Bulletin No. 307 Series F, Geography, 56

DEPARTMENT OF THE INTERIOR 

UNITED STATES GEOLOGICAL SURVEY

CHARLES D. VVALCOTT, DIRECTOR

MANUAL

TOPOGRAPHIC METHODS

G-ANNETT

WASHINGTON

GOVERNMENT THIN TING OFFICE

'1906

CONTENTS.

Page.

CHAPTER I. INTRODUCTION................................................ 1

Object of this manual................................................ 1

Glasses of maps...................i.................................... 1

Topographic maps...................................................... 2

Contours.............................................................. 2

Scales..............................................................l. 3

Cultural features....................................................... 3

Plan of the map of tho United States................................... 4

Diversity of conditions............................................. 4

Scale............................................................. 4

Contour interval............................................i....... 5

Features represented.......................^...................... 5

Size of sheets................................ .^.................... 6

CHATTER II. METHODS AND CLASSIFICATION OF WORK........................ 7

General methods...................................................... 7

Correctness of maps..................................................... 7

Accuracy of location............................................... 7

Number of locations................................................ 8

Distribution of locations............................................ 9

Quality of sketching................................................ 10

Classification of work.......................'........................... 10

CHAPTER III. ASTRONOMIC DETERMINATION OF POSITION..................... 12

Object and importance of astronomic determination............L.......... 12

Definitions............................................................. 12

Astronomic transit and zenith telescope.................................. 13

Chronograph........................................................... 14

Field work............................................................. 16

Observations for latitude ............................................... 17

Reduction of latitude observations....................................... 18

Measurement of a division ot the head of the micrometer screw ....... 19

Measurement of a level division.................................... 22

Computation of apparent declination of stars........................ 23

Observations for time................................................... 25

Reduction of time observations................................:........ 25

Correction for error of level.......................................... 26

Correction for inequality of pivots.................................. 26

Correction for error of collimation................................... 26

Correction for deviation in azimuth.................................. 27

Correction for diurnal aberration............................. L....... 27

Example......................................................... 27

Comparison of time................................................. 32

Observations for azimuth.............................................. 33

Reduction of observations for azimuth.................................. 35
IV CONTENTS.

Page

CHAPTER IV. BASE LINE, PRIMARY CONTROL, AND ELEVATIONS............... 38

Party organization...................................................... 38

Base-line measurement................................................. 38

Site.............................................................. 38

Methods and instruments used..................................... 39

Personnel of party................................................... 4.1

Reduction of base-line .measurement..................................... 42

Reduction to standard............................................. 42

Correction for inclination.......................................... 42

Correction for temperature. .......... ..................:........... 42

Reduction to sea level............................................. 42

ExampleX........................................................ 43

Primary triangulation.................................................. 45

Selection .of stations................................................ 45

Signals............................................................ 46

Heliotropes.................................................... 47

Theodolites for triangulation........................................ 48

Instructions for'measurement of horizontal angles..................... 49

Organization of parties and prosecution of work........................ 56

Reduction of primary triangulation..........................'............ 57

Reduction to center...........................«... r ................ 57

Spherical excess.................................................... 58

Station adjustment................................................. 58

Figure adjustment.................................................. 60

Computation of distances.......................................... 63

Computation of geodetic coordinates............................._ ... 64

Traverse lines for primary control....................................... 67

Precise and primary elevations......................................... " 69

CHAPTER V. SECONDARY TRIANGULATION AND TRAVERSE AND STADIA MEASUREMENTS ............................................................... 70

Secondary triangulation................................................ 70

The plane table.................................................... 70

The alidade....................................................... 73

Measurement of altitudes.......................................... 75

Traverse work.......................................................... 76

Traverse plane table............................................... 77

Measurements of altitudes in connection with traverse work.......... 78

Vertical angles................................................ 78

The aneroid..........................................!......... 79

Organization of parties and distribution of work.......................... 80

Stadia measurements.... .^............................................ 81

CHAPTER VI. SKETCHING, OFFICE WORK, AND PROJECTIONS................... 83

Sketching............................................................. 83

Scale of field work...................................................... 85

Office work............................................................ 85

Projections........ 1................................................... 85

INDEX. ................................................................... 87
ILLUSTRATIONS.

Page.

PLATE I. Astronomic transit and zenith telescope............................ 12

II. Baldwin tape-stretching device for base measurement................ 40

III. Framed signal for use where observing tower is not necessary......... 45

IV. Signal combined with tower for observer and instrument............ 46

V. Eight-inch theodolite............................................. 48

VI. Telescopic alidade................................................ 70

VII. Johnson plane table and telescopic alidade...................'..... 72

VIII. Traverse plane table and ruler alidade............................. .76

FIG . 1. Chronograph...................................................... 15

2. Johnson plane-table movement (section)............................. 72

3. Aneroid............................................................ 79

4. Works of the aneroid................................................ 79

MANUAL OF TOPOGRAPHIC METHODS."

By HENRY GANNETT.

CHAPTER. I.

INTRODUCTION.

OBJECT OF THIS MANUAL.

The object of this manual is a description of the topographic work, 

instruments, and methods of the United States Geological Survey, 

primarily for the information of the men engaged in this work. It 

is not intended to be an elementary treatise on surveying, as it presupposes a knowledge of the application of mathematics to surveying 

equivalent to that obtained in our professional schools. Neither is 

it intended to be a general treatise on topographic work, altho it 

may, to a certain extent, supply the need of such a work.

Surveys are made for two widely different purposes. One consists 

.in laying down upon the ground certain geometric figures, such as 

a town plat or the alignment of a railroad or an aqueduct. The other 

purpose is the making of maps, or miniature representations of the 

country. While the instruments and operations are to some extent 

the same in both, the purpose and the results are very different.

Most books on surveying have in view mainly, if not entirely, the 

first of these two purposes, operations incident to the making of 

maps being more or less slighted. Herein the author will attempt to 

describe the most approved methods of surveying as applied to the 

production of topographic maps, whether on large or small scales 

and whether of a high or a low degree of accuracy, including rough 

reconnaissance as well as accurate and detailed surveys.

CLASSES OF MAPS.

A map is a representation, in plan, of a part of the earth's surface. 

The variety of maps is legion, depending upon the class of features 

or phenomena which they represent or which is made prominent on 

them. Thus there are geologic maps, zoologic maps, botanic maps,

. a This manual was first published in 1893 as Monograph XXII of the United States Geological

Survey. _ .-

. -' s ' 1 ' '
2 MANUAL OF TOPOGRAPHIC METHODS.

cadastral maps (which represent, primarily, property lines), political 

maps, etc. For all these it is essential that there be a base upon 

which the specific phenomena can be represented. This base should 

show certain topographic features, as streams and other bodies of 

water, railroads, towns, and cities; while some sort of convention to 

represent the relief of the ground is usually a valuable adjunct.

From the point of view of scale, those maps which show only 

topographic features may be grouped in two classes: First, those 

on small scales, say 10 or more miles to an inch, which may be properly called geographic maps and which are usually compiled from 

other maps on larger scales; and, second, those on large scales, which 

may properly be called topographic maps. Maps of the latter class 

are often made directly from surveys, and being the first product of 

these surveys, are known as mother maps, because they constitute 

the source from which other maps are compiled. In this work are 

described the methods of surveying and the preparation of maps on 

large scales maps which can be conveniently made directly from 

surveys.

TOPOGRAPHIC MAPS.

A topographic map should represent rivers, creeks, lakes, ponds, 

and all other bodies of water, together with coast lines; all artificial 

features that are of sufficient prominence to be represented on a 

scale,' such as roads, boundary lines, cities, towns, villages, houses, 

and, in case of large-scale maps, fences and other such objects, 

bridges, fords, dams, canals, aqueducts, etc., and the relief of the 

ground its hills, valleys, mountains, canyons, etc.

The relief of features can be shown by different methods, which 

may be classed as qualitative and quantitive. Qualitative methods 

show relief by shading, either by means of crayon or color or by 

means of hachure lines lines which are drawn in the direction of the 

slope and which by their relative heaviness and closeness produce the 

effect of shading. These methods merely show the degree of slope; 

they give no information regarding the absolute amount of relief, or 

elevation above sea level. They have been in extensive use thruout 

the world, but their use is now rapidly diminishing.

CONTOURS.

The method of expressing relief now commonly in use, which 

has largely replaced the methods described above, is that known 

as the contour method. Contours are lines drawn at equal elevations above sea level, every point upon a contour being, or being 

supposed to be, at the elevation indicated by that line. These contours would become shore lines if the level of the sea were raised 

successively to the heights indicated by them. This method is con-
SCALES.AND CULTURAL FEATURES. 3

fessedly not so graphic as the others, but it has this great advantage, 

that it expresses absolute elevations; the height above the datum 

(usually sea level) of any portion of the area represented can be read 

directly from the contours. Moreover, it expresses the slopes, since 

where they are steep the contours must necessarily be close together, 

and where they are gentle the contours must be far apart. Modern 

maps, as a rule, are contour maps.-

The contour interval is the vertical scale of the map. This should 

be proportioned to the horizontal scale and to the degree of relief 

of the country. With the same horizontal scale the contour interval 

would naturally be large for a country of bold relief and small for one 

of slight relief, since, in the former case the same difference of altitude 

is of much less importance than in the latter case.

SCALES.

The scale of a map is the proportion which exists between the 

dimensions of the map and those of the area which it represents. It 

is designated in several ways:

By the number of miles or of feet represented by an inch on the map.

By what is called a fractional scale, the numerator representing a 

linear unit of the map and the denominator expressing the number of 

such units in a corresponding distance on the ground, as T^OTTBy a bar scale, wherein a measure on a map is marked with its 

corresponding measure on the ground.

On many maps all of these forms of designating the scale are given.

The scale of a map should be chosen so as to represent as much 

detail and as great accuracy as are needed for the purpose for which 

the map is made. It should be neither larger nor smaller than is 

required to suit this purpose. For a small amount of detail a large 

scale is unnecessary, while too small a scale does not allow the legible 

representation of the detail required. Thus the statement of the 

scale of a map should afford an idea of its accuracy and detail.

CULTURAL FEATURES.

The list of cultural features, or works of man, which should be 

represented on a map depends upon the scale of the map and the 

purposes for which it is made. A general topographic map, which 

is designed for use for many years without revision, should contain 

comparatively few of these features only those which are likely to 

be permanent and which are of importance to the community rather 

than to individuals. These features change with considerable rapidity, and keeping the map revised involves the expenditure of much 

labor and money if many such features are represented. For these 

reasons their introduction should be restricted so far as is compatible
4 MANUAL OF TOPOGBAPHIC METHODS.

with the usefulness of the" map. For maps on large scales and for 

temporary purposes the list may be enlarged to, any extent and in 

any desired direction, but the matter of inclusion and exclusion of 

these features should be considered carefully in view of all attendant 

circumstances. It is very easy so to clutter a map with useless 

information as to obscure its really valuable features.

PLAN OF THE MAP OF THE UNITED STATES. 

DIVERSITY OF CONDITIONS.

The field upon which the United States Geological Survey is at work 

is diversified. It comprizes broad plains, some of which are densely 

covered with forests, while on others trees, are absent; high and rugged 

mountains, plateaus, and low, rolling hills. In some regions its topographic forms are on a grand scale; in other regions the entire surface 

is made up of minute detail. Some parts of the country are densely 

populated, as much so as any other region on the earth, while in other 

parts of the country great areas are almost, without settlement. 

Geologically, portions of the country are extremely complex, requiring, for the elucidation of geologic problems, maps in great detail, 

while other areas are very simple.

It is obvious that, because of this diversity of conditions, maps of 

different areas should vary in scale, and that with such diversity in 

natural conditions and in scale there must be differences in the 

methods of work employed. The system which' is found to work 

advantageously in the high mountain regions of the West is more or 

less inapplicable to the low, forested plains of the Mississippi Valley 

and the Atlantic Plain.

SCALE.

The scales which have been adopted for the publication of the map 

are 1:62500, or very nearly 1 mile to an inch; 1:125000, or very nearly 

2 miles to an inch; and 1:250000, or very nearly 4 miles to an inch.

The following considerations have determined the selection of the 

above scales: They are believed to be sufficiently large to admit of the 

representation of all the details necessary to a picture of the country 

which shall show the proper relations of its features and shall be of the 

greatest possible service for industrial and scientific uses consistent 

with other requirements to be mentioned hereafter. These scales 

are also sufficiently large to enable geologists to present the details of 

nearly all geologic phenomena. The map represents the country in 

sufficient detail to admit of the selection of general routes for railroads 

and other public works, and of the indication of boundary lines in 

such way that their positions may be recognized upon the ground. 

On the other hand, the scales are not so large as to prevent the rep-
PLAN OF MAP OF UNITED STATES. 5

resentation of a considerable area on a single sheet, so that the relation 

between different regions can be seen at a glance.

A map on a larger scale would be more expensive and would require 

longer time for its completion. On the scales adopted the map of the 

United States, excluding Alaska, will cost about twenty-five million 

dollars and at the present rate of progress will require fifty years for its 

completion.

There is another objection to increasing the scale be}^ond that 

absolutely necessary, viz: Such a map, to be of value, must undergo 

revision at frequent intervals in order that changes in culture and 

possibly in natural features due to natural or artificial agencies may 

be incorporated. The larger the scale the more frequently such 

revision would be necessary, and hence the labor and expense of keeping such a map up to date would be greatly increased.

CONTOUR INTERVAL.

The relief of the earth's surface is now represented on maps almost 

entirely by contour lines, or lines of equal elevation. Until a comparatively recent date this relief, secondary in importance only to the 

horizontal location, or the plan, has not been exprest quantitatively.

The contour intervals which have been adopted for the map of the 

United States are as follows: For the scale of 1:62500 the intervals 

range from 5 to 50 feet; for the scale of 1:125000 the "range is from 10 

to 100 feet; and for the scale of 1:.250000 the interval is 100, 200, or 

250 feet..

FEATURES REPRESENTED.

The experience of European nations tends in the direction of reducing the number of features which should be placed on a map. In 

the preparation of the map of the United States it has been decided 

to go even beyond the present practise of European countries in this 

regard,- and to show only natural features of such magnitude that the 

scale warrants their representation and cultural features of general or 

public importance. Under these limitations the map will show cities, 

towns, and villages; roads, railroads, and other means of communication; bridges, ferries, tunnels, fords, canals, and acequias; and boundaries of civil divisions. Fences, property lines, and other objects of 

kindred nature will not be represented. The reasons for excluding 

private culture are apparent: First, because such features are not of 

sufficient general interest to pay the cost of surveying or representing 

them; second, because they change rapidly, and in order to keep the 

maps up to date constant resurveys and republication would be 

required; and, third, because their number and complexity confuse 

the map and render its more important features less intelligible.
b MANUAL OF TOPOGKRAPH1C METHODS.

SIZE OF SHEETS.

The atlas sheets are designed to be of approximately the same 

size 17$ inches in length by 12 to 15 inches in width, depending upon 

the latitude. Those on the same scale cover equal areas exprest in 

units of latitude and longitude; that is, each sheet on the 4-mile scale 

covers 1 degree of latitude by 1 degree of longitude; each sheet on the 

2-mile scale, 30 minutes of latitude and 30 minutes ot longitude; and 

each sheet on the 1-mile scale, 15 minutes of latitude and 15 minutes 

of longitude. The sheets are thus small enough to be conveniently 

handled, and if bound they form an atlas of suitable size. From the 

fact that each sheet covers either a full degree or a certain fractional 

part of a degree, its position with relation to adjacent sheets and to 

the area of the country may be readily ascertained.
C II A P T E R I I .

METHODS AND CLASSIFICATIONS OF WORK.

GENERAL METHODS.

Every map is essentially a sketch controlled by locations. It follows that the work of making a map consists of two parts that of 

fixing these locations, which is done by means of surveying instruments and is geometric, and that of sketching, which is done by the 

hand and eye and is artistic. This general description applies to the 

making of all maps, whatever their scales, their accuracy, their 

degree of detail, or the character of the organization making them. 

The methods and instruments employed in fixing the locations vary 

with different scales, kinds of country, and the ideas of the organizations engaged in the work.

Altho the sketching is all that appears on the finished map, a 

description of the methods of making the locations must of necessity 

form the bulk of a volume descriptive of surveying methods, for it is 

the essential part of the work. This volume will treat only of the 

most approved methods and instruments now used in topographic 

surveying; time and space will not be wasted upon methods or instruments that have been tried and discarded.

CORRECTNESS OF MAPS.

The correctness of a map depends upon four elements: Accuracy 

of location, number of locations per square inch of map, distribution 

of locations, and quality of the sketching.

ACCURACY OF LOCATION.

The greatest accuracy attainable is not always desirable, because 

it would not be economical. The highest economy is in properly 

subordinating means to ends, and it is not economical to do work of 

greater accuracy than is needed for instance, to execute triangulation of geodetic refinement for the control of maps on small scales. 

The quality of the work should be such as to insure against errors of 

sufficient magnitude to be appreciable on paper. While errors in 

triangulation tend to balance one another, still they are liable to 

accumulate, and this liability must be guarded against by maintaining a somewhat higher degree of accuracy than would be required

7
8 MANUAL OF TOPOGKAPHIC METHODS.

for the location of any one point. It is not difficult to meet this 

first condition of accuracy of maps. The maximum allowable error 

of location may be set at 0.01 of an inch on the scale of publication. 

This admits of an error upon the ground, on a scale of 1:62500, not 

greater than 50 feet, and proportionally on other scales.

NUMBER OF LOCATIONS PER SQUARE INCH OF MAP.

The second condition of correctness for the proper control of the 

work is not easily defined. The requirements differ with the character 

of the country. A region of great detail and of abrupt features 

requires more control than one of greater uniformity, of gentle slopes, 

or of large features, so that no general rule can be laid down. Furthermore, it depends upon the quality of the sketching. With indifferent sketching a greater number of locations is required in order to 

bring the map to the requisite quality. Examples of the amount of 

control upon maps are scarce, since little has been published on this 

subject.

There are two general methods of locating stations and points for 

the correction of the sketch one by angular measurements (triangulation) and the other by measurement of direction and distance, or 

what is known popularly as the traverse or meander method. In 

ordinary practise work may be done by either of these methods, 

or they may be used in- conjunction. The former method can be 

carried on with the plane table, with various forms of the theodolite, 

with a compass, or, indeed, with airy angle-reading instrument. The 

latter method can be pursued with the same instruments, supplemented by various forms of odometers, chain, steel tape, stadia, etc., 

for the measurement of distance. The first method, whenever it can 

be used economically, is the more accurate and, as a rule, the more 

rapid, and locations thus made are likely to be of greater service and 

to be distributed more uniformly than if made by the other method. 

It can be used economically where the country presents relief and 

where either natural or artificial points for location exist in sufficient 

numbers and are well distributed. These conditions are satisfied 

.almost everywhere in the western mountain regions, where peaks, 

summits of hills, plateau points, buttes, etc., furnish an abundance 

of natural points for stations and locations. It can be used to a considerable extent, tho not with the same ease and economy, in the 

Appalachian Mountains; but in this region it is necessary to supplement it/extensively by traverse lines, especially in tracing the courses 

of streams in valleys. It can be used in 'the hill country of New 

England, where objects of culture, such as churches," houses, etc., 

furnish plenty of signals. On the other hand, thruout the whole 

extent of the Atlantic and Gulf plains and the Mississippi Valley, 

where the country is level, or nearly so, and is covered with forests,
COKEECTNESS OF MAPS. 9

the traverse method of surveying must be resorted to. In these 

parts of the country there are no sharp natural objects, and extended 

views can not be obtained. For many reasons the second method 

of obtaining locations is inferior to the former. It is inferior not 

only in accuracy but in the facilities Avhich it affords for sketching 

.the country, and it should be so regarded and should be adopted 

only when it becomes necessary or when the former method can not 

be applied economically. For convenience traverse lines are generally run along roads or trails and thus the best points for commanding views of the country are avoided rather than sought, and there 

is danger lest the topographer neglect the areas lying between the 

roads. On account of the errors incident to running a traverse, it is 

necessary that in this class of work frequent locations be made by 

triangulation for checking and thereby eliminating the errors.

As a rule, triangulation locations are selected points chosen because 

each controls positions in a certain area. On the other hand, traverse locations, as a rule, are not chosen for their control value, but 

only for intervisibility on roads. Furthermore, the great majority 

of traverse stations are of no service whatever beyond carrying the 

line forward, so that in estimating the total amount of control in a 

certain area where the control is made up wholly or in part of traverse lines less weight should be given to them than to locations by

triangulation.

DISTRIBUTION OF LOCATIONS.

The third element of accuracy is a point concerning which it is 

equally difficult to speak definitely. Other things being equal, the 

distribution should be uniform over the area, but it will necessarily 

vary with the character of the surface. In general, in mountain 

regions, locations by angular measurements are frequent and follow 

the ranges or ridges, but in valle)rs such locations are few in number, 

being supplemented by traverses.

The subject of locations maybe treated from another point of view 

that of the features to be represented. All points for locations should 

be chosen for the purpose of guiding the sketch of certain features. 

Streams, roads, and other linear features ordinarily can be best and 

most economically located by means of traverse lines; that is, by 

establishing by traverse a series of points along their courses. Houses 

can usually be best located by. intersection, either in connection with 

the secondary triangulation or from a traverse.

The location of contour lines requires more than a passing mention. At the outset a very common impression should be corrected. 

It is generally supposed by those unacquainted with the subject that 

contours are located by finding and fixing points directly on each 

contour and sketching the contour between these points. This
10 MANUAL OF TOPOGRAPHIC METHODS.

method, however, is very rarely employed, and then, only in the most 

detailed surveys. In the topographic surveys of all nations, except 

the English Ordnance Survey, a different method is employed, or, 

rather, a different selection of points is made. For location and 

height measurement salient points are selected, such as the summit 

of a hill, the shoulder of a spur, the foot of a slope, the bed of a valley. The contours are then sketched with reference to these points. 

By judicious selection of such points comparatively few locations 

and height measurements serve to locate the intervening contours 

with sufficient accuracy. On the whole, the results are quite as good 

as if points had been located immediately on the contours, and the 

work of location is vastly reduced.

QUALITY OF SKETCHING.

The fourth of the elements of the correctness of the map depends 

on the artistic sense of the topographer, on his ability to see things 

in their proper relations, and on his facility in transferring his impressions to paper.

Every map, whatever its scale, is a reduction from nature and is 

generalized, and consequently departs from the original in greater or 

less degree. The smaller the scale the greater the generalization, 

and the further the map must depart from nature the less can it 

resemble the ground which it represents. Bends in streams, petty 

gulches, canyons, and hills must be omitted. Contours must cross 

such omitted gulches instead of turning up and down their courses. 

A searching examination of the map would therefore show countless 

places where the contours are widely out of place, owing simply to 

this generalization. To specify that the contours of a map must 

nowhere be in error more than a contour interval, for instance, is to 

require an impossibility.

