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values of a are much below those given

above, i.e. 49Â° and 30Â°, 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== 24Â° 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 = 24Â° 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

24Â° 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 24Â° 26'.

If we draw R' V, vertical angle V R' Q' =

R' Q' T' = 24Â° 26', and

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

-: 24Â° 26' + 24Â° 26'

= 48Â° 52'

as before,

a = angle V R' N',

and therefore

a :^ 48Â° 52' . . . (9)

For absolute total reflection to occur at the

first facet, the inclined facets must make an

angle of not less than 48Â° 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 -= 24Â° 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 = 90Â° â€” a

angle R S C = 90Â°â€” x

angle RCS == 2 X angle RCM

-=2 X ARQ =- 2(90Â°â€” a).

The sum of these three angles equals

two right angles,

90Â°â€” a +90Â°â€” :\; +180Â°â€” 2 a = 180Â°,

or

3a -\-x = 180Â°

3a = i8oÂ°â€” X.

Now, X is not less than 24Â° 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 24Â° 26'. Draw RV vertical

at R.

Fig. 27.

Then

angle Q R V = angle T Q R = 24Â° 26'

angle V R N = a.

As before, in triangle R S C,

angle SRC = go^-NRS = 9oÂ°-a-24Â°26'

angle RCS = 2(90Â°- a)

angle R S C = 90Â°â€” x.

70 DIAMOND DESIGN

Then

90â€” aâ€” 24Â° 26'+i8oÂ°â€” 2a+90Â°â€” ;t; = 180Â°

3a +% = 180^-24Â° 26' - 155Â° 34'

In the case now considered,

X = 24Â° 26'.

Then

3a = 155Â° 34'-24' 26' = 131Â° 8'

Â« == 43Â° 43' . â€¢ . . (10)

For absolute total reflection at the second

facet, the inclined facets must make an angle

of not more than 43Â° 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 45Â°

In the discussion of refraction in a

diamond, we have to consider two cases,

i.e. a is less than 45Â° or it is more than 45Â°.

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 == 24Â° 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< 24Â°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 24Â° 26', and (11) becomes

angle RSN == 24Â° 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 45Â°.

This gives for angle R S N

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

2-417 2-417

therefore

angle RSN = 17Â° 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 = %( = 17Â° 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 = 90Â°â€” a

angle Q R C = ^

angle Q C R == 2(90Â°â€” a),

therefore

(90 â€” a) +z +180Â°â€” 2a = 180Â°,

or

i = 3aâ€” 90Â°.

MATHEMATICAL 75

Introduce this value of i in (13),

2a = A;+6a â€” 180Â°

4a == 180Â°â€”%

and giving x its value 17Â°,

4a ^ i8oÂ°â€” 17Â° = 163Â°

a = 40Â° 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 24Â° 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 24Â° 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 45Â°

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 â€”

For first reflection, a must be greater

than 48Â° 52'.

For second reflection, a must be less than

43Â° 43'-

For refraction, a may be less or more

than 45Â°. When more, the best value is

49Â° 15', but it is unsatisfactory. When less,

the best value is 40Â° 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

40Â° 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 = 40Â° 45'.

B. Front of the Brilliant

When arriving at the value of a = 40Â° 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 : â€”

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 : â€”

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 â€”/"sin xdx ^= â€” cos x.

The total area = [â€” cos x] = i -fi =: 2,

therefore

"o area â€” "o.

The value of a corresponding to the vertical

dividing lines on the curve is thus given

by

cos x = I â€” I = I

cos:^ = Iâ€” t - â€” i

MATHEMATICAL 85

therefore

X = 7o|Â° approximately

and

X = 1091Â°.

Taking the value x = 90Â° as zero for

reckoning the angles of incidence,

and

z == 90Â°â€” 701Â° -= 19J

i = 90Â°â€” 1091Â° = â€” 19I,

The corresponding angles of refraction are

sin i sin iqi^ -333 3

sin r = = ^^ = 0000 _ . J07 7

;z 2-417 2-417

r-7Â°52'.

The range of the different classes is thus

as follows : â€”

Angle of incidence :

vertical rays â€”19!Â° to +19!Â°

oblique rays â€”90Â° to â€” I9jÂ°

and +191Â° to +90Â°. /

Angle of refraction :

vertical rays â€”7Â° 52' to +7Â° 52'

oblique rays â€”24Â° 26' to â€”7Â° 52'

and +f 52' to +24Â° 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 : â€”

Angle of incidence :

vertical rays oÂ°

oblique rays â€”42Â°

and +42Â°.

