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values of a are much below those given
above, i.e. 49° and 30°, as where the
material is thick enough to allow such
steep angles it is much better to cut it
into a brilhant.



64



DIAMOND DESIGN



The Brilliant

A . Back of the Brilliant

Let us now pass to the consideration of
the other alternative, i.e. where the top




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



MATHEMATICAL 65

' First Reflection

Let a vertical ray P Q strike A B. As the
angle of incidence is zero, it passes into the
stone without refraction and meets plane
A C at R. Let R N be the normal at that
point, then, for total reflection to occur,

angle NRQ== 24° 26'.
But

angle N R Q := angle Q A R === a,

as AQ and QR, AR and RN are per-
pendicular.

Therefore, for total reflection of a vertical
ray,

a = 24° 26'.

Let us now incline the ray P Q so that it
gradually changes from a vertical to a
horizontal direction, and let P' Q' be such
a ray. Upon passing into the diamond it
is refracted, and strikes AC at an angle
Q' R' N' where R' W is the normal to A C.
When P' Q' becomes horizontal, the angle
of refraction T' Q' R' becomes equal to
24° 26'. This is the extreme value attain-

5



66 DIAMOND DESIGN

able 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', and

angle V R' N' - V R' Q' + Q' R' N'

-: 24° 26' + 24° 26'
= 48° 52'

as before,

a = angle V R' N',
and therefore

a :^ 48° 52' . . . (9)

For absolute total reflection to occur at the
first facet, the inclined facets must make an
angle of not less than 48° 52' with the horizontal.

Second Reflection

When the ray of light is reflected from
the first inclined facet AC (fig. 26), it
strikes the opposite one B C. Here too the
light must be totally reflected, for other-
wise there would be a leakage of light
through the back of the gem-stone. Let us
consider, in the first instance, a ray of light
vertically 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 respec-
tively, then for total reflection,

angle N' S R -= 24° 26'.



P



Q



M



B



N



o(



N'



\oC



Fig. 26.

Let us find the value of a to fulfil that
condition :

angle Q R N = angle Q A R = a

as having perpendicular sides.

angle S R N =:= angle Q R N



68 DIAMOND DESIGN

as angles of incidence and reflection.
Therefore

angle NRS = a.
Now let

angle N' S R = a;

Then, in triangle R S C,

angle S R C = 90° — a
angle R S C = 90°— x
angle RCS == 2 X angle RCM

-=2 X ARQ =- 2(90°— a).

The sum of these three angles equals
two right angles,

90°— a +90°— :\; +180°— 2 a = 180°,
or

3a -\-x = 180°
3a = i8o°— X.

Now, X is not less than 24° 26', therefore
a is not greater than

3
Let us again incline P Q from the vertical
until it becomes horizontal, but in this case
in the other direction, to obtain the inferior
limit.



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,
TQR will be 24° 26'. Draw RV vertical
at R.




Fig. 27.

Then

angle Q R V = angle T Q R = 24° 26'
angle V R N = a.
As before, in triangle R S C,
angle SRC = go^-NRS = 9o°-a-24°26'
angle RCS = 2(90°- a)
angle R S C = 90°— x.



70 DIAMOND DESIGN

Then

90— a— 24° 26'+i8o°— 2a+90°— ;t; = 180°
3a +% = 180^-24° 26' - 155° 34'

In the case now considered,

X = 24° 26'.
Then

3a = 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 arrived at on page 66 are in
opposition. We will discuss this later, and
will pass now to considerations of refraction.

Refraction

First case : ql is less than 45°

In the discussion of refraction in 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,



MATHEMATICAL



71



if S N 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 will be broken too frequently and the
result will be disagreeable.
Therefore,

angleRSN< 24°26' . (11)

Suppose this condition is fulfilled and the
light leaves the stone along S T. It is re-
fracted, and its colours are dispersed into



72 DIAMOND DESIGN

a spectrum. It is desirable to have this
spectrum as long as possible, so as to disperse
the various colours far away from each other.
As we know, this will give us the best
possible ** fire."

This result will be obtained when the
ray is refracted through the maximum
angle. Ry (ii) the value for that angle
is 24° 26', and (11) becomes

angle RSN == 24° 26' for maximum dis-
persion.