The education of the topographer therefore consists of two parts 

the mathematical and the artistic. The first may be acquired largely 

from books, but this book knowledge must be supplemented by practise in the field. The second, if not inherent, can be acquired only 

by long experience in the field, and by many can never be acquired 

in high degree. In fact, the sketching makes the map, and it should 

be done by the best topogragher in the party, usually the chief, 

whenever practicable.

CLASSIFICATION OF WORK.

The work involved in making a map usually comprizes several 

operations, which in practise may be more or less distinct from one 

another. They are as follows:

(1) The location of the map upon the earth's surface, by means 

of astronomic observations.
CLASSIFICATION OF WOEK 11

(2) The horizontal location of points. This is usually of three 

grades of accuracy: Primary triangulation, or primary traverse in 

cases where triangulation is not feasible; secondary triangulation 

for the location of numerous points within .the primary triangulation; and ordinary traverse, for the location of details.

(3) The measurement of heights, which usually accompanies the 

horizontal location, and which may, similarly, be divided into three 

classes, in accordance with the degree of accuracy.

(4) The sketching of the map.

Nearly all of the geometric work of surveying the work of location is executed with few instruments, as follows:

Theodolites, of a powerful and compact form, used in the primary 

control.

Wye levels.

Plane tables, with telescopic alidades of the best type, used for 

secondary triangulation and height measurements.

Plane tables of simple form, with ruler alidades, compasses, .or 

solar attachment, used for traversing and minor triangulation.

Aneroids, for the measurement of details of heights.

These instruments, with which nine-tenths of the work is done, 

will be described in their proper places with such fulness of detail as 

seems necessary.

Other instruments, such as transits, surveyor's theodolites, compasses, hand and Abney levels, telemeters, chains, and tapes, are 

occasionally used. Most of these instruments, which are figured and 

described in nearly all works on surveying, are assumed to be well 

known to the readers of this manual, and will therefore receive no 

special attention.

Bull. 307 07 2
 CHAPTER III.

ASTRONOMIC DETERMINATION OF POSITION.

OBJECT AND IMPORTANCE OF ASTRONOMIC DETERMINATION,

The object of astronomic determinations of position is to locate 

the map upon the earth's surface. They.are made also for the purpose of checking and correcting positions determined by.primary 

triangulation and primary traverse.

With regard to the checking of primary triangulation by astronomic determinations, it should be understood that in the case of 

a single determination the work by triangulation is far more accurate 

than that by the astronomic method, even when made under the 

best circumstances. It is, therefore, desirable to introduce checks 

of this kind upon primary triangulation only when the latter has 

been carried for a long distance 200 or 300 miles, for instance in 

the course of which there may have accumulated errors greater than 

those incident to astronomic work.

The case is different with primary traverse. The great number of 

courses required in this work affords an opportunity for the accumulation of error, much greater than can arise in triangulation, and 

consequently it is desirable to introduce more frequent checks. It 

may be said that, in general, such work should be checked at every 

100 miles. . As has been suggested, the best astronomic determinations are 

none too good for the control of maps. Indeed, certain errors incident to this work, some of which as yet can not be corrected, may 

be of magnitude sufficient to show upon the map. It is necessary, 

therefore, in these determinations to use the best instruments and 

the most refined methods known to modern science, in order to 

reduce all avoidable errors to a minimum.

Determinations of position that have been made by the United 

States Coast and Geodetic Survey, the United States Lake Survey, 

or the Mississippi River Commission, whether by astronomic work 

or by triangulation, may be utilized for the above purposes.

DEFINITIONS.

Sidereal time is the time indicated by the stars, a sidereal day 

being the time which elapses between two passages of the vernal 

equinox across the meridian, Solar or apparent time is the time 

12
U. S. GEOLOGICAL SURVE' LLETIN NO. 307 PL. I

ASTRONOMIC TRANSIT AND ZENITH TELESCOPE.
ASTRONOMIC DETERMINATION OF POSITION. 13 '

measured by the sun's apparent movement, or the revolution of the 

earth with reference to the sun; and since the earth revolves at a 

differing rate in different portions of its orbit, the solar days are not 

of equal length. A mean day is the average solar day; mean time 

differs from solar time by an amount which varies with the time of 

year, and which, under the name "equation of time," is given in the 

Nautical Almanac. Mean time differs from sidereal time by about 

a day in the course of a )7ear, or about four minutes in each day, 

the mean day being longer than the sidereal day. To convert a 

given date of mean time into sidereal time it is necessary to obtain, 

from the Nautical Almanac, the sidereal time at noon immediately 

preceding the date in question. Then the interval after noon, exprest in mean time, is converted into sidereal time, and the result 

added to the sidereal time of mean noon. Local time, whether 

sidereal, solar, or mean, .is the time of the locality as distinguished 

from the time of any other locality. It must be distinguished from 

standard time, which is the local time of only certain meridians.

The right ascension of the sun or a star is the sidereal time which 

has elapsed between the passage of the vernal equinox and the star 

across the meridian. It is commonly exprest in hours, minutes, and 

seconds.

Declination is the angular distance of a heavenly body north or 

south of the equator. It is plus when north and minus when south 

of the equator.

The zenith distance of a heavenly body equals its declination 

minus the latitude of the place of observation.

Latitude is determined by what is known as Talcott's method, by 

measuring the differences of zenith distance at culmination of two 

stars which culminate on opposite sides of the zenith.

Longitude is determined by telegraphic comparison of local time 

at two stations, the longitude of one of which is known. This involves 

the determination of the errors of the clocks or chronometers used, 

which is done by observation of transits of stars across the meridians of the places of observation.

ASTRONOMIC TRANSIT AND ZENITH TELESCOPE.

A single instrument is used for the determination of both latitude 

and time. This is a combination of the transit and the zenith tele-. 

scope. The instruments in use by the United States Geological Survey were made by Saegmuller and-embody the latest improvements. 

(See PI. I.) The circular base rests upon three leveling screws. 

Upon this circular base the whole instrument can be made to revolve 

when using it as a zenith telescope. A circle is graduated around 

the base, having a micrometer screw for slow motion, for making 

settings and adjusting the instrument in azimuth. The frame of the
14 MANUAL OF TOPOGRAPHIC METHODS.

instrument is cast in one piece, and the standards are hollow in order 

to reduce the weight of the upper part. The telescope has a focal 

distance of 27 inches and a clear aperture of 2.5 inches. Its magnifying power with diagonal eyepiece is 74 diameters. The length of 

the axis of the telescope is 16 inches. For use as a zenith telescope 

the instrument is equipped with a vertical circle reading by vernier 

to 20 seconds, attached to which is a delicate level. In the focus of 

the object glass there is, besides the ordinary reticule for use in 

transit work, a movable thread, which is moved by means of a 

micrometer screw, by which measurements of differences of zenith 

distances are made. It is furnished with direct and diagonal eyepieces, the latter of which is commonly used in astronomic work.

For use as'a transit instrument the telescope is equipped with a 

delicate striding level for measuring the inclination of the pivots and 

a reversing apparatus for turning the telescope in the wyes. The 

reticule, as the stationary threads in the focus of the instrument are 

called, consists of five threads for observing the transits of stars. The 

reticule is illuminated by means of bull's-eye lamps, the light from 

which comes thru the hollow axis of the telescope and is reflected 

by a mirror placed at the intersection of the telescope with its axis.

CHRONOGRAPH.

The chronograph is used for the purpose of recording the time of 

transits of stars as observed with the transit instrument. It may be 

popularly characterized as an instrument for measuring time by the 

yard. It consists essentially of a drum upon which is wound a strip 

of paper and which is kept in revolution by a train of clockwork 

controlled by an escapement. A pen carried on a small car, which is 

moved very slowly in a direction'parallel to the axis of the cylinder, 

traces a spiral line upon the paper on the drum. This pen is held in 

place by a magnet carried on the car, and as long as the current from 

the battery passes thru the coil and thus hold's the armature, the 

pen traces an unbroken spiral line. If the current is suddenly broken 

and restored, the armature is set free for an instant and a jog is made 

in the line traced. The battery commonly used in connection with 

this outfit is the ordinary zinc copper, and copper sulfate ^battery, of 

which four cells are usually required. The ordinary dry battery can 

also be used and is much more convenient. With this apparatus 

break-circuit chronometers are used. These differ from ordinary 

chronometers in that they are arranged to break an electric circuit 

automatically at regular intervals. Those in use by the United States 

Geological Survey break the circuit every two seconds, and the end 

of the minute is indicated by breaking at the fifty-ninth as well as 

the fifty-eighth and sixtieth seconds. When one of these chronometers is connected with a battery and a chronograph is put in the same
CHKONOGRAPH. 15

circuit, the beginning of every even second is recorded upon the chronograph by a jog on the paper, and the distance between the jogs 

in each case therefore represents two seconds. The observer at the 

instrument is provided with a telegraph key, which may'also be put 

in the circuit with the chronometer and chronograph, and as a star

near the meridian crosses a thread in the telescope he records that 

fact by pressing on the key, which makes a record upon the chronograph along with the record of the chronometer.

An illustration of the form of chronograph in use by the United 

States Geological Survey is shown in fig. 1.
16 MANUAL OF TOPOGKA.PHIC METHODS.

FIELD WORK.

Since the observations for latitude and longitude, tho different, 

are made with the same instrument, at the same time, and by the 

same party, certain parts of the work apply equally to both determinations and may be described once for all.

In the selection of a station care must be taken to avoid a locality 

where for any cause the ground is liable to be seriously jarred, as, for 

instance, proximity to a railroad track or to a street over which 

heavy wagons pass. The station should have a clear view from the 

southern horizon thru the zenith to the northern horizon. It is 

desirable to locate at a convenient distance from a telegraph office, 

as it is necessary to bring in a wire from such office for the purpose 

of comparing chronometers. If possible, the station should be 

selected on a public reservation, in order that the permanence of 

the monument which is to be erected to mark the spot may be 

assured. In any event a locality in which such a monument is 

likely to be disturbed should be avoided.

The support of the instrument should consist of a brick, stone, or 

concrete pier sunk fully 3 feet in the ground and rising to the requisite height. Upon this should be placed a block of stone well set in 

mortar for the immediate support of the instrument. The chronograph may be set up on an ordinary table. Over all should be 

erected a wall tent with a slit in the roof, closed by flaps, which can 

be opened when observing. The instrument should be set up on 

the pier, collimated, and leveled, the vertically of the threads should 

be tested as accurately as possible', *and then it should be pointed 

upon the pole star. This places it somewhere near the meridian. 

Then taking the time of transit of a star which culminates close to

.the zenith, and comparing this time with the right ascension of the 

star, a sufficiently close approximation to the clock error is obtained 

for use in placing the instrument in the meridian. The instrument 

should then be turned in azimuth to point upon some close circumpolar star approaching upper or lower culmination, the instrument 

being moved in azimuth with the tangent screw, so as to keep the 

star under the middle wire to the instant of culmination. If this 

is done accurately at the first attempt the instrument is placed 

nearly in the meridian and is ready for work, but it commonly happens that more than one trial is required before the meridian is 

reached. In any case the result should be verified by a second star 

before proceeding with the observations.
ASTRONOMIC DETERMINATION OF POSITION. l 

OBSERVATIONS FOR LATITUDE.

As preliminary to this work it is necessary to prepare a list of pairs 

of stars, the two stars of each pair having such zenith distances that 

they will culminate at nearly equal distances from the zenith, one to 

the north and the other to the south of it. Such a list can be prepared 

from the Safltord Catalog of the Wheeler Survey. For this it is 

necessary to know the approximate latitude of the station and the 

right ascensions and the declinations of the stars. The zenith distance 

of a star is equal to its declination minus the latitude of the place 

of observation. The stars of each pair must be observed consecutively, and therefore those stars should be selected which culminate 

as nearly as possible together, leaving only a sufficient interval of 

time between them for setting the instrument.

On the approach of the first star of the pair to the meridian the 

instrument should be set for it, using the vertical circle for that 

purpose, and setting the spirit level upon the vertical circle as nearly 

level as possible. Then, as the star traverses the field of the telescope, the movable thread in the reticule should be kept upon it by 

means of the micrometer screw until it crosses the middle vertical 

thread. Then the micrometer and the two ends of the level bubble 

should be read and recorded. Without disturbing in the slightest 

olegree the setting of the telescope, the entire instrument should be 

turned 180 degrees upon its bedplate, when it will point north of the 

zenith at the same angle at which8 it formerly pointed south, or vice 

versa, as the case may be, and will be set for the other star on the 

opposite side of the zenith. As this approaches culmination it 

should be followed with the micrometer as before until it reaches 

the middle thread, then the readings of the micrometer and of the 

level should be recorded as before, whether changed or not.

This constitutes the observations upon a single pair of stars. For 

the determination of latitude twenty such pairs of stars should be 

observed each evening, if possible, and the same pairs of stars should 

also be observed on other evenings, assuming it to be possible. The 

following example, taken from observations at Rapid, S. Dak., shows 

a portion of the star list and the form of record:
18 MANUAL OF TOPOGBAPHIC METHODS.

List of stars for observation loith zenith telescope. 

[Station: Rapid, S. Dak. Approximate latitude: 44° 05'.]

Name or 

number. 

Safford's Catalog.

7 Lacertse. . . . 

10 Lacertse...

1539..........

1551..........

1565..........

1579. .........

1600. .........

1633..........

1676..........

1686..........

1702. .........

1722..........

Mag.

4.0 

5.0

6.5

6.5

6.5

5.0

6.0

6.7

5.6

6.5

4.5

6.6

Class.

A A 

A A

B

A

C

A

A

B

A

' A

A

B

R. A.

h. m. 

22 27 

22 34

22 41

22 47

22 52

22 59

23 08

23 18

23 42

23 47

23 52

24 00

Dec.

49 '43 

38 29

45 37

42 42

38 42

49 26

56 34

31 56

67 12

21 03

24 32

63 35

Zen. dist.

5 38 N. 

5 36 S.

1 32 N.

1 23 S.

5 23 S.

5 21 N.

12 29 N.

12 09 S.

23 07 N.

23 02 S.

19 33 S.

19 30 N.

Setting.

| 5 37 N.

| 1 27 N.

I 5 22 S.

Il2 19 N.

{ 23 05 N.

}-19 31 S.

t Example of record.

[Station: Rapid, S. Dak. Date: November 9, 1890. Instrument: Fauth combined transit and zenith 

telescope No. 534. Observer: S. S. G. Recorder: A. F. D.]

Star name or 

number.

7 Lacertfe. ... . 

10 Lacertse...

1539..........

1551 ..........

1565..........

1579..........

1600..........

1633..........

1676. .........

1686. .........

1702. .........

1722..........

N. or 

S.

N. 

S.

N. 

S.

S.

N.

'N. 

S.

N. 

S.

S.

N.

Micrometer 

reading.

Rev. 

26. 256 

24. 052

30. 432 

20. 095

25. 164 

26. 703

32. 214 

16. 033

26. 656 

17.684

25. 345 

. 23. 722

Diff.

Rev. 

- 2.204

-10.337 

+ 1.539 

-16.181 

- 8.972 

+ 1.623

Level.

N.

Div. 

39.9 

26.5

42.0 

21.9

14.1 

38.1

37.5 

19.9

51.0 

17.0

18.0 

36.0

S.

Div. 

16.7 

49.7

18.7 

45.0

37.6 

15.0

14.1 

43.1

28.0 

39.6

40.9 

13.2

(N+S) 

-(N'+S').

Div. 

+ 56.6 

-76.2 

-19.6

+60.7 

-66.9 

- 6.2

-51.7 

+53.1 

+ 1.4

+51. G 

-63.0 

-11.4

+ 79.0 

-56.6 

-22.4

-58.9 

+ 49.2 

- 9.7

Remarks.

Faint. 

Distinct.

Faint.

REDUCTION OF LATITUDE OBSERVATIONS.

Before proceeding with the reduction of latitude observations it 

is necessary to investigate the constants of the instrument and to 

ascertain the value of a division of the head of the micrometer screw 

and of a division of the latitude level.
ASTRONOMIC DETERMINATION OF POSITION. 19

MEASUREMENT OF A DIVISION OF THE HEAD OF THE MICROMETER

SCREW.

The value of a division of the head of the micrometer screw is 

measured by observing the transits of some close circumpolar star, 

when near elongation, across the movable thread, setting the thread 

repeatedly at regular intervals in advance of the star, and taking 

the time of its passage with the reading of the micrometer. The 

precaution should be taken to read the latitude level occasionally 

and correct for it if necessary. This correction, which is to be applied to the observed time, is equal to one division of the level, in 

seconds of time, divided by the cosine of the declination of the star 

and multiplied by the level error, the average level reading being 

taken as the standard.
Determination of value of one revolution of micrometer belonging to Z. T. No. 53$, by obser

1890, Rapid, S. Dak.

log cot 6.... 87 

log tan $.. .44

log cos f 0 . -.87

.«.

o 12, 

04

17

 w

50". 

45

57

8=8.6871826 

= 9.9860380

= 8.6732206

.-0.12385

. - 1. 31334

8

OOOQ1

 f>.h rnn

T

log 1. 82................... = 0. 26110

Time of observation (recorded 

on chronograph 

sheet).

h. m. s. 

00 40 19.7 

41 12.3 

(lost on chronograph sheet.) 

42 53 7 

43 44. 6 

44 35. 9 

45 31.9 

(mist) 

(mist) 

48 08. 0 ' 

48 57.1 

49 50.2 

50 47.0 

51 42. 3 

(mist)

Micrometer revolutions.

31.0 

30.5

30.0 

29 5 

29.0 

28 5 

28 0 

27.5 

27.0 

26.5 

26.0 

25.5 

25 0 

24.5 

24.0

Level.

IS1 .

Div. 

37.0

37.1

38.3

s.

Div. 

16.0

16.1

17.5

Level.

N+S.

D-n: 

53 0

53.2

55 8

Reduction 

to mean 

level.

Div. s. ' 

+ 0.8X1.82=

+ 0 6

/

-6 i>

Correction for 

change 

of level.

s. 

+ 1.5 

+ 1.3

+ 1.1 

+ 0.8 

+ 0.5 

+ 0.3 

+ 0.0 

-0.3

-0.5

-0.8 

-1.1 

-1.1 

-1.0 

-1.0 

-0.9c;

Time from 

elongation.

_ qi

m. s. 

- 19 40. 9' 

IS 48. 3

17. 06.9 

16 16.0 

15 24. 7 

14 28.7

11 52. 6 

.11 03 5 

10 10. 4 

9 13.6 

S 18.3

C

tio
Determination of value of one revolution of micrometer belonging to Z. T. No. 534, by observati

1S90, Rapid, S. Dak. Continued.

Time of observation (recorded 

on chronograph 

sheets.)

h. m. s. 

53 22.4 

54 IS, 8 

55 07.1 

56 01.2 

50 52.4 

57 41 2 

58 SO. 7 

59 25. 9 

1 00 17. 9 

01 10. 0 

(mist) 

02 52 8 

03 44 8 

04 34. 2 

05 25. 9 

06 IC.O 

07 14.8 

OS 05. 1 

OS 58. 2 

09 49.1 

10 41.0 

11 35.0

Micrometer revolutions.

23.5 

23.0 

22 5 

22 0 

21.5 

21.0 

20.5 

20.0 

19:5 

19 0 

18.5 

18.0 

17.5 

17.0 

16.5 

IC.O 

15 5 

15.0 

14.5 

14.0 

13.5 

13.0

Level .

N. S,

Div. 

38. 3

38. 3

38.0

37.8

37.2

Div. 

17.3

17.3

17.0

10 S

10.2

Level

N +S.

Div. 

55. 6

55 0

55 0

54.6

53.4

= 54.5

Reduction 

to mean 

level.

Div. s. 

-0 5

-0 5

-0 3

-00

40.5

Correction for 

change 

of level.

s. 

-0 9 

-0 9 

-0 9 

-0.9 

-0 9 

-0 9 

-0 9 

-0 8 

-0 8 

-07 

-0 6 

-0 6 

-0 5 

-05 

-04 

-0 3 

-02 

-0.1 

-00 

403 

40 6 

40.9

Time from 

elongation. 

= T.

TO. S.

6 38.2 

5 41 8 

4 53. 5 

3 59.4 

3 OS 5 

2 19. 4 

1 29.9 

- 0 £4 7 

4 0 17.3 

1 09. 4

2 52 2 

3 44. 2 

4 33. 6 

5 25.3 

0 15. 4 

7 14.2 

8 04 5 

S 57.6 

9 48 5 

10 40. 4 

11 34.4

Corr

tion 

T.

s. 

40.

40 

0

0 

0

0 

0

40.

-0 

0 

0

0

0 

0

0

0 

-0.

-0 

-0.

-0 

-0

-0.
22 MANUAL OF TOPOGRAPHIC METHODS.

The time from elongation of the star requires a correction in order 

to reduce the curve in which the star apparently travels to a vertical line. The hour angle of the star is first obtained from the 

equation,

cos t0 = cot d tan q>, 

d being the star's declination and <p the latitude.

The chronometer time of elongation, T0 = a 10 dt, a being the 

right ascension of the star obtained from the Nautical Almanac, and 

dt the error of the chronometer.

Having thus obtained the chronometric time of elongation, the correction in question is obtained from the observed interval of time of 

each observation before or after elongation, from tables in Appendix 

No. 14, United States Coast and Geodetic Survey Keport for 1880, 

pp. 256 and 257. A discussion of this subject will be found in the 

appendix above referred to and in Chauvenet's Practical Astronomy, 

vol. 2, pp. 360 to 364.

The times of observation thus corrected for level, and distance from 

elongation, are then grouped in pairs, selected as being a certain number of revolutions of the micrometer apart, and the time intervals 

between the members of each pair are obtained. The mean of these, 

divided by the number of revolutions which separate the members 

of each pair, is yet to be corrected for differential refraction, which 

is derived from the equation,

refraction = 57".7 sin R sec 2 Z,

R being the value of a division of the micrometer and Z the zenith 

distance of the star. Four-place logarithms are sufficient for computing this correction, as it is small. Herewith is an example of 

record and computation of the value of a revolution of the micrometer of combined instrument No. 534, one of the two in possession of 

the United States Geological Survey.

0

MEASUREMENT OF A LEVEL DIVISION.

The value of a division of the level is commonly measured with 

a level trier. The latitude level, however, may be easily tested by 

means of the micrometer, the value of a revolution being obtaine'd 

by the following method:

Point the telescope upon some well-defined terrestrial mark and 

set the level at an extreme reading near one end of the tube. Set 

the movable thread upon the object and read the micrometer and the 

level.

Now move the telescope and level until the bubble is near-the other 

end of the tube. Again set the movable thread upon the object and 

again read both micrometer and level. It is evident that the micrometer and the level have measured the same angle, and that the
ASTRONOMIQ DETERMINATION OF POSITION. 23

ratio between these readings equals that between a revolution of the 

micrometer and a level division. 

An example illustrative of this is appended.

Determination of value of 1 division of latitude level No. 534. 

[By comparison with micrometer screw 534.]

Micrometer.

r. 

