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```BOUGHT WITH THE INCOME
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1891

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http://www.archive.org/details/cu31 924031 221 1 65

-CHAPTER I.— GENEEAL INSTRUCTIONS.

Page

Field Work 1

Care of Field Equipment 2

Field Notes 6

OfBce Work 12

CHAPTER II.— THE CHAIN AND TAPE.

Units of Measure 13

Linear Measuring Instruments 14

Use of Chain and Tape 16

Perpendiculars 17

Parallels 18

Angles 19

Location of Points 19

Location of Objects 31

The Line Surveys 21

Ranging in Lines 21

Signals 33

Stakes and Stake Driving 23

Problem A 1. Length of Pace 34

A 2. Distances by Pacing 34

A 3. Axemen and Flagmen Practice 34

A 4. Range Pole Practice 34

A 5. Standardizing Chain or Tape 36

A 6. Distances with Surveyors' Chain 36

A 7. Distances with Engineers' Chain 37

A 8. Distances with 100-foot Steel Tape 38

A 9. Horizontal Distance on Slope 38

AlO. Angles of Triangle with Tape 30

All. Survey of Field with Tape 30

A 13. Area by Perpendiciilar Method 33

A13. Area by Three-Side Method 33

2 ix

Pago

A14. Area by Angle Method 33

A15. Area from Plat 34

A16. Survey of Field with Curved Boundary. . 33

A17. Area of Field with Curved Boundary. ... 33

A18. Area (of same) from Plat 36

A19. Passing an Obstacle with Tape 36

A30. Obstructed Distance with Tape 38

A31. Running in Curve with Tape 38

A33. Discussion of Errors of Chaining 40

A33. Testing Standard of Length 40

A34. Constants of Steel Tape 43

A35. Making a Standard Wire Tape 43

A36. Comparison of Chains and Tapes 43

CHAPTER III.— THE COMPASS.

Types of Magnetic Compass 45

Declination of the Needle 46

Variation of the Declination 47

Local Attraction 48

The Vernier .' 49

Use of the Compass 49

Adjustments and Tests of Compass 50

Problem B 1. Declination of Needle 51

B 3. Angles of Triangle with Compass 52

B 3. Traverse of Field with Compass 53

B 4. Area of Field with Compass 54

B 5. Adjustment of Compass 56

B 6. Comparison of Compasses 56

CHAPTER IV.— THE LEVEL.

Types of Level ' 57

The Telescope 58

Line of Collimatioh 58

Objective 5S

Chromatic Aberration 58

Spherical Aberration 60

Eyepiece 60

Definition 61

Illumination 61

Aperture of Objective 61

Page

Size of Field 61

Mag-nifying Power 61

Parallax 61

Cross-Hairs 63

The Bubble Vial 63

Leveling Eods 64

Use of tte Level 65

Differential Leveling 66

Profile Leveling 68

Eeciprocal Leveling , 68

Contour Leveling 69

Cross-sectioning 69

Running Lines 70

Practical Hints 70

Problem C 1. Differential Leveling with Hand Level. . . 76

C 3. Differential Leveling, Engineers' Level.. 77

C 3. Profile Leveling for Drain 77

C 4. Eailroad Profile Leveling 81

C 5. Vertical Curve 83

C 6. Establishing Grade Line : 83

C 7. Setting Slope Stakes ; 85

C 8. Calculation of Quantities 85

C 9. Staking Out a Borrow Pit 85

CIO. Levels for Street Paving : 86

Cll. Coiitour Leveling 88

C13. Use of Contour Map 1 89

C13. Eeciprocal Leveling '. 89

C14. Delicacy of Bubble Vial ; 90

C15. Comparison of Level Telescopes 91

C16. Tests of Wye Level 91

C17. Adjustment of Wye Level 93

C18. Sketching Wye Level 93

C19. Tests of Dumpy Level 93

C20. Adjustment of Dumpy Level 93

C21. Sketching Dumpy Level 93

C33. Stretching Cross-Hairs 93

C33. Error of Setting Level Target 94

C34. Making a Leveling Rod 95

C35. Comparison of Engineers' Levels 95

Page
CHAPTER v.— THE TRANSIT.