Angle of refraction :

vertical rays 0Â°

oblique rays â€”16Â°

and +16Â°.

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 = 40Â° 45') which we know

permits leakage, and we have to take that

leakage into consideration.

The angle where leakage begins is in-

clined at 24Â° 26' to the pavilion (fig. 24).

We have thus

Q' R' W = 24Â° 26',

MATHEMATICAL 87

therefore

Q' R' A' = 90Â°-24Â° 26'= 65Â° 34^

Now in triangle A Q' R',

Q' R' A + A Q' R' +R' A Q' - 180Â°,

therefore

A Q' R' = 180Â° -65Â° 34-40Â° 45'

= 73Â° 41'.

The Hmiting angle of refraction R'Q'T

is thus

= 90Â°-73Â° 41' - 16Â° 19',

corresponding to an angle of incidence of

sini = nsinr = 2-417 sin 16Â° 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 191Â° to 42!Â°, with an average

(obtained as before) of 30Â° 15'. The corre-

/Â«o^

Fig. 34.

sponding refracted rays are 7Â° 52', 16Â° 19',

and 12Â° 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 40Â° 45'.

V

[To face p. 88.

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

[To face p. 88.

A-

P P, P' P, P

2 Â»^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^Â° and

an angle of refraction of 16Â° 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 oÂ° and 12Â° 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 17Â° 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^Â° 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 29Â°

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

Hghtâ€”

for vertical rays TC^

for oblique rays -493(80^â€” QO).

The first group strikes the bezel at 17Â°

92 DIAMOND DESIGN

to the vertical, and the second at 69!Â° 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 29Â° 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 29Â° +17Â° == 46Â°, 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 42Â° 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 42Â°,

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 (42Â°)

had to be made smaller than the best angle

for refraction (46Â°) 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 25Â° to the vertical (instead

of 29Â°), and the best value for refraction

for the front half facets will be between

25Â°+i7Â°-42Â°.

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 2Â° 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 3Â° steeper, or

even more, than the pavilion.

above, i.e. 49Â° and 30Â°, 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== 24Â° 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 = 24Â° 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

24Â° 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 24Â° 26'.

If we draw R' V, vertical angle V R' Q' =

R' Q' T' = 24Â° 26', and

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

-: 24Â° 26' + 24Â° 26'

= 48Â° 52'

as before,

a = angle V R' N',

and therefore

a :^ 48Â° 52' . . . (9)

For absolute total reflection to occur at the

first facet, the inclined facets must make an

angle of not less than 48Â° 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 -= 24Â° 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 = 90Â° â€” a

angle R S C = 90Â°â€” x

angle RCS == 2 X angle RCM

-=2 X ARQ =- 2(90Â°â€” a).

The sum of these three angles equals

two right angles,

90Â°â€” a +90Â°â€” :\; +180Â°â€” 2 a = 180Â°,

or

3a -\-x = 180Â°

3a = i8oÂ°â€” X.

Now, X is not less than 24Â° 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 24Â° 26'. Draw RV vertical

at R.

Fig. 27.

Then

angle Q R V = angle T Q R = 24Â° 26'

angle V R N = a.

As before, in triangle R S C,

angle SRC = go^-NRS = 9oÂ°-a-24Â°26'

angle RCS = 2(90Â°- a)

angle R S C = 90Â°â€” x.

70 DIAMOND DESIGN

Then

90â€” aâ€” 24Â° 26'+i8oÂ°â€” 2a+90Â°â€” ;t; = 180Â°

3a +% = 180^-24Â° 26' - 155Â° 34'

In the case now considered,

X = 24Â° 26'.

Then

3a = 155Â° 34'-24' 26' = 131Â° 8'

Â« == 43Â° 43' . â€¢ . . (10)

For absolute total reflection at the second

facet, the inclined facets must make an angle

of not more than 43Â° 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 45Â°

In the discussion of refraction in a

diamond, we have to consider two cases,

i.e. a is less than 45Â° or it is more than 45Â°.

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 == 24Â° 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< 24Â°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 24Â° 26', and (11) becomes

angle RSN == 24Â° 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 45Â°.

