Charles Davies.

Elements of surveying and navigation : with a description of the instruments and the necessary tables online

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LIBRARY



^NSSACfft;^^^




1895



ELEMENTS



OF



SURVEYING,



NAVIGATION;

WITH A DESCRIPTION OF THE INSTRIFMENTS AND
THE NECESSARY TABLES.



BY CHARf.ES DAVIES, LL.D.

AOTUOROF ARITHMETIC, ELEMENTARY ALGEBRA, ELEMENTARY GEOMETRY, PRACTICAL
GEOMETRY, ELEMENTS OF SURVEYING, ELEMENTS OF DESCRIPTIVB
GEOMETRY, SHADES SHADOWS AND PERSPECTIVE, ANA-
LYTICAL GEOMETRY, DIFFERENTIAL AND
INTEGRAL CALCULUS.



REVISED EDITION



NEW YORK:

PUBLISHED BY A. S. BARNES &. CO.
No. 51 JOHN STREET.

1847.



DAVIES'

COURSE OF MATHEMATICS.



DAVIES' FIRST LESSONS IN ARITHMETIC— For Beginners.

DAVIES' ARITHMETIC— Designed for the use of Academies and Schools.

KEY TO DAVIES' ARITHMETIC.

DAVIES' UNIVERSITY ARITHMETIC— Embracing the Science of Num-
bers and their numerous Applications.

KEY TO DAVIES' UNIVERSITY ARITHMETIC.

DAVIES' ELEMENTARY ALGEBRA— Being an introduction to the Sci-
ence, and forming a connecting luik between Arithmetic and Algebra.

KEY TO DAVIES' ELEMENTARY ALGEBRA.

DAVIES' ELEMENTARY GEOMETRY.— This work embraces the ele-
mentaiy principles of Geometiy. The reasoning is plain and concise, but at the
same time strictly rigorous.

DAVIES' ELEMENTS OF DRAWING AND MENSURATION — Ap-
plied to the Mechanic Arts.

DAVIES' BOURDON'S ALGEBRA— Including Sturm's Theorem— Being
an abridgment of the Work of M. Bourdon, with the addition of practical examples.

DAVIES' LEGENDRE'S GEOMETRY and TRIGONOMETRY— Being

an abridgment of the work of M. Legendre, with the addition of a Treatise on Men-
suration OF Planes and Solids, and a Table of Logarithms and Logarithmic
Sines.

DAVIES' SURVEYING— With a description and plates of the Theodolite,
Compass, Plane-Table. and Level; also, Maps of the Topographical Signs
adopted by the Engineer Department — an explanation of the method of sui-veying
the Public Lands, and an Elementary Treatise on Navigation.

DAVIES' ANALYTICAL GEOMETRY — Embracing the Equations of
THE Point and Straight Line — of the Conic Sections — of the Line and Plane
IN Space ; also, the discussion of the General Equation of the second degree, and
of Surfaces of the second order.

DAVIES' DESCRIPTIVE GEOMETRY— With its application to Spher-
ical Projections.

DAVIES' SHADOWS and LINEAR PERSPECTIVE.

DAVIES' DIFFERENTIAL and INTEGRAL CALCULUS.



Entered, according to Act of Congress, in the year 1835, by Charles Davies, in the Clerk's
Office of the District Court of the United States, in and for the Southern District of
New York.



28652






PREFACE



The Elements of Surveying-, published by the author in
1830, was designed especially as a text-book for the Military
Academy, and in its preparation little regard was had to the
supposed wants of other Institutions.

It was not the aim of the author to make it so elementary
as to admit of its introduction into academies and schools, and
he did not, therefore, anticipate for it an extensive circulation.

It has been received, however, with more favor than was
anticipated, and this circumstance has induced the author to
re-write the entire work. In doing so, he has endeavored to
make it both plain and practical.

It has been the intention to begin with the very elements
of the subject, and to combine those elements in the simplest
manner, so as to render the higher branches of plane-survey-
ing comparatively easy.

