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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', S R' and the normal will

again be found equal.

In the first experiment S appeared to

lie 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 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 b}/ the head of pin P.

The angle N M P (position I) is called

OPTICAL 33

angle of incidence, and the angle N M S

angle of reflection.

The laws of reflection (verified by the

above tests) can now be formulated as

follows : â€”

I. 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.

From I it follows, as shown, that

in. 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.

. JRefraction

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,

3

34

DIAMOND DESIGN

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 fiat sheet

of paper (fig. 14) we place a fairly thick

Fig. 14.

rectangular glass plate with one of its

edges (which should be poHshed perpendi-

cularly to the plane of the paper) along a

previously 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 35

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 hne 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', L M and

K M' are measured and the ratio tfitt?

found.

The experiment is repeated for different

positions of P and Q and the corresponding

.. LM , ,

ratio j^^, calculated. It will be found that

for a given substance (as in this case glass)

this ratio is constant. It is called the

index j4 refraction, and generally repre-

sented by the letter n.

Referring to fig. 14, we note that as

P K = P L = radius of the circle.

36 DIAMOND DESIGN

we can write

LM

sin R P M

LM

LM

PL

KM'

~KM'

sin Q P M'

PK

Writing the angle of incidence R P M as i,

and the angle of refraction Q P M' as f , this

equation becomes

sin 2^ , X

n=-. â€” . . . I

sm r

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 f = 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.

OPTICAL

37

Of two substances with different index'

qf 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.

N'

A'

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 O A' which is

38 DIAMOND DESIGN

more divergent from the normal N O N'

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 N O A, 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 into 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 B O N 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.

OPTICAL 39

Substituting in (3)

n sin r = n^ sin ^'

sin r â€” I

sm ^ =- -7 . â€¢ â€¢ (4)

n

Or, if the less dense medium be air,

n = 1

sini' =^^ . . (5)

n

This formula (5) is very important in

the design of gems, for by its means the

critical angle can be accuratety 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.

40

DIAMOND DESIGN

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 : â€”

Water .

1-33

Crown glass

. .1-5 approx

Quartz

. I-54-I-55

Flint glass

. 1-576

Colourless strass

1-58

Spinel

172

Almandine

179

Lead borate

1.83

Demantoid

. 1-88

OPTICAL 41

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

sm^

I

n

I

= -4136

2-417

i = sin~^ '4136

i =- 24Â° 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 DESIGN

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 â€” 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 : â€”

Red, orange, yellow, green, blue, indigo,

violet.

Note. â€” 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))

# sin r = sin i.

OPTICAL

43

When passing out of the glass, the angle

of refraction is given by

n sm I = sm r

As the faces of the glass are parallel, i' â€” r.

Therefore, / = 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 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

44 DIAMOND DESIGN

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

Fig. 17.

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.^ 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.

OPTICAL

45

persion, on the whole, increases with the

refractive index, although with exception.

The dispersion of a number of gems and

glasses is given below : â€”

Quartz

â€¢ -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 gem-stones) that the diamond

owes its extraordinary '' fire." For when

a ray of light passes through a well-cut

diamond, it is refracted through a large

angle, and consequently the colours of the

spectrum, becoming widely separated, strike

46 DIAMOND DESIGN

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 : â€”

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

OPTICAL 47

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 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 : â€”

Let A B C be the section of a prism of the

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 a beam R I upon the face A B

of the prism. The beam RI is broken,

crosses the prism in the direction IF, is

again broken, and leaves it along F R'.

Supposing now that we rotate prism ABC

48

DIAMOND DESIGN

about its edge A. The direction of F R''

changes at the same time ; we note that as

we gradually turn the prism, V R' turns

in a certain direction. But if we go on

turning the prism, F R' will at a certain

moment stop and then begin to turn in

Fig. iS.

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. i8) are equal.

OPTICAL 49

Let A M be a line bisecting the angle A.

Then I T is perpendicular to A M. Let R I

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 between the original and

the final direction of the light passing

through the prism).

Therefore OIF-: \d.

Draw the normal at I, N N'.

Then

MIN' = IAM=-i^

if a be the angle B A C of the prism.

Now by equation (i)

sin i

n = - â€”

sm r

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

^ Id+la =^{d+a)

r - M I N' = 1^

therefore

sin Md-\-a) ,n\

n = ^A_^ â€” L . . (8)

sm fa

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.

50 DIAMOND DESIGN

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

colUmator 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 : â€” The position of the

stone is arranged so that the light after

OPTICAL

51

passing through the coUimator enters it

from one selected facet and leaves it by the

other. The telescope is moved mitil the

spectral image of the source of light is

found. The table and the stone are now

rotated in the direction of minimum devia-

F1G/19.

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

52 DIAMOND DESIGN

reading of the angle of deviation d is taken

on the graduated scale.

The values of a and d are now introduced

in equation (8) :

^^ _ sin|(a+^)

sin \a

and the value of n calculated with the help

of sine tables or logarithms.

