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PRACTICAL OPTICS FOR THE 
LABORATORY AND WORKSHOP 



J. 
PRACTICAL OPTICS 

FOR THE LABORATORY 
AND WORKSHOP 



BY 

B. K. JOHNSON 



WITH A FOREWORD BY 

PROFESSOR CHESHIRE, C.B.E. 

DIRECTOR OP THR OPTICAL ENGINEERING DEPARTMENT 
IMPERIAL COLLEGE OF SCIENCE 



LONDON: BENN BROTHERS, LIMITED 

8 BOUVERIE STREET, E.C 4 

1922 



Primed in Great Britain 
by Turnbull & Spears, Edinburgh 



PREFACE 

THIS little book has been written primarily as a 
course of instruction for the student in Practical 
Optics ; and secondly, to deal with the more recent prac- 
tical applications of Optics for use in the optician's 
\v< irkshop. 

The exercises contuinl in tin* book are compiled from 
a series of experiments through which students in the 
Optical Engineering Department of the Imperial College 
of Science (South Kensington) usually pass before pro- 
ceeding to the more advanced optieal work of the de- 
partinent. It commences with the juite elementary work 
and covers a considerable amount of ground, and -hould. 
I think, prove a useful laboratory course in " Light " 
lieges and Schools of Science. 

The experiment- involve as little expensive apparatus 
a> possible, hut at the present day >at i-faetnry experi- 
mental optics demands somewhat better apparatus than 
tli- rather old-time favoured piece of wood and . ard, and 
therefore it has heen partly the aim in these pages to 
suggest means of producing such apparatus in the best 
possil>l< 

Although some of the devices mentioned are not 
to be found on the market. I have given scale _:s of 

-u. h things (as, for example, optical I.enrhe-i. all <t \\hieh 
l"-'-n I'Mind thoroughly \> M that those who 

a -mall \\..rks hop available ma\ ooDBtntol D606MIJF 
apparat hem-elves. 

That the l.n,,k i- not entirely devoted to the UM f the 
hrought alMuit li\ tin- fad that some of tin- 
latter chapters such as Chapter VIII., for example deal 
\\iih practical testing of optical instmni* -m-. \\lnh I 



(5 PRACTICAL OPTICS 

hope may be of some interest and value to the person 
engaged on work in the testing department of the optician. 

All the diagrams are new and by the author. 

My thanks are greatly due to Mr L. C. Martin for very 
valuable assistance and advice during the process of com- 
pilation ; also to Prof. Cheshire, C.B.E., for kindness in 
writing the Foreword. 

B. K. JOHNSON 



OPTICAL ENG. DEPT. 

IMPERIAL COLLEGE OF SCIENCE 



CONTENTS 

PAGE 

PREFACE 5 

FOREWORD 11 



CHAPTER I 

REFLECTION AND REFRACTION OF LIGHT .... 13 

Verification of laws of reflection Formation of an image 
by a plane mirror Laws of refraction, and experimental \ n 
ficat ion Total internal reflection Ray plotter Path of 
rays through a 60 prism Minimum deviation Path of rays 
through a 45 prism Constant deviation prisms. 

CHAPTER II 
MIRRORS AND LENSES (OPTICAL BENCH EXPERIMENTS) . 30 

Optical l>cn< h (d-, Tij.ti. m of ne\v. inexpensive, and accurate 

bench) Measurement of the radius of curvature of a concave 

mirror or concave lens surface Radius of curvature of a 

convex mirror or n\. \ lens surface Focal length of 

lenses Focal length of " thin " concave lenses 

Relation between size of image and focal Im-th of a l-n- 

(graph) The relation between the conjugate distances an. I 

itures for thin positive and nega- - Simple 

t< leM-ojK' : (i) astronomical, (ii) ( ialil.an The simple compound 

niovMoi 

CHAPTKK Hi 

I'HuiuMl.l l:Y ......... 52 

"Richie pri-m photometer " llmnfonl " photometer 
Photoped " hummer- Brodhui- -Largo photometer 

benches (loss of light in a telcM im* nt) " Nutting " 

photometer " Lummer-Brodhun " MCI t 

7 



8 PRACTICAL OPTICS 

CHAPTER IV 

PAGE 

SPECTROMETER MEASUREMENTS 66 

The spectrometer Adjustments Measurement of prism 
angles Refractive index and dispersion Refractive index 
by immersion Determination of the wave-length of light by 
means of diffraction grating Calibration of the spectrum. 

CHAPTER V 
DETERMINATION OF RADII OF CURVATURE OF SURFACES . 81 

Spherometers : 3-legged, ring, Abbe, and Aldis types 
Curvature of small diameter surfaces Curvature by Newton's 
rings method Reflection method. 

CHAPTER VI 
MISCELLANEOUS ELEMENTARY EXPERIMENTS ... 93 

Use of a measuring microscope Appearances of " star " 
image at the focus of a lens : (i) single lens, (ii) achromatic 
lens Focal lengths of " eyepiece " systems Eccentricity of a 
divided circle Photographic tests on a lens. 

CHAPTER VII 

FOCAL LENGTHS OF " THICK " LENSES AND LENS 

SYSTEMS . . . . . . . . 105 

The " bar " optical bench Focal length of a thick lens by 
the magnification method " Cheshire " focal-length method 
Focal collimator " Lens rotation " method. 

CHAPTER VIII 

MISCELLANEOUS ADVANCED EXPERIMENTS . . .117 

Microscope objectives Focal length and numerical aperture 
Complete measurements of the optical system of the micro- 
scope for the microscopist The auto-collimating telescope 
Tests on a telescope The testing of prismatic binoculars. 



CONTENTS 9 

CHAPTER IX 

PAGB 

REFRACTOMETERS ...... .141 

The " Pulfrich " rt-fnu-tonu>ter The Abb6 refractometor 
Gas interferometric refractometer. 

CHAPTER X 

APPLICATIONS OF POLARIZED LIGHT . .152 

Detection of strain Microscope polarizer Sacchari meters. 



APPENDIX ......... 161 

The cleaning of optical surfaces .Silvering of glass Grind- 
ing and polishing a flat glass surface Babaming Developers 
f-.r photographic work: (i) a frosting Mention for glass, 
(ii) an optical cement Table of useful wav ! -n^ths also 
refractive indices Tables of : ]n^ ; rc< iprocals ; sines ; 
cosines ; tangent-. 



FOREWORD 

THE book which Mr Johnson has written deaU primarily 
with the experimental side of applied optics. Very 
little will be found in it about caustics, but a great deal 
about collimators. 

The title of the book has the merit of indicating not 
only its contents, but at the same time giving information 
as to the way in which the book differs from other books. 

Up to the present time there has been an unfortunate 
want of co-ordination between the practice of the labora- 
tories and that of the workshop, to the distinct disadvantage 
of both. Each has been in a position to assist the other, 
but for one reason or another has rarely done so. 

Laboratory work, on the other hand, has too often ignored 
everyday wants. The microscopist, however, who requin- 
simple methods within the compass of his equipment. 
for the determination of the focal lengths and aj> -mm-- 
of lii< lenses, and the magnifying powers of the various 
combinations of them, will find all the neee>-ar\ informa- 
tion given in this book. The owner of a telescope, too, 
who Mispects that only a part of the aperture of his object 
glass is operative, will now be able to test the matter for 
himself. He will learn something of the function and 
importance of that little stop in the erect 01 
ol \\hich mav not have ln-m suspected. 

F. .1 ^ IIKSHIKK 



n 



PRACTICAL OPTICS FOR THE 
LABORATORY AND WORKSHOP 



CHAPTER I 
REFLECTION AND REFRACTION OF LIGHT 

(a) VERIFICATION OF LAWS OF REFLECTION 

PLACE a piece of cartridge paper on a drawing board. 
On this place a mirror, preferably silvered on its 
"front" surface. A microscope "slip," 3 in. x 1 in.. 
>il\ered, makes an admirable mirror for the purpose ; 
it -hould be supported at the back so that it will stand 
with the silvered face at right 
angles to the paper. Place two 
pins A and B (Fig. 1) in 
front of the mirror in the 
j><itions shown. Look at the 
"images" of the pins, A' and 
I'. . in the mirror, and adjust two 
more pins C and D so as to 
appear in (lie same straight lint 
as these images. Let the lim- 
1 1m >ugh AB and CD intersect 
mi th<- mirror at O. Draw the 
normal at O and show that it 
m ikes equal angles with the incident m\ AB and the 
reflected ray CD. Repeat this experiment three or f >ur 
limes, using different positions for the pin A. and -how that 
in each case the angle of incidence Ifl c.pial to the angle 

Section, 
I li incident ray, the normal to the mirror at the point 

13 




I 



14 



PRACTICAL OPTICS 



of incidence, and the reflected ray all necessarily lie in 
one plane. Tabulate your values of the angles of incidence 
and reflection for each ray. 

(b) FORMATION OF AN IMAGE BY A PLANE MIRROR 

Place a pin P (Fig. 2) in front of the mirror as before, 
and place the eye in such a position that the lower part 

of the pin can be seen by 
A/> reflection. Behind the mirror 

adjust another pin so that its 
upper portion appears to be a 
continuation of the lower portion 
Y of the image, for all positions of 

the eye. The second is then at 
the image position of the first. 
Let P and P x (Fig. 2) be the 
positions of the object and image, 
< >P and let P P x cut the mirror XY in 

FIG. 2. 0. Prove by actual measure- 

ment that PO=P X and that 

the angle POX is a right angle. This will show that the 
image on the normal to the mirror is as far behind the 
mirror as the object is in front. 

In Fig. 1 produce DC back to A' and measure AE, A'E, 
BF, and B'F. 



(c) LAWS OF REFRACTION 

First Law. The incident and 
refracted rays, and the normal 
at the point of incidence, all lie in 
the same plane. 

Second Law. The ratio of the 
sines of the angles of incidence 
and refraction for the two media 
in question is constant. (See 
Fig. 3.) 



| Normal 




FIG. 3. 



REFLECTION AND REFRACTION OF LIGHT ir 



Explanation of the phenomenon of refraction by the 
" icave " theory of light. 

The " wave theory " is now a fully established fact, 
and refraction is very easily made clear by considering it 
on these lines. 

Let AB (Fig. 4) be the bounding line between two 
media, and suppose the lower half to be the denser 
medium. Let the velocity of light in the upper medium 
be v, and in the lower v v Let Cc, Dd, and Ee be three 
rays in an oblique parallel beam of light, and CDE the 





Fio. 4. 

wave front at any instant. This will advance parallel 
t" itself until it reaches cde. The ray Cc then enters a 
different medium, and its velocity is changed from v to 
Vj. Consequently, whilst the ray Ee is travelling from 

e to e lt Cc will move through a distance ( xe^J. With 

c as centre and ( Vl x eej as radius, describe a semi-circle 

in the lo\\cr medium. From e l draw a tangent to this semi- 
irele, touching it at c v Join cc v Then cc^ will be the 
direction of the ray Cc ; cfr will be the new wave 
h"ht ; and the disturbance at e will travel to e 1 in the 
same period ..t time " t " that the disturbance at c travels 



16 PRACTICAL OPTICS 

Now FG is a normal to AB at the point c, and the angle 
the beam was making with this normal was CcF. But 
having undergone this change of direction (i.e. refraction) 
in the denser medium the angle is now r 2 cG. 

The ratio of the sines of these two angles is constant 
whatever incidence is given to Cc, and this ratio is known 
as the " refractive index " between the two media. 

Refractive Index is usually denoted by the letter " n," 

so that the above may be written n = . . 

sin r 

It is also easily shown from Fig. 4 that the " refractive 
index " is also the ratio of the velocities of light in the 
two media : 

TT. sin i 

For n=~^ . 

sin r 

Now sin " i "=sin ece-,= , 



and sin " r " =sin ce 1 c l = 1 ; 



ee, 




CC 

But ee 1 = vt. 
and cc = vt 



(c) EXPERIMENTAL WORK FOR =A CONSTANT. 

Place a block of glass (about 4 in. x 3 in. x 1 in.) with 
parallel sides on a piece of drawing paper, and draw two 
fine lines along the two edges AB and DC (Fig. 5). Place 
a pin P in the position shown in contact with the edge 
of the block, and arrange a series of pins Pf^P^P^ on 
the circumference of a circle whose centre is P. The 
radius of this circle should be about 3 in. This gives a 
series of incident rays PJ?, P 2 P, P 3 P, P 4 P. Determine 
the paths of these rays through the glass by placing against 
the side DC pins P 5 , P 6 , P 7 , P 8 , which appear to be in the 



REFLECTION AM) RKFR.V HQN OF LIGHT 17 

same straight line as P 1 P 2 P 3 P 4 respectively. Remove 
the block and join PP 5 , PP 6 , PP 7 , PP 8 . Also draw a normal 
l'\ to the surface AB. 

-ure with a protractor the angle of incidence and 

the angle of refraction for each ray and show that ?* <t * , 

sin r 
is constant. 

This ratio is the " refractive index " of the material. 
It may be determined graphically from the figim 1,\- com- 
pleting the circle P^PaP,, thus cutting the refracted 
rys, Draw perpendiculars to the normal PN from tli. 




points at which an incident ray and a corresponding 

ted ray cut the circle. The ratio of these perpendicular 
lengths gives the required result. 

Second Method. Place the glass block as before on the 
; nir paper, and draw fine lines along the edges AH 
i CD (Fig. 6). Place a pin at P in contact with tin- 
edge AB. Insert other pin- .\,\ 2 X 8 on the other edge 
DC in the positions shown. On looking through the glass, 

further pin^ YjYgYj H< that thrv appear in tin- sainc 

irht lino as Xf, X 2 P, X 3 P respectively. Remove 
th block, and draw a normal P\ t. \B at P. Join ^ 
.in.l produce it back to cut the normal in V. v Produce 
each other ray back in the saim manner, 

B 



18 PRACTICAL OPTICS 

PX PX 

Show by actual measurement that 7 v 1 ~7~X 2> anc * so 

on, is constant. This ratio is the " refractive index " 
of the glass. 

Explanation of foregoing. Consider Fig. 7. PX X and 
Y^ are the same rays as lettered thus in Fig. 6. Draw 
a second normal OR at X x . Then Y^jR^ angle of 
incidence, and PX X O= angle of refraction. 




Air 



FIG. 8. 



But PN and OR are parallel. 

So 

and 



1C 

Now n = - n r,. 
sin r 



_ 
Therefore n= 



(d) TOTAL INTERNAL REFLECTION 

Total internal reflection is dependent on refraction. 
Fig. 8 shows how a series of rays coming from a point 
" " in a denser medium than air are refracted at the 
bounding surface XY (e.g. & stone in a pool of water). 

Let " n " be the refractive index, which in this case 
will be less than unity. Now, for any angle of incidence 
" i " (measured inside the denser medium) the angle of 



REFLECTION AND REFRACTION OF LIGHT 19 

ivt'rartion ** r " is calculated from the formula n = ~ -. 

sin r 

So that, in this case, " r " is always greater than 

A- / in. -reases the refracted rays get r md 

nearer the surface, until a position is reached such as ODY, 
\vhnv / =90. Now the sine of 90 is unity, and no 
angle has a sine greater than unity. >o that for our formula 

(-in i\ si; 

n = - ) to give any real value for r, - must be 

"i n f*/ w 

il to. or less than, unity. Thus, for a refracted ray 
to be formed, the greatest value of " i " is when sin * 
This angle is called the " critical angle." 

Tin- j u -st inn then arises What happens to tin- in. 




rays \\ln-n th< ml OI), as at K, making 

greater angles \Mtli the normal ' In this case no light 

?!. l.ui all \\hi.-h t.i!U ..n i!,, urface is reflected 

10 the first mi*diuin I DMOOO i- t< nned 

\ angle, however, a 

certain amount <>t mt< mal reflection takes place. 

' ' / I' .- .1 n-l.t an-!. 

1 rawing paper on a drawing board. The p uld 

be a large on,-. |( t|, th.- h\ pot.-nu-..- -urfaoe 

about 4 in. IOIIL: l.i\\ im- Inn - n>un<i the three faces of 

tin .- ji |,n, I 1 , an nhown in Y\\i. '. m r.i, 

tlicmirfarr \l % - With I' . oentfl md I'P, an radiiu 



20 PRACTICAL OPTICS 

(about 4 in.) describe a semi-circle about AB. Place a 
second Pj in some such position as indicated, and on 
looking at the hypotenuse face AC, insert further pins 
R! and S x so that they appear in the same straight line 
as PPi- This will give the refracted angle in air N 1 O 1 R 1 
corresponding to the incident angle PO^E^ in the glass. 
Move the pin P x into another position P 2 so that the angle 
POjEj is increased, and insert the pins R 2 and S 2 . In 
this way move P x continually towards M until the eye 
can only just see the two images of P and P 1 in line with 
the perpendicular edge C of the surface AC. This will 
give the last ray in the glass that is able to get outside 
the bounding surface AC. Remove the prism, and draw 
a normal PM. Join P C P. P c is this last position of P x . 
Measure the angle MPP with a protractor. With the 
refractive index of the prism given, calculate the refracted 
angle, and draw in PO, making this angle with the normal 
PM. This is the " critical angle " for this particular glass. 
Experiment II. Total Internal Reflection. Replace the 
prism on the paper as in Fig. 9, and place P in the same 




FIG. 10. 

position as before. Move P c further towards M so as to 
make the angle POE just greater than the critical angle. 
Then place the eye so as to look in the face BC, and 
position the pins R x and S x (Fig. 10) so as to appear in 
the same straight line as PP C (Fig. 10). You will now 
notice that as soon as the angle POE is made greater 
than the critical angle, the ray is totally reflected at the 
face AC. 



REFLECTION AND REFRACTION OF LIGHT I \ 

Repeat the experiment for other positions of P e as 
indicated at P 3 and P 4 , and show in each case that the 
ray undergoes "total internal reflection." 





X 

A E 


C 




\ 
v 1 


1 




^ Unity ^ ri J 

I 





(e) "SMITH'S" RAY PLOTTER (Trans. OpL Soc., 1919-20, vol. 
xxi., No. 3). 

In the case of all graphic experiments in connection 
uith refraction, it is continually necessary to draw re- 
fracted rays at the bounding surfaces of media. These 
angles have, in an ordinary way, to be calculated from 

the formula n= . and then drawn out with a pro- 
sin r 

tractor, which, if a number 
of surfaces are involved, be- 
comes very laborious and al>n 
occupies a great deal of time. 
Therefore it is of great ad- 
IL: it these angles can be 
obtained readily and easily : 
this " ray plotter " here de- 
scribed gives a means of doing 
this, and in a very simple 
manner. 

Procure a piee. ..!' thin sheet celluloid, 6 in. long by 
2.J in. \\ide <-<< FIL:. 11). On it scratch a fine straight 
line (H . ujth a marking point; also a line XY at right 
angles to this at O, as shown in the figure. Mark off 
-distances OA- - in.. AB = 1J in., and AC = 3 in. The 
relation between these distances is dependent n the re- 
fractive index, hut this is explained later. All glasses, 
of course, have not the *a x; but for 

graphic experiments such as would be done in the la bora- 
ton the refract x of glass would probably be taken 
as approximately 1*50. And this is the value on which 
the above figures are based. If a part i v pe of glass, 
of known refract m index, is in question, then of course 
above values will differ l>nt in every case OA must 






22 PRACTICAL OPTICS 

be equal to unity. AB equal to > and AC equal to " n " 

in some convenient unit. 

At the points A, B and C drill three very small holes, 
just sufficient in diameter to take the point of a pin. The 
" ray plotter "is now complete. 

How to use it. Suppose we wish to determine tilt- 
direction of the refracted ray EM (Fig. 12) in a piece of 

glass corresponding to an in- 
cident ray VE. Place the " ray 
plotter " on the paper, so that 
the line XY lies on the bounding 
line of the two media PR, and 
O of the " ray plotter " coincides 

with E on the paper. 

Insert a pin through the hole 

A and revolve the celluloid until 
Glass the hole C comes directly over 

the incident ray VE. Prick the 
paper through the hole B ; re- 
move the " plotter/' and join 

this point to E. This line produced EM gives the 
refracted ray. 

If the ray in the rarer medium is required to be traced 
from the ray in the denser medium, the method of pro- 
cedure is very similar, with the exception that when the 
celluloid is revolved about the point A,* the point B must 
be made to coincide with the refracted ray in the denser 
medium. The paper is then pricked through the hole ( ' 
and this point joined to E and produced. This gives the 
ray in the rarer medium. 

Proof of Method. Let VE (Fig. 13 a and b) represent 
the incident ray in the rarer medium incident at the point 
E of the denser medium, and EM the corresponding re- 
fracted ray. EA is the radius of a circle and equal to 
unity. Draw AC (Fig. 13 a) perpendicular to EA, and 
in Fig. 13 b draw it obliquely. Where the refracted ray 

* In this case the hole A will be in the position A] (Fig. 12). 




REFLECTION AND REFRACTION OF LIGHT 

KM cuts AC, describe a circle with radius AB ; and 
where VE produced cuts AC, describe a circle with radius 
AC. 

It is required to show, that for rcrr to be constant, 

Mil .M I'. \ 

AC must equal "n" and AT. i whore n = the refractiv. 

H 

index). 




n). '). 

Now the triangles AEC and AKB an -imilar (from 
the ratio of sides and a common angl 
so that </AEC=^EBA 
and ^ECA=Z.AEB 
i. in trianph 

\< 
sin i sin AEC EC AC 



_ 
" 



sin 



But EA i.s nn i 
crefore AC 
Smiil.ulv. in triangle AEB, 



KC 



KA 



-in sin EBA EB_EA 
""sinr^sin AEB "IB "IB" 



But I 
Th rrfore AB = 



24 



PRACTICAL OPTICS 



(/) PATH OF RAYS THROUGH A 60 PRISM 

On a sheet of drawing paper on a drawing board place 
a 60 glass prism. A large prism should be used for this 
experiment, preferably about a 3 in. face and refractive 
index about 1'52. Draw fine lines along the two sides 
AB and BC (Fig. 14). Remove the prism for a moment, 
and at the mid-point P x of AB draw a normal N X D. At 
P! set off a line PiP 2 at 40 to the normal with a protractor. 
Place two pins, one at P x and the other at P 2 ; then put 
the prism back into its former position. On looking in 
the face BC of the prism arrange two more pins P 3 and 
P 4 so that they appear in the same straight line as PiP 2 . 
The images of P 1 and P 2 will be fringed with colour owing 
to dispersion, but this will not interfere with the position- 
ing of the pins P 3 and P 4 . Remove the prism, join P 4 P 3 
and let it meet the surface BC in F. At this point draw 
a second normal N 2 D. Measure the angle of emergence 
P 4 FN 2 corresponding to the angle of incidence NjP^ 
(which was 40). Produce P 4 F and P 2 P X and let them 
meet at " 0." Then the angle P 4 OM is the deviation 
produced by the prism. 

Increase the angle of incidence NjP^ by 5 and repeat 
the experiment, and so on until N 1 P 1 P 2 is as large as 
possible. In each case measure the angle of emergence 
from the prism and the deviation, and tabulate the results 
as follows : 



Angle of 
Incidence. 


Angle of 
Emergence. 


Deviation. 


40 






45 






50 






55 






60 






65 






70 






75 







On a piece of squared paper then plot two curves, one 



REFLECTION AND REFRACTION OF LIGHT 25 



showing the relationship between the angle of incidence 
and the angle of emergence, and the other between the 
angle of incidence and deviation. Plot incidence angles 
in a horizontal direction and emergence and deviation in 




14. 

Fig. 15 shows the type of graphs 



a vertical direction, 
obtained. 

Observe from the curves you obtain that there is a 
lM>iti"ii where the " deviation " is at a minimum. The 
angle of incidence should be noted for this position ; also 




30 40 60 60 70 
Anttes in Degrees 
Incidence, 

Fio. 15. 

by i 'to the "emergence angle curve " the corre- 

sponding emergence angle will be obtained. If 

ves are plotted correctly these two angles will be found 
the same. 

This shows that win n ; nt .m<l 

are equal, the deviat . is at its minimum. 



26 PRACTICAL OPTICS 

To determine the position of minimum deviation. Draw 
a straight line P X P 2 (Fig. 14) on a piece of drawing paper, 
and place two pins in the positions P^. Place the 60 
prism as indicated so that P l touches the face AB. Look 
into the face BC and place the eye so that the images 
of P x and P 2 appear in the same straight line. Now* 
rotate the prism slowly, first in one direction and then 
in the other, moving the eye the whole time so that the 
two images always appear in the same straight line. A 
position will be noticed when the two images, moving 
in one direction, suddenly become stationar} r , and com- 
mence to move in the opposite direction. This stationary 
position of the images is the position of " minimum de- 
viation " for the prism. Insert two pins P 3 and P 4 so 
that they appear in the same straight line as P 1 and P 2 . 
Remove the prism, join P 4 P 3 , and show that the angles 
of incidence and emergence are equal. 

(g) PATH OF RAYS THROUGH A 45 PRISM 

(i) Place a 45 prism ABC (Fig. 16) on a piece of 
drawing paper (the prism should be large, from 3 in. 

C 

P+ 






P, P 2 A B 

FIG. 16. 

to 4 in. hypotenuse face). Mark fine pencil lines round 
the three faces. Remove the prism for a moment, and 
draw five lines to the left of AB parallel to the hypotenuse 
AC. Number these lines 1 to 5, and replace the prism. 
On line " 1 " place two pins P x and P 2 . Look into the 
face BC, and insert further pins P 3 and P 4 so that they 
appear in the same straight line as P X P 2 . Do this for all 
the five incident rays, and number each corresponding 
emergent ray. Remove the prism, and draw in the path 
of the rays through the prism, remembering the laws of 
refraction and reflection. 



REFLECTION AND REFRACTION OF LIGHT 27 

Note that " internal reflection " takes place at the face 
AB, also that an " up and down " reversal of the object 
takes place. It will also be seen that there is a limit to 
the "useful aperture" of the prism when used in this 
way ; for after No. 5 ray in the figure has got through, 
the portion FCG is no longer useful, as no more rays above 
F can get through the face BC. The figure AFGB is called 
an erecting prism. 

(ii) Place the prism on a fresh piece of paper and draw 
tine lines round the edges as before. Remove the prism 
and draw a series of parallel lines at right angles to AB 



(B) 



(4) 



(2) 



(D 



/; 





(SX4)(3)(2)(1) 
FlO. 17. 



12345 S4321 
Fio. 18. 



(Fig. 17). Number these* lin- an.l replace the pn-m. 
Insert pin- P x and P 2 on the No. 1 line, and looking in the 
face B< place P 8 and P 4 so as to be in the same straight 
lin. with the images of P l and P a . Do the same f i raya 
Nos. 2, 3, 4 and 5. Note that a- m incident 

normal to the face AB, no deviation takes place at tin- 
acting surfaces, hut that total reflection t ik<- pla.-e 
he face A< Also, observe that there is a "right and 

d ..t the object in this case. 

din Determine as befnn l-y means of the pin method 
of ray-tracing, the pal ^h-n thc\ are inddenf 

on t face AC (see Fig. i^ \.tc in tin- 

ca84 . total internal r. tl.-,-iion takes place at luth 

surfaces AB and !< md alto that there is again a right 



28 



PRACTICAL OPTICS 



and left reversal of the object. Right-angled prisms are 
used in this last manner in prismatic binoculars. 

(h) CONSTANT DEVIATION PRISMS 

(i) There are two special types of prism which should be 
noted in connection with the work of this chapter. 

They are at present in everyday use and involve principles 
dealt with here. The first of these is illustrated in Fig. 19, 
and is known as a Pentagonal Prism ; these prisms are 
used to a very great extent on military and naval " range- 
finders/' The figure shows the direction and path of the 

Thls t surface 
silvered, 



ThhS surrace\~~ 
silvered. \ 



Pentagonal Prism 
FIG. 19. 





FIG. 20. 



rays through the prism, and, as will be seen, " internal 
reflection " takes place at the silvered surfaces. This is 
not the same kind of internal reflection that has been 
dealt with before in this chapter, as that is dependent 
on the critical angle ; in this case it is essential that the 
two surfaces of the pentagonal prism indicated should 
be silvered. The importance of this prism, however, 
lies in the fact that the " deviation " between the inci- 
dent and emergent rays always remains constant, and 
also that this deviation is 90. If this type of prism is 
available in the laboratory, the above points should be 
proved by ray- tracing with pins. 

(ii) The second type of prism is illustrated in Fig. 20, 
and is used a great deal in connection with spectrometers. 



REFLECTION AND REFRACTION OF LIGHT 29 

This prism also gives " constant deviation " between the 
incident and emergent rays. The path of the rays are 
indicated in the figure, and, as will be seen, they undergo 
t\v> refractions and one total internal reflection. The 
prism is all one piece of glass, but the dotted lines indicate 
ho\\ it may be considered as built up from two 30 prisms 
and one 45 prism. 

If the laboratory has this type of prism, rays should be 
t raced through it by pin methods. 



CHAPTER II 



MIRRORS AND LENSES (OPTICAL BENCH 

EXPERIMENTS) 
(a) DESCRIPTION OF OPTICAL BENCH 

AN optical bench of the type here described is very 
convenient in a laboratory. Its combined simplicity 
and accuracy make it invaluable for both instructional and 



Cross Line 
Object 



Lens 
Holders 



Ground-Glass 
Screen 



Eyepiece in 
Holder 




FIG. 21. 

commercial work. Fig. 21 shows the general appearance of 
the bench, and, as will be seen, it consists of a Chesterman 
steel metre rule supported in a vertical plane, along which 
all other necessary fittings slide. These fittings are all 




FlQ. 2lA. 

very simple and inexpensive to construct. A group of these 
are shown in Fig. 22, such as the cross wire object, ground 
glass screen, lens holders, mirror, etc. It will be noted 
that the base of all these fittings is " cut away " in such 
a manner that readings may be taken direct from the steel 

3J 



MIRRORS AND LENSES 



rule without any appreciable error being introduced. Where 
more accurate results are necessary a " correction rod " 
may U employed. The lens holders are designed to 



Cross -Wire 



Ground- Glass 
Screen 




Correct/on 
Rod 



o 



, Mirror 
m Mount 



Achromatic Uns 
in Mount 



tt, 

carry len>es from any ordinary spectacle trial case, so 
for xperimental work a large range of lenses may 
be obtained. 

The fittings that >upport the >teel rule in a vertical 

Trial Case Lens is 

out in here , and clamped by this sliding fitting 



,n line with 
centre of Lens 
from which 
resd ing 53 re 
taken off 
5tee/ rule 




2. 8.A. Thread 



S/ot for 
Steel Rule, 



-IM.UII m I-'I L - L'I \ Th'-se are adaptable 
M..t .M,! in. -ire rule but to shorter lengt 

a foot rule, \\hni ,in ), nzperiineiits ..ul\ jnvlve -in. ill 
ranges. Scale dra\\ the lens holders and ground 



32 



PRACTICAL OPTICS 



glass screen holders are shown in Figs. 23 and 24. From 
these and Fig. 22 a general idea of all the fittings may be 
obtained. (See Article by Prof. Cheshire in Trans. Opt. 
Soc., vol. xxii., No. 2.) 



Watch Spring 



A Concave 
Mirror is 
shown dotted 




^Bevelled 

Slot for Steel Rule 

FIG. 24. 

(6) MEASUREMENT OF THE RADIUS OF CURVATURE OF A 
CONCAVE MIRROR OR CONCAVE LENS SURFACE 

The concave mirror * provided for this experiment should 
be held in one of the optical bench fittings so that the 





FIG. 2f>. , 

" pole " of the mirror is in the same plane as the edge of the 
mount from which readings are taken (see Fig. 24). 

Arrange on the optical bench the cross- wire object and 
the mirror whose curvature is required. Place a plane 

* These mirrors may be made very simply by silvering the surfaces of a 
convex and concave lens from an ordinary spectacle lens " Trial Case," and 
mounting them with the silvered surface " outwards." 



MIRRORS AND LENSES 33 

glass reflector G (micro cover slip) diagonally, as shown in 
Fig. 25, so that light from a lamp L (an electric lamp 
" frosted " or covered with a piece of tissue paper) 
illuminates the cross-wire object O. Place the eye in t Im- 
position shown and an " image " of the cross- wires will 
be seen near the "real" cross-wires reflected from tin 

ace M. It is at once evident that if the " image " 
and "real" cross- wires are in the same plane the di-- 
M() \\ill !>< the radius of curvature of the mirror, for 
all rays diverging from O will return back along ti 
original paths, and therefore they must strike the mi 
normally (the normal to a spherical surface at any part icular 
point is its radius). The method of ensuring that " image " 
and object are in the same plane is by employing tin- 
parallax method. By moving the head from side to side 
' image " of and "real" cross-wires will appear to 
move together \\ln-n the mirror is in its correct position ; 
it however, the "image" does not appear to move as 
fast as the " real " cross-wires the plane of the image will 

}>ehind the plane of the object, and vice rersri. As 
an alter the image" may be focussed direct!, 

on the white surface at the back of the cross-line 
object. 

When the curvature of a concave lens surface is re<|un 
exactly the same procedure is employed, with the ex< -eption 

iface not under t.-t must be covered in H 
manner in ..nld t ; :lccted ha 

If tlii-, hack surface of the lens is covered with a thin 

"I pi.i-ti. me." this serves the purpose very \\ell. 

hlotting ni<k to the back surface with 

vaseline does equally \\--ll image" of the cross- 

\\irc-, will not he -<> hnuht a- \\hen a -ilven-d surface i-* 

used, but sufficiently hriLrht fr taking measurements. 

(c) RADIUS OF CURVATURE OF A CONVEX MIRROR OR A 

CONVEX LENS SURFACE 

Arrange the apparatu tre "optical IM-IK -h " 

as shown m Kig. 26. O is the cross-line object at the 



PRA< TICAL OPTICS 

end of the steel rule. A is an achromatic lens * (held in 

>t the lens holder fittings) which forms an image of 

tlu> eross-wiivs on the ground glass screen S. G is a plane 

L 'lass retire -tor \\hich illuminates the object from a lamp 

I. 

Determine carefully the reading, on the optical bench, 
<>f the ground glass screen S when the image is sharply 
in focus. Interpose the convex mirror f to be tested 
.M. in the position indicated, and adjust its position until 
< n viewing the object as in the last experiment, the plane 
of the image " thrown back " by the mirror M is co- 
incident with the plane of the "object/' This is done 
ly the parallax method as before or by focussing the 




S M A O 

FIG. 20. 

image direct on the white surface at the back of the cross- 
line object. The reading of the mirror is then taken, 
and the distance SM is the radius of curvature of the 
surface. For, in order that the " rays " leaving and A 
should retrace their paths after reflection from the mirror 
and form an " image " at 0, they must strike the mirror 
" normally," and this is only the case when the distance 
S.M is the radius of curvature of the surface. A number 
of independent readings should be taken for the position 
of .M and the mean obtained. 

For the determination of the radius of curvature of a ' 
convex lens surface, the same method is adopted the 
back surface of the lens being covered by some such 
method as mentioned in the previous experiment. 

* An " achromatic " lens is used to prevent undue dispersion of the light, 
which would otherwise arise with a " single " Ion-. 
t See footnote on page 32. 



MIRRORS AND LENSES 35 

Curvature (Introductory i 

Thr curvature of a circle may be defined as being equal 
t the reciprocal of its radius. 

CD_ DA 
DA~CE-CD ; 
and \vhrn the angle DOA is small 

j j ^ =-^- (very nearly, r being the radius of circle). 

Whence CD <* - 
r 

Thus the length CD, known as the " sagitta " (trigo- 
nometrically the versed sine of the 
angle DOA) is a measure of the 
curvature of the arc ACB. This fact 
i- the foundation of the curvature 
method. 

So that, light waves as they reach a 
lens or mirror from a point source at 
a distance " u " have a curvature equal 

to and t his curvature has a negative sign 

M 

\\ IK n the waves are "convex-fronted " and thus expanding 
from a focus ; and a positive sign when they are " concave- 
fronted " and thus contracting to a focus. Similarly the cur- 
vature imparted or " impressed " by a positive lens of focal 

length / i- (jual to + , whilst in the case of a negative 

l n> it is equal to - 

The curvature "impressed" upon a plane-fronted \\.-m- 
by a mirror or lens is defined as its " focal power." 

This power is impressed upon all waves acted upon, 
DO matter at what distance the object may be. Thus 
tin in \ at ure of each wave, as it emerges from a lens, or it 
may be reflected by a mirror, is equal to the curvature 
..i th- in. id< nt wave added to the curvature impressed 
by the lens or mirror. In other words, final curvature 
-equal- initial < urvature-f that impressed. 




PRACTICAL OPTICS 

It // distance of object to lens 

v= image 
and f = focal length of the lens, 



V U 



(d) FOCAL LENGTH OF A CONVEX LENS (THIN) 

(i) Place a 5D lens from the " trial case " in one of 
the lens holders on the metre optical bench. Direct the 
optical bench at the furthest bright object that can be seen 
for instance, a street lamp, or an electric lamp placed 
in a long corridor, the distance should not be less than 
50 yards. Place also on the bench a ground glass screen 
in its holder and receive an " image " of the distant lamp 
produced by the lens on this. The difference between 
the readings of the lens holder and ground glass screen 
holder will give the " focal length " of the lens. Make 
a number of independent settings and measure the dis- 
tance in each case. See how nearly any one measurement 
is likely to be correct. 

(ii) After having used a distant object, use an object 
comparatively near to the lens. This method involves 

the use of the formula -,= --- > where "/" is the focal 

/ v u 

length of the lens, " u " the distance between the object 
and the lens, and " v " the distance between the " image " 
and the lens. Due respect must be made to the use of 
signs when employing this formula, and it should be 
remembered that divergent light is always reckoned as 
possessing negative curvature, whilst convergent light is 
positive. Set up the cross-line object (Fig. 28) at one 
end of the optical bench and illuminate it with a lamp. 
Place the 5D lens L (in holder) at a distance of about 
45 cms. from the object and receive an image of the cross- 
lines on the ground glass screen. Take a number of 
independent readings for the position of this screen. 
Measure the distance " u " (object to lens), in this case 
it will be a negative value. Also measure " v " (image 



MIRRORS AND LENSES 37 

t lii-. this will be a positive curvature. From these 
values calculate the result for "/." 

Move the lens to another position (say 55 cms. from 
tin- ohjeet) and repeat the experiment. 

. i ) A uto-collimation Method. It will be seen from 
Fig. 29 that if light diverging from the object O is rendered 

O L G 




u 4-*- - - +1; - 



Pic; 

parallel by the lens L. reflect rd hack by a mirror M, 
airain brought to a focus by the lens, the di-tan.. 
<>L will he the focal length of the lens. Set up the object 
O at the end of the bench as before and illuminate it ; 
pi c the lens about 20 cms. from the object, and further 
along the bench place the mirror M in position. ( 

O L M 

+ 




fully adjust tin- I'-n- holder until an " image " of the object 
1 1 ply focussed "ii tin ulntined back of the object.* 
Measure the focal length OL. Take a number of in- 
depend.-nt n-ad - the position ol I. Pnk< 

ii i. ult ^ tor -ach method and compare tin n 
result 

FOCAL LENGTH OF THIN CONCAVE LENSES 

up the cross-v iect O (Fig. 30) at one and 

<>pn<-;d l.rneh, and form an image of th; 

tiltmir 



38 PRACTICAL OPTICS 

of the achromatic lens A on the ground glass screen ftj. 
Place a -3D lens from the trial case in one of .the lens 
holders and insert this in the path of the convergent 
beam at L. Move the screen until the image is again 
focussed, as at S 2 . The image produced at Sj by the lens 
A serves as the object for the negative lens, so that the 
distance LS X is " u " and is positive, while the distance 
LS 2 is " v " and is also positive. Using the formula 

-: - J as before, the focal length of the negative lens 

may be determined. All values of readings taken from 
the " bench " should be the " mean " of a number of 




O 



independent settings. Move the negative lens L to a 
fresh position and repeat the experiment. 