But then the light leaves A B tangentially,
and the amount of light passing is zero.
To increase that amount, the angle of
refraction has to be reduced : the angle of
dispersion decreases simultaneously, but
the amount of light dispersed increases
much more rapidly. Now we know that
the angle of dispersion is proportional to
the sine of the angle of refraction. It is,
moreover, proved in optics that the amount
of light passing through a surface as at
AB is proportional to the cosine of the
angle of refraction. The brilliancy pro-



MATHEMATICAL ^z

duced is proportional both to the amount
of Hght and to the angle of dispersion, and
therefore to their product, and (by the
theory of maxima and minima) will be
maximum when they are equal, i.e. when
the sine and cosine of the angle of refraction
are equal. For maximum brilliancy, there-
fore, the angle of refraction should be 45°.
This gives for angle R S N

sin RSN = ^HI4_5 _ -7071 = .2930,

2-417 2-417

therefore

angle RSN = 17° for optimum brilliancy (12)

Let (fig. 28) Q X and R Y be the normals
at Q and R respectively, and let ZZ' be
vertical thr-ough R.

We know that

angle R Q X =- angle P Q X = a,

therefore

angle PQR == 2a.

Produce Q R to Q'.

Then, as P Q and Z Z' are parallel,

angle Z R Q' = angle P Q R = 2a.



74 DIAMOND DESIGN

Now, let

angle R S N = %( = 17° for optimum
brilliancy) .

Then, as Z Z' and S N are parallel,

angle Z R S = a;.

As they are complements to angles of
incidence,

angle Q R C = angle S R B = z (say),

but

angle Q' R B = angle Q R C,

therefore

angle S R Q' = 2i.

In angle ZRQ' we have _.^"

angle Z R Q' - angle Z R S +angle S R Q'

2a = X+2i , . (13)

\ In triangle Q C R

angle RCQ = 90°— a

angle Q R C = ^

angle Q C R == 2(90°— a),

therefore

(90 — a) +z +180°— 2a = 180°,



or



i = 3a— 90°.



MATHEMATICAL 75

Introduce this value of i in (13),

2a = A;+6a — 180°
4a == 180°—%

and giving x its value 17°,

4a ^ i8o°— 17° = 163°
a = 40° 45' . . . (14)

If we adopt this value for a, the paths of
oblique rays will be as shown in fig. 29,
P Q R S T when incident from the left of the
figure, and P' Q' R' S' T' when incident from
the right.' Ray P Q R S T will leave the dia-
mond after the second reflection, but with a
smaller refraction than that of a vertically
incident ray, and therefore with less **fire."
Oblique rays incident from the left are,
however, small in number owing to the acute
angle Q R A with which they strike A C ; the
loss of fire may therefore be neglected^____



Ray P'Q'R'S'r will strike AB at a
greater angle of incidence than 24° 26', and
will be reflected back into the stone. This
is a fault that can be corrected by the
introduction of inclined facets D E, F G ;
ray P' Q' R' S' T' will then strike F G at an



76



DIAMOND DESIGN



angle less than 24° 26', and this angle can
be arranged by suitably inclining FG to
the horizontal so as to give the best possible
refraction. The amelioration obtained by
thus taking full advantage of the refraction

U




is so great that the small loss of light caused
by that arrangement of the facets is in-
significant : the leakage occurs through the
facet CB, near C, where the introduction
of the facet D E allows hght to reach C B
at an angle less than the critical. In a



MATHEMATICAL 77

brilliant, where CB is the section of the
triangular side of an eight-sided pyramid,
the area near the apex C is very small, and
the leakage may therefore be considered
negligible.

Second case : a is greater than 45°

In this case the path of a vertical ray
will be as shown by P Q R S T in fig. 30, and
the optimum value for a, which may be
calculated as before, will be

a=:49^i3' . . (15)

As regards the vertical rays, this value
gives a fire just as satisfactory as (14)
(a = 40"^ 45') ; let us consider what happens
to oblique rays.

Rays incident from the left as pqrstu
may strike B C at an angle of incidence less
than the critical, and will then leak out
backwards. Or they may be reflected along
s t, and may then be reflected into the stone.
Both alternatives are undesirable, but they
do not greatly affect the brilliancy of the
gem, because, as we have seen, the amount
of light incident from the left is small.



78



DIAMOND DESIGN



That incident from the right is, on the
contrary, large.

Let us follow ray P'Q'R'S'T'. It will
be reflected twice, and will leave the




diamond after the second reflection, like
the vertically incident ray, but with a
smaller refraction, and consequently less
fire ; most of the light will be striking
the face A B nearly vertically when leaving
the stone, and the fire will be very small.



