances with different index' qf refraction, that which has the greaterindex of refractionto glass, whichis

called optically denser.

In the experiment the light passed from airis

of greater optical density.i.e.

Let us

now

consider the reverse case,N'A'

when

lightless

passes

another

from one medium to dense optically. Suppose a

beamair.

of light

AOit is

(fig.

15)

with a small

angle of incidence passes from water into

At the surface

of separation a small(as

proportion of

reflected to A''.

weis

have seen underis

reflection)

The remainder

refracted in a direction

O A'

which

38

DIAMOND DESIGN

more divergent from the normalthan

N O N'gradu-

AO.

Suppose now that the angleally increases.light also increases,

AON

The proportion

of reflected

and the angle

of re-

fraction

N'OA'valueof

increases steadily

and

at

a more rapid rate thancertain

N O A,angle

until for a

the

of

incidence

BONis

the refracted angle will graze theIt is clear that

surface of separation.

under

these conditions therefractedIf

amount

of light

whichairisstill

and passes into theangleat

zero.

theas

of incidence is

greater,

CON,

there

is

no

reis

fracted ray, and the whole of the lightreflected into the optically denseror,

medium,

as

it

is

termed, total reflection thenis

occurs.angle,

The angle B O Nand caneasily

called critical(3)

be calculated by

when theknown.angleofi' ,

refractive indicesIt will

n and n' are be noted that when theattainsits

incidence

critical

value

the angle of refraction becomesi.e.

a right angle,to unity.

its sine

becomes equal

OPTICALSubstituting in(3)^'

39

n

sin r

= n^ sin sin r I^

smOr,if

=- -7

n

.

(4)

the less dense

medium be1

air,

nsini'

=

=^^ nis

.

.

(5)

This formula

(5)

very important in

the design of gems, forcritical

by

its

means the

angle can be accuratety calculated.especiallylike the

A

precious stone,

a colourless

and transparent onebest possible effect

diamond,

is

cut to the best advantage and with the

when

it

sends to the

spectator as strong and as dazzling aof light as possible.

beam

Now

a gem, not being

in itself a source of light, cannot shine with

other than reflected light.

amount*

of light willis

The maximum be given off by the gem

here of double refraction, as the diamond is a singly refractive substance, and it was considered unnecessary to introduce irrelevantmatter.

No mention

made

40if

DIAMOND DESIGNthe whole of the light that strikesit is

reflected

by the back of the gem, i.e. by that part hidden by the setting, and sent out into the air by its front part. The facets of the stone must therefore be soit is let

disposed that no light that enters

out throughreflected.

its

back, but thatis

it is

wholly

This result

obtained by having

the facets inclined in such a the light that strikes

way

that

all

them does

so at an

angle of incidence greater than the criticalangle.

This point will be further dealt withare a few indices of refrac-

in a later chapter.

The followingtion

which

may.

be useful or of interest

:

Water

OPTICALLeadsilicate.

41.

.

2-12

Diamond

.

.

2*417

.

(6)

These indices have, of course, been found by methods more accurate than the testsdescribed.

One

of

these

methods,

one

particularly suitable for the accurate de-

termination of the indices of refraction in gems, will be explained later.

Withfraction

this value

for

the the^

indexcritical

of

re-

of

diamond,

angle

works out at

sm^

I

nI

2-417i i

=.

-4136

= sin~^

'4136.

=- 24 26'

.

(7)

This angle will be found very important.

Dispersion

What we

call

white light

is

made up

of

a variety of different colours which produce white by their superposition. It is to the

42

DIAMOND DESIGNits

decomposition of white light into

com-

ponents that are due a variety of beautiful

phenomenaof the soap

like the

rainbow or the coloursit

bubble

and,

mayis

be added,

the

''

fire

" of a diamond.of refraction

The index

found to be

different for light of different colours, red

being generally refracted least and violetmost, the order for the index of the variouscolours being as follows:

Red, orange, yellow, green, blue, indigo,violet.

Note.

Inof

the

list

given above theis

indexyellow

refraction

that

of

the

light

obtained

by theit

incan-

descence of a sodiumis

salt.

This colouris

used as a standard, asdefinite,

verypro-

bright, very

and

easily

duced.If

white light strikes a glass plate with(fig.

parallel surfaces

20) the different colours

are refracted as

shown when passing

into

the glass.

Now#

for every colour the angle

of refraction is givensin r

by (equation

(2))

= sin

i.

OPTICAL

43

When

passing out of the glass, the angle

of refraction is given

by

n sm

I

= sm ri'

As the facesTherefore,

of the glass are parallel,

r.