(/) RELATION BETWEEN SIZE OF IMAGE AND FOCAL LENGTH 
OF A LENS 

Set up the metre optical bench with a lens holder 
mounted on it. Arrange at the zero end of the steel rule 
a piece of ground glass screen (4J in. x 3J in.) in a vertical 
plane so that the ground surface lies flush with the end of 
the rule. As far away as it is possible to arrange, set 
up two light sources at the same height as the optical 
bench. Make the distance apart of these two lamps 
about 6 or 8 feet, so that they subtend a small angle at 
the lens. In the lens holder place, in turn, lenses from 
the trial case ranging from a +2D to + 12D, varying by 
ID every time. In every case measure the distance 
between the centres of the two images produced on the 
ground glass screen. This is most easily done by laying 
a short millimetre rule on the ground glass and observing 



MIRRORS AND LENSES 

with a watchmaker's eyeglass. The position of tin- len- 
holder mi the optical bench when the images are sharply 
in focus on the screen must be taken for each individual 
len-. Tin- will jzive the focus of the lens (appmximai. 
Tabulate the values fnr the distance apart of the in. ages 
and the corresponding focal lengths for eaeh lens, and 
plot these values on squared paper. On the same p 
of paper plot the reciprocal of the focal length auain-t 
the distance apart <>f the images, Write do\\n the mean- 
ing of your graphs thus obtain d. 

2 Metre Steel Rule supported 
in Wooden Base forming 
extra long Optical Bench 

Stee/ Rule 




Wood Clamps 



THE RELATION BETWEEN THE CONJUGATE DISTANCES 
AND CURVATURES FOR THIN POSITIVE AND NEGATIVE 
IMAGING LENSES 

bhfa experiment the nptieal helieh 18 employed. 

l.ut in plaee nf the metre steel rule .1 two-metre -leel 
rule is used, as a larger \\.>ilm_ range is necess t \ 

tWO -teel rule can be obtained from .Messrs 

Chester.! ;lle. hut it i- 

pOSwil'N" to UM- tun o . ; ule- pi 

In either OM6 it i- hetter to mntint them 
wooden base, a por; uhn h > .'U. so 

that I Meel t ' d 

periment. To obtain and j l<.t the <ur\e ^h<. \\m:_ tin- 

\e,-II the p. Ill 

Positive Lens (< 
Cose 1. A ' K.I! object moving a] iee 

' point < 



4i PRACTICAL OPTICS 

of the incident light-waves is negative and varies from O 

to - - (where " / " is the focal length of the lens). 

In this case the image is always real, and can therefore 
be focussed on a ground-glass screen. 

Place the cross-wire object at the extreme left-hand 
end of the bench, and illuminate it by means of a lamp 
placed behind it. Place a 5D lens L (Fig. 32) in one of the 
optical bench lens holders, and adjust its position on the 
bench so that the distance 10 (I is the " image plane " 
and recorded by the ground-glass screen) is the maximum 
obtainable under the conditions. 

Adjust the screen I so that the image is sharply focussed. 
Then measure the distance L0=w (-) and TL=v ( + ). 




FIG. 32. 

Move L a short distance (say 5 cms.) nearer to and 
repeat the experiment. In this way obtain a series of pairs 
of values for " u " and " v." Plot these values on a piece 
of squared paper, remembering that when the incident 
waves are diverging, " u " is plotted negative ; when 
converging, positive. A graph should be obtained similar 
to the one shown in the top left hand quadrant of Fig. 38. 
On a second sheet of squared paper plot the curvatures 

and - for the same experimental data (see Fig. 40). 

Case II. (see Fig. 33). A " real " object moving from 
the first focus to the lens ; i.e. the curvature of the incident 

waves is negative and varies from - j to - oc . 

In this case the image is virtual and cannot therefore 
be focussed on a ground-glass screen. So that, for this 
part of the experiment the optical system is arranged 



MIRRORS AND LENSES 



41 



as shown in Fig. 33. First set up a simple telescope by 
employing the achromatic lens A and an eyepiece (these 
should be standard fittings for the optical bench). Focus 
tin- telescope for "parallel light" on son distant 

object, and situate it near the middle of the bench. Place 



Telescope 




sa, 

the "!) 1 n- L with its holder near the object (i.e. within 
ins.), and the beam now passing out from L will l< 
rgent. By inserting a further lens C (from the trial 
case) of knu\\n focal length, say a + 2D, in this divergent 
beam, the light will be rendered parallel, so that looking 
thnmgh the telescope a virtual image I of O will be seen : 
tin- image is situated at the principal focus of C. Then 
LO = M, and v = focal length of C - CL. 

< ham:'- the pnHtion of L and repeat. In tin- \\a\ obtain 
a series of pairs of value- tor " // " and " *-." Diving that 




t th- (WYe let\\ern the limits U= -/ and tt = O. 

\\ill be seen in the ] \\.-r N tt )i,.nd < ( uadrant 

;*> \U<> |>l<t the corresponding em-\ani e curve 

/// (e l-'i. 34). A \irtualobject moving fi 

lell- t> the IlL'llt. /.'. tin- riilAatlire .if (he lll-ldmt 

in positive and varies from + to O. 



PRA< TICAL OPTICS 



tlu' achromatic lens A to the left of the bench 
and adjust it so as to give an image O' of the cross-wires 
mar tlu- riirht-hand end of the bench. Insert the +5D 
Irns L in the path of the convergent beam and receive the 
image I on the ground glass screen. Then LO' u and 
LI=r. Obtain a series of pairs of values for "u" and 
as before, commencing with " n " as about 110 cms. 
and moving L step by step until " u "is about 5 cms. 
Plot these values as a continuation of the last curve (see 
Fig. 38, top right-hand quadrant), also the curvature 
graph as for Cases I. and II. (see Fig. 40). 

L C Telescope 




FIG. 35. 



Curves for Negative Lens (concave) 

Case I. (see Fig. 35). A " real " object moving up to 
the lens from the left, i.e. the curvature of the incident 
light is negative, and varies from O to - . 

Place the cross-wire object at the extreme left-hand 
end of the bench. Arrange the telescope with the 
achromatic lens and eyepiece (as for Case II. of the posi- 
tive lens) at the right-hand end. In a lens holder place 
the -5D lens L, and make its distance from the object 
(i.e. " u ") 100 cms. Between this lens and the 
telescope insert an auxiliary positive lens of known focal 
length (from the trial case), about a + 2D, and adjust it 
until the object is brought sharply into focus when looking 
through the telescope. Then OL= -u, and focal length 
of C-CL = v. Move the position of L and repeat. Make 
a series of pairs of values for " v " and " u " as before, 
and plot them on a fresh piece of squared paper (see Fig. 39, 
both on left-hand quadrant), also the curvature values, 

1 and - (see Fig. 41). 
v 



MIRRORS AND LENSES 



43 



Case 11. (see Fig. 36). A " virtual " object moving 
from the len> to the second focus of the lens, i.e. the 
curvature of the incident light i> positive and varies from 

+ cc to . Iii this case the image is real and can be 
focussed on a ground-glass screen. 

A L 

O 




Pto, 



Place the object at the left-hand end of the bench and 
arrange the achromatic lens A to form an image 0' of 
O at about the middle of the bench. Place the -5D 
lens L in the convergent beam about 3 cms. to the left 
of 0' and adjust the screen until the Image is again 
sharply in focus. Then 0'L= + ', and IL=+r. Move 
L a short distance, say 1 cm., and repeat the experi- 
ment. In this way obtain as before a scries of pairs 
of values for " u " and " v" Plot these as a continua- 



Tclescope 




h- 



tion of the curve lor the last case (see Fig. 39, top right- 
hand quadrant). Also plot the corn '-ponding and 

" V 

curve (see Fig. 41). 

Case III. (see Fig. 37). A "virtual" objen n , 

from the lir-t focus of the leu-, to the ri-jln. /.< the 
eurvatui-e ot the incident liL'ht i- p..viti\e ,-ind \aries from 

' toO. 



44 



IMJACTK'AL OPTICS 



Retain the same positions of the object 0, the 
achromatic lens A, and consequently the image 0'. 
The -5D lens L should then be placed a short distance 
to the right of A, so as to make LO' u as large as possible. 
The image I now being virtual, obtain its position by 
means of the telescope and auxiliary lens as before. 
Then L0'= +u, and LI= +v. 



20 

C/775.1SO 160 HO 120 100 80 60 40 20 



cms, 
160 



140 



120 



+11 



cms, 
FIG. 38. 

Move L further to the right by, say, 5 cms., and repeat 
the experiment. Obtain, as before, a series of pairs of 
values for " u " and " v " and plot them (see Fig 39, right- 
hand lower quadrant). Also curvature graphs - and 
(see Fig. 41).* 

* In all the above experiments the sharpness of the "images" may be 
improved by using a yellow "colour filter" in front of the object, in order to 
cut out the blue rays. 



M I RRORS AND LENSES 



45 













+u 




















1 








































/ 










00 6 


6 


4 


2 





L 


4 


6 


8 


3 1 


-w, 

^ ^ 








- 


^ 










-f-M, 
















/* 


^ 


i 














/ 




















/ 


















-V 






















+ 




/ 














/ 














/ 








) I 


_ 1 


5 


X 


i 


> 1 


3 1 


S 2 


-& 




/ 










3 




/ 














/ 
























-^ 












These six experiment- give tin full data for plot t MIL 
M -lm\Mi in I-'IL'-. ."' s . .'{'. 40 and -I I 



46 



PRACTICAL OPTICS 



(h) SIMPLE TELESCOPE 

Thr experiment Consists in setting up a simple astro- 
nomieul or inverting telescope and taking measurements 
in ennneetion with the "system/' and then repeating 
the measurements for a Galilean telescope. 

At^trnin, mil-ill. Use a metre optical bench for the ex- 
periment. At the left-hand end place a positive lens 
(from the trial case) of fairly long focal length, e.-. 
a + 2D, in one of the lens holders. Receive an in age 









1 ^ 
















10 








/ 














/ 




1 


3 1 


5 


> 


c 


z 


D 1 


5 2C 










/ 














; 














/ 


i r 












/ 




7.0 











;. 41. 



of some very distant object (such as a lamp), produced 
by the lens on the ground glass screen (in its holder). 
Place in a second holder, and on the other side of the ground 
glass screen, a short focal length positive lens, such as 
a+12D. Turn the optical bench completely round, and 
again focus the distant object on to the ground glass 
screen by adjusting the position of this lens holder. Then 
remove the ground -glass screen, and look at the distant 
object through the system of the two lenses. This is 
a simple form of inverting or astronomical telescope ; 
the +2D lens would be known as the object glass, while 
the -f 12D is the eyepiece (see Fig. 42). 



MIRRORS AND LENSES 



17 



that : 
the image is larger than the object as seen direr tly. 

i.e. it subtends a greater angle at the e 
the imaire i> inverted and reversed. 
that the edge of the field of view is indefinite and 

ill-defined optically. 

the distances, off the optical bench, from the 
glass to the image, and from the eyr l< -us to the 
and compare these values with the nominal focal 



Object 
Glass 



Eye 




Plane of 



- J 1 /7T^ 

- ^^k-4-l 

^y~ 



Focal Length 



of O.G. 



Focaf Length 
o f Eye Lens 






: i .. if. 



lengths of the two lenses as given by the focal power en- 
graved on the lens ring (focal length =p r in cms.^ . 

Repeat these measurements with two <>tli< T telescopes 
in a-!( up from different pairs of lenses, and tabulate 
resutte, a< follows : 





nial 


nee of 


Nominal 


Ob 


ol 




Focal 


\*n* 


tad 




mal 


from 
gfc 


Length <>f 


from 
Image. 


Length of 


Kv, 

LMM, 


food 
Ln|tka 















Observe that the distance apart of the lenses \\lu-n th- 
telescope is focussed for parallel light is equal to the MIH, 



48 PRACTICAL OPTICS 

of the focal lengths. In this condition the telescope is 
-aid to be in " normal " or " afocal " adjustment. 

Find the position with a ground glass screen of the 
image of the O.G. aperture projected by the eye lens. This 
image is variously known as the Ramsden circle, the eye- 
ring, or the exit-pupil. Note that for comfortable vision 
this image must fall on the pupil of the eye of the observer. 

Field of View. Note that only when the eye is placed 
in the plane of the " eye-ring " will the whole available 
field appear fairly well defined. 

Measurement of Field of View 

Direct Determination. Place two candles at the far 
end of the room and adjust their distance apart until the 
images of the flames as seen in the telescope are just 
simultaneously visible one in either edge of the field of 
view. Measure the distance from the O.G. of the telescope 
to the mid-point between the two candles L, and let 
the distance apart of the candles be D. Then the field 

of view of the telescope in degrees is : tan = -^- (approx., 

as long as the angle is small). 

(i) Magnifying Power. Use the telescope with the 
+ 2D lens as "object glass/' and the +12D lens as eye 
lens. Observe through it with one eye a distant vertical 
scale pinned to a wall (the divisions should be about 
10 in. apart), whilst with the other eye the scale is seen 
directly. Note how many divisions of the scale, seen by 
the unaided eye, are covered by a single division as seen 
through the telescope. The number of divisions thus 
seen in the space of one magnified division is equal to 
the magnifying power of the telescope. Compare this 
result with the calculated value of the magnifying power 
obtained by dividing the focal length of the O.G. by that 
of the eye lens. 

(ii) Determine the Magnifying Power from the diameters 
of the entrance and exit pupils. Illuminate the O.G. with 
diffused light, by placing a frosted lamp close to it. Place 



MIRRORS AND LENSES 49 

a millimetre scale on glass * in one of the optical bench 
tit tings, and receive an image of the O.G. aperture pro- 
jected by the eye lens on to it. Measure the M/< 
thi> image with the scale. Also measure the diameter of 
the <).<;. .with a pair of dividers). Then, the magnifying 
_ diameter of entrance pupil 
diameter of exit pupil 

l>raw a sketch to illustrate how the "magnified image 
is formed in the astronomical telescope. 

Galilean Telescope. Set up, on the metre optical len< h 
as before, a +2D lens in a holder at about the middle 
of the bench. Receive an image of a distant lamp pro- 

Object Eye 

Gtess L ens (concave) 




duced I v tl Hi a ground glass screen. Put a - !L'|> 

tens in a holder, and place it on the l>< n h between the O.G. 
and the ground glass screen. Inn the latter. Ob- 

serve the distant object through the telescope and adju-t 

thr e\r leli- Illltil thr ..lijrrt is -harplx 111 

focus. This i- now a -implr form of Galilean telescope 

(see I'll*, n, 
Otorrc tint 

the image is larger than the object as seen 
i.e. it subtends a greater angle at the eye. 
the image is erect and <T*ed, as m the oaae 

rhr -irnpl.- t. I. ., p.- 

\ i> IIH: ill- 

tincd opt 
Thwe gUuM cak may be obuincd from MCM Rhcinbrrg * Co., 23 Tho 

D 



50 



PRACTICAL OPTICS 



tlic same experiments with the Galilean telescope 
aa mentioned before with the astronomical telescope, and 
tabulate all the results. 

Draw a sketch to illustrate how the magnified image is 
formed in a Galilean telescope. 

(0 THE SIMPLE COMPOUND MICROSCOPE 

Place the cross-wire object at one end of the metre 
optical bench, and the ground glass screen (in its holder) 
about three-quarters of the way to the other end of the 
bench. Place a short focus lens, say a + 10D trial case 
lens, in one of the lens holders, and adjust its position, 
not far from the cross-wires, so that a " magnified " image 



Objective 



Eye Lens 




\ i ;. 44. 

of the latter is given on the screen. Now, take another 
fairly strong lens, say a + 12D, and mount it in a holder 
on the other side of the ground glass. Adjust the position 
of this lens until a very distant object is focussed sharply 
on the screen, but do not move the ground glass screen. 
The screen may new be moved and the " aerial " image 
of the cross-wires observed by looking through this second 
lens. 

This is now a simple form of con. pound microscope 
(see Fig. 44). 

Measuring the " First Magnification." This is the ratio 
of the sizes of the first " real " image produced by the 
first lens, and the object. Place one of the millimetre 
scales on glass (paragraph (h) of this chapter) against 
the cross-wire object, and a second centimetre scale at 



M I RRORS AND LENSES 5 1 

tin- p. ,-ition where the " aer al " image is forim-d ly the 
lir-t lens. See how many divisions of this latter scale 
i. division of the magnified image; then determine 
how many eover two magnified divisions, and so on ; thus 
obtain the " first magnification " of the microscope. 

M Compare the image of a definite 

number of divi>ions of the millimetre scale against the 

-wires, as seen through the microscope, with the 

e number of divisions on a second scale as seen directly 

with the other eye at a distance of about 10 in. (the 

ar point" of the eye). Of course, in making this 

:son the microscope must be so focussed that the 

image of the first scale seen through it is formed apparently 

at a distance of H in., and not at infinity, as was the case 

Again, see how many divisions of the scale seen directly 

i one division of the "magnified" scale, and thn- 
obtain the niatrnifyiiiL: p^-r. Draw a >keteh in illu>trate 
the formation of the magnified images. 

>ips. Try the effect on the image of euttini: d<\\n 
the .1 pert u re of the front lens : 
>t to half the diameter. 

nd to (jiiaiter the diameter. 
iill\ d-erile the effects produced. 

that tlie eye n.nst be placed at the .\r-ring" 

in order that tin- \\ hole a\ailal|e field >hall he taill\ well 

defined. 



CHAPTER III 
PHOTOMETRY 

FOR the theory of Photometry, text-books should be 
consulted ; a good book on this subject is " Illumina- 
tion and Photometry/' by Wickenden. 

Introduction. The basis of all photometric comparisons 
between light sources is the law that the intensity of light 
given out by a source varies inversely as the square of 
its distance. Suppose a luminous point is giving out 
light in all directions. It is at once obvious that a 
sphere, whose centre is the luminous point, will be equally 
illuminated over its entire interior surface. Let " r " be 
the radius of any particular sphere, then the area covered 



Suppose L to be the amount of light emitted by the source 
per second, then the illumination per unit area= 



Thus, the illumination at a given distance from a source 
of light is inversely proportional to the square of the 
distance. This law is known as the " Inverse Square 
Law/' 

A photometer is a means of measuring the relative 
luminosities of two light sources by the simple expedient 
of estimating (with the human eye) the quality of two 
illuminations thrown on a white screen by the two light 
sources, and by being able to measure accurately the 
distance between the screen and the lamps, when equality 
of illumination due to the two lamps is secured. A standard 
lamp, such as the Vernon-Harcourt Pentane Lamp or 
the Hefner Alteneck (see text-books), of known candle- 
power, may be employed as one of the sources of light, 

52 



PHOTOMETRY 53 

so that the other may be determined in candle-pov 
This is obtained from the distances "r" and 
measured from the lamps to the screen when the intensities 

matched," for : 

It L is the amount of light emitted by one source per 
sec., and L 1 is the amount emitted by the other, the 

illirninations per unit area are _. and 

tiv-ly, but these intensities are "matched" or equal, so 



s-* 



If L is the standard of kno\\n cam He-power, by simply 
measuring "/ and , the candle-power of the lamp 
L 1 under te-t can be obtained. Tin- i- the principle 
\\hi--li all photometers are based. 

There an many types of bench photometers, but they 
all involve the nee. a darkened room \\ith 

walls painted with a "dull Mack " \arni-h. in order 
to stop any reflections, which would otherwise interfere 
with the results obtain, d. A -mail n.<.m -hnuld be chosen 
for the purpose, and a good coat of "dull Mack" sj 
varnish given to the walls. 

(a) " RICHIE " PRISM PHOTOMETER 

This photometer consists of two right-angled prisms 
of about } in. face " balsamed " to the polish i 
a piece of ussing son- 

so that the two edges of i 
(see Fig. 45) touch one another. The 
piece of ground glans \\ith the j. 

m.\\ attached i> mounted in a vertical 

position m a -m.ill u...-d-n framework 
(see Fig. 46). -. nh 

Phis can then i.. 

mteil on ..IK- of the metal tittmifH BO ES U> - 

cal bench referred to in I 
M use, the light sources, i.e. the " standard 




54 



PRACTICAL OPTICS 



and the lamp being tested, are placed in the same straight 
line as the steel rule, preferably at each end, and on ob- 
serving through the circular aperture of the photometer 
the ground glass screen will be seen to be illuminated, 



r 



2-0' 



Right Angled 

Prisms GroundGlass 



j- 




\ 



v 



FIG. 46. 



1 cr 



half the aperture from one source of light and the other 
half from the second source. This is brought about by 
the manner in which the prisms are arranged (Fig. 45), 
so that total internal reflection takes 
place and illuminates the ground 
glass. 

Thus by moving the photometer 
backwards and forwards along the 
optical bench, a position will be 
found where the two halves of the 
- aperture are of equal intensity, and 
the distances between the photo- 
meter and the two lamps obtained. 
If necessary the " standard " and the 
lamp under test may be mounted 
on fittings *to slide on the optical 
bench, in order to attain more accurate results. 

A very good standard lamp for early experiments in 
photometry is the Hefner- Alteneck. This lamp is shown 
in Fig. 47 ; the height of the flame can be adjusted and 
measured ; and this standard may be trusted to within 



FIG. 47. 



PHOTOMETRY 55 

about '2 per cent.. pro vided that correction for pre. 
and humidity of the air have been made. 

/ , riment. Set up the photometer just described 
on the "two-metre" optical bench, and place at one end 
tin- standard t% Hefner " lamp, and at the other 

:1\ so, place an electric lamp (preferably "carbon" 
filament) of about 16 candle-p<>\u -r. Arrange the 
d connections for this lamp as shown in Fig. 48, so 

Lamp 

Voltmeter 



Mains 



Iwvw/w 

Variable 
ties/stance 

that th- \ariahle re>i>tniK'C * ill the circuit and 

also a voltmeter across the la:n|> tern inaU. In tin- 
the candle-power <t tin- lamp can be md in 

case determined by the photometer \\hiUt . 
sponding voltage from the voltmeter may le read off 
in each case. About ten different candle-powers of tin- 
lamp should be taken, and a graph plotted showing the 
relationship be'tween the voltage and candle-p, ,\\,T 

(b) RUMFORD PHOTOMETER 

. ..I thi- t \ pr nt phtm< ~h.\\n m 

Fig. 49. \ i-.| \ ; cd a short d 

intc Mum D I 1 - 1 the t\\o sources of lii:I 
be compared are placed at 1 S t so that sha- 
ll,.- rod tall 00 til'' -T.cn a! d -l\ . 

A -. - r\ -t tuple and convmimt type of variable mfeUaoe b made by wing 
a Uige (12 in. x 12 in.) photognphio dervloping dkh which ban it, 

um of wat '.' wire* fitun t ! main," 

and to the lamp, nhould bare small plate, aoldeml to them, and thru ,..,i 

the aohitkm. By varying the dintance apart of tbeM two plate*, 
immereed. a vrry fine adjuntrornt (,, r , , r ,. -.mny : .l.tl.-rmee in voltage k obtained. 
The amount of acid hypo put in b umall. l.ut ,- (.. un d on trial ; the tolution 
.houW be well -tirmL 



56 



PRACTICAL OPTICS 



and are coincident at "a." The distances of S x and S 2 
are adjusted, usually, by allowing one of them to move 




FIG. 49. 

along a divided scale, until the two shadows appear 
equally dark. Then, as before, the intensities of the two 
lamps will be proportional to the inverse square of their 
distances from the screen. 

Experiment. Set up a two-metre optical bench as re- 
ferred to in Fig. 31, in a darkened room, and on one of the 
sliding fittings for this bench mount 
an electric lamp (about 16 candle- 
power carbon filament). At the zero 
end of the bench place the white 
screen. This may be constructed in 
the following manner : cut cut a 
block of wood to the shape shown 
in Fig. 50, and on its front face 
attach a small strip of brass about 
| in. wide, and then cover the rest 
of the wood on this face with black 
velvet. Take a piece of " mag- 
nesium ribbon " about 6 in. long and ignite one end ; 
hold the block immediately above the flame and allow 
the brass strip to be well coated with the oxide thus 
produced. The velvet should be covered with a piece 
of card which has an aperture cut in it to allow the 
brass plate to project through. This gives a very good 
screen for photometric work. 

The standard (Hefner) lamp should then be set up on 




FIG. 50. 



PHOTOMETRY 57 

tin- table in some such position as shown at S 2 in Fig. 49, 
and the electric lamp " wired " as before (see Fig. 48). 
A -mall circular rod (such as a pencil) should be mounted 
: the screen and its distance adjusted until the edges 
he two shadows produced by the lamps are coinridmt. 
Having set the "standard lamp" at a known distance 
from the screen (measured with a steel tape), move the 
Irrtrir lamp along the optical bench until the two 
shadows on the screen appear equally dark. Take tin- 
distance giv<n 1>\ the optical bench lrt\\-i-n the screen 
and this lamp and obtain its candle-power from tin- 
formula given before; also note the voltage from tin- 
voltmeter. Repeat this a number of time-, in < ach case 
altering the voltage by separating the two (rim in tin 
hvpo -(.hit ion (see variable n --i-tam-e in last experin - 
and thus plot a graph showing tin n -lation>hip l>et\\ 
IN -power and voltage. 

PHOTOPED 

The construction of this photon etei i- illu-t rated in 
51. It consists of two tubes A and B. about H in 
diameter, sliding one in-ide tin- other. In-ide the ; 
B is a metal "stop" with a rectangular aper 




n. I in.) -ut in it. At tin- mil of tin- tulx' A i- 
attached a tran>lu. -nt ion 'u as a piece of oiled or 

greased pap< r Ih* tu<. lit to be compared 

are placed at 8| and S 2 and tin light proceeding through 
tin- .ijHirure in B illiiinin.iti-H the 8creen attached to \ 
\Mth two rectangular patchen of light, as nhown 1 



58 



PRACTICAL OPTICS 



and " ac " in the figure, the edges of which are made to 
coincide at " a " by the adjustment of the tube B either 
towards or away from the screen. 

B}' arranging the two patches of light to appear equally 
bright, the intensities of the lamps may be obtained as 
before. 

Experiment. Precisely the same experiment as per- 
formed with the other two types of photometer may be 
done with the " Photoped," by setting it up at the end 
of the two-meter optical bench and carrying out the same 
instructions. This type of photometer is used a great 
deal in actual practice by " gas referees/' 

LUMMER-BRODHUN PHOTOMETER 

The Lummer-Brodhun type is a rather better and 
more accurate photometer. The instrument is shown 
in Fig. 52. Two screens of magnesium oxide (as applied 



from 




Fm. 52. 



before), or zinc oxide C and C 2 , are illuminated by the 
two sources of light S x and S 2 . Light from each of these 
two screens is then brought into the field of view of the 
telescope T by means of two mirrors Mj and M 2 and a 



PHOTOMETRY 






Lummer-Brodhun cube A. Such a cube is shown in 
Fig. 53, and consists of two right-angled pri-m> which 
are put in "optical contact over a small circular ana 

in the centre of their hypotenuse faces. The reman 
>t tin- face of the prism 1 is ground away a< indi<-.t : 
this allows light from both mirrors M, and M 2 to enter 
the telescope. The appearance seen i- that <>t t\\<> con- 
centric circles of light, the centre patch of light 




Lummer - 
Cube 




Field as seen 
in Eyepiece. 



from the source S, and the outer ring of light tn.m S f ; 
\\itl. arrangenient dr in e.pialitx <>f tin- 

intn^ities is v<-r\ ea-ily detected. \Vln-n in us<- tin- 
light sources an- u-uall\ Urj.t M..I .n.l th.- jl 
meter moved until the mtcn tin- \\\ parts of 

th- Held are ecju.il ; \\li.-n in tin- |>n Mil. .11 thr IHUMI.: 
line between tin- tu. }>;\n^ \\i\\ not !. \i-ihl.-. Tlii^ 

adapted \' 
larger t 'I me in>tnnnents 

n. th. ..)- .,| dl\ idll 



60 PRACTICAL OPTICS 

LARGE PHOTOMETER BENCHES 

The application of the " steel-rule optical bench " for 
use as a photometer bench is quite suitable for early and 
introductory experiments in photometry, but for more 
advanced work and general use a larger type of mounting 
for photometers is desirable. 

For this purpose a double-lined track (similar to that 
shown in Fig. 54) is usually employed, which supports 
the carriages for the lamps, screens, etc. Such a track 



Divided Rail- 



Rails on which 
(fittings 




FIG. 54. 

should be straight, level, and firmly supported. The 
front circular rail of the track should have a scale of equal 
divisions on it to permit distances apart of the various 
fittings to be read. The length of the track should be 
from 10 to 15 ft. long. 

With such an apparatus more satisfactory photometric 
measurements may be made. 

Experiment I. Using this bench and the Lummer- 
Brodhun photometer, the candle-power of an electric 
lamp should be obtained. For this purpose it is well 
to mount the lamp on a suitable fitting (as shown in 
Fig. 55), in order that the lamp may be rotated and the 
candle-power measured for various positions of the lamp, 
from which a " light-flux " diagram can be plotted. It 
is of the greatest importance that when using electric 
sources of light in photometric work the state of current 
passing should be known ; to this end, therefore, either 



PHOTOMETRY <>l 

an " ammeter " should be put in series in the circuit, or 
a voltmeter across the lamp terminals. 

Tin- candle-power of a lamp determined in this way 




Fio. 65. 

would be considered as that measured from the centiv <>t 
rotation of the lamp serving as a reference point. 

periment II. As a practical application of photomot r\ . 
the following experiment may be performed. It consists 
in measuring the loss of light in a telescopic in-tnimrnt 

i'/) For this purpose a collimator should be used a- an 
accessory to the photometer bench in order to produce 




a parallel beam of light for passing through the telescopic 

*yM-Mi under tr-t. Tin- m-m-ral arraini'-iiirnt <>t tin- 
apparatus for the experiment is shown in Fig. 50. 

I irst, receive the parallel beam from the collim 

i to the photometer I* <>n one nde (i.e. without the 
telescope in position), and liulit foom an aii\iliar\ l.i!ii|> A 

on ' M tll<- |Mv|t| ( ,n .,( tlir |llOtO- 



PRACTICAL OPTICS 

im-UT until a " balance " is obtained. Take the distance 
'/! from the photometer to the auxiliary lamp. Focus the 
telescopic instrument supplied for "infinity/ 5 and support 
it in the position T on the bench so that it receives 
the parallel beam into the eyepiece of the instrument. 
\M\\. the ratio of the intensity of the emergent beam to 

that of the incident beam should be ^ 2 (neglecting in- 

ternal losses), where M is the magnification of the instru- 
ment ; the student should prove this for himself. The mag- 
nification should be found as explained in Chapter VIII. 
With the instrument now in position the position of 
the photometer should again be adjusted until a second 
" balance " is obtained ; let the second distance between 
photometer and auxiliary lamp be d 2 - 
Then, 

Intensity of final beam (d-^)' 2 
Intensity of original beam~~(c? 2 ) 2 ' 

By the theory above, if instrumental losses were non- 
existent, we should find 

&)*=! 

(dtf M" 
but in practice we shall have 



where K is the transmission coefficient of the instrument. 

The above description gives the outline of the experi- 
ment ; the student should suggest and carry out all 
necessary experimental precautions, such as the deter- 
mination of the current in the lamps, repetition of 
readings, etc. 

NUTTING PHOTOMETER 

This instrument is made as an attachment to ordinary 
spectroscopes for spectro-photo metric work. It is used 
for the comparison of light sources as to their intensity 



PHOTOMKTHY 



of radiation for the various wave-lengths of the spectrum : 
it may also be used for absorption work. The optieal 
m of the instrument is shown in Fig. 57. Light 
from the two sources are admitted through the apertim-- 
A, and A,. (For absorption work it is better to use a 
" -pi it " beam from one source.) Light through A , 
passes through a stationary Nicol (or Glan Thompson) 
pri-m Xj ; that through the aperture A, is brought in 

L L 




Hypothenusc 

Face of 

Prism P 2 



Slit 1 of 
Spectroscope 



tin- direction indicated by means of the pri-m 1*,. The 
inner >urface of the prism P 2 (\Uii<li is balsamcd to PI) 

Ivered \\ith t\v<> lini-i/niiial -trip-, so that light t'rmn 
ill l. it fleeted along the path \ _.!.,. and tlu- light 
\, \\ill pa-- -traJL r lit thrmiL'li tin- uu-il\ rjvd -tup. 
li-u- L, irndciv j|, ( . | !L rj,t pai-allrl I .. : ! 

ill.- rotating Nir..i N, 'I'hr tani L usses an 

_ .t tin- in-partite " !i-ld -n tin- slit 

rotation <t th. \i,-,,| \ \\\ l>e measured 

l>\ .1 di\id<-d curie C and pointer. I in 



<u PHOTOMETRY 

the eyepiece of the spectroscope should be that of three 
-'imply defined spectrum " bands/' the centre one of 
which is varied in intensity by rotation of the Nicol N 2 . 
The source or specimen (if for absorption) to be tested is 
placed in front of the aperture A x , and the intensity of 
the centre band varied, until a " match " with the outer 
bands is obtained. Since only one of the incident beams 
is polarized, the intensity of the light varies as the square 
of the cosine of the angular turn of the Nicol N 2 . 

Experiment. With the Nutting photometer and spec- 
troscope, the intensities of illumination for various parts 
of the spectrum, of, say, a piece of cobalt glass or 
some solution, may be determined, and a graph plotted 
showing the relationship between intensity and wave- 
length. 

LUMMER-BRODHUN SECTOR 

This piece of apparatus is frequently used in connection 
with photometric measurements. It serves as a means 
of varying the intensity of a beam of light by a known 
amount, by inserting a revolving sector (whose apertures 
are adjustable) in the path of the said beam. Fig. 58 

Sector 

, ,^_ \ Motor 

Arm 




FIG. 58. 

illustrates the apparatus, and, as will be seen, an electric 
motor is employed for driving purposes, whilst an ad- 
justable arm will be noticed for varying the aperture of 
the sector whilst in motion. 

When in use the speed of the sector should be arranged, 
so that on looking into the instrument with which the 
sector is being used no flicker of the field is noticeable. 



PHOTOMETRY 65 

Under this condition, the intni>ity f the light trans- 
mitted by the sector may be taken as being proportional 
to the angular aperture. 

The instrument may be used on the photometer ' 
in the path of one of the beams of light, and serves as a 
means of varying it^ intensity without the necessity of 
rig the source of light n -l.it i\- t< tin- photom< 



CHAPTER IV 
SPECTROMETER MEASUREMENTS 

(a) THE SPECTROMETER 

THE spectrometer is an instrument of fundamental 
importance for the measurement of refractive index 
(see Chapter I., section (c)). The essential parts of the 
instrument comprise a " collimator " SL X (Fig. 59), a 
rotating prism table T, and a telescope L 2 E on a movable 
arm. The collimator consists of a metal tube, at one 
end of which is an achromatic lens L x and at the other 

V* 




FIG. 59. 

end a vertical " slit " S in the focal plane of the lens. 
Light diverging from this slit is rendered parallel (or 
collimated) by L x and " parallel light " falls on the prism. 
The light having passed through the prism, the spectrum 
thus produced is brought to a focus by means of the lens 
L 2 of the telescope in its focal plane, and this image is 
viewed by an eyepiece E. The telescope rotates so that 
it is always directed towards the axis of rotation of the 
prism table, and is provided with a vernier V l5 which 
moves over a divided circle concentric with the prism 

66 



S1>K( TROMKTKR .MKASfKKMKNTS 



67 



table. To the latter there is also a vernier V, attached, 
which moves over the inner edge of the dividing of the 
circle. It is necessary that the instrument should be 
thoroughly rigid, and precision must be exercised in the 
lilting of the bearings, verniers, and circle. It will be 
found less expensive if such an instrunn -nt is bought out- 
right rather than to try and construct such an instrument 
in one's workshop. A selection of numerous "makers" 
"ill be found in the "Dictionary of British Scientific 
Instruments." 



Dirtied Circie attached 
Telescope^ to Prism table 
Lye-piece 

ff 



Collimator 



Slit, 




Slow Motion 
Adjustment 



Main Divided 
Circle 



Two opposite Verniers 
Fio. 60. 

Kg. 60 shows a convenient type of spectrometer for 
laboratory use i ts). 

Adjustments. The following adjustments are necessary 
before commencing an experim< -m with the spectromct* 

1. To adjust the eyepiece. The eyepiece lens syst 
movable in th< ml., \\ln.-h carries the cross-webs. I 
a piece of white paper in front of the telescope objective 
so 88 to reflect light into the t< 1,,<,>|>, . then move the 
eyepiece in or out until the cross-lines are sharply 
defined 

I'o adjust the telescope. Din < t the telescope 
towards some distant object, sin- 1, .0 ., < i 
and move the tube carrying the eyepiece and cross-wires 
(usually by a rack-motion until tin- m,. :_ 
tant object is seen sharply defined at the same time aa 



68 PRACTICAL OPTICS 

the cross-lines. To be sure of this see that there is no 
parallax between the two. The telescope is now in 
normal adjustment. 

3. To adjust the collimator. Illuminate the "slit" 
of the collimator. Swing the telescope into such a posi- 
tion so that it and the collimator tube are in the same 
straight line ; and then, while looking through the tele- 
scope, move the slit in or out until there is no parallax 
between its image and the cross-lines. Set the slit 
vertical. 

4. There are two alternative methods of focussing the 
collimator for parallel rays which should be taken note 
of. First, Schuster's method : the prism is set in the 
position of minimum deviation, and the telescope turned 
so that the image of the D line or some other convenient 
line is seen. The telescope is then turned a little to one 
side of the image ; it is evident that there are now two 
positions of the image, one on each side of that of 
minimum deviation, which will bring the image of the 
line again into the centre of the field of view of the tele- 
scope. The prism is turned to these two positions in 
succession, and the line observed in each case ; if the 
line appears in perfectly good focus at each time, then 
the telescope and collimator are both adjusted for 
parallel rays. If, as is more probable, the focus of the 
line appears better at one position than at the other, the 
following procedure should be adopted. The prism is 
first turned to one position, and then the collimator is 
focussed until the line is seen perfectly sharp ; after 
turning the prism to the other position the telescope is 
focussed until the line is again sharp. After one or two 
repetitions it will be found that the condition will be 
obtained so that the line remains in perfect focus which- 
ever way the prism is turned. When this is obtained, 
telescope and collimator will be in adjustment for parallel 
light. 

The second method is by Lippmann, who employs two 
strips of " plane parallel " glass, which are set one above 



SPECTROMETER MEASUREMENTS 



50 



th* other and at right angles to one another (see Fig. 61). 

- apparatus is set in the path of the beam from the 

collimator ; if these rays be truly parallel no effect will be 



GttNtattar 




Telescope 



81, 

produced, but if they are convergent or divergent, the 
upper and lower halves of the image of the slit will appear 
relatively displaced. 

