International Correspondence Schools.

The elements of railroad engineering (Volume 2) online

. (page 9 of 35)
Online LibraryInternational Correspondence SchoolsThe elements of railroad engineering (Volume 2) → online text (page 9 of 35)
Font size
QR-code for this ebook


subtractive.

1329. Traverse Tables. The latitude and departure
of any distance for any bearing can be found by a table
of natural sines and cosines, but for facilitating work
special tables, called traverse tables, have been prepared.
They usually give the latitude and departure for any bear-
ing to each quarter of a degree and for distances from
1 to 9.

To use the tables (see traverse tables, or Latitudes and
Departures of Courses), find the number of degrees in the
bearing in the left-hand column if the bearing be less than
45, and in the right-hand column if the bearing be greater
than 45. The numbers on the same line running across-
the page are the latitudes and departures for that bearing
and for the respective distances, 1, 2, 3, 4, 5, 6, 7, 8, 9,
which appear at the top and bottom of the pages, and which
may be taken to represent links, rods, feet, chains, or any
other unit. Thus, if the bearing be 10 and the distance
4, the latitude will be 3.939 and the departure .695; with
the same bearing, and the distance 8, the latitude will be
7.878 and the departure 1.389, or double the latitude and
departure for the distance 4. Any distance, however great,
can have its latitude and departure readily obtained from
this table, since, for the same bearing, the latitude and
departure are directly proportional to the distance because
of the similar triangles which they form. Hence, the lati-
tude and departure for 80 is ten times the latitude and
departure for 8, and is found by moving the decimal point
one place to the right; that for 500 is 100 times the latitude
and departure for 5, and is found by moving the decimal



LAND SURVEYING. 719

point two places to the right, and so on. By moving the decimal

point one, two, or more places to the right the latitude and

departure may be found for any multiple of any number

given in the table. In finding the latitude and departure

for any number such as 453, the number is resolved into

three numbers, viz. : 400 and the latitude and departure

5 for each taken from the table

3 and then added together.



We thus obtain the following

Rule. Write down the latitude and departure, neglecting
the decimal points, for the first figure of the given distance;
write under them the latitude and departure for the second
figure, setting them one place further to the right ; under
these place the latitude and departure for the third figure,
setting them one place still further to the right, and so
continue until all the figures of the given distance have been
used ; add these latitudes and departures and point off on the
right of their sums a number of decimal places equal to the
number of decimal places to wliicli t/ie tables being used are
carried; the resulting numbers will be the latitude and de-
parture of the given distance in feet, links, chains, or whatever
unit of measurement is adopted.

EXAMPLE. A bearing is 16 and the distance 725; what is the lati-
tude and departure ?

Distances. Latitudes. Departures.

700 6729 1929

20 1923 0551

5 4806 1378



725 696.936 199.788

SOLUTION. Taking the nearest whole numbers and rejecting the
decimals, we find the latitude and departure to be 697 and 200.

When a occurs in the given number the next figure must
be set two places to the right, as in the following example:

EXAMPLE. The bearing is 22 and the distance 907 feet; required,
the latitude and departure.



720



LAND SURVEYING.



SOLUTION.

Distances.
900
Jf
907



Latitudes.
8345



Departures.
3371

2622
339.722




FIG. 320.



Here the place of in both the distance column and in the latitude
and departure columns is occupied by a dash . Rejecting the deci-
mals, the latitude is 841 feet
and the departure 340 feet.
When the bearing is more
than 45, the names of the
columns must be read from
the bottom of the page. The
latitude of any bearing, as
60, is the departure of its
complement, 30 G ; and the
departure of any bearing, as
30, is the latitude of its
complement, 60'. This will
be readily understood from
an inspection of Fig. 320, in
which if IV S be the mag-
netic meridian and BOA 60 the bearing, then A O is the latitude
and A B the departure. If, now, O C be made the meridian and
BOC=W (the complement oi,BOA) the bearing, then O C (the
equal of A B) is the latitude, and B C (the equal of A O) the departure.

EXAMPLE. Let OB 1,326 feet, and its bearing = 60.
SOLUTION.

Distances. Latitudes. Departures.

1,000 0500 0866

300 1500 2598

20 1000 1732

6 3000 5196

1326 , 663.000 1148.316

The required latitude is 663 feet and the departure 1,148 feet!

Where the bearings are given in smaller fractions of
degrees than is found in -the table, the latitudes and
departures can be found by interpolation.