MATHEMATICAL 79

This time it is impossible to correct the
defect by introducing accessory facets, as
the paths S'T' of the various obHque rays
are not locaHsed near the edge B, but are
spread over the whole of the face ; we are
therefore forced to abandon this design.

Summary of the Results obtained for a

We have found that —
For first reflection, a must be greater
than 48° 52'.

For second reflection, a must be less than

43° 43'-

For refraction, a may be less or more

than 45°. When more, the best value is
49° 15', but it is unsatisfactory. When less,
the best value is 40° 45', and is very satis-
factory, as the light can be arranged to
leave with the best possible dispersion.

Upon consideration of the above results,
we conclude that the correct value for a is
40° 45', and gives the most vivid fire and
the greatest brilliancy, and that although
a greater angle would give better reflection,
this would not compensate for the loss due



So



DIAMOND DESIGN



B



to the corresponding reduction in dispersion.
In all future work upon the modern brilliant
we will therefore take

a = 40° 45'.

B. Front of the Brilliant

When arriving at the value of a = 40° 45',
we have explained how the use of that

angle introduced
defects which
could be cor-
rected, by the use
of extra facets.
The section will
therefore be
shaped somewhat
as in fig. 31. It will be convenient to give
to the different facets the names by which
they are known in the diamond-cutting
industry. These are as follows : —

A C and B C are called pavilions or quoins
(according to their position relative
to the axis of crystallisation of the
diamond) .



p . E

c



Fig. 31.



MATHEMATICAL 8i

AD and EB are similarly called bezels

or quoins.
DE is the table.
FG is the culet, which is made very

small and whose only purpose is to

avoid a sharp point.

Through A and B passes the girdle of
the stone.

We have to find the proportions and in-
clination of the bezels and the table. These
are best found graphically. We know that
the introduction of the bezels is due to the
oblique rays ; it is therefore necessary to
study the distribution of these rays about
the table, and to find what proportion of
them is incident in any particular direction.

Consider a surface A B (fig. 32) upon which
a beam of light falls at an angle a. Let
us rotate the beam so that the angle
becomes ^ (for convenience, the figure
shows the surface A B rotated instead to
A' B, but the effect is the same). The light
falling upon AB can be stopped in the
first case by intercepting it with screen



82



DIAMOND DESIGN




Fig. 32.



B C, and in the second with a screen B C
where B C C is at right angles to the

direction of thebeam.
And if the intensity
of the light is uni-
form, the length of
B C and B C wiU be
a measure of the
amount of light fall-
ing upon A B and A B' respectively.
Now

BC^ABsina
BC'-3 A'BsiniS = ABsin^g.

Therefore, other things being equal, the
amount of light falling upon a surface is pro-
portional to the sine of the angle between
the surface and the direction of the light.
We can put it as follows : —

If uniformly distributed light is falling
from various directions upon a surface A B,
the amount of light striking it from any
particular direction will be proportional
to the sine of the angle between the surface
and that direction.



MATHEMATICAL



83



If we draw a curve between the amount
of light striking a surface from any parti-
cular direction, and the angle between the
surface and that direction, the curve will
be a sine curve (fig. 33) if the light is equally
distributed and of equal intensity in all
directions.

M




O 4-2 70''2 90 l09'/^ 158 180''

M'

Fig. 33.

For calculations we can assume this to
be the case, and we will take the distri-
bution of the quantity of light at different
angles to follow a sine law.

It is convenient to divide all the light
entering a diamond into three groups, one
of vertical rays and two of oblique rays,
such that the amount of light entering
from each group is the same. Now in the



84 DIAMOND DESIGN

sine curve (fig. 33) the horizontal distances
are proportional to the angles between the
table of a diamond and the direction of the
entering rays ; the vertical distances are
proportional to the amount of light entering
at these angles. The total amount of light
entering will be proportional to the area
shaded. That area must therefore be
divided into three equal parts ; this may
be done by integrals, or by drawing the
curve on squared paper, counting the
squares, and drawing two vertical lines on
the paper so that one-third of the number
of the squares is on either side of each line.
By integrals,

area —/"sin xdx ^= — cos x.
The total area = [— cos x] = i -fi =: 2,



therefore



"o area — "o.



The value of a corresponding to the vertical
dividing lines on the curve is thus given

by



cos x = I — I = I
cos:^ = I— t - — i



MATHEMATICAL 85

therefore

X = 7o|° approximately



and



X = 1091°.