The values for diamond are

;^ = 2'4i7 for sodium light

Dispersion = ^redâ€” ^vioiet == â€¢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

53

54 DIAMOND DESIGN

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.

MATHEMATICAL 55

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 :

:^ sin f = n' sin i . . (3)

where r = angle of refraction.

i â€” 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

nsinr = 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

56 DIAMOND DESIGN

value. This critical angle is given by

equation

' smt = - . . (4)

n

Or, if the less dense medium be air,

sin.-' = ^, .â€¢ . (5)

3. The paths of the incident and of the

refracted ra}^ 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

sodium light)

dispersion : S =

= -044

critical angle : i' =

= 24Â° 26'

(for a

(7)

MATHEMATICAL 57

DETERMINATION OF THE BEST ANGLES

AND THE BEST PROPORTIONS

, Postulate. â€” ^The design of a diamond or

of any gem-stone 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 S P striking face

A B. It will be refracted along P Q and

leave by Q R, parallel to S P (as we have

seen in studying dispersion) . We also know

that if NN' is the normal at Q, angles

58

DIAMOND DESIGN

N Q P and Q P M' are equal. Therefore, for

total reflection,

O P M' == 24Â° 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 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 rose-cut

diamond and in the second case that of a

MATHEMATICAL

59

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 A B 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 A C. Now for this to be the case,

the minimum angle of incidence upon AC

must be 24Â° 26'. Let us draw such an

incident ray P Q. To ensure that no light

6o DIAMOND DESIGN

is incident at a smaller angle, we must make

the angle of refraction at entry 24Â° 26' and

arrange the surface of entry as shown, A B,

for we know that then no light will enter

at an angle more oblique to A B or more

vertical to A C. This gives to a a value of

twice the critical angle, i.e. 48Â° 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, S P Q R T

(fig. 22). Let i and r be the angles of

incidence upon and of refraction out of the

diamond.

At Q, P Q N - R Q N, and therefore in

triangles A P Q and R C Q

angle A Q P == angle COR.

Also by symmetry A == C,

therefore

angle A P Q - angle C R Q ;

it follows that i = r.

MATHEMATICAL

6i

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 hght reflected which-

ever way we arrange it ; if we steepen the

facet near the edge, there is a large propor-

tion of Hght projected backwards and being

62

DIAMOND DESIGN

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 49Â° and

30Â° for the base and the apex respectively.

The rose cut, however, is fundamentally

MATHEMATICAL

63

wrong, as we have seen above, and should

be abohshed 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

Fig. 24.

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

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', S R' and the normal will

again be found equal.

In the first experiment S appeared to

lie 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 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 b}/ the head of pin P.

The angle N M P (position I) is called

OPTICAL 33

angle of incidence, and the angle N M S

angle of reflection.

The laws of reflection (verified by the

above tests) can now be formulated as

follows : â€”

I. 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.

From I it follows, as shown, that

in. 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.

. JRefraction

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,

3

34

DIAMOND DESIGN

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 fiat sheet

of paper (fig. 14) we place a fairly thick

Fig. 14.

rectangular glass plate with one of its

edges (which should be poHshed perpendi-

cularly to the plane of the paper) along a

previously 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 35

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 hne 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', L M and

K M' are measured and the ratio tfitt?

found.

The experiment is repeated for different

positions of P and Q and the corresponding

.. LM , ,

ratio j^^, calculated. It will be found that

for a given substance (as in this case glass)

this ratio is constant. It is called the

index j4 refraction, and generally repre-

sented by the letter n.

Referring to fig. 14, we note that as

P K = P L = radius of the circle.

36 DIAMOND DESIGN

we can write

LM

sin R P M

LM

LM

PL

KM'

~KM'

sin Q P M'

PK

Writing the angle of incidence R P M as i,

and the angle of refraction Q P M' as f , this

equation becomes

sin 2^ , X

n=-. â€” . . . I

sm r

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 f = 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.

OPTICAL

37

Of two substances with different index'

qf 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.

N'

A'

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 O A' which is

38 DIAMOND DESIGN

more divergent from the normal N O N'

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 N O A, 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 into 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 B O N 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.

OPTICAL 39

Substituting in (3)

n sin r = n^ sin ^'

sin r â€” I

sm ^ =- -7 . â€¢ â€¢ (4)

n

Or, if the less dense medium be air,

n = 1

sini' =^^ . . (5)

n

This formula (5) is very important in

the design of gems, for by its means the

critical angle can be accuratety 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.

40

DIAMOND DESIGN

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 : â€”

Water .

1-33

Crown glass

. .1-5 approx

Quartz

. I-54-I-55

Flint glass

. 1-576

Colourless strass

1-58

Spinel

172

Almandine

179

Lead borate

1.83

Demantoid

. 1-88

OPTICAL 41

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

sm^

I

n

I

= -4136

2-417

i = sin~^ '4136

i =- 24Â° 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 DESIGN

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 â€” 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 : â€”

Red, orange, yellow, green, blue, indigo,

violet.