Types of Transit 97

Use of the Transit 99

Prolongation of Lines 99

Horizontal Angles 100

Azimuth 100

Deflection 100

Vertical Angles 100

Traversing 1 00

Compass Bearings 101

Leveling with Transit 101

Problem D 1. Angles of Triangle with Transit 106

D 2. Prolongation of Line with Transit 106

D 3. Intersection of Lines with Transit 108

D 4. Referencing Out a Point 109

D 5. Triangulation Across River 110

D 6. Passing Obstacle with Transit 110

D 7. Traverse of Field with Transit 113

D 8. Area of Field with Transit 113

D 9. Staking Out Building 114

DIO. Height of Tower with Transit 114

Dll. Survey of Line Shafting 116

D12. Survey of Race Track 117

D13. Angles of Triangle by Repetition 118

D14. True Meridian by Polaris at Elongation 119
D15. True Meridian by Polaris at Any Time.. 121

D16. True Meridian by Solar Transit 127

D17. True Meridian by Direct Observation ... 131

D18. Comparison of Transit Telescopes 132

D19. Test of a Transit 132

D20. Adjustment of a Transit 133

D21. Sketching a Transit 133

D22. Error of Setting Flag Pole 134

D33. Comparison of Engineer's Transits 135

CHAPTER VI.— TOPOGRAPHIC SURVEYING.

Topography 137

The Plane Table 143

The Sextant 146

Problem E 1. Stadia Constants, with Fixed Hairs 148

E 2. Stadia Reduction Table 143

E 3. Azimutt Traverse witli Stadia 150

E 4. Plane Table Survey by Radiation 151

E 5. Plane Table Survey of Traversing 152

E 6. Plane Table Survey of Intersection 152

E 7. Three Point Problem with Plane Table 153

E 8. Angles of Triangle with Sextant 153

E 9. Coefficients of Standard Tape 153

ElO. Measurement of Base Line 155

Ell. Calculation of Triangulation System . . . 156

E13. Sketching Topography 156

E13. Topography with Transit and Stadia . . . 157

E14. Topography with Plane Table and Stadia 159

E15. Topographic Survey 159

CHAPTER VII.— LAND SURVEYING.

Functions of a Surveyor 161

United States Rectangular System 163

Surveys by Metes and Bounds 173

Problem F 1. Investigation of Land Corner 173

F 3. Perpetuation of Land Corner 174

F 3. Reestablishing Quarter-Section Corner.. 175

F 4. Reestablishing Section Corner 176

F 5. Resurvey of Section 176

F 6. Resurvey of City Block 179

F 7. Resurvey by Metes and Bounds 179

F 8. Partition of Land 180

F 9. Design and Survey of Town Site 180

Organization 183

Transit Party 184

Level Party 191

Topography Party 194

Office Work 197

Cross-Sectioning Party 203

Land-Line Party 307

Bridge and Masonry Party 207

Resurvey Party 209

Problem G 1. Review of Instrumental Adjustments ... . 208

G 2. Use of Field Equipment 209

Page

G 3. Preliminary Field Curve Practice 310

G 4. Indoor Curve Problems 310

CHAPTER IX.— EEEOES OF SURVEYING.

Probable Error 811

Tests of Precision 315

Linear Errors 315

Angular Errors 316

Traverse Errors 316

Leveling Errors 333

CHAPTER X.— METHODS OF COMPUTING.

Consistent Accuracy 333

Logarithmic Calculations 234

Arithmetical Calculations 335

Reckoning Tables 334

Computing Machines 334

CHAPTER XI.— TOPOGRAPHIC DRAWING AND FREE-
HAND LETTERING.

Practice Plates 237

Freehand Titles 245

Topographic Symbols 349

FIELD AND OFFICE TABLES.

Table 1. Logarithms of Numbers 254

Table 2. Logarithmic Functions of Angles 277

Table 3. Natural Functions of Angles 323

Table 4. Squares, Cubes and Roots 367

Table 5. Trigonometric Functions 380

Explanation of Tables 382

Index 385

SPECIFICATIONS FOR A GOOD ENGINEER.

" A good engineer must be of inflexible integrity, sober,
truthful, accurate, resolute, discreet, of cool and sound
judgment, must have command of his temper, must have
courage to resist and repel attempts at intimidation, a firm-
ness that is proof against solicitation, flattery or improper
bias of any kind, must take an interest in his work, must
be energetic, quick to decide, prompt to act, must be fair
and impartial as a judge on the bench, must have experi-
ence in his work and in dealing with men, vsrhich implies
some maturity of years, must have business habits and
knowledge of accounts. Men who combine these qualities
are not to be picked up every day. Still they can be found.
But they are greatly in demand, and when found, they are
worth their price ; rather they are beyond price, and their
value can not be estimated by dollars." — Chief Engineer
Starling's Report to the Mississippi Levee Commissioners.

" Be sure you are right, and then go ahead." — D. Crockett.

CHAPTER I.
GENERAL INSTRUCTIONS.

FIELD WORK.