Taking the value x = 90° as zero for
reckoning the angles of incidence,



and



z == 90°— 701° -= 19J
i = 90°— 1091° = — 19I,



The corresponding angles of refraction are

sin i sin iqi^ -333 3
sin r = = ^^ = 0000 _ . J07 7

;z 2-417 2-417

r-7°52'.
The range of the different classes is thus
as follows : —

Angle of incidence :

vertical rays —19!° to +19!°
oblique rays —90° to — I9j°
and +191° to +90°. /

Angle of refraction :

vertical rays —7° 52' to +7° 52'

oblique rays —24° 26' to —7° 52'

and +f 52' to +24° 26'.

The average angle of each of these classes



86 DIAMOND DESIGN

may be obtained by dividing each of the
corresponding parts on the sine curve in two
equal parts. The results are as follows : —

Angle of incidence :

vertical rays o°
oblique rays —42°
and +42°.

Angle of refraction :

vertical rays 0°
oblique rays —16°
and +16°.

For the design of the table and bezels,
we have to know the directions and positions
of the rays leaving the stone. The values
just obtained would enable us to do so if
all the rays entering the front of the gem
also left there. We have, however, adopted
a value for a (a = 40° 45') which we know
permits leakage, and we have to take that
leakage into consideration.

The angle where leakage begins is in-
clined at 24° 26' to the pavilion (fig. 24).
We have thus

Q' R' W = 24° 26',



MATHEMATICAL 87

therefore

Q' R' A' = 90°-24° 26'= 65° 34^
Now in triangle A Q' R',

Q' R' A + A Q' R' +R' A Q' - 180°,
therefore

A Q' R' = 180° -65° 34-40° 45'

= 73° 41'.
The Hmiting angle of refraction R'Q'T
is thus

= 90°-73° 41' - 16° 19',
corresponding to an angle of incidence of

sini = nsinr = 2-417 sin 16° 19'
= 2-417 X -281 = -678.

i = 42^.

Upon referring to the sine curve, we find
that the area shaded (fig. 34), which repre-
sents the amount of light lost by leakage,
although not so large as if the same number
of degrees leakage had occurred at the
middle part of the curve, is still very ap-
preciable, forming as it does about one-
sixth of the total area. Just under one-
half (exactly -493) of the fight incident
obhquely from the right (fig. 25) is effective,



88



DIAMOND DESIGN



the other half being lost by leakage. Still,
the sacrifice is worth while, as it produces
the best possible fire.

The oblique rays incident from the right
range therefore 191° to 42!°, with an average
(obtained as before) of 30° 15'. The corre-




/«o^



Fig. 34.

sponding refracted rays are 7° 52', 16° 19',
and 12° o'.

We have now all the information necessary
for the design of the table and the bezels.

Design of Table and Bezels (fig. 35)

Let us start with the fundamental section
ABC symmetrical about MM', making the
angles A C B and A B C 40° 45'.

V



[To face p. 88.




E. & F. N. SPON, LTD., LONDON.



[To face p. 88.



A-



P P, P' P, P



2 »^3




M



R



Fig. 35.



Tolkowsky, Diamond Design.]



E. & F. N, SPON, LTD., LONDON.



MATHEMATICAL 89

The bezels have been introduced into the
design to disperse the rays which were
originally incident from the right upon the
facet AB. To find the limits of the table,
we have therefore to consider the path of
limiting oblique ray. We know that this
ray has an angle of incidence of 42^° and
an angle of refraction of 16° 19'. Let us
draw such a ray P Q : it will be totally
reflected along Q R, if we make P Q N =
N Q R, where Q N is the normal. Now Q R
should meet a bezel.

If the ray PQR was drawn such that
M P = M R, then P and R will be the points
at which the bezels should meet the table.
For if PQ be drawn nearer to the centre of
the stone, QR will then meet the bezel,
and if P Q be drawn further away, it will
meet the opposite bezel upon its entry into
the stone and will be deflected.

The first point to strike us is that no
oblique rays incident from the left upon
the table strike the pavilion AB, owing to
the fact that the table stops at P. We
will, therefore, treat them as non-existent,



90 DIAMOND DESIGN

and confine our attention to the vertical
rays and those incident from the right.

Let us draw the limiting average rays
of these two groups, i.e. the rays of the
average refractions o° and 12° passing
through P, P S, and P T. The length of the
pavilion upon which the rays of these two
groups fall are thus respectively C S and C T.