This gives for angle R S N

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

2-417 2-417

therefore

angle RSN = 17Â° 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 = %( = 17Â° 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 = 90Â°â€” a

angle Q R C = ^

angle Q C R == 2(90Â°â€” a),

therefore

(90 â€” a) +z +180Â°â€” 2a = 180Â°,

or

i = 3aâ€” 90Â°.

MATHEMATICAL 75

Introduce this value of i in (13),

2a = A;+6a â€” 180Â°

4a == 180Â°â€”%

and giving x its value 17Â°,

4a ^ i8oÂ°â€” 17Â° = 163Â°

a = 40Â° 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 24Â° 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 24Â° 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 45Â°

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 â€”

For first reflection, a must be greater

than 48Â° 52'.

For second reflection, a must be less than

43Â° 43'-

For refraction, a may be less or more

than 45Â°. When more, the best value is

49Â° 15', but it is unsatisfactory. When less,

the best value is 40Â° 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

40Â° 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 = 40Â° 45'.

B. Front of the Brilliant

When arriving at the value of a = 40Â° 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 : â€”

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 : â€”

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 â€”/"sin xdx ^= â€” cos x.

The total area = [â€” cos x] = i -fi =: 2,

therefore

"o area â€” "o.

The value of a corresponding to the vertical

dividing lines on the curve is thus given

by

cos x = I â€” I = I

cos:^ = Iâ€” t - â€” i

MATHEMATICAL 85

therefore

X = 7o|Â° approximately

and

X = 1091Â°.

Taking the value x = 90Â° as zero for

reckoning the angles of incidence,

and

z == 90Â°â€” 701Â° -= 19J

i = 90Â°â€” 1091Â° = â€” 19I,

The corresponding angles of refraction are

sin i sin iqi^ -333 3

sin r = = ^^ = 0000 _ . J07 7

;z 2-417 2-417

r-7Â°52'.

The range of the different classes is thus

as follows : â€”

Angle of incidence :

vertical rays â€”19!Â° to +19!Â°

oblique rays â€”90Â° to â€” I9jÂ°

and +191Â° to +90Â°. /

Angle of refraction :

vertical rays â€”7Â° 52' to +7Â° 52'

oblique rays â€”24Â° 26' to â€”7Â° 52'

and +f 52' to +24Â° 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 : â€”

Angle of incidence :

vertical rays oÂ°

oblique rays â€”42Â°

and +42Â°.

Angle of refraction :

vertical rays 0Â°

oblique rays â€”16Â°

and +16Â°.

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 = 40Â° 45') which we know

permits leakage, and we have to take that

leakage into consideration.

The angle where leakage begins is in-

clined at 24Â° 26' to the pavilion (fig. 24).

We have thus

Q' R' W = 24Â° 26',

MATHEMATICAL 87

therefore

Q' R' A' = 90Â°-24Â° 26'= 65Â° 34^

Now in triangle A Q' R',

Q' R' A + A Q' R' +R' A Q' - 180Â°,

therefore

A Q' R' = 180Â° -65Â° 34-40Â° 45'

= 73Â° 41'.

The Hmiting angle of refraction R'Q'T

is thus

= 90Â°-73Â° 41' - 16Â° 19',

corresponding to an angle of incidence of

sini = nsinr = 2-417 sin 16Â° 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 191Â° to 42!Â°, with an average

(obtained as before) of 30Â° 15'. The corre-

/Â«o^

Fig. 34.

sponding refracted rays are 7Â° 52', 16Â° 19',

and 12Â° 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 40Â° 45'.

V

[To face p. 88.

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

[To face p. 88.

A-

P P, P' P, P

2 Â»^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^Â° and

an angle of refraction of 16Â° 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 oÂ° and 12Â° 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 17Â° 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^Â° 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 29Â°

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

Hghtâ€”

for vertical rays TC^

for oblique rays -493(80^â€” QO).

The first group strikes the bezel at 17Â°

92 DIAMOND DESIGN

to the vertical, and the second at 69!Â° 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 29Â° 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 29Â° +17Â° == 46Â°, 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 42Â° 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 42Â°,

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 (42Â°)

had to be made smaller than the best angle

for refraction (46Â°) 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 25Â° to the vertical (instead

of 29Â°), and the best value for refraction

for the front half facets will be between

25Â°+i7Â°-42Â°.

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 2Â° 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 3Â° steeper, or

even more, than the pavilion.