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LIBRARY
OF THF.
UNIVERSITY OF CALIFORNIA,
GIFT OF"
Jl
Accession No. 836.9-U-.
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Received ,190




J


TRAVERSE TABLES
WITH AN INTEODUCTOEY CHAPTEE ON
CO-ORDINATE SURVEYING
BY
HENEY LOUIS, M.A, A.E.S.M., F.I.C., F.G.S., ETC.
i\
PROFESSOR OF MIXING AND LECTURES ON SURVEYING, DURHAM COLLEGE OF SCIENCE, NEWCASTLE-ON-TYNE EXAMINER IN MINE SURVEYING TO THK CITY AND GUILDS OF LONDON INSTITUTE
AND
GEOKGE WILLIAM CAUNT, M.A.
LECTURER IN MATHEMATICS, DURHAM COLLEGE OF SCIENCE, NEWCASTLE-ON-TVNE
LONDON
EDWAED ARNOLD
37 BEDFORD STREET. STRAND
1901


a


PKEFACE.
THE publication of this little work is due to the writer's
conviction, gained in many years of miscellaneous surveying
practice, as well as in some spent in the teaching of surveying, that the co-ordinate method of plotting traverses is far preferable
to any other, on the score of both accuracy and expedition. There are, of course, several traverse tables already in existence ; but
whilst some of these are calculated with a degree of accuracy
greater than is required in ordinary surveying, and more
especially in ordinary mine surveying, others are not accurate enough, inasmuch as their calculations are not extended to
every minute of the degree. The price of the former works is,
moreover, somewhat prohibitive, at any rate as far as the
ordinary mine surveyor is concerned.
At the present day it is usual to employ, in ordinary underground and surface work, instruments divided into single
minutes, so that the tables must be calculated for this unit to be
of any real use. In ordinary chaining it may be taken that it
is rare for any traverse to exceed ten chains in length, whilst the
limit of accuracy for such lengths is about one link. The tables
are therefore calculated to five significant places (four places of
decimals), so that their accuracy is about ten times as great
as is attained in ordinary actual work. This limit is therefore
sufficiently near for all practical purposes, and, at the same time, does not involve any undue amount of arithmetical work.


iv PREFACE.
These tables are not intended for cadastral surveys, for which
seven decimal places are required, or at times even more. The
arrangement of the tables is one which the writer finds in
practice to be convenient for rapid work, all the figures needed
for any given angle being found at one opening of the pages and in one line. The tables have been entirely recalculated by
Mr. Gaunt, and checked in all possible ways, and every
precaution has been taken to ensure accuracy in printing, so as to warrant the hope that they may be found free from
error. The writer ventures to hope that their publication may
serve to popularize this most convenient method of working out
traverse surveys in this country, HENEY LOUIS.
NEWCASTLE-ON-TYNE, December, 1900,


TBAVEESE TABLES.
CO-ORDINATE SURVEYING.
CO-ORDINATE surveying, or, to speak more precisely, co-ordinate plotting, is the name given to a method of recording the results
of traverse surveys in which the draughtsman represents each draft of the survey by means of its rectangular co-ordinates. It cannot well be applied to any other than traverse surveying, hence its utility is mainly restricted to such forms of survey work
as depend upon traverses, that is to say, mine surveys, surveys of roads, rivers, or railways, and surveys of areas. In a traverse survey the lengths and directions of the various traverses or drafts are determined in the field by methods with which all
surveyors are familiar; it need only be here observed that, however determined, the direction of any traverse is the angle which it makes with any determinate direction ; the latter may
either be an absolutely fixed direction, such as the terrestrial
meridian, or it may be comparatively fixed, as the magnetic meridian, or it may be purely arbitrary, such as the direction of the first draft of the survey, or of any other traverse, or of one of
the main directions in which the survey extends, e.g. the main
road of a large colliery, the principal street of a town, etc. In mining surveys it is customary in this country to refer all directions to the magnetic meridian, in spite of several obvious
inconveniences to which this practice is subject; although it is certainly better to use the terrestrial meridian, yet the
method of plotting by co-ordinates is exactly the same whatever a2