(6) MEASUREMENT OF PRISM ANGLES 

First Method. Let ABC (Fig. 62) be a prism, of which 
the an-le A is required to be measured. The prism is 
placed on its table and levelled so that the faces AC and 
AB are vertical. Adjust 
its position and that of the 
telescope so that an image n 
of the slit is formed on the 
en> -uires by means of 
light reflected from the face 
AB. Without MIM\ HILT tin- 
telescope, rotate the prism 
table until the face AC 
acquires such a position 
that light reflected from 
this face forms an image 
of the slit on the cross- wires. In order tr this to be so, 
it is obvious that \< ,,,,; , t take up a position parallel to 
that previously occupied by AB, which is equivalent to 
rotating the |n-m through the angle CAD. Therefore the 
angle of the the supplement of this angle, which 

equals 180 - / CAD. The angle ( \ I > obtained by the 
readings taken from the divide* I he two positions 

lie prism table 







70 PRACTICAL OPTICS 

Second Method. Arrange the prism with angle to be 
measured towards the axis of the collimator so that the 
parallel beam from this falls partly on the face AB (see 

Fig. 63) and partly on AC. 
Move the telescope round 
until an image of the slit is 
seen by light reflected from 
one of these surfaces. Set 
this image on the intersec- 
tion of the cross-wires and 
take the reading of the 

~ "~#\. telescope position from the 

divided circle. Leaving the 
prism table stationary, move 
the telescope round until the 
image of the slit is again 

seen, but this time after reflection from the second surface, 
and take a second reading from the circle. The difference 
between these two readings is equal to twice the angle BAC. 
Proof : 

Produce EA to D, GF to M, and PO to M. 
Then, because HF and ED are parallel 




Andzl GFC = ,/AFM (vert, opposite angles). 
But z.HFA= Z.GFC (by reflection). 



Z.GMD= ^FAM+ Z.AFM (two interior and opposite 
angles). 



Similarly, 

So that L GMP (the angle moved through by the telescope) 
= 2 BAC. 

Experiment. Measure the angles of the prism supplied 
to you by the two methods described. Test the accuracy 
of your results by seeing if the sum of the three angles 
added together equal 180. 



SPE< TROMETEB MEASUREMENTS :i 

(c) MEASUREMENT OF REFRACTIVE INDEX AND DISPERSION 

\Vitli any pri-m thnv is an important relation between 

th<- ; \<- Imlrx i/o. the Vertical Angle of the prism 

and the Angle of Minimum Deviation (D) (see 

Chapter I.). The equation connecting these three quan- 

- is written as : 

. (A + D) 

>' > 



-in 



Before going any further it is well to look at the proof of 
this formula. 

Let the angl. MAC (Fig. 64) be the measured vi 




angle of the prism. Call this angle ** A," ami -uppi^r 
tli- pri-iii t. IM- in the position >t M minimum <lr\iati..n 

(see Chapt I i.e. L EGN t = L DF^ = i, and L\\ ( I- 

HFG=r. 

Now. in tin- tiL'in. \nn; the angles \l"M tad AGH 
are right angles, so that the angles FAG and V \ \ < n 
together equal tun right angles, from \vhi< h we see t)> 

^lFAG= Z.HFG+ </ HGF = 2^HFG = L 

A 

i.e. Z.BAC = 2rorr 

r DKN, Mil. 
But IIKLsGFL+ (!'!!. uhi.h. () ual 



A 
4-ror 



72 



PRACTICAL OPTICS 



(A + D) 



sin i 



siir 




So that n = 



Making use of this formula the refractive index of the 
prism may now be determined. First of all, however, it 
must be remembered that a prism produces a spectrum 
and that the various colours or wave-lengths are deviated 
by different amounts on passing through the glass, the 
red being the least refrangible and the violet the most 
refrangible. Therefore " n " will vary, depending on 
the wave-length of the light ; thus it is that certain de- 
finite wave-lengths in the spectrum have to be decided 
on in order that some standard of comparison may be 
formed for identification of all glasses, also liquids. These 
wave-lengths are : 



Wave-length. 


Produced by 


Notation. 


00005893 cms. 
6563 
4861 
4102 


Sodium Flame 
Hydrogen Tube 

5 J 5 ) 
J ) J > 


Dline 

c 

F 

GI >j 



Experiment. Place a sodium flame * in front of the 
slit of the spectrometer open the slit fairly wide. Put 
the prism on its table and observe the image of the slit 
with the telescope after the light has been refracted through 
the prism. If the telescope and collimator have been 
carefully set for parallel light previously, the slit-image 
should be well .defined ; close the slit down gradually 
until a very narrow image is obtained, and if necessary focus 
it sharply by means of the rack motion of the telescope. 

To set the prism at minimum deviation. Rotate the 

* A very suitable " sodium flame " may be made by employing a " Meker 
Burner," as supplied by Messrs Baird & Tatlock, Hatton Garden, and by placing 
common salt on the " grid " at the top of the burner. 



SPECTROMETER MEASUREMENTS 73 

prism table and observe the movement of the image; 
if it goes outside the field of view of the telescope, 
move the latter round the circle so as to keep it in vi- 
but on continuing to move the prism table in the same 
direction thr image will reach a limiting position and 
then commence moving in the opposite diiv-ti.n. When 
the image reaches this position set the intersection of 
-s-wires on it : this is the position of minimum 
iation for the sodium line, i It two sodium lines are 
seen, set the cross-wires midway between the two, for 
th< -re are actually two lines with six Angstrom units 
between them.) The reading from the circle should be 

Hydrogen Slit of 

Tube n n .Spectroscope 



Secondary 
Terminals 




taken, thru n-nmvc the pri-m and take the reading when 

telescope is set t..i th< -In image as viewed dire< ti\ 
i.e. in thr same straight line as th <<>ilimator. The 
difference in these two readings will give the 
D. The angle of th< p n , m A has already been deter- 
mined, so that the refractive index for sodium light (de- 
noted n D ) can h ted. 

placing a hydrogen tube* in front of the 
imiices of the prism for the C (red), 
F (green), a> tained. 

Dispersion noted by the difference in refractive 

\ N voooraient type of" Hydrogen Tube "for thfcwoifcfe the" Ct 
form. Thk u .hown in Fig. 06. with the electric*! ooOMcUoM wbfn 
ftccumuUtore and " coil " MB employed 



74 



PRACTICAL OPTICS 




index between the two wave-lengths in question, and is 
written usually : 

C to D = -00481 (for instance) 

D to F^ 00970 

F to G! = -01741 
Dispersive Power of the prism is given by the formula 

T>= Ua> ? _ ?c > where n Gl and n c are the values for the re- 
n D - 1 

fractive index for the G x and C lines respectively. 

Refractive Index of Liquids. The 
refractive index and* dispersive powei 
of liquids may be found by the 
above methods by using a hollow 
glass prism with the sides of "plane 
parallel " and optically flat glass, 
and filling the prism with the liquid 
under test. Such a prism is shown in 
Fig. 66. Plaster of Paris makes a good 
FIG. CG. cement to secure the sides and base. 

(d) REFRACTIVE INDEX BY IMMERSION. (See Trans. Opt. 
Soc., vol. xvii., No. 3, Dec. 1916. Mr L. C. Martin on 
" Refractometry and Identification of Glass Specimens.") 

A very useful means of obtaining the refractive index 
of specimens of glass in a rough or unpolished state, or 
of lenses, is by immersing the specimen to be tested in 
a liquid of the same refractive index contained in one of 
the hollow prisms shown in Fig. 66. The whole can then 
be mounted on the spectroscope and the usual necessary 
measurements taken. 

For this purpose, however, it is necessary to have a 
liquid of variable refractive index. Carbon disulphide 
and alcohol mixed together provide a readily adjustable 
solution ; in practice it is found best to start with pure 
carbon disulphide in the prism, immerse the specimen, 
and then dilute the solution with the alcohol. The 
strength of the liquid should be adjusted so that its index 
is very slightly higher than the value required to focus 



SPECTRUM KT I :ii MEASUREMENTS 

ply (on looking through the telescope) any particular 
line <f the spectrum K.V. the sodium linos), and the 
evaporation of the carbon disulphide, which usually occurs 
faster than the alcohol, will presently bring the line into 
focus very slowly and distinctly. At the moment of 
sharpest focus, the anirlr of M miniiiunu deviation " is 

n in the usual waj% and the refractive index \\, 
out iii the usual way from the formula. 

One of the most important factors of the \\hole ex- 
periment is that the liquid in the prism should be kept 

niogeneous. To this end it is necessary to have the 
liquid mechanically stirred ; a small " propeller blade " 
driven at a suitable speed in the liquid by a small elec- 
tric motor will secure this condition. The motor must not 
be mounted on the same table as the spectroscope, as the 
\il>ration \\ill interfere with the readings. The method is 
suitable for any rough small pieces of glass, except forms 
approximating to plane parallel plates. 

'periment. Find the refractive index of the specimen 
supplied to you by the above method for " D " light 
(X 5893), and then for the C, F and Gj lines (hydrogen) ; 
also determine the dispersion and dispersive 



() DETERMINATION OF THE WAVE-LENGTH OF LIGHT BY 
MEANS OF A DIFFRACTION GRATING 

This experiment again involves measurements \\ith 
the spectroscope, but instead <>f u-ing it for the d< 

i is to be used fW ti- 
the wave-length of certain lines in the spectrum 

purpose a "Grating' d tin- eon-i-u of 

a piece of speculum metal uhich has its surface ruled 
a great numtx tallel lines very close togethei 

rulings are about 14,000 lines table 

transmission 'grating is made 1> 
gelatine from a 

The full theory of the grating must be revin text- 

books (Edser's \.\x\\\ for Students" or Baly'n " Spe 



76 PRACTICAL OPTICS 

scopy "), and cannot be dealt with in this book, but it 
will be sufficient to say here that the spectrum produced 
by a grating is due to the " interference " of waves 
passing through the spaces in the grating. Let (Fig. 67) 
AB and CD be two adjacent apertures in the grating, 




FIG. 67. 

and that parallel light is incident in the direction in- 
dicated by the arrow, e.g. from a collimator SL, S being 
a slit parallel to the apertures of the grating. Now the 
supposed " ether particles " lying in the apertures AB 
and CD become sources of vibration which proceed 
chiefly in the direction towards N l5 N 2 , N 3 and N 4 , but 
also, however, in other directions, as towards O l5 O 2 , O 3 
and O 4 . If the former rays are brought to a focus by 
means of a lens (i.e. the telescope objective), they will 
produce a bright image of the slit without any mutual 
interference taking place, whereas the case with the 
diffracted rays O lt etc., is rather different. In order to 
investigate the " interference " among the latter, the 
straight line BE is drawn perpendicular to D0 4 , when 
the line DE will represent the difference in path travelled 
by the two outside rays D0 4 and BO 2 and also between 
the two outside rays C0 3 and A0 l5 and, therefore, also 
the difference in path travelled by every pair of corre- 
sponding rays in the two " pencils/' If now DE is 
equal to any odd number of half wave-lengths, it follows 
that for every ray in one pencil there is a corresponding 
ray in the other " pencil " at opposite " phase/' and, 
therefore, total interference takes place when the rays 
are combined at the focus of the lens. The same holds 
good for every adjacent pair of apertures of the grating. 



SPK< TimMKTKi; MEASUREMENTS :: 

But if DE be e<|iial to any even number of half \\ 
length- tli- corresponding ray in the two pei 

will be at equal "phase," and. therefore, the rays from 
these two apertures and every adjacent pair will combine 
the focus of tin- lens to give a bright image of tin ^\\\. 
Thus it will be seen that from a "grating" a spectrum 
will be formed on * 'cither -id. of the direct image of 
the slit, and the deviation of lines in th -JM< truin from 
the direct image is dependent on tin wave-length, i.e. 
the length DE decides the angle Dl'.K which equals 
Z.N 4 DO 4 . This gives a means of determining experi- 
mentally the wave-length particular line in the 
spectrum, for the deviation N 4 D0 4 can be measured with 
spectroscope, and the di-tanee DB can be obtained 
from knowledge of the number of lines per inch of the 
rulings: 

/. '; -m /_DBE. 

! hi: .\a\r-lenirth the spectrum seen in t In- 

direction O 4 i- km.\vn as the " first order spectrum." If DE 
equals "two" wave-lengths a "second order spcetrum 
\sill be seen, and so on. 

Experiment. Perform all necessary adjustments to th 

spectroscope, and then, illuminatim: the slit, take the 

reading of the telescope when the image of the slit is 

on the cross-wires as seen "directly/' iV. m the same 

straight line. Set up the M grating " in a vertical POM 

over th- of the prism table It ;- important . tir-t 

of all that the grating is set n to the .,\i* of 

I > do tin-, move the telescope round 

le until it i- exactly 90 from its previous reading, 

rotate the prism table with the grating on it until light 

ii the collimator is reflected off the plane glass surface 

the t lescope and an image of the slit 

tiade to cor ith the mt-r--cti.,n of the cross- 

ui!l then be at 46 to ctascope 

or eoli Take the vernier readings of the |H 

e and then rotate it to a position 45 ft ious 



78 PRACTICAL OPTICS 

reading. The grating will then be at right angles to 
tlu- axis of the collimator ; the plane glass side of the 
grating should be towards the O.G. of the collimator. 

For this experiment a very good source of light to use is 
a mercury vapour lamp, as it has a few prominent and well- 
spaced lines in its spectrum. These lamps can be obtained 
from the Cooper, Hewitt Co., and are very suitable for the 
laboratory. However, if this is not available, the sodium 
flame and hydrogen tube may be used as before. 

Direct the collimator towards the source, and on 
moving the telescope to about 18 from the direct reading, 
the spectrum (first order) will be seen in the field of view. 
Set the cross-wires on some definite line (if the mercury 
spectrum is used two yellow, one bright green, and one 
violet line will be seen), and take the reading of the 
telescope verniers, take also a reading when the telescope 
is on the other side of the " direct " position ; these two 
values should be the same, of course. Calculate the wave- 
length of the line from the formula : 

\=d sin 6, 
where X=the wave-length, 

d = the mean distance apart of the rulings, 
and 0=the angle between the direct and diffracted image 
of the particular line in question. 

Repeat the experiment for the other lines in the 
spectrum, then move the telescope still further round, 
when the spectrum will be seen to repeat itself, this being 
the " second order." Take readings for the same lines 
in this spectrum and again determine their wave-lengths ; 
in this case, from what has been said before, the formula 
will be : 

X = -d sin 6, 

and X = d sin 6 for the third order, 
o 

Tabulate all your results. 



SPK< TR< ).M KTKK M KASl'H KM KNTS 



79 



of Prominent Line* in the Sodium, Hydrogen, and 
Mercury Spectra 



Line. 



\Vu\'-l, 'IlL'tll ill (Ml-. 



D! Sodium 

D 2 

C Hydrogen 

F 

G! 

Mercury lino 



6563 red 
4861 blue 
4102 violet 



5461 green- 
4359 violet 



(/) CALIBRATION OF THE SPECTRUM 

The use of the spectrum for the purpose of analysis is 
now well known, gases and metallic substances each having 







a characteristic spectrum when seen through the spectro- 
scope. Thus it is that the spectrum may be " mapped out " 



80 PRACTICAL OPTICS 

by simply measuring the various deviations (with one of 
the previously described spectroscopes) for certain lines of 
the spectrum of known wave-length, and plotting a curve 
one against the other. By the aid of this curve we can 
find the wave-length of any unknown line. 

Experiment. Determine the values and draw out such 
a curve. 

There is a certain type of spectroscope, however, which 
gives the wave-length of any spectrum line direct, without 
the necessity of having to make a calibration curve. It 
is known as the Constant Deviation Spectrometer, and 
employs a prism of the type shown in the last section of 
Chapter I. A plan of the instrument is shown in Fig. 68. 
SL X is a collimator and EL 2 a telescope set accurately 
at 90 to one another. P is the " constant deviation " 
prism through which the light from the collimator travels 
as indicated in the figure, and becomes dispersed. This 
prism rests on a circular table T which is rotated by 
means of a micrometer screw M ; to this screw is 
attached the drumhead D, which is engraved in wave- 
lengths. To use the instrument, all that is necessary is 
to set the drum to read a known wave-length, such as 
X5890, then move the prism on its table by hand until 
that particular line comes coincident with the inter- 
section of the cross- wires in the telescope. Clamping 
the prism in this position, the instrument is now adjusted. 
By bringing any other line of the spectrum on to the 
cross- wires, its wave-length may be read off direct from 
the drum, the calibration of which has been carried out 
once for all by the makers. 



CHAPTER V 

DETERMINATION OF RADII OF CURVATURE OF 
SURFACES 

(a) *TpH K most usual instrument that is employed for de- 
JL tennininir the radius of curvature of lens surfaces 
is a "spherometer." There are numbers of types of this 
instrument for example: (i) the "three-legged," (ii) the 
"ring" type, (iii) the " Aldis type, 
(iv) A Me type, etc., but all spherometers 
are dependent on a certain formula. 

This formula is deduced in the follow- 
ing manner (see Fig. 69) : 
Let ADB be part of the rinumfer- 

of a spherical surface (e.g. a lens) 09. 

in section, a require the radm- 

< \ or OD of the surface. Draw AB perpend i -ular to 
OD. Then the A OAC is a right-angled triangle, and 

OA^OCP + AC* ; alsoOC = OD-CD. 
Call OA = R, CA = r, and CD-A 
nR*=R 




.mil R = 

Z/I 

the spher< is a means of obtaining the dis- 

i* CD=A and CA=r, from \\hirh K (the reqn 

oaloolatecL 
Fig. 70 shows a tin 

be j" Mnall tripod, in the centre of \\hxh 

is mounted a very finely pitched mi. r. -meter sen 
a divided disc attached to it. T 1 disc reads 

r mediate values of di \i-n.n-. <>f the vertical scale 
v ii 



82 



PRACTICAL OPTICS 



shown in the figure on one of the legs. When using the 
instrument, it must first be placed on a flat surface (such 
as an optical " flat "), and the micrometer screw moved 



Vertical 
Scale 




Circular Divided 
Plate 



Micrometer 
Screw 



T V 



FIG. 70. 

up or down until all four " feet " are exactly in contact 
with the surface at the same moment ; the reading on the 
scale and divided disc should then be taken ; this should 
be the zero of the instrument. After this, place the in- 
strument on the surface whose radius is required, and 
again move the micrometer screw up or down according 
as the surface is convex or concave and take a second 
reading. (All readings should be a mean value of a 
number of settings.) The difference between the readings 
taken on the flat and on the curved surface will give the 
value " h " in the formula. The value " r" which is 
the radius of the circle on which the three legs lie, is very 
often engraved on the instrument ; however, for accurate 
measurements this should always be checked by measure- 
ment with a travelling microscope. This may be done 
in two ways * : either by measuring the distance between 

* Proof. In Fig. 71, A, B, and C are the three ''feet" of the spherometer 
in plan, and the distance AB, BC, and CA are measured. Call their mean 
value "p." 



Then 



: ~p . cos 30 C 
3 

: 2 P ^!- 



This method for obtaining "r" is more espe3ially useful when the "feet " of 
the spherometer are worn flat. 



ItADII OF CURVATURE OF SURFACES 83 




the centre leg and each outside leg in turn, and taking 
the mean value, or by obtaining the 
mean distance l>et \\een each outside 
leg and dividing ly >/3. 

Experiment. Determine the radius 
of eurvature of the convex and con- 
cave surfaces supplied to yu \\ith 
the three-legged spherometer. Check 
the value for " r " by means of a B~~ T 

measuring microscope. 71. 

"Ring" Type Spherometer. This 

of spherometer is very similar to the three-legged, 
but it involves the use of a metal " ring " in place 
of the three legs. The micrometer 
screw and drumhead are used in a 
similar manner, but to determine the 
value "r" in the previously men- 
tioned formula the maximum in- 
ternal diameter of the ring must be 
measured for convex surfaces and 
tin- maximum external diameter for 
ive surfaces. The instrument is 
ho\\n in Fi^r. 72, 

Abbe Type. A rather better and 
more accurate type of spheromet* r is 
the Ahhe t\jn-. The instrument is 
shoun in KiL r . 7.T It U8C8 a veil- 
tit ting steel plunger sliding up and 
down in a vertical direction The 
surface to be tested is placed on 
an accurately mrm-d ting situated 
at the top of the instrument. \\hiUt the spherical nose 
of tin plunger is kept in contact with the surface by 
means of a con ^ht suspended over small pulleys. 

Attached to tli- plunder is an enuraxed seale divided in 

tenths of a millimetre (-1 mms.) t which is observed by a 
microscope with micrometer eyepiece, and readings may 
be taken to th of a millimetre. As in the last case, 




84 



PRACTICAL OPTICS 



the internal and external diameters of the particular ring 
in use must be measured. A series of rings of various 



on which 
Lens rests "- 



Steel 










M 


j 


_L 




1 1 






1 | 











ft- h 

i/r 
i/i 

( i 

i 
i 
i 

i_i_ij 





N 

y 


P 


DIL^ 

Reading Microscop 

Balar 
Wei& 


e 

ice 
ht 






f 




\ 




( 




V 

FIG 


\ 
. 73. 



diameters is supplied with the instrument for use with 
corresponding sizes of lenses. 

Aldis Type. A still better and probably the most 
accurate instrument of its kind is the " Aldis " Sphero- 
meter (an illustration of it is given in Fig. 74). The 
surface to be tested is allowed to rest on three small 
spheres, and the micrometer screw is screwed up to touch 
the surface. Opposing the screw is a weighted plunger 
which rests on the other side of the lens ; by this means 
the instrument is rendered extremely sensitive, for contact 
between the point of the micrometer screw and the surface 
is at once detected by touching the edge of the lens with 
the finger-tips and judging the ease of -rotation. If the 
lens revolves freely the micrometer screw is too high, 
and if the lens will not revolve the screw is not touching 
the surface ; a position will be found when the lens will 
just and only just revolve, this will be when the point 
of the screw is in correct contact. The drum attached 
to the micrometer screw is 2 in. in diameter, and readings 



RADII OF CURVATURE OF SURFACES 



Bfi 



may be taken to -00001 of an inch. In using the sphero- 
meter formula with this instrument the value " R " is 
the radius of curvature of the surface + the mean radius 
of the spheres ; therefore on arriving at the calculated 





Pic 

value of " R," to obtain tin- true radius of curvature of 
tin- surface the radius of the spheres must be subtracted. 
a sketch is drawn this will become evident : it is 
equivalent to working on a sphere of radius R + x, where 
x is the radius <>f the spheres.) 

URVATURE OF " SMALL DIAMETER " SURFACES 
It is obvious that the use of the spherometer is limited 

the diameters of lenses ul.. n N uses are from 1 m. 

liamcter downwards, and more especially microscope 
objective lenses, some other method than the sphero- 
meter has to be employed. The following method gives 
a good and very accurate way of determining the radii 
of CM; 1 diameter lens surfaces. 



86 



PRACTICAL OPTICS 






jbfifo 



Fig. 75 gives a diagrammatic explanation of the method. 
Light from a distant lamp is reflected into the eyepiece 
of a microscope by means of a plane glass reflector G. 
(For this experiment it is better to remove the field lens 
of the eyepiece.) Then an image of the lamp will be 

formed in the focal plane of the 
eye lens at I 1? and also a second 
image by the micro, objective O 
at I 2 . Now, if the surface to be 
tested is placed at I 2 , light will 
be reflected from it, and, returning 

along its original path, will form 

another image at I l5 so that an 
eye placed at E will see this 

Concave image ; the first image will, of 
Surface , , 

course, not be seen, as the light 

is travelling in the wrong direc- 
tion. Similarly, by placing the 
surface in a second position, as at 
I 3 (when all the rays from strike 
the surface normally), another 
image will again be seen at Ij. 

The distance between these two positions of the surface, 
namely, at I 2 and I 3 , will give the radius of curvature of 
the surface. Refer to method of determining the curvature 
of a convex mirror (Chapter II., section (c)). 

For measuring this distance I 2 I 3 accurately, either the 
microscope must remain fixed and the lens move in a 
vertical direction, or the lens remain stationary and the 
microscope move on a vertical axis. In the latter case 
the experiment is simplified by employing a measuring 
microscope with a special adapter (made by the Cambridge 
& Paul Scientific Instrument Co., Cambridge), described 
in Chapter VI., section (a), as this instrument can be 
used very conveniently in a vertical direction, and 
measurements taken to a thousandth of a millimetre. 
In the former case a simple piece of apparatus may be 
made up by adapting a Brown & Sharpe micrometer 



75. 



IlADII OF CURVATURE OF SURFACES 



87 



1 to the stage of a student's microscope, as depicted 
in Fig. 76; in this case the' lens would be attached to 
the movable head and moved up 
and down with it. readings being 
taken from the micrometer drum 
for the two positions of the lm> 

ice I 2 and I 3 in Fig. 65. 
Experin - Determine the 

radius of curvature of the convex 
and concave surfaces supplied to 
you by one of the above methods. 
(The method applies equally well 
to concave surfaces as well as 
< -on vex.) 



CURVATURE : NEWTON'S RINGS 
METHOD 




Micrometer 
Head. 



Thoroughly clean a long focus 
convex lens and a piece of plate 
glass (flat), press them together 

and examine the reflection of th ^k\ n< n tin point of con- 
tact. A dark spot surrounded by a series of colour. -d rings 
"ill be seen. By using monochromatic light, such as a 
sodium flame or im-miry vapour lamp, many more rings, 
alternately light and dark, may be seen It will he found 
igs are closer together as they are larger, also 
it \\ill IM- noticed that the rings are closer for \rll..\\ than 
d light, and still closer for green or him- light. Th< n 
formation is due to the interference between the light 
in tit* ti'.nt and back surface* of the air film 
.n the lens and glass plate. The i 
be seen \>\ tted light . m this case, however, they 

are mn<-l. 

Let ua con*id< i thr theory; and to Himj.hfy this n i> 
are cone< t \\ith ih. 

< the \*> the 

i. -tm- <>n a j.imr Hurfucc OMN (Fig. 77). In the 
figure, O is the point of contact of the lens and surface, to 



88 



PRACTICAL OPTICS 



the complete figure is symmetrical about the point O. 
Consider light coming in the direction LO normal to the 
surface OMN. At a given point, A, the thickness of the 
air film between the two surfaces is AM. Part of the 
light incident at A passes straight through the film at this 
point without reflection ; another part is reflected at M, 






r, 






N 



FIG. 77 



and again at A, and finally passes out at M in the 
direction MM.^ It therefore suffers a retardation in 
path equal to 2AM. If 2AM is equal to half a wave- 
length of the light considered, or any multiple of half 
a wave-length, the two portions of light differ in phase 
by half a period and " interfere/' producing a dark band 
at A. If, however, 2AM = a whole wave-length (h.) or 
any multiple of h, the two portions of light combine at M 
in the same phase, and A is the middle point of a bright 
band. At 0, where there is no difference of phase, there 
is a bright spot. On passing outward from the thick- 
ness of the air film increases until it becomes equal to 

At this point there is a dark ring : still further out 

the thickness has increased to ~> and at this point there 

3h 
is a bright ring : then when the thickness is -r there is 

a second dark ring : and so on. 

If M is the position of a dark ring and R is the radius 
of curvature of the surface OAB, then by a property of 
the circle 



RADII OF CURVATURE OF SURFACES 89 

I'll x AM = r l 2 nearly ; 
or - AM = ? R' 

If B is the position of the next dark ring, 2BN = -=|~ Hence 

r 2 , X 3X 5X (2n+l)X t 
,. must equal > > , etc., or generally r 2 = ^ = - R, 

I . Z Z Z - 

where " n " is any integral number or zero. 

The radii of successive dark rings, therefore, increases 
as the square roots of the odd natural numbers, and the 

areas of the annul! hetween successive rin^s are the same. 

Also iMiN i'AM=X: 

therefore X = gW - r t *) . 

If B is not the next but the n th dark ring after A, we have 

1 



In this experiment the wave-length of the light being 
used would be known (that of sodium light being 
0000589 cms. or that of a mercury vapour lamp passing 
through a green filtn 1> -ing 000054 <> ems.), so that the 
expression may be made, with a km.\\l-dge of the radii 
't th- rings, to give the value of R. 

When the rings are viewed by reflected light dark bands 
are seen \\h-n tin retardation within the film is X or any 
multiple of X. 

This is due to the t\\> reflections not taking place under 
MM- same condition-, lii the transmitted light both re- 
fl<( tions are from -urfaceR of the glass, but for reflected 
light on< ntl,n>n i. it A i i- from a surface of air and 
one (at M) from a surface of glass. IYm tin- MOM th -n- 

i- produced a retardation of phase ', \\ln.-h must be add< d 

to that due to t in paths. 

With th< |, -us and glass "flat" supplied 
to \ nigs by placin- tin- t\\ m oontaot, th- 

flat surface resting on the curved one (see Fig. 78). The 



90 



PRACTICAL OPTICS 







system may be made <]iiiu* stable by small pieces of soft 

,it A ami l. 

The measi: of the rings formed by reflected light 

is effected by means of a measuring 
microscope. The Cambridge & Paul 
Scientific Instrument Co/s type, as de- 
scribed in Chapter VI., section (a), is 
very suitable. 

The point of contact of the two sur- 
faces is viewed with the microscope, 
and is illuminated by means of a 
" vertical illuminator " in the micro- 
scope. This piece of apparatus is shown in Fig. 79, and 
consists of a small plane glass plate placed diagonally 
between the objective and the microscope body tube ; in 
this way light from the monochromatic source is reflected 
down normally upon the " flat " and lens. As a mono- 
chromatic source, light from a mercury vapour lamp filtered 
tli rough a green gelatine filter gives best results for this 
experiment, although, of course, a sodium flame may be 
oaecL 

By means of the microscope measure the diameter of 
the 3rd, 5th, 6th, and 7th dark rings also the 15th, 16th, 
and 17th or even of three rings further 
from the centre, say the 25th, 26th, and 
27th if possible. 

Calculate an approximate value of the 
radius from the 3rd and 7th rings say 
correcting this value from calculations 
made from the radii of the most widely 
separated pairs measured, say the 5th and 
25th, the 6th and 26th, etc. 

The determination from the 3rd and 
7th riiiL^ will prevent mistakes being 




FIG. 79. 



made if a wrong number of rings is counted in the further 
work. 

In this way, applying the formula, the radius of curvature 
of the surface R may be obtained. 



RADII OF CURVATURE OF SURFACES 91 

(d) RADII OF CURVATURE : REFLECTION METHOD 
(KOHLRAUSCH) 

This experiment gives another convenient method of 
determining the radius of curvature of lens and mirror 
surfaces ; moreover, the method is applicable to both 
large and small surfaces. 

Fig. 80 shows the method employed. Two light sources, 



.--" i 
&*~ 



FIG. 80. 

such as candle flames, or preferably two illuminated vertical 
slits, are placed at L t and L 2 . At the mid-point between 
these two is situated a telescope T, so that the object-glass 
lies in the same straight line as the two lights. The surface 
to be tested, either convex or concave, is placed at S, at a 
distance not less than 3 metres, so that on looking through 
the telescope two images of the light sources will be seen 
by reflection from the surface under test. If, now, a glass 
scale G is placed in contact with the surface, the separation 
of the t\v<> iriiai^-s may !>< nic.i-invd. l<Y<>m this and a 
knowledge of the distances ST and LjL 2 (these can be 
measured with a steel tape), the radius of curvature of ih- 
surface may be obtained from the following formula : 

r- "' for a convex surface 
and r- ~' (/ for a concave surface 



PRACTICAL OPTICS 

\\ here r = the radius of curvature of the surface, 
d = the distance ST, 
/ = the measured separation of the images on the 

scale, 
L = the distance apart of Lj and L 2 . 

The student should prove these formulae for himself 
from previous knowledge ; however, the proof for a convex 
surface is given below : 

The line L gives an image behind the spherical surface 

1121 
at a distance x y by the rule - = -,+ - (-r is the focal length). 

X u> T 

The length y of this image is also given by 

y _x 
L~d 

From these two formulae we find 

dr Lr 



The length between the two images seen in the surface and 
measured with the glass scale is " I " and equals y -* - , 
from which, by substituting the above values of x and y, 

1 rL 

~2dT? 

2dl 
r = L-U 

In exactly a similar way is deduced the formula for 
concave surfaces. 



CHAPTER VI 
MISCELLANEOUS ELEMENTARY EXPERIMENTS 

(a) THE MEASURING MICROSCOPE 

THE measuring microscope is an instrument of funda- 
mental importance, and therefore its use should be 
familiar to all students. A very good type of instrument, 
especially for laboratory work, is made by the Cambridge 
& Paul Scientific Instrument Co., and is shown in Fig. 81. As 




81. 

will be seen, it consists essentially of a microscope M, which 
i- made to travel by means of an accurate, finely-pitched 
nii< rometer screw S. The microscope is attached to the 
tube T, along which it may be adjusted at will, hut can be 
ii\<d rigidly when measurements are being taken. The 
tube T slides in two V's at V t and V 2 , in which it is held by 
two opposing springs ; at the end of the tube is situated 
the micrometer drum D, by means of which intermediate 
values of the whole divisions on the scale C are read off. 
Usually C is divided into millimetres and the drum D into 
one hundred parts, so that with careful estimation readings 

93 



94 PRACTICAL OPTICS 

may be taken to one-thousandth of a millimetre. F is the 
stage on which the object to be measured is placed. An 
advantage of this type of instrument is that it may be used 
cither in a horizontal (as shown in the figure) or a vertical 
position. 

Experiment. Examine carefully the measuring microscope 
supplied to you, noting its mechanical construction, the 
arrangement of the optical parts, and the adjustments, 
etc., and draw a sketch of the instrument. 

Adjust the eyepiece of the microscope to view the " cross- 
wires " clearly when the eye is " at rest/' i.e. so that the 
" accommodation " is not strained. Place the object (a 
"graticule" or "spectrogram") to be measured on the 
stage, and carefully focus it by means of the milled 
head O until the " image " is seen sharply defined at the 
same time as the cross- wires. Arrange the cross- wires 
diagonally so that a line of the object may be set accurately 
on their intersection. In this way measure the distance 
between consecutive lines of the object by readings obtained 
from the scale C and drum D. Care should be taken in 
making a " setting " always to rotate the milled head R 
in one direction for each independent reading, in order to 
overcome any error due to " backlash " of the micrometer 
screw. 

As an additional experiment, the " pitch " of a screw 
may be measured in a similar manner. Measurement of 
(say) three threads will give the interval very nearly ; a 
large number may then be measured without counting, 
the actual number of threads being found by the first 
approximate result. The length divided by the number of 
threads then gives a value for the pitch. 

(b) APPEARANCES OF " STAR " IMAGE AT THE FOCUS OF 
A LENS 

One of the best ways of testing the performance of a 
lens or lens system is by viewing the image of a distant 
star produced by the lens under a high power, such as a 
microscope or high-power eyepiece. 



M I SCELLANEOUS ELEMENTARY EXPERIMENTS 95 

As actual stars are not always available, a very good 
artificial star may be made by allowing light from a circular 
aperture to fall on to a small steel ball about J in. in diameter 
(one from a " Hoffmann " ball-bearing acts extremely 
well) at right angles to the direction in which the tests are 
to be made. The extremely small image seen in this 
spherical surface affords a very satisfactory " point " source. 
The distance of the lens under test from the artificial star 
should not be less than 50 feet ; it is advisable also to have 
a black non-reflecting background immediately behind and 
in the neighbourhood of the steel ball. 

Experiment. Mount a single lens of about 25 cms. 
focal length in one of the optical bench lens holders (see 
Chapter II.), and place it on a one -foot steel rule made 
up as an optical bench, as described in that chapter. 
Place a high-power eyepiece (in its fitting) also on the 
steel rule, direct the latter towards some distant object, 
and arrange the position of lens and eyepiece until the 
object is clearly seen. Now direct the optical bench 
towards the artificial star, carefully centring the system 
so that the image of the star as seen in the eyepiece appears 
perfectly central. 

Focus the image until it appears at its best focus and 
make a coloured sketch of what is seen. Then move the 
eyepiece about 2 cms. inside the " best focus " position, 
observe the appearance, and again draw and colour the 
rings seen. Do the same when the eyepiece is moved 
- < IMS. outside the " best focus " position. 

Explain with a sketch the reason why "inside" the 
best focus a red ring is seen on the edge and blue in tin- 
centre, and why " outside " the best focus a blue ring is 
seen on the edge and red in t he centre. 

Fig. 82 shows what actually happens light from the 
star on reaching the lens L is refracted, and exactly as 
in the case of a pri-m i> >|>lit up into its various eom- 
ponents, blue it will In <1 \yeing deviated more 

than red ; so that \\h-n the rays are brought to a focus, 
l>Iue rays will focus at a point nearer the lens than the 



96 PRACTICAL OPTICS 

red, as shown in the diagram. Consequently when the 
appearance is viewed inside the focus, as at I, a red ring 
will be seen on the edge and blue in the centre ; and 
conversely for outside the focus. 




position 
FIG. 82. 

This appearance of colours at the focus of a lens or lens 
system is known as " chromatic aberration/' 

An " achromatic " lens should now be substituted in 
place of the " single " lens and the difference in appear- 
ance noted. Unless the achromatic lens is an extremely 
good one the coloured rings will still be detected inside 
and outside the focus, only on a very much smaller scale, 
and from these it will be possible to tell whether the lens 
is "over-" or "under-corrected." In connection with 
spherical aberration, the most noticeable effect seen with 
an achromatic lens, more especially when a microscope 
is used to view the star image, is the appearance of a 
series of concentric dark and light rings ; these are due 
to diffraction. With an " over-corrected " objective 
the ring system outside the focus will be clearer and better 
defined than that inside the focus. With an under- 
corrected objective the reverse will be the case. If the 
lens is satisfactorily corrected the appearance will be the 
same both inside and outside the focus. 



(c) DETERMINATION OF THE " FOCAL LENGTH " OF EYEPIECE 
SYSTEMS 

The equivalent focal length of an eyepiece system 
may be determined very conveniently in the following 
manner : 



MISCELLANEOUS ELEMENTARY EXPERIMENTS 97 



Tin i -\v piece to be tested should be held in some 
suitable mount (a retort stand) on the table at a distance 
of about 15 to 20 feet from the wall. To the wall >hould 
be attached a piece of paper or cardboard on which are 
painted two bold Indian ink lines about 2 metres apart. 
If now the eyepiece is directed toward- the mid-point 
of these two lines, images of them will be formed by the 
eyepiece, as at I x and I 2 (Fig. 83), and whose distance 




i i 

apart can be measured either \\ith a dynameter " * or 
measuring microscope. In the case of "negative" 

M n i^ hetter la n>e the latter \\ith about a 2 in. 
objective. 

In the figure (83) ASB i- the cardboard with the t\\. 
hnrs at A and B. P l and P 2 represent the t\\< prinrip.il 
plain-s of the < \^i.in ,md IJ, the images of B 

and A. It is at < <irnt that tin- triangles ABPj 

and IiI s P s an- -imilar. BO that : 

SP = fT^* ( FP i8tllr '''!""rd tncal length.) 

_I 1 I 8 xM 
P " Ah 

The distances AB and SP, can be measured \\iti 

or | itad t..p. . it these distances are large, 

a small error in thnr measuremt>nt \\ill n<>t cause any 

/'>/""""'". A Mimll j.Hirc.f ftjijiarntusconniiitingof a R*mJHlrn or jx 

eyepi.. . ;n tl f... .1 i.l.i mounted A finely divided gl*M Male, 

usually I . in. ih it*. It prove* very tweful in many experiment*. 
O 



98 PRACTICAL OPTICS 

appreciable error in the focal length of the eyepiece. The 
three values on the right of the equation having been 
obtained, the equivalent focal length of the eyepiece may 
thus be found. 

Second Method. Another very satisfactory method of 
determining eyepiece focal lengths is by using a colli- 
mator (see Fig. 84) having two lines A and B subtending 



Microscope 




FIG. 84. 

a certain angle at the object glass C. (This angle is 
carefully obtained beforehand.) The eyepiece E to be 
tested is placed in the path of these two parallel beams, 
and two images are formed at I x and I 2 and their separation 
measured. 

Then if I = this separation and "/" the required focal 
length of the eyepiece, 

-, = (in angular measure, when is small, as it is). 



So that when once I is measured it need only be multiplied 
by a constant (i.e. the reciprocal of " " in angular 
measure), and the focal length of the eyepiece is obtained. 

Sometimes a microscope is used to view the image I 
and which has a scale in its eyepiece. In this case 



where M is the " first " magnification of the microscope, 
and I 3 is the separation of the two images measured by 
the scale in the eyepiece. 