/

=

i,

and the ray when leaving

the glass

is

parallel to its original direction.

The various

colours will thus follow parallel

Fig.

I 6.

paths as shown in

fig.

i6,

and as they areis

very near together (the dispersion

very

much

exaggerated), they will strike the eye

together and appear white.in the pin experiments

This

is

why

on refraction, dispersion was not apparent to any extent.If,

instead of using parallel surfaces as in

a glass plate,

we

place

as in a prism, light

them at an angle, falling upon a face of

44

DIAMOND DESIGN17and,

the prism will be dispersed as shown infig.;

when,

leaving

by another

face, the light, instead of

combining to form

white

(as in a plate) is still further dispersed

and forms a ribbon

of lights of the different

Fig. 17.

colours,is

from red to

violet.

Such a ribboncoloursof

called

a spectrum.

The

a

spectrum cannot be further decomposed by the introduction of another prism. The difference between the index ofrefraction of extreme violet lightof1

and thatDis-

extreme red

is

called dispersion.^

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

OPTICALpersion,

45

on the whole, increases with theof a

refractive index, although with exception.

The dispersionglasses is given

number:

of

gems and

below

Quartz

46

DIAMOND DESIGNso

a spectator's eye separately,

that

at

one moment he sees a ray of vivid blue, at another one of flaming scarlet or one ofshining green, while perhaps at the nextinstant a

beam

of purest

white

mayall

be

reflected in his direction.

And

these

colours change incessantly with the slightest

motion

of the

diamond.:

The effect of refraction in a diamond can be shown very interestingly as follows

A

piece of white cardboard or fairly

stiff

paper with a hole about half an inch indiameter inoflight.its

centre

is

placed in the

direct rays of the sun or another source

The stoneit

is

held behind the

paper and facingof spots of the

in the ray of light

which

passes through the hole.

A

great

number

most diverse colours appear then upon the paper, and with the slightest motion of the stone some vanish, others appear, and all change their position andtheir colour.If

the stone

is

held with the

hand,

its

slight

unsteadiness will give a

startling appearance of life to the

image

upon the paper.

This

life

is

one of the

OPTICALchief reasons of the

47

diamond's attraction,factors of its beauty.

and one

of the

main

Measurement of Refractionwas pointed out that the manner by which the index of refraction was calculated there, although the simplest, was both not sufficiently accurate and unsuitable for gem-stones. One of the best methods, and perhaps the one giving the most correct results, is that known as method of minimum deviation. Owing to the higher index of refraction of diamond it is especially suitable in its case, where others might not be convenient. The theory of that method is as follows Let A B C be the section of a prism of theit:

In the study of refraction

substance the refraction index of which

we want

to calculate

(fig.

i8).

A

source

of light of the desired colour is placed at

R, and sends aof

the prism.

beam R I upon the face A B The beam RI is broken,and leavesit

crosses the prism in the direction IF, is

again broken,

along

F

R'.

Supposing now that we rotate prism

ABC

48

DIAMOND DESIGNits

about

edge A.

The

direction of;

F

R''

changes at the same time

we noteif

that as

we

gradually turn the prism,

V R'

turns

in a certain direction.

But

we go ona certain

turning the prism,

F R'

will at

moment

stop and then begin to turn in

Fig. iS.

the reverse direction, although the rotationof the prism

was not

reversed.

We

alsois

note that at thesmallest value.

moment whenIt is

the ray

stationary the deviation has attained

its

not

difficult to

prove

that thispassesi.e.

is

the case

when the rayi'

of light

through the prism symmetrically,anglesi

when

and

(fig.

i8) are equal.

OPTICALLet

49

AMI

be a

line bisecting the angle A.

Then

T

is

perpendicular to

A M.

Let

R

be produced to

Q and R' I' to O. They meet on A M and the angle Q O R' is the deviation d[i.e.

the angle betweenfinal

the original andthelight

the

direction

of

passing

through the prism).Therefore

OIF-:

\d.I,

DrawThen

the normal at

N N'.

MIN' = IAM=-i^BAC(i)

if

a be the angle

of the prism.

Now by

equation

n

=

smsin -

i

r

i::::.NIR-:OIN'-OIM+MIN'

^ Id+la r - M I N' = 1^therefore

=^{d+a)

n

=

sin Md-\-a)

^A_^ sm fa

L

.

.

,n\ (8)

The indexculatedif

of refraction

can thus be

cal-

the angles d and a are known.of a spectroscope.