8. 025 

8.508

8.509 

7.984

8.511 

8.045

9.076 

8.604

9.442 

9.009

10. 055 

9.574

10. 661 

10. 212

11.771 

11. 252

12. 328 

11. 872

12. 869 

12. 438

13. 468 

13. 080

14. 146 

13. 702

14. 758 

14. 282

Sum.

Level.

N.

d. 

47.3 

20.7

18.9 

49.8

18.5 

47.2

18.7 

46.0

23.7 

48.0

21.8 

48.0

24.0 

50.7

18.3 

48.3

20.0 

46.1

22.2 

47.7

23.0 

44.5

20.1 

45.4

22.3 

48.6

S.

d.

29.2 

02, 7

01.0 

31.0

00.6 

29.1

00.8 

28.0

08.0 

30.0

04.0 

30.1

06.1 

33.0

00.7 

31.9

02.3 

28.5

04.-6 

30.0

05.3 

26.9

02.4 

27.8

04.8 

31.0

Difference.

Microrn.

6. 

d. 

48.3

52.5

46.6

47.2

43.3

48.1

'44.9

51.9

45.6

43.1

38.8

44.4

47.6

Level.

a. 

d. 

26.55

30.45

28.60

27.25

24.15

26. 15

26.80

30.60

26.15

25. 45 .

21.55

25.35

26.25

aa.

704.9

927.2

818.0

742.6

583.2

683.8

718.2

936.4

683.8

647.7

464.4

642.6

689. 1

9241.9

ab.

1283.

1599.

1333.

1286.

1046.

1258.

1203.

1588.

1192.

1097.

836.

1125.

1249.

16095.

log...................16095. =4.20669.

a. c. log............. 9241. 9-6. 03424.

log 1 div. micrometer........ =9. 87966.

1 div. level..... .=lx/ .320 log.=0.12059.

COMPUTATION OF APPARENT DECLINATION OF STARS.

After the determination of the constants of the instrument used, 

the next step is to obtain the apparent declinations of the stars 

used. Whenever possible, these should be taken from the Nautical 

Almanac or the Berliner Jahrbuch. In other cases they must be 

computed. The positions of stars are given in Safford's Catalog for 

the epoch 1875.0, together with the annual precession and proper 

motion. The declinations there given should be revised by the aid 

of more recent catalogs, particularly with reference to stars of class C. 

The annual precession and proper motion multiplied by the number 

of years which have elapsed, and applied, together with the effect of 

secular variation in precession, give the declination at the beginning
24 MANUAL OF TOPOttJRAPHIC METHODS.

of the year. Further corrections to bring the positions down to the 

date of observation are exprest by the symbols Aa', Bfr', Ccf , Ddf . 

Logarithms of a', V, c', d' are given in Safford's Catalog, and A, B, 

C, and D are given in the Nautical Almanac. A slight additional 

correction is also to be made for proper motion for the elapsed portion 

of the year. This reduction is illustrated below:

Example of reduction. Computation of apparent declination of star 1539. 

[From Satford's Catalog, p. 40.]

Star No. 1539

Declination 1875.0 

45 33 29.20

Annual 

,p recession.

+ 18.87

Proper 

motion.

-.03

+ 4 43.05= Precession for 15 years.

 0 00. 45= I'roper motion for 15 years.

+ 0 00.07= Secular variation in precession.

45 38 11. 87= Declination 1890. 

+ 9. 38= Ao?

- 0.78=IW 

+ 0. 88= Cc' 

+ 10.10= Dd'

- 0.03= Proper motion, Jan. 1 Nov. 9,1890.

45 38 37. 48= Declination Nov. 9, 1800.

log a'=l. 2757 log b'=9. 5294 log c'=9. 7307 log d'=9. 8273 

Nov. 9. log A=9. 0900 log 13=0. 3049n log C=l. 1000 log D=l. 1790

1.0090

With' all this preliminary work done, the reduction proper of latitude observations is comparatively simple. The observations being 

grouped by pairs, the mean declination of each pair is obtained, the 

corrections for difference of micrometer readings and levels are 

applied, with a small correction for differential refraction, and the 

result is the desired latitude.

Following is an example of the reduction of six pairs of stars 

observed for latitude at Rapid, S. Dak. :

Example of reduction. 

[Station: Rapid, S. Dak. November 9, 1890. Half rev. micrometcr=37".900. One div. level=l".33.]

Date.

Nov. 9.

Star Nos.

/7 Lacert and 

\ 10 Lacert.

1539 155.1

1565 1579

1600 1633

1670 1080

1702 1722

5i

J49 42 87. 3J

45 38 37. 48

38 43 39. 78

56 34 00. 00

07 12 10.93

24 32 09. 04

S 2

38 29 04. CO

42 44 04. OS

49 27 41.04

31 55 50. 91

21 03 54. 02

03 35 27. 34

i- ffii+s*)

44 00 15.97

11 21. 00

05 40. 41

15 01. 78

08 02. 48

03 48. 19

Corrections.

Microm.

- 1 23. 53

- 6 31. 77

- 0 58. 33

-10 13.25

- 3 08. 43

+ 1 01. 51

Level.

-6.51

-2.06

+0.46

-3.78

-7.44

-3.22

Refr.

-0.03

- .11

- .03

- .19

- .07

+ .02

Latitude

n.

44 04 45. 90

47.12

42.51

44.50

40. 54

46. 50

.d

a

^

0.98

.90

.79

.90

.93

.90

5. <0

R

p.

 5.78

0.41

1.98

4.10

6.08

5.85

30.20

November 9. Weighted mcan=44° 04' 45".59.
ASTEOKOMIC DETEEMINATION OF POSITION. 25 

OBSERVATIONS FOR TIME.

With the transit mounted, leveled, and adjusted in the meridian, 

the chronograph set up and running and connected in a circuit with 

the battery, and the chronometer and observing key connected in the 

same circuit, the observer is prepared to begin time observations.

The list of. stars which should be used is that given in the Berliner 

Jahrbuch, as the list is fuller and more accurate than that in any 

other catalog which gives day places. Stars should be so selected 

north and south of the zenith that the azimuth errors will balance 

one another as nearly as possible, as is explained hereafter. On the 

approach of the selected star to the meridian, the telescope should 

be set by means of the vertical circle upon the altitude of the star 

above the horizon, deduced from the declination and the latitude. 

As the star crosses each thread in the reticule, the fact should be 

recorded by pressing the observing key, which produces, as previously described, a record, upon the chronograph sheet. In this 

way four time stars, as stars between the equator and zenith are 

designated, and one circumpolar star, or a star so near the pole that 

it is constantly in sight, should be observed. Then the telescope 

should be reversed in the wyes and a similar set of stars observed. 

Between observations upon any two stars the striding level should 

be placed upon the pivots of the instrument and readings taken to 

ascertain the departure of the axis from a horizontal position.

In order to avoid unequal expansion of the pivots from unequal 

heating both bull's-eye lamps must be lighted and placed in their 

stands.

After the comparison of chronometers at the two stations, to be 

hereafter' described, a similar set of stars should be observed, if 

possible.

REDUCTION OF TIME OBSERVATIONS.

Certain constants of the transit should be measured before proceeding with the reduction of time observations. The value of a 

division of the striding level should be measured by means of a level 

trier. The equatorial interval of time between each of the threads 

and the mean of all the threads should be obtained, as it ]s not 

infrequently needed in utilizing broken or imperfect observations. 

These can best be obtained from observations on slow-moving stars, 

but any stars may be used for the purpose. The intervals as observed 

are reduced to the equator by multiplying them by the cosine of the 

declination of the star observed.

The object of these observations is specifically the determination 

of the error of the chronometer. This error equals the right ascension of a star minus its observed time of .transit, corrected for certain 

instrumental errors. These errors are as follows:
26 MANUAL OF TOPOGKAPHIC METHODS. 

CORRECTION FOR ERROR OF LEVEL.

The level error, designated ' by &, is ascertained from the readings 

of the striding level. The value of a division of the level in seconds 

of time must have been previously ascertained by means of a level 

trier. The effect of the level error is greatest at the zenith and 

diminishes to zero at the horizon. The correction in seconds of time 

is given by the equation,

cor. = 6 cos (q> 8} sec 6 = &B.

When the declination is north it is to be regarded as having a 

plus sign for upper and a minus sign for lower culmination; when 

 south it is negative.

CORRECTION FOR INEQUALITY OF PIVOTS.

This correction can be made a part of the level correction. 

Let p = the inequality of pivots,

B = inclination of axis given by level for clamp west, 

B' = inclination of axis given by level for clamp east, 

Z> = true inclination of axis for clamp west, 

V = true inclination of axis for clamp east ;

l> = B + p for clamp west, 

V =B' p for clamp east. 

(Chauvenet, vol. 2, p. 155.)

CORRECTION FOR ERROR OF COLLIMATION.

This correction, designated by c, is the departure of the mean of the 

threads from the optical axis of the telescope. For stars at upper culmination, with clamp west it is plus when the mean of the threads 

is east of the axis, and minus when it is west of it; for stars at 

lower culmination the reverse is the case. The value of c is onehalf the difference between the clock error indicated by stars observed 

before and after reversal of the instrument, divided by the mean 

secant of the declinations of the stars. This is slightly complicated 

with the azimuth, altho the effect of that is largely eliminated by the 

proper selection of stars ; consequently it is to be obtained by approximations in conjunction with the azimuth errors. The correction to 

be applied to each star equals c sec d = cC, which is plus for a star at 

upper culmination and minus for a star at lower culmination. It is 

least for equatorial stars and increases with the secant of the declination.
EEDUCTION OF TIME OBSERVATIONS. 27 

CORRECTION FOR DEVIATION IN AZIMUTH.

This correction, designated by a, represents the error in the setting 

of the instrument in the meridian. Its effect is zero at the zenith and 

increases toward the horizon. Since the instrument is liable to be 

disturbed during the operation of reversal, it is necessary to determine 

the azimuth error, both before and after reversal, separately. A comparison of the clock error, determined from observations upon north 

and south stars, will furnish the data necessary for the determination 

of azimuth. Practically, it fs determined by elimination from equations involving the mean of all these stars observed in each of the two 

positions of the instrument, after correcting for level, and as it is 

slightly complicated with collimation it must be reached by two or 

more approximations. The error is essentially positive when the telescope points east of south, and negative when west of south. The 

correction applicable to any star is exprest in the equation,

cor. = a sin (<p 6} sec # = a&.

It must be understood that the declination when north is positive 

for upper and negative for lower culmination, and that with south 

declination it is negative.

CORRECTION FOR DIURNAL ABERRATION.

The right ascension of stars, as taken from the Berliner Jahrbuch, 

must be corrected for diurnal aberration, which equals 0s . 021 cos <p 

sec 8. This correction is positive for upper and negative for lower 

culmination.

These corrections are summarized in the equation,

A, B, C, as seen above, are constants, depending upon the latitude 

of the place of observation and the declination of the star. Tables 

for these quantities will be found in a special publication of the 

United States Coast and Geodetic Survey for 1874.

EXAMPLE OF REDUCTION OF TIME OBSERVATIONS.

The following is an example of the form for record of observation 

and reduction of time observations, taken from a campaign for the 

determination of the position of Rapid, S. Dak.

Bull. 307 07 3
Time determination: Example of rec

'[Rapid, S. Dak., November 20,1890. Fauth transit, No. 534. Sidereal chronometer: Bond & Sons, No. 187

Star ....................................................

Thread I ...............................................

II..............................................

III.............................................

IV .............................................

V...............................................

Tabular R. A (a) ......................................

W.

Telescope north. 

W. Sum. E.

19.8 -88.1 08.3 

68.2 +87.6 19.4

- 0.5

h. m. s. 

23 34 52. 25

35 11.40

29.41

46.78

36 05.00

4.84

23 35 28.97

-.07

-.22

+ .05

23 35 28.83

23 34 53. 13

-35.70

<t> Pegasi.

W.

Telescope south. 

W. Sum. E.

68. 0 +87. 1 19. 1 

20.2 -89.2 69.0

- 2.1

h. m. s. 

23 47 24.00

28.55

32.72

36.75

41.09

3.11

23 47 32.62

 .02

-.06

+ .03

23 47 32.57

46 55. 67

-36.90

W

Telescope 

TT7. Sum

68.8 +87

-2

h. m

23 5

23 5

23 5

5

div.

202 s- s- 

Mean of levels = - j- X .118= .0596 = 6. Ineq
Time determination: Example of record Co

Star. ....................................................................

Level ................................................................. J

Thread V ...............................................................

IV ..............................................................

III..............................................................

11...............................................................

I................................................................

a t

E.

Telescope south. 

W. Sum. E.

19.2 -88.3 69.1 

68.9 +87.8 18.9

n ^

*

h. m. s.

09.30

-1 0 c A

7.74

-.02

-.02

-.02

00 07 36.59

-36.90

Br. 6.

E.

Telescope s

W. Sum. 

68.7 +87.

19.4 -88.7 

1.

, h. m

00 10 

10 

div.

.925 s- ? 

;Mean of levels= -r- X .118 = .027=6. Inequality of pi
Example of reduction. 

[Rapid, S Dak., November 20,1890. After exchange of 

Clamp.

W.

.<

1.

wt.

«t0 

1

1

1

1

1

1

1

1

1

1

2. 

Star.

33P<scium..............

(1) Mean............

Bradley 6.. .............

i Ceti ..................

12 Ceti.... ..............

(2) Mean. .......... 

3. 

A.

-2.42

+0.45

+0.62

+0.77

+ 0.30

+ 2.14

-2.42

-0.06 Aw

+ 0.51

-2.26

+ 0.81

+0.68

+ 0.75

+ 2.75

-2.26

+0. 10 Ae

4. 

C,

+4.45

1.05

1.01

1.01

+ 1.14

+ 8.66 . + 1.73C

-1.03

4.23

1.01

1.00

-1.00

-8.27

-1.65C

5. 

a-t

s. 

- 35.70

36.90

37.03

37.13

- 36.90

-183.66

- 36.732

- 36.90

36.17

36.99

36.89

- 37.03

- 183'. 98

- 36.796

6. 

Cc.

S.

c=-.019 

s. 

-.08

-.02

-.02

-.02

-.02

+ .02

.08

.02

.02

+ .02

7. 

a-t 

Corrected 

for Cc.

s. 

-35.78

36.92

37.05

37.15

-36.92

-36.88

36.09

36.97

36.87

-37.01

8. 

Aaw .

s. 

-1.00

+0.19

+0.26

+0.32

+0..12

Aa. 

+0.15

- .64

+ .23

+ .19

+ .21
Subtracting (2) from (1), ignoring azimuth terms which are small, we have:

s. 

4 3.38c 4 .064 =0

s. 

Approx ..... c = - . 019

s. 

From below.c' = 4.015 Adopted..c = .004

s. s. 

-2. 42 Aw -35.78 =0 -2.26 Ae -36.09=0 

40.54 -37.01=0 40.69 -36.93=0

S. «. 

42.96 Aw - 1.23=0 42.95 Ae - 0.84 =0

s. s. 

Aw =4. 415 Ae=4.285 

Forming equations to determine c' from columns 4 and 9:

s.

41.73C' -36.786=0 

-1.65 -36.736=0

s. 

+3.38C'- 0.050=0

s. 

C* =4.015

Forming equations fro

-2

4-0

4-

Ad

Ad

Normal equation, for

105t - . 28A

St -.028Aw 

5t
32 MANUAL OP TOPOGRAPHIC METHODS. 

COMPARISON OF TIME.

After time has been thus observed the chronometers at the two 

stations should be compared by telegraph.

Chronometers are compared in the following manner: The chronometer at one station being in circuit with the chronograph and 

recording upon it, the chronometer at the other station is switched 

into the general telegraphic circuit, by which it is brought to the 

first -station and switched into the local circuit there, so that the two 

chronometers register upon, the same chronograph, their beats being 

marked side by side by the same pen. After this has gone on for a 

minute or more the operation is reversed, the chronometer at the 

first station being switched into the telegraphic circuit and made 

to record upon the chronograph with the chronometer at the second 

station. Of course the observers are informed of the hour and minute 

at which the joint record upon the several chronographs begins.

This method constitutes what is known as the automatic exchange 

of signals.

The arbitrary exchange of signals is made as follows:

Each chronometer recording on its own chronograph as usual, and 

each local circuit being connected with the main-line circuit, the 

observer at one station breaks the circuit by means of the main-line 

talking key, which break is recorded on the chronograph sheets at 

both stations. The breaks are repeated every two seconds for at 

least one full minute. The operation is then reversed by the observer 

at the second station making the breaks, which are recorded at both 

stations as before.

The differences of time between the chronometers at the two 

stations are read from the chronograph sheets at each station and 

corrected 'for error of the chronometers. The results from the two 

chronograph sheets differ by an amount equal to twice the time 

occupied in transmission of signals. The mean of the two is therefore 

the approximate difference of longitude.

This result is yet to be corrected for personal equation, or the 

difference between the errors of observing of the two observers. 

Every observer has the habit of recording a transit a little too early 

or too late, the difference between two observers not infrequently 

being as great as a fourth of a second. To measure this difference, 

the observers usually meet, preferably at the known station, both 

before and after the campaign, and observe for a time each with 

his own instrument or with one similar in all respects to that used in 

the campaign. A comparison of the time determinations made by 

the two observers gives an approximation to the personal equation.

A better method, but one not always practicable, is for the observers, 

having completed half of the observations for time and longitude, to
OBSERVATIONS FOR AZIMUTH. ' 33

exchange stations for the remainder of the^work. The mean of the 

results before and after exchange of stations will eliminate personal 

equation.

There is one error incident to this work which can not be eliminated. 

This is the unequal attraction of gravity, or local attraction, or, as it is 

sometimes called, station error. A neighboring mountain mass will 

attract the plumb line and deflect the spirit level to such an extent 

as to cause serious errors in astronomic determinations of latitude 

and time. The same result is frequently produced by a difference in 

density of the underlying strata of rock, so that station errors of 

magnitude often appear where they are not expected. Indeed, the 

station error can not be predicted with any certainty as to amount 

or direction.

The only practical method of even partially eliminating this error 

is to select a number of stations for astronomic location under conditions as widely diverse as possible, connect them by triangulation, 

and by this means reduce all these astronomic determinations to one 

point, thus obtaining for this point a number of determinations each 

having a different station error. The mean of these gives for this 

point a position from which in part, at least station error has been 

eliminated, and this mean position can be transferred back to the 

several astronomic stations by means of the triangulation, thus 

giving each of them a position similarly comparatively free from station error.

OBSERVATIONS FOR AZIMUTH.

The initial direction from which the directions of other lines in 

primary triangulation and in primary traversing are computed is 

obtained by means of astronomic observations. Such observations 

should be taken not only upon the initial line, but at intervals thruout 

the work for its verification. Such intervals should not exceed' 100 

miles in the primary triangulation and 10 to 20 miles in primary 

traversing.

Xzimuth observations are made with the theodolite used in primary 

triangulation or traverse. The observations consist in the measurement of the horizontal angle between some close circumpolar star, 

usually Polaris, and a terrestrial mark, generally a lantern set at a 

distance of half a mile to a mile from the observing station. The 

time of observation on the star should be noted by a chronometer or 

a good watch. As the star is at a much higher angle of elevation 

than the lamp, it is necessary not only to level the instrument carefully but to measure the error of level and to correct for it. It is 

therefore essential that the value of a division of the level bulb be 

known. These observations for azimuth ma;f be made at any time
34 MANUAL OF TOPOaKAPHIC METHODS.

of the night, but it is very important that they be made at or near the 

time of elongation of the star, as it is then moving most slowly in 

azimuth, and any error in the time of observation has the least effect 

on the resulting azimuth. If such observations be taken at elongation, the reduction of the observations is simplified. When such 

observations are made at any other time than at elongation, the time 

must be noted, as it forms an element in the reduction. The error of 

the clock or watch used can be obtained by comparison with standard 

time, making correction for the difference in longitude between the 

station and the meridian of the standard time. A form of observation 

and record is appended.

Example of record of azimuth observations.

[Station: West base, near Little Rock, Ark. Fauth 8" theod. No. 300. December 27,

micr.=2". 1 div. level=3".]

1 div.

Object.

East base (mark) . 

East base (mark) .

East base (mark) . 

East base (mark) .

Time P. M.

h. m. s. 

11 00 18

11 09 20 

11 17 14

11' 26 22

Level.

West 

end.

Div. 

13.0 

50.5

64.44-

East 

 end.

Div. 

47.1 

10.2

57.3 

7.1

50.4 

13.8

64.2 

+

50.5 

12.9

63.4

+

10.3 

46.5

56.8 

7.4

10.1 

46.6

56.7 

6.7

14.3 

50.1

64.4

+

46.3 

10.5

56.8 

7.6

Micrometer.

A.

Telescoj

0 ' Div. 

346 00 14. 8

!)

101 32 18. 1 

101 32 19. 8 

345 58 22.0

Telescop 

211 28 29. 0

327 05 06. 7 

327 04 26. 3 

211 27 10. 7

B.

e direct.

0 ' Div. 

165 58 25. 1

281 31 21.8 

281 31 19. 7 

165 57 01. 4

3 reverse. 

31 27 23. 4

147 03 09. 5 

147 03 00. 6 

31 26 07. 4

Mean.

345 59 39.9

101 32 09. 9 

101 32 09. 5 

345 57 53. 4

211 28 22. 4

327 04 16. 2 

327 03 56. 9 

211 26 48. 1

Angle.

115 32 30. 0 

J115 34 16. 1

115 35 53.8 

J115 37 08. 8
REDUCTION OP OBSERVATIONS FOE AZIMUTH. 85

Summary of results of azimuth observations. 

[Station: West base, Arkansas. December 27,1888.]

Individual results.

Second set.......;.....'

Grand mean

294 10 34. 2|35 25 D 

34. 1}42' 35 Kg; 8}37. 65 D.

49 1\

Combined results.

38.80 

39. 38- 

38. 75 

40. 10

294 10 39.26

REDUCTION OF OBSERVATIONS FOR AZIMUTH.

The time of observation of a star should be first corrected for the 

difference in longitude, assuming that standard time has been used, 

and for the error of the watch. It is then reduced from mean to 

sidereal time. From the sidereal time of observation should be subtracted the right ascension of Polaris, if that star is used, which is 

given in the Nautical Almanac, the result being the hour angle or 

the sidereal time which has elapsed since it past the meridian of the 

place'of observation, given in hours, minutes, and seconds. This 

result should be converted into degrees, minutes, and seconds.

Then tan A = :

a sin t

1 & cos t ' 

where a = sec <£> cot d (< = latitude),

& = T -ps (6 = the declination of star) , -

2 = hour angle, 

and A = angle between true north and the star.
36 MANUAL OF TOPOGRAPHIC METHODS.

This angle should be corrected for level as follows:

level correct!011= '4-{w-\-w') (e-f-e')}tan li;

d being the value of a division of 'the level; 

w + w', readings of west end of level bubble; 

e + e', readings of east end of level bubble; 

Ji, the angular elevation of pole star.

An example of reduction is as follows:

Example of reduction of azimuth observations.