A " f ocometer " of this kind may be very easily made 
by attaching such a collimator as shown in Fig. 84 to 
the underside of the stage of any ordinary microscope. 



MISCELLANEOUS ELEMENTARY EXPERIMENTS 99 

'11i- lens C should be achromatic and about 1| in. to 2 in. 
focal length : the twn lin. - A and B should be about 1 mm. 
apart. 

See also Chapter VII., section (d). 

Searle's Goniometer. In connection with these experi- 
ments a piece of apparatus known as Searle's Gonio- 
meter will be found useful. It consists, as will be seen 
from Fig. 85, of an arm A, on which are mounted 
a len^ L and a single vertical line object O, the latter 




Fio. 85. 

being at the focus of this lens. This arm swings about 

the centre L, and the amount of rotation is read off a scale 

S by means of a fine wire W. A strip of mirror M is 

situated at the side and slightly below the scale, in order 

to ensure a directly vertical observation of the reading 

being made. This is done by moving the eye until the 

urn- and it- iniaL'e I'nmi the mirror appear coincident. 

So that )>y the use of this appaiai uigular subtense 

-t the object may be obtained at will. 

As an example, this goniomct- r may l>< n place 

lie scale on the wall, mentioned in the first method 

for determinim: the focal lengths of eyepieces in this 

ter. 

(d) ECCENTRICITY OF A " DIVIDED CIRCLE " 
The testing of the ecceni a di\i ,-le is 

iys a necessary experiment in order to obtain a 
\vledge of the error of readings taken from such a 

circle \\hen in use. More especially is this im- 



100 



PRACTICAL OPTICS 



portant when only one vernier is employed on the circle. 
In the case of more accurate instruments, where micro- 
meter microscopes are used instead of verniers, besides 
the systematic error brought about by eccentricity, the 
individual error of each division of the circle must be 
taken into account. For such a circle a " calibration 
curve " is made out, so that error for any part of the circle 
may be read off from the graph. 

Fig. 86 will illustrate effects on the readings of the 
circle due to eccentricity. Let D be the " dividing 
centre " of the graduations, C the centre of the alidade 

(i.e. the arm on which the 
verniers are carried), and V x and 
V 2 the zeros of the verniers. 
Suppose, in this case, that the 
circle remains stationary and the 
verniers move round the circle. 
Evidently, when VxCVg coincides 
with the diameter through C 
and D, the readings of the two 
opposite verniers will differ by 
exactly 180 (this assumes that 
the zeros of the two verniers a re- 
in one and the same straight line as C), and when at right 
angles to that diameter the difference will be a maxi- 
mum. In the figure, "V^CVg represents this position, and the 
" angular eccentricity " will be half the difference in the 
readings, that is, the angle VjDA. 

Experiment. The student should be supplied with 
some instrument fitted with a divided circle with two 
opposite verniers fitted, such as a spectrometer or 
theodolite. 

Take the readings of the two opposite verniers at twelve 
or more points round the circle and obtain their differ- 
ences, care being taken to subtract these values always 
in the same direction. Then plot the differences on 
squared paper against the angle ; from this the position 
of zero or minimum departure from the ideal difference 




MISCELLANEOUS ELK.M i:\TARY EXPERIMENTS 101 



of 180 may be found. In i\\'\> way a diagram may be 
<lra\\n >1 n>\ving the relative eccentricity, and a table of 
values drawn up from the graph, giving the angular error 
eiitrieity at any point on the circle. 

If the difference of the vernier readings is never exactly 
180, the zeros of the verniers and the point of rotation 
C are not in the same straight line (such as at V 3 instead 
of V They -hould be adjusted to be so. This error 

can be obtained from the graph by the difference between 
the minimum eeeentrieity >hown and 180 exactly. 

Fig. 87 show- typical eccentricity curves. Angular 

readings of the circle are plotted laterally, and the dif- 

!ice -f or between the two vernier readings are 



Difference in Seconds of the tyo Verniers 
from true I8O reacting, 
I i r i i + +<* + * 

8. S. S. g. 5. o 5. 8. S. 8. . 


















*/ 


x^ 


N 


V 










/ , 


/^ 


N 


A 










// 






\\ 










2 






V 


V 








y 






> 


\ 






/ 










v\ 






// 










\\ 




/ 


7 










\ 


^ 


H / 


/ 










A 


X. 


s 





































90* 



160 

th 



270" 



plotted tlx>ve and below th /.cro posit i"i 

pcctivi-ly. ( urvr A iiulicates that the alidade V^V, 

t \\iih th- li.i Kig. 86) 

at on i' ' the greatest ece< 



'' 



was at 90 nnl 'I WM -'i" ; 'l to ,, -i \I--. 

M tin- error wa the same at each <>f these last two 



102 PRACTICAL OPTICS 

mentioned positions, the zeros of the verniers must have 
been set at exactly 180. 

Curve B shows that the greatest angular eccentricity is 

again -^ seconds, but that as the two exact 180 differences 

of the verniers occur at 200 and 340 on the circle (i.e. 
not at 180 apart), it indicates that the verniers are not 
set exactly opposite one another, as illustrated by an 
alidade VjVg in Fig. 86. 

(e) PHOTOGRAPHIC TESTS ON A LENS 

Apart from the tests for spherical and chromatic 
aberrations of a lens or lens system, as mentioned in 
section (d) of this chapter, it is sometimes necessary to 
test the performance of a lens by the actual results given 
on a photographic plate when a photograph is taken with 
the lens. 

For this purpose the lens should be mounted in some 
type of camera which has a fairly large " rack adjust - 




FIG. 88. 

ment " for movement of the focussing screen. Two 
" test-charts " should be made similar to the one shown 
in Fig. 88, one small and one large. This type of chart 
is extremely good, as it is designed to bring out every 



MISCELLANEOUS ELEMKV1 A K Y EXPERIMENTS 103 

effect of error that the lens can produce. The size of 
the two charts depend somewhat on the focal length of 
the lens under test ; the small one can be drawn on a 
piece of white card with Indian ink. of such dimensions 
so that when it is placed at the same distance in front 
of the lens as the image is behind (i.e. when u =v) the 
image of the chart will cover the whole of the focussing 
screen or photographic plate. The second chart will have 
to be very much larger, as the distance from the chart 
t< tin- lens in the second case is made about ten times 
that from lens to image (i.e. u = lQv). It is better if 
this chart is painted with Indian ink on a flat white 
wall or board. Card is not advisable, as it is very liable 
to bend when of large dimensions ; and " flatness " is 
essential. 

When tln-M- two charts have been prepared. they 
should be illuminated either with daylight or by a carbon 
arc and photographs of them taken with the lens. As 
mentioned before, one photograph should be taken \\lnii 
u=v and one when u = \Qv. The point of making " u " 
equal to " v " is that defects due to the lens will be more 
pronounced and are an aid for judging the other ivMilt. 
Of course, the images must be focussed carefully on the 
ground glass screen of the camera before any photograph 
is taken; it is best to use a fairly high pow< i eyep 
for this purpose. Exposure should be found by trial 
lltord " ordinn \ plates are good for such a test. 

\\lirn tin- plates are developed. ti.\-d. washed and 
dried they should be examim-d u tull\ and th< following 
points looked for : 

i ' ntral Definition. 

(2) AstigiiMii in 

(3) Distort) 
Coma. 

(5) Flatness of I'M M. 

As regards No i < Ultra] iMimtinn u ^. sharp- 

rie* of th< lm< -s (supposing tlm th< plate is at its best 



PRA< TICAL OPTICS 

ia position) would indicate tliat aberrations, either 
chromatic or spherical, are presented by the lens. 
It is a good thing to use a yellow screen in front of 
the lens and so cut out the blue rays, which will to 
a great extent do away with chromatic aberration. 
A-tiirmatism would be detected by the lack of 
definition on certain of the "radial" lines at right 
Bfi to those on which the definition appeared 
good. 

i'>) Distortion, if present, would be most evident at 
the edge of the plate, where the straight lines of 
the square would appear curved (a straight-edge 
should be laid along them). Distortion would be 
either " barrel-shape " or " pin-cushion/' 
I "Coma" would be indicated by the appearance of 
the small white circles in the large central cross 
as being blurred or diffused on one side. Having 
the effect of a " tail of a comet/' 

) If the definition is equally bad at all four edges 
of the plate, and if by taking a photograph slightly 
in-ide or outside the best focus position for the 
centre of the plate, the definition at the edges im- 
proves, roundness of the field would be indicated. 
In this way the photographic test on a lens may be 
carried out ; this, combined with the " visual star test " 
already described for spherical and chromatic aberrations, 
will give a very good idea as to the performance of the 
lens. 



CHAPTER VII 

FOCAL LENGTHS OF " THICK " LENSES AND 
LENS SYSTEMS 

(a) THE " BAR " OPTICAL BENCH 

IX connection with experiments dealt with in this 
chapter it is important that a good type of optical 
bench be available. The one described in Chapter II. 
is extremely good for early and more preliminary ex- 



Lerrs 
Holder 

\ 



Eyepiece wth a 




Fio. 89. 



, but for first-class work it is essential to have 
i larger and somewhat more serviceable type. Therefore 



Rectangular Steel Bar. 
'( Hnch * '* inch section) 



Supports. 



' 



Divisions engrdved on top of Rod. Circular Steel Rod. 

Length ISO Cms. divided into Millimetres. (//nsfi diam.) 

it will l.- \\rll here to describe an \; good bench 

which, although not on tin- m.irkn. \\ill lo found suitable 



KM; 



1 TACTICAL OPTICS 




for the experiments suggested ; therefore scale drawings 

are given for those who may 
have the opportunity of making 
this type of bench for them- 
selves. 

Fig. 89 shows the general 
appearance of the optical 
bench with the holders and 
various fittings. Figs. 90 and 
91 illustrate rather more clearly 
the construction of the " bed " 
of the bench. It consists of a 
steel rod and vertical bar 
mounted side by side and 
parallel to one another, the 
former being divided in milli- 
metres. Along these two slide 
the " holders/' one of which 
is shown in Fig. 92 ; the design 
of these holders makes them 
quite rigid and free from any 




FOCAL LENGTHS OF " THICK " LENSES 107 



tendency to turn on a vertical axis when placed on the 
" bed " of the bench. The cylinder underneath is of lead, 




!'!:. 

in order to make the holder steady, 
engraved on a knife-edge on j. & 
the holder with which read- 
ings are taken from the 
di vided rod (constituting 
part of the " bed " of the 
l>ench). The milled head at 
the top of the hollow "pillar" 

t he holder is for clamping 
stems of the various 
fitting which fit into tin-" 
pillars. 

Almost any fitting can. in 
tin- way, be adapted to the 
optical bench ; the more 
e.-ntial ones, hn\\r\rr, are 
(i) l"iis carriers, (ii) object 



FIG. 94. 
An index line is 

l"sep.Ramscten 
eyepiece 

,. 1 1 







108 



PRACTICAL OPTICS 



or scale holders, and (iii) a scaled eyepiece (/..tli millimetre 
scale mounted in the focal plane of an eyepiece). Drawings 
are shown of these fittings in Figs. 93, 94, and 95. Other 
iiM't'ul fittings can be made up as desired. 

Such a " bench " as this will be found an invaluable 
piece of apparatus for almost every type of experiment. 



(b) FOCAL LENGTH OF A " THICK LENS " BY THE MAGNIFICA- 
TION METHOD 

Revise the theory of the method and prove the following 
formula : 

/= 2^j 

m l m 2 
where / = required focal length, 

u 1 and u 2 = the readings taken from the optical 
bench for the two positions of the object 
scale, 

and m l and m 2 the two corresponding magnifications 
measured with the micrometer eye- 
piece. 

A glance at Fig. 96 explains the formula. The well- 
known Gauss construction is used, and if this be remem- 




FIG. <w. 

bered the formula may be re-derived. Pj and P 2 represent 
the two " principal planes " of a thick lens, and AB an 
object on the left-hand side of the axis. Draw a ray 
from A passing through the first principal focus F and 
cutting the first principal plane at M ; then this particular 
ray will emerge from the lens parallel to the axis, and 



FOCAL LENGTHS OF " THICK " LENSES 109 

the image of A must fall somewhere along MN. Therefore, 
the size of the image of AB must be MO ; so that the 
triangles ABF and MOF are similar, and thus 

OM FO FO 
m, (the magnificat* = = = 



Similarly m 2 (the magnification when the 
object is moved to some other = 

j. -*r\ 

position, as at X) 

These two ('({nations may be written : 

i JBO-FO 

///, FO 
1 XO-FO 



and 
m, 




Subtracting, 
J. 
m 

and therefore 



in 

Experiment. Place a photographic lens (to be tested) 
in one of the lens holders on the optical bench. At a 
distance of about 50 cms. set up a millimetre scale on 
glass in one of the scale carriers, and then focus the image 
of this scale on the other side of the lens with a micrometer 
eyepiece. Measure the size of a number of di\i>ions of 
the scale as seen through the eyepiece, and thus get tin- 
magnification for this position of the object. Make a 
note of the reading of the " object holder " on the optical 
bench scale. Move the glass object scale to a fresh 
lion on the bench, focus up the image again and 
determine the second maLnifi'-ation. From the value-, 
nl.t.iiiMd .ind using the formula, the focal length of the 
\- be obtain. -l. Tin- experiment should be 
repeated for a number of posit inn- ,,t the object and tin- 
' in. MII " rc-ult ohtainnl. 

Negative Lens. In the case of a "thick" negative 

. or l.-ns system, the same formula hold- r.jually well. 



110 



1'liACTICAL OPTICS 



but an auxiliary positive lens has to be used, in the same 
way as in Chapter II., section (e). Form an image I x 
of the scale by means of the positive lens (see Fig. 97) 
this serves as the " object " for the negative lens- 
measure the size of a number of divisions of the scale with 
the micrometer eyepiece, this value is then the " object/' 



Scale 




Pos.Lens 



FIG. 97. 



Insert the negative lens between the positive lens and 
this last image, and move the micrometer eyepiece until 
an image of the scale is again seen (say at I 2 , Fig. 97). 
Measure the size of the same number of divisions of the 
scale ; this value divided by the last will give the first 
magnification. Then move the negative lens to another 
position and repeat the procedure. In this way and using 
the formula the focal length may be determined. 

(c) " CHESHIRE " FOCAL LENGTH METHOD 

A simpler and perhaps more accurate method of deter- 
mining focal lengths of lens systems has been developed 
by Professor Cheshire recently. A and B (Fig. 98) are 




FIG. 98. 



two lines of known separation or a millimetre scale on glass. 
L is the lens to be tested and E the micrometer eyepiece. 
S is a piece of metal with a narrow (1 mm.) vertical slit 
cut in it ; this piece of apparatus is known as a telecentric 



FOCAL LENGTHS OF " THICK " LENSES J 1 1 

-top and increases the exact ness with which A^ may 
be focussed. The slit S is set at the first principal focus 
ol the lens under test by placing a mirror M behind the 
lens and adjusting the latter until a sharp image of the 
-lit i- seen reflected back near the "real" slit, When 
thi- i- the case S will be at the principal focus of L. As 
the ray- AS and BS pass through the first principal focus 
of the lens, the images A l and B x must lie on parallel lines 
and EA X , so that the triangles ABS and EDS are 

/" can be 



x f 
-imilar, and therefore r~ = -.- -r , f rom which 



found, for A 1 B l is measured with the micrometer eyepiece, 
AB is known, and the distance x is obtained by a " measuring 
rod." A metre or half-metre steel scale set up on the 
optical bench serves admirably. 

This method can be performed very satisfactorily on the 
" Bar " optical bench described previously. 

(d) " FOCO-COLLIMATOR " METHOD (Trans. Opt. Soc., vol. xxii.. 
No. 1, 1920-21). 

Thi< method of determining "focal lengths** is 
accurate (to -2 per cent.), extremcl y nmple, but chiefly 
a quick method. It is this last point which makes the 
" foco-collimator " very suitable as a " workshop tool." 

The principle of the method will be seen from Fig. 99. 
A and B are two fine diamond lines on glass, situated 




Fio. 00. 

accurately in the focal plane of an achromatic lens " C," 
in<l -nl >tending a certain definite angle at the first 
prim-ipal plane of thi- len-. Tims, two parallel beams 
emerge from the 1m- inclined at the -aid angle to I 



II:.' PRA< IK AL OPTICS 

so that the len> \<> be tested L placed in the path of the 
t\\o beams will form an image of the two lines at A 1 and 
B! ; and their separation is measured with a micrometer 
eyepiece E, or " dynameter." 

It i^ (jiiite obvious that the two triangles ABC and 
AjBtL are similar, so that the angle A 1 LE 1 = the angle 

Now the angle ACB is previously determined accurately 
ly a method described later, and as A^ is found by the 
micrometer e} r epiece, it follows, therefore, that the distance 
" / " (i.e. the focal length) may be obtained. For 

l . l = (in angular measure) 

or/=A 1 B 1 xg; 
but T\ is a constant, so that all that is necessary to deter- 

\7 

mine the focal length of a lens is to measure the distance 
A^! accurately and multiply it by the previously worked- 
out " factor." Thus the operation becomes a very quick 
one and is ideal for the workshop or testing department. 

The graticule AB and the lens C, constituting the 
" foco -collimator," are mounted in metal cells at the 
ends of a suitable tube and fixed permanently with " set- 
screws " when finally adjusted. The " multiplying factor " 
should be engraved on the tube. The lens C should be 
about 8 in. focal length, and the distance between A 
and B about 4 mms. 

Focussing and Measurement of Angle. The accurate 
setting of the two lines A and B in the focal plane of the 
lens C, and the measurement of the angular subtense of 
these two lines at the lens, are both of extreme importance. 
These two settings can be done very completely using the 
same apparatus in each case. Set up the foco-collimator 
in a horizontal position and illuminate the graticule from 
a lamp by means of a microscope cover slip or a piece 
of mica, as shown in Fig. 100. A mirror M is then placed 



FOCAL LENGTHS OF "THICK" LENSES 113 

as shown, and a microscope (using a 2 in. objective) is 
arranged to view the graticule. A back reflected " image " 
of lines of the graticule will thus be produced by the mirror 
M. It at once becomes evident that when the " image " 




Spectrometer or 
Theodolite Circle 



FIG. 100. 

of the lines and the " real " lines themselves are in focus 
simultaneously, as seen on observation with the micro- 
scope, the graticule lines must be in the focal plane of 
the lens C. The distance between the graticule and the 
lens should be adjusted until correct. 

In order to determine the angular subtense of the two 
lines at C, the apparatus can be used exactly as it is, 
with the exception that the mirror M should be mounted 
on the centre of the prism table of a spectrometer or some 
m-trument on which angular rotation of the mirror may 
be measured. All that is necessary then is, on observing 
through the microscope, to adjust the mirror until the 
images of the two lines are exactly coincident with the 
" real " lines ; take a reading of the vernier from the circle 
on which the mirror rotates, then rotate the prism table 
i \\ith mirror on it, of course) until the "first" line of 
the image has become coincident \\ith the "second" 
"real " line, and read the circle again. This value will 
give just half the angular subtense. 

A Workshop Tool. A very convenient and useful 
"tool" for use commercially or in a testing department 
may be made by a simple adaptation of the principle of 
the " foCD-collimator." It i- an ni-iiuuent for deter- 
mining the focal length- of .-u< Imses or leu- 



114 



I'UACTK'AL OPTICS 



systems (such as eyepieces) <|iii< -kl\ -. All that is necessary 
is to attach a " foco-collimator on a much smaller scale " 
to the stage of a simple upright microscope. Fig. 101 
shows such an instrument in side elevation. C is the 
small foco - collimator, employing an achromatic lens of 
about 1J in. to 2 in. focus and a graticule with the separa- 




Graticule 



FIG. 101. 

tion of the two lines equal to about 1 mm. This is mounted 
to a metal case which carries a right-angled prism P. 
The lens to be tested L is rested on the microscope stage, 
and the images of the two lines formed by this lens are 
viewed by means of the microscope (which has a tenth- 
millimetre scale in its eyepiece), with which the separation 
of the images are measured. Therefore, taking into 
account the " first " magnification of the microscope 
(which must be determined beforehand and called here 
44 M "), the separation of the two images will now be : 
MxA 1 B 1 = (say) A 2 B 2 , 



so that 



AlBl ~ M 2 



Substituting in the previous formula at the beginning 
of this section, namely, 



FOCAL LENGTHS OF "THICK" LENSES 115 



AB 



(in angular measure); 



x J is constant and will be the multiplying factor. 

So that on measuring the separation of the two lines 
\vith the scale in the eyepiece of the microscope it 
is only necessary to multiply this separation by the 
" factor " and the focal length of the lens under test is 

obtained. 



(e) " LENS ROTATION " METHOD 

This method employs the rotating of the lens system 
about a vertical axis and can be performed very suitably 
on the " bar " optical bench. The theory of the method 
will be seen from Fig. 102. Let N x and N 2 be the nodal 
points of the lens system, which we will 
suppose has been rotated through an angle 
0. Now, a ray ANj entering the system 
and passing through the first " nodal " 
point will emerge from the lens parallel to 
it- original direction from N 2 . If, then, 
tin- lens be rotated about any point other 
than .V. the ray N 2 B will shift from side 
to side. It is using this fact that the 
following method is based : A collimator /-^. 
i \\itli it li< r a slit or small circular aperture 
as ohjeet) is set up on one of the "V 1 
>upports on the optical bench. The lens (((> 

to be tested is held in one of the lens 
holders nniilar to that >h<>\\n in KiiT. '.'.'I. \\ith the 
exception of a " rack motion " being fitted for movement 
back \\arl- or forwards of tin- upper portion of the holder. 
The image produced by the l-n- ifl \ ie\\ <l \\ith a mi. 
MI-MIL: I in. objective). Tic- leu- li..|<lrr i- thru rotated 
through a small annle and hark again to the Other 
\\hrn the image of the slit or aperture <as the case may 




IK; PRACTICAL OPTICS 

be) will be seen to move across or perhaps right out of 
tlu' field of view of the microscope. The lens should be 
moved either backwards or forwards by means of the 
iimt inn on the lens holder and again rotated. When 
the image remains stationary the second nodal point of 
the lens will he over the centre of rotation of the lens holder. 
This will be recorded on the optical bench by the index 
line on the holder. The focal length of the lens will be 
the distance between this last position and the focal plane 
of the microscope. This focal plane may be recorded on 
the bench by resting a set-square on the dividing and 
moving it backwards or forwards until its edge is sharply 
focussed when observing through the microscope. 



CHAPTER VIII 
MISCELLANEOUS ADVANCED EXPERIMENTS 

(a) FOCAL LENGTH AND NUMERICAL APERTURE OF MICRO- 
SCOPE OBJECTIVE 

THE focal length of a microscope objective may be 
determined in a very simple manner with no 
apparatus other than a microscope itself, and by the 
adaptation of an alternative formula used in the " mag- 




'n;. lo:i. 



nification method " of finding the focal length of any 
ordinary objective (as in the last chapter). The formula 
is deduced as follows (see Fig. 103) : 

A 1 B 1 is the size of an " image " of AB produced by 
a lens at a distance v l from the second principal plane 
of the lens, so that the (first) magnification 



Similarly, if the image is made to fall at a second position 
A 2 B 2 at a distance v 2 from the second principal plane, 
thr magnification (w ,) in this caae \\ill be 



Subtracting, 



117 



" 



w 2 - m l = a "T 1 



118 PRAiTlrAL OPTICS 

It is using this formula and the fact that the distance 
between the two images (i.e. the distance v 2 - vj is required, 
that the microscope itself can be used for the determina- 
tion of the focal length, for (v 2 ~ v i) can be measured from 
readings taken on the side of the draw-tube of the 
microscope. 

i; i periment. Screw the micro-objective to be tested 
in position on the microscope : if a Huygenian eyepiece 
is fitted, the field lens should be removed as it introduces 
a slight error in the magnifications. With a Ramsden 
eyepiece, which should be used if possible, this is not 
neee--ar\ . Whichever type is used, a " tenth-millimetre " 
glass scale should be fitted in its focal plane for the 
experiment. 

Place a second " tenth-millimetre " scale on the stage 
of the microscope, draw out the " draw-tube " of the 
microscope to its full extent and focus this scale. De- 
termine the magnification by estimating the number of 
divisions in the eyepiece scale covered by one or a number 
of divisions of the " image/' 

Reduce the tube length by a known amount (say 
4 cms. either by taking a reading from a " divided " draw- 
tube or with a pair of calipers, and measure the second 
magnification. 

Having thus obtained the value for (t^-t^), also m 2 
and w 1 , " / " may be determined from the formula. 

Repeat the experiment for various tube-lengths and 
take a mean of the values calculated. 

(a) Numerical Aperture (N.A.) of a microscope objective. 

Numerical Aperture (usually written " N.A/') is de- 
fined as Ix-ing equal to the product of the refractive index, 
" n," of the medium immediately outside the objective, 
and the sine of half the apical angle of the cone of light 
taken up, i.e. 

w sin "a/ J 



Numerical Aperture, in connection with the " resolving 



MISCELLANEOUS ADVANCED EXPERIMENTS 119 

power " of a microscope, is even of more importance than 
the magnification. 

Determination of N.A. This experiment may be per- 
formed very conveniently on the " bar optical bench " 
described in the last chapter. A microscope mounted 
on a horizontal axis should be placed in one of the holders 
on the bench. At the extreme end of the bench should 
be mounted a metre steel rule held in. one of the clips on 
the optical bench. Two pieces of white paper with straight 
edges should be cut and folded so that they slide con- 



Ramsden 
Circle 




FIG. 104. 



vrniently along the edge of the rule. The principle of 
the method will be seen from Fig. 104. M is the microscope, 
whose working distance is at A.* CBD is the steel scale 
\\itli the pieces of paper at C and D. The Huygenian 
eyepiece of the microscope should be of low power (about 
50 m / m sep.), and in place of the ordinary stop a 2 m / m 
diameter stop should be inserted. The Ramsden circle 
produced by this eyepiece should be viewed by a positive 
Ramsden eyepiece placed behind it. The pir.-o <>t paper 
on the steel scale should then be moved outwards from 
the centre until their edges can only just be seen in the 
extreme edges of the Ramsden circle. We then have 

* 'I'!..- point \ may be fixed relative to tin- <li\i.ling of the o|>tiral l><-n. h l.y 
resting a set-square - .n- <>n th ,1* or 

forwards until t I edge is seen sharply in f 

tin mi, i.... ..-.-... it,,, reading of the bottom edge of the square may then be 

; fn.ni tli. .,ns. 



L20 PRACTICAL OPTICS 

a means of determining the angle " a," for CB or DB 
(which -In >ul<l be the same) can be obtained from the steel 
scale and AB from the optical bench. So that 



Various distances of AB should be taken for the same 
objective, the experiment repeated, and a mean value 
of " a " obtained. 

Various quick methods whereby the numerical aperture 
may be read off " direct " have been devised by Prof. 
< hc-shire, which give very good results. One of these 
methods consists in placing on the stage of the microscope 
a piece of card on which is painted the design shown 
in Fig. 105. This design, when seen in the plane of the 
Ramsden circle of the microscope, projects as a number 




FIG. 105. 

of straight lines of equal thicknesses. The distance of 
the card from the front of the objective is of importance ; 
to obtain this correct distance, a small metal or hard 
wood block is made of the right length * ; this is rested on 
the card and the top surface of the block focussed with 
the microscope. The block is then removed, the positive 
Ramsden eyepiece placed so as to view the Ramsden 
circle as before, when the number of lines corresponding 
to the N.A. will be seen just to fill the diameter of the circle. 

(b) COMPLETE MEASUREMENTS OF THE OPTICAL SYSTEM OF 
THE MICROSCOPE FOR THE MICROSCOPIST 

This section is written for the microscopist who wishes 
to take measurements on the optical system of his own 

* This length should be the distance between the " working distance " of the 
objective and the position at which the card was calibrated ; in this case 
25 mms. These cards are obtainable from Messrs Baker, 244 High Holborn. 



MISCELLANEOUS ADVANCED EXPERIMENTS HM 



instrument, and, therefore, naturally does not want to go 
to the expense of having to obtain much auxiliary 
apparatus for the purpose. 

(a) Considering first, then, the Numerical Aperture of his 
objective ; this is best done by using the Cheshire 
Apertometer * shown in Fig. 105, the method of 
using being described in the preceding section. 
(6) The focal length of the objective may be determined 
by the magnification method mentioned in sec- 
tion (a) of this chapter, i.e. using the draw-tube 
extension. 

(c) The focal length of the eyepiece can be obtained in 
a similar way by making up a simple adaptor (see 
Fig. 105A) to carry the eyepiece, and which can 

Microscope 



Adaptor 
which screws 

into 
Microscope 




Small 
Stop 

yepiece 
in its 
Adaptor 



^$ca/e on 
Microscope Stage 

Fio. 105*. 

be screwed in position in place of the objective. 
This is, in (T< ct, using the eyepiece as an objec- 
tive, \\liicli incidentally must be stopped <l\\n. 
Another eyepiece (which has a tentli-millimrt n> 
scale in its focal plane) is then used at the eye- 
piece end of tli- -cope, and by tin- maLrnifira- 

timi method described previously the focal length 
may be obtained exactly as before. 

'/) Magnifications. The fir t " magnification may be 
See footnote on page 1 



122 



I'll ACTICAL OPTICS 



determined by placing a tenth-millimetre scale 
on the stage of the microscope and comparing the 
size of the image of a certain number of divisions 
of this scale, projected by the objective, on a second 
scale situated in the focal plane of the eyepiece. 

The total " magnifying power " may be very conveni- 
ently obtained by the method shown in Fig. 105B. A 
piece of neutral tint glass G (if this is not available, a 
piece of ruby or cobalt glass will do, or plane glass) 




FIG. 105s. 

is placed at about 45 to the axis of the microscope. 
If now the eye is placed at E, the magnified image of 
the scale S as seen through the microscope can be viewed 
so that it appears on a piece of card S 1 S 2 at about 10 in. 
away (i.e. at the " near point " of the eye). Whilst thus 
observing, two lines can be drawn with a pencil at the 
positions where two particular lines of the magnified scale 
are seen, and then this distance S^g measured, from which 
the magnifying power of the microscope may be obtained. 

(c) THE AUTO-COLLIMATING TELESCOPE 

This instrument is, as its name implies, a combination 
of a collimator and telescope, and plays an important 
part in the testing department of the optician. Its ap- 
plications are many, but it is used chiefly in connection 



MISCELLANEOUS A I >\ A \< ED EXPERIMENTS 123 



with the measurement of prism angles and the testing 
of parallelism of glass plates. It will be well, first of all, 
to look at the optical system of the instrument ; this is 
shown in Fig. 106. O is the object glass (usually about 
1- in. focal length), in the focal plane of which is mounted 
a graticule G. One of the best types of graticules is that 
shown in the figure, the horizontal line on the left being 




Graticule 
Bloc* l/nes on ctoargla** 



Gratiqule 



Ramsden 
Eyepiece 

L 




Object Glass 



45Prism 



FIG. 106. 



covered by a small 45 prism as indicated by the dotted 
lines, and the spaced lines on the right correspond to a 
definite angular subtense at the object glass. If a " tenth- 
millimetre " scale is used, each division may be made to 
<ni respond to 1 min. angular subtense by choosing an 
object glass of suitable focal length, so that by estimation 
reading may lie taken to 6 sec. of arc. 

F and E are the field lens and eye lens respectively 
of a Ramsden type eyepiece. By means of an aperture 
in the side of the telescope tube light is admitted from 
a lamp, and thus the previously mentioned horizontal 
line becomes illuminated. Tin- line serves as the object 
for the collimator. Rays from thi^ collimator go out 
" parallel," and it a mirror or plane glass surface is placed 
in the path of the beam normal to the axis, the rays will 
return ;dni: their ori-mal path and come to a focus again 
in the plane of G, when an image of the hnri/ontal line 

u viewed l>\ meant ot the eyepiece. In this way an\ 

di-plaerment of the image from the centre line .i" the scale 
may he mea-ured in aii'jular amount 



124 



PRACTICAL OPTICS 



Sometime^ an eyepiece with a plane glass reflector 
in it is used (see Fig. 107), and a graticule of the design 

G F D 

OifT 




FIG. 107. 



shown in Fig. 108 instead of the graticule and 45 prism, 
but owing to scattered light from the plane glass reflector 





FIG. 108. 



FIG. 109. 



it is not nearly so successful as the type of auto-collimating 
telescope shown in Fig. 106. A very suitable mount and 
stand for the auto-collimating telescope is shown in 
Fig. 109 ; the arm A can be swung into any position within 

the 180 and can be clamped at 
will by a " winged " nut at the 
back of the instrument. 

Parallelism of a Glass Plate. 
If, now, a glass plate is placed 
in front of the objective, in most 
cases (unless the two faces are 
absolutely parallel) two images 
of the horizontal line will be seen. 




FIG. 110. 



These are due to reflection from the two surfaces of the 
I .late, the brighter of these two images being the reflection 
from the first surface. The angular separation of the two 
images can then be measured on the graticule. 



MISCELLANEOUS ADVANCED EXPERIMENTS 125 

Fig. 110 illustrates the path of the rays in the plate; 
the angle NOP is the one measured with the auto- 
collimating telescope, from which, with a knowledge of 
the refractive index of the glass (it is near enough to take 
n = l-5) the inclination of the two surfaces may be obtained. 

For^iSAR (the required angle) = 1 80 - zASR - 



= 180 -90 -(90 - 



But L SRO - L ROM = l x L NOP. 

ft 

NQP 
/. z. SAB = 180 -90 -90 + n 



NOP 
= _ // . 

2 
If n is taken as 1-5 



Testing the Angles of a Right-angled Prism. The auto- 
colli mating telescope may be used to great advantage 
for the testing of the angles of right-angled prisms, and 
becomes an extremely simple and quick method when the 
observer is once acquainted with his instrument. 

It is general to determine the error of the 90 angle 
fir-t. a- this aids the determination of the 45 angles. 
I MI tli is purpose the auto-collimating telescope may be 
n-ed in two ways: one as shown in Fig. H!A and the oth< T 
as in Fig. Ills. In the first case, if the angle between 
tin prism face and the "flat i- exactly 90, only one 
im.iL'f of the hori/.ontal line would be seen, and that 
(.incident \\ith the "zero" line of the graticule. Tin-. 
houever. i- not u-ual : more often two images \\ill l>e 
seen equally displaced each >ide of the /.< -n. Thi> in- 
dicates that tli- rn i in the !m . and tin- error .f 

say ) will be represented by an angular di-|.la< ement of 
t he t \\o imagM ot " 4a " on the graticule. 

In the second case, \\hen the liL'ht travel- m-ide the 






l'RA< TICAL OPTICS 



prism, the <1< -\iation is increased to w(4ct), where n is the 
. index df the prism. The student should prove 
these for himself. 
To test the 4.1 angles, the auto-collimating telescope 




Prism 



Flat 



Prism 



(a) 




Fio. 111. 



should be adjusted until its axis is " normal " to the face 
A I! (see Fig. 112), when the face BC is put carefully in 

contact with the flat. The 
prism should then be carefully 
taken off and the face AC put 
in contact with the flat. On 
looking into the telescope, but 
without altering its position in 
any way, it will be observed (in 
all probability, unless the 45 
angles are exactly equal) that 
the horizontal line image has 
moved a certain number of 
divisions. This angular move- 
ment will be just twice the difference in angle between the 
two 45 angles. Let this difference be /3. Then the 
< AB (supposing that it is the greater of the two 45 
angles) 




Optical 'Flat' 

in-. 



MISCELLANEOUS ADVANCED EXPERIMENTS 127 

/180-C 



and the 



2 

u here < ' is the actual value of the 90 angle. 

/) TESTS ON A TELESCOPE 

Tests on the performance of a complete telescope are 
of the greatest importance. They may be divided into 
t u n sections : 

(i) Geometrical Tests (such as angular field of view, 

magnification, etc.), and 
(ii) Definition Tests. 

In dealing with the first section, the focal lengths of 
the object glass and eyepiece may be determined by 
methods described in previous chapters. The magnifica- 
ma\ l>e determined very accurately by the following 
method: Focus the telescope on some very distant 
object (parallel light). Then support it in a vertical 
position on the table, with a frosted lamp immediately 
beneath the object glass. In front of the object glass 
place one of the millimetre glass scales, and over the 
piece place a " dynameter " (see page 97), and focus 
-harply the 1 Jam-den circle, when the divisions of the 
glass scale in front of the object glass should also be 
in tncii-. In tin- u.iy the -i/.e of both scale and image 

n ay l< measured m nltaneously, and the magnification 
obtained ; 

M _ Size of Scale 
~~ Size of Image* 

/ >fld of View. The angular extent of the field of view 

n i\ I.e l.r-t nhtaimd l.y observing two distant objects 
appear at the ext rem edge of tin field as seen \\ ln-n 

thnniL'li the trloenpe, .ind atteruai'ds measuring 



[28 I'KA< TICAL OPTICS 

the angular >ul>ten-e to the naked eye of these two objects 
l.v means of a theodolite or sextant 

If a permanent seale can be set up at some distance 
(such as might be done in connection with any optical 
testing dt }' which has been previously divided 

according to known angular subtenses, it is possible to 
read off the angular field of any telescope directly from 
the scale. 

Owing to the (possible) finite distance of the scale, 
however, it i> necessary to place the instrument under 
test, so that the front or anterior focus of the object glass 
coincides with the point from which the angular subtense 
of the scale divisions were previously measured. 

Effective Aperture of Object Glass of a Terrestrial Telescope. 
The determination of the position and size of the stop 




L L 3 L, 



in the ''erecting eyepiece" of a terrestrial telescope is of 
considerable importance, as this is frequently found to 
be incorrect, with the consequence that the " effective 
aperture " of the object glass is reduced. 

O in Fig. 112A is the object glass, and L! and L 2 are 
the lenses of the "erector." From the paths of the rays 
proceeding from the object glass shown in the figure, it 
becomes evident that the stop S must have a definite size 
and position between the lenses L x and L 2 in order to 
ensure that all the light regularly transmitted by the 
object glass passes to the image and ultimately to the 
eye. At the same ti i e any stray light reflected by the 
sides of the telescope tube are prevented by the stop 
u passing to and thus confusing the image. Makers 
frequently take advantage of this point and place tin's 
stop in some position such that the definition of their 



MISCELLANEOUS ADVANCED EXPERIMENTS 129 

instrument is increased, but which in effect decreases the 
aperture of the object glass. This is unfortunate for the 
customer, who always has to pay for aperture ! 

To test the Position of the Stop. Focus the telescope 
for infinity. Illuminate the aperture of the object glass 
with a piece of paper and measure the size of the " exit 
pupil," as mentioned in Chapter VIII., section (d). Then 
take out the complete eyepiece and remove the stop S 
altogether. Replace the eyepiece and again measure 
the size of the "exit pupil." If this latter exit pupil 
(which is the true one) is found to be larger in diameter 
than the previous one, it is obvious that the stop is cutting 
off some of the aperture of the object glass. The position 
of the stop should then be adjusted until the true diameter 
of the exit pupil is obtained. 

(ii) Definition Tests (Test Objects). For these tests 
it is essential to have certain definite " test objects." 
They should preferably be illuminated by daylight, and 
should be situated at not less than 150 feet from the 





+ 



113. 

position at which observations are to be taken. The most 
important of these objects is an "artificial star"; this 
in i\ be made up very easily, as explained in section (6), 
( h.iptrr VI.. by employing a small steel ball, on to 
\\liidi light from a < ircular aperture is allowed to fall, 
at right angles to the direction in which observations are 






n: v TICAL OPTICS 



to be taken Such a device gives quite a >ati>i'actnry " point 



tot ol.jrct can he made by painting with 
Indian ink a sketch of a tree (without foliage), showing 
branches and tuiirs. upon a piece of "opal" glass, and 
illuininatinLT it fmni l)chind. This serves admirably, as 
the "degree of blackness" of the branches and twigs, as 
D through the telescope, serves as an all-important test 
tor tlic presence of " spherical aberration." 