These are found by means

50

DIAMOND DESIGN:

This instrument consists of three partscollimator, the table,light enters

the

and the telescope.slit

The

by the

collimator (a long brass

and a lens) passes through the prism which is placed on the The table, and leaves by the telescope. colUmator is usually mounted rigidly upontube fitted with athe stand of the instrument.is

Its function

to determine the direction of entry ofits

the light and to ensure

being parallel.

Both the table and the telescope are movable about the centre of the table, and are fitted with circular scales which are graduated in degrees and parts of a degree, and by means of which the anglesare found.

Now

two

facets of a stone are selected,is

and the stonetable.

placed upon the table soof the prism,

that these facets are perpendicular to the

The angle a

i.e.

the

angle between these facets, can be found

measurement with a goniometer The angle d or also by the spectroscope. The position of the is found as followsdirect:

by

stone

is

arranged so that the light after

OPTICALpassing

51it

through the coUimator entersit

from one selected facet and leavesother.

by theis

TheThe

telescopeof

is

moved

mitil the

spectral

image

the source of light

found.

table

and the stone are now

rotated in the direction of

minimum

devia-

F1G/19.

tion,is

same time the telescope moved so that the image is kept in view.at the

and

We know

that at the point of

minimum

deviation the direction of motion of thetelescope changes.is

When

this exact point

reached the movements of the stoneof the telescope are stopped,

and

and the

52

DIAMOND DESIGNis

reading of the angle of deviation d

taken

on the graduated

scale.

The values

of a

and d are now introducedsin|(a+^) sin \a

in equation (8)^^

_

and the value

of

n calculated with the help

of sine tables or logarithms.

The values

for

diamond are

;^

Dispersion

= 2'4i7 for sodium light == 044= ^red^vioiet

Part III

MATHEMATICALIn the survey of the history of diamond

most remarkable fact is that so old an art should have progressed entirely by trial and error, by gradual correction and slow progress, by the almost accidental elimination of faults and introduction of ameliorations. We have traced the history of the art as far back as 1375, when the earliest recorded diamond manufactory existed, and when the polishers had already attained a high degree of guild organisation. We have every reason to believe that the process of diamond polishing was known centuries before. And yet all these centuries, when numerous keen minds were directed upon the fashioning of the gem, have left no single record ofcutting, perhaps the53

54

DIAMOND DESIGN

any purposeful planning of the design of the diamond based upon fundamental optics. Even the most bulky and thorough contemporary works upon the diamond or upon gems generally rest content with explaining the basic optical principles, and do no more than roughly indicate how these principles and the exceptional optical properties of thebrilliancy;

gem

explain

its

extraordinary

nowhere has the author seen calculations determining its best shape andItis

proportions.

the

purpose

of

the

present chapter to establish this shape andthese proportions.

The diamondif

treated essentially asless

it

be were a worthwill

crystal

in

which the desiredi.e.

results

areto

to

be obtained,great

without regardtherelation

the

value

which

between

a

great

demand

and

a

very

small supply gives to the least weight of

the material.It is useful to recall here the principles

and the properties whichthe calculations.

will

be used

in

MATHEMATICALReflection1.

55

The angles

of

incidence and

of

re-

flection are equal.2.

The paths

of the incident

and

of the

reflected ray lie in the

same

plane.

Refraction1.

When

a ray of light passes from one

mediumit is

into a second of different density,

refracted as:^

by the following equation

:

sin f

=

n' sin

i

.

.

(3)

where

= angle of refraction. i angle of incidence. n = index of refraction of the secondr

medium.n'

= indexfirst

of refraction of the first

medium.If

the

mediumnsinr

is

air,

n'

=

1,

and

equation becomes

=

sin

i

.

.

(2)

2.

When

medium

a ray of light passes from one into another optically less dense,

total reflection occurs for all values of the

angle of incidence above a certain critical

56value.

DIAMOND DESIGNThiscritical

angle

is

given

by

equation'

smt

=n

.

.

(4)

Or,

if

the less dense

medium be

air,

sin.-'

= ^,same

..

(5)

3.

The pathslie

of the incident

and

of the

refracted ra}^

in the

plane.

Dispersion

Whena band

a ray of lighti.e.

is

refracted,is split

dis-

persion occurs,or

the ray

up

into

spectrum of various colours, owing to the fact that each colour has a The disperdifferent index of refraction.sionforis

the difference between these indices

extreme rays on the spectrum.

Data

MATHEMATICAL

57

DETERMINATION OF THE BEST ANGLES AND THE BEST PROPORTIONS,

Postulate.

^The

design of a diamond or

any gem-stone must be symmetrical about an axis, for symmetry and regularityof

in the disposition of the facets are essentialfor a pleasing result.