Traverse tables are chiefly employed in testing the accuracy
of surveys, platting them, and calculating their content.

133O. Testing a Survey. When a surveyor has
completed the survey of a field or farm by taking bearings



LAND SURVEYING.



721



and measuring courses, it is evident that he has gone as far
north as south and as far east as west. The sum of the
north latitudes shows how far north he has gone, and the
sum of the south latitudes shows how far south he has gone.
The sum of the east departures shows how far east he has
gone, and the sum of the west departures shows how far west
he has gone. Hence, if the survey has been correctly made
these sums will be equal or will balance.

The entire operation of testing a survey is illustrated in
the following example :









Latitudes


Departures.


Q, ,..


fy


TV^t












N +


S -


E +


W -


1


N34iE


273


226




154




2


N 354 E


128


10




128




3


S 56f E


220




121


184




4


S 34 W


353




292




199


5


N"56i W


320


177






267



413



413



1653 1126
5786 3940
2480 1688



225.640 153.688



56f
220



1097 1673
1097 1673



128



84*'
353



466

0079
01 57
0628



466

0997
1994
7975

10.098 127.615



120.67



184.03



2480 1688
4133 2814
2480 168!



291.810 198.628



564
320



1656 2502
1104 1668

176.64 266.88



Adding up the north and south latitudes we find them to
exactly balance each other, as do the east and west departures,
which proves the survey to be correct. On account of the
inherent defects of the compass and the errors which are



722



LAND SURVEYING.



liable to occur in measurement, especially on rough and
extensive areas, it is but rarely that the survey will exactly
balance. A moderate discrepancy, which would indicate
what may be called unavoidable errors, will be allowable,
and the survey accepted as correct. How great a difference
in the sums of the columns may be allowed is a doubtful
question. Every surveyor of experience knows the average
degree of accuracy of his work, and will readily distinguish
between a serious error and an allowable inaccuracy.

1331. Balancing a Survey. When the sums of the
latitudes and of the departures do not equal each other, and
yet the difference does not indicate any error, the different
latitudes and departures are modified so that their sums
shall be equal. This process is called balancing the survey.

The error is distributed among the different courses in
proportion to their length by the following

Rule. As the sum of all the courses is to any separate
course, so is the whole difference in latitude to the correction
for that course. A similar proportion corrects the departures.

An example illustrating the process of balancing a survey
is given below. In this example four separate columns are
given for the corrected latitudes and departures. In prac-
tice, however, the corrected latitudes and departures are
written in red ink directly above the original ones, which
are crossed out with red ink. The distances given are in
chains:



Sta-


Bearings.


Dis-


Latitudes.


De-
partures.


Corrected
Latitudes.


Corrected
De-


tions.




tances.
















N +


S


B4-


W-


N + l S-


B +


w-


1


N 52 E


10.63


6.54




8.38




6.58




8.34




2


S 29fE


4.10




3.56


2.03






3.55


2.01




3


S 31fW


7.69




6.54




4.05




6.51




4.08


4


N 61' W~


7.13


3.46






6.24


3.48






6.27






29.55


10.00


10.1010.41


10.29


10.06


10.06


1 <).:!.->


10.35



LAND SURVEYING.



723



The corrections are made by the following proportions:



For Latitudes.

29.55 : 10. 63:: 10 : 4 links.
29.55: 4.10::10: 1 link.
29.55: 7. 69:: 10: 3 links.
29.55: 7. 13:: 10: 2 links.



For Departures.
29.55 : 10. 63:: 12 : 4 links.
29.55 : 4.10::12 : 2 links.
29.55: 7.69-12 : 3 links.
29.55: 7.13-12 : Slinks.



10 12

This rule should not always be strictly followed, especially
if one line has been measured over rough and broken coun-
try, while the others have been measured over smooth and
open ground. In such a case the greater part of the error
will probably lie in the rough line, and, consequently, it
should receive the larger share of the correction. A slight
alteration of a bearing will sometimes balance a survey.
This may be done where an obstructed sight has probably
caused an error in the bearing.

1332. Application of Latitudes and Departures
to Platting. Rule three columns, one for stations, the
next for total latitudes, and the third for total departures,
as shown in the following diagram.

To obtain the total latitudes, begin at any station, the
extreme east or west one is preferable, and add up algebra-
ically the latitudes of the following stations, observing that
north latitudes are plus (+), and south latitudes minus ( ).
In the same manner find the algebraic sum of the depar-
tures for the different stations, placing each successive sum
opposite its proper station.