Tin- third object should be one of some such design as 

-hown in Fig. 113. It consists of a metal plate with 

squares and circles of varying size cut in it. It should be 

illuminated behind either by artificial daylight or real 

iiirht. in the latter case by means of a mirror at 45, 



Circular 
aperture. 

Steel 




.*.-)' 

-"I 




=- 4 



I, I , I i I i I i I , I i I , I, I i I 

Kiu. 114. 



and in the former by using a " Chance " artificial daylight 
aeon in trout of a MVatt electric lanip. 

Fig. 114 shows a useful set of test objects which may be 
mounted together in some form of wooden casing. They 
should each have a hinged door which can be swung in 
front at will, in order that any one object may be used 
without interference from any of the others. Such a set of 
test objects as illustrated in Fig. 106 is very simple to make, 
and introduces everything that is essential for telescope 
testing. 

Performance Sheet for a Telescope. The procedure for 
i '-ting a telescope will be as follows : 

Determine 



MISCELLANEOUS ADVANcKD EXPERIMENTS 131 

Magnification (including size of "Exit Pupil"). 
Angular field of view. 

Set the telescope on the " artificial star " and 
observe the appearances of the image at the centre 
of the field : 

First. At the best focus. 
Second. Inside the best focus. 
Third. Outside the best focus. 

A properly corrected instrument should show a clearly 
defined diffraction " ring system " on each side of the 
best focus. If the rings are " harder " on one side than 
on the other, " spherical aberration " is indicated. If 
they are more clearly defined inside the focus, "under- 
correction " of the system is indicated ; and if more 
clearly defined outside the focus, " over-correction." 

If the rings seen are " elliptical," as shown in Fig. 115, 
" astigmatism " is present (due probably to some cj-lin- 
'ririty of one of the refracting surfaces). "Coma" is 
identified by the appearance of the ring system shown 
in Fig. 116. 

Colour Correction. Observations should then be taken 
on the " colour correction " of the instrument. This is 





115. Lift 

best seen by directing the telescope towards the edge 

of one of the bright larp iquaree -li\\n in Fig. 114. The 

appearance of this edge inside and outside the focus \\ill 

be that it i- fringed with colour : it it has a red fringe 

<Ie the focus and a blue fringe outside the focus, " under- 

ction " will be indicated. If " red " outside the focus 

;ind hlue inside, "over-correction" \\ill be indicated. 



PRACTICAL OPTICS 

A further test for spherical aberration may be given 

using the telescope on the black tree test object. 

Spherical aberration would be indicated by any lack of 

ickness" of the image seen. For any considerable 

difference in focus between the marginal and par-axial 

- will cause a considerable " scattering " of light, and 

consequent 1\ tin- imuirc will appear " greyish " instead 

of black. 

The angular extent of " good " field should then be 
determined for each of the above tests, i.e. " the star/' 
the bright edge, and the black tree. 

Therefore the " Performance Sheet " may be tabulated 
in the following manner : 

PERFORMANCE SHEET FOR A TELESCOPE 

Form to be filled up for each instrument tested 

1. Description of Instrument. 
-. Mairnifyinjr Power. 

3. Angular Field of View. 

(Including size of " Exit Pupil.") 

4. Angular extent of " good " field. 

First. For " star " object = 
Second. For " bright edge " (i.e. colour) = 
Third. For " black tree " object = 
Effective Aperture (stop). 

' Artificial Star " Tests. 

(a) State whether the system is " over-corrected " 
or " under-corrected " as regards " spherical 
aberration." 

(6) Is " astigmatism " present ? 
(c) Is " coma " present ? 

6. Colour Correction. 

State whether the system is " over ' 
corrected as regards " colour." 

7. Centring. 

8. General Remarks. 



MISCELLANEOUS ADVANCED EXPERIMENTS 133 



A'lto-cottimation Test for testing Telescope Objectives. 
Another very convenient way of testing the object glass 
of a telescope is by using the following auto-collimation 
method. It is in effect a " star " test, using an artificial 
star by means of light reflected from a small steel ball. 

Light from a lamp (Fig. 117) is reflected by means of 
a steel ball (situated at the focus of the lens), which travels 

Lamp 



GoodMirror 




^'diameter 
Cycle 




Lens being tested 

KM;. 117. 



F 

High Power 
Ram set en Eyepiece 



towards the lens in the direction indicated. A " good " * 
mirror is placed as shown and the reflected beam brought 
to a focus F in the same plane as the steel ball, but slightly 
to one side of it. The appearances of the " star " image 
may then be viewed by a high-power eyepiece, and the 
performance of the lens judged therefrom. 

It is of great importance also in this test that the lens 
be properly " centred " before the star-images may be 








Ms. 

fairly judged. For this purpose a piece of apparatus 
known as a " self -centring eyepiece" should be used. 

* It i-i ..f the greatest importance that this *ilvered mirror w 11 i>< rf.vtly 
good "flat." 



PI: \< TICAL OPTICS 



-Imuii in Fig. 118, and was devi>l 

hv PP> h consists of an outer tube A, into 

\\hirh sli<lr< tin- portion B. B consists of a tube, at one 
ml of \\hic! Jily-pnlislird >i Ivor or german silver plate 

P sweated on at l~> with a hole (about J in. in diameter) 
bored mit rally in it. whilst at the other end is a knurled 
ring II \\hich has a pin-hole H drilled in it. S is a stop 

he aperture shown. 

In use tin- "eyepiece " is set up approximately on the 
axis of the object glass to be tested. A lamp is then 
arranged so as to illuminate the reflector P through the 
( -nt -a way portion of the tube A, so that an annulus of light 
will be sent towards the object glass, and to an observer's 
placed at H an effect such as shown on the left of the 
figure will be seen reflected in the first surface of the object 
glass. When the object glass is satisfactorily " centred " 
the annuli thus seen should all appear concentric. 

. TESTS ON PRISMATIC BINOCULARS 

The testing of the prism binocular is inevitably an all- 
important subject, and it is the aim of this " section " to 
give a complete description of how this should be done. 

The tests may be divided up under various headings. 

Treating each half of the binocular as a telescope : 

1. iMinition, 

2. Magnification, 

3. Field of View, 

may be determined in exactly the same manner as de- 
scribed in the previous section (c). 
Other tests are : 

4. Parallelism of axes in all positions. 

5. Strain in prisms. 

6. Inversion produced by the prisms. 
7 Stray light. 

8. Angular subtense of graticule (if fitted). 
Parallelism of Axes. This is the most important test 
of all in connection with binoculars. The apparatus 



M IS( KLI ANEOUS ADVANCED EXPER I M KXTS 135 



needed for this test is essentially of a somewhat special 
nature, but as it also serves as a means for adjusting 
binoculars, it is well worth while having such a device 
constructed. A diagram of the apparatus is shown in 
Fig. 119. It consists of two collimators parallel to one 
another at a distance apart equal to the average separation 
of the binocular object glasses, an adjustable table on 

Mounts with 3" 'Capstdn' screws 
for adjustment of cross- fines 



Supports on 
which table 
swivels 



Milled Head 
for raising^ 
or lowering 
the table 




Adjustable 
/table 

MilledHead 
/for lateral 
adjustment 
of table 

, Slide for 
Telescope 



Telescope 
(F-6"dbout) 

II!'. 

\\hich to rest the binoculars, and a small telescope which 
travels along a geometric slide at right angles to the axes 
of thr collimators (see Fig. 120). 

I'irst of all, the axes of the collimators are adjusted 
parallel to one another l>\ -luiing the telescope in front 
of each collimator object glass in mm. and adju- 
the " adjustable " cross-lines of the collimators to coincide 
with the cross-lint m th- eyepiece <>t the telescope. \\ hm 
tin- ha- hrrn (l<nr tin- hinoculars to be tested are placed 
nn th table and adjusted until tin image" \i<\M<l 



LSI 



PRA< TIGAL OPTICS 



!i the telescope) through one-half of the binocular 
made coiiu id. nt \\ith the cross-line in the telescope. 
On sliding the telescope so as to view the " image " through 
the other half of the binocular, any deficiency in coin- 
cidence of the image will at once be seen, and this is a 
measure of the want of parallelism of the axes of the two 




halves of the binocular. The actual displacement of the 
image may be determined in angular amount by the 
graticule in the focal plane of the eyepiece (a suitable 
type of graticule is shown in Fig. 108). In this way both 
the vertical and horizontal angles between the image 
forming rays from the two halves of the instrument may 
be determined. Below is given the maximum allowance 
in angle between the axes of the two halves and the 
corresponding magnification : 







Magnifying 
Power. 


Horizontal 
Allowance. 


Vertical 
Allowance. 


3x 


30min. 


lOmin. 


6x 


12min. 


4min. 


10 x 


6min.40sec. 


2min. 12 sec. 


12 x 


5min. 30 sec. 


1 min. 50 sec. 



This " binocular testing bench " is also convenient for 
the ordinary adjusting of binoculars, for the quickly ad- 
justable table allows the binocular to be placed in position 
and tested with the least amount of trouble possible. 
Adjustments in binoculars are effected, either by move- 



MISCELLANEOUS ADVANCED EXPERIMENTS 137 



ment of the prisms or by rotation of the object 
glasses. 

Strain in Prisms. Owing to the method by which the 
prisms are held in the binocular, excessive pressure is 
sometimes exerted on them. This is an extremely bad 
fault, as the double-refracting effect thus produced will 
appreciably affect the definition of the instrument, and 
sometimes if the binocular is accidentally given a sharp 
" jar " a piece of the prism will chip off owing to the 
strain. 

Strain may be quite easily detected by holding the 
binoculars in a clamp stand and allowing light reflected 
from a " blacked glass " polarizer (at the polarizing angle, 
see Fig. 121) to enter them. The "exit pupil" may 



Diffusing 
Screen 





V 

'Prism 



Binoculdr 



Lamp 



Blackened Glass Plate 

Vl\. 

then be examined with a Xieol prism, whieh has been 
rotated to give "extinction" before tin binoculars are 
inserted in the path of the polarized beam Any strain 
in the instrument will be shown up* by the appearance 
of " light patches" among the darkened field. IV 
and lenses of all kinds should be held sufficiently tightly 
\\ 1 1 hout any undue strain being imposed on them. 

Inversion produced by Prisms. For this test the binoculars 
are supported horizontally and focussed on a vertical 
lint \\hich is not less than Inn teot away. A theodolite 
\\hieli has been previously made to "transit" over this 
vertical line satisfactorily (after levelling, etc.) then views 



li; \. TK'AL OPTICS 




Pun vie* of 2 right-angled Prisms 
K used in a Binocular showing 
an error O 'which produces 
bad inversion effect. 






the lump' 't the lin- through each half of the binocular 
in turn. In this \\ay the perpendicularity of the image 

the line may be estimated, 
and hence the perfection of the 
inversion produced by the prisms. 
Any lack of " inversion " is due 
to error in the setting of the two 

l )ris!iw at ri S ht an ^ le to one 
another (see Fig. 122). 

Stray Light. A square frame 
should be made up, with tissue 
paper stretched across it and 
having a black circular disc of 

r in the middle, such as is shown in Fig. 123. 

tram* should be brightly illuminated from behind, 

one-half of the binocular directed towards the black 

disc. The distance of the binocular from the disc should 

be such that the disc rather more than fills the field of view. 



Tissue 
Paper 



Wooden 
Frame 




Circular 
Disc of 
BlackPaper 
(V/i'diam.) 



FIG. 123. 

On examining the exit pupil, either with an eyepiece or 
l.y taking a photograph, any stray light in the instrument 
will be made manifest. Bright reflections are distinctly 
detrimental to the action of the binocular, especially in 
" night glasses." 

Test of the Graticule. A graticule is very often fitted 



MISCELLANEOUS ADVANCED EXPERIMENTS 139 



in the focal plane of one ocular of a binocular, for purposes 
of " range-taking/' and it is necessary that the angular 
subtense of the graticule divisions (usually 30 min>.) 
should be tested. 

For tin's purpose the binocular should be supported in 
a horizontal position, with some source of illumination 
(preferably diffused) placed in front of the eyepiece. At 
tin 1 object glass end a theodolite should be arranged so 



Mirror 






L 






Fi. . li'l. 

as to view the linage of the graticule lines. The angular 
separation may then be measured by setting the cross- 
win^ in the theodolite on the images of the lines in turn 
and taking readings from the theodolite circle. 

Second Method. Another method may be adopted, 
\\ hieh involves the use of a " mirror mounted on a theodolite 
table"* (a piece of apparatus invaluable to the testing 
room), and is sho\\ n in Fig. 124. Light from a lamp is 
reflected into the eyepiece of the l.inocular l\ UK an- !' 
a plane glass reflector. The theodolite table, with a good 

Tlii- >]>< <>f tli<Mxlolit- i.ii.li- with"-: 1 unting may be 

obtu-i ,l.-r\\.-ll. 



140 PRACTICAL OPTICS 

mirror mounted on it, is then placed as shown in the figure, 
and u(Iju>u-d until an image of the graticule is seen in 
the same plane as the " real " graticule on observing 
thiMiiirh the eyepiece. On rotating the theodolite table 
the image of the central graticule line may be made to 
travel across the divisions of the real graticule, when 
readings from the theodolite table may be taken which will 
give just half the value of the actual angular subtense. 



CHAPTER IX 
REFRACTOMETERS 

THE subject of determination of refractive index by 
the spectrometer was dealt with in a previous 
chapter, and this instrument in reality is the fundamental 
instrument for such determinations. There are, how- 
ever, other instruments designed solely for refractomctiv 
which either give " refractive index " direct or by the 
simple determination of one angle and the use of tables. 
As these instruments are in considerable use at the present 
day, this chapter has been devoted to the explanation 
of the more important types. 

(a) PULFRICH REFRACTOMETER 

The principle of this type of refractometer will be seen 
from Fig. 125. The substance or liquid whose refractive 




index is to be measured is placed on the top of a glass 
block of known refractiv m<l. -\. In tin- MM >t a solid, 
a thin layer of liquid of high refractive index is placed 

between the two surfaces. The angle A iK'twrm tin- 
in 



I i_ PRA< TICAL OPTICS 

vertical and hori/ontal Miriace- of the prism is usually 

accurately !n . 

It. then, li^rht niters the substance or liquid of unknown 
'idex from a position L, that entering above 
the normal LO will enter the Pulfrich prism and pass out 
again as indicated along the path NP ; a telescope placed 
at P would then see a band of light with a sharp bounding 
line on the upper side. The rays which enter normally 
along LO will graze the two surfaces in contact and will 
be the limiting rays of the band of light observed at P. 
Any rays entering below the normal LO will not be able 
to enter the Pulfrich prism at all. 

So that, the sharp line observed in the telescope of 
the refractometer represents the rays which have just 
been able to enter the prism ; the angle through which 
these " grazing " rays have been refracted is the com- 
plements of (90 - r) .... [see Fig. 125], 
which is the " critical angle " of the Pulfrich block with 
respect to the substance above it, and depends solely on 
the refractive indices of the two materials. (An inter- 
mediate medium if of greater refractive index than the 
one above it has no appreciable effect.) 

The angle " i " at which the beam emerges into the 
air depends on the magnitude of the angle " r," and is 
measured with the refractometer. 

Considering the refraction at the two prism faces in 
turn, we have 

sin <:o ^ sin (90 - r) ) 
and sin " 1"=^ sin "r" I 

where n^ and n 2 are the refractive indices of the Pulfrich 
block and substance or liquid to be tested respectively. 

Combining these equations, the unknown refractive n 2 is 
calculated from the expression 



n 2 = Jnj 2 - sin 2 i. 
The instrument is usually supplied, however, with a 



RKFHA( TH.MKTKRS 



143 



tal>l<- \\hirh iv prepared for all values of the angle " * " 
from tlie above formula, so that refractive index may be 

ni lined directly from the table. 

Fig. 126 gives a general illustration of the instrument. 
Light (usually from a hydrogen tube) is sent into the 
substance or liquid being tested, through the condenser 
" C," which renders the light convergent. The beam on 



Thermometer 



Clamp for 

Hydrogen 

Tube 




iil'i. 

< merging from the Pulfrich prism face F is received by 
tin- telescope T, \\lii* -h is attached to the rotating circle S. 
With tliis circle, by meanfl of a vernier and tangent screw 
tin- angle i- MM -a-im -d. The telescope is auto- 

collimating, in order that the normal to the face F of 
the Pull rich Unrk may be obtained by back reflection. 
Kin ins:* W for a water circulation are provided, so that 
Mil-tan<<> may be investigated at raised trni| atures ; 
also it is particularly useful in the case of substances, 
-IK h as fats and waxes, vlm-h mily become liquid and 
transparent at these temperatures. D is a right-angled 



144 PRA< IK AL OPTICS 

I'MMii \\hirh can be swung in and out of the path of light 

IK I -user so that sodium light may be used when 

!vd without having to remove the hydrogen tube. 

K i- -imply a device for limiting the aperture of the 

incident beam. 

The Instrument in Use. When testing a solid it is 
essential that the specimen has two surfaces nearly at 
right angles, and the one which is placed in contact with 
the Pulfrich prism should be well polished and reasonably 
flat, while the other need only be sufficiently polished 
to allow light to enter. It is important, however, that 
the edge at which the two surfaces join should be very 
sharp. For measurements on liquids a small glass cell 
is cemented on the top of the Pulfrich block,* into which 
a small amount (a layer of about 3 mms. deep) of the 
liquid can be held (see Fig. 125 (6)). 

First of all, the reading of the circle when the telescope 
is " normal " to the face of the " block " should be 
obtained (i.e. the zero setting checked). 
For this purpose a lamp should be 
arranged to illuminate the small prism 
near the eyepiece of the telescope, and 
the circle rotated until an image of two 
small lines will be seen in the field of 
view. When they are in such a position 
that the one line of the real graticule is 
midway between them, this should give 
the zero setting of the instrument. The type of graticule 
generally used is shown in Fig. 127. 

This being done, the specimen to be measured should 
be placed on top of the " block." This has to be done 
with great care. The two surfaces to be put in contact 
should first be thoroughly cleaned. A small quantity of 
liquid f (of higher refractive index than either the specimen 
or the Pulfrich prism) is then placed between the two 

* Two Pulfrich prisms are supplied with the instrument, a "light" and 
44 dense," the former for use with liquids and the latter for solids. 

f A suitable liquid to use, and of high refractive index, is " Monobrom- 
naphthaline." 




REFRACTOMETERS 145 

surfaces, and the specimen pressed firmly on to the prism. 
The surface of contact should then be examined by re- 
flecting from it monochromatic light (a sodium flame), 
\v I it'ii alternate light and dark interference bands will be 
seen. The bands should be made as broad as possible 
1>\ pressing on the specimen, as the surfaces are then most 
nearly parallel. The number of bands seen should not 
be more than six, and with " well-worked " specimens 
having flat surfaces it will be found possible to bring the 
>urfaces so parallel that one band fills the whole surface 
of contact. 

The instrument is now ready for taking readings. 
Sending " sodium light " first, therefore, through the 
specimen, the telescope should be moved round until the 
graticule is brought on to the sharp bounding edge between 
the sodium coloured and dark part of the field. The 
difference in readings taken at this position and that of 
the "normal" or "zero" reading will give the value "'" 
(Fig. 125). 

By a simple reference to the tables supplied with the 
instrument, the refractive index (for D light) of the 
specimen can be obtained corresponding to the value 
of " ." 

Similarly, by using the hydrogen tube* with the in- 
Mrumrnt the values of "t" may be obtained for the C, 
F, and G t lines, and with the use of the tables the refractive 
index for each line. Also mean and partial dispersions 
may be obtained. 

Tables are also supplied for temperature variation \\lini 
such are needed. 

(b) THE ABBE REFRACTOMETER 

The principle of this instrument again depends on the 
use of a stan <l.ii<l prism and the border line between the 

hi m.-t -uit;il.lr i tube to use is the type nn-iiti..nr.l in 

< h i|ter IV., as the large side bull* all..wH a much heavi to be put 

th- tul>< u Mil.. ut the great rise in pressure, and thus increases the 



146 



l'i:.\< TICAL OPTICS 



light and dark parts of the field, due to "grazing inci- 
dence " illumination. 

Its general arrangement will be seen from Fig. 128. 
It consists of two 30 prisms A and B, mounted in a 
metal casing which can be rotated on a horizontal axis 
immediately beneath a telescope T. To this metal case 



Graduated 
Scale 



Object 6/ass 
Amid Prisms. 




FIG. 128. 

is attached an arm R, at the end of which is a graticule 
line ; this moves over a scale graduated directly in terms 
of " refractive index." 

The general principle of the use of the standard prism 
is the same as in the case of the Pulfrich refractometer, 
but it should be noted its angle is no longer 90. B in 
Fig. 117 is the standard prism and is usually of "dense 
flint " ; the auxiliary prism A is solely for the purpose 
of leading light at " grazing* emergence " into the liquid 
film, when of course it will fall on the main prism face 
at " grazing incidence." 

Fig. 129 (a) shows the path of "grazing incidence " light 
on the face of the standard prism ; Fig. 129 (b) shows the 



REFRACTOMETKRS 



147 



use of a prism when testing a solid or when using a test 
prism; and Fig. 129 (c) shows the prisms as generally used 
when the liquid being measured is spread out as a thin 
film between the flat glass surfaces. 

Light is admitted into the prism system by means of 
a mirror M (Fig. 128). This may be either light from the 





(b) 



sky or from a lamp ; monochromatic light is not necessary, 
as the colour of the " bounding line," as seen in the tele- 
scope, is annulled by the use of two " direct-vision " prisms 
(known as Ainici prisms), situated in front of the object 
glass, and wliieh can be rotated in opposite directions by 
means of a rack and pinion. Only \\lnn the "bounding 
line " is properly achromatized can readings be taken. 

Theae Amid prisms (see Fig. 130) are so constructed 
that they have no deviation for "D" light, but \\ill 
produce deviation for all 'other colours ; so that two 
such pi i itmly inverted will be achromatic, but 

similarly plaeed \\ill produce approximately double the 
di-persion due to one alone. For details of construction 
a text-book on Geometrical Optics should be consult. I 




148 PRACTICAL OPTICS 

The exact calculation "t tlic dispersion due to two 

such prisms when placed at any 
relative angle is a very awkward 
one, and it is probably better 
to calibrate the instrument ex- 
perimentally. It should be 
remembered, however, that the 
Crown Crown dispersion value furnished by 
this test is only approximate and 
is only meant for rough identification purposes. 

The ln*trtinn at in Use. The two surfaces of the prisms 
betwt < n \\hich the test liquid is to be put should be 
thoroughly cleaned. The instrument is then swung into 
such a position so that the hypotenuse face of the standard 
PHMII is horizontal ; a few drops of the liquid are then 
put on, and the other prism in the other half of the metal 
case swung over and clamped. The telescope of the in- 
strument should then be brought into its most convenient 
position, and on looking in through the eyepiece the mirror 
M (Fig. 128) should be adjusted until good illumination 
is present in the field. The arm R should then be moved 
round until the " bounding line " between the light and 
dark part of the field comes into view. This must then 
be made quite free from any colour fringes by rotating 
tin- Amici prisms by means of the milled head provided 
for that purpose. The cross- wires in the eyepiece should 
be sharply focussed and the bounding line set accurately 
"ii to the intersection of the former. The reading given 
by the graticule index line on the graduated scale will 
give the " refractive index " of the liquid. The scale 
is graduated from 1'3 to 1'7 and is divided to the third 
decimal place of refractive index, the fourth place being 
obtained by estimation. As with the Pulfrich refractometer, 
temperature precautions are of the greatest importance 
with liquids, and therefore the prisms are surrounded 
with a water-jacket to secure constancy in this respect. 
If available, a " thermostat " should be used to ensure 
uniform circulation of the water. 



REFRACTOMETERS 149 

(c) REFRACTOMETER FOR GASES 

The determination of the refractive indices of gases is 
obviously a more delicate operation than that of liquids 
and solids. To obtain the required sensitiveness of the 
instrument a method employing "interference" of two 
beams of light is used. The original principle was from 
Lord Rayleigh, but the instrument described here is a 
modification of this principle. A diagrammatic sketch of 
the optical system is shown in Fig. 131. Light from a 
Miiall electric lamp L illuminates a slit " S," which is in 



PLAN 



TJ 



c 



3 





tin- tneal plane of an achromatic lens O (about Gin. focal 
length) The parallel beam emerging from this objective 
travels on until it reaches a mirror M, which has two 
parallel slits of km>\\n separation in front of it. The light 
then returning along the same path will come to a focus 
again in the plain- of the slit S, where a bright imaire <>t 
the slit will be seen with a number of diffraction bands 
on either side of it. Two gas cells G (of km\\n length) 
side by side are situated in one halt <>t the parallel In -am. 
a- illustrated hrtueen i he object-glass O and the mirror M, 
so that half the IMMHI thrmi L 'h the -elU and the 

<M her half over the top of them. C 1 and C a are two " com- 
pensators," one of Avhieh is adjustable by means of a slow 
motion with micrometer screw. P 8 is a block of glass 
>nuated in the top part of the beam and equal in thick- 



160 



PRA( TICAL OPTICS 



t tin- combined thicknesses of the plates P l and P 2 
and one thickness of a compensator C\ ; this is done in 
order that both halves of the beam shall travel in equal 
amounts of glass. The appearance, therefore, as seen 
\\ith the eyepiece E,* will be that of two sets of diffrac- 
tion bands, one set from the lower half of the beam, the 
light of which will have passed through the gas cells and 
back again ; whilst the other set will be formed by light 
passing over the top of the cells and back again. 

When the instrument is adjusted correctly the central 
bright band of light of the lower set of bands should be 
coincident with the central bright band of the upper 
set. If, now, the gas to be tested is allowed to flow through 
one of the gas cells, and air is still kept in the other, any 




FIG. 132. 

in refractive index will be represented by a re- 
tardation of the beam through the gas cell. This retarda- 
tinn in one of the halves of the lower beam will result in 
a lateral displacement of the lower set of diffraction bands. 
The upper set of bands, of course, will not move at all, and 
thus they serve as a reference mark. By bringing the 
central bright band of the lower set of bands back to its 
original position by tilting the compensator C t with the 
micrometer slow motion, the retardation produced by the 
gas cell may be calculated for (see Fig. 132). 

Let AC be the second principal plane of the object glass 
and also the distance apart of the two slits, DE the focal 
length of the objective ; and let B be the position of the 

* The eyepiece is made up from a piece of cylindrical glass rod about 3 mms. 
in dininrtcr ; this gives a very high magnification, which is necessary in order 
to be able to see the diffraction bands at all. 



REFRACTOMETERS 1 5 1 

central bright band of the lower set of bands, after being 
displaced a distance EB. 

Then for a bright band to be formed at B, the difference 
in path between AB and BC must be an even number of 
half wave-lengths. This difference in path is obtained 
from the following : 

Call AD =b, FB =d, and EB =x, 
then 
and 



Subtracting AB 2 - BC 2 = 

or (AB + BC)(AB - BC) = 4bx : 
but (AB + BC)^= 2d (sufficiently near). 



So that 

a 

From this the actual retardation of the beam passing 
through the gas cell may be obtained ; with this and a 
knowledge of the length of the gas cells, 

2Z (AB-BC) 
"=X + X 
21 
X 
where w = the refractive index, 

l = the length of the gas cells, 
X =the wave-length of light. 

AB and BC the distances referred to in Fig. 132. 

Pressure and drying precautions of both air and gas 
should, of course, be taken. 



CHAPTER X 
APPLICATIONS OF POLARIZED LIGHT 

TIIK theory of the subject of polarization should be 
revised from other text-books, as this chapter deals 
with useful applications of polarized light. 



(a) DETECTION OF STRAIN 

One of the most convenient ways of producing a beam 
of " polarized " light is by reflection. If skylight is re- 
flected from a blackened glass plate so that the reflected 



Extraordinary 
Kay 




Ordinary 
Ray 



'ilm of 
Canada Balsam 



\\\:\. 



beam leaves the plate at the correct angle (viz., 56 J with 

the normal), the light thus reflected will be plane polarized. 

Another very usual method is by employing a " Nicol " 

in. Such a prism is shown in Fig. 133; it consists 

of a rhomb of Iceland spar cut and cemented together 



APPLICATIONS OF POLARIZED LIGHT 153 

along the face AB with Canada balsam. A ray entering 
the face AC will be split up into its two components, the 
" ordinary " and " extraordinary " rays, the former of 
which has a greater refrangibility. Canada balsam having 
a refractive index between that of the ordinary and ex- 
traordinary rays, and the length of the prism ABC being 
suitable, the " ordinary " ray is " totally reflected " at 
the face AB, whilst the extraordinary ray passes almost 
straight on and leaves the prism with its original direction. 
So that there will no longer be two beams coming out of 
the spar with vibrations at right angles to one another, 
but one beam with vibrations only in one direction or 
plane. Thus the Nicol prism is a very suitable means of 
obtaining plane polarized light. 

The double refracting effect produced by Iceland spar 
is also present when ordinary glass is under any stress, 
due either to applied pressure or to bad annealing of the 




Fm. 



glass setting up internal stresses. Such strain becomes 
very evident when a suspected specimen is examined by 
|M.|,ui/<d light. It is this fact that makes the use of 
polarized light of such importance. Suppose two Nicols 
Nj and N 2 (Fig. 134) to be " crossed " so that all light is 
extinguished, and let the specimen to be tested be placed 
in het \\ern the two. If any strain is present it will be 
n -presented by the appearance of patches of light in the 
previously dark field, and colour effects will be seen \\lu-n 
great pressure is present . 

A con vr nil -lit piece of apparatus for detecting sti 
may In- arranged as shown in Fig L3fl \.\n\\\ from the 
sky or from a diffused an \\\<\i\ source strikes a blackened * 
glass reflector B. The reflected beam is then viewed \\ .th 

vamiith. 



UU PRA< IK AL OPTICS 

a Nienl in hont of the eye. On rotating the Nicol and 

l>\ moving the head up or down a position will be 

.(1 when the retleeted beam is almost entirely ex- 

dflhed, The Nienl >hnuld tlu-n be rigidly held in this 

position with a i lamp of some kind. The object to be 

tested is thru placed between the reflector and the Nicol 




as shmvn. when any strain in the specimen will at once 
be detected. If a piece of glass is held in a small vice, 
placed in the beam as before and the vice gradually 
tightened, the effects due to increased pressure will at once 
become obvious. 

Almost any optical work, both mounted and unmounted, 
may be examined for strain in this way. More especially 
is this test essential to ascertain whether object glasses 
are held too tightly by their counter-cells, the over-clamping 
of prisms, and numerous other cases. 

(6) MICROSCOPE POLARIZER 

A simple application of the blackened glass reflector 
is to form a polarizer for the microscope, for use in con- 
nection with petrological work. For observation of rock 
sections, etc., polarized light is greatly advantageous in 
the microscope. In place of the somewhat expensive 
I pri<m usually used as the polarizer, a 3 in. x 1 in. 
cover slip may be blackened with varnish and stuck with 
soft wax to the tilting mirror beneath the microscope 



APPLICATIONS OF POLARIZED LIGHT 155 



stage. On observing through the analyser and rotating 
same the tilt of the slip polarizer can be adjusted until 

40- 




a 16 32 

NUMBER OF PLATES 
Fio. 136. 

the best position of " extinction " is obtained. This 
makes quite an efficient polarizer. The polarization of 



* 



Showing adaption of 
'Pola riser ' to a fixed 
vertical Microscope. 



Glass 311 ff 




Block of wood 






tin- beam, however, may be increased if necessary l>\ 
superposing a second :; in. 1 in -lip on the face of 



156 PH\< TK'AL OPTICS 

tin- first. Fig. 136 shows a graph by Stokes giving the 
relat inn-hip between the percentage of polarized light re- 
flected from a number of plates from 1 to 32, the light 
U-mg incident at the polarizing angle (i.e. 56J). From 
tli is curve it will be seen that practically the maximum 
amount of polarized light which can be obtained by re- 
flection is from eight plates. In practice, however, one, or 
at most two, will be sufficient for most work. Fig. 137 
shows a " slip-polarizer " used with a simple vertical 
microscope, where it is necessary to use an auxiliary mirror 
lying flat on the table in order to get light into the 
instrument. 



C SACCHARIMETERS 

One of the most important applications of polarized 
light at the present day is saccharimetry. 

Certain transparent substances possess the property 
that when plane polarized light is passed through them 
it emerges plane polarized, but in a different plane to 
tli.tt of polarization at incidence. These substances are 
said to rotate the plane of polarization ; such a substance 
is quartz. The effect is also produced by solutions of 
certain substances ; for instance, a solution of sugar in 
water rotates the plane of polarization. The rotation 
which a substance produces is the key to the determina- 
tion of the degree of concentration of that substance in 
-"lution. The instrument for measuring this rotation is 
kimun as a saccharimeter or polarimeter ; they are used 
to a very great extent commercially in testing " sugar " 
solutions. 

It has been determined that the rotation produced is 
proportional to the " mass " of substance in a given volume 
of the solution. 

Now, suppose a mass, " w" of a substance to be con- 
tained in each cubic centimetre of an inactive solvent 
(i.e. one that does not rotate the plane of polarization), 
and let plane polarized light of a definite wave-length 



APPLICATIONS OF POLARIZED LIGHT 157 

(say sodium) traverse a length " / " of the solution. Then 
the rotation R is proportional to " lw." 

So that R=Klw . (i) 

w IK re K is known as the " specific rotation " of the 
substance. K is dependent to some extent on the wave- 
length of light, also temperature and concentration. To 
determine K by experiment, suppose " x " grammes of 
the substance to be dissolved in " y " grammes of the 
solvent, and let the density of the solution be " d." Then, 
the volume of the solution is 

y + x 

d ccs - 

Therefore, the mass of the substance contained in a cubic 
centimetre is : 



d 
xd 



Thus (from (i)) R Klxd 



., . . . (ii) 

So that, from formula (i) it is clear that if for any sub- 
stance the value of K is known, and the rotation R, pro- 
duced by a known length "I" of the solution, can be 
determined, the concentration " w " of the solution may 
be obtained. 

The optical system of a very usual type of saccharimeter 
is shown in Fig. 138. The source of light S is placed at 
the focus of the lens L lf so that a parallel beam of light 
enters the polarizing Nicol N t . Two auxiliary Ni I 
A and B (known as Lippich prisms) are situated im- 
ni'diately behind N r Thus the field as seen with the 
telescope consists of three parts : the central part Nj 
corresponds to light which has passed through the analyser 



L68 



I TACTICAL OPTICS 



milv. \\hile the two outer parts A and B correspond to 
light which has passed in addition through the two 
auxiliary Nicols. It is found that dividing the field into 
three parts in this way facilitates the accuracy with which 
tin* Nicol N 2 can be set. N 2 is the second Nicol which 
together with its mount rotates with the divided circle C, 
from whirl i readings of rotation are taken. The solution 
to be investigated is placed between the polarizer and 
analyser ; it is enclosed in a tube of known length,* at 
the ends of which are plates of optically worked glass. 




Vernier 



FIG. 138. 

These plates are held against the ends of the tube 
with metal caps, so that the plates may be removed 
for cleaning and filling the tube. T is a low-power 
telescope which focusses on the sharp edges E x and E 2 
of the auxiliary Nicols, thus giving a sharp dividing line 
to the three parts of the field as seen through the 

telescope. 

lid " Type. In some forms of saccharimeter the 
angle through which the plane of polarization is rotated 
i- measured by interposing a certain thickness of some 
substance which rotates the plane in the opposite direction, 
and thus neutralizes the rotation produced by the solution 
under investigation, instead of measuring the rotation 
with a divided circle. 

* They are usually 10, 20 or 30 cms. in length. 



APPLICATIONS OF POLARIZED LIGHT 159 

Soliel devised a means involving the use of two quartz 
wedges ABC and DEF (Fig. 139), which by means of a 
rack and pinion are caused to move in opposite direc- 
tions, thus enabling varying thicknesses of quartz to be 
obtained. The wedges have equal angles and are cut 
with the optical axis of the quartz perpendicular to the 
faces BC and DF. 

When the wedges are immediately behind one another, 
a scale mounted above should read zero. The solution 
to be tested is then placed in the instrument, and the 
wedges moved so as either to increase or decrease the 
thickness of quartz until the two halves of the field (i.e. 



K 




ED 



when using a bi-quartz) again appear equally dark. The 
reading from the scale can then again be taken. The 
scale is calibrated beforehand with solutions of kim\\n 
rotation, so that any reading on the scale may at once 
be converted into angular rotation from the graph. The 
optical system of the Soliel saccharimeter is shown in 
Fig. 139. 

N! is a polarizing Nicol ; Q a bi-quartz prism; T the 
tube emit Dining the solution ; Qj is a right-handed qua it/ 
plate cut at right angles to the axis ; ABC and DEF are 
the two wedges of left-handed quartz. N 2 is the k ' ana 1 yser," 
and L a lower power Galilean telescope \\hich can be 
focussed on tin li-juartz Q. 

\\hen th< wedges are set at zero, they with (^ produce 
no effect, and the tint of passage is seen on both sides 
of the bi-quart/. When an "active" -olution is inter- 



LOO PRACTICAL OPTICS 

posed in T, the change in tints is neutralized by bringing 
the wedges into play. N and Q 2 are a Nicol and quartz 
plate to be used if the solution in T is coloured. By their 
means the light emerging from N\ is made to be com- 
plfUH-ntury in colour to the solution, and then the appear- 
ance is as if the solution were colourless. 



APPENDIX 

(a) THE CLEANING OF OPTICAL SURFACES 

THE cleaning of surfaces of optical glass is a subject 
which cannot be too fully emphasized. Not only 
is it of importance in the laboratory, but still more so 
in the optician's assembling or testing room. 

One of the best methods of " thoroughly cleaning " 
an optical surface is to wash it well with soap and hot 
water, using a perfectly clean linen cloth, then rub it well 
with a cloth dipped in alcohol, finally rinsing it in distilled 
water and drying with a piece of " grease-free " chamois 
leather. Great care should be taken not to let the hands 
or finger-tips come into contact with any surface ; it will 
be found advisable to wear a pair of chamois leather gloves 
when cleaning. 

When mounting optical work into instruments it will 
be found advantageous to immerse the glass in a 20 per 
'nt. solution of nitric acid for about two hours before the 
cleaning (as mentioned above) is begun, as this prevents 
to some extent the very objectionable " filming " that 
occurs on the optical surfaces when optical work remains 
in an instrument for some considerable period. In in- 
struments that are finally sealed and made air-tight it 
is advisable to do all mounting in a perfectly dry atmo- 
sphere. All particles of dust should be removed with a 
small camel-hair brush. Such brushes should be con- 
tinually washed out in distilled water to prevent grease 
clinging to the small hairs. 

For surfaces of ordinary glass (i.e. non-optical) a paste, 
made up of " rouge and ammonia," serves extremely \\cll 
for cleaning purposes, and should be applied \\ith apiece 
of chain.. i, I, , it her or a " Selvyt " cloth. 

161 



PRA< TICAL OPTICS 

pith " in Mirk- of "elder" an- very useful for re- 
moving " tarni>h " tn>m surfaces of the denser flint glasses. 



(b) SILVERING OF GLASS 

In >il\vriiiL r . rlcanliiK Is again the all-important factor 
for success. 

KiiM ot all prepare two solutions : 

1. Di <>l\e silver nitrate in distilled water, and add 
ammonia till the precipate first thrown down is 
almost entirely redissolved. Filter the solution, 
and dilute it so that 100 c.cs. contain 1 gramme of 
silver nitrate. 