Let us

now

consider a block of

bounded by polished surfaces, consider the effect on the patha gradual change in shape;

diamond and let usof light ofwill also

we

observe the postulate and keep the block

symmetrical aboutLet us take asparallel faces(fig.

its axis.first

section one having

20),

and

let

MM'

be

its

axis of

symmetry.

Let us for convenience

place the axis ofall

symmetry

vertically init

future work, so that surfaces crossing

are horizontal.

Consider a ray of light S

P P

striking face

A B.leave

It will

be refracted alongparallel to S.

P Q and

by Q R,if

(as

seen in studying dispersion)that

We

we have also know

NN'

is

the normal at Q, angles

58

DIAMOND DESIGNand

NQP

Q P M'

are equal.

Therefore, for

total reflection,

O P M'

== 24 26',

but at that angle of refraction the angle of incidence S P M becomes a right angle and

no

light penetrates into the stone.

It is

thus obvious that parallel faces in a

gem

are very unsatisfactory,

as

all

the lightpasses out

passing in by the front of a

gem

again by the back without any reflection.

can avoid parallelism by inclining either the top or the bottom faces at anangle with the directioncase

We

AB.of

In the

first

we obtain the shapein the

a rose-cut

diamond and

second case that of a

MATHEMATICALbrilliant

59rose'

cut.first

We

will

examine the

cut in the

instance.

The RoseConsider(fig.

21)

a section having thelet

bottom surface horizontal, and

us incline

an angle a with it. To maintain symmetry, another surface BC is introduced. We have now to find the value of a for which total reflectionthe top surfaceat

AB

occurs at

A C. Now26'.

for this to be the case,

the

minimum

angle of incidence upon

AC

must be 24incident ray

Let us draw such an

P Q.

To ensure

that no light

6ois

DIAMOND DESIGNincident at a smaller angle,

we must make

the angle of refraction at entry 24 26' and

arrange the surface of entry as shown,for

A B,moreSuch

we know

that then no light will enter

at an anglevertical to

more obliquei.e.

to

AB

or

A C.is

This gives to a a value of48 52'.

twice the critical angle,asection

very satisfactory indeed asas,

regards reflection,

owing to

its

deriva-

tion, all the light entering it leaves

by the

front part.refraction?

Is it also satisfactory as regards

Let us follow the path of a ray of light of any single colour of the spectrum, S P Q R T

and r be the angles of incidence upon and of refraction out of the(fig.

22).

Let

i

diamond.

R Q N, and therefore triangles A P Q and R C Q angle A Q P == angle COR. Also by symmetry A == C,At Q, P Q N--

in

therefore

angleit

APQ i

angle

C

RQ

;

follows that

=

r.

MATHEMATICALAs the anglewilli is

6iall

the same for

colours

of a white ray of light, the various colours

emerge

parallellight.

out of the diamond

and give whitemental reasonrose;

This

is

the funda-

of the unpopularity of the

there

is

no

fire.

This effect 23 and

may

be remedied to a small(figs.

extent by breaking the inclined facet24), so that

the angle be not the same

at entry as at exit.ful to the

This breaking

is

harm-

amount of hght reflected whichif we steepen the ever way we arrange it;

facet near the edge, there is a large propor-

tion of Hght projected

backwards and being

62lost, for

DIAMOND DESIGNwe may take(fig.

it

that the spectator

will

not look at the rose from the side of23).If,

the mounting

on the other(fig.

hand,

we flattenis

the apex of the rose

24)

(which

the usual method), a leakage willits base.

occur through

There

is,

of course.

no amelioration in the refraction if the light passes from one facet to anothersimilarly placed (asS' P' Q'

shown

in

fig.

23,

path

Taking the effect as a whole, the least unsatisfactory shape is as shown in fig. 24, with the angles a about 49 and 30 for the base and the apex respectively. The rose cut, however, is fundamentallyR'T').

MATHEMATICAL

63

wrong, as we have seen above, and should be abohshed altogether. It is the highcost of the material thatstill

is

the cause of

its

being used in cases where the rough shape is especially suitable, and then only in small sizes. In actual practice the

Fig. 24.

proportions of the cut rose depend largely upon those of the rough diamond, thestone being cut with as small a loss of material as possible. Generally the values of a are much below those given

above,material

i.e.

is

and 30, thick enough49it

asto

where theallow

suchcutit

steep angles

is

much

better to

into a brilhant.

64

DIAMOND DESIGNThe Brilliant

A Back.

of the Brilliant

Let us

now

pass to the consideration ofi.e.

the other alternative,

where the top

surface

is

a

horizontal

plane

AB

and

where the bottom surface A C is inclined (fig. at an angle a to the horizontal As before, we have to introduce 25). a third plane B C to have a symmetricalsection.