All the instruments needed for plotting have been carefully
described ; and the uses of those required for the measurement
of angles are fully explained.

The conventional signs adopted by the Topographical Beau-
reau, and which are now used by the United States Engineers
in all their charts and maps, are given in plates 5 and 6.

Should these signs be generally adopted in the country, it
would give entire uniformity to all maps and delineations of
ground, and would establish a kind of language by which
all the peculiarities of soil and surface could be accurately
represented.

An account is also given of the manner of surveying the
public lands; and although the method is simple, it has,
nevertheless, been productive of great results, by defining,
with mathematical precision, the boundaries of lands in the
new States, and thus settling their titles on an indisputable
basis.



MITI i EHPA



4 PREFACE.

The method was originated by Col. Jared Mansfield, whose
great acquirements in science introduced him to the notice
of President Jefferson, by whom he was appointed surveyor-
general of the North-Western Territory.

May it be permitted to one of his pupils, and a graduate of
the Military Academy, further to add, that at the organization
of the institution in 1812, he was appointed Professor of Nat-
ural and Experimental Philosophy. This situation he filled
for sixteen years, when he withdrew from the academy to
spend the evening of his life in retirement and study. His
pupils, who had listened to his instructions with delight, who
honored his learning and wisdom, and had been brought near
to him by his kind and simple manners, have placed his por-
trait in the public library, that the institution might possess
an enduring memorial of one of its brightest ornaments and
distinguished benefactors.

At the solicitation of several distinguished teachers here is
added, in the present edition, an article on Plane Sailing, most
of which has been taken, by permission of the author, from an
excellent work on Trigonometry and its applications by Pro-
^*~^or Charles Hackley.

Hartford,

March, 1841.



CONTENTS.



INTRODUCTION





CHAPTER L




Of Logarithms,
Table of Logarithms,





Page.
9


Multiplication by Logarithms,
Division by Logarithms,
Arithmetical Complement,




14
15
16



CHAPTER H.

Geometrical Definitions, 17

CHAPTER in.

Description of Instruments, 21

Of the Dividers, " . .22

Ruler and Triangle, 22

Scale of Equal Parts, . . . . • 23

Diagonal Scale of Equal Parts, 24

Scale of Chords : : 25

Semicircular Protractor, : . . . . 26

Sectoral Scale of Equal Parts, 27

Gunter's Scale, 28

Solution of Problems, 29

CHAPTER IV.

Plane Trigonometry, .......... 34

Table of Logarithmic Sines, 37

Solution of Right Angled Triangles, .49



ELEMENTS OF SURVEYING



CHAPTER L
Definitions and Introductory Remarks,



51



CHAPTER II.

Of the Measurement and Calculation of Lines and Angles, .... 53

To Measure a Horizontal Line, 54

Of the Theodolite, 55

Heights and Distances, .... ..... 66

Of Measurements with the Tape or Chain, , . . . . 74

Surveying Cross, 76



CONTENTS.



CHAPTER III



Of the Content of Ground,

Of Laying Out and Dividing Land,



Page.
79
89



CHAPTER IV.

Surveying with the Compass, 91

Of the Compass, 92

Field Notes 96

Traverse Table, 98

Of Balancing the Work, 100

Of the Double Meridian Distances of the Courses, 102

Of the Area, 104

First Method of Plotting, 107

Second Method of Plotting, 107

Method of Finding the Content of Land by Means of the Table of Natural

Sines, 120

Method of Surveying the Public Lands, . . • 126

Variation of the Needle, 127

Of the Plain Table, . 133

CHAPTER V.

Of Levelling, 137

Of the Level 140

Of the Level Staves, 143

CHAPTER VI.

Of the Contour of Ground, 148



CHAPTER VI.

Of Surveying Harbours, .159

To fix the Principal Points, . # 159

Manner of Using the Compass, 163

Of the Circular Protractor, ......... 165

First Method of Plotting, 166

Second Method of Plotting, 167

Surveying a Harbour for the Purpose of Determining the Depth of Water, &c., 168



CHAPTER VII.