-. hi--olve 2 grammes of silver nitrate in a little dis- 
tilled water and pour it into a litre of boiling 
illed water. Add 1-6 grammes of Rochelle 
-alt. and boil the mixture for a short time, till the 
precipitate contained in it becomes grey ; filter the 
solution whilst it is still hot. 

The glass should then be " thoroughly " cleaned, with 
tin- same precautions taken as mentioned in the previous 
section, and whilst still wet from the lastly applied dis- 
t illed water, should be placed in a clean glass vessel (e.g. a 
t;i]'i/m<4 dish), with the surface to be silvered placed 
uppermost. 

Kijual (jiiantities of the solutions 1 and 2 should then 
be mixed together and poured into the vessel so as to 
cover the glass, the solutions should be cold. After about 
an hour the silvering will be completed. The liquid can 
t hen be poured off and the glass removed ; any of the silver 
d< -posit can be rubbed off where it is not required, and 
that which is required may be coated with some black 
\ ;i rnish for preserving purposes when the silver has dried. 

(c) GRINDING AND POLISHING A FLAT GLASS SURFACE 

The fact of being able to grind and polish a flat surface 
on a piece of glass is of great importance both for instruc- 



APPENDIX 163 

tional purposes in the laboratory and for commercial 
purposes in the workshop. 

Such a subject is of too large a scope to deal with very 
fully in these pages, as practical experience is the chief 
key to success ; but a general outline of the methods 
employed will no doubt be of use. 

It will be presumed that some sort of machine for re- 
volving the tools is available, either the treadle type of 
" grinder " or the type fitted with a small power unit. 

First of all screw the " roughing tool " to the spindle 
of the machine, and take a little emery (about grade 90 *), 
mix it with water, and use a little at a time on the tool. 
Hold the piece of glass in the fingers of both hands firmly, 
and revolving the " rougher," press the glass down on 
the tool, giving it a backward and forward motion. In 
due time all the prominent irregularities of the glass surface 
will be removed and a smooth ground surface will be left. 
Another tool for finer grinding is now used. This tool 
should have already been made a fairly correct Hat surface, 
and therefore may be used for the more exact work. 
" Fine grinding " can then be done by using 10-minute,* 
15-minute, 20-minute, and 60-minute emery in succession 
in the same way, the grinding being continued with each 
grade until all bits and scratches left from the coarser 
grades are removed. The surface must be continually 
viewed with a fairly high-power eyepiece in order to detect 
such scratches. 

As each grade of emery is used care must be taken to 
remove any particles of a previous grade ; this is best done 
with a small soft sponge, and by rubbing a rough piece of 
flat glass known as a " bruiser " on the tool prior to using 
the actual glass surface on the tool. 

When the surface has been successfully brought to the 
finest condition, the polisher may then be prepared. 

* 111' numbered grades of emery, such as 90, refer to the number of meshes 
INT in> h of a sieve through which ul.tr emery has passed. The 

10, 20, 60, etc., minute emery rrf- r t<> tin; particles that an 1. ft m -us|., 
in irafea Mowed to stand for the respective number of 

minutes. 



n; I PRACTICAL OPTICS 

Preparing the Polisher. The polisher is made up of a 
r of pitch melted on to the surface of one of the iron 
tools. The pitch, which may be softened by the addition 
of tallow or lard, is freed from grit by straining it through 
a piece of fine muslin on to the tool while the pitch is 
molten and hot. The tool is heated sufficiently to keep 
the pitch plastic, and its surface is then flattened Im- 
pressing the pitch down on a cold iron plate. Before the 
pitch is quite hard a number of grooves may be cut in it, 
in order to give places which will accumulate the polishing 
medium. 

The polishing can then be commenced. The " polisher " 
should be screwed to the spindle of the machine, warmed 
slightly, and moistened with a little " rouge and water." 
The glass surface should be rubbed over the " polisher " 
as during the grinding process, but in this case the speed 
of relative movement between glass surface and polisher 
should be very much slower. 

After about twenty minutes' polishing the glass surface 
will be ready for the " test plate/' This is placed on the 
uice and the interference fringes viewed by reflected 
light from a Mercury Vapour Lamp ; from this is ascer- 
tained the relative roundness of the surface and its form, 
u hcther convex or concave. 

If the surface is convex, the best procedure to try and 
correct this tendency is to increase the " stroke " and 
to press harder on the polisher. The result will be to 
increase the wearing of the surface in the centre and thus 
give a tendency towards concavity. If the surface is con- 
cave, however, the stroke should be shortened and some 
of the pressure on the tool relaxed. There are various 
ways of varying the relative amounts of wear in different 
regions of the surface, such as cutting grooves in certain 
parts of the polisher to alter the glass surface in the same 
part ; but experience is the only master which can teach 
all the devices used in practice for the correction of surfaces 
in such a manner. 

The period necessary to complete the polishing will, 



APPENDIX 165 

of course, depend on the time taken entirely to remove 
all trace of " grey " from the surface and to produce the 
best flat possible. 

(d) BALSAMING 

When balsaming it is of first importance that the surfaces 
to be put in contact are absolutely clean and " dust-free." 
The surfaces should be cleaned as mentioned in section (a) 
and carefully dusted with a soft camel-hair brush. All 
44 balsam " should be carefully filtered before use. 

The two optical parts which are to be cemented together 
should first be slightly warmed in the balsaming oven. 
A very suitable oven for this purpose is the small (9 in. 
cube) copper oven supplied by Messrs Baird & Tatlock 
of Hatton Garden, and is fitted with gas heating. Failing 
this, an ordinary biscuit tin may be converted into an 
oven, the heating being provided by a carbon filament 
electric lamp in the circuit of which is arranged a vari- 
able resistance. The lamp should be placed inside the 
tin, and means for fitting a thermometer and adjustable 
air regulation provided in the lid. Such a device works 
extremely well. 

A small amount of balsam should then be placed in 
the centre of one of the surfaces which is to be balsamed, 
and the other surface pressed carefully but firmly (with a 
piece of cork) on to the first until the balsam spreads out 
as a thin film over the entire surface. Any small bubbles 
should be removed by pressure with the cork. The parts 
being balsamed should then be placed on a glass plate 
covered with paper and supports placed at the sides to 
prevent any sliding movement of one surface relative to 
the other. The parts are then put into the oven and the 
temperature slowly raised until it reaches 77 C., where 
it -houN I..- k-pt for four Imurs, and then slowly redu< I 
until the temperature of the room is again attained. The 
parts can then be removed I mm tin- oven and all super- 
fluous balsam cleaned off with benzol. The operation is 
thru complete. 



1,,,, PRA< TICAL OPTICS 

There are various grades of Canada balsam, known as 

td "soft" balsam, but all except the "very 

soft" should be taken to 77 C. The "very soft" will 

be sufficiently mobile to be put on without any heat and 



3-jaw adjustable 
mount similar to 
that shown in 
i. 86. M 



Light from 
lamp or 



'Groove for 
mount to slide 
in 



Mirror 
i i 140. 

will set when left exposed to the atmosphere for an hour 
or two. 

When achromatic lenses are being balsamed it is 
necessary to " centre " the two lenses while the balsam 
is still " plastic." For this purpose a piece of apparatus 
similar to that shown in Fig. 140 will be found of great 
convenience. It consists of a cross-line object 0, an ad- 
justable mount M for the lens L, and a telescope T, all 



APPENDIX 167 

mounted on the same rigid base and supported in a vertical 
position.* The lens is rested in the recessed mount M, 
which is adjusted so that O is in the focal plane of the lens. 
Observing through the telescope the lens is then rotated, 
when any centring defect will be shown up by movement 
of the image. The lens and lens mount can then be 
heated while in this position until the balsam becomes 
sufficiently plastic to move one lens relative to the other, 
when the test can again be repeated until the centring is 
correct. 



(e) DEVELOPERS FOR PHOTOGRAPHIC WORK 
Hydroquinone Developer 
FOR PLATES 

Solutions A and B to be mixed in equal quantities 
when required for use. They should be kept in separate 
bottles. 

Solution A . 

Hydroquinone ..... 25 gms. 

Potassium Metabisulphih . . 25 gms. 

Potassium Bromide . . . . 12 gms. 

Water . . . 1000 c.cs. 

Solution B. 

Potassium Hydrate .... 150 gms. 
Wai. , . . . 1000 c.cs. 

Fixing />'///> 

Hypo . . . 150 gms. 

Water . . . 1000 c.cs. 

f cross-line and positive component <>f tho achromatic lens is 
first - r with the axis of the telescope, before the negative or flint 

component is put on to the first. 



his PRACTICAL OPTICS 

Pyro Developer 

FOR PLATES 

Solutions A and B to be mixed in equal quantities 
when required for use. They should be kept in separate 
bottles. 

Solution A. 

I'vn.LMllir Acid . . . . 10 gms. 

Potassium Metabisulphite ... 2-4 gms. 

Water 1000 c.cs. 

Solution B. 

Sodium Carbonate . . . .100 gms. 
Sodium Sulphite ... 100 gms. 

Potassium Bromide ... 12 gms. 

Water 1000 c.cs. 

Developer 

FOR GASLIGHT PAPER 
Sodium Carbonate . . . .170 gms. 

Sodium Sulphite . . . . 30 gms. 

II vdroquinone ..... 8 gms. 

Mrtol ...... 2-5 gms. 

Potassium Bromide . . . . 1 gm. 

Water 1000 c.cs. 



(i) A FROSTING SOLUTION FOR GLASS 

Such a solution is very convenient for frosting electric 
lamp bulbs, instead of using tissue paper over a bulb, a 
much practised method in opticians' workshops. 

Dissolve : 

25 grammes of (leaf) gelatine and 120 grammes of either 
;il< ium carbonate or magnesium oxide in 250 c.cs. of hot 
distilled water. 

Let the solution cool to 34 C. and dip the glass into 



APPENDIX 



169 



it. Allow to dry and then immerse the glass a second 
time. 

Two coats will in general be enough, but more may be 
Driven if required. 

(ii) A CEMENT FOR OPTICAL PURPOSES 

For cementing glass cells, glass windows to metal cells, 
etc., etc., one of the best cements will be found by mixing 
equal quantities of " beeswax " and " rosin " (whilst molten), 
and on cooling make it into thin " sticks." It should be 
applied with a small heated rod, and then, placing all 
parts to be cemented into a hot-air oven, should be left 
until the cement becomes " plastic." At this stage the re- 
<|iiired surfaces should be put in contact, and then allowed 
to cool. 

This cement will resist the action of aqueous solutions 
and organic solvents for a very considerable time. 



(/) TABLE OF USEFUL WAVE-LENGTHS 



Snl stance. 


How emitted. 


Wave-length 
in 10 -8 cms. 


Colour. 


Sodium 


Bunsen Flame 


5890-2 


Orange 








5896-2 





Lithium 


On pole of "Arc" 


r.Tns.L' 


Red 


Rubidium 


if 


7947-0 


Far red 








7806-1 


it 


Hydrogen 


Vacuum Tube 


li.Vi.'Mi 


Red 





ii 


4861-5 


Blue-green 




ii 


4340-7 


Violet 


Mercury 


Mercury Lamp 


5790-7 


Yellow 





it 


5769-6 


ii 


' 




5460-7 


Green 


ii 


ii 


4078-1 


Violet 


Cadmium 


Vacuum Tul>< 


6438-5 


Red 


ii 




5085-8 


Green 





> 


!7'.9-9 


Wur 


Strontium 


Bunsen Flame 


4607-5 


Blue 



PRACTICAL OPTICS 
REFRACTIVE INDICES FOR SODIUM LIGHT (A=589 





Refractive Index. 


Fluorspar 


1-4339 


Quartz 


1-5442 ordinary 





1-5533 extraordinary 


Rocksalt 


1-5443 


Water 


1-3329 


Carbon Bisulphide 


1-6277 


Benzene 


1-5004 


Iceland Spar 


1-6584 ordinary 





1-4864 extraordinary 



TABLES 



LOGARITHMS 













Mean DilTctvi 


ires. 
















1 2 3 


456 


7 8 9 


























10 


0000 


0043 


0086 












0874 


4 8 12 


17 21 2.1 


29 33 37 


11 


0414 


...V, 








0607 


0648 


<n;*2 n7l;i .17.1.1 


4 8 11 


M 111 2." 


j.; :;n :u 


12 


0793 


'*2* 










1004 


IOM M72 lino 


3 7 10 


It 17 21 




13 


1139 


1 1 73 




ISM 


1271 


L808 


1MB 


i:;o7 i",:i'.i 1 i:;n 


3 6 10 


13 16 19 


2", 20 29 


14 


1461 










1614 


1'iil 


]i;7:; I7n:i 17:12 


.", (i li 


12 15 18 


21 21 27 


15 


1761 


1790 




L841 


1*71 


L808 


L981 


I'.'.v.' 1987 


9014 


:; o s 


11 11 17 


._.,, -,._, ._,- 


16 


3041 


:,> 


MM 


2122 


8148 


8178 


2 2n I 


2227 22-1." 


2279 


358 


11 13 10 


Is 21 21 


17 


n 'i 


::.:;- 


MM 


8880 


8408 




9458 


1 2.121. 


2 .1 7 


in 12 1.1 


17 2i 22 


18 


2553 




2601 




- 


M79 


20 -.1.1 


27 IS 2712 270.1 


257 


9 12 14 


n; in 21 


19 


2788 


2810 


8888 







2900 


2:i2:i 


211 1.1 -J'107 2'.IS1 


2 7 


9 11 13 


16 18 20 


20 


3010 


3032 






3096 


3118 


3139 


SlOn 3181 


3201 


2 6 


8 11 13 


1.1 17 19 


21 




3243 


M68 




3304 


889 i 


8846 


3365 


8886 


8404 


2 6 


8 W 12 


1 1 10 IS 


22 




3444 




- 


8809 


.".122 


8841 


3560 


M79 


3598 


2 6 


8 10 12 


14 15 17 


23 




3636 




M74 


3092 


",711 


3729 


3747 


8766 


37SI 


2 6 


7 9 11 


l.'i M 17 


24 


3802 


3820 




- 


J874 




:{'.iu'. 


3927 


3945 


3962 


2 5 


7 9 11 


12 14 16 


25 


-.-:, 


3997 


!! 1 


1081 


4048 




1089 


4099 


4116 


4133 


235 


7 9 10 


12 14 15 


28 


4150 


4166 




1800 


4 2 Hi 


1989 


1249 


4205 


198] 


4298 


235 


7 8 10 


11 13 15 


27 


4314 


4330 




1869 


4378 


4393 


1409 


4496 


(440 


1 I.K; 


235 


689 


11 13 14 


28 


1471 


4487 




4518 




1648 


1864 


4579 


i.v.M 


4609 


235 


689 


11 12 11 


29 






16*4 


ir.r.'.i 


4683 




4728 


4742 


4757 


134 


679 


in 12 i:; 


30 


1771 


17--; 


4800 


IMi 




1848 1857 


4871 


4886 


4900 


134 


6 7 U 


10 11 13 




. 


























31 


4914 


4928 


1941 






4983 


1997 


6011 


5024 


5038 


134 


678 


10 11 12 


32 




MM 


5079 


5099 


5105 


5119 


5189 


6145 


5159 


5172 


134 


578 


9 11 12 


33 




6198 


.1211 


.122 1 




5250 


5263 


59 76 


5289 


5302 


134 


568 


9 10 12 


34 


5315 










5378 


5391 


5408 


5416 


5428 


134 


508 


10 11 


35 


.'-111 






- 


MM 


5502 


.1.11 1 


,1.127 .1.i:Hl 


5551 


124 


.1 7 


9 10 11 


38 


1.1.;:; 






5599 


Mil 


M98 


-,635 


r,oi7 5668 5670 


] 2 1 


5 7 


8 10 11 


37 


1'>2 






5717 


172:1 


5740 


5769 


:.703 1775 


5786 


1 2 :; 


.1 7 


s 11 in 


HS 


5798 


5809 


MS] 


- 






.ISM; 


-<8 5899 


1 2 3 


507 


s 9 10 


3f 


5911 










59M 


5977 




l 2 :$ 


1 .1 7 


S !l 111 


40 








0...13 


1064 


6075 


ins;, 


iniifi ;in7 HI 17 


l 2 :; 


450 


8 9 10 


41 


6128 




8149 


6160 


H70 




0191 


!2'il .1212 0222 


l 2 :; 


1 .1 o 


789 


42 








021; :t 


1271 




i'.--".i 1 


;:;n i;:;i i i;:i2.i 


123 


l .1 i; 


7 s : 


43 


' .:.:, 






o:n;.i 


(876 




M96 


i|i i.l 11.1 012.1 


1 2 3 


1 .1 


789 


44 




.III 




<; n;i 


; i: i 


8484 


8498 


1508 6518 0.122 


1 2 .", 


l .1 r, 


789 


45 




IM9 


M61 


6561 


m 




6590 


(099 0609 


0018 


123 


l :, i; 


789 


46 


...... 








1665 




.;.,->! 




6702 


0712 


123 


1 .1 i; 


7 7 .s 


47 








r,7lti 




8767 


6776 


-7S.1 07SM 


6808 


123 


455 


7 8 


48 














.is.;.; 


is 7.1 (iSS| 


OK 93 


123 


1 1 .1 


078 


49 




mi 




092* 






6955 


;:'!. I 8979 


0981 


1 2 .'! 


445 


ii 7 s 


50 


, ,;, , 






7016 






7049 


7050 


7n.V.i 


7007 


] 2 :; 


345 


r, 7 s 






























51 


7'.;o 




7093 


7101 


nio 


7118 


7120 


71:1.1 7li:i 


71.12 


123 


345 


078 


52 


710-. 




7177 




71;.:; 


72D2 


72 in 


721.* 7220 723.1 


122 


3 4 5 


7 7 


53 








. 






7292 


~:;nn 731 >s 7:110 


122 


:i l .1 


7 


54 


....I 




7340 


7848 




7864 


7372 




1 2 2 


3 4 5 


r, r, 7 



LOGARITHMS 



173 



55 

56 
57 
58 
59 
60 

61 
62 
63 
64 
65 

66 
67 
68 
69 
70 

71 
72 
73 
74 
75 

76 
77 
78 
79 

80 

81 

8J 
83 
84 
85 

86 
87 
88 

8!) 

90 
91 

98 

0< 





1 


2 

7419 

7497 
7574 
7649 

7796 

7868 
7938 
8007 
8075 
8142 

8209 
1374 

8338 
8401 
8463 

8525 
8585 
8645 
8704 
8762 

8820 
8876 
8932 
8987 
9043 

9096 
9149 
M01 

9253 
9304 


3 


4 

7435 

7513 
7589 
7664 
7738 
7810 

7882 
IBM 

8021 
8089 
8156 

8222 
8287 
8351 
8414 
8476 

8537 
8597 
8657 
8716 
8774 

8831 
8887 
8943 
8998 
9053 

9159 

:'_ 1 _ 
MM 
9315 

MM 

MM 
Mil 
MM 


5 

7li:. 

7745 

7818 

7889 
7959 
8028 
8096 

8162 

8357 

8482 

8543 
8603 
8663 

8779 

8837 
8893 
8949 
9004 
MM 

9112 
;'!,:, 
9217 
MM 
MSO 

M70 
MM 
MM 
MU 
MM 

9614 
Mtl 

MOO 

M4I 

MM 
Ml 
M7I 


6 

7451 

7528 
7604 
7679 

77.VJ 
7825 

7896 
7966 
8035 
8102 
8169 

8235 
8299 
MM 

8426 
8488 

U48 

8609 
8669 
8727 
8785 

8842 

SSJ.JI 
SU.-.l 

1 '.' 
;:{ 

9117 
9170 
9222 
9274 
9325 

.:;;:. 
.'.'.- 

.'171 

;..vj:t 

'.'.-.71 

9619 

;i;i;r, 

i7.vj 

;,,.-, 

I.V..I 
.vt 
:.'..:; :i 

r.'v; 


7 


8 

7:,):; 
7619 

7.:;' 1 
7 7. .7 

71M" 
TWO 
M4I 

sin; 
8181 

>-J|s 

s:;r_- 
BSTfl 

-::. 
BMO 

SMi 

BM1 
8681 
B7M 
B7t7 

8854 
M10 
8965 
MM 
9074 

9S85 
94S5 

9581 

9628 
9675 
9722 
9768 

9899 
9903 

9991 


9 


Diflerenon. 














71"! 

74S3 

7. :.. 

. 

- 
- 

7993 
8062 
8129 

- 

8388 
MM 

- 

8633 
8692 
8751 

8808 
MM 

8976 
9031 

9085 
9138 

IMIU 

'.'.".' 1 

MM 


74U 

7566 

7 ,!_ 
771G 

7860 

8000 
8069 
8136 

8331 
8395 
MM 

Ifll 

8639 
8698 

-M I 
-71 

MSI 
MM 

MM 

.-1 c; 

M4I 


71.7 

T.'.s-j 
7.;:. 7 
7731 
7803 

7-7--. 
7945 
8014 
8082 
8149 

S^'15 
8280 
-:;il 
MO? 
8470 

8531 
8591 
8651 

8768 

8882 
8938 
8993 
9047 

9101 
9154 
9206 
9258 
9309 

9360 

1'." 


MM 

7536 
761'2 
IfU 

77f." 

ran 
rwM 

7;- 7:; 
8041 

not 

8176 

U 1 1 
MM 

8370 
843'J 
8494 

8555 
8615 
8675 
B7M 
B7f] 

8848 
M04 
MM 
1018 
MM 

9122 
9175 

'.."-7 
9279 
9330 

9380 
9430 
9479 
9528 
9576 

MM 

'.'.;71 
9717 
..;..:. 
MM 

MM 
MM 
M4I 

9987 


7171 

7551 
76S7 
7701 

7771 
7846 

7917 
7987 
8055 
Blf] 

8189 

3264 

8319 
8382 
8445 
8506 

8567 
8627 
8686 
8745 
8802 

8859 
8915 
8971 
9025 
9079 

- 
9238 
9289 

i:;in 

9390 
9440 

'.Ms'J 

i.VIS 

y.iso 

963S 

M;X,I 
9727 

9818 

-,,;i 
..,., n 
..-.. :.j 
r.....j 






:> ; 7 

567 
567 
567 
.567 
566 


l -J -.- 
1 1 2 
112 
1 1 2 


345 
345 
344 
344 
344 


112 
1 1 2 
112 
1 1 2 






















1 2 
1 2 
1 2 
1 2 

1 2 
1 2 
1 2 
1 2 
1 2 

1 L' 

1 2 
1 2 


334 
334 
234 


556 

1 :, i; 
4 5 C 






234 
234 


j :, :, 
l :, :, 


233 


i :, :, 








445 








1 2 










1 . 
1 _' 


























1 1 
1 1 
1 1 

-1 1 1 


















<) 1 1 
Oil 


223 
I | | 


344 

a i i 


' 1 1 
" 1 1 
t 1 1 


238 


344 


338 


3 4 4 









171 



AMMLOGARITHMS 











.Mean DilVei-eiu-e-. 


00 1000 


1002 


1005 1007 


1009 




1014 


1016 


1019 










1021 


001 


111 


2 2 2 


01 1023 


1026 




1033 




1038 


l"l" 1"I2 


1045 


001 


111 


2 2 2 


02 1047 


1050 




1057 




1".H H..17 


L069 


001 


111 


2 2 2 


-08 1072 




1076 1079 


1081 




L089 L09] 


L094 


001 


111 


2 2 2 


04 1096 




11"2 IK.'I 


1107 


li"n 1112 


1111 


1117 


L119 


Oil 


112 


222 


-06 1122 




1127 11."" 


1132 


1135 




111" 1113 


1146 


Oil 


1 1 2 


2 2 2 


08 1148 


1151 


1153 1156 


1159 


nc.i in;i 


1H17 lit;'.' 1172 


Oil 


1 1 2 


2 2 2 


07 1175 


- 




1186 


n-.' n '.'1 


ll'.'l 11-.I7 11U9 


Oil 


112 


222 


08 1202 






1213 


1210 


1219 


1-22 1226 1227 


Oil 


112 


2 2 :; 


09 1230 




1236 1239 


1212 


1246 


1217 


12:," 12.-.:; ii'.-n; 


Oil 


112 


223 


10 125'.. 


:_ . 




1271 


1271 


1276 


1271. 1281 L286 


Oil 


112 


223 


11 12-- 






1300 




L809 1:112 


1315 


Oil 


122 


223 


12 1318 


1321 




1330 


i:;:;i i:;:;7 


i:il" 1343 134G 


Oil 


122 


223 


18 1349 




- 


1361 


l. ;.;:, L86S 


1371 


i:i 71 i:i77 


Oil 


122 


233 


14 1380 




1881 1 ''.'" 


1393 


l :.'."'. 1 l"" 


1403 


1 lor, 11"'.) 


Oil 


122 


233 


16 1413 


in.; 


M1H 1122 


1426 


1I2H Ii:i2 


1435 


1439 1442 


Oil 


122 


233 


16 1445 




11. V2 ll.V. 


1459 


11(12 lltltl 


1469 


1 172 


1-17(1 


1 


122 


233 


17 11 71* 




1486 1489 


1 in:; 


1496 1500 


1503 


15ti7 151" 


1 


122 


233 


18 1514 


1811 


1 52 1 1 52 1 


1528 


1531 1535 


1538 


1512 1515 


1 


122 


233 


19 1549 


1882 




15G3 


L687 


1570 


1574 


1578 1581 


1 


122 


333 


20 1585 


1888 




1600 


n ;..,:; io"7 


1611 


n; n IGIS 


1 


122 


333 


21 1622 






1637 


Hill Hill 


1648 


L662 


1656 


Oil 


222 


333 


22 1660 


L66J 




1675 


1679 1GK3 


L687 


1690 


1694 


Oil 


222 


333 


23 1698 




1706 1710 


1714 


1718 


1722 


172G 


1730 


1734 


Oil 


222 


334 


24 1738 


17.2 


1746 1750 


1 7:. 1 




1762 


17GG 


1770 


1774 


Oil 


222 


334 


28 1778 




1786 1791 


1795 


1 7n it 


1803 


1807 


1811 


1816 


Oil 


222 


334 


28 1820 






1837 


L84J 


1845 


1849 


1854 


1858 


Oil 


223 


334 


27 1862 




1871 1875 


1879 


1884 


1888 


1892 


1897 


1901 


Oil 


223 


334 


28 1905 


L910 


inn nun 


L982 


1928 


1932 


1936 


1941 


1945 


1 


223 


344 


29 1950 








1977 


1982 


1986 


1991 


1 


223 


344 


80 1995 






2014 


2018 


2023 


2028 


2032 


2037 


1 


223 


344 




2041 





2061 


2065 


2<>7ii 


2075 


2080 


2084 


1 


223 


244 


82 2089 


2094 




2109 


2113 


2118 


2123 


2128 


2133 


1 


223 


344 


88 2138 






2158 


2163 


2168 


2173 


2178 


2183 


1 


223 


344 


84 2188 






2208 




2218 


2223 


2228 


2234 


112 


233 


445 




2844 




2259 


2265 


2270 


2275 


2280 


2286 


112 


233 


445 










2:1 17 -j:;2:; 


2328 


2333 


2339 


112 


233 


445 


37 ...II 






2366 2371 2:1 7 7 


2889 


2388 


2393 


112 


233 


445 


88 2399 




211" I'll:, 


2421 


2127 


2482 


2438 


2443 


2449 


112 


233 


445 


89 2455 


Ml 








2 Ilt5 


2600 


2506 


1 1 2 


233 


455 










2 Ml 2517 


2553 


2559 


2564 


112 


234 


455 








2694 




2612 


2618 


2624 


112 


234 


455 


l.'i 






MM 

2 7 1 u 


2M-.I 2MI7 
_'7L':i 272H 


2871 

2735 


2679 
2742 


2685 
2748 


112 

1 1 2 


234 


4 5 G 


44 2751 






2780 






- i l - 

2806 


2812 


L X z 

112 


334 


456 


45 -'-I- 






2844 


28 


2-5- 


2884 


2.S71 


2877 


112 


334 


5 5 G 


48 2884 


2891 


2881 2804 




21U7 2H21 


2931 


2988 


2944 


112 


334 


556 




2968 






2992 


21)1111 


3006 


3013 


112 


334 


B B ; 


48 3020 


3027 


2084 204] 




8066 


3062 


;";:. 


3076 


:;">:; 


112 


344 


566 


49 3090 


. 181 




1119 


312(1 




nil :;ns 


3155 


112 


344 


5 G 6 























\\PILOGARITHMS 



175 








123 












Differences. 




















50 31C-J 

53 MM 

-54 3167 

56 3631 
58 MOJ 

61 "7 
62 41y 
63 Jl'GG 
64 65 
85 no; 

66 : ''l 

68 7SO 
69 -'"> 
70 ''"!- 

71 .'.l-J'.t 
'-.' 

si '-i"'; 

.go M01 

83 '"''-I 

86 72441 

*88 ' * " ' 1 
'89 77'>'-'l 


:;i7" 

3243 
3319 

:;:;...; 

3556 

3811 
3899 
MM 

urn 

UTS 

r_'7; 
4477 

tin 

1688 

4797 

5768 

MM 

6180 
6324 

6471 


.". 1 7 7 

un 

3327 
3404 
3483 
3565 

3648 
3733 
3819 
3908 
3999 

4093 
4188 
4285 
4385 
L487 

1091 
MM 
4808 
IMQ 
MM 




3192 

3266 
3342 
3420 

:;i n'j 
3581 

3664 
3750 
3837 
3926 
4018 

4111 

MM 
1KM 

4721 
4831 

I'.M:; 
5058 

ii?< 

IM1 
MM 

'.:. ic 
'it; 7."p 

5808 
5943 
6081 
6223 
6368 

6516 

,,;,> 
>_:; 
6982 
7146 

7311 
7482 
KM 

v.i 

."l 


3199 
3273 
MM 
3428 
IMC 
MM 
1671 
3758 
3846 
3936 
4027 

US1 
1217 

iin; 

4624 

4842 
MM 

5070 

5188 

not 

MM 
MM 
MM 

5821 
5957 
ION 
6237 
IM1 

6531 
MM 
1881 
IMI 

71-.1 

rest 

7499 
-'.71 
7862 
MM 
Ml 
8414 


3206 

3281 
3357 
lift 
UK 

:;:... 7 

M81 

B767 
UN 

4036 
4130 

4529 

4634 
4742 
4853 
L964 

5082 

5200 

5445 
5572 
5702 

>r.l 
5970 

;],,., 

;_:._ 
6397 
; ;,|,; 
;.;.... 
>:,:, 
7015 
7178 
;:;,., 
7516 

;<,'.n 
-.;. 

- .,! 

v: 1 1 
si:;:i 

V.I 


::-JMI 
:;:;.;:, 
:;u:; 
3524 
3606 

3690 
3776 
3864 

:;:.-, I 
4046 

4140 
4236 
4335 
4436 
4539 

4645 
4753 
4864 
4977 
5093 

5212 
5333 
5458 
5585 
5715 

U4I 

MM 

fill l 
MM 

; 1 1 - 

IM1 

1711 
6871 
7031 
7194 

7362 
7634 
7709 
7889 
8072 

MO 
IN 
MO 

-.-.I 
Oil 

Ml 

i*i 


3221 

:;-.".M 
:;:;7: 
3451 
3532 
1614 

MM 

B784 

3873 
3963 
IOM 

I 1 :,. ' 

LS4J 

IMC 

AMO 

MM 

J7.U 

^7;. 
ISM 
5105 

5470 
5598 
5728 

5861 
6998 
6138 
M8I 

'.I-J 7 

6577 
6730 
6887 
7047 
7211 

7379 
7551 
7727 
7907 
8091 

8279 
8472 
8670 
8878 
9078 

9290 
MM 
9727 
MM 


MM 

MO 






567 

6 7 
1 7 

6 7 
6 7 


3258 
3334 
:: 1 1 1 
MM 
U7I 

MM 

:; 7 1 1 

3917 
4009 

4102 
4198 

4498 

4603 
4710 
4819 

:.MI 7 

.V.>1 


1 2 2 


345 


MM 

3540 
3622 

3707 
3793 
3882 
3972 
4064 

4159 
4256 
4355 
4457 
4560 

1667 
1771 

4887 
5000 
5117 

5236 
5358 
5483 
5610 
5741 

5875 
6012 
6152 
6295 

;n-j 

,:.'._- 
;;i:. 
;...<_ 
7063 
7228 
-:,.M; 
-:> 
;i:. 
.'::. 
8110 

^yj 
-r.: 

,;-., 
-:.: 
,-.,-., 

:.ll 
:.-- 






1 2 2 


345 


1 2 3 


345 


7 8 


















7 8 










7 8 
7 8 
7 8 






1 2 3 


456 


123 
I 2 3 


456 
457 


7 9 10 
8 9 10 


1 -J :: 
124 

124 
124 
134 
134 
1 3 4 

134 
1 3 4 
134 
1 3 4 


5 7 

~ 

5 7 
5 7 
5 8 
5 8 
5 8 

578 
578 
678 
679 


8 9 10 
8 9 11 

8 10 11 
9 10 11 
9 10 11 
9 10 12 
9 10 12 

9 11 12 
10 11 12 

10 11 1." 
10 11 13 

10 ]_ i:; 

11 12 14 
11 12 14 
11 13 14 
11 13 15 
12 13 16 

12 13 15 
12 14 16 
12 14 16 
13 14 16 
13 16 17 

13 15 17 
14 15 17 
i LI LI 
14 16 18 
16 17 19 

16 17 19 
U LI N 
16 18 20 
16 18 20 


5781 
Mil 
MM 
SIM 
MM 
MM 


5794 

:.._".' 
.;..,;: 
;_.... 
.;:;:,:; 

.;:.., 1 
; ,:. 


235 


689 






235 

- :; 1 

236 
| :, 
246 
246 
246 

1 '. 
4 6 
4 6 
4 6 
1 I 

4 r, 
4 7 
4 7 

:. 7 


6 8 10 
7 8 10 

7 8 10 
7 9 10 
7 9 11 
7 9 11 
7 9 11 

8 9 11 
8 10 12 
8 10 12 
8 10 12 
8 10 12 

8 11 13 
9 11 13 
9 11 13 
9 11 14 



176 



NATURAL SINES 











Mean l>irtViviuv>. 




0' 


6 


12' 


18' 


24 30' 36' 


42' 


48' 


54' 



























r 2' 3' 


4' 5' 





0000 


0017 




0052 


0070 


0087 


0105 


0122 


0140 


0157 


3 f. y 


l-j ir, 




0175 


0192 




0887 


"j|| 


0262 


0279 


< i-_".. 7 


0314 


0332 


3 (1 '.i 


12 I.'. 




0141 




0401 


0419 


0436 


0454 


0471 


0488 


0506 


369 


1 _ 1 :. 




0523 


0541 


MM 


0576 


0593 


0610 


0628 


0645 


0663 


0680 


3 6 9 12 Ifi 




MM 


0715 




0750 


0767 


0785 


0801 


0819 


0837 


0854 


369 


12 11 




0872 


0889 




Q994 


0941 


0958 


0976 


0993 


1011 


1028 


369 


I-.' 11 






1063 


1080 


1097 


1115 


1132 


1119 


1167 


1184 


1201 


369 


12 1! 




l.'l'.t 


1236 


1851 


1-J71 


1288 


1305 


1323 


1340 


1357 


1*74 


369 


1 _' 11 




1392 


L409 


1426 


1444 


1461 


1478 


1495 


1513 


1530 


1547 


369 


12 ] 1 




1564 


1582 


1599 


1616 


1633 


1650 


1668 


1685 


1702 


1719 


369 


12 11 


10 


1736 


1754 


1771 


1788 


1805 


1822 


1840 


1857 


1874 


1891 


369 


11 14 


11 


1908 


1925 


1942 


1959 


1977 


1994 


2011 


3088 


2045 


2062 


369 


11 14 


12 


2079 


2096 


2113 


2130 


2147 


2164 


2181 


2198 


2215 


2233 


369 


11 14 


13 


SSM 


2267 


2284 


2300 


2317 


2334 


2351 


2368 


2385 


2402 


368 


11 11 


14 


2419 


2436 


2453' 


2470 


2487 


2504 


2521 


2538 


2554 


2571 


368 


11 11 


15 


2588 


2605 


2622 


2639 


2656 


2672 


2689 


2706 


2723 


2740 


368 


11 14 


16 


2756 


2773 


2790 


2807 


1888 


JS10 


2857 


2S7I 


2890 


2907 


368 


11 14 


17 


2924 


2940 


2957 


2974 


2990 


3007 


3024 


3040 


3057 


3074 


368 


11 14 


18 


3090 


3107 


3123 


3140 


3156 


3173 


3190 


3206 


3223 


3239 


368 


It 11 


19 


3256 


3272 


3289 


3305 


3322 


3338 


3355 


3371 


3387 


3404 


358 


11 14 


20 


3420 


3437 


3453 


3469 


3486 


3502 


3518 


3535 


3551 


3567 


358 


11 11 


21 


3584 


3600 


3616 


3633 


3649 


3665 


3681 


3697 


3714 


3730 


358 


11 14 


22 


3746 


3762 


3778 


3795 


:isn 


3827 


3843 


3S59 


3875 


3891 


358 


11 1 1 


28 ; '3907 


3923 


3939 


3955 


3971 


3987 


4003 


4019 


4035 


4051 


358 


11 14 


24 


4067 


ins:; 


4099 


4115 


4131 


4147 


4163 


4179 


4195 


4210 


358 


11 13 


25 


i-j.i; 


4242 


4258 


4274 


ll'S'.l 


4305 


4321 


4337 


4352 


4368 


358 


11 13 


26 


4384 


ISM 


4415 


4431 


1116 


4462 


4478 


4493 


4509 


4524 


358 


10 13 


27 


r.l'i 


l :,:>:> 


4571 


4586 


16(12 


4617 


4633 


4648 


4664 


4679 


358 


10 13 


28 


4695 


4710 


4726 


4741 


4756 


4772 


4787 


4802 


4818 


4833 


358 


10 13 


29 


4848 


4863 


4879 


4894 


4909 


4924 


4939 


4955 


4970 


4985 


358 


in i:; 


30 




5015 


5030 


5045 


5060 


5075 


5090 


5105 


5120 


5135 


358 


in i:; 


31 


5150 


5165 


5180 


5195 


5210 


5225 


5240 


5255 


5270 


5284 


257 


10 12 


32 


-. " -".)'. 


5314 


5329 


5344 


5358 


5373 


5388 


5402 


5417 


5432 


257 


10 12 


33 


MM 


5461 


5476 


5490 


5505 


5519 


5534 


5548 


5563 


5577 


257 


lit ]- 


34 


MM 


MM 


5621 


5635 


5650 


5664 


5678 


5693 


5707 


5721 


257 


10 12 


35 


5736 


B7M 


5764 


5779 


B7M 


5807 


5821 


5835 


5850 


5864 


257 


it 12 


36 


5878 


MM 


5906 


.V.rjo 


5934 


5948 


5962 


5976 


.V.I'.IO 


6004 


257 


'.1 I L- 


37 


6018 


6032 


6046 


5060 


6074 


6088 


6101 


6115 


6129 


6143 


257 


II 1 -' 


38 


6157 


6170 


6184 


6198 


6211 


6225 


6239 


6252 


6266 


6280 


257 


9 11 


39 


6JM 


6307 


6320 


6334 


6347 


6361 


6374 


6388 


6401 


6414 


247 


t I! 