vi TRAVERSE TABLES.
be the line of reference that is used. For the sake of sim
plicity it will here be supposed that a method of surveying
has been adopted by which the angle which each traverse makes with the terrestrial meridian can be accurately determined.
Thus, in Fig. 1, let the traverse line of length OA (= I)
make an angle a with the meridian line YY' through 0, and
let XX' be drawn at right
angles to TOY'. Draw Aa
and Aa' perpendicular respec
tively to the lines XX' and
YY' ; then Aa = Oa' = I cos a,
and Aa' = Oa = I sin a, are
**f
"" the co-ordinates of the point A referred to its origin 0, or the co-ordinates of the traverse OA.
Oa', the meridian co-ordinate,
*' is usually spoken of as the difference of latitude (generally abbreviated into latitude), and Oa, the equatorial co-ordinate, is generally called the departure of the traverse OA, and these words, latitude and departure, are adhered to even when the
reference lines, YY' and XX' , do not correspond with either the terrestrial or the magnetic meridian and equator respectively. It is obvious that the co-ordinates of any traverse can
be calculated by the aid of a table of sines and cosines, but this is a laborious and slow operation. Traverse tables are merely tables in which the results of these calculations are recorded so as to save time; in the present traverse tables the latitude and departure are given for every minute of angle and for all lengths from 1 to 10, so that the co-ordinates for any desired length can be taken out by simple addition, at the same
time moving the decimal point as may be required. Thus, required the co-ordinates of a traverse 1638 links long, making an angle of 27 49' with the meridian line. Entering the table headed 27, the minutes are found in the column at the left
hand (headed Min.), and looking horizontally along the line corresponding to 49', the several figures are taken out under the unit distances (Dist.) which head each double column of lati
tude (Lat.) and departure (Dep.), thus


TRAVERSE TABLES. vii
27 ,49' Dist. Lat. Dep. 1000 884-5 466-6 600 530-7 280-0 30 26-5 14-0 8 7-1 3-7
1638 1448-8 764'3
So that the required latitude is 1449 links, and the required departure 764 links. It will be noticed that the figures are
only taken out to the first decimal place ; in ordinary surveying fractions of links are not recognized, so that the co-ordinates are merely required to be correct to the nearest unit ; there is
therefore no object in using more than one decimal place. Whenever the angle given exceeds 45, the angle must be sought for at the bottom right-hand corner, and the minutes
read upwards in the last right-hand column; care must be taken also to read the latitudes and departures upwards in accordance with the respective designations at the bottom of the page. In this connection it is worth remembering that
when the angle is less than 45, latitudes are greater than departures, and when the angle exceeds 45, latitudes are less
than departures, A useful check is also obtained by noting
that
(Distance)2 = (Latitude)2 + (Departure)2
When one traverse only has to be plotted, but little is gained by the use of co-ordinates ; but when a number of successive traverses have to be laid down, as is the case in an ordinary
traverse survey, the advantage is evident, as the various latitudes
and departures can be added together arithmetically, and thus the exact position of the end point determined before the survey
is plotted. Thus, let OA, AB, and BC (Fig. 2) be three traverses, of which the lengths and the angles which they make with the meridian are known. Then Oa' and Oa are, as before, the
co-ordinates of the traverse OA or of the point A referred to
its origin ; similarly, J93/ and AM are the co-ordinates of B
referred to its origin A. But Ob = Oa + ab = Oa -J- AM, and
Ob' = Oa' -f- a'V = Oa' -f MB; therefore the co-ordinates of
the point B referred to the origin are the sums of the re
spective latitudes and departures of the two traverses OA and AB. In the same way Oc and Oc', the co-ordinates of the point


Vlll TRAVERSE TABLES.
C referred to the origin 0, are the sums of the co-ordinates OA,
AB, and BC, it being noted that BG runs in the opposite
direction to OA and AB} and its latitude is therefore negative.
In other words, the meridian and equatorial co-ordinates of any
point that is reached by a series of traverses, are the algebraical sums of the respective latitudes and departures of each one of the component traverses. It is usual to treat the directions
OJC and OY as positive, and OX' and OY' as negative; in other
words, northerly latitudes and easterly departures are treated
as + quantities, and southerly latitudes and westerly departures as quantities. In loose-needle surveys either meridian or quadrant angles
may be read at the will of the surveyor. In ordinary theodolite
surveys and in " racking " or " fixed-needle " surveys with the Vernier dial, the angles are determined that any given traverse makes with the meridian (or other arbitrary direction), so that
any angle may be registered from to 360. The fii'st step is therefore to reduce these meridian angles (or azimuths, as they
are often called) to quadrant angles by the following rules : If the meridian angle is letween and 90, the quadrant is
N.E., and the quadrant angle = meridian angle.