Habitual Correctness. — Habitual correctness is a duty.
Error should be looked upon as probable, and every precau-
tion taken to verify data and results. Unchecked work may
always be regarded as doubtful. A discrepancy which is
found by the maker in time to be corrected by him before
any damage is done is not necessarily discreditable, pro-
vided the error is not repeated. However, habitual error
is not only discreditable but dishonorable as well, and noth-
ing except intentional dishonesty injures the reputation of
the engineer more quickly or permanently.

Consistent Accuracy. — The degree of precision sought
in the field measurements should be governed strictly by the
dictates of common sense and experience. Due considera-
tion of the purposes of the survey and of the time available
will enable one to avoid extreme precision when ordinary
care would sufBce, or crudeness when exactness is required,
or inconsistency between the degrees of precision observed
in the several parts of the survey. It is a very common
practice of beginners, and of many experienced engineers
as well, to carry calculated results far beyond the consistent
exactness.

Speed. — Cultivate the habit of doing the field work
quickly as well as accurately. True skill involves both
quantity and quality of results. However, while the habit
of rapid work can and should be acquired, the speed at-
tempted in any given problem should never be such as to
cast doubt upon the results. Slowness due to laziness is
intolerable.

Eamiliarity with Instructions. — The instructions for
the day's work should be read over carefully, and prelim-
inary steps, such as the preparation of field note forms,
should be taken so as to save time and make the work in

2 GENERAL INSTEUCTIONS.

the field as effective as possible The ability and also the
desire to understand and obey instructions are as essential
as the skill to execute them.

Inferior Instruments. — Should a poor instrument or
other equipment be assigned, a special eilort should be made
to secure excellent results. In actual practice, beginners
often have to work vifith defective instruments, but they
should never seek, nor are they permitted, to justify poor
results by the character of the field equipment. The stu-
dent should therefore welcome an occasional opportunity to
secure practice with poor instruments.

Alternation of Duties. — The members of each party
should alternate in discharging the several kinds of service
involved in the field problems, unless otherwise instructed.
Training in the subordinate positions is essential whether
the beginner is to occupy them in actual practice or not,
for intelligent direction of work demands thorough knowl-
edge of all its details.

Field Practice Decorum. — The decorum of surveying
field practice should conform reasonably to that observed
in other laboratory work.

THE CAEE OF FIELD EQUIPMENT.

RESPONSIBILITY. — The student is responsible for the
proper use and safe return of all equipment. All cases of
breakage, damage, loss or misplacement must be reported
promptly. The equipment should be examined when as-
signed and a report made at once of any injury or de-
ficiency found, so that responsibility may be properly
fixed.

PRECAUTION'S.— Careful attention to the following
practical suggestions will save needless wear to the equip-
ment and reduce the danger of accidents to a minimum,
besides adding to the quality and speed of the work.

Tripod. — Inspect the tripod legs and shoes. The leg is
of the proper tightness if, when lifted to an elevated posi-
tion, it sinks gradually of its own weight. The tripod
shoes should be tight and have reasonably sharp points.

Setting' Up Indoors. — In setting up the instrument in-
doors press the tripod shoes firmly into the fioor, prefer-
ably with each point in a crack. Avoid disturbing other
instruments in the room.

Instrument Case. — Handle the instrument gently in re-
moving it from and returning it to the case. It is always

THE CAKE OF FIELD EQUIPMENT. 3

best to place the hands beneath the leveling base in hand-
ling the detached instrument. Considerable patience is
sometimes required to close the lid after returning the in-
strument ; if properly placed the lid closes freely.

Mounting the Instrument. — See that the instrument
is securely attached to the tripod before shouldering it.
Undue haste in this particular sometimes results in costly
accidents. When screwing the instrument on the tripod
head, it should be turned in a reverse direction until a slight
jar is felt, indicating that the threads are properly engaged.

the kind of weather. The sunshade is a part of the telescope
tube and the adjustment of a delicate instrument naay
sometimes be affected by its absence. In attaching or re-
moving the sunshade or object glass cap, always hold the
telescope tube firmly with one hand and with the other
twist the shade or cap to the right to avoid unscrewing the
object glass cell.

Carrying the Instrument. — Do not carry the instru-
ment on the shoulder in passing through doors or in climb-
ing fences. Before shouldering the instrument, the prin-
cipal motions should be slightly clamped ; with the transit,
clamp the telescope on the line of centers ; and with the
level, when the telescope is hanging down. In passing
through timber with low branches, give special attention
to the instrument. Before climbing a fence, set the instru-
ment on the opposite side with tripod legs well spread.