40 


6428 


0111 


6455 


6468 


6481 


6494 


6508 


6521 


6534 


6547 


247 


9 11 


41 




9674 


6587 


6600 


6613 


;r,2<; 


6639 


6652 


6665 


0678 


247 


9 11 


42 


6691 


6704 


6717 


6730 


6743 


6756 


6769 


6782 


6794 


6807 


246 


9 11 


43 




6833 


6845 


6858 


6871 


I5S.X1 


6896 


6909 


6921 


6934 


246 


8 11 


44 


6947 


6959 


6972 


6984 


6997 


7009 


7022 


7034 


7046 


7059 


246 


8 Kt 



NATURAL SINES 



177 





A 


81 .1 1Q- 


OA' fit}' 


00' 


49" 




* 


Mean Differences. 




u 


U lo 


^ >U 


uO 


4 . 1 -3 


o* 


1 2 3 4 5 


45 




7083 




not 






7i:.7 


7169 


7181 


246 


8 10 


46 


7193 


LI 7 _:; 




7266 






246 




47 








7385 


- 




246 




48 




UU 71 'if, 




7501 




7.-.:..; 


246 


8 10 


49 


-7.-.I7 


~ > >.<*n 1 


7593 7604 

. i" i - 1 


7615 




mi 


246 


8 9 




,.!.' l 


7**OO 


IOOJ 
-(\n 


ion 

WOA 1 


1 tOo 


< t 10 


7727 


~Q 1 U TOKO 


77'' 1 ' 
- fc ., . 






62 


till 

7880 


7701 

7891 




, Ml J 

7912 


folO 




7 S .">7 
7'.. 1 


, i - Jon 


, Mi'.l 

7976 


245 




53 


TtM 


7997 




8018 


8028 


8039 


SllJil 




8080 


j : 


7 


54 

rr 


MM 

v 1 '*' 


8100 


Bill 


8121 


8131 


Mil 


H5i 




8181 


j 1 : 


7 




56 


s 1 .*- 


8300 


8211 


8221 

- 




8251 

v;i> 




BZ/J 


8281 
8377 




i; - 


57 


8887 


8396 


8406 


8415 




1441 




8471 


235 


6 8 


58 

CQ 


8480 

^7* 


aroi 


8499 

QKCIA 


8508 




v.-,:;,; 




BM| 


235 


6 8 


9 
60 


8660 


8669 


OW 

8678 


>">'.''.' 
8686 




l',l'."i 

8712 




i;.v_' 
8738 


134 


6 7 
























61 


8746 




8780 8788 


-7'...; 




8821 


134 


f. 7 


62 


8829 




8864 




8878 




8902 


1 3 4 


:. 7 


6'J 


8910 




8934 








- 


134 


5 6 


64 


.0901 


8996 








Mil 


1M|1 <II|S 


MM 


1 3 4 


;, . 


65 


9063 


9070 




9085 


9092 




9107 


9114 


III'JI 


9128 


124 


;, .. 




























66 






9157 


'.- 1 1: l 


'.171 


9178 




919S 


i _ r, 


5 6 


67 






'.'..:. 








MM 


1 2 3 


4 6 


68 


an 












9323 


MM 


i _ :i 


4 5 


69 


9336 




9354 




9373 


9379 


9385 




i _ :; 





70 

M*1 


MM 






.. 1 1 :. 








9438 


'..III 


9449 


1 2 3 


4 5 


71 

72 
73 

* 


.' 1 > > 

9511 

Ma 

'it i " 


MCI 
Mil 
MM 

, , . - 




'..-,7* 


9583 


'.' I *;; 
MM 




'.. I 1 ' I 
PMJ 
MM 


9600 
9563 


'.i.". ' i.'i 
9558 
9608 

... - 


i - :; 

i _ -j 


3 4 
8 4 


/* 

75 


."-... 

MM 


Po] i 
MM 


''. . 


'."'-' 7 
9673 


MM 

.,77 


MM 


9686 


'.., |i. 
MM 


'.ll'i.'d l 

9694 


.tt,,,., 

MM 


1 I 2 


3 4 


76 


9703 


9707 


'..711 


9715 


9720 






mi 


9736 




1 1 2 


1 | 


77 




9748 




tnt 


'7.V 


9763 




9770 


9774 


9778 


1 1 2 


8 8 


78 


9781 


9785 


9789 




MM 


9799 


9803 


MM 




1 1 


:. 


79 


'Mil 


9820 


MM 


MM 


MM 


9838 


MM 


MM 




M4I 


1 1 


1 


80 


M4I 


9851 


9854 


MM 


9860 


9863 


MM 


MM 




" 1 1 




81 


9877 




9882 


HH:, 


9888 


9890 




9895 




9900 


H 1 1 


I 


82 


..:...: 


MM 


9907 


..In 


9912 


9914 


9917 




1 1 


2 2 


83 


MM 


MM 


9930 


...:;: 


9934 


9936 


993H 


B940 


9942 


9943 


1 1 


1 | 


84 


9946 


9947 


9949 




,,-,- 


9964 


MM 


9957 






II 1 1 


1 3 


85 


MM 


..... 


9966 


...,.. 


... 


MM 


9971 






(1 .. 1 


I I 


86 


>M7fl 




9978 


9979 








, ... 




M II | 


I 1 


87 


MM 




MM 








.... 




1 1 


88 


MM 


9096 999A 


,, 








9998 


, , 





89 


MM 




,-.,..:, 


i i i i 




1 ,,.,., 






" 






178 



NATURAL COSINES 








6 


12- 


18- 


24 


30 


38 


42' 


48' 


54' 


Mean Differences. 


1' 2' 3' 


4 5 


Q 


1-000 


1*000 


1-000 




1-000 


1-000 


-MM 


MM 


9999 


9999 


000 







MM 


9998 


9998 






9997 


9996 


MM 


9995 


9995 


000 







MM 


r....:: 


9993 


'.'.'.'_' 


9991 


MM 


MM 


M89 


9988 


9987 


000 


1 1 




....-. 


'.'-.-. 


9984 


.'-:; 


M8S 


9981 


9980 


M79 


9978 


9977 


001 


1 1 




H7I 


9974 


9973 


9972 


9971 


9969 


MM 


9966 


9965 


9963 


001 


1 1 




MM 


;,.;,, 


9959 


9957 


MM 


9954 


H'.i.Vj 


9951 


M49 


9947 


Oil 


1 2 


g 


Ml 


r.'i:; 


9942 




MM 


9936 


;i;i." \ 


9932 


9930 


9928 


Oil 


1 2 


7 


MSI 


9923 


9921 


nn in 


9917 


9914 


9912 


9910 


9907 


9905 


Oil 


_' - 


g 


MOI 


9900 


MM 


-.-.:, 


9893 


MM 


9888 


M86 


9882 


9880 


Oil 


_.' 


9 


9877 


9874 


9871 


9869 


9866 


9863 


9860 


MS? 


9854 


9851 


Oil 


_' L' 


10 


9848 


9845 


Mti 


9839 


9836 


9833 


9829 


9826 


9823 


9820 


112 


2 3 


11 


9816 


9813 


9810 


9806 


9803 


9799 


9790 


9792 


9789 


9785 


112 


2 3 


12 


9781 


9778 


9774 


9770 


767 


9763 


9759 


9755 


9751 


9748 


112 


3 3 


13 


9744 


9740 


9736 


9732 


9728 


9724 


9720 


9715 


9711 


9707 


112 


:; :; 


14 


9703 


9699 


'.";;' 1 


9690 


9686 


9681 


9677 


9673 


9668 


9664 


112 


:; l 


15 


9659 


9655 


9650 


9646 


9641 


9636 


9632 


M27 


9622 


9617 


122 


3 4 


16 


9613 


9608 


9603 


9598 


9593 


9588 


9583 


9578 


9573 


9568 


122 


3 4 


17 


9563 


9558 


9553 


9548 


9542 


9537 


9532 


9527 


9521 


9516 


123 


3 4 


18 


9511 


9505 


9500 


9494 


9489 


9483 


9478 


9472 


9466 


9461 


123 


4 5 


19 


9455 


9449 


'.-III 


9438 


MM 


9426 


9421 


9415 


9409 


9403 


123 




20 


9397 


9391 


9385 


::7'.i 


9373 


9367 


9361 


9354 


9348 


9342 


123 


4 5 


21 


9336 


9330 


9323 


9317 


9311 


9304 


9298 


9291 


9285 


9278 


123 


4 5 


22 


9272 


9265 


MM 


9252 


M48 


9239 


9232 


9225 


9219 


9212 


123 


4 6 


23 


9205 


9198 


9191 


9184 


9178 


9171 


9164 


9157 


9150 


9143 


123 


r, r, 


24 


9135 


9128 


9121 


9114 


'.M 07 


9100 


9092 


9085 


9078 


9070 


124 


5 6 


25 


9063 


9056 


9048 


9041 


9033 


9026 


9018 


9011 


9003 


8996 


134 


5 6 


f)n 


8988 


8980 


8973 


8965 


8957 


8949 


8942 


8934 


8926 


8918 


134 


r, ; 


:.'7 


8910 


8902 


MM 


8886 


8878 


8870 


8862 


ssr.i 


8846 


8838 


134 


5 7 


:.'8 


8829 


8821 


8813 


8805 


8796 


8788 


8780 


8771 


8763 


8755 


134 


G 7 


29 


8746 


8738 


8729 


8721 


8712 


8704 


8695 


Hf.Sfi 


8678 


8669 


134 


6 7 


t>0 


8660 


MM 


8643 


8634 


8625 


8616 


8607 


8599 


8590 


8581 


134 


G 7 


31 


8572 


8563 


8554 


8545 


8536 


86J6 


8517 


8508 


8499 


8490 


235 


<> s 


.'{:.' 


8480 


8471 


8462 


8453 


8443 


84S4 


si _'.-, 


8415 


8406 


8396 


235 


6 8 


33 


8387 


8377 


8368 


S.'S.-.s 


BUS 


8339 


8329 


8320 


8310 


8300 


235 


6 8 


'' l 


8290 


8281 


8271 


si';i 


886] 


8241 


8231 


8221 


8211 


8202 


L' :i 5 


7 8 


35 


8192 


8181 


8171 


8161 


8151 


8141 


8131 


8121 


8111 


8100 


235 


7 8 


36 


8090 


8080 


8070 


* i.V.i 


Si i J'l 


8039 


8028 


8018 


8007 


7997 


235 


7 '. 


37 


79M 


7976 


7965 


7'..:,:, 


7944 




7923 


7912 


7902 


7891 


2 5 


7 9 


38 


7880 


7869 


7859 


7-1* 


7837 


7826 


7815 


7804 


7793 


77SL' 


2 5 


7 9 


w 


7771 


7760 


7749 


7738 


7727 


7716 


7705 


76M 


7683 


7672 


2 6 


7 9 


40 


;.,.,,, 


7M8 


7638 


7627 


76U 


7604 


7593 


7581 


7570 


7559 


2 6 


8 9 


41 


7547 


7536 


T.VJI 


::.!:; 


7501 


7490 


7478 


7406 


7455 


7443 


2 6 


8 10 


42 




7420 


74M 


7:;'."'. 


7385 


7373 


73G1 


7349 


7337 


7325 


2 6 


8 10 


43 


7314 


7302 


7290 


7278 


7266 


7254 


7242 


7SM 


7218 


7206 


2 6 


8 10 


44 


7193 


7181 


7169 


7157 


7145 


7133 


7120 


7108 


7096 


7083 


246 


8 10 



NATURAL COSINES 



179 



M.'Mll I>i! V . 











Mean DJfft ; 


A' 


r-i .) 10 


.11 on Ofi 


4O' AO' K.A' 







IX lo 


I 


M 


ow 


Vb 


VOJ 


* 


r v r 


4' 5 


45 -7071 


\>\ 7"."l 


7m.' L > 7,Hi:i 




6984 


6971 




246 


8 10 


1 






6858 


M4B 




i' i 6 


8 11 


47 '6820 




>:756 




6730 


R717 


6701 


246 


11 


48 '6691 






Mil 


MOO 


M07 




.' 1 7 


11 


49 -6561 


1547 U 




6481 




544J 


J 1 7 


11 


50 '6428 


f.lll I'T.l r,:;ss 




M47 


6334 




6307 


2 -i 7 


11 


51 


6293 


._*. .;-,.; .;_>:,_ 




61M 


6184 


;i7' 


l' :> 7 


11 


52 -6157 


.ii:> f,i-.".t tin:, 


c.lul i;n88 6074 


BOM 


BOM 




l' 5 7 


12 


53 -6018 




,!)48 


:>.:; I 


5920 


5906 


5R92 


i' 5 7 


I] 


54 -5878 


:>M; i 


5850 




5821 




:,7'.':; 


6779 




257 


'.' ] _' 


56 -5736 


vm 


5707 


MM 


5678 


:,.;.; i 


MM 




MM 


-' 5 7 


10 12 


























56 "5592 


.'...77 


MOI 


:,:,ls 


5534 


.-.-> 19 


5505 


5490 


M7< ">i'''i 


267 


10 12 


57 '5446 


5432 


5417 


MM 


5388 




5358 


:.:; ) i 




267 


in 12 


58 


MM 


5284 


5270 




5240 




5210 


5195 


:,lsu .-,1.;:, 


257 


10 12 


59 


5150 


5135 


r>r.'<i 




.-.Him 


M7B 


5060 


5045 


5030 


r,oi ;, 


368 


10 13 


60 




IMI 


1070 




i;.:;;. 


I'.'L'I 


4909 


ism 


IS T;I 


1801 


358 


10 13 


61 


4848 


4833 


IMS 


1001 


4787 


4772 


I7M 


1711 


I7M 


171" 


358 


10 13 


62 


l'il'0 


4679 


IM;| 


MM 


M8J 


4617 


4602 


4586 


LOT] 


IMI 


o 5 8 


10 13 


63 


I.-.IM 


4524 


IBM 




1471 




1 1 ir, 


4431 


1415 


UM 


358 


10 13 


64 


4384 




I:;IM 1808 


4289 


I'.' 71 


LSM 


4MJ 


:; : 


11 13 


65 


4226 


ino 


ir.i.i H7'. 


4163 


1117 


1 1 :; i 


4115 


IOM 


IOM 


358 


11 13 


66 '4067 


4051 


IM:;:, 


1011 


4003 


M07 


M71 


3955 


IMI 


MM 


358 


11 14 


67 || '3907 


3891 


M7I 


MM 


3843 


18*7 


3SM 


I7M 


Ml- 


1701 


358 


11 14 


68 |i -3746 


3730 


::7i I 


M07 


M61 


MM 


M4I 


3633 


Ml 


MOO 


:; ;. s 


11 14 


69 "3584 


3567 


Iff] 


MM 


3518 


MOI 


MM 


MM 


MM 


M07 


358 


11 14 


70 i- 


MM 


1069 


M71 


MM 




MSI 


MOB 


3289 


3272 


368 


11 14 


71 


DM 


3239 


MSI 


MM 


noo 




3156 


3140 




3107 


:; ; s 


11 14 


7:2 


3090 


3074 


1007 


1040 


MM 


3007 


MOO 


-".'71 




J'.'lH 


:: .; s 


11 14 


73 


MM 


MOT 


ION 




2857 






2807 




2773 


:: .; ^ 


11 14 


74 


_:.-., 


2740 


ITU 


.'7".; 


2089 




2656 


2639 


MOI 


2605 


368 


11 14 


75 


flOOfl 




MM 


-:.:> 


Mil 




2487 




2453 


MM 


S 6 8 


11 14 


78 


2419 


2402 


2S85 


2368 


2351 






MOO 


MM 


M07 


3 6 


11 14 


77 


SSM 


2233 




Jin- 






2147 


noo 


mi 


MM 


:; ; 


11 14 


78 


1071 


H61 




_,,_.. 


MU 


1994 


1077 


LOM 


I'.'U 


1 ::-, 


:; .; 


11 14 


79 


1SOC 






1857 




1822 


1805 


L7M 


1771 


1754 


:; .; 


11 14 


80' 


17M 






1685 


LOM 


LOM 


1.;:;:; 


i',|., 


1 Ml 


I :.--. 


| .; 


12 14 


81 


i M i 




1530 


1513 


1495 


1478 


IP,] 


mi 


MM 


1409 


8 6 


IS 14 


82 


1 :::.: 






1340 


LOM 


LOM 


IV- 




LOM 


i .:..: 


8 6 




83 


1219 


ISO] 






1149 


M.-.J 


Ull 




Ins,, 


in,;;; 


8 6 


18 14 


84 


1046 


LOM 


lull 


..:.<.,;; 


1071 


OOM 


.Hi 




MM 


,,ss;. 


3 6 


IS 14 


85 


0671 


iM 


oosi 


..sin 


Ml 


OfM 


0767 


1700 


Ofn 


0715 


8 6 


IS 14 


86 


06M 


I06Q 




<,.!.-, 


)6S8 


0010 


..V.i.1 


,:,;,-, 


MM 


-Ml 


.1 r, 


IS 15 


87 


out 


I6M 


MM 








n||'.' 


MOI 


MM 


..;;,,,, 


.1 r, 


IS 1ft 


88 


0349 


ion 


0014 


'.".'7 


171 


OOM 


..II 


001 


MOO 


.. 1 ..'.- 


| : 


1 1 1 1 


89 


0175 


mi 


0140 


il'.-J 


001 


0007 


1111711 


IOM 


OOM 


,,,,17 


3 


13 15 



ISO 



NATURAL TANGENTS 













Menu DillViviii'i M. 




0' 


6' 


11 


18 


24' 30' 36' 


42' 48' 54' 


r 2- 3' 


4' 5' 





o-ooo 


mi 


MM 


,.,,:,_ 


070 


0087 "in:, 


1188 "1 lo nir, 7 


:; c, it 


1 _ 1 :, 








0227 


_ 1 1 




1891 ".".11 ":;:;_ 


c, n 


l-j i:, 




0349 








iin 041 


. 17'J (Us 1 .. 11.107 


:i t; it 


l _ l .-, 




MM 


41 




2 or.-jit 


n;.|7 <.i;i;i i.r.si' 


:; c, j 


i _ 1 :, 




MM 


717 






s __ 084 


:; r, 9 


!_' 1.1 




171 




0910 


,,,-JS 


0945 




0981 


098 L016 i":;:! 


:; r, 9 


1 -J 1 .1 




























1061 




1104 


11 _'.' 


L189 


1157 


1 7--. 1 1 icj 


L210 


:i c. '.. 


12 15 




;_.> 




l -".''.< i::i7 mi 


:;.vj r,7" i::ss 


:; (i 9 


12 15 




1405 


11 H.V.I 


1177 mi:, i:,l-j 


:,.-," i.-.is i:,t;i; 


a r. 9 


12 15 




1584 


.0 1638 


L6M n;::-, ir.sn 


7o;i 17:.' 7 17i:> 


369 


1 2 1 1 


icr 


1763 


1731 1 


1817 


J68 1871 


isiio 1908 


1926 


3 r, it 


1 _ 1 .1 
























11 


]'.'! 




Ht'.tS 


jolt; 




2071 


8089 


2107 


369 


1 1' 1 -> 


12 


2126 


Jill 2 K.- 1 


2180 


J 1 1'H 


_'_' 17 _'-:',:. 


2254 


2272 


2290 


3 K li 


1 L' 1 r, 


13 


not 




2364 


1 'Jill* 


2438 


2456 


2475 


3 6 9 


U' i:, 


14 


MM 




2549 


M M05 


8688 


_M;|-J 


2661 


3 6 9 


12 16 


15 


1671 




2736 


27.11 -J77: 1 , 27112 


2811 


8880 


2849 


369 


13 16 


16 


M61 


I __ | 


-."..1:1 


2962 


2981 


3000 


3019 


3038 


:; c, g 


13 16 


17 


3057 


:;u96 


3115 


3134 


3153 3172 


3191 


3211 


3230 


:; r, 10 


13 16 


18 


3249 


3269 


3288 


3307 


3327 


3346 3365 


8888 


3404 


3424 


3 6 10 


1 :; 1 r, 


19 


3443 


3463 


3482 




3522 


:;:.n :;:,.;! 


3581 


3600 


3620 


3 7 10 


13 16 


20 


3640 


3659 


3679 


3699 


3719 


3739 


3759 


3779 


3799 


3819 


3 7 10 


13 17 


21 


MM 


3859 


3879 


3899 


3919 


:;:.:;;. 


3959 


3979 


4000 


4020 


3 7 10 


i :; 17 


L':.' 


4040 




4081 


4101 


4122 


11-12 in;:; 


4183 


4204 


4224 


:! 7 10 


1 1 17 


23 


4245 




4286 


4307 


4327 


1848 1869 


4390 


4411 


4431 


3 7 10 


1 1 17 


24 


4452 




4494 


\:>\:> 


4536 


1.1.17 


4578 


Ifilili 


4621 


4642 


1 7 11 


14 18 


25 


4663 


4684 


4706 


4727 


1748 


177 


4791 


4813 


4834 


4856 


4 7 11 


14 18 


:.'.; 


I '4877 


4899 


r.'-ji 


11U2 


4964 


4986 


5008 


5029 


5051 


5073 


4 7 11 


15 18 


27 


MM 


r-117 


5139 


r.ir.i 


5184 


5206 


5228 


5250 


5272 


5295 


4711 


1.1 is 


28 


5317 


:,:;i" 


5362 


:.:;s i 


:.t"7 .vi:;n 


5452 


5475 


5498 


5520 


4 8 11 


1 .1 1 H 


2! 


5543 


MM 


U89 


B6U 


5635 


5658 


5681 


5704 


5727 


5750 


4 8 lli 


1 :, 1 it 


30 




5797 


5820 




.isr,7 




5914 


5938 


5961 


.M.sr, 


4 8 12 


16 20 


31 


00| 


BOM 


ION 


aow 


6104 


6128 


6159 


6176 


6200 


6224 


4 S 1 2 


16 20 


32 


'. !.' 


6273 


.',.".'7 








6420 


6445 


r.ic.ii 


4 8 12 


it; 211 


:>,:>, 




.-..Ml. 


6544 


.;:,.;;. 


6594 


Of! Ill 


6644 


6669 


(it;:, i 


6720 


4 8 13 


17 21 


34 




6771 


6794 




i!M7 


6878 


6899 


6924 


6950 


6976 


4 li 1 :; 


17 _ 1 


35 




7028 


7".1I 


7080 


7107 


7188 


7159 


7186 


7212 


7L':w 


4 9 13 


IS -2-1 


:;*, 




7292 






7373 


7KKI 


7127 


7454 


7481 


7r,o.x 


r. 9 u 


IK W 


37 


>7M 


7.-,.,:; 


7690 


7618 


7646 


7r, 7:1 


7701 


7789 


7787 


7785 


r, i. i 1 


is -j:; 


:;s 


7813 


7841 


7869 


7898 


7926 


7'.:, 7ns:; 


8012 


SOIO 


8069 


5 It 1 1 


19 24 


:;. 




8127 


SIM 


8186 


8214 


8243 


8878 


8302 


8332 8361 


5 10 15 


jo -j| 


40 


8391 


8421 


8461 


148] 


8511 


8541 


S571 


8601 


8632 


8662 


5 10 15 


jo 2.1 


41 


MM 






8786 


-i.; 


8847 


8878 


S'.MO s'.MI 


8973 


5 10 16 


21 26 


42 -9004 




BOM 


11 1 :; 1 


0168 919! 


HL'L'S li'jc.ii '.._".: 


5 11 16 


Jl 27 


43 -MSI 






m:, 7 in:". '.:,_: 


9556 9590 9623 


6 11 17 


22 2S 


44 '9657 






HS-.M; HUSO 9965 


6 11 17 


j:; -".. 



\ ATI' HAL TANGENTS 



181 






6' 12' 18' 


24 


30 36 


42' 48' 54' 


Mean Differences. 


1' 2* 8- 4' 5' 


45 I'OOOO 


0035 


0070 


0105 


0141 


0176 


0212 


0247 




6 12 18 24 30 






















46 1*0355 


0392 


0428 


0104 




1611 




0686 


6 12 18 25 31 


47 l-"72i 


0761 


0799 


iiss; 


>*"> .IVU3 '0951 


0990 1028 


1067 


6 13 19 


25 32 


48 1-1 IM.; 


1145 


1184 


12 24 




. 


1383 142:; 1 i<;:; 


7 13 I'" 


27 33 


49 




1544 


1585 


1626 


IM1 




1750 


17'.2 1833 


1875 


7 It 21 


28 34 


IH 


1-1918 


1960 


2002 


2045 


2088 


2131 


2174 




2305 


7 11 


29 36 


51 


1-2349 


2393 


2437 


2482 


2527 


2572 


2617 


_'.;;-.' 27'is 


2753 


8 15 23 


30 38 


52 




2846 


2892 


2938 


MM 


3032 


3079 


3127 3175 


MM 


8 16 2 i 


31 39 


53 




;:;i'.' 


3367 


3416 


3465 


3514 


3564 


3613 


3663 


3713 


8 10 _>:, 


33 41 


54 


1-3764 


3814 


3865 


3916 


3968 


4019 


4071 


1124 


4176 


ISM 


9 17 26 


34 43 


55 


1*418] 


4335 


4388 


1112 


4496 


4550 


4605 


MM 


4715 


4770 


9 18 27 


:;.; :, 


56 


1-4826 


4882 


4938 


4994 


5051 


5108 


5166 


5224 


5282 


5340 


10 19 2 ii 


38 48 


57 


1-5399 


5458 


5517 


5577 


MSI 


5697 


5757 


5818 


5880 


5941 


10 20 30 


40 50 


58 


1-6003 


; .;.; 


6128 


6191 


6255 


6319 


6383 


6447 


6512 


6577 


11 21 32 


43 53 


59 




6709 


6775 


6842 


6909 


6977 


7045 


7113 


7182 


7251 


11 23 34 


i.-) :..; 


60 


l-ToiIl 


7391 


7461 


7532 


7603 


7675 


7747 


7820 


7893 


7'.";.; 


12 24 36 


48 60 


61 


1-8040 


8115 


8190 


s .';:, 


8341 


8418 


8495 


8572 


8650 


8728 


13 26 38 


51 64 


62 


1-8807 


8887 


SM7 


9047 


9128 


9210 


'.'_.': 


8375 


9458 


M4I 


11 27 41 


;,:, .* 


63 


i '.;_; 


9711 


797 


.--:; 


9970 


0057 


ill 45 


0233 


0323 


0413 


15 29 44 


68 73 


64 




0594 


MM 


0778 


0872 


0965 


1000 


1155 


1251 


1348 


10 31 17 


-.:; 7< 


65 




1543 


1642 


L74J 


1842 


1943 


2045 


2148 


2251 


I 


17 34 51 


68 85 


66 


M4M 


HM 


M7I 


2781 


2889 


2998 


:uo-.i 


ISM 


MM 


1448 


18 37 55 


73 92 


67 


- MM 


;,::; 


1780 


:;'..,; 


ion 


4142 


t 2H2 


4383 


4504 


I.-.-J7 


10 60 


7'.' i.y 





2 I7.M 


4876 


5002 


list 


mi 


5386 


5517 


5649 


5782 


.VI.; 


, 65 


87 108 


60 




6187 


HM 


f.lf.l 


MM 


6746 


;x-j 


7034 


7179 


I 


21 17 71 


'.'.'. 11'.' 


70 




7625 


7776 


7 '.'.;. 


8083 


UM 


8397 


8556 


8716 


8878 


^ 


104 131 


71 


rt04i 


9208 


9376 


9544 


ITU 


9887 


0061 




0415 


0596 


29 58 87 


11.', I);, 


72 




OM1 


1 1 If. 


1334 


1524 


i7i.; 


1910 






uoa 




12 H I'.l 


73 




2914 






MM 


3759 


:r..77 


11 '.'7 


4420 


4646 


36 72 108 


111 1*0 


74 


3-4874 


5106 


5339 


6576 


5S16 


6059 


.;:;..:, 


6554 


MM 


IOM 


11 -1 122 


i'.:; 104 


75 


:;;:; 21 


7583 


7848 


nu 


8391 


8667 


-'..17 


9232 


MM 


Mil 


46 93 139 


lr. _:;: 


78 
77 


4-0108 
43316 


0408 
8662 


0718 
4015 


LOS1 

I.-.7I 


1335 
4787 


LMI 

5107 


L976 
5483 


1801 

6864 


MM 
8MS 


M7I 
MM 




78 


4-7046 




8716 


9152 


...'...I 


0041 


0604 






79 


5-l44 




:;:..-. 


MM 


MM 


:,'..; 






80 


6-6713 


7894 


8602 


9124 


MM 


ftM 


1066 


17'.' 2I3-. 




81 


6-3128 


>.v. 


4506 


6860 


6122 


Mil 


7720 


M41 


MM 




J/rt differences no 


82 7-1164 
83 8-1443 


:">, 
..,;;,, 


8002 8962 
3863 6126 


4 -.it 7 

,i;7 


MM 


MM 
9161 


HM-,2 
(.:,;.. 


IU:,H 
i06l 


fen 


longer fufflcUutlj 
accurate. 


84 




.,77 






LOW 


LO-M 


IM ;, 


10-99 






85 


11-43 








12-71 


i :; 


1 :;:;.. 


1 .V..;' 


18-96 




86 


14-80 


14-67 


15-06 16-46 


I.V-. 


16-Sfi 


\ ,.*:: 


17 :.i 


17-89 


18-46 




87 




I-..-7I 






.'.'." 


tr 


,, ,, 


:.;-. 


27-27 




88 


28-64 


... 11 


31-82 33-69 


86-80 






47-74 


MDI 




89 


67-29 


.,:,-..,. 


71-62 81-86 


J :.!.' 1 


r..i .' 









182 



IJM, \KITHMIC SINES 








6' 


12' 


18* 


24' 


150 


36' 


42' 


48 


54' 


Mean DilTriviH-t-.-. 


r 2- 3- 


4' 5' 





- 00 






7190 


MS9 


9408 


5so 


0870 


1450 


LM 








- : 1 1 '.- 








3880 


4179 


1409 


4723 


4971 


5206 








- MM 


5640 


5842 


6035 






6567 


6731 


8889 


7041 








8-7188 


7330 


7468 


r< - 


7731 


7857 


7979 


8098 


8213 


8326 








- MM 


8543 


8647 


-71'.' 


SSl'l 


8946 


9041 


9188 




.:; i : 


16 32 48 


64 80 




8-9103 


9489 


9573 


9655 


9736 


9816 


9894 


9970 


0046 


01* 


13 26 39 


52 65 




9-0192 


0264 


0334 


0403 


0472 


0539 


0605 


0670 


0734 


0797 


11 22 33 


1 1 :,:, 




9-0859 


0920 


0981 


1040 


L099 


1157 


lL'1 1 


1171 


1326 


1381 


10 19 29 


38 48 




9-1436 


1489 


1542 


1594 


1646 


1697 


1717 


1797 


1847 


1895 


8 17 25 


34 42 




9-1943 


1991 


2038 


20S5 


2131 


2176 


211] 


2266 


2310 


2353 


8 15 23 


30 38 


10' 


9-2397 




2482 


..VJI 


2563 


2606 


1647 


2687 


2727 


2767 


7 14 20 


IT :;t 


11 


9-2806 


2845 


2883 


-".'Jl 


2959 


2997 


3034 


3070 


3107 


3143 


6 12 19 


25 31 


12 


9-3179 


3214 


3250 


1384 


3319 


88M 


3387 


3421 


3455 


3488 


6 11 17 


23 28 


13 


9-3521 


3554 


3586 


3618 


3650 


3682 


3713 


3745 


3775 


3806 


5 11 16 


21 i'i; 


14 


9-3837 


3867 


3897 


3927 


3957 


3986 


4015 


4044 


4073 


4102 


5 10 15 


20 24 


15 


9-4130 


4158 


4186 


I-J11 


[243 


4269 


4296 


4323 


4350 


4377 


5 9 14 


18 23 


16 


9-4403 


4430 


4456 


L481 


4508 


4533 


4559 


4584 


4609 


4634 


4 9 13 


17 21 


17 


9-4659 


4684 


4709 


4733 


4757 


4781 


4805 


4829 


4853 


4876 


4 8 12 


16 20 


18 


>4800 


4923 


I'.Mii 


4969 


4992 


5015 


5037 


5060 


5082 


5104 


4 8 11 


15 19 


19 




5148 


5170 


5192 


5213 


5235 


5256 


5278 


5299 


5320 


4 7 11 


14 18 


20' 


9-5341 


5361 


5382 


5402 


3i _>.-! 


5443 


5463 


5484 


5504 


5523 


3 7 10 


14 17 


21 


9-5543 


5563 


5583 


5602 


5621 


5641 


5660 


5679 


5698 


5717 


3 6 10 


13 16 


22 


9-5736 


:.7--.i 


5773 


5792 


5810 


5828 


5847 


5865 


5883 


5901 


369 


12 15 


23 


PHU 


5937 


5954 


5972 


5990 


6007 


6024 


6042 


6059 


6076 


369 


1 2 1 r. 


24 


9-60M 


6110 


6127 


6144 


61G1 


6177 


6194 


6210 


6227 


6243 


368 


11 14 


25 


...;_'.-,'.. 


6276 


8191 


6308 


6324 


6340 


6356 


6371 


6387 


6403 


358 


11 13 


26 


9-6418 


6434 


6449 


8465 


6480 


6495 


fi3 10 


6526 


6541 


6556 


358 


10 13 


27 


9-6S70 


6585 


6600 


6615 


6629 


6644 


6639 


6673 


6687 


6702 


257 


10 12 


28 


.'; : i; 


6730 


6744 


6759 


6773 


6787 


0801 


6814 


6828 


6842 


257 


11 1 2 


29 


'.,>.-,,; 


6869 


6883 


6896 


6910 


;:_':; 


6937 


6950 


6963 


0977 


247 


9 11 


80 


-6t90 


7003 


7016 


7029 


7042 


7055 


7008 


7080 


7093 


7100 


246 


9 11 


81 


9-7118 


7i:;i 


7144 


7156 


7168 


7181 


7193 


7205 


7218 


7230 


246 


8 10 


32 


9-7242 


7254 


7266 


7278 


7190 


7302 


7314 


7326 


7338 


7349 


2 4 c, 


8 10 


38 


'.- ::,'. i 


7373 


7384 


7396 


7407 


7419 


7430 


7442 


7453 


7464 


246 


8 10 


84 


9-7471 


7487 


7498 


7509 


7 3 -jo 


7531 


7542 


7553 


7564 


7575 


: -1 (i 


7 9 


85 


j ;.>.; 


7597 


7607 


7618 


7629 


7640 


7050 


7661 


7671 


7082 


245 


7 9 


86 


9*ntl 


7703 


7713 


7723 


7734 


7744 


7731 


7764 


7774 


7785 


235 


7 9 


87 


9-7795 


7805 


7815 


7825 


7835 


7844 


7854 


7864 


7874 


7884 


235 


7 8 


38 


9-7893 


7903 


7913 


7922 


7932 


7941 


7951 


7960 


7970 


7979 


235 


8 


88 


IBM 


7998 


8007 


8017 


BOM 


8035 


8044 


053 


8068 


8072 


235 


6 8 


40 


9-8081 


8090 


8099 


8108 


8117 


8125 


8134 


143 


8152 


8101 


134 


6 7 


41 


: 8169 


8178 


8187 


8195 


8204 


8213 


8221 


230 


8238 


8247 


134 


7 


42 


9-8255 


8264 


8272 


s-.-so 


8389 


8397 


8305 


313 


8322 


8330 


134 


6 7 


48 


9-8338 


8346 


8354 


8362 


8370 


8378 


8386 


394 


8402 


8410 


134 


5 7 


44 


9-8418 


8426 


8433 


sill 


8449 


8457 


8464 


472 


8480 


8487 


134 


5 G 



LOGARITHMIC SINES 



183 








6' 


12- 


IP 


24' 


30 


36 


42' 


43 


54 


Mean Differences. 


r 2 s 


4 5 


45 


9-8495 


8502 


8510 


8517 


8525 


8532 


8540 


8547 


-.-,:,:, 


8562 


124 


.; 


46 


! 9-8569 


-.-.77 


MM 


8591 


8598 


8606 


8613 


8620 


MSI 


8634 


1 2 4 


| .. 


47 


9-8641 


8648 


MM 


MtJ 


8669 


MTfl 


8683 


^,;:,,, 


-.'..7 


8704 


1 2 3 


5 6 


48 


9-8711 


8718 


8724 


8731 


s;.-{s 


8741 


8751 


8758 


8765 


8771 


1 2 3 


4 6 


49 


| 9-8778 


-7-1 


8791 


8797 


ss,,| 


8810 


--17 


- 


8830 


8836 


1 1 


4 5 


50' 


9-884J 


--!'.' 


8855 


8862 


8868 


8874 


8880 


8887 


8893 


8899 


1 2 3 


4 5 


51 


9-8905 


8911 


8917 




8929 


8935 


8941 


8947 


8953 


MM 


1 2 3 


4 5 


52 


9-8965 


8971 


8977 


M8I 


s-iix-i 


8995 


: 


'. ; 


9012 


9018 


123 


4 5 


53 


9-9023 


9029 


9035 


9041 


9046 




9057 


9063 


9069 


M74 


1 2 3 


1 


54 


|>908Q 


Mfcl 


9091 




9101 




'.HI-.' 


9118 




lltt 


1 2 3 


4 5 


55 


9-9134 


1119 


9144 


9149 


9155 


9160 


9165 


9170 


9175 


9181 


1 2 3 


3 4 


56 


.'..I-.'. 


9191 


9196 


Hl'nl 


9206 


9211 


9216 


9221 






1 2 3 


3 4 


57 




M4] 


M44 


9251 


MM 


M6Q 


9265 


9270 






1 2 2 


I I 


58 




M8f 


..-..I 


9298 


MOI 


MM 




9317 






1 I 


:; i 


59 


:..:;:;! 


MM 


M40 


9344 


M49 


..:;:,:; 


'..:;:,- 


M<I 


..:;.; 7 


9371 


1 1 -' 


3 4 


60 


9-9375 


9380 


9384 


.'.-;-- 


9393 


.:;: 7 


9401 


9406 


9410 


'.>lll 


1 1 2 


:; i 


61 


9-9418 


I-.".' 


Mil 


1 


Mtl 


MM 


'.MI:; 


9447 


9451 


MM 


1 1 2 


I :; 


62 


'.''.'I.V.. 


M| 


MM 


'-171 


17.-, 


.'17'.' 


.-I-:; 


'.'1-7 


I'.M 


MM 


112 


3 3 


63 


,,.,,,,,,, 


9503 


9507 


...'.In 


9514 


9518 


9522 


9525 


1619 


9533 


1 1 I' 


I :; 


64 


.::.::: 


9540 


9544 


..'.is 


9551 


MM 


1..V.S 


9562 


1161 


.'.-..:.. 


1 1 2 


j :; 


65 


'.'.'.-,::; 


9576 


9580 


..:>-:; 


9587 


9590 


9594 


9597 


MU 


9604 


112 


2 3 


66 

, - 


9-9607 


9611 


9614 


.".17 


9621 


9624 


9627 


9631 


9634 


...',:; 7 


1 1 2 


2 3 


o7 
68 


'.' '."'. I 1 ' 
-..- '...;:_ 


"'. i .. 
9675 


."'> 1 7 

M71 


.':',' ' 

.';-i 


MM 
9684 


.it',.",r, 
9687 


."'..".'.' 

H-.-...I 


H'.t;-.' 
9693 


u'.i'.i; 
9696 


."'.'.. 
...;;.;. 


Oil 


J . 


69 


> m 


9704 


9707 


.71" 


9713 


9716 


9719 


I7SI 


9724 


n 


1 1 


-.' i' 


70 


9-0730 


9733 


9735 


.'7:;- 


9741 


9743 


.'71'. 


'71'.' 


9751 


9754 


1 1 


- i' 


71 


.."..7:,; 


7.-.'.' 


9762 


-7--.I 


9767 


9770 


77-J 


9775 


9777 


,7-.. 


1 1 


J I' 


72 


'.'. : _ 


9785 


9787 


7s-.i 


9792 


9794 


':'! 


9799 


9801 


,SM, 


1 1 


J _ 


78 




, 


9811 


si:: 


9815 


9817 


,,-..,, 


Ml 


MM 


is-.v. 


