Arthur Winslow.

Stadia surveying : the theory of stadia measurements, accompanied by tables of horizontal distances and differences of level for the reduction of stadia field observations online

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ISmo, Boards. Price 50 Cents Each.

'ractice by Richard IT

wiin results oi:
Buel, C.E.

No. 11.— THEORY OP ARCHES. By Prof. W. Allan

VV. E. Cain.

Atkinson. Third edition, revised and enlarged
by Edward H. Williams, jun.
No. 14.-FRICTION OF AIR IN MINES. By J. J. Atkinson
No. 15.— SKEW ARCHES. By Prof. E. W. Hyde C E Il-

L. Vose.

:r. .. 17.— WATER AND WATER- SUPPLY. By Prof W H
Corfleld of the University College, London.
Prof. W H. Corfield, M.A., of the University
College, London.



Accompanied by Tables of Horizontal Distances

and Differences of Level for the Reduction

of Stadia Field Observations.



Assistant Geologist, Second Geological Survey of Pennsylvania.

Republished by permission from First Report of Progress in
the Anthracite Region, Second Geological Survev of




23 Murray and 27 Warren Street.



« -c c c« <

" J c c <

c C '■f c c c<

c c t c c ' <

^f V

c c c c c c
, c c c
etc '
c c c c c

r'r C C C , C,


The rapid extension of the practice of
stadia measui'ements has naturally created
a demand for a guide to the method.
The present hand-book contains a com-
plete exposition of the theory, with di-
rections for its application in the field.
The tables for reduction of observations
have been in use by the author on the
Geological Survey of Pennsylvania.

To increase the serviceableness of the
book the trigonometrical four place tables
have been added.

Editor of Magazine.


The fundamental principle upon which
stadia measurements are based, is the
geometrical one that the lengths of par-
allel lines subtending an angle are pro-

portional to their distances from its apex.
Thus if, in Fig. 1, a represents the

* Tbe credit of having first introduced this method
of measurement into this country would seem to be-
long to Mr. John R. Mayer, a French Swiss. It was
used by him as early as 1850 ; and subsequently, during
bis connection with the United States Lake Survey,
he did much towards perfecting the instruments and
improving the methods of work. An essay by him in
the Journal FranMin Institute for January, 1865, con-
tains a short historical sketch of the development of
topographical surveying and a brief discussion of the
general principles of btadia measurements.


leng'.v'K ot a line subtending an angle at a
distan<9e d from its apex, and a' the length
of line, parallel to and twice the length of
«, subtending the same angle at a dis-
tance d' from its apex, then will d' equal

This is, in a general way, the underly-
ing principle of stadia work ; the nature
of the instruments usedg however, intro-
duces several modification::, and these will
be best imderstood by a. consideration of
the conditions undc:; which such measure-
ments are generally made.

In the telescopes of most instruments
fitted for stadia work, thore are placed
either two horizontal wires (usually ad-
justable) or a glass with two etched hori-
zontal lines at the position of the cross
wires, and equidistant from the center

A self -reading stadia rod is further
provided, graduated according to the
units of measurement used.

In a horizontal sight with such a tele-
scope and rod, the stadia wires seem to
be proiected upon the rod and to inter-

cept a distance which in Fig. 2 is repre-
sented by a.

In point of fact there is formed, at the
position of the stadia wires, a small con-


jugate image of the rod which the wires
intersect at points b and c, which are re-
spectively the foci of the points B and G
on the rod. If, for simplicity's sake, the
object glass be considered a simple bi-
convex lens, then, by a principle of oj)tics,
the rays from any point of an object con-
verge to a focus at such a position that a
straight line, called a secondary axis, con-
necting the point with its image, passes
through the center of the lens. This
point of intersection of the secondary axes
is called the ojDtical center. Hence, it
follows that lines such as c C and b B, in
Fig. 2, drawn from the stadia wires
through the centre of the object glass
will intersect the rod at points corre-
sponding to those which the wires cut on
the image of the rod. From this follows
the proportion:

dap .^

Where :

c?=the distance of the rod from the
center of the objective ;


j9=the distance of the stadia wires from
the center of the objective ;

«==the distance intercepted on the rod
by the stadia wires ;

I — the distance of the stadia wires

If p remained the same for all lengths
of sight, then ~ could be made a desir-
able constant and d would be directly
proportional to a. Unfortunately, how-
ever, for the simplicity of such measure-
ments, p (the focal length) varies with
the length of the sight, increasing as the
distance diminishes and vice versa. Thus
the proportionality between d and a is

The object, then, is to determine ex-
actly what function a is of d and to
express the relation in some convenient

The general formula for bi-convex
lenses is:

- + -^=>. ... (2)


/ is the principal fo:al lengtli of the
lens, andji? and jt>' are the focal distances
of image and object, and are approxi-
mately the same as p and d, respectively,
in equation (1) :

therefore, - +- =-, approximately.