OSIation: West base; December 27,1888. Observer, S. S. G. Latitude=34 45 20.8. Longitude 92 13 31.5.]

h,. jn. s.

Time of observation Tw=ll 00 18

Correction; ninetieth, meridian time to 92°.215= 8 54

Watch slow; ninetieth meridian time + 02

Local mean time Tm=W 51 26

Correction; mean to sidereal time = +1 47

Right ascension mean sun 18 26 36

Sidereal time of observation =29 19 49

R. A. Polaris - 1 18 25

i=28 01 24 -24

h. m. s. 

t (time)= 4 01 24

t (arc) =60 21 00

a sin t tan <6 

tan A°~ 1-6 cost whero a=sec * cot S and 6== tolTs

0=34 45 26*8 log sec =0.0853539 log tan..34 45 26.8=9.8413076 

8=88 43 11.9 log cot.=8. 3491690 log tan. .88 43 11.9=1.6508310

log a ° ' " =8.4345229 log b =8.1904766 

log sin t 60.21 00 =9.9390515 log cos t =9.6943423

log a silU =8.3735744 log-. 0076704 =7.8848189 

log (1-6 cos t) o f =9.9966559 + 1.0000000

log tan A 178 38 08.0 =8.3769185 0.9923296 =l-&cos«

angle to +115 32 30.0.

mark , . .

Level correction 3.8 level corr.= -.1 (w+w'} (e. e') xtan h.

Az. of mark =294 30 34.2

When observations for azimuth are to be made at elongation, it is 

necessary to know the mean time of elongation. This is computed 

by the following method: »

The hour angle at elongation is obtained from the equation,

cos te = tan <p cot d.

The hour angle plus the right ascension of the star gives the sidereal 

time of its western elongation, which, reduced to mean time, gives 

the local -mean time in question.
REDUCTION OF OBSERVATIONS FOR AZIMUTH. 37

The azimuth, of a pole star at elongation is determined by the use 

of the equation,

sin A = sec </> cos d.

The following is an. example of these computations:

Example of the computation of the azimuth at elongation, and thv local mean times of

both elongations of Polaris.

[LatitiKle=4>=40°. Meridian of Washington. November 28, 1891.]

Sine azimuth at elongation=sec <j> cos 8. 

log sec 40° =0.1157400

log cos 6 88 44 05.5 =8.3439803 

log sin A 1 39 05.8 =8.4597203

Cos hour angle at elongation. /«,=tan <|> cot 8. 

log tan 40° ' =9.9238135

log cot S 88 44 05. 5 =8. 3440802

log cos le 88 5C 17.5 =8.2078997

h. m. s. 

te=5 55 45.2.

Sidereal time, western elongation, Ts=R. A. Polaris+fe.

h. m. s. 

R. A. Polaris = 1 19 35.2

te = 5 55 45.2

Sidereal time, western elongation,T«= 7 15 20.4 

R. A. mean sun= a«=16 29 14.4

Sidereal interval before noon, = 9 13 54.0 

Correction sidereal to mean interval = 1 30.7

Mean interval before noon = 9 12 23.3 Nov. 28.

Local mean time, western elongation^ 2 47 30.7 A. M., Nov. 28.

n. m. s.

Sidereal time, eastern elongation= 24h +a /«= 19 23 50.0

as=16 29 14.4

Sidereal interval after noon, = 2 54 35.6

Correction sidereal to mean interval = 0 28.0

Local mean time, eastern elongation = 2 54 07.0 -P. M., Nov. 28. 

Local mean time,, western elongation = 2 47 30.7 A.M..Nov. 28.

For longitudes west of Washington, decrease times of elongation 

0.66 second for each degree.
CHAPTER IV. 

BASE LINE, PRIMARY CONTROL, AND ELEVATIONS.

The primary control or geometric work is, in the ordinary case, 

effected by triangulation. Wherever this is not practicable or not 

economical, resort is had to what is known as primary traversing, but 

wherever the country presents sufficient relief for the purpose triangulation is employed, as it is more accurate and cheaper. In some 

parts of the country triangulation of sufficiently accurate character 

for controlling the topographic map of the United States has been 

executed by other organizations, notably by the Coast and Geodetic 

Survey and the Lake Survey. . Wherever such triangulation is available the results should be utilized.

PARTY ORGANIZATION.

The primary triangulation is generally carried on by a special party. 

Under certain circumstances, however, it is economical and advisable 

for one party to do all the work. The disadvantage is that it divides 

the time and attention of the topographer, requiring him to turn from 

one thing to another; the advantage, that it insures the selection of 

such points as are heeded by the topographer. If the work is done 

by a special party, the points selected are more likely to be chosen on 

account of their forming good figures in the triangulation than on 

account of their convenience and usefulness to the topographer. 

The secondary triangulation, the traversing, and the sketching should 

be carried on by different men, but under a single party organization. 

The sketching should be done by the chief of party, the secondary 

triangulation and height measurement by his most experienced 

assistant, and the traversing, with height measurement, .by the other 

assistants.

BASE-LINE MEASUREMENT. 

SITE.

The measurement of the base line is ordinarily the first of the preparatory steps toward map making. Upon the proper selection of its 

site and its correct measurement depends all the subsequent work of 

triangulation. The site must be reasonably level. It is not essential 

that it be absolutely so, but the more closely it approaches a plane

38
BASE-LINE MEASUREMENT. 39

the less difficulty will be experienced in making an accurate measurement. The site should afford sufficient room for the measurement of 

a base about 5 miles in length. A base much less than 5 miles in 

length is not always an economical one, inasmuch as it is less costly 

to extend the base than to complicate the expansion. A much 

greater length than 5 miles is unnecessary, because this length permits 

of easy expansion, and if the length be greater than this it may be 

difficult to construct intervisible signals at the ends of the base.

The ends of the base must be intervisible, and they must be so situated with regard to suitable points for expansion and triangulation as 

to form well-proportioned figures. Whenever possible, the base line 

^should form a side or diagonal of a closed quadrilateral or pentagonal 

figure.

While it is unnecessary to devote time to obtaining extreme accuracy in the measurement of a base, this measurement should be so 

nearly accurate that its errors can not affect the map altho multiplied 

many times in the associated triangulation. All necessary precaution 

should be taken to secure this result.

METHODS AND INSTRUMENTS USED.

Various methods and instruments have been employed in the measurement of base lines by the United States Geological Survey. At 

first wooden rods were employed,- varnished and tipt with metal. 

When used in measuring, these were supported upon trestles and contacts were made between them with considerable refinement. The 

advantage of using these rods consisted in the fact that their length 

was but slightly affected by temperature, which is the main source of 

error in base-line measurement, and being thoroly Varnished they 

were not greatly affected by moisture.

Subsequently bars of metal were employed of the pattern known as 

the Coast Survey secondary bars. These consisted each of a steel rod 

between two zinc tubes. As the two metals expand at different rates 

under changes of temperature, their relative lengths at any temperature as compared to the relative lengths at a normal temperature was 

theoretically an indication of the temperature of the bars at any time. 

The arrangement for indicating their relative lengths formed part of 

the apparatus, and was intended to indicate the temperature of the 

bars, and thus to afford means of reducing their lengths to a normal 

temperature. These bars have not been found, however, to work 

well in practise. Besides, there are other objections to the use of 

bars of any kind, which may be summarized as follows: First, their 

use is expensive; a considerable number of men are needed, and as the 

measurement proceeds slowly it often requires from a month to six 

weeks to measure and remeasure a base 5 miles in length. Again,
40 MANUAL OF TOPOGKAPHIC METHODS.

since these bars are but 4 to 6 meters in length, there are many contacts to be made in each mile of measurement, and each contact offers 

the possibility of a slight error.

In view of these objections and of certain positive advantages 

which the change would produce, it was decided to drop the use of 

bars in the measurement of base lines and to adopt long steel tapes. 

By their use it has been found easy to attain the required degree of 

accuracy in measurement, inasmuch as the number of contacts is 

much smaller than when bars are used, while the uncertainty in 

regard to the temperature of the measuring apparatus is reduced to a 

minimum by carrying on the measurement at night or in cloudy 

weather. The expense of the measurement is greatly reduced, since, 

fewer men are required, the work of preparing the ground and the 

work of measuring are much lessened, and the rapidity of measuring 

is increased manyfold. The diminished cost makes it practicable to 

measure much longer bases, thus decreasing the number of stations 

required in the expansion. This method allows, also, the measurement of base lines at shorter intervals in the triangulation.

The tape in use has a length of 300 feet. It should be carefully 

compared, at an observed temperature, with a standard, both before 

and after its use in base measurement. Preferably, the site for the 

base line should be selected along a railway tangent, as such location 

is approximately level, and the railway ties afford an excellent support for the tape. If such a location is not available, one should be 

selected that will fill the requirements previously mentioned, should 

be cleared of brush and undergrowth, and if necessary its sharp inequalities should be leveled. The tape should be supported at intervals of not more than 25 feet by a series of low stools whose legs are 

prest into the ground, or by a series of laths driven into the ground, 

each bearing a horizontal support made of a wire nail, while similar 

stools or laths should sustain each end of the tape.

The measurement of a base with the steel tape is a simple matter. 

Provision must, however, be made, first, for the proper alignment of 

the base; second, for the proper tension of the tape; and, third, for 

the measurement of temperature.

The alignment also is simple, and should be marked upon the 

ground in advance of the work of measurement. In cases where a 

railway tangent furnishes the site for the base line, no alignment is 

needed beyond the provision for keeping the tape always at a uniform 

distance from one of the rails.

For insuring a uniform tension of the tape, an ordinary spring 

balance should be attached to the forward end of the tape, where a 

tension of 20 pounds is applied. In order to apply this uniformly, 

and to insure against slip of the tape, an apparatus has been devised 

by Mr. H. L. Baldwin, jr., of the U. S. Geological Survey. It is
U. S. GEOLOGICAL SURVEY BULLETIN NO. 307 PL. II

-^^--jir-Tn \ / --p- MH^ ri-T-^i-^'-r*- t k - n

i :: «

i* --^

jj

r \

' J!4<

!, 1 «

"1-,.

A.

C

\ !! 1 _

*-Q

e.

o

<t\ 

^iii i

; [

ms iteiiiij

Mill

! ,' MIIIIII]

o

&K

^ 

' 

h.

c

1 '"' ° 1 ' ' 1

V

. forward enA ,profHe

" '.' plan. 

Tiear " profile

a..Tape b. Spring balance c Zinestrtps d Cogrrheel for str-curung tape e.Ratchet f. -WaiLfbrlwUling 

yearinplace (p.RaiLroajd. tie h,.-73ine strip i SLock support for vine,strip

DEVICE FOR STRETCHING TAPE IN BASE MEASUREMENT. 

Designed and used by H. L. Baldwin, Jr.
BASE-LINE MEASUREMENT. 41

shown in PL II. For its use, it is necessary to obtain strips of board 

about 5 feet long and 4 inches wide, in number equal to the number 

of lengths of tape of which the base line consists. Numbered strips 

of zinc, each about 8 inches long and 1 inch wide, are tacked to the 

boards, and the boards are nailed to the railway ties if measurement 

is to be made along a railway tangent. These boards are designed 

to support the devices for maintaining the tension, and the contacts 

are marked upon the strips of zinc. Mr. Baldwin's apparatus consists 

essentially of a wheel worked by a lever and held by .ratchets in any 

desired position. This wheel is attached to the spring balance in such 

way that by turning it the strain is put upon the spring balance, which 

is held at the desired tension by the ratchets. A small mechanism 

at the rear end of the tape holds the zero of the tape at the opposite 

mark. The great length of the tape, 300 feet, allows considerable 

friction or drag when the supports are frequent, and in order to insure 

a reasonably uniform distribution of the strain upon the tape it 

should be raised and allowed to fall with the strain on.

The measurements should be made at night or during cloudy days, 

in order that the temperature of the air, which is that indicated by the 

thermometers, and that of the tape shall be as nearly as possible the 

same. The temperature must be carefully observed by at least two 

thermometers at each tape length, in order that the best possible data 

for temperature correction may be obtained.

The base should be measured at least twice, and the two results 

compared by sections of 1,200 feet, or four tape lengths. The ends 

of the base should, if possible, be permanently marked by means of 

stone monuments set into the ground so that their surfaces are but a 

few inches above ground level, and the exact position of the ends 

should be indicated by a cross cut in a copper bolt embedded in the 

head of the stone, in order that the base may be preserved for future 

references.

A line of levels must be run over the site or over the stools which 

support the tape, for the purpose of obtaining its profile and thereby 

the means for deducing its horizontal length.

PERSONNEL OF PARTY.

The personnel required in the measurement of a base line should 

be, in an ordinary case, as follows:

The chief of the partjr, who exercises a general supervision over 

the work, marks the extremities of the tape, and provides the necessary precautions against errors in the measurement, as hereafter 

stated.

The rear tapeman, who adjusts the rear end of the tape to the 

contact marks and who carries and reads one of the thermometers.

Bull. 307 07 4 .
42 MANUAL OF TOPOGRAPHIC METHODS.

The head tapeman, who adjusts the forward end of the tape, 

exerts the requisite tension upon it, and carries and reads a second 

thermometer.

A recorder.

REDUCTION OF BASE-LINE MEASUREMENT. 

REDUCTION TO STANDARD.

The first correction to be applied is that of reduction to a standard. 

The correction for this should be obtained by comparison with the 

standard of the Bureau of Standards. The correction for the entire 

line is in proportion to the correction as obtained by comparison with 

the standard. If the tape be longer than the standard the correction 

will be positive; if shorter, negative.

CORRECTION FOR INCLINATION.

Data are obtained by running a line of levels over the base line. 

This line of levels gives the rise or fall, in feet and decimals of a foot, 

between the points of change in inclination. From this and the 

measured distance the angle of inclination is computed from the

formula, sin 0 = ~; R being the distance and ~h the difference in height,

both given in feet. The correction in feet to the distance is then 

computed by the equation,

sin 2 1' 

correction = ~ 02 R or 0.00000004231 02 R, 0 being exprest in

Zj

minutes. (See Lee's Tables, p. 83.)

 CORRECTION FOR TEMPERATURE.

Steel expands for each degree of temperature .0000063596 of its 

length. This fraction multiplied by the average number of degrees 

of temperature at the time the base line was measured above or below 

62 degrees, which is taken as the normal temperature, gives the proportion in which the base line is to be diminished or extended on 

account of this factor. Care must be taken to obtain correctly this 

average temperature. It must be the mean of all the thermometric 

readings, taken at uniform intervals of distance during the measurement. If the temperature be above the normal the correction is 

positive, and vice versa.

REDUCTION TO SEA LEVEL.

The base line is measured on an arc of a circle parallel to the sea 

surface and raised above it, at an elevation which is known at least 

approximately. This arc with radii drawn from its extremities to
REDUCTION OF BASE-LINE MEASUREMENT. 43

the center of the earth forms approximately a triangle similar to that 

formed by the radii of the earth with the sea surface. The length 

at sea level is derived with a sufficient approximation to correctness 

by the proportion,

R: li: : K: correction,

R being the radius of the earth, li the mean height of the base line 

above sea level, and K its measured length. (See Report United 

States Coast and Geodetic Survey, 1882, Appendix 9, p. 196.)

EXAMPLE OP REDUCTION OF BASE-LINE MEASUREMENT. '

The following is an example taken from the records of measurement of a base near Spearville, Kans., together with the reduction 

of this base for inclination, temperature, and elevation above sea 

level:

Record of measurement and reduction of Spearville base, Kansas. 

[Section!. Stations 0-10. October-10, 1889. Light rain falling.]

No. of 

tape.

1.. .......

2.........

3.........

4.........

5.........

6.........

7.........

8.........

9.........

10.........

Time, 

a. m.

h. m. 

10 13 

20 

26 

31 

37 

42 

47 

51 

55 

58

Tension.

Lbs. 

19.75 

20.00 

20.00 

20. 25 

20.00 

20. 125 

20.25 

20.00 

20. 125 

20.00

Thermometers.

A.

0

50.5 

50.5 

50.5 

50.5 

50.7 

51.5 

51.0 

50.8 

50.8 

50.7

B.

0

50.0 

50. 0 

50.0 

50.0 

50.5 

50.6 

50.8 

50.2 

50.0 

50.5

Temperature 

correction.

0

Mean temp. =50. 51

00 0

62-50.51=11.49

-11.49X3000. 

X- 000006

= -.207 foot.

Total length of section.

Feet. 

1 tape length.. .......... -300.0617

10X300.0617. ............. -3,000. 617

Temperature correction . . 207

Result first measurement = 3, 000. 410

[Second measurement, October 17, 1889.]

. No. of 

tape.

1... ......

2.........

.3.........

4.........

5.........

0.........

7.........

8.........

9.........

10.........

'

Time, 

p. m.

h. m. 

12 13 

21 

25 

29 

33 

36 

38 

41 

45 

50

Tension.

Lbs. 

20.00 

20.25 

20. 00 

19.75 

20.00 

20.00 

20.00 

20.12 

19.75 

20.13

Thermometers.

A.

52.3 

53.3 

53.8 

55.0 

55.0 

53.8 

54.0 

54.5 

55.1 

54.5

B.

0

52.4 

52.9 

54.0 

54.8 

53.2 

54.0 

54.0 

54.0 

54.4 

54.1

Temperature 

correction.

Mcan= 53.96

000

62 -53.90=8.04

-8.04X3000. 

X -000006

= -.145 foot.

Total length of section.

Tape set back from sta. 0 .85 inch. 

= .071 foot.

Feet. 

10X300.0617. .............. .-3, 000. 617

Set back................... .071

Temperature correction.. . .145

Result second ineasurement=3,000.401
44 MANUAL OF TOPOGKAPHIC METHODS.

Correct'! on for inclination, Sptarville base, Kansas.

Correction sins]/. Distance.

Approximate distance.

Feet.

200

4,200

4,000

1,000

2,000

4,200

2,800

1,000 

1,000

4,200

3,800

2,000

5,400

2,000

135

Difference of 

elevation.

Feet.

0.8

4.2

12.0

1.0

3.0

22.0

7.0

0.0 

1.0

20.0

6.0

4.0

31.4

2.0

0.05

Angle 0

, ,,

13 34

2 22

10 08

3 23

5 04

12 23

8 27

0 00 

 3 23

11 16

5 20

6 45

19 39

4 24

1 18

Log0

1. 1326

0. 3674

1.0052

0. 5250

0. 7024

1. 0917

0. 9263

0.0000 

0. 5250

1.0504

0. 7267

0. 8293

1.2934

0. 6437

0. 1072

2 log 6

//

2. 2652

0. 7348

2. 0104

1.0501'

1. 4049

2. 1834

1. 8527

0. 0000 

1.0500

2. 1008

1. 4535

1. 6586

2. 5867

1. 2874

0. 2144

Log ' 

sin 2 1'

2

2. 6264

 Constant'

Log 

dist.

2. 3010

3. 6232

3.6021 .

3.0000

3. 3010

3. 6232

3. 4472

3.0000 

3. 0000

3. 6232

3. 5798

3. 3010

3. 7324

3. 3010

2. 1303

Log correction.

7. 1926

6.9844

8. 2389

6. 6765

7. 3323

8. 4330

7. 9203

0. 0000 

6. 6764

8. 3504

7. 6597

7. 5860

8. 9455

7. 2148

4.9712

Correction.

.0015

.0010

.0173

.0005

.0021

.0271

.0084

.0000 

.0005

.0224

.0040

.0038

.0882

.0016

.0000

' . 1790

Reduction to sea level.

Kft 

...............--.....................................................= R

....................................................................... =4.05956

....................................................................... =2. 87599

.......................................................................= 3.19660

....................................................................... =0.13215

.......................................................................= 0.51599

Log 4.448 feet................................................................................. =0. C4814

Correction......

Log K (meters). 

Log ft (meters). 

CologR.........

Log 1.356 meters.. 

Log meters to feet.

Spearville base: Summary by sections.

' [Corrected for temperature.]

Stations.

Ito 10

10 20

20 30

30 40

40 50

50 60

60 70

70 80

80 90

90 100

100 110

110 119

119 126

First 

measure.

3,000.410

.418

.431

.426

.437

.417

.369

.366

.955

.676

3,000.899

2, 700. 581

2, 100. 244

37,806.629

Second 

measure.

3,000.401

.393

.431

.446

.478

.455

.392

.356

.938

.667

3,000.898

2, 700. 571

2, 100. 234

37,806.660

Difference.

First Second.

+.009

+.025

+ .000

-.020

-.041

-.038

-.023

+ .010

+ .017

+ .009

+.001

+ .010

+.010

-.031=. 372

Mean of 2 measurements.......... =«37,806.645

Reduction from S. W. base to A- - -168. 235 

Reduction from N. E. base to A- - . 2. 864 

Correction for inclination......... 0.179

Reduction to sea level............. 4.448

Corrected length.................. = 37,630.919

o Corrected for temperature.
BULLETIN NO. 307 PL. Ill

AL FOR USE WHERE OBSERVING TOWER IS NOT NECESSARY.

1
BASE LTNE, PRIMARY CONTROL, AND ELEVATIONS. 45

PRIMARY TRIANGULATION.

The base line having been measured, the next step is the expansion. 

This work, as well as the body of the triangulaticn, consists in the 

selection of stations, the erection of signals, and the measurement of 

angles. Each triangle from the base line outward will, when the 

angular measurement is completed, have one side and the three angles 

known, from which the other two sides can be computed by means of a 

simple trigonometric formula.

The expansion differs from the body of the triangulation only in the 

fact that the average length of the sides of the triangles is less. As 

the expansion progresses away from the base line the sides of successive 

triangles become gradually longer, until the average length of side of 

the triangulation is reached. Since the sides are increasing in length, 

and hence since any inaccuracy in the measurement of the base is

multiplied, thisswork must be planned and executed with greater care 

than the body of the triangulation requires.

A base line measured as above prescribed requires little expansion, 

since from the extremities of a 5-mile base one can observe directly 

on points 8 to 10 miles away, a distance as great as the average side 

of a triangle. Ordinarily, from the ends of the base, the surveyor 

can observe directly upon stations in his scheme of triangulation.

In the Western mountain region of the United States, where the 

sides of triangles may be 20 to 50 miles in length, an expansion is 

required.

SELECTION OF STATIONS.

In the selection of triangulation stations two sets of requirements 

must be observed.

(1) Such stations must be so selected as to afford what is known 

as strong figures, in order to reduce to a minimum the errors which 

will creep into an extended system. In order to insure intervisibility, they should, if possible, be located upon hill or mountain summits', the most commanding in the neighborhood. No triangle on 

which dependence is placed for the location of a station should have 

at that station an angle of less than 30 degrees or more than 150 degrees.

The stations should, if practicable, be grouped into simple figures, 

as quadrilaterals or pentagons, with an interior station, etc. Where 

an area is being covered with triangulation such groupings naturally 

occur, but in certain cases the triangulation takes the form of narrow 

belts of figures, and then the belt may consist of simple triangles or 

quadrilaterals, as more complex figures are rarely desirable.