The rays of the first group P' Q' R' S' are
all reflected twice before passing out of
the stone, and make, after the second
reflection, an angle of 17° with the vertical
(as by eq. (12)). Of the rays of the second
group, most are reflected once only (Pj Qi Rj)
and make then an angle of 69^° with the
vertical (this angle may be found by
measurement or by calculation). Part
of the second group is reflected twice
(P3Q3R3S3), and strikes the bezel at 29°
to the vertical. This last part will be
considered later, and may be neglected
for the moment.

We have to determine the relation be-
tween the amount of light of the first
group and of the first part of the second



MATHEMATICAL 91

group. Now we know that the amount
of obHque Ught reflected from a surface on
paviHon A C is -493 of the amount of verti-
cal light reflected (cp. fig. 34 and context).
If we take as limit for the once-reflected
oblique ray the point E (as a trial) on
pavilion B C, i.e, if it is at E that the girdle
is situated, then the corresponding point
of reflection for that oblique ray will be
Q2 (fig. 35). The surface of pavilion upon
which the oblique rays then act will be
limited by S and Q2, and as in a brilliant
the face A C is triangular, the surface will
be proportional to

sc2-q;c^

'Similarly, the surface upon which the
vertical group falls will be proportional to



TO.

Thus we have as relative amounts of
Hght—

for vertical rays TC^

for oblique rays -493(80^— QO).

The first group strikes the bezel at 17°



92 DIAMOND DESIGN

to the vertical, and the second at 69!° to
the vertical. The average inclination to
the vertical will thus be



17 xTg+691 x^93(Sg - QC^)
TC^+.493(SC2-QC^)

Let us draw a line in that direction
(through R, say), and let us draw a perpen-
dicular to it through R, R E ; then that
perpendicular will be the best direction for
the bezel, as a facet in that direction takes
the best possible advantage of both groups
of rays.

If the point E originally selected was not
correct, then the perpendicular through R
will not pass through E, and the position
of E has to be corrected and the corre-
sponding value of CQ2 correspondingly
altered until the correct position of E is
obtained.

For that position of E (shown on fig. 35),
measures scaled off the drawing give

CS =- 2-67 CS2 = 7-12 __

CT =2-13 GT^ = 4-54 CS^-CQ^ = 4-57.
CQ, = i-6o CQ,^ = 2-56



MATHEMATICAL 93

Therefore the average resultant incHna-
tion will be

17 X CT^+691 X^3(Cg-CQ^)
(CT2+.493(CS^-CQ^)
_ 17 X 4-54 +69-5 X -493 X 4-57
4-54 +-493x4-57

to the vertical.

By the construction, the angle /S, i.e. the
angle between the bezel and the horizontal,
has the same value

^ = 34r-.

The small proportion of oblique rays
which are reflected twice meet the bezel
near its edge, striking it nearly normally :
they make an angle of 29° with the vertical.
Facets more 'steeply inclined to the hori-
zontal than the bezel should therefore be
provided there. The best angle for re-
fraction would be 29° +17° == 46°, but if
such an angle were adopted most of the
light would leave in a backward direction,
which is not desirable. It is therefore



94 DIAMOND DESIGN

advisable to adopt a somewhat smaller
value ; an angle of about 42° is best.

Faceting /

The faceting which is added to the
brilliant is shown in fig. 43. Near the
table, '* star " facets are introduced, and
near the girdle, ''cross'' or *' half '' facets
are used both at the front and at the back
of the stone.

We have seen that it is desirable to intro-
duce near the girdle facets somewhat steeper
than the bezel, at an angle of about 42°,
by which facets the twice-reflected oblique
rays might be suitably refracted. The
front " half " facets fulfil this purpose.

We have remarked that the angle (42°)
had to be made smaller than the best angle
for refraction (46°) to avoid light being
sent in a backward direction, where it is
unlikely to meet either a spectator or a
source of light.

To obviate this disadvantage, a facet two
degrees steeper than the pavilion should
be introduced near the girdle on the back



MATHEMATICAL 95

side of the stone ; for then the second
reflection of the obHque rays will send them
at an angle of 25° to the vertical (instead
of 29°), and the best value for refraction
for the front half facets will be between

25°+i7°-42°.

These values are satisfactory also as
regards the distribution of light ; for now
the greater part of the light is sent not in
a backward, but in a forward, direction.

The facet two or three degrees steeper
than the pavilion is obtained in the brilliant
by the introduction of the back '* half ''
facet, which is, as a matter of fact, generally
found to be about 2° steeper than the
pavilion in well-cut stones. Where the
cut is somewhat less fine and the girdle is
left somewhat thick (to save weight), that
facet is sometimes made 3° steeper, or
even more, than the pavilion.


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