Note. â€” 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))

# sin r = sin i.

OPTICAL

43

When passing out of the glass, the angle

of refraction is given by

n sm I = sm r

As the faces of the glass are parallel, i' â€” r.

Therefore, / = 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 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

44 DIAMOND DESIGN

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

Fig. 17.

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.^ 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.

OPTICAL

45

persion, on the whole, increases with the

refractive index, although with exception.

The dispersion of a number of gems and

glasses is given below : â€”

Quartz

â€¢ -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 gem-stones) that the diamond

owes its extraordinary '' fire." For when

a ray of light passes through a well-cut

diamond, it is refracted through a large

angle, and consequently the colours of the

spectrum, becoming widely separated, strike

46 DIAMOND DESIGN

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 : â€”

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

OPTICAL 47

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 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 : â€”

Let A B C be the section of a prism of the

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 a beam R I upon the face A B

of the prism. The beam RI is broken,

crosses the prism in the direction IF, is

again broken, and leaves it along F R'.

Supposing now that we rotate prism ABC

48

DIAMOND DESIGN

about its edge A. The direction of F R''

changes at the same time ; we note that as

we gradually turn the prism, V R' turns

in a certain direction. But if we go on

turning the prism, F R' will at a certain

moment stop and then begin to turn in

Fig. iS.

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. i8) are equal.

OPTICAL 49

Let A M be a line bisecting the angle A.

Then I T is perpendicular to A M. Let R I

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 between the original and

the final direction of the light passing

through the prism).

Therefore OIF-: \d.

Draw the normal at I, N N'.

Then

MIN' = IAM=-i^

if a be the angle B A C of the prism.

Now by equation (i)

sin i

n = - â€”

sm r

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

^ Id+la =^{d+a)

r - M I N' = 1^

therefore

sin Md-\-a) ,n\

n = ^A_^ â€” L . . (8)

sm fa

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.

50 DIAMOND DESIGN

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

colUmator 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 : â€” The position of the

stone is arranged so that the light after

OPTICAL

51

passing through the coUimator enters it

from one selected facet and leaves it by the

other. The telescope is moved mitil the

spectral image of the source of light is

found. The table and the stone are now

rotated in the direction of minimum devia-

F1G/19.

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

52 DIAMOND DESIGN

reading of the angle of deviation d is taken

on the graduated scale.

The values of a and d are now introduced

in equation (8) :

^^ _ sin|(a+^)

sin \a

and the value of n calculated with the help

of sine tables or logarithms.

The values for diamond are

;^ = 2'4i7 for sodium light

Dispersion = ^redâ€” ^vioiet == â€¢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

53

54 DIAMOND DESIGN

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.

MATHEMATICAL 55

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 :

:^ sin f = n' sin i . . (3)

where r = angle of refraction.

i â€” 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

nsinr = 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

56 DIAMOND DESIGN

value. This critical angle is given by

equation

' smt = - . . (4)

n

Or, if the less dense medium be air,

sin.-' = ^, .â€¢ . (5)

3. The paths of the incident and of the

refracted ra}^ 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

sodium light)

dispersion : S =

= -044

critical angle : i' =

= 24Â° 26'

(for a

(7)

MATHEMATICAL 57

DETERMINATION OF THE BEST ANGLES

AND THE BEST PROPORTIONS

, Postulate. â€” ^The design of a diamond or

of any gem-stone 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 S P striking face

A B. It will be refracted along P Q and

leave by Q R, parallel to S P (as we have

seen in studying dispersion) . We also know

that if NN' is the normal at Q, angles

58

DIAMOND DESIGN

N Q P and Q P M' are equal. Therefore, for

total reflection,

O P M' == 24Â° 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 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 rose-cut

diamond and in the second case that of a

MATHEMATICAL

59

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 A B 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 A C. Now for this to be the case,

the minimum angle of incidence upon AC

must be 24Â° 26'. Let us draw such an

incident ray P Q. To ensure that no light

6o DIAMOND DESIGN

is incident at a smaller angle, we must make

the angle of refraction at entry 24Â° 26' and

arrange the surface of entry as shown, A B,

for we know that then no light will enter

at an angle more oblique to A B or more

vertical to A C. This gives to a a value of

twice the critical angle, i.e. 48Â° 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, S P Q R T

(fig. 22). Let i and r be the angles of

incidence upon and of refraction out of the

diamond.

At Q, P Q N - R Q N, and therefore in

triangles A P Q and R C Q

angle A Q P == angle COR.

Also by symmetry A == C,

therefore

angle A P Q - angle C R Q ;

it follows that i = r.

MATHEMATICAL

6i

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 hght reflected which-

ever way we arrange it ; if we steepen the

facet near the edge, there is a large propor-

tion of Hght projected backwards and being

62

DIAMOND DESIGN

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 49Â° and

30Â° for the base and the apex respectively.

The rose cut, however, is fundamentally

MATHEMATICAL

63

wrong, as we have seen above, and should

be abohshed 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

Fig. 24.

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