(2) Since the sole object of triangulation is the control of the 

topographic map, the location of stations should be, adjusted to the 

needs of the topographers as far as is consistent with accuracy. 

This requirement affects most seriously the distance between stations.
46 MANUAL OF TOPOGRAPHIC METHODS.

Every plane-table sheet should contain at least three primary stations, and a fourth is desirable. Thus, for controlling the sheets on 

the scale 1:62500, the stations should not be more than 10 or 12 miles 

apart, and should be located with direct reference to the control of 

certain sheets. Again, since the primary stations must be occupied 

by topographers for intersecting on numerous points, they should be 

selected with reference to this requirement. They should command 

an extended view, especially of points suitable for cutting in, such as 

hill and mountain summits, houses, churches, etc.

Wherever possible, the instrument should be accurately centered 

under the signal. Whenever it is necessary to set up off center the 

direction and distance to the signal should be carefully measured 

and recorded.

SIGNALS.

While signals should be as simple and inexpensive as possible, 

their form and material must.depend upon the requirements and the 

materials at hand.. In a mountainous country where the summits 

are treeless, simple cairns of stone 7 to 10 feet in height are employed. 

Where the summits are wooded it is frequently convenient to clear 

them, leaving a single tree to serve as a signal. In such cases it is 

advisable to trim the tree of branches, with the exception of a tuft 

at the top. Where the station is clear but with green timber easily 

accessible, it is advisable to make a tripod of small trees, each with a 

tuft at its top. In undulating and hilly country it is often necessary 

to erect scaffolds. These should be built of sawed lumber and framed 

in simple fashion. If the lines are short a pole with a flag may be 

set in the top. If the lines are long the tower itself may serve as a 

signal, in which case its upper part should be clothed in black and 

white cotton.

PI. Ill shows a form of framed signals adapted to the treeless plains 

of Kansas and the rolling open hills of New England and elsewhere 

where observing towers are not necessary.

It is frequently necessary to raise the instrument to a considerable 

elevation in order to overlook surrounding obstacles. In such cases 

the structures for supporting the instrument should be combined 

with the signals, and hence they may properly be described and 

figured here. These observing towers. should be in two parts an 

interior structure, solidly built of sawed lumber if available, for the 

immediate support of the instrument, and a framework surrounding 

it and supporting a platform just below the instrument stand for the 

observer. The two should be separate, in order that the jarring 

incident to moving about on the platform may not be communicated 

to the instrument. Such a type of observing tower is shown in PI. IV.

In the Sierra Nevada of California, among the sugar-pine forests, a
U. S. GEOLOGICAL SURVEY BULLETIN NO. 307 PL. IV

SIGNAL COMBINED WITH TOWER FOR OBSERVER AND INSTRUMENT.
SIGNALS. 47

support for the instrument is frequently obtained by sawing off the 

top of a high tree and setting the instrument upon the stump, 50 or 

75 feet above the ground, the tree being guyed by wire cables to prevent swaying. Neighboring trees, sawed off, support the platform 

for the observer. Similar devices are resorted to in the forests of 

West Virginia, Kentucky, and Tennessee. In secondary triangulation in these regions the instrument's support in many cases is provided as above described, while the observer's platform is attached 

to the same tree instead of having an independent support. This 

is objectionable, but is often the best plan available.

In other cases it is more economical to support the instrument 

upon the ground, and to have openings made thru the forest upon 

the station hill, in the directions of the sight lines, or even to have 

the whole summit cleared.

HELIOTROPES.

It is often necessary to use more elaborate forms of signals, especially when the point observed upon is below the horizon line, so that 

the background, instead of being the sky, consists of forests or brown 

plains. In such cases resort is had to heliotropes. These are nothing 

more than devices for reflecting the sunlight to the observer at the 

instrument. The simplest form is a circular mirror with a screw 

hinged at the back, giving a universal motion. This is screwed into 

a stake or tripod over the center of the station to be observed upon, 

and a ray of sunlight is thrown thru a small hole in a board nailed 

to a stake 10 or 15 feet away and in the direction of the observer at 

the distant station. This form has the advantage of simplicity, as 

the simplest backwoodsman can manage it; and the triangulator can 

firmly fix all range stakes upon one visit to the station, and be sure 

of seeing the flash as he observes from each of the surrounding stations 

in turn.

Two other forms are in use the Coast Survey type and the Steinheil. The former consists of a telescope which is provided with a 

screw for fastening it into any convenient support or upon the theodolite. Upon the telescope are a mirror and two rings, the axis of 

the rings as well as the center of support of the mirror being 

parallel to the line of sight of the telescope. The telescope being 

directed upon the observing station, the mirror is so turned as to 

reflect the sunlight thru the rings and necessarily to the observing 

station. In many cases the use of a second mirror is necessary, 

owing to the relative position of the two stations and the sun, and 

such a mirror forms a part of the outfit. This form is little used, 

on account of its liability to get out of adjustment. The Steinheil 

heliotrope is a compact little instrument which can be easily carried 

in a case. It consists of a small sextant mirror, the two surfaces of
48 MANUAL OF TOPOGRAPHIC METHODS.

which are as nearly absolutely parallel as possible. This mirror has 

a small hole in the center of the reflecting surface. Below this hole 

is a small lens in the shaft carrying the mirror, and below the lens 

is some white reflecting material, as plaster of Paris. The mirror is 

so mounted that it has four motions, two about its horizontal axis 

and two about its vertical axis, each of which can be separately 

bound or controlled by clamps or friction movements. To use the 

Steinheil, it is screwed into some wooden upright, as the side of a tree 

in such a position that the main axis carrying the lens and plaster of 

Paris reflector shall be parallel with the sun's rays. The observer, 

standing behind the mirror, receives from the rear surface of the glass 

a reflection of the sun, producing an imaginary sun. The mirror 

should not be moved until this imaginary sun, moving with it, 

appears to rest on the object to which it is desired to cast the flash, 

as the hill on which the triangulator is standing. As the surfaces of 

the mirror are parallel, the true reflected rays of the sun will also be 

cast on the object sighted to.

This instrument is in great favor, especially with the Western 

parties, where portability is of moment, first, because it is light and 

convenient to carry and use, and, second, because there are no movable parts to get out of adjustment by jarring. The latter is a serious 

defect in the Coast Survey instrument, since it requires frequent 

testing to make sure that neither of the two rings has been moved, 

as such movement would cause the reflections to be cast out of parallelism with the line of sight of the telescope.

The use of heliotropes presupposes the employment of men to 

operate them, thus increasing materially the expense of- the work. 

Misunderstandings continually arise between the heliotropers and 

the observer, causing vexatious delays, and therefore their employment should be avoided whenever possible.

THEODOLITES FOR TRIANGULATION.

Instruments differing widely in power and degree of accuracy have 

been in use by the United States Geological Survey for the measurement of angles in the primary triangulation. Formerly theodolites 

having circles 6, 7, 8, 10, and 11 inches in diameter and reading by 

vernier to 10 seconds were employed, and the results were reduced 

and adjusted by least squares. Subsequently it appeared desirable 

to employ instruments of a higher class and thus obtain more accurate results, which would render unnecessary this tedious adjustment. Pursuant to this decision the use of these vernier theodolites 

has been largely discontinued, and theodolites having 8-inch circles, 

reading by micrometer microscopes, have been substituted almost 

universally in the primary work." (See PL V.)
U. S. GEOLOGICAL SURVE ULLETIN NO. 307 PL. V

EIGHT-INCH THEODOLITE.
MEASUREMENT OF HORIZONTAL ANGLES. 49

The circle, as above stated, has a diameter of 8 inches, and is subdivided to 10 minutes. The object glass is 2 inches in diameter and 

its focal distance is 16^ inches. The telescope with the eyepiece 

commonly used has a power of about 30 diameters.

The circle is read by means of two microscopes placed opposite 

each other. Within the field of the microscope is a comb stretching 

over the space of 20 minutes. This comb has ten teeth, divided 

into two parts by a depression, each corresponding to 2 minutes. 

Parts of a minute down to 2 seconds are read by means of a micrometer screw moving a pair of fine threa.ds in the field of the microscope.

INSTRUCTIONS FOR MEASUREMENT OF HORIZONTAL ANGLES.

 The following general precautions should be observed in the measurement of all horizontal angles in the primary triangulation:

. The instrument should have a stable support, which may be a stone 

pier, a wooden post, or a good tripod. If a portable tripod is used 

its legs should be set firmly in the ground.

The instrument should be protected from the direct rays of the sun 

by means of an umbrella or a piece of canvas like a tent fly. It 

should also be shielded from winds that might jar or twist either it 

or its support.

The foot screws of the instrument, after it is leveled for work, 

should be tightly clamped. Looseness of the foot screws and tripod 

is a common source of error, especially with small instruments.

The alidade, or the part of the instrument carrying the telescope 

and verniers or microscopes, should move freely on the vertical axis. 

Clamps should likewise move freely when loosened. Whenever either 

of these moves tightly the instrument needs cleaning, oiling, or 

adjusting.

The observer should always have a definite preliminary knowledge 

of the objects or signals observed. The lack of it may lead to serious 

error and entail cost much in excess of that involved in getting such 

knowledge.

Great care should be taken to insure correctness in the degrees 

and minutes of an observed angle. The removal of an ambiguity 

in them is sometimes a troublesome or expensive task.

The errors to which measured angles are subject may be divided 

into two classes those dependent on the instrument used, or instrumental errors; and those arising from all other sources, which, for 

the sake of distinction, may be called extrainstrumental errors;

The best instruments are more or less defective, and all adjustments 

on which precision depends are liable to derangement; hence the 

general practise of arranging observations in such a manner that the 

errors due to instrumental defects will be eliminated in the end results.
50 - MANUAL OF TOPOGRAPHIC METHODS.

The principal errors of this kind and the methods of avoiding their. 

effects are enumerated below.

Measurements made with a graduated circle are subject to certain 

systematic errors commonly called periodic. Certain of these errors 

are always eliminated in the mean (or sum) of the readings of the 

equidistant verniers or microscopes, and both of the latter should be 

read with equal care in precise work. Certain other errors of this 

class are not eliminated in the mean of the microscope readings, and 

only these need consideration. Their effect on the mean of all the 

measures of an angle may be rendered insignificant by making the 

number of individual measures with the circles in each of n equidistant positions separated by an interval equal to , where m

is the number of equidistant verniers or microscopes. Thus, if m = 2, 

the circle should be shifted after each measure by an amount equal

180° 

to , which, for example, is 45° if ft = 4 and 30° if ft =6. The

n ' ' ^ '

degree of approximation of this elimination increases rapidly with n. 

The effect of errors of this class is always nil on an angle equal to the 

angular distance between consecutive microscopes or a multiple 

thereof. Other things being equal, therefore, we should expect the 

measures of such special angles to show less range than the measures 

of other angles.

Besides the instrumental errors of the periodic class, there are also 

accidental errors of graduation. These are in general small, however, in the best modern circles and their effect is sufficiently eliminated by shifting the circle in the manner explained in the preceding 

paragraph.

The effect of an error of collimation on the circle reading for any 

direction varies as the secant of the altitude of the object observed. 

The effect on an angle between two objects varies as the difference 

between the secants of their altitudes. This effect is eliminated 

either by reversing the telescope in its wyes, or by transmitting it 

without changing the pivots in the wyes, the same number of measures being obtained in each of the two positions of the telescope. 

The latter method is the better one, especially in determining azimuth, 

since it eliminates at the same time errors due to inequality of pivots 

and to inequality in height of the wyes.

The effect of the error of inclination on the circle reading for any 

direction varies as the tangent of the altitude of the object observed. 

If the inclination is small, as it may always be by proper adjustment, 

its effect will be negligible in most cases. But if the objects differ 

much in altitude, as in azimuth work, the inclination of the axis must 

be carefully measured with the striding level, so that the proper cor-
MEASUREMENT OP HORIZONTAL ANGLES. 51

rection can be applied. The following formula includes the corrections to the circle reading on any object for collimation and inclination 

of telescope axis:

c sec Ji -f 1) tan Ji;

c collimation in seconds of arc,

b = inclination of axis in seconds of arc,

Ti = altitude of object observed.

Parallax of wires occurs when they are not in the common focal 

plane of the eyepiece arid objective. It is detected by moving the 

eye to and fro sidewise while looking at the wires and the image of 

the object observed. If the wires appear to move in the least, an 

adjustment is necessary. The eyepiece should always be first adjusted to give distinct vision of the cross wires. This adjustment 

is entirely independent. of all others and requires only that enough 

light to illuminate the wires enter the telescope or microscope tube. 

This adjustment is dependent on the eye and is in general different 

for different persons; hence maladjustment of the eyepiece can not 

be corrected by moving the cross wires with reference to the objective. 

Having adjusted the eyepiece, the image of the object observed may 

be brought into the plane of the cross wires by means of the rack-andpinion movement of the telescope. A few trials will make the parallax 

disappear.

When circles are read by micrometer microscopes it is customary 

to have them so adjusted that an even number of revolutions of the 

screw will carry the wires over the image of a graduation space. If 

the adjustment is not perfect an error of run will be introduced. This 

may in all cases be made small or negligible since, by means of the 

independent movements of the whole microscope and the objective 

with respect to the circle, the image may be given any required size. 

In making this adjustment some standard space, or space whose error 

is known, should be used. At least once at each station where angles 

are read observations should be made for run of micrometers.

Tangent and micrometer screws should move freely, but never 

loosely. In making a pointing with the telescope the tangent screw 

should always move against or push the opposing spring. Likewise, 

bisections with the micrometer wires should be made always by making the screw pull the micrometer frame against the opposing spring 

or springs.

Extrainstrumental errors may be divided into four classes errors 

of observation, errors from twist of tripod or other support, errors 

from centering, and errors from unsteadiness of the atmosphere.

Barring blunders or mistakes, the errors of observation are in general relatively small or unimportant. With practised observers in
52 MANUAL OF TOPOGRAPHIC! METHODS.

angular measurements, such errors are the least formidable of all the 

unavoidable errors, and their elimination in the end results is usually 

well-nigh perfect. The recognition of this fact is A^ery important, 

for observers are prone to attribute unexpected discrepancies to bad 

observation rather than to the much more probable cause. After 

learning how to make good observations the observer should place 

the utmost confidence in them, and never yield to the temptation 

of changing them because they olisagree with some preceding observations. Such discrepancies are in general an indication of good 

rather than poor work.

Stations or tripods which have been unequally heated by the sun 

or other source of heat usually twist more or less in azimuth. The 

rate of this twist is often as great as a second of arc per minute of 

time, and it is generally nearly uniform for intervals of ten to twenty 

minutes. The effect of twist is to make measured angles too great 

or too small according as they are observed by turning the microscopes 

in the direction of increasing graduation or in the opposite direction. 

This effect is well eliminated, in general, in the mean of two measures, 

one made by turning the microscopes in the direction of increasing 

graduation and followed immediately by turning the microscopes in 

the opposite direction. Such means are called combined measures 

or combined results, and all results used should be of this kind. As 

the uniformity in rate of twist can not be depended on for any considerable interval, the more rapidly the observations on an angle can be 

made the better will be the elimination of the twist. The observer 

should not wait more than two or three minutes after pointing on one 

signal before pointing on the next. If for any reason it should be 

necessary to wait longer, it will be best to make a new reading on the 

first signal.

The precision of centering an instrument or signal over the reference 

or geodetic point increases in importance inversely as the length of 

the triangulation lines. Thus, if it .is desired to exclude errors from 

this source as small as a second, one must know the position of the 

instrument within one-third of an inch for lines a mile long or within 

6 inches for lines 20 miles long. The following easily remembered 

relations will serve as a guide to the required precision in any case:

1 second is equivalent to 0.3 inch at the distance of 1 mile.

1 second is equivalent to 3.0 inches at the distance of 10 miles.

1 second is equivalent to 6.0 inches at the distance of 20 miles.

1 minute is equivalent to 1.5 feet at the distance of 1 mile.

The notes should always state explicitly where the instrument and 

signals are and give their coordinates (preferably polar coordinates) 

if they are not centered.

Objects seen thru the atmosphere almost always appear unsteady, 

and sometimes this unsteadiness is so great as to render the identity 

of the object doubtful. It is usually greatest during the middle of
MEASUREMENT Ob' HORIZONTAL ANGLES. 53

the day, and generally subsides or ceases an hour or two before sundown. There is also frequently a short interval of quietude about 

sunrise, and on cloudy days many consecutive hours of steady atmosphere may occur. For the best work observations should be made 

onty when the air causes small or imperceptible displacements .of 

signals. In applying this rule, however, the observer must use his 

discretion. Errors of pointing increase rapidly with increase of 

unsteadiness, but it will frequently happen that time may be saved 

by counterbalancing errors from this source by making a greater 

number of observations. Thus, if signals are fairly steady it may be 

economical to make double the number of observations rather than 

wait for better conditions.

The best results in a triangulation are to be obtained by measuring 

the angles separately and- independently. Thus, if the signals in 

sight around the horizon are in the order A, B, C, etc., the angles 

A to B B to C, etc., are by this method observed separately, and 

whenever there is sufficient time at the observer's disposal this 

method should be followed.

Besides measuring single angles, it is desirable to measure independentl}7" combined angles i. e., angles which consist of the sum of two 

or more single angles. Thus, supposing.0 to be the observing station 

and A, B, and C stations sighted on, the observer should measure not 

only the angles AOB and BOG, but the combined angle AOC. This 

is necessary not only because this angle may be used directly in the 

triangulation, but because it will be needed in forming conditions for 

adjusting the angles about the observing station, or the station 

adjustment, as it is called.

In order to secure the elimination of the errors mentioned above, 

the following procedure must be strictly adhered to:

Point on A and read both microscopes.

Point 011 B and read both microscopes.

Transit telescope and turn microscopes 180°.

Point on B and read both microscopes.

Point on A and read both microscopes.

] 80° 

Shift circle by -" -- - and proceed as before until n such sets of measIV

ures have been obtained.

Then measure the angles B to C, C to D, etc., including the angle 

necessary to close the horizon, in the same manner.

A form for record and computation of the results is given on page .

When repeating instruments are used; the same procedure will be 

followed, except that there should be five pointings instead of one on 

each of A and B, the circle being read for the first pointing on A and 

the fifth on B, and again for the sixth pointing on B and the tenth 

on A.
54 MANUAL OF TOPOGRAPHIC METHODS.

The importance of having the measures of a set follow in quick 

succession must be constantly borne in mind. Under ordinarily 

favorable conditions an observer can make a pointing and read the 

microscopes once a minute, and a set of five repetitions should be 

made in five minutes or less.

When several stations or signals are visible and a nonrepeating 

instrument is used, time can be saved without material loss of precision in the angles, by observing on all the signals successively 

according to the following procedure, the signals being supposed in 

the order A, B, C, etc., as above:

Point on A and read microscope.

Point on B and read microscope.

Point on C and read microscope.

Point on A and read microscope.

Transit telescope and turn microscopes 180°.

Point on A and read microscope.

Point on B and read microscope.

Point on C and read microscope.

Point on A and read microscope.

180° 

Shift circle by and proceed as before until n such sets have been

obtained.

The angles A to B, B to C, etc., read in this way may be computed 

as in the first method, always combining the measure A to B with 

the immediately succeeding measure B to A to eliminate twist. 

There is a theoretic objection to this process of deriving angles, 

founded on the fact that they are not independent, but in secondary 

work this objection may be ignored as of little weight.

For the 11-inch theodolite and for the 8-inch instruments made 

by Fauth & Co., all of which read by micrometer microscopes, four 

sets of measures on as many different parts of the circle will be 

required; and for the repeating theodolite six sets of measures will 

be required, all measures being made according to the procedures 

given above.

Under ordinary circumstances and with due care in centering, 

angles measured as specified. above should show an average error of 

closure of less than 5 seconds. Under specially unfavorable conditions the number of sets of measures should be increased, care being 

always taken to shift the circle so as to eliminate periodic errors.

The practise of starting the measurement of an angle or series of 

angles with the microscopes reading 0° and 180°, 90°, and 270°, etc., 

should be avoided; otherwise the errors of these particular divisions 

will affect many angles. In shifting the circle it is neither necessary
MEASUREMENT OF HORIZONTAL ANGLES. 55

nor desirable to have the new positions differ from, the preceding

180° 

one by exactly .. A difference of half a degree either way is unimTb

portant as respects periodic errors, and it is advantageous to have 

the minutes and seconds differ for the different settings.

Field notes should be clear and full. The date, place, name, and 

number of instrument used, and the names of observer and recorder 

should be recorded at the beginning of each day's work at a station. 

The positions of the instrument and signals observed should be 

denned, either by a full statement or by a reference to such, in each 

day's notes. The time of observations should be noted at intervals, 

to show that the instrument does not stand too long between 

pointings.

A preliminary reduction of all observations should be made in the 

field, in order to detect and correct gross errors. A plat of the 

triangulation should be made as the work progresses.

When mistakes are made in the record, the defective figures should 

not be erased, but simply crost out, and an explanation furnished 

in the column of remarks. Great care should be taken not only to 

avoid "cooking" or "doctoring", notes, but to avoid suspicion thereof.

The following example of form of record is taken from the primary 

triangulation executed in western Kansas:

Record of measurement of horizontal angle.

Station: Township corner, Kansas, July 1, 1889. Fauth 8-inch theodolite No. 302, one division of

micrometer head 1 seconds.]

Station.

Walton.......................

Walton.......................

Walton.......................

Walton.......................

Walton.. .....................

Micr. A.

Telescor. 

0 ' Div. 

93 12 11.3 

129 41 11.9 

129 41 15. 0 

93 12 10.0 

Telescope 

138 27 .03.2 

174 50 02.8 

174 50 06.2 

138 27 05.2 

Telescope 

183 07 03.0 

219 30 05. 0 

219 30 08. 1 

183 07 06.4 

Telescof 

228 24 28. 1 

264 53 27. 4 

264 54 01.1 

228 24 29. 3

Micr. B.

e direct. 

0 ' Div. 

273 12 09.9 

309 41 13.2 

309 41 12.1 

273 12 09.1 

reversed. 

318 26 28.0 

354 55 28.9 

354 55 29.5 

318 26 27. 4 

reversed. 

3 06 27.2 

39 35 29.8 

39 35 29.5 

. 3 06 28.1 

e direct. 

48 24 22.6 

84 53 26.1 

84 53 '26.1 

48 24 22.1

Mean reading.

93 12 21.2 

129 41 25.1 

129 41 27.7 

93 12 19.7

138 27 01.2 

174 . 56 01. 7 

174 56 05.7 

138 27 02.0

183 07 00.2 

219 30 04.8 

219 30 07.0 

183 07 04.5

228 24 50.7 

264 53 . 53. 5 

264 53 57.2 

228 24 '51. 4

Angle.

36 29 03.9 

08.0

00.5 

03.1

04.6 

03.1

02.8 

05.8

Mean.

05.9

01.8 

03.9 

04.3

. ,4 | 15". 9 

.Mean of 4 combined measures a.......................................................... =36° 29' 03". 98

<* Instrument over center of station.
56 MANUAL OF TOPOGRAPHIC METHODS.