In the example given in Art. 133O, beginning at Station
1, we obtain the fol-
lowing results.

The work is proved
to be correct by the
latitudes and depar-
tures for Station 1
coming out equal to
0. To apply these
total latitudes and de-
partures in platting,



Stations.


Total Latitudes
from Station 1.


Total Departures
from Station 1.


1


0.00


0.00


2


+ 2.26


+ 1.54


3


+ 2.36


+ 2.82


4


+ M 5


+ 4.66


5


-1.77


+ 2.67


1


0.00


0.00



724



LAND SURVEYING.



we draw a meridian through the point taken as Station 1,
Fig. 321. Scale off from Station 1 upwards on this meridian
the latitude 2.26 chains to A and to the right from A, and
perpendicularly lay off the departure 1.54 chains to Station
2. Join 1-2. From 1 again lay off the latitude 2.36 ( =
2.26 + 10) chains to B, and to the right perpendicularly
the departure 2.82 (= 1.54 + 1.28) chains to Station 3.
Join 2-8, and proceed in like manner to locate Stations




FIG. 321.

4 and 5, laying off + latitudes above Station 1 and +
departures to the right of the meridian, and latitudes
below Station 1 and departures to the left of the merid-
ian. The principal advantages of this mode of platting
are rapidity of work, the fact that each course is platted
independently, and the certainty of the plats closing, pro-
vided the latitudes and departures have previously been
balanced.

1333. Calculating the Content. The survey of a
field or farm having been made and platted, the content can
always be found by dividing the plat into triangles, and



LAND SURVEYING.



725



scaling off their bases and perpendiculars from which the
contents are calculated. This and other methods previously
mentioned are only approximate, the degree of accuracy
depending upon the largeness of the scale and the skill of
the draftsman. The method of calculating content by
latitudes and departures is perfectly accurate, and does not
require the previous preparation of a plat.

1334. Definitions. If a meridian be passed through
the extreme east or west corner of a field, the perpendicular
distance from any station to that meridian is the longitude
of that station, additive or plus if east and subtractive or




minus if west. The distance of the middle point of any line,
such as the side of the field, from the meridian is called the
longitude of that side. The difference of the longitudes
of the two ends of a line is called the departure of the line;
the difference of the latitudes of the two ends of a line is
called the latitude of the line.



720 LAND SURVEYING.

1335. Longitudes. Let N S, Fig. 322, be the
meridian passing through the extreme westerly station of
the field A B C D E. From the middle and ends of each
side draw perpendiculars to the meridian. These perpendicu-
lars will be the longitudes and departures of the respective
sides. The longitude F G of the first course A B is evi-
dently equal to one-half its departure H B. The longitude
/ K of the second course B C is equal to ./ L + L M -\- M K
equal to the longitude of the first course plus half the de-
parture of the first course plus half the departure of the
course itself. The longitude Y Z of some other course E A,
taken anywhere, is equal to W X V X U V, or equal
to the longitude of the preceding course minus half the de-
parture of that course minus half the departure of the course
itself, i. e., equal to the algebraic sum of these three parts,
remembering that south latitudes and west longitudes are
negative, and, therefore, to be subtracted when the
instructions are t,o make an algebraic addition.

To avoid fractions, the preceding expressions are doubled,
whence we deduce the following

Rule for Double Longitudes :

The double longitude of the first course is equal to its
departure.

The double longitude of the second course is equal to the
double longitude of the first course phis the departure of that
course plus the departure of the second course.

TJie double longitude of the third course is equal to the
double longitude of the second course plus the departure of
that course plus the departure of the course itself.

The double longitude of any course is equal to the double
longitude of the preceding course plus the departure of that
course plus the departure of the course itself.

The double longitude of the last course (as well as of the
first} is equal to its departure. This result, when obtained by
the above rule, proves the accuracy of the calculation of the
double longitudes of all the preceding courses.



LAND SURVEYING.