^d d ^
and-=-— 1
p f

_ ,^. d a

From(l), -=-

2) I

a _d

■'■ i=r

whence d=-a+f . . . (3)

In this formula, it will be noticed that,
as f and Z remain constant for sights of
aU lengths, the factor by which a is to be
multipHed is a constant, and that d is
thus equal to a constant times the length
of «, plus /. This formula would seem,
then, to express the relation desired, and
it is generally considered as the funda-
mental one for stadia measurements. As
above stated, however, the equation


1 1_1

V ^"/
is only approximately true and the con-
junction of this formula with (1) being?
theref jre, not rigidly admissible, equation
(3) does not express the exact relation.*
The equation expressing the true rela-
tion, however, though differing from (3)
in value, agrees with it in form and alsa
in that the expression corresponding to

:=r IS a constant and that the amount ta

be added remains, practically, f. The

constant corresponding to - may be

called ^t and thus the distance of the
rod from the objective of the telescope i&
seen to be equal to a constant times the
reading on the rod, plus the principal
focal length of the objective. To obtain
the exact distance to the center of the in-

* This is demonstrated on page 21.

+A;is dependent upon I and can, therefore, be made-
a convenient value in any instrument fitted with ad-
justable stadia wires. It is generally made equal to
100, so that a reading on the rod of 1 corresponds to a.
distance of 100+/.


strument, it is further necessary to add
the distance of the objective from that
centre, to /; which sum may be called c.
The final expression for the distance,
with a horizontal sight, is then

d=ka-\-€ . . (4)

The necessity of adding c is somewhat
of an incumbrance. Tn the stadia work
of the United States Government sur-
veys an approximate method is adopted
in which the total distance is read di-
rectly from the rod. For this method
the rod is arbitrarily graduated, so that,
at the distance of an average sight, the
same number of units of the graduation
are intercepted between the stadia wires
on the rod, as units of length are con-
tained in the distance. For any other
distance, however, this proportionality
does not remain the same ; for, according
to the preceding demonstration, the
reading on the rod is j^J'oportional to its
distance, not from the center of the in-
strument, but from a point at a distance
" c " in front of that center ; so that,


when the rod is moved from the posi-
tion where the reading expresses the
exact distance to a point, say half that
distance from the instrument center, the
reading expresses a distance less than
half ; and, at a point double that distance
from the instrument center, the distance
expressed by the reading is more than
twice the distance. The error for all
distances less than the average being
minus, and for greater distances plus.
The method is, however, a close approxi-
mation, and excellent results are obtained
by its use.

Another method of getting rid of the
necessity of adding the constant was de-
vised by Mi*. Porro, a Piedmontese, who
constructed an instrument in which there
was such a combination of lenses in the
objective, that the readings on the rod^
for all lengths of sight, were exactly pro-
portional to the distances.* The instru-

* A notice of this instrument will be found in an
article by Mr. Benjamin Smitii Lymon, entitled " Tele-
scopic Measurements in Surveying," in Jour. Frank-
lin Inst., May and June, 1868, and a fuller description
is contained in Armaks des Mines, Vol. XVI, fourth


ment was, however, bulky and difficult to
construct, and never came into extensive

For stadia measurements with inclined
eights there are two modes of procedure.


One, is to hold the rod at right angles to
the line of sight ; the other, to hold it
vertical. With the first method it will
be seen by reference to Fig. 3, that the
distance read is not to the foot of the
rod, E, but to a point, f, vertically under
the point, F^ cut by the center wire. A
correction has, therefore, to be made for
this. An objection to this method is the
difficulty of holding the rod at the same
time in a vertical plane and inclined at a
definite angle. Further, as the rod
changes its inclination with each new po-
sition of the transit, the vertical angles
of back and foresight are not measured
from the same point.