Setting XTp in the Field. — When setting up in the field,
bring the tripod legs to a firm bearing with the plates ap-
in windy vsreather or in places where the instrument may
be subject to vibration or other disturbance. On side-hill
work place one leg up hill. With the level, place two
tripod shoes in the general direction of the line of levels.

Exposure of Instrument. — Do not expose the instru-
ment to rain or dampness. In threatening weather the
water proof bag should be taken to the field. Should the
instrument get wet, wipe it thoroughly dry before return-
ing it to the case. Protect the instrument from dust and
dirt, and avoid undue exposure to the burning action of the
sun. Avoid subjecting it to sudden changes of tempera-
ture. In cold weather when bringing an instrument in-
doors cover the instrument with the bag or return it to
the case immediately to protect the lenses and graduations
from condensed moisture.

4 GENERAL INSTRUCTIONS.

Guarding tte Instrument. — ^Never leave an instrument
unguarded in exposed situations such as in pastures, near
driveways, or where blasting is in progress. Never leave
an instrument standing on its tripod over night in a room.

Manipulation of Instrument. — Cultivate from the very
beginning the habit of delicate manipulation of the instru-
ment. Many parts, when once impaired, can never be re-
stored to their original condition. Rough and careless
treatment of field instruments is characteristic of the un-
skilled observer. Should any screw or other part of the in-
strument work harshly, call immediate attention to it so
that repairs may be made. Delay in such matters is very
destructive to the instrument.

Foot Screws. — In leveling the instrument, the foot screws
should be brought just to a snug bearing. If the screws are
too loose, the instrument rocks, and accurate work can not
be done ; if too tight, the instrument is damaged, and the
delicacy and accuracy of the observations are reduced. Much
needless wear of the foot screws may be avoided if the
plates are brought about level when the instrument is set
up. With the level, a pair of foot screws should be shifted
to the general direction of the back or fore sight before
leveling up.

Eyepiece. — Before beginning the observations, focus the
eyepiece perfectly on the cross-hairs. This is best done by
holding the note book page, handkerchief, or other white
object a foot or so in front of the object glass so as to illum-
inate the hairs ; and then, by means of the eyepiece slide,
focus the microscope on a speck of dust on the cross-hairs
near the middle of the field. To have the focusing true for
natural vision, the eye should be momentarily closed sev-
eral times between observations in order to allow the
lenses of the eye to assume their normal condition. The
omission of this precaution strains the eye and is quite cer-
tain to cause parallax. After the eyepiece is focused on the
cross-hairs, test for parallax by sighting at a well defined
object and observing whether the cross-hairs seem to
move as the eye is shifted slightly.

Clamps. — Do not overstrain the clamps. In a well de-
signed instrument the ears of the clamp screw are purpose-
ly made small to prevent such abuse. Find by experiment
just how tight to clamp the instrument in order to prevent
slipping, and then clamp accordingly.

Tangent Screws. — Use the tangent screws for slight
motions only. To secure even wear the screws should

THE CAKE OF FIELD EQUIPMENT. 5

be used equally in all parts of their length. The use of the
wrong tangent movement is a fruitful source of error with
beginners.

Adjusting Scre'ws. — Unless the instrument is assigned
screws.

XEagnetic Keedle. — Always lift the needle before should-
ering the instrument. Do not permit tampering with the
needle. If possible, avoid subjecting the needle to mag-
netic influence, such as may exist on a trolley car. Should
the needle become reversed in its polarity or require re-
magnetization, it may be removed from the instrument and
brought into the magnetic field of a dynamo or electric
motor for several minutes, the needle being jarred slightly
during the exposure; or a good horseshoe magnet may
be used for the same purpose. The wire coil counterbalance
on the needle will usually require shifting after the fore-
going process.

Lenses. — Do not remove or rub the lenses of the tele-
scope. Should it be absolutely necessary to clean a lens, use
a very soft rag with caution to avoid scratching or marring
the polished surface. Protect the lenses from flying sand
and dust, which in time seriously affect the definition of
the telescope.

Plumb Bob. — Do not abuse the point of the plumb bob
and avoid needless knots in the plumb bob string.

Cleaning Tripod Shoes. — Eemove the surplus soil from
the tripod shoes before bringing the instrument indoors.

Leveling Rods. — Leveling rods and stadia boards should
not be leaned against trees or placed where they may fall.
Avoid injury to the clamps, target and graduations. Do not
mark the graduations with pencil or otherwise. Avoid
needless exposure of the rod to moisture or to the sun.

Flag Poles. — Flag poles should not be unduly strained
and their points should be properly protected.

Chains and Tapes. — Chains should not be jerked. Avoid
kinks in steel tapes, especially during cool weather. When
near driveways, in crowded streets, etc., use special care to
protect the tape. Band tapes will be done up in 5-foot
loops, figure 8 form, unless reels are provided. Etched tapes
should be wiped clean and dry at the end of the day's work.