727




it is equal to the product



1 336. Areas. The following is an application of the
rule for finding areas by double longitudes. See Fig. 323.
Let A B C be a three-sided field,
of which A is the most westerly
station. Through A draw a
meridian, and from the stations
B and C and the middle points
of the three sides of the field
draw perpendiculars to the
meridian. It is evident that the
area of the field A B C is equal
to the area of the trapezoid
D B C E less the triangles
A D B and A E C. The area of
the triangle A D B is equal to
the product of A D by F G, i. e.
of the latitude of the first course by its longitude. The area
of the trapezoid D B C E is equal to the product of D E by
half the sum of D B and E C, or H K, i. e. , it is equal to the
product of the latitude of the second course by its longitude.
The area of the triangle A E C is equal to the product of A
by half E C, or L M, i. e., it is equal to the product of the
latitude of the third course by its longitude. The bearing
of the course A B is N E, and that of C A is N W. Their
latitudes are, therefore, north. The bearing of the course
B C is S E, and its latitude is south. Calling the products
in which the latitude is north, north products, and the
products in which the latitude is south, south products, we
find the area of the trapezoid to be a south product and the
areas of the triangles to be north products. The difference
of the north products and the south products is, therefore,
the area of the three-sided field ABC.

Using double longitudes, to avoid fractions, in each of
the preceding products, their difference will be double the
area of the field ABC.

Take, now, a four-sided field, A B C D, Fig. 324, and
drawing a meridian through its most westerly station A,



728



LAND SURVEYING.



M



N



and longitudes as in the preceding case, it will be evident
from inspection that the area of the field A B C D is equal
to the trapezoid F C D G, diminished by the area of tri-
angles A G D,A B, and
the trapezoid E B C F.
The area of the triangle
A E B is equal to the
product of the latitude
A E of the first course by
its longitude H K. Its
product is north. The area
of the trapezoid E B C F
is equal to the product of
the latitude E F of the
second course by its longi-
tude L M, and is also a
north product. The area
of the trapezoid F C D G
is equal to the product of
the latitude F G of the
third course by its longi-
tude N O, a south product.
The area of the triangle
A G D is equal to the prod-
uct of the latitude A G of the fourth course by its longi-
tude P Q, a north product. Subtracting the sum of the
north products from the sum of the south products the
difference is the area of the field A B C D. If double longi-
tudes had been used, as in the previous case, the difference
would have been double the area of the field.




1337. The Application of Double Longitudes to
the Finding of Areas. Whatever the number or direc-
tions of the sides of a field or any surface enclosed by
straight lines, its area will always be equal to half the dif-
ference of the north and south products arising from multi-
plying together the latitude and double "longitude of each
course or side, whence the following



LAND SURVEYING. 729

General Rule for Finding Areas :

1. Prepare ten columns, headed as in the following exam-
ples, and in the first three write the stations, bearings,
and distances.

2. Find the latitudes and departures of each course by the
traverse table, as directed in A rt. 1329, placing them in the
four following columns.

3. Balance them as in A rt. 1331, correcting them in red
ink.

4- Find the double longitudes as in Art. 1335, with refer-
ence to a meridian passing tlirougli the extreme east or west
station, and place them in the eighth column.

5. Multiply the double longitude of each course by the
corrected latitude of that course, placing the north prod-
ucts in the ninth column and the south products in the tenth
column.

6. Add the last two columns ; subtract the smaller sum
from the larger, and divide the difference by two. The
quotient will be the content required.

1338. To Find the Most Easterly or Westerly
Station of a Survey. Make a rough hand sketch of the
tract, giving the sides, their approximately true direction, and
length. The most easterly or westerly station may then be
determined from an inspection of the sketch.

Example 1 of this article refers to the five-sided field, a
plat of which is given in Fig. 321, and the latitudes and de-
partures of which were calculated in Art. 133O. Station
1 is the most westerly in the plat, and the meridian will be
passed through it.

The double longitudes are found by applying the rule for
double longitudes, given in Art. 1335. As the additions'
are made algebraically, due attention must be paid to the
signs. The double longitudes are marked D. L., as shown in
the marginal diagram. These double longitudes are ob-
tained by the following operation. As stated in the rule,
the double longitude of the first course is equal to the



730



LAND SURVEYING.



departure.



Stations.


D. L.


1
2
3
4
5


+ 1.54D.L.
+ 1.54
+ 1.28


+ 4.36D.L.

+ 1.28
+ 1.84


+ 7.48 D.L.
+ 1.84
-1.99


+ 7. 33 D.L.
-1.99
-2.67


+ 2. 67 D.L.