The method usually adopted is the-
second, where the rod is always held ver^
tical. Here, owing to the oblique view
of the rod, it is evident that the space in-
tercepted by the wires on the rod varies,
not only with the distance, but also with
the angle of inclination of the sight.
Hence, in order to obtain the true dis-
tance from station to station, and also it»
vertical and horizontal components, a


correction must be made for this oblique
view of the rod. In Fig. 4,

AB=«=the reading on the rod ;
MF^t^=the inclined distance=c + GF


MP = D = the horizontal distance = d

cos 71,
FP — Q = the vertical distan ce = D tan n
?2 = the vertical angle,

It is first required to express d in terms
of a, n and m.

From the proportionality existing be-
tween the sides of a triangle and the sides
of the opposite angles,

AF sin 711

GF sin [90° + (n — w)]

or, AF=:GF sin^Ti-


cos {n — m) '

BF sin ni

GF' "^ sin~[90^^^(^M^m)]


or, BF = GFsin m

cos {n + rn)


{n — m)



cos (n + m)y



i -\-BF = ff, and GF=

2 tan m

CD cos m
'Z sin in

^y substituting and reducing to a com-
mon denominator,

_CD Gosm [cos (n-[-m)-\- COS (71—971)2

2 cos {71 -\- ?n) cos {)i — m)

Reducing this according to trigono-
metrical formulae,

cos n cos ni
as c?=MF=c + ^\CD,

clz=zc + h a

cos'^ n cos'' m — sin^ ?^ sin^ m

cos ?^ cos ???

The horizontal distance, T>^=zd cos n.

/. D=(3 cos n + ^05 cos^/2 — i^ a sin"?z tan^ ??2..

*'The third member of this equation,
may safely be neglected, as it is very
small even for long distances and large
angles of elevation (for 1500', n=4t^° and
^— 100, it is but 0.07'.) Therefore, the
final formula for distances, with a stadia


kept vertical, and with wires equidistant
from the center wire, is the following :"

'D=c cos n-\- a k Gos^n . . (5)

The vertical distance Q, is easily ob-
tained from the relation : Q==D tan n.

.-. Q=c sin 71 + a k cos 7^ sin ?i

^ . , sin 2?2 ,^,^

orQ,=G smn + a/c — - — . (6)*

With the aid of formulae (5) and (6) the
horizontal and vertical distances can be
immediately calculated when the reading
from a vertical rod, and the angle of ele-
vation of any sight are given ; and it is
from these formulae that I have calcu-
lated my stadia reduction tables. The

values of ak cos^?i and ak — ^ — were sep-

arately calculated for each two minute™
up to 30 degrees of elevation ; but, ag
the value of c sin n and c cos n have

* The above demonstration is substantially that
^ven by Mr. George J. Specht, in an article on Topo-
graphical Surveying in Tan Nostrand's Engineering
Magazine for February, 1880, though enlarged and cor-


quite an inappreciable variation for 1 de-
gree, it was thought sufficient to de-
termine these values only for each de-
gree. As c varies with different instru-
ments these last two expressions were
calculated for three different values of c,
thus furnishing a ratio from which values
of c sin 71 and c cos n can be easily deter^
mined for an instrument having any con-
stant (c).

Similar tables have been computed by
tT. A. Ockerson and Jarecl Teeple, of the
United States Lake Survey. Their use
is, however, limited, from the fact that
the meter is the unit of horizontal meas-
urement while the elevations are in feet.
The bulk of the tables furnish differences
of level for stadia readings up to 400
meters, but only up to 10° of elevation.
Supplementary tables give the elevations
up to 30° for a distance of one meter.
For obtaining horizontal distances refer-
ence has to be made to another table^
which is somewhat an objectionable fea-
ture, and a multiplication and a subtrac-
tion has to be made in order to obtain


the result. Last, but not least, these
tables are, apparently, only accurate
when used with an instrument whose
constant is 0.43 meters.

As stated in the preceding discussion
(p. 11), the generally accepted formula
expressing the relation between the dis-
tance in a horizontal sight, the reading on
the rod, the distance of the stadia wires
apart, and the focal length of the object-
ive is

d=:j a + f . . . (3)

where d, a, I and /' represent these fact-
ors respectively.