Of Navigation,
Of Plane Sailing, .
Of Traverse Sailing,
Parallel Sailing,
Middle Latitude Sailing,
Mercator's Sailing, .
Mercator's Chart, .



171
174
176
179
181
184
187



INTRODUCTION.



CHAPTER I.

Of Logarithms.

1. The nature and properties of the logarithms in common
use, will be readily vuiderstood, by considering attentively the
different powers of the nmiiber lo. They are,

10'' = i
10' = 10
10* = 100
10^=1000
10^ = 10000
10^ = 100000

&c. &c.
It is plain, that the indices or exponents 0, l, 2, 3, 4, 5, &c.
form an arithmetical series of which the common difference is
1 ; and that the numbers 1, 10, 100, 1000, 10000, 1 00000, &c.
form a geometrical series of which the common ratio is 10,
The number 1 0, is called the base of the system of logarithms ;
and the indices, 0, 1, 2, 3, 4, 5, &c., are the logarithms of the
numbers which are produced by raising 10 to the powers de-
noted by those indices.

2. Let a denote the base of the system of logarithms, m any
exponent, and M the corresponding number : we shall then
have, a'^=M

in which m is the logarithm of M.

If we take a second exponent n, and let JST denote the cor-
responding number, we shall have,

in which n is the logarithm of JV.

If now, we multiply .the first of these equations by the
second, member by member, we have



8 INTRODUCTION.

but since a is the base of the system, m+n is the log-arithm
Mx»N*; hence,

The sum of the loganthms of any two numbers is equal to the
logarithm of their product.

Therefore, the addition of logarithms corresponds to the mul-
tiplication of their numbers,

3. If we divide the equations by each other, member by
member, we have,

but since a is the base of the system, m—n is the logarithm
of — hence :

jsr

If one number be divided by another, the logarithm of the quo-
dent will be equal to the logarithm of the dividend diminished by
that of the divisor.

Therefore, the subtraction of logarithms corresponds to the di-
vision of their numbers.

4. Let us examine further the equations

10^ = 10
10« = 100

io'=iooo
&c. &c.
It is plain that the logarithm of 1 is 0, and that the loga-
rithms of all the numbers between 1 and 10, are greater than
and less than 1. They are generally expressed by decimal
fractions : thus,

log 2=0.301030.

The logarithms of all numbers greater than 10 and less
than 100, are greater than 1 and less than 2, and are gen-
erally expressed by 1 and a decimal fraction : thus,
log 50 = 1.698970.

The logarithms of numbers greater than 100 and less than
1000, are greater than 2 and less than 3, and are generally
expressed by uniting 2 Avith a decimal fraction ; thus,
log 126=2.100371.

The part of the logarithm which stands on the left of the
decimal point, is called the cluiracterislic of the logarithm.



OF LOGARITHMS. 9

The characteristic is always one less than the places of integer
figures in the number whose logarithm is taken.

Thus, ill the first case, for numbers between i and 10,
there is but one place of figures, and the characteristic is 0.
For numbers between 10 and 100, there are two places of
figures, and the characteristic is 1 ; and similarly for other
numbers.

TABLE OF LOGARITHMS.

5. A table of logarithms, is a table in which are written
the logarithms of all numbers between 1 and some other given
number. The logarithms of all numbers between l and
10,000 are written in the annexed table.

6. The first column on the left of each page of the table,
IS the column of numbers, and is designated by the letter JV;
the logarithms of these numbers are placed directly opposite
them, and on the same horizontal line.

To find, from the table, the logarithm of any number.

7. If the number is less than 100, look on the first page of
the table, along the column of numbers under JV, until the
number is found : the number directly opposite, in the column
designated log, is the logarithm sought. Thus,

log 9=0.954243.

When the number is greater than 100, and less than 10,000.

8. Since the characteristic of every logarithm is less by
unity than the places of integer figures in its corresponding
number (Art. 4), its value is known by a simple inspection
of the number whose logarithm is sought. Hence, it has not
been deemed necessary to write the characteristics in the table.