By reference to the given example, we find
that the departure of the first course is
1.54 chains, an cast departure, and, there-
fore, positive. We record this in the
column headed D. L., opposite Station 1.
The D. L. of the second course is equal
to the D. L. of the first course plus the
departure of that course plus the de-
parture of the second course. Accord-
ingly, we place under the D. L. of the
first course, the departure of that course,
viz., -f 1.54, and the departure of the
second course, viz., -f- 1.28, as given in
the east departure column of the exam-
ple. This sum, viz., -f 4.36 is the D. L.
of the second course and placed opposite
Station 2. The D. L. of the third course
is equal to the D. L. of the second course
plus the departure of that course plus the departure of
the third course. Accordingly, we place under the D. L.
of the second course the departure of that course,
viz., +1.28, and the departure of the third course, viz.,
+ 1.84. This sum, + 7.48, is the D. L. of the third course,
which we place opposite Station 3. In a similar manner
we find the D. L. for the fourth and fifth courses. The
double longitude of the last course is equal to its de-
parture, which proves the work. The double longitudes of
the courses are then multiplied by their corresponding lati-
tudes, and the content of the field obtained as directed in
the given rule.

Had the meridian been supposed to pass through Station
4, the most easterly station, all the longitudes would have
been west or minus, but the difference in the double areas
would have been the same, giving the same content as
before.

The following examples will give the student some prac-
tice in the use of traverse tables, and in applying latitudes
and departures in the calculation of areas:



LAND SURVEYING.



731




I



+ + I + I I +



S 8



00



80-



CO
Cfi H



s



732



LAND SURVEYING.















t-

[


?

i


Q Q Q Q

co co co os co x c: x ic co i.-

co co ^ ?< ~ ~ o; os T-I o i-


Q

5 O* O
- X O


c

91 X T

X 1O C


3 Q CJ Q
pxoo ^xco i coLO m

*in<7<J O5L- Tft-X X '






T-l T-H ,-0 -* l-H t- T-


IT-: d


TH


H THrHrH^H^H^HTjiTj:






III III Ml 1


1 1


1 +


+ 1 1 1 + 1 ++ 1


i


i


0,





b


- X OS TH






m co co o
o i- co m


5






g




I-H -* m i-


X






I




IO <N <ft r-t

TH


OJ






1


X


C007CS


?x


J) 44. 0512
22.0256




1


?l


lococsococ-t - '* o

XCOCJ0500070S Tf




c




O

_


32


1 1 I 1 1 II 1




o

V












ft






I


coeoxiow x


osos


3

o*




.


fe


" i-H


t- t-


C/}




1












P












E












a






xcs






2


w


5^ ^ TH


t-i-






^


i

(A


co *K.TH c- 10

TH *-<X t- CO



coco






j


+


lOgXlOCO^













MM .




dl




gj
E


^


t-oo^coosin^


10









2


C-CJOCsOJ-rHOiO TH


^









o




^









j


U^*U H




<




*


i


in co co t- o in co 10 co
c/3'/!c/)t/5H^^ ^i




II

c
c




i


8


TH o* eo * 10 co t- oo os




o

o





LAND SURVEYING.



733



The notes of the survey given in Example 3
by total latitudes and total
departures from Station 1.
A plat of the survey is given
in Fig. 325 and the total lati-
tudes and departures in the
accompanying table. From
an inspection of the plat it
will be seen that Station 2 is
the most easterly, and the
double longitudes given in
Example 3 are reckoned from
a meridian passing through
that station.



are platted





Total


Total


Stations.


Latitudes
from


Departures
from




Station 1 .


Station 1 .


1


0.00


0.00


2


-3.13


+ 4.85


3


-4.94


+ 3.52


4


-5.71


+ 2.89


5


-6.06


+ 1.91


6


-5.61


+ .76


7


-4.39


-1.06


8


-3.51


- .48


9


-2.66


-1.76


1


-0.00


0.00




TOWN SITES AND SUBDIVISIONS.

1339. First Considerations. In laying out town

sites the consideration of first importance is the location of

the streets rather than the greatest number of lots obtainable.

The custom of laying out town sites in rectangular lines,



734 LAND SURVEYING.

without reference to topographical conditions, prevails
almost universally throughout the United States. This is
largely owing to two principal causes, viz., first, the sup-
position that the rectangular method or plan will yield the
greater number of lots, and, hence, the greater profit, and,
second, the haste in surveying, platting, and placing the
property on the market does not admit of a thorough study
of the ground.

The town site should be considered as a whole, the loca-
tion of its main streets and thoroughfares being determined
by traffic considerations chiefly. These considerations will
necessarily involve the questions of grades, drainage, and



Online LibraryInternational Correspondence SchoolsThe elements of railroad engineering (Volume 2) → online text (page 9 of 35)