ORGANIZATION OF PARTIES AND PROSECUTION OF WORK.

A party for carrying on primary triangulatiori usually comprizes 

only the chief and an assistant, with the addition of a driver and a 

cook in case the party is living in camp. Frequently, however, it 

is found economical to employ a man to superintend the construction of signals. The chief of party is expected to select the stations 

and direct what forms of signals shall be erected, and to measure 

angles. In a mountainous country the selection of stations is usually 

a simple matter. From the summit of a mountain the chief of party 

may be able to select stations for considerable distances ahead and 

to order the erection of signals, leaving to the man employed for that 

purpose the erection of them. On the other hand, in a densely 

wooded region such, as the Cumberland Plateau, where the summits 

have approximately the same elevation, the selection of stations is. 

extremely difficult, requiring great ability and experience and involving an immense amount of labor. In such a region the chief of party 

finds it necessary to travel great distances, visit many hills, and 

even to climb to the summits of the highest trees, in order to select 

intervisible stations.

The selection of stations must be kept in advance of the reading 

of angles, but it is not advisable to keep it too far ahead, on account 

of the danger of the destruction of signals before angles have been 

read upon them. Therefore the chief of party finds it necessary to 

alternate between the two kinds of work, selecting and preparing 

three or four stations, then returning and measuring the angles.

When it is necessary to use heliotropes the party must be increased 

by one man for each such instrument employed. The proper management of such a party then calls for the exercise of much judgment 

on the part of the triangulator. If it is convenient for the chief of 

party to place each heliotroper before observing angles, and to show 

him where to direct his instrument, men of ordinary intelligence 

may be employed, and the work is one calling for time rather than 

for skill. Where, however, the party is moving frequently the 

observer and heliotropers occupying different stations nearty every 

day, as is possible in the dry atmosphere usually prevailing in the 

West, the chief of party has to arrange a schedule for each man, 

showing the order in which he is to occupy the stations and in what 

direction he is to flash from each. In this case the heliotroper must 

be a man having some topographic and technical skill, so that he 

may find his point, set up on center, and direct his flashes to the right 

place, besides exercising a goodly amount of judgment. A simple 

code of signals having been agreed upon, it becomes easy for the triangulator to let the heliotroper know that the work is completed, 

when he at once moves to the next designated station.
J3ASE LINE, PRIMARY CONTROL, AND ELEVATIONS. 57

REDUCTION OF PRIMARY TRIANGULATION. 

KEDUCTION TO CENTER.

In case any station is occupied off center, the directions as read 

must first be reduced to center. In the diagram, let x be the point

occupied, y the station, / the distance between them, A the point 

to which the direction is laid and the angle at that point, and R its 

distance, approximately known. Then, from the relations between 

the sides and the angles of the triangle, '

R :r :: sin x : sin A,

. r sin x , . ,. , . r sin x sin A = -p , and A (in seconds) =

sin

correction in seconds of arc.

The following example taken from the triangulatioii in Kansas will 

illustrate the form of effecting this reduction. The references are to 

the diagram on page 59 :

. Reduction to center of station at Walton A, Kansas.

[Sec explanation: Appendix No. 9, page 107, U. S. Coast and Geodetic Survey report for 1882.]

Distance, inst. to center=0'. 48 log = 9.0812 

log feet to meters = 0. 5160

Distance, inst. to center log meters = 9.1652=logr.

Direction. x to n

7°.

9. 0859

5. 9321

9. 1652,

= (>1 AA

9. 4970 

0". 31

x to o 

' 73°.

9. 9800

5. 9182

9. 1652

5

Q1X/1

0. 3784 

2". 39

x to p 

105°.

9. 9849

0. 4228

9. 1652

C 01 AA

0. 8873 

7". 71

x to q

185°.

8. 9403

0. 2434

9. 1652

'C O1 AA

9. 6633

x to r 

273°.

9. 9994

6.0079

9. J652

5 *}ld4

0. 4869 

3". 06

x to s 

306°.

9. 9080

' 0.2514

9. 1052

5 O1 A A

 0. 6390

Correction to angle a = n to o - 0.31 +2. 39 = + 2.08 

6=6 to p -2. 39 +7. 71= +5.32 

g = n to p -0. 31 +7. 71 = +7. 40 

c= p to q -7. 71 -0.40= -8.17 

d=<? to r +0.46 -3.06 =-2. 60 

e = .r to 5 +3.06 -4.36= -1.30 

h=q to s+0.46-4. 36= -3.90 

t=s to 71+4.36+0.31 =+4.67

Bull. 307 07 5
58 MANUAL OF TOPOGBAPHIC METHODS.

SPHERICAL EXCESS.

The angles are measured on a spherical surface, and the sum of the 

three measured angles of each triangle should equal 180° plus the 

spherical excess. The latter, however, needs to be computed and 

subtracted from the sum of the angles only for the purpose of testing 

the accuracy of closure of the triangle, as in the reduction the angles 

are treated as plane angles. When the area of the triangle is large 

the spherical excess in seconds (E) should be computed by the 

equation,

_

r'smi'"

where S is the area of the triangle in square miles, and r the radius of 

curvature of the earth in miles. When the triangle (being within the 

United States) has an area less than 500 square miles, r may be 

assumed as constant, and the spherical excess may be obtained by 

dividing the area in square miles by 75.5.

STATION ADJUSTMENT.

The next step is the adjustment of the angles about the observing 

station, or the station adjustment, as it is called. Referring to the 

diagram, which represents the angles read at Walton station, in 

Kansas, it is seen that eight angles were measured, as follows:

Record of angles measured at Walton, Kansas.

g Dunkard Newt ................................

Obs. angle.

0

65 

31

97 

97

45 28.37 

47 58. 50

33 26.87 

33 28.39

-1. 52

87 

34

121 

121

44 57.41 

00 03.35

44 60.76 

44 59.05

+ 1.71

61 

97 

79 

121

359

09 26.17 

33 28.39 

32 06.25 

44 59.05

59 59.86 

-0.14

Station adjustment.

+.51 

+ .52

-.49

-.56 

 -.50

+.59

+.02 

-.49 

+.02 

+.59

Correction to 

center.

+2.08 

+5. 32

+7.40

-2.60 

, -1.30

-3.90

+4.67 

+ 7.40 

-8.17 

-3.90

Angles locally 

adjusted and 

reduced to 

center.

65 45 30.96 

31 48 04.34

97 33 35.30 

97 33 35.30

' 00.00

87 44 54.25 

34 00 01.49

121 44 55.74 

121 44 55.74

00.00

01 09 30.86 

97 33 35. 30 

79 31 58.10 

121 44 55.74

-360 00 00.00 

00.00

Of these a + 1 should = g, d + e should = h, and g + c + li +f should 

= 360°. Thus are formed in this case three conditions affecting eight 

unknown quantities. The method by which are found the correc-
STATION ADJUSTMENT. 09

tions which fulfil these conditions is that known as the method of 

least squares, for a full discussion of which, showing its application to 

triangulation, see " The Adjustment of Observations," by T. W. Wright 

and J. F. Hayford, pp. 180-260, (New York, D. Van Nostrand, 1906). 

It is unnecessary to explain the theory of this method, but it is desirable to show how it is applied in the class of cases under consideration,

which can best be done by tracing a case thru. There are here 

three equations of conditions, as follows:

(1) a + Z>-<7-!".52 -0;

: (2) d + e-li + l"..7l =0;

(3)

in which the letters represent, not angles, as in the diagram, but 

unknown corrections to the angles.' The coefficient of each of these 

corrections is unity. Arrange them in tabular form, the letters at 

the top referring to the equations, thus forming what is called a table 

of correlates. Now multiply each coefficient by itself and every 

other in -the same horizontal line and sum them. Three normal 

equations result as follows:

a 

b 

c 

d 

e 

f 

(1 h

1 

1

_ ^

1 1

 1

1

1 1 1

1 +3. OOw - 1. 002 - 1". 52 = 0 

2 +3. QOy -1.002 +1". 71 = 0 

3 -l.OOw -LOO?/ +4.QOZ-0". 14=0
60 MANUAL OF TOPOGEAPHIC METHODS.

These three equations involving three unknown quantities are then

solved by elimination, with results as follows:

//

w= +.515. ' 

y =-.562. 

2= +.023.

These values can now be substituted in the table or correlates, 

columns 1, 2, 3; the algebraic sum of lines a, I, c, d, etc., giving corrections to the angles a, 1), c, d, etc.

1 9 ., Corrections to 

' angles.

+. 515 

+ .515

-.562

-.562

+ .023

+ .023

+ .023

+ .023

+ .515

+ .515

+ .023

-.562

-.562

+ .023

-.492

+ .585

FIGURE ADJUSTMENT.

The measurement of the angles having been executed by instruments and methods much better than the needs of the map require, it 

is not ordinarily necessary to make any figure adjustment further 

than an equal distribution of the error of each triangle among the 

three angles. Still, as the necessity for a more elaborate adjustment 

may arise, a description of the method of applying the least squares 

to adjustment of geometric figures in triangulation is here given, with 

a simple example of its application.

Each geometric figure in a system of triangulation is composed of a 

number of triangles. The measured angles of each triangle should 

equal 180° plus the spherical excess. Each triangle, therefore, furnishes an equation of condition, which is known as an angle equation. 

The number of angle equations in any figure is equal to the number of 

closed triangles into which it can be resolved. But since certain of 

these are a consequence of the others, the number of angle conditions 

which it is desirable to introduce is less than the number of triangles. 

The number of angle equations in any figure is equal to the number of 

closed lines in the figure plus one, minus the number of stations. Thus, 

in a closed quadrilateral, the number of angle equations is 6 + 1 4 = 3. 

There is another class of conditions, known as side equations, which 

can be best explained by reference to a figure. In the diagram, suppose the figure 0, 1, 2, 3 to represent the projection of a pyramid, of which 1, 2, 3 is the 

base and 0 the apex. ^ A geometric condition 

of such figure is that the sums of the logarithmic 

sines of the angles about the base, taken in one 

* ^ direction, must equal the similar sums taken in 

the other direction i. e., the product of the sines must be equal. In
FIGURE ADJUSTMENT. 61

the present case, log sin 0, 1, 2 -|-log sin 0, 2, 3 + log sin 1, 3, 0 should 

equal log sin 1, 2, 0 + log sin 2, 3, 0 + log sin 0, 1, 3.

The number of side equations which can be formed in any figure is 

equal to the number of lines in the figure, plus 3, minus twice the 

number of stations in it, or I + 3 2n. In a quadrilateral, 6 + 3 8 = 1.

The numerical term in each angle equation is the difference between 

the sum of the observed angles on the one hand and 180° plus the spherical excess on the other. This is positive when the sum of the observed 

angles is the greater and vice versa. The coefficients of the unknown 

corrections are in each case unity, unless weights are assigned.

The numerical term in each side equation is the difference between 

the sums of the logarithmic sines, taken in the two directions. The 

coefficients of the unknown corrections are the differences for one 

second in the logarithmic sines of the angles.

The method of making up and solving these equations and applying 

the corrections to the angles can best be shown by means of an 

example. That here given is the simplest case involving both angle 

and side equations, namely, the case of a quadrilateral. The method 

of forming correlatives and normal equations, and their solution, is 

similar to that employed in station adjustment, and therefore the 

details are omitted.

In the equations of conditions and correlatives the angles are 

designated by directions to which the corrections are finally applied. 

Thus the angle of 3, 0, 2 is designated 3/0 + 2/0, the sign being 

given to the left-hand and the sign + to the right-hand direction.

EXAMPLE OK1 FIGURE ADJUSTMENT BY LEAST SQUARES.

(«) J

(>>) . .'!

f CO--- 

I

3-0-1 

0-1-3 

1-3-0

Spherica 

Closure e

0-1-2 

1-2-0 

2-0-1

Closure e

1-2-3 

2-3-1 

3-1-2

Closure e

2-3-0 

3-0-2 

0- 2- 3

Closure e

Observed angles.

120 

21 

37 

180 

excess 

rror

81 

<i2 

35 

180

rror

91 

28 

CO 

180

rror

65 

84 

29 

180

rror

39 14. 781 

26 17 SOU 

54 37.180 

00 09.767 

 =- 0.148 

+ 9.619

52 51.222 

22 38. 500 

44 45. 861 

00 15.583 

- 0.189

+ 15.394

28 38.000 

95 10.360 

26 33. 416 

00 21.776 

- 0.234

+21. 542

59 47.540 

54 28.920 

05 59. 500 

00 15.960 

- 0. 193

+ 15.767
62 MANUAL OF TOPOGRAPHIC METHODS.

Side equation. 

[Taking 0 as the pole.]

(rf).

Angle.

0.1.2.........

0.2.3.........

1.3.0.........

1.2.0.........

2.3.0......... 

0.1.3.........

From above . . .

Difference . . .

Log sines of 

spherical angle

9. 9956249. 7

9. 6869340. 0

9. 7884705. 9

29. 4710295. 6

9. 9474437. 5

9.9607184.9 

9. 5628859. 2

29. 4710481. 6

29. 4710295. 0

00. 0000186. 0

Tabular 

difference 

for 1".

+3.0

37.9

27.0

11.0

9.4 

53.7

. Corrections to 

log sines.

- 25.0

-127.9

- 1.2

- 59.4

- 77.7 

-203.0

Corrected log 

sines of spherical angles.

9. 9956224. 7

9. 6869212. 1

9. 7884704. 7

29. 4710141. 5

9. 9474378. 1

9.9607107.2 

9. 5628656. 2

29. 4710141. 5

000.0

Spherical 

excess.

-.063

.065

' -.050

-.053

-.034 

-.049

Log sines of 

plane angles.

9. 9956224

9 6869210

9. 7884703

29. 4710137

9. 9474378

9. 9607107 

9. 5628653

29. 4710137

0000.

(d). .0=4-186.0 - 3.0 00+03.0 (?)-37.9 (£)+37.9 (3)-27.0 (J)+27.0 (§) 

-[-11.0 (J) + 11.0 (§)- 9-4 (§)+ 9.4 (|)-53.7 («) + 53.7 (?)

Equations of condition.

(a). 

(&)- 

(c).

. 0= + 9".619-g+&-?+?-$ 

.0= + 15 -394 ?+?-£+§-£ 

. 0= + 15 .767- S+S-S+g-r

Collecting terras in (d) and dividing thru by 100 so as to avoid dealing with large numbers.

(d). . 0= + 1.86+ .507 (-?) + .030 ? -!489 (g) + .379 (J)-.270 Q) 

+ .176 (S) + .110(4) + .094 (§) -.537 (-?).

Table of correlatives.

Direction.

0-1

0-2

0-3

1-0

1-2

1-3

2-0

2-1

2-3

3-0

3 1

3-2

a.

-1

+ 1

+ 1

-1

-I

+ 1

6.

-1

+1

+ 1

-1

-1

+1

c.

-1

+ 1

+ 1

-I

_ ^

+ 1

d.

+ .507

-.489

+ .170

+ .110

-.270

+ .030

+ .094

. 537

+ .379

Forming the normal equations in the usual manner, we have:

(a)............... 

(6).. .............

(c).... ...........

(d).. .............

0=+ 9.019 

0= + 15. 394

0= + 15. 767

0=- 1.860

+6.000 

+2.000

+2.000

-0.598

+2.000 

+6.000

-2.000

-1.070

+2.000 

-2.000

+6.000

+0.950

-0. 598 

-1.07C

+0.950

+1.054

Solving, wo find the following values:

a= + 1.900 

6 = - 4.386 

c = - 5.208 

d= + 3.059
COMPUTATION OF DISTANCES.

Substituting the values of a, b, c, d, in the table of correlatives.

63

Direction.

?

0

1

4

2

v \

2

a

f

3

a.

-1.900

+1.900

+ 1.900

-1.900

-1.900

+ 1.900

b.

+4.386

-4.386

-4.386

+4.386

+4.386

-4.386

c.

 +5. 208

-5.208

-5.208

+5.208

+5.208

-5.208

d.

-1.496

+0. 538

+0. 336 

-0. 826

+0.092

' +0.288

-1.643

+ 1.159

Correction 

to each 

direction.

+4.037

 0 674

-2.770

2.486

+ 4.722 

"-2.726

-0.822

-4.294'

+5. 496

+3.308

+0. 257

-4.049

3.0.1 

0.1.3 

1.3.0

0.1.2 

1.2.0 

2.0.1

1.2.3 

2.3.1 

3.1.2

2.3.0 

3.0.2 

0. 2. 3

Observed angles.

120 39 14.781 

21 26 17.806 

37 54 37.180

81 52 51.222- 

62 22 38.500 

35 44 45.861

91 28 38.000 

28 05 10.360 

60 26 33. 416

65 59 47. 540 

84 54 28.920 

29 05 59.500

Corrections.

-3.308-2.486 

-4.037+0.257 

+2.726-2.770

-4.037-4.294 

-4.722-0.674 

+0.822-2.486

-4.722-4.049 

-5.496-2.726 

-0.257-4.294

-5.496-2.770 

-3.308-0.822 

+0.674-4.049

Corrected spherical angles.

120 39 08.986 

21 26 14. 026 

37 54 37.136

180 00 00.-148

81 52 42.891 

62 22 33.104 

35 44 44.194

180 00 00.189

. 91 28 29. 229 

28 05 02.138 

60 26 28.865

180 00 00.232

65 59 39.274 

84 54- 24.794 

29 05 56.125

181 00 00.193

Sph. excess.

-.049 

-.049 

-.050

-.148

-.063 

-.063 

-.063

-.189

-.078 

-.078 

-.078

-.234

-.064 

-.064 

-.065

- . 193

Plane angles.

0 / //

120 39 08.94 

21 26 13.98 

37 54 37.08

180 00 00.00

81 52 42.83 

62 22 33.04 

35 44 44.13

180 00 00.00 '

91 28 29.15 

28 05 02.06 

60 26 28. 79

180 00 00.00

65 59 39. 21 

84 54 24.73 

29 05 56. 06

180 00 00.00

COMPUTATION OF DISTANCES.

In each triangle, starting with the base line, there is known at least 

one side and the three angles. The remaining sides .are computed 

by the well-known proportion of sides to sines of opposite angles,

or, exprest mathematically, a = -. -. In this computation distances should be used in meters, and seven-place logarithms should be 

employed.

The following is an example of the correction of the angles and 

the- computation of the sides of triangles, taken from the work in 

Kansas:
64 MANUAL OF TOPOGRAPHIC METHODS.

Correction of angles and computation of sides of triangles.

Station.

Angles locally 

adjusted and 

reduced to 

center.

30 29 04.0

63 58 56.2

79 31 58.1 '

179 59 58.3 

Error=-1.7

^error.

+ .5

+ .6

+ .6

Plane angles.

36 29 04.5

63 58 56.8

79 31 58.7

Log sines.

0. 2257704

9.9535952

9.9927124

Log dist. Newt-Walton......................................... ........... ; .3.5771611

Log sin Newt.......................................................................... 9.9535952

a. c. log sin Township comer.......................................................... 0.2257704

Log dist. Township corner-Walton.................'............................:...... 3.7565267

Log dist. Newt-Walton............................................................... 3.5771611

Log sin Walton....................................................................... 9.9927124

a. c. log sin Township corner......................................................... 0.2257704

Log dist. Township corner-Newt...................................................... 3.7956439

COMPUTATION OF GEODETIC COORDINATES.

The next step is the computation of the latitude and longitude of 

the stations and the azimuth or direction of the lines connecting them. 

Initially, the latitude and longitude of some point are determined by 

astronomic observations, and this point is connected with the triangulation. The azimuth, or angle with a south line, of a line connecting 

this point with some station in the triangulation is also determined by 

astronomic observations. These, with the observed angles and the 

computed distances between the stations, form the data from which 

the latitudes ari*d longitudes of the stations and the azimuths of the 

lines connecting them are computed. The difference in latitude 

between two adjoining stations is obtained from the following equation, based upon the Clarke spheroid:

-cZL = K cos a! B + K2 .sin2 a' C+ (£L) 2D-7tK2 sin 2 a' E, 

in which

dL is the difference in latitude;

K, the distance between the stations in meters;

a', the fore .azimuth of the line connecting them, measured around 

clockwise from the south thru the west;

Ji, the first term;

#L, the approximate difference in latitude, being the sum of the first 

two terms;

B, C, D, and E, constants derived from the dimensions and figure of 

the earth (these are given for various latitudes in tables in Appendix 

9 of United States Coast and Geodetic Survey Report for 1894).
\

COMPUTATION OF GEODETIC COORDINATES. 65 

The difference in longitude is obtained by means of the formula,

'

-- ; 

cos L ;

in which

dM. is the difference in. longitude;

U, the newly determined latitude;

A', a constant, from tables; and the others as above.

The azimuths at the ends of a line differ from each other because of 

the convergence of the meridians. That first determined is known as 

the fore azimuth, the other as the back azimuth. All azimuths are 

measured from the south point around clockwise.

The back azimuth is computed from the formula,

sin (L + L')

- da =

cos i dL ' 

in which

M is the longitude of the first station;

L, the latitude, and

L', the latitude of the second station.

The constants used are those of the Clarke spheroid of 1866.

These formulas are derived and explained in Appendix No. 9, 

Report United States Coast and Geodetic Survey for 1894.

The following are examples of the use of the formulas, taken from 

the triangulation in New Mexico:

Examples of compvl.at.ion of geodetic, coordinates.

(1) Azimuth a:

Spherical angle:

Azimuth a':

Azimuth (a):

Nell Chnsca.

Nell Zuni.

Zuni Noll.

159 29 08.728

120 54 13.980

38 34 54.748

179 50 02 124

218 24 50.872
/ 66 MANUAL OF TOPOGBAPHIC METHODS.

GEODETIC COORDINATES.

LONGITUDE.

L: 35 25 13.443 Nell. M: 108 37 24.925 

dL -17 47.540 Geo. Pos. No. 5. d M +17 15.360

L' 35 07 25.927 

Geo. 1 

Computation for 

latitude:

log K ' 4. 6236305 

" B 8.5111933 

" cos a' 9.8930500

log (T) 3.0278738

log R2 9. 24726 

" C 1.25696 

" sin 2 a' 9.58986

log (II) 0.09408

log D 2. 3679 

" [I+*I1]2 6.0568

log (III) 8. 4247

log E 6. 0124 

" K2 Sin2a' 8.8371 

" (I) 3.0279

log (IV) , 7.8774

(I) 1000. 286+ 

(II) -1.242+

(III) .026+ ' [1+11 

(IV) .008- log "

 V I L ' ' *

Zuni. M' 108 54 40.285 

'os. No. 6. 

Computation for longitude:

log K 4. 6236305 

" sin a' 9.7949286 

" A' 8.5092394 

" secL' 0.0872944 

Corr. for cliff, arc & sine= 15

log (V) 3. 0150914 

d M 1035". 300

Computation of azimuth: 

log (V) 3.015091 

" sin (~~T~) 9- 761522

"sec ( dL } 0- 000001

log (VI) 2. 776614 

da - 597". 876 

-9' 57". 876

Azimuth check. _

1067. 528 -

3.0283792 Check: 

2 0. 0507584 Spher. angle 

at 

Computation of Azimuth a, in Book , page . 