This formula is derived from the con-
junction of the two equations :

d=^^a', . . . . (1)

and -+—-=>; . . . (2)
P P f '

p and jt> in (2) being considered as equal
to p and d in (1). p and d in (1), it will
be remembered, are the distances from
the center of the objective to the image


and object respectively. But the general
formula for lenses, (2), is derived on the
supposition that p and p' are measured
from the exterior faces of the lens, and
therefore p and d m (1) are each greater,
by half the thickness of the lens, than p
and p' in (2). Further, this formula is
derived on the supposition that the ob-
ject glass of the telescope is a simple, bi-
convex lens, whereas, in fact, it is a com-
pound lens composed of a piano concave
and a biconvex lens. Now, though these
points may seem insignificant in them-
selves, they may greatly influence the
final result, as a difi'erence of only 1 in
the denominator of such a fraction as

-^ — ^ may alter the result by as much

as 500,000. Considerable thought and
time has, therefore, been given to the
consideration of the effect of these cor-
rections, and, as a result, it was found
that the formula (3) does not express the
true relation even within practical limits ;
and that if it were attempted to calculate
the distance, c?, by this formula, when


tlie factors /*, ^ and a were given, a re-
sult would be obtained which would dif-
fer considerably from the real distance.
The inaccuracy lies in the expressioni


-. The one to be substituted for it is,

however, like it, a constant for each in-
strument ; and, as we determine the value
of this constant by actual trial and not
from a knowledge of the values of f and
I, the correction to be made will not af-
fect the practice.

Considering first the case of a tele-
scope with a simple, biconvex lens, the
optical center being, here, in the center
of the lens, d andjo, in equation (1), as
before stated, are measured from the
center of the lens, while, in equation (2),
p and p' are measured from the exterior
faces. If the thickness of the lens be-
taken as 2cc, then

p in equation (l)=ii:> in equation (2),.
minus x ; and

p' in equation (l)=d in equation (2),
minus x.


Therefore, while (1) remains

d= -^a, or p~~d . (la)

J- \Af


by substitution, (2), becomes,
I 1 1

/)— 93 d — X~f

Substituting d- for p in (2a)


1 1 1



,1 d — X f

d X '^


^\ d—x-hd x=-ld—x] Id a?. I

—~ ~ (jj — — ~ (aj~'X ~ Cv X -\ — ~^ X

f a f a f f

whence, — 2cc — % cc^ = - d"^-
f f (^

Multiplying both sides by - -,

d T




11/^ 1 .\ 1 .,

«/V fit

Adding to both sides

^ squared,

4 af\fl


/*«/ /«


Extracting the square root of both terms,
f a\ J I fa



f 'If,.

or, d 21

This is the exact formula correspond-
ing to (3), for biconvex lenses. This can,
however, be considerably reduced with-
out materially affecting its value. With
B, telescope of the dimensions of that of
an ordinary engineer's transit, the term

:p {x^ + 2a!/) diminishes the result by about

J of an inch and, therefore, may be neg-
lected. Formula (3a), then becomes :

(I + ^)(^+/)
cl-.A ^i

'lx-{-^f^a.ic + af

Tik e addition of x (half the thickness
of the object glass) would be inappreci-
able in the length of any ordinary sights
and may be omitted. The final expres-
sion becomes, then,

dJ^-^a+f . (3b)

This formula, it will be observed, dif-
fers from (3) in that the reading on the
rod {a), is multiplied b}^ x-\-f instead of
A The numerical difference between the
results is seen in the following examples:

Consider first the case with a one-foot,
reading on the rod, and let x=^ .18",.
/•=:9.00", and I=.08".*

Formula (3) becomes, then:

^^^ 12.00'' + 9.00" = 1359"r=113.25';;

* These are very closely the dimensions in Heller &
Brightly's large Surveyor's Transit (5-inch needle), a&
kindly furnished me by Mr. Heller.


Formula (3b) becomes-


+ 9.00"=1386 = 115.50'


Differences 2.25'

When the reading on the rod is 5 feei
(or 60") then, (3) becomes:

9 00"
r7=—_- 60.00'' + 9.00= 563.25';

«iid (3b) becomes:

^ .18'' + 9.00'' ^ ,
^= ?T577 60.00'' + 9.00=574.50'

Difference =11.25'*

The above demonstration shows, then,
that, with a simple biconvex object glass,
the nsuall J accepted formula expressing
the relation between the distance, the
reading on the rod, the distance of the
stadia wires apart, and the focal length
of the objective, is not accurate even

* As the difference is evidently proportional to the
length of sight, with a lOCO' sight it would amount to
22.5', etc.