To obtain the decimal part of the logarithm, find, in the
column of numbers, the first three figures of the given number.
Then, pass across the page, along a horizontal line, into the
columns marked 0, 1, 2, 3, 4, 5, &c., until you come to the
column which is designated by the fourth figure of the given
number: at this place there are four figures of the required
logarithm. To the four figures so found, two figures taken
from the column marked 0, are to be prefixed. If the four
figures thus found, stand opposite to a row of six figures in the
column marked 0, the two figures from this column, which
are to be prefixed, are the first two on the left hand : but if



10 INTRODUCTION.

the four figures found are opposite a line of only four figures,
you are then to ascend the column till you come to tlie line
of six figures ; the two figures, at the left hand, are to be
prefixed, and then the decimal part of the logarithm is ob-
tained ; to which prefix the characteristic, and you have the
entire logarithm sought. For example,

log 1122 = 3.049993

log 8979===3. 953228
In several of the columns, designated 0, 1,2, 3, 4, &c., small
dots are found. When the logarithm falls at such places,
a cipher must be written for each of the dots, and the two
figures, from the column 0, which are to be prefixed, are then
found in the horizontal line directly below.

Thus, .... 'log 2188 = 3.340047
the two dots being changed into two ciphers, and the 34 to
be taken from the column 0, is found in the horizontal line
directly below.

The two figures from the column 0, must also be taken from
the horizontal line below, if any dots shall have been passed
over, in passing along the horizontal line : thus,

log 3098 = 3.491081
the 49 from the column 0, being taken from the line 310.

When the number exceeds 10,000, or is expressed by Jive or
more figures.

9. Consider all the figures, after the fourth from the left
hand, as ciphers. Find from the table the logarithm of the
first four figures, and to it prefix a characteristic less by unity
than all the places of figures in the given number. Take
from the last column on the right of the page, marked D, the
number on the same horizontal line with the logarithm, and
multiply this number by the figures that have been considered
as ciphers : then cut oflf from the right hand as many places
for decimals as there are figures in the multiplier, and add the
product so obtained to the first logarithm, and the sum will be
the logarithm sought.

Let it be required, for example, to find the logarithm of
672887.

log 672800 = 5.827886
the characteristic being 5, since there are six places of figures.
The corresponding number, in the column J9 is 65, which



OF LOGARITHMS. Jl

being multiplied by 87, the figures regarded as ciphers, gives
for a product 5655 ; then pointing off two decimal places, we
obtain 56.55 for the number to be added.

Hence . . log 672800 = 5.827880
Adding .... +56.55

gives . log 672887=5.827943.

In adding the proportional number, we omit the decimal
part ; but when the decimal part exceeds 5 tenths, as in the
case above, its value is nearer unity than ; in which case,
we augment by one, the figure on the left of the decimal
point.

10. This method of findmg the logarithms of numbers
which exceed four places of figures, does not give the exact
logarithm ; for, it supposes that the logarithms are propor-
tional to their corresponding numbers, which is not rigorously
true.

To explain the reason of the above method, let us take the
logarithm of 672900, a number greater than 672800 by 100.
We then have,

log 672900 = 5.827951
log 672800 = 5.827886
Difference of numbers 100 6 5 =difrerence of loga-

rithms.
Then, 100 : 65 :: 87 : 56.55

In this proportion the first term 100 is the difference be-
tween two numbers, one of which is greater and the other
less than the given number; and the second term 65 is the
difference of their logarithms, or tabular difference.

The third term 87 is the difference betAveen the given num-
ber and the less number 672800; and hence the fourth term
56.55 is the difference of their logarithms. This difference
therefore, added to the logarithm of the less nimiber, will give
that of the greater, nearly.

Had there been three figures of the given number treated
as ciphers, the first term would have been 1000 ; had there
been four, it would have been 10000, &c. Therefore, the
reason of the rule, for the use of the column of differences, is
manifest.

To find the logarithm of a decimal number.