Spherical angle and distance=K, in Book , page , Triangle No. .

(2) 

Azimuth a: 

Spherical angle:

Azimuth a': 

(la+18Q°

Azimuth («) :

Chusca Nell. 339 21 40.150 

25 11 38.601

Chiisca-Znni. 4 33 18.751- 

; 179 57 25.650

Zuni Chusca. | 184 30 44.401
TRAVERSE LINES FOR PRIMARY CONTROL. 67

GEODETIC COORDINATES.

LONGITUDE.

L: 35 53 06.746 Chusca. M: 

 d.L - 45 40.818 Geo. Pos. No. 4. dM

L' 35 07 25.92S

Computation for 

latitude:

log K 4. 9280530 

" B S. 5111594 

" cos a' 9.9986260

log (I) 3. 4378393

log K2 9. 85610 

" C 1.26435 

" sin" a' 7.79982

log (II) 8. 92027

log D 2. 3703 

" [I+II? 6.8757

Zuni. M' 

Geo. Pos. No. 6. 

Computati

logK " sin a' 

" A' 

" sec L' 

Corr.fordiff

log (V) 

dM

Computai

log (V)

.. /L+

, . / d

log (III) 9. 2400 ' '\ 2

logE 6.0214 log (VI) 

" K2sin2a' 7.6559 da 

" (I) 3.4378

log (IV) 7. 1151

(I) 2740. 500+ 

(II) . 083+

(III) . 176+ 

(IV) .001-

-dL +2740.818

Azim

[I+II] 2740.643 

log " 3.4378525 Check: 

" [I+I1]2 6.875705 Spher. angle 

at Zuni

108 50 14.518 

+ 4 25.768

108 54 .40.286 

on for longitude:

4. 9280539 

8. 8999280 

8. 5092394 

0. 0872944

2. 4245028 

+205". 768

ion of azimuth: 

2. 424503

J \ n finnnno

2. 188514 

- ' 154". 350 

-2' 34". 350

uth check:

218 24 56.872 

184 30 .44.401

33 54 12.471 

33 54 42.469

Computation of Azimuth a, in Book 67, page 4. 

Spherical angle and distance=K, in Book 64, page 12, Triangle No. 3. 

 Station; Computed by H. M. W.

When the lines are not more than 20 miles in length the equation 

for latitude may be simplified without appreciable error by dropping 

the last two terms.

TRAVERSE LINES FOR PRIMARY CONTROL.

In a level country, especially if covered with forests, it is very 

expensive' to carry on triangulation, and in some cases practically 

impossible to do so. Under such circumstances traverse lines afford 

 the only means of obtaining adequate control for maps.

A traverse line consists of a series of direction and distance measurements. Each course, as the direction and the accompanying distance are called, depends upon the one immediately preceding it, 

and a continuous chain is thus formed. Traverse lines are largely 

used in topographic work proper for making minor locations. The 

primary traverse differs from these only in the fact that it is much 

more elaborately executed.

The initial point of a primary traverse must be located either by 

triangulation or by astronomic determinations. The end of the line 

should, if possible, be a point similarly well located. The line should, 

if practicable, follow a railroad, in order to obtain the easiest possible 

grades, and thus avoid errors incident to slope.
68 MANUAL OF TOPOGRAPHIC METHODS.

The instruments used should be a 20- or 30-inch transit, a 300-foot 

steel tape, a spring balance, two flag poles, a thermometer, and a 

watch.

The party should consist of one chief to act as transitman, one 

recorder, two tapemen, and a flagman. At each station the transitman should proceed as follows:

The telescope should be set on the rear flag, both verniers read, 

the telescope transited, pointed on the front flag, and both verniers 

read. The circle should be shifted and the same angle remeasured 

with the telescope reversed.

Along a railroad the operation of measuring should be conducted 

as follows:

The front tapeman should attach to the front end of the tape a 

20-pound tension with the spring balance. He should make a check 

mark on the rail or drive a tack or nail in the tie. The distance 

which he records should be checked by the transitman and by at 

least one. other member of the party. The rails should be counted 

as a check upon the distance.

Along highways or across open country the tape should be kept 

level. The plumb bob should be used on steep slopes to bring the 

tape vertically over the point. Tape lengths should be marked on 

the measuring board with the marking needle. Where slopes are 

too steep for leveling with the 300-foot tape a shorter tape should be 

used. The temperature of the tape should be taken every hour 

during progress of the work.

Observations for azimuth should be made at the close of each 

day's work, if possible, and azimuth stations should be not more 

than 10 miles apart except on long tangents.

Where the line traversed is very crooked the instrument should be 

fitted for observation with a solar attachment, and this should be 

used at least twice each day, weather permitting, in addition to the 

observations on Polaris/

Careful notes should be taken,0 with a description of the starting 

point of the line, at the beginning and ending of each day's, work. 

The location of each railroad station should be noted, also each milepost, switch, wagon road, stream, land or county line crost, and 

connection made with corners of the public-land surveys. Permanent marks should be placed at each end of the line and also at prominent junction joints, from which other primary-control lines may be 

started.

Lines of traverse exceeding 100 miles in length should be reduced 

by computation. The distances should be corrected for error of 

tape and for temperature and slope, and should be reduced to sea
PRECISE AND PRIMARY ELEVATIONS. - 69

level, in the same manner as described in treating of the reduction 

of base lines, in case these corrections are of sufficient amount to 

affect the length appreciably upon the map.

The courses should be corrected for convergence of meridians. 

Then, commencing at the initial point, the latitude and departure 

of each station, one from another, should be computed in feet. The 

sum of the latitudes converted into seconds of latitude gives the 

olifference in latitude, and the sum of the departures converted into 

seconds of longitude gives the difference in longitude.

Short lines of traverse may be platted with minute-reading protractors, but in this platting the utmost care should be exercised.

PRECISE AND PRIMARY ELEVATIONS.

For the control of elevations level lines of three orders of accuracy 

are employed:

(1) Precise levels, which are run from the seacoast or from previously established precise benches to distant fields of work. These 

are of the highest order of accuracy obtainable.

(2) Primary levels, which are run across quadrangles or around 

their boundaries for the immediate purpose of furnishing primary 

points for heights. Under the law at least two primary bench 

marks should t be established in each land-office township, and an 

equal number in regions where no land surveys have been made.

(3) Flying levels, which are run over traverse lines for the purpose 

of locating contours.

The instruments used in precise and primary levels are in all 

respects similar to those employed by the Coast and Geodetic Survey, 

which are figured and described in the United States Coast and 

Geodetic Survey Reports for 1899 and 1903.

In connection with these descriptions are general instructions for 

the use of these instruments in running precise lines, which differ 

only in minor details from the instructions followed by the United 

States Geological Survey.

In- regions where secondary triangulation Ss practicable heights 

may be measured with the plane table directly from datum points 

established by the. primary leveling and be carried thruout the work 

by means of this instrument.

When the control of the map is effected by means of primary 

traversing, such traverse should be accompanied by a level line.
CHAPTER V.

SECONDARY TRIANGTJJLATION AND TRAVERSE AND

STADIA MEASUREMENTS.

SECONDARY TRIANGULATION.

o

The work.of making secondary locations by intersection is done 

mainly by plane table. The use of the theodolite for this purpose is 

restricted to.those cases where but little of this kind of location can 

be effected, and where, therefore, it seems scarcely worth while to 

prepare plane-table sheets.

By means of the primary triangulation, three or four points are 

usually located upon each plane-table sheet. Within this primary 

triangulation, and depending upon it, are then located a large number 

of points, either by intersection, by traverse, or by both methods, 

forming a geometric framework upon which the sketching of the map 

depends.

Location by intersection should be carried as far as practicable 

that is, all points capable of being located in this manner should be. 

so placed, in order to afford the most ample control possible for the^ 

traverse lines, by which the intervening areas are to be filled in, it 

being understood that location by intersection is more accurate and 

more rapid than location by traverse, and consequently in every way 

more economical.

THE PLANE TABLE.

Much misapprehension exists regarding the character and application of the plane table. This apparently arises from the fact that it 

is little known. It is applicable to all kinds of country, to all methods 

of work, and to all scales. It is a simple, direct, and economical instrument, rendering possible the making of a map directly from the country 

as copy, and doing away with elaborate notes, sketches, photographs, 

etc., which not only are more expensive, but produce inferior results. 

(See PI. VII.)

The plane table consists of a board upon which is fastened a sheet 

of drawing paper. This board is mounted upon a tripod, which, in 

the more elaborate forms of the instrument, possesses great stiffness 

and stability. It should be capable of being leveled, of being turned 

in azimuth, and of being clamped in any position. On the paper is 

produced directly in miniature a representation of the country.

70
U. 8. GEOLOGICAL SURVEY 

jwrt^ Ml VIHV **£ isL^-"* 

TELESCOPIC ALIDADE.
THE PLANE TABLE. 71

When set up at different places within the area in process of being 

"mapped, any edge of the board must always extend in the same direction that is, a certain edge of the board must always be set at the 

same angle with the north and south line. This is called orienting 

the board.

Directions are not read off in.degrees and minutes, but platted 

directly on the paper. The instrument used for this purpose is 

known as the alidade, and consists of a ruler with a beveled edge, to 

which are attached for rough work two raised sights, and for the 

higher class of work a telescope, whose axis is ,fixt parallel to the 

edge of the ruler. This telescope carries also a delicate level and a 

vertical arc for the measurement of angles in the vertical plane, from 

which relative heights are obtained. The method of using this instrument is extremely simple in principle and not difficult in practise. 

(See PI. VI.) -

The making of locations from intersections obtained by means of 

the plane table requires that the instrument have the utmost stability 

consistent with lightness and portability. It demands an alidade 

equipped with a telescope of considerable power and good definition. 

In short, it is necessary that the plane table be in every respect of the 

best modern type in order that the highest degree of accuracy may be 

attained. Various forms of plane-table movement have been in use, 

including the heavy and cumbersome but'stable movement employed 

by the United States Coast and Geodetic Survey, and the light but 

unstable movement used by the same organization in its less important 

work. At present a table is in general use which was invented by 

Mr. W. D. Johnson, of the United States Geological Survey, which 

combines, in a remarkable degree, the elements of stability, lightness, and facility of operation. (See PI. VII and fig. 2.) The movement is essentially an adaptation of the ball-and-socket principle, so 

made as to.furnish the largest practicable amount of bearing surface. 

It consists of two cups, one set inside the other, the inner surface of 

one and the outer surface of the other being ground so as to fit 

exactly. The inner cup is in two parts, or rather consists of two 

rings, one outside the other, one controlling the movement in level 

and the other that in azimuth. From each of these rings a screw 

projects beneath the movement, and upon each of these screws is a 

nut by which it is clamped. There is no tangent screw for either the 

leveling or the azimuth motion, as none is required. The movement 

is sustained by a light hard-wood tripod with split legs. The boards 

used generally accommodate a full-sized atlas sheet, but they necessarily differ in size, owing to the different scales of field work adopted. 

The largest board used for this movement holds an atlas sheet on a 

scale of 1:45000, and is 24 by 36 inches in size.
72 MANUAL OF TOPOGRAPHIC METHODS.

The kind of paper for plane-table sheets is of great importance, 

especially in intersection work, as paper which expands and contracts 

differently in different directions under varying conditions of moisture will easily produce errors of magnitude in the work. It matters 

little if the paper contracts and expands, provided it does so uniformly 

in all directions, but all paper is made with more or less fiber, and 

accordingly expands and contracts more in one direction than in 

another. To counteract this, two thicknesses of paper are used 

JOHNSON PLANE-TABLE HEAD

a.:Plane Table boarcL 

b .Bearing Plate 

c. TY-ipotLHecut

ft. ''

f. UpperLevel Cup

g. liorr&r " 

ft,. IseveL Clamp 

t JlzimuuLh. CLamp

FIG. 2. Johnson plane-table movement (section).

preferably that known as Paragon paper mounted with the grain of 

one sheet at right angles to that of the other, and with cloth between 

the layers. In sheets so prepared it has been found that there is practically no distortion, even under the most severe tests.

The board is generally made of seasoned white pine, from one-half 

to five-eighths of an inch thick, with cleats across the ends, fastened 

in such way as to allow the body of the board to contract and expand 

freely, thereby preventing warping. Into the comers of this board
U. 8. GEOLOGICAL SURVEY 

JOHNSON PLANE TABLE AND TELESCOPIC AL
THE ALIDADE. 16

and on the edges at points halfway between the corners are set female 

screws for holding the paper to the board. At corresponding points 

in the plane-table sheet are punched holes half an inch in diameter, 

thru which pass screws with broad heads fitting into the female screws 

in the board. The holes in the paper, being larger than the screws, 

allow the paper to expand or .contract freely when the screws are 

loose; when tightened, the broad heads of the screws bind the paper 

firmly in place.

THE ALIDADE.

The alidade used witli the plane table (see Pis. VI, VII) consists of a 

ruler of brass or steel 18 inches to 2 feet in length, sometimes graduated upon a chamfered edge to suit the scale of .work, and carrying 

upon a column a telescope having a focal distance of 12 to 15 inches 

and a power'of about 15 diameters. It has a vertical arc of 60 degrees, 

reading by vernier to single minutes, and a delicate level upon the 

telescope, the latter turning in a sleeve for the^ adjustment of vertical 

collimation. The arc is graduated in one direction only, reading from 

0 to 60 degrees,"and the reading is approximately 30 degrees when 

the telescope is horizontal. This is to obviate any possible error in the 

sign of the correction for level.

Upon the plane-table sheet is constructed a projection on the scale 

of the field work, and upon that are platted such of the primary points 

as fall on the sheet, each plane-table sheet being made to correspond 

to an atlas sheet. These primary points are first occupied by the 

plane tabler. ' 

- The instrument is set over one of these stations, leveled, and 

clamped in level. The ruler edge of the alidade is then laid upon the 

line connecting this station with a neighboring one on the sheet, and 

the table turned-in azimuth until the other station is upon the vertical 

wire in the telescope. The instrument is then oriented and, after 

clamping in azimuth, is ready for work. Keeping the ruler upon the 

occupied station on the sheet, the telescope is then turned upon other 

objects which it is desirable to locate, and lines are drawn in turn 

toward them. The instrument is then taken up and moved to a second station, where it is again set up, leveled, and oriented,-as before. 

A sight is then taken, and a line drawn in the direction of each point 

sighted from the first station. The intersection of each pair of sight 

lines is the true position of the corresponding point on the map. In 

this way station after station is occupied by the plane table, and 

numerous points are located by intersection. If possible, each point 

thus located should be intersected from at least three stations, in 

order to verify its location.

Any point located on the map may be used afterwards as a station. 

For the location of points which have not been fixt many different

Bull. 307 07 6
74 MANUAL OF TOPOGKAPHIC METHODS.

methods have been proposed, all of which are graphic solutions of the 

three-point problem. Probably the simplest and easiest of application is the following, which requires that three located points be visible, and that their distances be approximately known:

Set up, level, and clamp the instrument in level and azimuth, after 

orienting it as nearly as possible. Set the alidade successively on the 

three points and draw lines from their platted positions. Then the 

desired location will be found at a distance from each of these lines, 

measured at right angles to it, proportional to the distance on the 

ground between the point occupied and the known point. Thus the 

point of intersection of these lines at right angles to the lines of sight 

is the desired position. Were the distances from the known points 

to the occupied point accurately known, its position on the paper 

would be obtained at the first trial. Practically, "however, two or 

three approximations are commonly found necessary.

Another method often used is as follows:

Fasten upon the plane-table board, which necessarily has not yet 

been oriented, a piece of tracing linen. Assume a point on this linen 

to represent the station, take sights upon and draw lines to all located 

points within the range of vision, and then, loosening the linen from 

the board, move it about over the map until these sight lines fall upon 

the proper points on the map. Then prick thru the position of the 

station from the linen to the map underneath. This location should 

then be tested by sighting from the point thus found to the various 

objects, to see if the sight lines fall upon the points as marked on the 

map. This method has the advantage of permitting the use of all 

located points that may be in sight, but it is not regarded as so accurate as the method first described.

In case one line of sight upon the required station has been obtained, 

that sight line maybe utilized in making the location as follows, by 

resection: Having leveled the table, place the alidade upon this sight 

line already drawn, with the telescope pointing toward the object from 

which, the sight was taken. Then turn the table .in azimuth until the 

telescope falls upon this point, and clamp it. .The table is now 

oriented, but the position of the present station is unknown further 

than that it is on this line. Then select some station whose direction 

makes a wide angle with this line, and move the alidade until the cross 

wire falls upon this selected station, while the ruler at the same time 

is upon the representation of the station on the map. The ruler will 

then cross the sight line at the point desired, ^y way of check, 

repeat the process with another station or located point. For this 

purpose a point in a suitable direction is valuable in proportion to its 

proximity.

Using'the instrument as described above, the topographer locates 

all possible points. Then visiting in turn such of them as he finds
'MEASUREMENT OF ALTITUDES. * 75

necessary, perhaps a dozen or twenty, he locates by intersection points 

all over the sheet in as great number and as well distributed as possible, and with, special reference to the needs of the traverse men, who 

will come after him and whose work will be located by means of his 

determinations. All this work must be done with the utmost nicety 

and precision. The setting of the alidade upon the station must 

bisect the needle hole by which it is marked and the lines of direction 

must be drawn with a sharp-pointed pencil.

The necessity for precision will be recognized when it is understood 

that any error introduced in the early part of the plane-table triangulation will be not only perpetuated but increased many times over as 

the work progresses, and as soon as an error becomes appreciable it 

produces difficulties and uncertainties in making locations, which may 

lead to embarrassing delays and ultimately require that all the work 

be repeated. .

MEASUREMENT OF ALTITUDES.

While making horizontal locations of points with the plane table, 

their heights also, relative to that of the point occupied, must be 

measured. This is done by means of the vertical arc of the alidade 

and the level on the telescope. Pointing upon the object whose relative height is to be measured, the telescope must first be brought to a 

horizontal position and the index error read. The telescope is then 

raised or deprest to the point and the reading obtained. This 

reading of the index error must be done for each point, as the table 

can not be leveled with sufficient accuracy and can not be expected to 

maintain its level so as to dispense with it. Knowing the horizontal 

distance to the point and the angle of elevation and depression, the ' ( ' 

difference in height is obtained by the solution of a right-angled tri- \ *\ 

angle, thus: j

Ti = d tan a,

li being the difference in height, d the distance, and a the angle of 

elevation or depression. This distance is then to be corrected for 

curvature of the earth and for refraction by the atmosphere. The 

correction for the former is obtained with sufficient accuracy by 

the following empirical rule: The curvature in feet equals two-thirds 

the square of the distance in miles. It is always positive in sign, 

whatever may be the sign of the difference in height.

Refraction is an uncertain and variable quantity. It is usually 

greatest at morning and night and least at midday. It is greater 

the nearer the line of sight is to the ground. Often in desert regions 

it is excessive. It is usually assumed as one-seventh the curvature 

and is negative.
76 MANUAL OF TOPOGRAPHIC METHODS.

Tables for the solution of vertical-angle work are used. These 

give differences in height for all angles and distances which should 

be employed, with corrections for curvature and refraction.

Differences of height should not be measured at greater distances 

than 10 miles if it can be avoided. An error of 1' in the measurement of the angle is at this distance about 15 feet, while the uncertainty of refraction in such a length of line is necessarily great.

TRAVERSE WORK.

As stated under the head of traverses for primary control (p. 67), a 

traverse line consists of a series of direction and distance measurements depending upon one another. These lines should be connected 

wherever possible with triangulation points in order to check up 

accumulated errors. If it were practicable or economical to carry 

on all the work of location-by intersection, this would be the most 

accurate and on most accounts the best way to effect it, but it is 

only in limited localities, such as high-mountain regions where bold 

topographic forms predominate and where there is little or no culture, 

that the method of intersection is practicable for locating all necessary 

points. It is probable that in nine-tenths of the area of the United 

States it will be found necessary to locate the details of topography, 

culture, and drainage by means of traverse lines. In different parts 

of the country the relative extent to which the two methods can be 

applied depends upon various circumstances, principally the amount 

of relief of the surface and the prevalence of forests. Thus on the 

Atlantic plain,which is denselycoveredwith forest and is practically 

level, it is necessary to use the traverse method exclusively, including 

even the primary control. Passing from this as an extreme case, 

thru rolling and hilly country, to the high, sharp mountains of the 

West, the triangulation method becomes more and more applicable, 

while the traverse method finally becomes used but little except in 

the details of roads and other cultural features.

For executing traverse work various instruments have been in use 

for measuring directions and distances. For direction there have 

been used theodolites of various forms and prismatic compasses, and 

for distances the stadia and the wheel.

All secondary traverse work should be done with plane tables upon 

which the directions and distances are platted directly. The Johnson 

plane table, with telescopic alidade, may be used for this work. 

When so used, a compass should be set on the edge of the board or of 

the alidade, for orienting, as orientation by backsighting is not 

sufficiently accurate.
U. S. GEOLOGICAL SURVEY BULLETIN NO. 307 PL. VIII

TRAVERSE PLANE TABLE AND RULER ALIDADE.
TBAVEKSE WOEK. 77 

TRAVERSE PLANE TABLE.

Another plane table convenient for this purpose is of the simplest, 

possible form, consisting of a board about 15 inches square, into one 

edge of which is set a narrow box containing a compass needle 3 inches 

in length. (See PL VIII.) This table is supported by a tripod of 

light construction, without leveling apparatus, the leveling of the 

instrument being effected with sufficient accuracy by the tripod legs. 

A single screw fastens the board to the tripod head, and the adjustment in azimuth is made by simply turning the board with the hand. 

It is held in place by friction. The table is adjusted in azimuth, or 

oriented, by means of the compass needle; that is, it is turned until 

the needle rests opposite the zero marks in the compass box, and is 

thus always, made approximately parallel to itself, provided the 

magnetic declination remains constant.

The alidade consists of a brass ruler, 6 to 12 inches long, with 

folding sights. The edge of- the ruler is graduated to facilitate 

platting of distances. Ordinary drawing paper backed with cloth is 

used for plane-table sheets, and is attached to the board by thumb 

tacks.

Distances may be measured by stadia, by tape, or by counting the 

revolutions of a carriage wheel. When traversing is done along 

roads, as is commonly the case, the distance is frequently measured 

by counting the revolutions of a wheel, usually one of the front wheels 

of a buggy or buckboard. For counting the revolutions, various 

automatic devices have been in use.

A traverseman is generally assigned a tract of country within which 

he is instructed to run traverses of all public roads and of such private 

roads as appear to be necessary in order to control the entire tract. 