J. V J^JTVOJ.
TRAVERSE TABLES. ix
If the, meridian angle is between 90 and 180, the quadrant
is S.K, and the quadrant angle = 180 meridian angle.
If the meridian angle is between 180 and 270, the quadrant
angle is S. W., and the quadrant angle = meridian angle 180.
If the meridian angle is between 270 and 360, the quadrant
is N. W., and the quadrant angle = 360 meridian angle.
For example, to find the quadrant angles corresponding to
the following meridian angles : (a) 17 23' ; (6) 141 44' ; (c) 250
21'; (d) 339, 08'.
Meridian angle. Quadrant angle. (a) 17 23' N. 1723'E.
(6) 141 44' S. (180 - 141 44') E. = S. 38 16' E.
(c) 250 21' S. (250 21' - 180) W. = S. 70 21' W. ,
(d) 339 08' N. (360 - 339 08') W. = N. 20 52' W.
Sometimes these angles are simply written +17 23' + , -38 16'+, -70 21'-, +20 52'-, this method being specially convenient when any arbitrary line is selected as the direction of reference in preference to a meridian ; it is under
FIG. 3.
stood that the first sign always refers to the latitude and the
last to the departure. The successive stages in working out a traverse survey by
co-ordinates, preparatory to plotting, are best illustrated by an
example. Let Fig. 3 represent a traverse survey of an area


x TRAVERSE TABLES.
bounded by straight lines, executed by the " double fore-sight
"
method with an ordinary theodolite, the area forming a seven
sided polygon. The first two columns (see p. xii), namely the measured lengths of the sides and the observed theodolite readings, are obtained in the field and taken from the field-book in which they were entered.1 At the beginning of the survey
the theodolite is supposed to be pointed due north; the first reading gives therefore the meridian bearing of the first traverse
OA; the meridian bearings, or azimuths, of the subsequent traverses are obtained by the well-known rule : Add the observed theodolite reading to the last meridian bearing and subtract 180 from, or add 180 to, the sum, according as that sum is greater or less than 180. The result in this ca.se is as
follows :
Meridian bearing of OA 295 12'
Theodolite reading of AB 72 13'
367 25' 180
Meridian bearing of AB 187 25'
Theodolite reading of B C 135 37'
323 02' 180
Meridian bearing of B C 143 02'
Theodolite reading of CD 87 20'
230 28' 180
Meridian bearing of CD 50 28'
Theodolite reading of DE 240 05'
290 33' 180
Meridian bearing of DE 110 33'
Theodolite reading of EF 41 26'
151 59' 180
Meridian bearing of EF 331 59'
Theodolite reading of FO 79 10'
411 09' 180 Meridian bearing of FO 231 09'
The meridian bearings thus obtained are entered in their
1 It goes without saying that the closing angle at 0, which in this case
should be equal to 244 03', is observed and noted in the field-book as a check, though it is not required for these calculations.