11. If the given number is composed of a Avhole number



GMT U^^^'"-^



12 iNtnODUCTIOW.

and a decimal, such as 36.78, it may be put under the form
»_e_7_8. But since a fraction is equal to the quotient obtained
by dividing the numerator by the denominator, its logarithm
will be equal to the logarithm of the numerator minus the
logarithm of the denominator. Therefore,

log 3_6_7_«3=log 3678 — log 100 = 3.565612 — 2 = 1.565612
from which we see, that a mixed number may be treated
as though it were entire, except in fixing the value of the
characteristic, which is always one less than the number of the
integer figures.

12. The logarithm of a decimal fraction is also readily
found. For,

log 0.8=log j\=\og 8 — l = -l+log 8. But,
log 8=0.903090
which is positive and less than 1. Therefore,

log 0.8 = -l+0. 903090 = — 1.903090
in which, however, the minus sign belongs only to the charade^
ristic. Hence it appears, that the logarithm of tenths is the
same as the logarithm of the corresponding whole number,
excepting, that the characteristic instead of being 0, is— 1.

If the fraction were of the form 0.06 it might be written yVo J
taking the logarithms, we have,

log -0/-=log 06— 2 = -2+log 06 = — 2.778151
in which the minus sign, as before, belongs only to the char-
acteristic. If the decimal were 0.006 its logarithm would be
the same as before, excepting the characteristic, which would
be — 3. It is, indeed, evident, that the negative characteristic
will always be one greater than the number of ciphers be-
tween the decimal point and the first significant figure.
Therefore, the logarithm of a decimal fraction is found, by
considering it as a whole number, and then prefixing to the deci-
mal part of its logarithm a negative characteristic greater by
unity than the number of ciphers between the decimal point ana
Ihe first significant figure.

That we may not, for a moment, suppose the negative sign
to belong to the whole logarithm, when in fact it belongs only
to the characteristic, we place the sign above the characte-
ristic, thus,

log 8=1.903090, and log 0.00=2.778151.







OP


LOGARITHMS.










EXAMPLES









log 2756 .






is






. 3.440270


log 3270






is






. 3.514548


log 287.965






is






. 2.459340


log 1.004 .






is






. 0.001734


log 0.002 .






is






. 3.301030


log 0.000678






is






. 4.831230



13



To find in the table, the number answering to a given logarithm.

13. Search in the columns of logarithms for the decimal
part of the given logarithm, and if it can be exactly found,
set down the corresponding number. Then, if the character-
istic of the given logarithm is positive, point off from the left
of the number found, one more place for whole numbers than
there are units in the characteristic of the given logarithm,
and treat the figures to the right as decimals.

If the characteristic of the given logarithm is 0, there will
be one place of whole numbers ; if it is — 1, the number will
be entirely decimal; if it is— 2, there will be one cipher
between the decimal point and the first significant figure ;
if it is — 3, there will be two, &c

The number whose logarithm is 1.492481, is found at page
5, and is 31.08.

But when the decimal part of the logarithm cannot be
exactly found in the table, take the number answering to the
nearest less logarithm; take also from the table the corres-
ponding difference in the column D, Then, subtract this
less logarithm from the given logarithm, and having annexed
any number of ciphers to the remainder, divide it by the dif-
ference taken from the column 2?, and annex the quotient to
the number answering to the less logarithm : this gives the
required number, nearly. This rule, like that for finding
the logarithm of a number when the places of figures ex-
ceed four, supposes the numbers to be proportional to their
corresponding logarithms.

1. Find the number answering to the logarithm 1.532708.
Given logarithm is . . 1.532708

Next less tabular logarithm is 1.532627

Their difference is . . 81



14 INTRODUCTION.

The number corresponding to the tabular logarithm is 34.09
And the tabular difference is . • . . 128 :

and, 128)81.00(63

The 63 being annexed to the tabular number 34.09 gives

34.0963 for the number answering to the logarithm 1.532708.

