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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 : —

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=-. — . . . 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 — I

sm ^ =- -7 . • • (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 : —



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 =- 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 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 — and, it may be added,
the '' fire " of a diamond.

The index of refraction is found to be
different for light of different colours, red
being generally refracted least and violet
most, the order for the index of the various
colours being as follows : —

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

Note. — In the list given above the
index of refraction is that of the
yellow light obtained by the incan-
descence of a sodium salt. This colour
is used as a standard, as it is very
bright, very definite, and easily pro-
duced.
If white light strikes a glass plate with
parallel surfaces (fig. 20) the different colours
are refracted as shown when passing into
the glass. Now for every colour the angle
of refraction is given by (equation (2))
# 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' — 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 : —



Quartz


• -013


Sapphire


. -018


Crown glass


. '019


Spinel


. -020


Almandine .


. -024


Flint glass .


. -036


Diamond


. -044


Demantoid .


. -057



The greater the dispersion of a medium,
other things being equal, the greater the
difference between the angles of refraction
of the various colours, and the further
separated do they become. It is to its
very high dispersion (the greatest of all
colourless 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 : —
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 : —
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 = - —

sm r

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

^ Id+la =^{d+a)

r - M I N' = 1^



therefore



sin Md-\-a) ,n\

n = ^A_^ — 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 : — 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 = ^red— ^vioiet == •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 — 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.-' = ^, .• . (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' =


= 24° 26'



(for a



(7)



MATHEMATICAL 57

DETERMINATION OF THE BEST ANGLES
AND THE BEST PROPORTIONS

, Postulate. — ^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' == 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 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 24° 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 24° 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. 48° 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 49° and
30° 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


2 4

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