TRAVERSE TABLES. xi
proper column, and then the column of quadrant bearings is at once filled in (see p. xii), in accordance with the rules already given. By reference to the tables, the latitudes and departures are then determined by simple addition ; the first two may be
given in full by way of example :
64 48' DUt. Lat. Dep. 1000 425-78 904-83 400 170-31 361-93 8 3-41 7-24
1408 599-5 1274-0
7 25' 800 793-31 103-27 40 39-67 5-16 7 6-94 0-90
847 839-9 109-3
The latitudes and departures are entered in their re
spective columns. As this is a closed survey, returning to the starting point 0, the total north and south latitudes and the
total east and west departures ought to be respectively equal to each other, and it will be seen that such is practically the
case, fractions of a link being disregarded. The last two columns, headed total latitudes and departures, are really the successive co-ordinates of each of the survey stations ; they are obtained by the successive algebraical additions of the latitudes and departures respectively, the sum or difference taking the same sign as the larger of the two
figures. Thus, to take the latitudes, the latitude of the point
A
is evidently 599*5 links N; then we have :
Latitude of point B = 599'5 N. + 839-9 S. = (839-9 - 599-5) S. = 240-4 S.
Latitude of point C = 240'4 S. + 1428*6 S. = 1669*0 S.
Latitude of point D = 1669-0 S.+ 1059-2 N. = (1669-0 - 1059-2) S. = 609'8 S.
Latitude of point E = 609-8 S + 347-9 S. = 957-7 S.
Latitude of point F = 957-7 S. + 1743-6 N. = (1743-6 - 957-7) N.= 785-9 N.
Latitude of point = 785'9 N. + 786-0 S. = (786-0 - 785-9) S. = 0-1 S.
The total departures are calculated in precisely the same way. The clerical work of the addition is checked by adding up the two columns of latitude and the two of departure ; the differences between these respective pairs should be equal to the final total latitudes and departures.


Xll TRAVERSE TABLES.
3s
FABCDEOODCABEF


TRAVERSE TABLES. xiii
the irregular expansion and contraction of even the best drawing paper 1 is more than enough to introduce grave inaccuracies into the best drawn plan. As an example of the method of calculation, let it be
required to determine the distance and bearing of station E
from station B. From the column of total latitudes and departures we have
Station B. Lat. S. 240-4. Dep. W. 1383-3 Station E. Lat. S. 957-7. Dep. E. 1903-2
Therefore E is 717-3 links S. and 3286'5 links E. of B.
Bearing of line BE = S. tan -1 -~j E. = S. 77 41' 15" E.
Distance EB = . = .
*ep'. = */tep* + lat.* = '
.* . ,
cos bearing sin bearing cos 77 41 15
sn + 3286-5* = 3364 links
These calculations are best made in the usual way by the aid of tables of logarithms. In case of need, the traverse tables
can be used for them, as the departure column for distance
=
1, is practically a table of natural sines, whilst the corresponding latitude column is practically a table of natural
cosines, and these evidently give all the elements required for the calculation. In ordinary practice it is, however, far better to use any good table of logarithms for this portion of the work. The above survey is an imaginary one, and there is therefore no closing error. In actual closed traverse surveys there is
of course usually some error. By working out the co-ordinates,
and by adding up the observed angles (including the closing angle), it is at once obvious whether the error is in the linear
or in the angular measurements ; in the latter case, if it is only
the closing angle that has been read wrong, the co-ordinates will close, though the angles do not. If the error falls within the required limits of accuracy, it is easily distributed between the co-ordinates, and the plotting is done from the co-ordinates thus rectified.
The following is an example from actual practice, of a survey
in a coal mine. A survey was started from a peg in the " flat,"
1 Those interested in this matter should consult an important paper by
Mr. C. C. Leach. Transactions of the North of England Institute of Mining
Engineers, Vol. xxxiv., 1884-85, p. 175.
I


XIV TRAVERSE TABLES.
and was extended to a point in a " back place
" to which it was desired to drive a road from the peg in the flat. The theodolite was set up in the flat, using the centre line of the flat as the axis
of direction to which the survey was to be referred. A copy of the field-book is given below, many of the minor details
being, however, omitted.
=


TRAVERSE TABLES. xv
The co-ordinates are worked out as previously explained, and the total latitudes and departures obtained as follows :


XVI TRAVERSE TABLES.
becomes one of quite secondary importance. All that is required to be known is already determined before the plotting is commenced. It may also be remarked that all the opera
tions up to and including the taking out of the total latitudes
and departures are of the utmost simplicity, involving no higher
S cede % inch - 1 Chcdrv
FIG. 4.
arithmetical knowledge than the addition and subtraction of decimals, and may hence be entrusted to any moderately intelligent lad, instead of occupying the time of the surveyor himself. The advantages of the use of co-ordinates are, however,