1 1 


_ - 


74 


MM 


9831 


9833 


-:;:, 


9837 


MM 


.-11 


9843 


M4I 


si; 


1 1 


1 9 


75 


'.''.'-r.. 


9851 


-:,:; 


,.:,:, 


9857 


9869 


.-.,i 


9863 


9865 


.-..; 


1 1 


1 2 


76 




9871 


9873 


.-7.'. 


9876 


9878 


<-- 


9882 


9884 


HS.-, 


1 1 


1 2 


77 


9-9887 


9889 


9891 


-.'. 


9894 


MM 


.-..7 


9899 


9901 


.-...: 


I 1 


1 1 


78 


:. MM 


.>..; 


9907 


,;,.,.., 


9910 


9918 


.:M:: 


9915 


9916 


.MS 


1 1 


1 1 


79 


9-9919 


Mtl 


,,_.., 


-. 1 


Mi 


'.-: 7 


.:.;- 


Ml 


9931 


.:...: 


1 


1 1 


80 1 


-.. MM 


9935 


9936 


.,... 


9939 


9940 


j.'ll 


9943 


9944 


.i:. 


001 


1 l 


81 


: MM 


9947 


9949 


..,.-.., 


9951 


9952 


... ,; 


'.'M 


9966 


..... 


M Q 1 


1 1 


82 




Ml 


MM 


.': 1 


.'M,; 


....,; 


, , 


Ml 


MM 


..... 


1 


1 1 


83 


MM1 


...,, 


MM 


,,,-,, 


9971 


M7I 




9974 


9976 


...;:, 


000 


1 1 


84 




M71 


9978 


-," 


9979 


MQ 


...-] 


9981 


MM 




000 


. 1 


85 




..-i 


9986 


r..-:, 


9986 


9987 


r..-r 


,,,.- 


9988 


..,-, 


000 





86 




, , , 




,..,.., 1 


mi 






9993 




..'.. 1 





,, ,, 


87 


I MM 


... .1 


MM 


.... 


...... 


....... 


........ 


....... 


9997 


.,.,.,; 


000 





88 


9-9997 


,.. 


MM 


,.,. 


.,.,.,. 


MM 


,,,, 


;...... 


MM 


.,., 


000 





89 


Mm 


....... 


too 







oooo 


UOOO 


(j ( 


MOO 




000 


t 








LOGARITHMIC COSINES 

r,ict MTUi DilTorn 








6 


12- 


18 


24' 


30 


36' 


42' 


48 


54 


Mean DitTi'i. 


r 2- v 


4' 5' 





10-0000 


0000 


0000 


0000 


oooo 


0000 





oooo 


OOOO 


9.9999 


000 





1 


'.''.".''.'.' 


9999 


9999 


ww 


MM 


9999 


9998 


9998 


9998 


9998 








2 


9Mti 


9997 


9997 


mmr. 


9996 


9996 


9996 


9995 


9995 


:''.<'.' 1 





II .1 


3 


,,.,,,,,,, 


M9 i 


.'.".'.-; 


vw:\ 


I''.".'-.' 


MM 


9991 


9991 


9990 


9990 


000 


II (1 


4 


9-9989 


9999 


9988 


.,i>S 


9987 


9987 


M86 


9985 


9985 


9984 


000 





5 


Ftttl 


9981 


9982 


9981 


9981 


9980 


9979 


9978 


9978 


9977 


000 


1 


6 


'.'".'.' 7'-. 


9975 


9975 


9974 


9973 


9972 


9971 


9970 


9969 


9968 


000 


1 


7 


.,;,.;> 


M61 


9966 


9965 


9964 


9963 


9962 


9961 


9960 


9959 







8 


9-9958 


MM 


MM 


9954 


9953 


9952 


9951 


9950 


9949 


9947 







9 


9-9946 


9945 




9943 


9941 


9940 


9939 


9937 


9936 


9935 







10 


9-9934 


9932 


9931 


9929 


9928 


9927 


9925 


9924 


9922 


9921 







11 


9-9919 


MU 


9916 


9915 


9913 


9912 


9910 


9909 


9907 


9906 


1 




12 


9-9904 


9902 


9901 


9899 


9897 


9896 


9894 


9892 


9891 


9889 


1 




13 


9-9887 


9885 


9884 


9882 


9880 


9878 


9876 


9875 


9873 


9871 


1 


2 


14 


9-9869 


9867 


9865 


9863 


9861 


9859 


9857 


9855 


9853 


9851 


1 


2 


15 


9-9849 


9847 


9845 


9843 


9841 


9839 


9837 


9835 


9833 


9831 


1 


2 


16 


9-9828 


9826 


9824 


9822 


9820 


9817 


9815 


9813 


9811 


9808 


1 


, ,, 


17 


9-9806 


9804 


9801 


9799 


9797 


9794 


9792 


9789 


9787 


9785 


1 


2 _' 


18 


9-9782 


9780 


9777 


9775 


9772 


9770 


9767 


9764 


9762 


9759 


1 


2 2 


19 


9-9757 


9754 


9751 


9749 


9746 


9743 


9741 


9738 


9735 


9733 


Oil 


2 2 


20 


9-9730 


9727 


9724 


9722 


9719 


9716 


9713 


9710 


9707 


9704 


Oil 


2 2 


21 


9-9702 


'.'.;;.;. 


B6M 


9693 


9690 


9687 


9684 


9681 


9678 


9675 


Oil 


2 2 


22 


9-9672 


9669 




966S 


9659 


9656 


9653 


9650 


9647 


9643 


i i i 


2 3 


23 


9-9640 


9637 


9634 


9631 


9627 


9624 


9621 


9617 


9614 


9611 


1 1 J 


2 3 


24 


9-9609 


9604 


9601 


9597 


9594 


9590 


9587 


9583 


9580 


9576 


112 


2 3 


25 


9*M7I 


9569 


9566 


9562 


9558 


9555 


9551 


9548 


9544 


9540 


11-2 


2 3 


26 


9-9537 


9533 


9529 


9525 


9522 


9518 


9514 


9510 


9507 


9503 


112 


3 3 


27 


,,.,,,.,,, 


9495 


9491 


9487 


9483 


9479 


9475 


9471 


9467 


9463 


112 


3 3 


28 


9-9459 


9455 


9451 


9447 


9443 


9439 


'.U.V. 


9431 


9427 


9422 


112 


3 3 


29 


9-9418 


'.Mil 


9410 


9406 


9401 


9397 


9393 


9388 


9384 


9380 


112 


3 


80 


9-9375 


9371 


9367 


9362 


MM 


9353 


9349 


9344 


9340 


9335 


112 


3 


31 


9-9331 


'.:;_'; 


9322 


9317 


9312 


9308 


9303 


9298 


9294 


9289 


122 


3 


32 


9-9*84 


9279 


9275 


9270 


M68 


9260 


9255 


9251 


9246 


9241 


122 


3 


33 


9-9236 


9231 


9226 


9221 


9216 


981] 


9206 


930] 


9196 


9191 


123 


8 


34 


9-9186 


9181 


9175 


9170 


9165 


9160 


9155 


'.'! I'.t 


9144 


9139 


1 2 3 


3 


35 


9-9134 


9128 


9123 


9118 


9112 


9107 


9101 


9096 


9091 


9085 


123 


4 5 


36 


9-9080 


'."71 


9069 


9063 


M57 


9052 


9046 


9041 


9035 


9029 


123 


4 5 


87 


9-9023 


in 1 - 


9012 


9006 


9000 


8996 


8989 


8983 


8977 


8971 


123 


4 5 


38 


9-8965 


*'..:,;. 


MM 


8947 


8941 


MM 


8929 


8923 


8917 


8911 


123 


4 5 


M 


9-8905 


-:.:. 


8893 


sss7 


8880 


8874 


HXI5S 


8862 


8855 


8849 


123 


4 5 


40 


9-8843 


8836 


8830 


8823 


8817 


8810 


ssoi 


8797 


8791 


8784 


123 


4 5 


41 


9-8778 


8771 


8765 


8758 


8751 


8745 


8738 


8731 


8724 


8718 


123 


5 6 


42 


9-8711 


8704 


8697 


8690 


8688 


8676 


8669 


8662 


MM 


8648 


123 


5 6 


43 


9-8641 


8634 


8627 


8620 


8618 


8606 


8598 


8591 


8584 


8577 


124 


5 6 


44 


9-8569 


8562 


8555 


8547 


8540 


8532 


8525 


8517 


8510 


8502 


124 


5 6 



LOGARITHMIC COSINES 

Subtract Mean Differences. 



185 





0' 


6' 


\9> 


18' 


04 


TO' 


Oft' 


4O' 


AQ 


r 1 


Mean L)iff< 


:rences. 








MM 


AO 


>* 


ov 


oo 


Ji 


VJB 


ov 


V V V 


4' 5' 


D 




0407 


<*. 1 VI I 


x 1 " 


V 1 .' \ 


V 1 - - 


g j IQ 


Q 4 4 1 


o f Q 


.)/* 






46 


'.* i '.'"> 
9-8418 


- * < 
8410 


s l x " 
8402 


i s 1 i - 

8394 


> 11. 1 
8386 


x 1 i 

8378 


o44 
8370 


0441 

8362 


MM 

8354 


x I -* 

8346 


134 


5 7 


47 


9-8338 


8330 


8322 


8313 


8305 


8297 


3*81 


8280 


8'.' 72 


8264 


1 3 4 


6 7 


48 


9-8255 


8247 


8238 


>.:;< 


- 


8213 


8204 


sm:, 


8187 


8178 


134 


6 7 


49 


9-8169 


8161 


8152 


U4I 


8134 


8125 


BUI 


8108 


S099 


8090 


1 3 4 


6 7 


50 


9-8081 


8072 


MM 


BOM 


- 


8035 


8026 


8017 


8007 


7998 


i' 3 5 


6 8 


51 


'.'7 .<--.' 


7979 


7970 


;'.M;M 


7951 


7941 


7932 


7922 


7913 


7903 


2 3 5 


.; s 


b'2 


9-7893 


7-M 


7874 


7-r,| 


7s:.l 


7844 


7835 


7825 


Tsi.-i 


7805 


2 :\ :, 


7 8 


53 


9-7795 


1 - 


7774 


77.; i 


77.M 


7744 


7734 


7723 


7713 


7703 


:! 3 5 


7 y 


54 


9-7692 


7682 


7671 


7661 


7650 


7640 


7629 


78U 


7607 


7597 


245 


7 9 


55 


9-7586 


7575 


7564 


7553 


7542 


7531 


7520 


TIM 


7498 


7487 


246 


7 9 


56 


9-7476 


7444 


7453 


7442 




7419 


7409 


r.:\: 


7384 


7373 


1 6 


8 10 


57 


9-7361 


7349 


7338 


7326 


7314 


7302 


7'-".", 


7J78 


7266 


7254 


246 


8 10 


58 






7218 


7.".-. 


7193 


7181 


7168 


7156 


7144 


7131 


t 6 


8 10 


59 


9-7118 


7106 


7093 


7080 


7061 


7055 


704J 


7029 


7016 


7003 


a 4 Q 


9 11 


60 


M9M 


6977 


6963 


MM 


6937 


6923 


6910 


6896 


6883 


6869 


247 


9 11 


61 


M8M 


6842 


6828 


814 


6801 


6787 


6773 


B7M 


-,711 


6730 


257 


9 12 


62 


9-6716 




M81 


6673 


MM 


6644 


6629 


Mil 


6600 


6585 


257 


10 12 


63 


9-6570 


,;.-,.-; 


6541 


6526 


6510 


6495 


6480 


6465 


6449 


MM 


358 


10 13 


64 


9-6418 


MM 


6387 


6371 


MM 


6340 


6324 


6308 


6292 




358 


11 13 


65 


MSM 


Ml 




6210 


194 


6177 


6161 


6144 


6127 


6110 


368 


11 14 


66 


9-6093 


6076 


6059 


6042 




6007 


:,'.''.> 


M7I 


MM 


5937 


6 9 


12 15 


67 


9-5919 


5901 


5883 


.>,;:, 


5847 


5828 


5810 


5792 


5773 


B7M 


6 9 


12 15 


68 


:.-,;:..; 


5717 


5698 


5679 


5660 


M 1 1 


:.._ i 


5602 


5583 


5563 


6 10 


13 16 


69 


MMI 


5523 


5504 


:.IM 


5463 


M4J 


5423 


1401 


5382 


5361 


7 10 


14 17 


70 


.-:.:. il 


1880 


5299 


:..-> 


5256 


MM 


5213 


5192 


5170 


5148 


7 11 


14 18 


71 


9-5126 


5104 


6082 


:,..,;,, 


BM1 


1011 


I '."._ 


UM 


I'.'l', 


4923 


8 11 


15 19 


72 


Ml 


4876 


I8M 


1 .-. 


MM 


4781 


4757 


4733 


L7W 


4684 


S 12 


!; M 


73 


MMI 


MM 


MM 


l.-.-l 


MM 


4533 


4508 


MM 


MM 


4430 


9 13 


17 1M 


74 


M40I 


4377 


1880 


\:.-:: 


4296 


4269 




4214 


4186 


4158 


5 9 14 


i- M 


76 


9-4130 


4102 


4073 


4044 


4015 


3986 


;.:,; 


8927 


3897 


;sr.7 


5 10 15 


20 24 


78 


9-3837 




1771 


:;7i:, 


8713 


3682 


n;:,. 


8618 


3586 


C..1I 


5 11 16 


21 26 


77 


9-3521 


MM 


MM 


.TI-JI 


8387 


3363 


;:;!.' 


8284 


3260 


3214 


6 11 17 


23 28 


78 


B*817I 


U4I 


3107 


:;..;.. 


8034 


2997 


:.'.-.'.. 


2921 


MM 


jt:. 


6 12 19 


25 31 


79 


:,..; 


2767 


2727 


:>; 


2647 


MM 


:;,.;;, 


BM 


2482 


.'I.V.I 


7 14 20 


27 34 


80 


.:,. .7 


2363 


2310 


-M; 


2221 


2176 


: i :; i 


2086 


2038 


i '.-., i 


8 16 28 


30 38 


81 


r.'i:; 


1895 


1847 


1797 


1747 


1697 


! 1' 


1694 


1542 


!!-, 


8 17 26 


34 42 


:.' 


9-1436 


1381 


1826 


1-J71 


1214 


1167 


i '-.. 


1040 


0981 


'.'I' 


10 19 29 


38 48 


83 


M8M 


0797 


0784 




M 


UM 




0403 


0384 


'..,, 


11 22 88 


44 65 


84 


H)1M 


0120 


1041 


B81Q 


i8M 


Mil 


9736 


too 


Mil 


.,s., 


18 26 89 


M M 


85 


9-9403 


9816 


'-.'; 


.-: . 


1041 


MM 


--I-.. 


8749 


8647 


v,i:, 




64 80 


Si 


B*84M 


8326 


8213 




7979 


7867 


. 


-',,; 


7468 


;:;:,< 






87 


|718I 


7041 


.--'.. 


....i 


8881 


6397 


.-"-" 


DM 


6842 


,r,|.i 






88 


B-MM 


5206 


4971 


ITM 


4459 


4179 


>KM 


IMI 


3210 


v.: 






89 


8-2419 


1961 


1460 


OS70 


M8 


9408 


MM 


7190 


6429 


2419 







ISti 



LOGARITHMIC TANGENTS 

























Mr. HI DillVi. 







6 


12* 


18 


24' 


30 


38 


42' 


48 


54' 




























r 2- 3' 


4 5' 


r 


00 




5429 


7190 


8439 


9409 


0200 


0370 


1450 


ins:; 






i 




2833 


3211 




1881 


4181 


IK'.l 


LTSfi 


1071 


5208 








8-5431 


5643 


5845 


BOM 


6223 


6401 


6571 


6786 


6894 


7046 












7475 


7f."'.' 


7739 


7865 


7988 


8107 


8223 


8336 








8-8446 


8554 


MM 


8762 


8862 


8960 


M)5 


9150 


9241 


9331 


16 32 48 


64 81 




t-MJO 


9506 


PHI 


9674 


9756 


9836 


9915 


9992 


0068 


0143 


13 26 40 


53 66 




9-0216 


0289 


0360 


0430 


0499 


0567 


0633 


OG99 


0764 


0828 


11 22 34 






9-0891 


0954 


1015 


1076 


1135 


1194 


1252 


1310 


1367 


1423 


10 20 29 


."'.i I'.i 




9-1478 


1533 


1587 


1640 


1693 


1745 


1797 


1848 


L898 


1948 


9 17 26 


35 43 




9.1997 




2094 


Jll-J 


2189 


2236 


2282 


2328 


2374 


2419 


8 16 23 


31 39 


HP 


9-2463 


2507 


2551 


MM 


2637 


2680 


-'7l'2 


2761 


2805 


2846 


7 14 21 


28 35 


11 


9-2887 


2927 


2967 


3006 


3046 


3085 


3123 


31G2 


3200 


3237 


6 13 19 


26 32 


12 


9-3275 


3312 


3349 


3385 


3422 


M58 


3493 


3529 


3564 


3599 


6 12 18 


24 30 


is 


9-3634 


3668 


3702 


3736 


3770 


3804 


3837 


3870 


3903 


3935 


G 11 17 


22 28 


14 


9-3968 


4000 


4032 


4064 


4095 


4127 


4158 


4189 


4220 


4250 


5 10 16 


21 26 


15 


9-4281 


4311 


4341 


4371 


4400 


4430 


4459 


4488 


4517 


4546 


5 10 15 


20 25 


16 


9-4575 


4603 


4632 


4660 


4688 


4716 


4744 


4771 


4799 


4826 


5 9 14 


19 23 


17 


9-4853 


4880 


4907 


4934 


4961 


4987 


5014 


5040 


5066 


5092 


4 9 13 


18 22 


18 


9-5118 


5143 


5169 


5195 


5220 


5245 


5270 


5295 


5320 


5345 


4 8 13 


17 21 


19 


9-5370 


5394 


5419 


5443 


5467 


5491 


5516 


5539 


5563 


5587 


4 8 12 


16 20 


20' 


9-5611 


5634 


5658 


5681 


5704 


5727 


5750 


5773 


5796 


5819 


4 8 12 


15 19 


21 


9-5842 


5864 


5887 


5909 


5932 


5954 


5976 


5998 


6020 


6042 


4 7 11 


15 19 


22 


1*064 


6086 


6108 


6129 


6151 


6172 


6194 


6215 


<;:.':>(; 


G257 


4 7 li 


14 18 


28 


9-6279 


6300 


6321 


6341 


6362 


6383 


6404 


6424 


6445 


6465 


3 7 10 


14 17 


24 


9-6486 


6506 


6527 


6547 


6567 


6587 


6607 


6627 


6647 


6667 


3 7 10 


13 17 


25 


9-6687 


6706 


6726 


6746 


6765 


8788 


6804 


6824 


6843 


6863 


3 7 10 


13 16 


26 


9-6882 


6901 


6920 


6939 


6958 


6977 


6996 


7015 


7034 


7053 


369 


13 16 


27 


9-7072 


7090 


7109 


7128 


7146 


7165 


7183 


7202 


7220 


7238 


369 


12 15 


28 


'.'71' :.7 


7275 


7293 


7311 


7330 


7348 


7366 


7384 


7402 


7420 


369 


12 15 


29 


9*7438 


7455 


7473 


7491 


7509 


7526 


7544 


7562 


7579 


7597 


369 


12 15 


80 


9-7614 


7632 


7649 


7667 


7684 


7701 


7719 


7736 


7753 


7771 


3 G 9 


12 14 


81 


9-7788 


7805 


7822 


7839 


7856 


7873 


7890 


7907 


7924 


7941 


369 


11 14 


82 


'.'7'.<:.* 


7976 


7992 


Si MIS 


8025 


9041 


8059 


8075 


8092 


8109 


368 


11 14 


88 


9-8125 


8142 


8158 


8175 


8191 


MM 


8224 


8241 


8257 


8274 


358 


11 14 


84 


1*8990 


8306 


8323 


8339 


B855 


8371 


8388 


8404 


8420 


8436 


358 


11 14 


85 


1*84*1 


8468 


8484 


8501 


-.-.17 


8533 


8549 


8565 


8581 


8597 


358 


11 13 


36 


'..: I. ", 


8629 


-',11 


8660 


8676 


8692 


8708 


8724 


8740 


8755 


358 


11 13 


87 


9-8771 


8787 


8803 


8818 


8834 


8850 


SSI5.1 


8881 


ss'.i? 


8912 


358 


10 13 


38 


9-8928 


8944 


-...v.i 


8975 


MM 


9006 


9022 


9037 


9053 


9068 


358 


10 13 


39 


MOM 


9099 


'..li:- 


9130 


9146 


9161 


9176 


9192 


9207 


9223 


358 


10 13 


40 


9-9238 


9254 


M6S 


MM 


MOO 


9315 


9330 


9346 


9361 


937G 


358 


10 13 


41 


'.':.:;'.'. 


9407 


9422 


Mia 


9453 


9468 


9483 


9499 


9514 


9529 


358 


10 13 


42 


'.Kir, 11 


9560 


'...-,7.-. 


9590 


9605 


9621 


9G36 


9651 


9666 


9681 


:; r, 8 


10 13 


43 


Mtl 


9712 


tni 


9742 


9757 


9773 


9788 


9803 


9818 


9833 


358 


10 13 


44 


9-9848 


9864 


9619 


'.<-.. 1 


.Hi. in 


9924 


9939 


9955 


9970 


9985 


358 


10 13 



LOGARITHMIC 1 TANGENTS 



187 



















Mean Differences. 




ft' 




12' 


18* 


24' 30' 36' 


42' 


48' 


54' 






u 














*l* 


1' 2 3" 4' V 
























45 


10-0000 


0015 


0030 


0045 


'076 


0091 


0106 


0121 


0136 


358 


10 13 


46 


10-0152 


0167 


0182 


0197 


Mil 




0243 


0258 


OT1 


MM 


358 


10 13 


47 


10-0303 


0319 


0334 


0349 




0396 


0410 


04JI 


0440 


358 


10 13 


48 


10-0456 


"171 


0486 


0501 


n.-.i; M 


0562 


0578 


MM 


358 


10 13 


49 


10-0608 




0639 


0654 


0670 






0716 


0731 


0748 


:; :, > 


10 13 


50 


10-0762 




0793 


0808 


0824 


OM9 


0854 


0870 


0885 


0901 


358 


10 13 


51 


10-0916 


0932 




0963 


0978 




loin 


LOM 


l"ll 


1056 


:; :, ^ 


10 13 


52 


10-1072 


1088 


1103 


1119 


1135 


1150 


1166 


1182 


1197 


1213 


358 


10 13 


53 


LO*1SM 


LS4I 


1260 


1-J7.1 


mi 




1340 


1356 


1371 


358 


11 13 


54 


10-1387 




1 1 1 ; 


1 l.'i.-i 


ll.-.l II.;; 11-:; 


i ism 


1516 


i. -:;- 


358 


11 13 


55 




1564 


1580 


1596 


Mil 


L6M 


it;i:, 


1661 


1677 


1694 


358 


11 14 


56 


10-1710 


1726 


1743 


1759 


1776 


1 ::_ 


IS.l'l 


1825 




1858 


358 


11 1J 


57 


10-1875 


1891 


1908 


1925 


194J 




1992 


2008 


2025 


:; r. s 


11 14 


58 


10-2042 


2059 


2076 


2093 


2110 


J1-J7 -.'Ml 


2161 


2178 


I1M 


1 


11 11 


59 




2229 




-'.', 1 


2281 


. 


2333 


M*1 


2368 


3 6 


1 _ 11 


60 


i 10-2386 


2403 


2421 


2438 


2456 


1474 


jiyi 


MM 


2527 


M4B 


3 6 


1 L 1 1 :. 


61 


10-2562 


BMQ 


2598 


2616 


2634 




2670 


1689 


2707 


I7M 


3 6 


12 15 


62 




2762 


2780 


2798 


2817 




js:,l 


2872 


2891 


2910 


3 6 


1 _ 1 :, 


63 


10-2928 


2947 


MM 


MM 


1004 




., !_ 


B061 


3080 


;{ msi 


3 6 


13 16 


64 


10-3118 


3137 


3157 


3176 


3196 




MM 


MM 


3274 


MM 


3 6 10 


13 16 


65 


10-3313 


:;:;:;:; 


3353 


3373 




MM 


3473 


3494 


3 7 10 


LI 17 


66 




1MB 


:;.-,:,:, 


lift 


3596 3f.I 


MM 


3679 


3700 


3 7 10 


11 17 


67 




3743 


3764 


:;7-:, 




3871 


3892 


1914 


4 7 11 


14 18 


68 


10-3936 


:;..:,- 


3980 


1001 




4091 


in:; 


4136 


4711 


15 19 


69 




11-1 


4204 


Itfl 




4319 


tMI 


4366 


4 8 11 


15 19 


70 


10-4389 


ill:; 


4437 


IP, i 


MM 


4509 


i:,:.:; 


4557 


4581 


4606 


4 8 12 


16 20 


71 


LO-4MQ 


MM 


4680 


4705 






1780 


1801 


IMI 


4857 


4 8 13 


17 21 


12 


10-4882 


MM 


4934 


I960 




:,'.:;., 


1064 


MM 


5120 


4 9 13 


i- M 


78 




:.17J 


.'-.'-I 


.-._'_". 




Mil 


5340 


MM 


:,::.- : 


6 9 14 


LI M 


74 




MM 


:,\*:: 


:.:,i-j 




:,r,.m 


5629 


MM 


:,.;-.:. 


5 10 15 


20 25 


75 


10-5719 


6750 


5780 


5811 




5873 


5905 


6936 


-,;.,> 


i: 


6 10 16 


:! 


76 


10-40*1 


6065 


6097 


KM 


net 




r.-.Ti.. 


Mfl I 


MM 


.;:;:;: 


6 11 17 


M M 


77 


1<>. ,.;;,;,; 


M01 


MM 


r.iTI 


6507 




6578 


6615 


MS] 


I',.;HS 


6 12 18 


24 30 


78 


10-0725 


ITU 


MOO 


.>:;- 


;>77 




6954 


(Ml 


7033 


7"::; 


6 13 19 


M M 


79 


i"-7in 


7154 


7196 


MM 






::;.;:; 


7406 


rut 


7I3 


7 11 21 


28 35 


80 


10-7537 


7681 


7626 


7672 






7-11 


nu 


not 


79.', 1 


8 16 23 


31 39 


81 


10-8003 


mi 


8102 


8162 


MM 


8266 


8807 


MM 


Mil 


-n ; 


9 17 26 


36 43 


8L> 


LO*MM 


-:,;; 


MM 


.,;;, 


8748 


8806 


ss,,, 


MM 




.,,, 


10 SO 29 


39 49 


HI! 


10-9109 


9172 


MM 


.'.-.Ml 


...; 






9570 


...i , 


'.'711 


11 12 34 


i , 


84 


LOfTM 


Mil 


Mil 


,H)U, 


MM 


(114 


. 


4M 


MM 


I 'i 


13 26 40 


H , | 


85 


ii > 


.,,,, 


0769 


..,:,.. 








IMI 


i ..11 


1 1 h. 


16 32 48 




H 


11-1504 


1664 


1777 


1893 


2012 




S261 


ftM 


:.-..:. 


,-.,.-. 






87 


L1>MM 


MM 


8106 


::v. i 


MM 


IM1 




. " ' 


I! .. 


I . .. 






88 




UN 


&027 


5275 




II! 


.>: 






89 


11-7681 


MM 


8660 


9130 




:-i 


4571 


7.-.-1 








i:i:< IPROCALS OF NUMBERS 

























20 


4 R ft 


70 g 
























o 


Tt u D 


o u 


10 




Ml 


M04 


>!".' 


815 


'.vj i 


9434 


9346 


9259 


171 


9 18 27 


;.; j:, :,:, 




11 






MM, 


*>:, 


772 


MM 


8621 


8547 


8475 


8403 


8 15 23 


30 38 45 


53 61 68 


1-2 


> 




8197 





8065 




7987 


7874 


7813 


7752 


6 13 19 


26 32 38 


i:, ;,i :,s 


13 








1 


463 


7407 


7:;:,:; 


"299 


7246 


7194 


5 11 16 


22 27 33 


:;s 11 in 


14 








<::'.':. 


944 


1897 


6849 


6803 


6757 


-.711 


5 10 14 


19 24 29 


33 38 43 


15 


. . . : 






6536 


6494 


6452 


6410 


6369 


6329 


6289 


4 8 13 


17 21 25 


29 33 38 


16 








.,;:;:, 


6098 


6061 


6024 


5988 


5952 


5917 


4 7 11 


15 18 22 




17 










5747 


5714 


5682 


(650 


5618 


5587 


3 6 10 


13 16 20 


-'." L'C, -JH 


18 










5435 


5405 


5376 


5348 


5319 


5291 


369 


12 15 17 


20 23 26 


19 








1181 


5155 


5128 


(101 


5076 


5051 


5025 


358 


11 13 16 


is L'l _'! 


20 











4902 


4878 


4854 


4831 


4808 


4785 


257 


10 12 14 


17 19 21 


21 




4739 


4717 


if,-..:. 


4673 


4651 


4630 


4608 


4587 


4566 


247 


9 11 13 


i:. 17 _'!' 


22 


1 :. I :. 


4525 


uoi 


MM 


4464 


HI) 


4425 


4405 


4386 


4367 


246 


8 10 12 


14 16 18 


23 


i :.!> 


4329 


4310 


ISM 


4274 


4255 


US 7 


4219 


4202 


4184 


245 


7 9 11 


13 14 16 


24 




4149 


4132 


an 


4098 


4082 


4065 


4049 


1039 


4016 


235 


7 8 10 


12 13 15 


25 


i ' 


3984 


3968 


3953 


no, : 


3922 


3906 


3891 


3876 


3861 


235 


689 


11 12 14 


26 


:;-|i 


3831 


3817 


3802 


3788 


3774 


3759 


3745 


3731 


3717 


134 


678 


10 11 13 


27 




3690 


3676 


3663 


B660 


3636 


3623 


3610 


3597 


3584 


134 


578 


9 11 12 


28 




,.-,:,'.. 


3546 


3534 


3521 


3509 


3497 


3484 


3472 


3460 


124 


567 


9 10 11 


L>'t 


- 


3436 






3401 


3390 


3378 


3367 


3356 


3344 


i i- :; 


567 


8 9 10 


30 


3333 


usi 


3311 


3300 


:;_'-;' 


3279 


3268 


3257 


3247 


3236 


123 


456 


7 9 10 


31 


M 


mi 


3205 


3195 


3185 


3175 


8165 


3155 


3145 


3135 


123 


456 


789 


32 


3125 


mi 


UOfl 


3096 


3086 


3077 


3067 


3058 


3049 


3040 


123 


456 


789 


83 


:;..:;.. 


Mm 


3012 


:;">:; 


. 


2985 


2976 


2967 


2959 


2950 


123 


445 


678 


34 


'.'941 


...:.:, 




2915 




2899 


2890 


2882 


2874 


2.S65 


123 


345 


678 


35 


2857 


1841 


2841 


2833 


2825 


2817 


JSII'.I 


2801 


2798 


2786 


123 


345 


667 


36 


2778 


2770 


2762 


J7:.:. 


.'717 


2740 


2732 


2725 


2717 


2710 


122 


345 


567 


37 


270S 


MM 


2688 


2681 


2674 


2667 


IT.IHI 


2653 


2646 


2639 


112 


344 


566 


38 


2632 


2625 


2618 


2611 


1604 


2597 


L'.V.tl 


2584 


2577 


2571 


1 1 2 


334 


556 


:.'. 


2564 


2558 




2545 


2538 


2532 


L-.VJ: 


2519 


2513 


2506 


112 


334 


456 


40 




2494 


2488 


2481 


2475 


1468 


2463 


2457 


2451 


2445 


1 1 2 


234 


455 


41 


MM 


2433 


2427 


2421 


2415 


'.'tin 


2404 


2398 


2392 


2387 


1 1 2 


233 


1 ~> :, 


42 


n*i 


.:.;:. 


2370 


2364 


-:;:,-> 


j:i:,:; 


2347 


2342 


j:;:n; 


2331 


112 


233 


445 


43 


.... 


::::' 


mi 


2309 


2304 


__".'.' 


2294 


2288 


8988 


2278 


112 


233 


445 


44 


..;: 


:.. - 


MM 


2257 


8S58 


2247 


2242, 


2237 


2232 


2227 


1 2 


233 


445 


45 


.... 


..'17 


1 1 1 1 


not 




2198 


2193 


2188 


2183 


2179 


1 


L' :.' :; 


344 


46 


.:; 


. : - :< 


nei 


2160 


2155 


2151 


2146 


2141 


2137 


2132 


1 


223 


344 


47 


.:. 


.:... 


2119 


2114 


2110 


2106 


2101 


2096 


Jiilrj 


2088 


1 


223 


344 


48 


. -. 


: . . 


I .;:. 


2070 


.;, 


2062 


2058 


2053 


JO 111 


.'()!.- 


1 


223 


334 


49 


. i 


. . 


: ,:;:; 


2028 


. 


2020 


2016 


2012 


2008 


20H4 


1 


222 


334 


60 


. 


; .... 


1 ..:._< 


1988 


1984 


!'."< 


1976 


1972 


1 :';:. 


1965 


1 1 


222 


334 


51 


1961 


1 '.- -, 7 


1958 


1949 


1946 


1942 


1938 


1934 


1931 


1927 


Oil 


222 


333 


.v.' 


: . 


l-.-l'.. 


L914 


1912 


].,,,, 


1905 


1901 


1898 


IS:M 


IHW 


Oil 


122 


3 3 3 


53 


:-- 


1--:, 


!-- 


1876 


1873 


1869 


1866 


1862 


1859 


1855 


Oil 


122 


233 


54 


:- 


1-1- 


1845 


1842 


1V> 


L8M 


1832 


1828 


1825 


1821 


Oil 


122 


233 



RECIPROCALS OF Nf.MUKKs 



189 



















7' 




Q 




























9 








55 


1818 


1815 


1812 


lsn> 


is,,.', 


1802 


1799 


1 7'.':. 


1792 


1789 


1 1 




2 :; I 


56 


IT-: 


1783 


1779 


1776 


1773 


177" 


17-; 7 


1 70 1 


1761 


1757 


(i 1 1 


2 2 


2 1 I 


57 


17M 


1751 


1748 


1745 


1742 


1739 


1736 


1733 


1730 


175.' 7 


1 1 




2 2 1 


58 


i :_ i 


1721 


1718 


171* 


1712 


1709 


1706 


1704 


1701 


ir.'.i.S 


(111 


1 -2 


223 


59 


1695 


1692 


H>'.' 


l;sr, 


1684 


1681 


1678 


1675 


L671 


1669 


1 1 


1 2 


2 i' :. 


6C 


1GG7 


1 i:.; l 


1661 


1658 


1656 


1653 


1650 


1647 


L64J 


1642 


II 1 1 


1 2 


223 


61 


1639 


1637 


1634 


1631 


1629 


1626 


1623 


Itl-Jl 


1618 


1616 


(1 1 1 


1 1 2 




62 


1613 


1610 


1608 


1605 


1603 


1 .; .. i 


1597 


1595 


LMI 


1590 


< 1 1 


112 


2 2 2 


63 


1587 


1585 


1582 


1580 


1577 


1575 


1071 


1570 


LM7 


1565 


.1 II 1 


1 1 1 




64 


1563 


1560 


1558 


IMI 


1553 


LMO 


i:.i> 


1546 


LMI 


1 :. u 


1) II 1 


1 1 1 




65 


1538 


1586 


1534 


i:.:;i 


I :.-.".' 


1.VJ7 


!524 


LMI 




1.-.17 


o 1 


1 1 1 




66 


K.i:. 


1513 


1511 


1508 


L5M 


1504 


1502 


1 in-.* 


1 l'7 


1495 


(I 1 


111 


2 -' 2 


67 


i '.':; 


1490 


1488 


1 1-1; 


1 1-1 


1481 


1 1 IV 


1 177 


1 17:. 


1473 


I) 1 


1 1 1 


2 2 2 


68 


1471 


1468 


MM 


Itr.i 


1462 


1460 


1 :.> 


1 l.ir, 


1453 


1451 


u n i 


1 1 1 




69 


111'.' 


1447 


1445 


1443 


1111 


1439 


1437 


i :;:. 




1431 


u n i 


1 1 1 


2 2 _' 


70 




1427 


1425 


1 22 


1 12" 


1418 


1416 


L414 


1112 


1410 


1 


1 1 1 




71 


1408 


1406 


11"! 


MM:; 


1401 


1399 




LSM 


1393 


1391 


(1 1 


1 1 1 


1 2 2 


72 


1389 


i:;-7 


1385 


- 


1S81 


1379 


1377 


L876 


1 :; 7 1 


1372 


.1 (1 1 


1 1 1 




73 




1368 


1366 


LMI 


1362 


1361 




LS57 


1355 


1353 


001 


1 1 1 


1 2 


74 




1350 






1344 


1342 


LMO 


Lilt 


1337 


LSM 


1 


1 1 1 


1 


75 


1333 


1332 




1328 


1326 


1325 




1321 


1319 


1318 


001 


1 1 1 


1 2 


76 





1114 




111] 


1309 


1307 


l :;.:, 


L804 


LJ02 


1300 


1 


1 1 


1 


77 


1299 


12 '.'7 




i _".' i 


H 




1189 


L287 


1285 


1284 


000 


1 1 


1 


78 




- 


1171 


1177 


1276 


U74 


LI 71 


1171 


1269 


I2t;r 


000 


1 1 


1 


79 




r.-r.i 




L261 


1259 


1258 


i .:.t; 


12:.:. 


1253 


12:. 2 


000 


1 1 


1 


80 






LM7 


1141 


L 144 




124] 


L1S9 


L1M 


i-.'.-u; 


ii n u 


1 1 


1 


81 










1229 


L117 


1221 


1224 


1222 


1221 


u it n 


1 1 


| 


82 






1117 


1215 


1214 


ini 


111] 




1208 


1206 


u ii n 


1 1 


1 


83 


1205 






1200 


1199 




1 1 I"'. 


1 1 .:. 


1193 


1192 


it u it 


1 ! 


1 


84 


1190 


L18f 




1186 


L188 




1181 


1181 


1179 


1178 


n n n 


1 1 


1 


85 




1 1 7:. 


1114 


1 1 7'.' 


1171 




L1M 


L167 


1166 


1164 





1 1 


1 1 


86 




il'-.i 


1160 


! 1 N 


II. -.7 




1 1:.:. 




1 1 :.-.' 


L1M 




1 1 


1 1 


87 


11 in 




1147 


1145 


Illl 




1 1 12 


11 i" 


1139 


li:: 




1 1 


1 1 


88 


1136 




1134 


1133 






Hi".. 


L117 


L1M 


1125 


000 


1 1 


1 


89 




1122 


1121 


1120 


1119 


1117 


ill*. 


in:. 


111 1 


1 II 2 


< i n 


1 1 


1 


90 


mi 


111" 


1109 


1107 


LlOi 


L1M 


ll"| 




11"! 







1 1 


1 


91 


LOM 




LON 


1005 








L0] 


1089 


IMSX 


M n ii 


'i 1 


1 


9-J 






1085 


1088 








L07I 




I".', 




1 


1 


93 


1076 


1"7I 


1071 


1072 


L071 










1 1 


ii n n 


1 


1 


94 


: ",i 




LOU 


LOOO 










1055 




n ii (i 


1 


1 


<5 


LOM 


1052 


1050 


: MM 










1044 


L041 




1 


1 


96 


L04J 




1040 


LOU 
















1 


1 1 


97 


LOU 




ion 


LOM 








1 




1021 


n n n 


1 


1 1 


!.H 






1018 


!! ; 






I'M | 




I"!'- 1 


1"11 




1 


' ' 


99 


1010 




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