2. Required the number answering to the logarithm
3.233568.

The given logarithm is . . 3.233568

Next less tabular logarithm of 1712 is 3.233504
Their difference is ... . 64

Tabular difference . 253)64.00(25
Hence the number sought, is 1712.25, marking four places
for integers since the characteristic is 3.

MULTIPLICATION BY LOGARITHMS.

14. When it is required to multiply numbers by means of
their logarithms, we first find from the table the logarithms
of the numbers to be multiplied ; we next add these loga-
rithms together, and their sum is the logarithm of the pro-
duct of the numbers (Art. 2).

The term sum is to be understood in its algebraic sense ;
therefore, if any of the logarithms have negative charac-
teristics, the difference between their sum and that of the
positive characteristics, is to be taken, and the sign of the
greater prefixed.

EXAMPLES.

1. Multiply 23.14 by 5.062.

log 23.14 = 1.364363
log 5.062=0.704322.
Product 117.1347 . . . . 2.068685

2. Multiply 3.902, 597.16 and 0.0314728 together.

log 3.902 = 0.591287

log 597.16=2.776091

log 0.0314728 = 2.497936

Product 73.3354 .... 1.865314

Here the 2 cancels the + 2, and the 1 carried from the
decimal part is set down.



OF LOGARITHMS. 15

3. Multiply 3.586, S.1046, 0.8372, and 0.029 4, together.

log 3.586=0.554610
log 2.1046 = 0.323170
log 0.8372 = 1.922829
log 0.0294 = 2.468347
Product 0.1857615 . . 1.268956.
In this example the 2, carried from the decimal part, can-
cels 2, and there remains T to be set down.

DIVISION OF NUMBERS BY LOGARITHMS.

15. When it is required to divide numbers by means of
their logarithms, we have only to recollect, that the subtrac-
tion of logarithms corresponds to the division of their num-
bers (Art. 3). Hence, if we find the logarithm of the divi-
dend, and from it subtract the logarithm of the divisor, the
remainder will be the logarithm of the quotient.

This additional caution may be added. The difference of
the logarithms, as here used, means the algebraic difference ;
so that, if the logarithm of the divisor have a negative
characteristic its sign must be changed to positive, after
diminishing it by the unit, if any, carried in the subtraction
from the decimal part of the logarithm. Or, if the charac-
teristic of the logarithm of the dividend is negative, it must
be treated as a negative number.

EXAMPLES.

1. To divide 24163 by 4567.

log 24163=4.383151

log 4567 = 3.659631

Quotient 5.29078 . . 0.723520.

2. To divide .0631*4 by .007241

log 0.06314=2.800305
log 0.007241=3.859799
Quotient . . 8.7198 . . 0.940506
Here, 1 carried from the decimal part to the 3 changes it to
2, which being taken from §, leaves for the characteristic.

3. To divide 37.149 by 523.76

log 37.149 = 1.569947

log 523.76=2.719133

Quotient . 0.07092T4 . 2.8508l~4



I(? INTRODUCTION.

4. To divide 0.7438 by 12.9476

log 7438 = 1.871456

log 12.9476 = 1.112189
Quotient 0.057447 . . 2.759267
Here, the l taken from f, gives 2 for a result, as set down.

ARITHMETICAL COMPLEMENT.

16. The Arithmetical complement of a logarithm is the num-
ber which remains after subtracting this logarithm from 10.
Thus . . 10—9.274687 = 0.725313.

Hence, 0.725313 is the arithmetical complement

of 9.274687.

17 We will now show that, the difference between two logo-
rithms is truly found, by adding to the first logarithm tfie arith-
metkal complement of the logarithm to be subtracted^ and then
diminishing the sum by 10.

Let a=the first logarithm

b=the logarithm to be subtracted
and c = io — 6=the arithmetical complement of b,

P^ow the difference between the two logarithms will be
expressed by a—b.

But, from the equation c = lO — b, we have
c~10 = — 6
hence, if we place for— 6 its value, we shall have



Online LibraryCharles DaviesElements of surveying and navigation : with a description of the instruments and the necessary tables → online text (page 1 of 45)