TRAVERSE TABLES. xvn
most evident when it is necessary to determine the area included in a closed traverse. Unless co-ordinates are used the
only method of determining such areas accurately is by an involved trigonometrical method, consisting of cutting the area up into triangles the apices of which meet in any assumed point.
The angles of each triangle have then to be calculated, the triangles solved, and the sum of their areas thus determined. This method is so laborious that it is never used in practice. Unless this or the method of co-ordinates is used, however, the determination of the area can only be made by first plotting the survey, by which a number of errors of more or less importance are necessarily introduced.
By the use of co-ordinates, all these difficulties are avoided,
and the area of any closed traverse can be calculated directly
and easily from its latitudes and departures, without any
plotting at all. The principle of the calculation is best seen from a simple example :
Let the five sided figure OABCD (Fig. 5) be the plan of
a traverse survey situated wholly on one side of the meridian
through the point 0, and let it be required to determine the area of the figure. The total latitudes and departures are calcu
lated in the usual manner, and we have for the respective survey stations :


xviii TRAVERSE TABLES.
Total latitude. Total departure.
00
A Oa'( = -y) Oa( = 'x)
B Ob'( = - yi) Ob( = *:)
C Oc'( = ya) Oc( = x,}
D Od'( = #3) Od( = a?8)
Then the area of the figure
OABCD = d'DCSAa' - d'DO - OAa'
d'DCBAa' = d'DCc' + c'CBV + VBAu'
d'DCc' = UDd' + Cc'}d'c'
= H+0)(-y-0) Hence
= J[(0 + 3 )(0 - 2/ 3 ) + (aJ 3 + a; a)
/(dep. of + dep. of D)(lat. of - lat. of D)
(dep. of D 4- dep. of (7)(lat. of 7) - lat. of (7)
/. the area OABCD = J ( (dep. of (7 + dep. of 5) (lat. of C' - lat. of 5)
(dep. of 5 + dep. of -4)(lat. of B - lat. of A)
((dep. of A + dep. of 0)(lat. of A - lat. of 0)
The rule for the calculation of the area contained by a closed traverse is therefore as follows :
The algebraic sum of the total departures of each pair of adjacent angular stations is multiplied by the algebraic difference of their total latitudes ; the products thus obtained are added
together, and the sum divided by two gives the area required. In applying this rule it must be borne in mind that the station points must always be taken in strict order.1 To each
total latitude or departure the correct algebraic sign must be prefixed, and regard must be had to it in the arithmetical
1 It makes no difference whether the points be taken in the order in which they have been surveyed, or in the opposite order; the essential point
is that one regular order shall be adhered to. If the points are taken in the
opposite order the only difference will be that the area will have a instead
of a 4- sign. This is easily seen; for in the above calculation if the points
be taken in the opposite order, the signs of the sums of the departures will be
unaltered, and the signs of the differences of latitudes will be changed (e.g.
(y2 ?/3 ) instead of (y3 y2 )> etc.), so that the sign of the area will be
changed, its numerical value being unaffected. The sign obtained for the area of any closed traverse depends upon the direction of the first traverse,
and upon that in which the points are taken ; it is always considered as positive.


TRAVERSE TABLES. xix
operations involved. The result will be expressed in squares of the unit of measurement employed, square links if the survey was made in links, square feet or square metres if the distances were measured in feet or metres, etc. The above rule is occasionally stated in a different way, which is sometimes more convenient for calculation. The total latitude of each station is multiplied by the
algebraic sum of the departure of that traverse, and of the one next
following ; 1 the sum of the products thus obtained, divided by 2,
gives the area required. The departure here referred to is not the total departure
referred to the origin of the survey, but the departure of the traverse referred to its own starting station. It can easily be shown that these two rules are practically the same. For in Fig. 5, taking the values given above, we shall have for the departures of each traverse
Departure of point D referred to O
C
1 *7')
> ^" ^^
Then according to the second rule
Twice area OABCD =
x
l - x.2
X -Xxl
Again, according to the first rule, we have seen that
Twice area OABCD =
*s - s/3 - + 0*3 + s2)(ys - 2/2) + (a?2 + i)(y
- XIJ\ + Xl*J the same result as that given by the second rule. All that has been said of the first rule holds equally good of the second. In both of them the words latitude and
departure may also be interchanged without altering the result, so that there are really four different arithmetical operations
that can be employed indifferently. Yet another method is sometimes employed, known as that of
the " double meridian distance." In this the successive latitudes
are multiplied by multipliers obtained from the departures ; a
1 It is evident that the suras of the departures of any two traverses is
equal to difference between the total departures of the point before and the
point after the one being worked.