within the limits of accuracy of such
measurements. With the usual combi-
nation of lenses in objectives this error
would still remain. The derivation of a
formula similar to (3^), for such lenses,
would, however, be extremely difficult,
and would only hold for the special Ifens
in question. For, with such a combina-
tion of lenses, the optical center would
no longer remain in the center of the
lens, but would vary its position accord-
ing to the relative thicknesses of the two
glasses, their radii of curvature and their
indices of refraction ; and, after its posi-
tion had been determineu. by abstruse
calculation and refined experiment, its
distance from the two exterior faces of
the compound lens would be expressed
by tioo different values {x and x') instead
of two equal values {£) ; and this would
very much complicate further calcula-

It was seen that, in the newly deduced
formula, for biconvex objectives, like that
heretofore accepted, the factor by which
the reading on the rod is multiplied is a

constant for each instrument, and that
the practical method of adjusting the
instrument remains the same. The
question now arises, does this remain tha
case with a compound objective ?

In view of the dif&culty of demonstrat-
ing this mathematically it was decided to
make a j^ractical test of this point with a
carefully adjusted instrument. A dis-
tance of 500 feet was first measured off
on a level stretch of ground, and each 50
foot point accurately located. From one
end of this line three successive series of
stadia readings* were then taken from
the first 50 foot and each succeeding 100-
foot mark. The following table contains
the results :

* The readingrs were taken from two targets, set so
that the sight should be horizontal and thus also pre-
venting any personal error or prejudice from affect-
ing the reading









iC QO J>



10 10 iO {>• Tfi «3



00 00 00 00 00 00


•^ Oi 05 Cti 05 OS

' th (7;i CO '^






10 co-^ t- C5



00 0000 00 00 Ci


tH C5 05 05 05 05


' tH <M CO-^l*




00 iO


1:0 J>- CD J> *0


00 00 UO 00 00 00


■^ Oi 05 05 CS Ci


* th (^^ CO Tii







10 lO iO 05 CO



00 00 00000000




* T-i oi CO -^




10 o o o o o
T-i C? CO -* to


Multiplying the mean of these readings'
by 100, and subtracting the result from
the corresponding distance, we obtain^
the following table :

CO r?

s s

C3 f^


CCj j> }> ^ i> o

•OO O O^ tH tH

*S s



> 2







lO O O 05 O CO



Ttl O iO 05 O CO











et-i "^ O

O c5 '^


^ g rH


lo i-o o t- "<^ CO

00 OO OO 00 GO oo

S ^ 2i


"^ CI C5 CI CI Oi


-r-lC^i COtJH





o o o o o o


o o o o o o


o o o o o o




T-f O^ CO '^i o


o co-
co -^

00 tH


o o

Si ;-»

s s

'S o


The variations between the numbers
of the column of differences are slight,
the maximum from a mean value of 1.43
feet being only .21 feet. A study of the
tables will show that these variations have
no apparent relation to the length of the
sight ; in the maximum case, the variation
corresponds to a reading on the rod cf only
.0021 feet (an amount much within the
limits of accuracy of any ordinary sight).
'We are, therefore, perfectly justified in
concluding that these variations are acci-
xiental, and that the " difference " is, for
all practical purposes, a constant value.

"We thus see that with a telescope hav-
ing a compound, plano-convex objective,
whatever the formula may be expressing
ihe relation between d, /, x, etc., the
horizontal distance is equal to a constant
times the reading on the rod plus a con-
stant^ and may, as in the other cases, be
expressed by the equation,

* This may seem a statement of what was already a
-well-known fact. But, heretofore, it has been as-
sumed to be a direct deduction from optical principles.


The many advantages of stadia meas-
nrements in surveying need not be dwelt
upon here, both because attention has
been repeatedly called to them, and be-
cause they are self-evident to every engi-
neer. Neither will it be within the com-
pass of this article to describe the various
forms of rods and instruments, or the
conventionalities of stadia work.

A few precautions, necessary for accu-
rate work, should, however, be empha-
sized. First, as regards the special ad-
justments : care should be taken that in
settiDg the stadia wires* allowance be

and as, according to the preceding article, this is not

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Online LibraryArthur WinslowStadia surveying : the theory of stadia measurements, accompanied by tables of horizontal distances and differences of level for the reduction of stadia field observations → online text (page 1 of 13)