MATHEMATICAL'

65

First Reflection

Let a vertical rayangle of incidence

PQ

strike

A B.

As the

is zero, it

passes into the

stone without refraction and meets plane

AC

at R.

Let

RN

be the normal at that

point, then, for total reflection to occur,

angle

NRQ==

24 26'.

Butangleas

NRQ

:= angle

QAR

===

a,

AQ

and QR,

AR

and

RN

are per-

pendicular.

Therefore, for total reflection of a verticalray,

a

=

24 26'.

Let usgradually

now

incline the ray

PQ

so that

it

changes from a vertical to a horizontal direction, and let P' Q' be sucha ray.is

Upon

passing into the diamond

it

and strikes AC at an angle Q' R' N' where R' is the normal to A C. When P' Q' becomes horizontal, the angle of refraction T' Q' R' becomes equal torefracted,

W

24

26'.

This

is

the extreme value attain5

66 able

DIAMOND DESIGN;

by that angle also, for total reflection, angle Q' R' N' must not be less than 24 26'. If we draw R' V, vertical angle V R' Q' = R' Q' T' = 24 26', andangle

V R' N' - V R' Q' +-: 24 26'

Q' R'

N'

+ 24 26'

= 48 52'as before,a

= angle V R' N',.

and thereforea :^ 48 52'.

.

(9)

For

absolute total reflection to occur at thethe inclined facets

first facet,

must make an

angle of not less than 48 52' with the horizontal.

Second Reflection

Whenthefirst

the ray of lightinclinedfacet

is

reflected(fig.

fromit

ACB C.

26),

strikes the oppositelight

one

Here too the

must be totally reflected, for otherwise there would be a leakage of lightLet us

through the back of the gem-stone.vertically

consider, in the first instance, a ray of light

incident

upon the

stone.

The

MATHEMATICAL

67

path of the ray will be P Q R S T. If R N and S N' are the normals at R and S respectively, then for total reflection,angle N' S

R -=M

24 26'.

P

QNo(\oC

B

N'

Fig. 26.

Let us find the value of a toconditionangle

fulfil

that

QRN

=

angle

QAR

=a

as having perpendicular sides.

angle S

RN

=:=

angle

QRN

68

DIAMOND DESIGNandreflection.

as angles of incidence

Thereforeangle

NRS = a.R=a;

Now

let

angle N' S

Then, in triangleangle angle

R S C, S R C = 90 a R S C = 90 x

angle

RCS

== 2

X

angle

RCMa).

-=2 X

ARQ =- 2(90=

The sum of these three angles equals two right angles,

90 a +90 +180 2 a:\;

180,

or

3a -\-x 3a

= =

180

i8o

X.

Now, Xais

is

not less than 24

26', therefore

not greater than

3

Let us again inclineuntil it

P Q from the verticalin this case

becomes horizontal, but

in the other direction, to obtain the inferiorlimit.

MATHEMATICAL

69

Then (fig. 27) the path will be PQRS. Let QT, RN, SN' be the normals at Q, R, and S respectively. At the extreme case,

TQRat R.

will

be 24

26'.

Draw

RV

vertical

Fig. 27.

Thenangle angle

QRV =

anglea.

TQR

=

24 26'

VRN =

Asangle

before, in triangle

R S C,

SRC = go^-NRS = 9o-a-2426' angle RCS = 2(90- a)angle

R S C = 90 x.

70

DIAMOND DESIGN

Then

90 a 24 26'+i8o 2a+90 3a +% = 180^-24 26' - 155In the case

;t;

=34'

180

nowX

considered,

=

24 26'.

Then3a

=

155 34'-24' 26'.

=

131 8'..

== 43 43'

(10)

For

absolute total reflection at the second

facet, the inclined facets

must make an angle

of not more than 43 43' with the horizontal.

We

will

note here that this condition

and the one arrivedopposition.will

We

will

on page 66 are in discuss this later, andat

pass

now

to considerations of refraction.Refraction

First case

:

ql

is less

than 45refractionin

In

the

discussion

of

a

diamond, we have to consider two cases, i.e. a is less than 45 or it is more than 45. Let us take the former case first and let P Q R S T (fig. 28) be path of the ray. Then,

MATHEMATICALif

71

SN

is

the normal at

S,

we know

that for

total reflection at S angle

RSN

== 24 26'.

We

want to avoid total reflection, for if the light is thrown back into the stone, some of it may be lost, and in any case the

Fig. 28.

ray

be broken too frequently and the result will be disagreeable.will

Therefore,

angleRSN