XX TRAVERSE TABLES.
column of " double departures " is formed by adding each
departure to the preceding one ; from these double departures the multipliers are obtained by adding each double departure to the last multiplier, the first multiplier being always zero.
By way of example, the area of the figure OABCD may be
calculated, the values of the departures and latitudes being as follows :


TRAVERSE TABLES. xxi
In both cases the results are of course identical, namely 49,910 square links or 0*4991 acre. It will be seen that the second method of calculation here produces a negative sign ; this would have been positive had the points been taken in the opposite order.
The following is an example of the application of the second rule to the same area :
Total latitudes.


XX11 TRAVERSE TABLES.
Latitude.


TRAVERSE TABLES. xxin
Stations.


XXIV TRAVERSE TABLES.
to select the traverse lines so as to equalize as nearly as possible the offset areas on either side ; this has to be done by inspection,
on the ground, and of course requires a good deal of practice. It occasionally happens that some of the points in a survey are determined by methods of triangulation instead of by
traversing. Broadly speaking, the term triangulation may be applied to the determination of any point by angular measure
ments from, the two ends of a base-line of known length ; the triangle is then solved, and the lengths of the two unknown sides calculated. This calculation is simply and easily performed by means of co-ordinates.
The problem in its most general form is shown in Fig. 7.
Suppose the points A,B have been already determined, their
CL'
Y' FIG. 7.
departures Oa and 01 being x and ^ and their latitudes Oa'
and Ob' being y and ^ respectively. The angles AC(= a)
and ABC( = &) are determined by observation ; from these data the latitude and departure of have to be calculated. Let
the quadrant bearing of the line BA be N". a E. ; then the
quadrant bearing (]3) of the line EC = 1ST. (a + 6) E., and the
quadrant bearing (y) of the line A C = S. (a a) E. Care must be taken in every case that the signs are correct according to
the particular quadrant.
Then tan 7=-^-=
a'c' y 
tan * ~

b'c Oc' yl
Whence Oc' = * tan ? + x
tan 7 + tan y8


TRAVERSE TABLES. xxv
The departure Oc may be calculated from the corresponding formula : QC _ *i cot & + V + x cot 7 yt
cot /3 + cot 7
The above is the method generally employed, and is perhaps the most convenient when the ordinary mathematical tables are
available. It is, however, possible to use a method to which the traverse tables can be applied, and the work thus consider
ably simplified. For in Fig. 7 Oc = Oa + ac = x + OK
... Qc = x sin (a + b)
which may be written
n AB sin b . 10 sin y
10 sin (a + b)
All these values can now be taken from the traverse tables
because
AB sin b is the departure of distance AB for the angle b
10 sin 7 is the departure of distance 10 for the angle 7
10 sin (a 4- &) is the departure of distance 10 for the angle a -f Z>; if
a + b is greater than 90, the angle 180 (a + b) should be used instead.
Another form for the above expression is
n
, AB sin a . sin
:a?1+ *in(a + b)
For the latitude Oc', either of the two following expressions may be used:
^
, , AB sin a . cos fl
1
sin (a + i) s^n ^ C08
sin (a + b)
Any of these may be used with the traverse table as above
indicated, by multiplying numerator and denominator by 10, or by any other convenient number so that in the last case the second term of the formula for Oc would read
(dep of AB for angle &) x (lat. of 10 for angle 7)
dep. of 10 for angle (a + 6)
As an example of these calculations let the co-ordinates of the two points A,B of a traverse survey be as follows :