If practicable, he is furnished with the positions of the points located 

within his tract, properly platted on his plane-table sheet, or, if these 

can not be furnished, with such descriptions of them as are necessary 

to enable him to recognize them and close his lines upon them or connect with them by triangulation. He is furnished with a horse and 

buggy or buckbo'ard, traverse plane table, and aneroid. He has no 

rodman, but is expected to sight natural objects. Setting up his 

instrument at his initial station, he levels it roughly by means of the 

tripod legs, orients it by turning the table until the compass needle 

is on the zero marks in the compass box, and then, marking a point 

on the paper to represent his initial station and placing his alidade 

upon it, he points it to an object selected.as his second station and 

 draws a line in that direction. Driving along the road, he passes the 

point sighted at, noting the distance to it by a count of the revolutions of the wheel, and the height as recorded by the aneroid, and 

passes on, selecting some point from which he can see the. point
78 MANUAL OF TOPOGRAPHIC METHODS.

sighted at. There he stops, sets up his table as before, orients it, 

and sights upon the same signal which he sighted from his initial 

station. He plots the distance to the signal along the sight line 

from his initial station; then from the location of the signal as thus 

established he plots his second staiion by the distance measurement 

and the reverse-of the observed direction. In this way the work 

progresses, a hundred stations or more being occupied in the course of 

the day. . In this work one should aim to make as few stations and to 

take as long sights as possible consistent with accuracy. Bends of 

the road between stations can be sketched with all needful accuracy.

During the progress of the work all points off, the line which are 

capable of being located by intersection must be located by sights 

taken from stations, and special care must be taken to connect 

them with the points located by the secondary triangulation, in order 

to afford as many checks as possible to the accuracy of the traverse 

line.

Traverse lines should close with but trifling error 1 per cent being 

as great an error as should be permitted and all errors of closure 

should be shown. No line should be arbitrarily closed on the traverse 

sheet.

The traverseman should sketch or locate all country houses,"and 

should note all road intersections and all railroad crossings, specifying 

by simple conventions whether over,, under, or grade 'crossing. He 

should similarly describe all stream crossings, distinguishing fords, 

ferries, and bridges.

MEASUREMENTS OF ALTITUDES IN CONNECTION WITH TRAVERSE WORK.

VERTICAL ANGLES.

Height measurements in connection with traverse lines are effected 

in one of two ways either by vertical angles with the telescopic 

alidade or by the use of the aneroid.

In regions where little or no secondary triangulation can be done, 

it becomes necessary to accompany certain of the traverse lines by 

profiles determined by vertical angles. Such profiles should be surveyed at intervals of 4 or 5 miles where the contour interval is 20 

feet, and at intervals of 8 or 10 miles where it is 50 feet.

The plan of the traverse is run precisely as above sketched, except 

that often a rodman is employed. In running the profile, which is 

done coincidently with the plan, the points sighted for elevation may 

be the same as those used for the plan. If a rodman is employed, 

the target on the rod should be set at the height of the instrument, 

to simplify record and computation.

It must not be understood, however, that it is at all necessary 

that the survey of the profile should establish the height of all the 

points located by the traverse. The profile should give the eleva-
THE AWEBOID. 79

tions of all valleys and summits and of all road crossings. The line 

should be carried forward and these points measured by as few and 

as long lines of sight as possible. Often the roof of a house will 

furnish a datum point for use for a mile or two. Indeed, in an open, 

settled country the line can frequently be carried forward continuously by using housetops as targets.

The reduction of the profile must keep pace with the field work, 

so that on arriving at a check point the amount of the error may be 

shown at once. If this is not more than one-fourth or one-fifth of 

the contour interval, it is not considered as of material account. If, 

however, it reaches half a contour interval, the work should be examined, and if the error be not discovered the line should be resurveyed.

The heights as determined should be written in ink on the planetable sheet in their proper places.

THE ANEROID.

In some traverse work heights are measured with aneroids. The 

aneroid consists of a vacuum box of thin corrugated metal, which is 

comprest by an increase and expanded by a decrease in the pressure

FIG. 3. Aneroid. FIG. 4. Works of the aneroid.

of the atmosphere. A train of mechanism magnifies this trifling 

movement enormously and moves an index upon a graduated dial. 

This dial is graduated to feet of elevation and also to inches of barometric pressure. (See figs. 3 and 4.) Several sizes of aneroids are 

made; that having a diameter of 2| inches is on the whole found the 

most satisfactory.
80 MANUAL OF TOPOGRAPHIC METHODS.

Owing mainly to it's extreme delicacy the aneroid is a very uncertain instrument. It should be used differentially only, and for small 

differences in height and small intervals of time. Its indications 

should be checked by reference to known elevations whenever opportunity is afforded during the day, and at the beginning and end of 

each day's work.

On commencing work the movable scale on the aneroid should be 

set at the known height of the starting point and a note made of its 

reading on the inch scale. Elevations should then be read directly 

from the scale of feet. The heights of all points along the line of 

traverse that will be required in making the contour sketch should 

be read and written upon the traverse. Every depression and elevation, road crossing, etc., should thus be measured. There is, however, 

no necessity for reading the aneroid at every station in the traverse; 

that would merely encumber.the work with a mass of useless data.

Upon reaching a check point, comparison should be made with the 

indications of the aneroid. If the difference is considerable more 

than a contour interval the error should be distributed backward 

along the line in proportion to distance. If it is small, it may be 

neglected.

In all this work notebooks are not required except as a convenient 

form of carrying paper on which to. make the trifling computations 

required. The plane-table sheets comprize all the records necessary. 

The work, as it progresses, criticizes itself by its closures in position 

and elevation, and wherever necessary is revised immediately.

ORGANIZATION OF PARTIES AND DISTRIBUTION OF WORK.

Secondary triangulation, traversing, .measuring of heights, and 

sketching are commonly carried on by one party, consisting of the 

chief, who directs all operations and does most of the sketching; an 

assistant who carries on the secondary triangulation, selected as possessing special fitness for that work; and one, two, or more assistants 

who are engaged in traversing, the' number depending upon the 

rapidity with which the country can be sketched relative to the rate 

at which the traversing progresses. If possible, the different items 

of work of such a party should follow one another in a certain order. 

The secondary triangulation should be done first, in order that the 

traversemen may be furnished with positions and heights for locating 

and checking their traverse lines. The traversing should follow, in 

order that all the control may be furnished to the chief of party for 

his use in sketching. This order requires that the members of the 

party be scattered over a considerable area, and if they are living in 

camp it necessitates their remaining away from it a considerable part 

of the time, or else that a large amount of traveling be done in order
STADIA MEASUREMENTS. 81

to reach camp at night. Where they are not living in camp, the most 

economical disposition is to scatter them at various places within 

their fields of work. In any case constant communication must be 

had between the chief of party and his assistants, in order that they 

may work in accord.

STADIA MEASUREMENTS.

Under certain circumstances it is found advisable to use the stadia 

method for measuring distances in place of the wheel. This is the 

case where lines are to be run without reference to roads, and consequently where the wheel can not be employed with advantage. It 

has been used, too, in southern Louisiana, where peculiar methods 

of work imposed by the nature of the topography have made its 

employment economical. Any instrument carrying a telescope may 

be used for the stadia or telemeter method of measuring distances. 

To the reticule of the telescope are added two or more fixt horizontal wires placed at certain distances apart. A rod or board subdivided to suit the interval between the wires and painted in glaring 

colors forms part of the outfit. When this rod is set up at a distance 

from the telescope, that distance is ascertained from the number of 

subdivisions of the rod which are included between the wires of the 

telescope, the value of each division of the rod being known. All 

alidades are equipped with stadia wires. These wires are three in 

number, the intervals between them being equal. The rods are 14 

feet in length and hinged so as to close to 7 feet. The intervals on 

the rods are 1 foot each. For convenience, the wires in the telescope 

should be so spaced that when the rod is at a distance of 100 feet the 

space between the two extreme wires will subtend 1 foot on the rod. 

At a distance of 1,400 feet, therefore, this space will subtend the 

entire length of the rod, while at a distance of 2,800 feet two adjacent wires in the telescope will subtend the entire length of the rod. 

Distances less than 100 feet are estimated by means of the fractional 

part of a foot on the rod which is included between the wires. The 

distances are read off on the rod by the surveyor at the instrument.

In measuring distance on slopes correction must be made to reduce 

the inclination measured to horizontal distance. Where the slope is 

slight, it is not regarded as necessary to make this reduction, especially where there are frequent points for checking and correcting 

the line.

The rod may be used also for measurement of the profile of the 

traverse line. For this purpose a point should be marked upon it at 

the same height as the telescope of the instrument and vertical angles 

taken to this point.
82 ' 'MANUAL OF TOPOGKAPHIC METHODS.

The foregoing matter applies most directly to maps on scales ranging from 1 to 8 miles to an inch and covering large areas of country. 

It applies also, tho less fully and directly, to maps on larger scales 

and those made for specific purposes. Among the latter class may 

be mentioned maps of mining districts, of watersheds, for drainage 

purposes or water supply, of railroad and canal lines, of parks, town 

sites, etc. In making such maps, whose scales are exprest in hundreds of feet to an inch instead of miles to an inch, we deal with inches 

instead of feet in both horizontal and vertical measurements, and 

consequently greater accuracy must be obtained as exprest in 

units of the ground surveyed, but in units of the map the standard 

of accuracy is unchanged. An error of one-fiftieth of an inch on the 

map is as large as should be permitted. The enlarged scale and 

reduced area under survey permits the use of certain instruments 

which are not economical for smaller scales, and also permits more 

extended use of certain others. The use of the stadia instrument in 

connection with the plane table here becomes economical, and it 

should be ifsed extensively. After executing a primary triangulation the work should be relegated to the plane table, with' which 

such secondary triangulation may be effected as seems expedient. 

Then the stadia comes into play, and, using one, two, or more- rodmen, as may be economical, the topographer will locate points by 

direction and distance measurements from his station, reaching out 

perhaps one-fourth of a mile in all directions and measuring heights 

by vertical angles on the rod at the same time that he locates the 

horizontal positions of the points. In this way he will locate points 

which control the contour or points on the contours directly, as seems 

expedient. If the country is nearly level, it may be best to do the 

latter, but if there is much relief he will find it more expedient to 

locate only controlling points and to sketch the contours with reference to them.

From one station he will pass to another, and so cover the country 

mainly by stadia 'locations.

This is by far the most accurate, thoro, and rapid method of making detailed maps of small areas.

In making ajocation map of a railroad or a canal line, the method 

is necessarily varied to some extent. The line itself is measured 

with accuracy with a tape, and should be platted on the plane table 

as measured, and will thus serve as a continuous base line. The 

topography of either side should be taken by stadia, two rodmen 

being employed, one on either side of the line.
CHAPTER VI.

SKETCHING, OFFICE WORK, AND PROJECTIONS.

SKETCHING.

Since sketching is by far the most important part of the work of 

map making, it should be done by the most competent man for this 

work in the party as a rule, by its chief. Besides the fact that he 

is presumably the best sketcher in the.party, there is another reason 

for requiring that he should execute the sketching he is held responsible for the work, not only for the quality of the sketching but 

also for the accuracy and sufficiency of the control, and in the sketching of the map he has the best possible opportunity for examining 

into the .condition of the control and of remedying any weaknesses.

Upon completing the. secondary triangulation, the traverse work, 

and the measurement of heights in short, the control within an 

area, which may be large or small according to convenience, but 

preferably should comprize a quarter sheet, he should cause all this 

control to be assembled on one sheet. The traverse lines with all 

points located from them should be adjusted to the secondary locations, and all measurements of height should be platted upon this 

skeleton, thus presenting in complete form all the control within the 

area. With this sheet upon a sketching board the chief of party 

should go over the ground, sketching the drainage, culture, and 

forms of relief, and generalizing the features to suit the scale of his 

map. The relief should be sketched .in actual continuous contours 

directly from the country as copy, so that upon leaving the sketching stations the only work remaining to complete the map will be 

inking and lettering. »' In heavy country, however, where the contours follow one another closely, it will often be sufficient to put in 

at the stations only a part of the contours every fifth one, for instance in order to economize time in the field. Stations for sketching may be selected with the utmost freedom. An exact location 

is unnecessary. Any point on or off the road which affords an out-. 

look will serve. As a rule, frequent stations should be made, and 

one should not attempt to sketch at great distance unless the conditions are favorable, as in a country of large, bold features. It 

may be necessary to travel over all the roads that have been traversed and to climb many hills in order to sketch the entire area satisfactorily. On the other hand, in a different region the entire area

83
84 MANUAL OF TOPOGRAPHIC METHODS.

may be sketched by a limited amount of travel or from a few elevated 

points. In a low country of small features much travel will l:e 

required, as these details must be sketched from near points. . In 

a bold country of high relief, which may be sketched entirely from 

a few points, care must be exercised in the selection of sketching 

stations. From a great altitude the lower details will be dwarfed 

and will measurably disappear, while from low points the relations, 

forms, and masses of the greater elevations can not be properly 

seen. In such a country stations at different elevations must be 

selected in order to see all parts of the country to the best advantage. The extreme" summits will prove of little service as sketching 

stations.

Sketching is artistic work. The power of seeing topographic 

forms in their proper shapes and proportions and of faithfully transferring these impressions to paper is one of the most difficult acquirements to obtain. The difficulty is increased by the necessity of 

expressing form by means of continuous contour lines at fixt intervals. This work involves knowledge of the elements of structural 

geology and good judgment in applying them.

Whatever its scale may be, every map is a representation reduced 

from nature, and consequently there is more or less generalization. 

It is therefore impossible to make any map an accurate, faithful 

picture of the country it represents. Moreover, the smaller the 

scale the higher must be the degree of generalization, and the further must the map necessarily depart from the original.

Now, it is in this matter of generalization that the judgment of 

the topographer is most severely tested. He must be able to take 

a broad as well as a detailed view of the country, he must understand 

the meaning of its broad features, and then must be able to interpret 

details in the light of those features. Only such a man is competent 

to make just generalizations and to decide what details should be 

omitted and what preserved, and, where details are omitted, what 

to put in their places in order to bring out the dominant features.

It is not possible to define the degree of detail which maps should 

represent. The limit commonly given that imposed by the scale of 

the map is not always the best. In representing country which 

has little plan or system, such as moraines or sand dunes, it is well 

to work in as much detail as the scUle will bear; but where the country shows system in its structure, to which the minor detail is subordinate, the omission of some of this detail will give greater prominence to the larger features. The amount of detail thus omitted 

must necessarily be left to the judgment of the topographer, but no 

more should be omitted than is necessary to give full expression to 

the general features of the county.
SKETCHING, OFFICE WORK, AND PROJECTIONS. 85 

SCALE OF FIELD WORK.

The scale 011 which the field work is executed is generally larger 

than that on which the maps are to be published, because the drawing in the field is necessarily coarser than is possible in the office. 

The field scale is commonly greater than the publication scale; that is, 

plane-table sheets which are to be published on the scale of 1:62500 

are made in the field on the scale of 1:48000 and those to be published 

on the scale of 1:125000 are made in the field on the scale of 1:96000.

OFFICE WORK.

The office work of the topographers consists in the reduction and 

transfer of the work from plane-table sheets to the original maps. 

The reduction is usually effected.by photography, this method having been found the most accurate and economical. It is often the 

practise simply to complete the plane-table sheet by inking and lettering it, making it serve as the original sheet. On many accounts 

this practise is advisable.

The original sheets are to serve as the original record of the work 

and as manuscript for publication. To answer these purposes they 

are made complete in all respects. Every original sheet should 

contain within itself all matter which is to be engraved or placed on 

record.

The method by which the map is to be published should have much 

influence upon the preparation of the original map. In case it is to 

be photolithographed or photo-engraved, it must be drawn by a skilled 

draftsman on a scale much larger than that of publication. If it is 

to be engraved, however, it is unnecessary to waste the time of a 

skilled draftsman upon it, as the engraver requires merely clearness 

and legibility.

PROJECTIONS.

The projection almost universally used is the polycoiiic.

The construction of a projection for a limited area is a simple 

matter, but requires care and accuracy and the use of the best drafting 

instruments.

Tables for this projection are published by the United States 

Coast and Geodetic Survey, which are applicable to all scales of map, 

the measurements given being the absolute lengths on the earth's 

surface, which must be reduced to the scale of the map. Tables are 

published in Bulletin No. 234 of the United States Geological Survey, 

which are directly applicable to certain scales without reduction.

First draw a line down the middle of the sheet. Lay off on this line 

the length of the several projection spaces in latitude by taking from the 

projection table for the scale the length of, say, 5 minutes of latitude
86 MANUAL OF TOPOGRAPHIC METHODS.

and lay it off repeatedly, thus establishing the points of intersection of 

parallels at each 5 minutes with the middle meridian. Thru these 

points draw lines across the sheet at' right angles to the middle meridian, using beam compasses for this purpose. Lay off on these lines the 

elm's corresponding to the latitude for 2' 30" and 7' 30" from the 

middle meridian on each side, and at these points erect short perpendiculars. On these lay off the dp's corresponding to the din's, and 

thru the points thus obtained draw and ink the projection lines.

For other areas the process is similar, but when a large area, such 

as that of the United States, is to be projected the mechanical difficulties greatly increase. '
INDEX.

Alidade, ruler, description of............... 77

plate showing.......................... « 76

Alidade, telescopic, description of.......... 73

plates showing......................... 70,72

use of................................... 73-75

Altitudes, measurement of........... 75-76,78-79

Aneroids, description of.................... 79-80

figures showing......................... 79

Angles, horizontal, adjustment of.......... 58-63

measurement of......................... 49-55

records of........................... 55,58

Angles, vertical, measurement of........... 78-79

Astronomic transit and zenith telescope,

checking of................?..... 19-23

description of........................... 13-14

plate showing.......................... 12

Astronomy, determination of position by.. 12-37 

Atlas sheets, size of........................ 6

Azimuth, deviation of, correction of......... 27

observations for........................ 33-35

example of......................... 34-35

reduction of........................ 35-37

example of...................... 36-37

Baldwin, IT. L., jr., tape-stretching device

for, plate showing............. 40

Bars, use of, for measurement.............. 39-40

Base line, character of...................... 38-39

measurement of......................... 38-42

instruments for..................... 39-40

methods of.......................... 39-41

personnel required for.............. 41-42

reduction of......................... 42-44

example of...................... .43-44

tape-stretching device for, plate

showing......................... 40

Chronograph. description of................ 14-15

figure showing:......................... 15

Chronometers, comparison of............... 32-33

error of, determination of.............. 25

Collimation, error of, correction of......... 26

Contour intervals, difference in, reason for . 3 

explanation of.......................... 3

size of, used by United States Geological

Survey.......................... 5

Contours, accuracy of...................... 10

explanation of.......................... 2-3

location of............................ 9-10,11

use of................................... 2-3

Cultural features, delineation of............ 3-4/5

Declination, computation of................ 23-24

example of.......................... ' 24

definition of............................. 13

Bull. 307 07 7

Page. 

Distances, computation of.................. 63-64

computation of, example of............. 64

measurement of......................... 77

Diurnal aberration, correction for.......... 27

correction for, example of.............. 27-31

Elevations, ascertainment of............... 69

Field work, methods of, description of..... 16

scale of.................................. 85

Figure, adjustment of...................... 60-63

adjustment of, example of.............. 61-63

Geodetic coordinates, computation of...... 64-67

computation of, examples of............ 65-67

Heliotropes, description of................. 47-48

Inclination, correction of base line for...... 42

Instruments, surveying, checking of........ 19

description of.................. 11,13-15,47-48

use of................................... 16

Johnson, W. D., plane table of............. 71

plane table of, figure showing........... 72

Latitude, calculation of..................... 64

definition of.'............................ 13

observations for........................ 17-18

example of.......................... 18

reduction of......................... 18-24

Level, division of, measurement of......... 22-23

example of.......................... 23

error of, correction of................... 26

Levels, running of.......................... 69

Locations, accuracy of ...................... 7-8

distribution of.......................... 9-10

methods of making of................ 8,10-11

number of............................... 8-9

sketching of....................... '...... 10,11

Longitude, computation of................. 65

definition of............................ 13

Map of United States, cost of .............. 5

plan of.................................. 4-6

sheets of, size of........................ 6

Maps, classes of............................. 1-2

scale of.................................. 2

Maps, mother, explanation of.............. 2

Maps, topographic, correctness of........... 7-10

making of, methods of.................. 7-10

classification of..................... 10-11

objects shown on..................... 2,3-4,5

scale of................................. 2

. surveying methods for................. 1,7-10

See also Relief; Contour; Scale; Sketching. 

Micrometer screw, division of, measurement

of............................... 19-22

division of, measurement of, example of 20-21

87
INDEX.

Page. 

Mother maps, explanation of............... 2

Office work, character of.................... 85

Organization, party, character of.... 38,56,80-81

Party organization, character of..... 38,56,80-81

Plane table, description of............... 70-73,77

paper for................................ 72

plates showing.......................... 72,76

section of, figure showing............... 72

use of................................ 70,77-78

Primary control, methods of obtaining..... 38

See also Triangulation; Primary traverse. 

Primary traverse, running of ......... 9,67-69,76

See also Traverse lines. 

Projections, making of, method of.......... 85-86

Relief, delineation of, methods of .......... 2-3

Right ascension, definition of............... 13

Scales, explanation of....................... 3

size of, used by United States Geological

Survey.......................... 4-5

Sea level, reduction of base line to.......... 42

Signals, character of........................ 46-?7

plates showing.......................... 46

Sketching, method of....................... 83-84

quality of............................... 10,83

Spherical excess, reduction for.............. 58

Stadia, measurement by.................... 81-82

Stations, angles observed.from, adjustment

of............................... 58-60

reduction of, to center.................. 57

selection of.............................. 45-46

Surveying, instruments for. 1............ 11,13-15

instruments for, use of................. 16

objects of............................... 1

Tapes, correction of........................ 42

stretching of, in base measurement,

plate showing................... 40

use of................................... 40-41

Time, comparison of........................ 32-33

Page. 

Time, definitions of......................... 12-13

observations for........................ 25

example of.......................... 28 -29

reduction of......................... 25-33

example of...................... 30-31

Temperature, correction for................. 42

Theodolites, character of................... 48-49

plate showing........................... 48

Topographic maps. See Maps, topographic. 

Tower, observing, -plate showing........... 46

use of.................................... 46

Traverse lines, checking of.................. 12,70

explanation of.......................... 9,67

instructions for......................... 68-69

running of......................... 9,67-69,76

Triangulation, accuracy of................. 12

accuracy of, checking of.......... ..... 12

required degree of................... 7-8

points for, selection of.................. 9

use of .................................. 38

Triangulation, primary, expansion of...... 45

instruments used in.................... 47-48

making of............................... 45-56

party organization for ................. 38,56

reduction of............................. 57-67

^signals in............................... 46-47

plates showing...................... 45,46

stations for, selection of ................ 45-46

work of, sequence of.................... 56

See also Angles; Figures; Distances;

Geodetic coordinates. 

Triangulation, secondary, methods of...... 70-76

party organization for.................. 80-81

See also Alidade; Plane table; Altitudes. 

United States, map of, plan of.............. 4-6

Zenith distance, definition of............... 13

Zenith telescope and transit. See Astronomic transit and zenith telescope.