XXVI TRAVERSE TABLES.
Lat. ofJ. ... N. 87 links Dep. of^L ... W. 204 links
Lat. of ... S. 85 Dep. of B ... E. 89
Quadrant bearing of AB = S. 59 35' E. Length of AB - 340 links.
From A and B, Fig. 8, the angles between the direction AB
and the lines joining these points with the two points C and D,
Scale inch,<-1 Chain.
which it is desired to fix, have been observed, and found to be as follows :
Angle CAB = 40 28'
= 69 13'
Angle ABC = 68 59'
Angle A BD = 38 05'
Then, to determine the point C, we have by the first method
Meridian bearing of AC = 180 00' - 59 35' - 40 28' = 79 57'
Quadrant bearing of AC = N. 79 57' E.
Meridian bearing of BC = 360 00' - 59 35' + G8 59' = 369 24'
Quadrant bearing of BC = N. 9 24' E.
T} , (87 x tan 79 57') + (85 x tan 9 24') + 89 + 204
tan 79 57' - tan 9 24' = N. 146 And Oc = ( 2Q4 x cot 7 9 57') + (89 x cot 9 24') + 87 + 85
= E. 127
cot 9 24'^- cot 79 57'
As a check upon the arithmetical work of the calculations
we have
Tan 9 24' = I27
n
"" 89 = 0-1 G5
14G + 85
To determine the point D, we have


TRAVERSE TABLES. xxvii
Meridian bearing of AD = 180 - 59 35' + 69 13' = 189 38'
Quadrant bearing of AD = S. 9 38' W.
Meridian bearing of BD = 360 - 59 35' - 38 05' = 262 20'
Quadrant bearing of BD = S. 82 20' W.
T
, n r _ (87 x tan 9 38') + (85 x tan 82 20') + 204 + 89
tan 82 20' - tan 9 38' = S. 129 . d , = (204 x cot 9 38') + (89 cot 82 20?
) + 87 + 85 cot 9 38' cot 82 20' = W. 240
In applying these formulas special attention must be paid to the signs of the departures and latitudes. Using now the second method given above
O -201 + (34 ' sin 68 59/ )( 10 sin 79 57/)
10 sin (68 59' + 40 28')
From the tables
Departure of 300 = 280-04 for angle G8 59'
Departure of _40 = 37-34
Departure of 340 = 317-38
Departure of 10 for angle 79 57' = 9'8466
Departure of 10 for angle 109 27' = departure for angle 70 33' = 9-4293
= -204 + 331-4 = +127-4
, = 87 , (340 . sin 68 59') x (10 cos 79 57')
10 sin (68 59' + 40 28')
From the tables
Latitude of 10 for angle 79 57' = 1-7451 , . 7 317-38 x 1-7451 9-4293 = 87 + 58-7 = +145-7
Od = -904 - (34 sin 38 05') x (10 . sin 9 38')
10. sin (69 13' + 38 05') 9fu _ 209-71 x 1-6734 9-5476 = -(204 + 36-7) = -240-7
Od' = 87 (340 . sin 38 05') x (10 . cos 9 38')
10 . sin (69 13' + 38 O5'j~~ 209-71 x 9-859 9-5476 = 87-216-5 = -129-5
UNIVERSITY
CALIF01


xxviii TRAVERSE TABLES.
The results obtained are of course practically the same if the calculations are made with a sufficient degree of accuracy, but the use of the traverse tables even in such triangulation
problems is seen to very much shorten the calculations.


TRAVERSE TABLES.


Degrees.
Dist.


e


1 Degree.
Dist.


6


2 Degrees.
Dist.


8


3 Degrees.
Dist.


6


4 Degrees.
Dis


t


5 Degrees.
Dist.


6


6 Degrees.
Dist




7 Degrees,
Dist


6


8 Degrees.
Dist


6


9 Degrees.
Dist.


e


10 Degrees.
Dist


e


11 Degrees.
Dist.


e


12 Degrees.
Dist.




13 Degrees.
Dist.


6


14 Degrees.
Dist.


e


15 Degrees.
Dist.


e


16 Degrees.
Dist


(


17 Degrees.
Dist.


6


18 Degrees.
Dist.


6


19 Degrees.
Dist.


6


20 Degrees.
Dist.


6


21 Degrees.
Dist.


6


22 Degrees.
Dist.




23 Degrees.
Dist.


6


24 Degrees.
Dist


6


25 Degrees.
Dist


6


26 Degrees.
Diet.


I UNIVERSITY I
Vog . -*\*-S


27 Degrees.
Dist


6


28 Degrees.
Dist.


6


29 Degrees
Dist.


6


30 Degrees.
Dist,


(


31 Degrees.
Dist.


6


32 Degrees.
Dist.


6


33 Degrees.
Dist.