Axes and Hatchets. — Axes and hatchets will be em-
ployed for their legitimate purposes only. Their wanton
use in clearing survey lines is forbidden, and their use at all,

6 GENERAL INSTEUCTIONS.

for such purpose, on private premises must be governed
strictly by the rights of the owner.

Stakes. — The consumption of stakes should be controlled
by reasonable economy, and surplus stakes returned to
the general store. For the protection of mowing machines
in meadows, etc., hub stakes should be driven flush with
the surface of the ground, and other stakes should be left
high enough to be visible. Whenever practicable, stakes
which may endanger machines should be removed after
serving the purpose for which they were set.

FIELD NOTES.

Scope of Field Notes. — The notes should be a complete
record of each day's work in the field. In addition to the
title of the problem and the record of the data observed,
the field notes should include the date, weather, organiza-
tion of party, equipment used, time devoted to the prob-
lem, and any other information which is at all likely to be
of service in connection with the problem. No item prop-
erly belonging to the notes should be trusted to memory.
Should the question arise as to the desirability of any item,
it is always safe to include it. The habit of rigid self criti-
cism of the field notes should be cultivated.

Character of Notes. — The field notes should have char-
acter and force. As a rule, the general character of the
student's work can be judged with considerable certainty
by the appearance of his field notes. A first-class page of
field notes always commands respect, and tends to estab-
lish and stimulate confidence in the recorder. The notes
should be arranged systematically.

Interpretation of Notes. — The field notes should have
one and only one reasonable interpretation, and that the
correct one. They should be perfectly legible and easily
understood by anyone at all familiar with such matters.

Original Notes. — Each student must keep complete notes
of each problem. Field notes must not be taken on loose
slips or sheets of paper or in other note books, but the
original record must be put in the prescribed field note
book during the progress of the field work.

Field Note Book. — The field record raust be kept in the
prescribed field note book. For ease of identification the
name of the owner will be printed in bold letters at the
top of the front cover of the field note book.

FIELD NOTES. 7

Pencil. — To insure permanency all notes will be kept
with a hard pencil, preferably a 4H. The pencil should be
kept well sharpened and used with sufficient pressure to
indent the surface of the paper somewhat.

Title Page. — ^An appropriate title page will be printed
on the iirst page of the field note book.

Indexing and Cross Referencing. — A systematic index
of the field notes will be kept on the four pages following
the title page. Eelated notes on different pages will be lib-
erally and plainly cross referenced. The pages of the note
book will be numbered to facilitate indexing.

Methods of Recording Field Notes. — There are three
general methods of recording field notes, namely : ( 1 ) by
sketch, (2) by description or narration, and 1[3) by tabula-
tion. It is not uncommon to combine two or perhaps all
three of these methods in the same problem or svirvey.

Porm of notes. — All field notes must be recorded in a
field note book ruled as shown below, except where cir-
cumstances require modification. If no form is given, the
student will devise one suited to the particular problem.

Lettering. — Field notes will be printed habitually in the
" Engineering News " style of freehand lettering, as treated
in Eeinhardt's " Freehand Lettering." The body of the field
notes will be recorded in the slanting letter and the head-
ings will be made in the upright letter. The former slants
to the right 1 : 2.5 and the so-called upright letter is made
to slant to the left slightly, say 1 : 25. Lower case letters
will be used in general, capitals being employed for initials
and important words, as required. In the standard field
note alphabet the height of lower case letters a, c, e, i, m,
n, etc., is %o ™ch, and the height of lower case b, d, f,
g, h, etc., and of all capital letters and all numerals is
I^Q (1^) inch; lower case t is made four units (%o) inch
high. This standard accords with best current practice and
is based upon correct economic principles. Sample pages
of field notes with letters and figures drawn full size are
' given on page 9. The student is expected to make the most
of this opportunity to secure a liberal amount of practice
in freehand lettering.

Field Note Sketches. — Sketches will be used liberally
in the notes and will be made in the flcU. If desired, a ruler
may be used in drawing straight lines, but the student is
urged to acquire skill at once in making good plain free-
hand sketches. The field sketches should be bold and clear,
in fair proportion, and of liberal size so as to avoid con-

8 GENERAL INSTKUCTIONS.

fusion of detail. The exaggeration of certain details in a
separate sketch sometimes adds greatly to the clearness of
the notes. The sketches should be supplemented by de-
scriptive statements when helpful, and important points of
the sketch should be lettered for reference. The precise
scaling of sketches in the field note book, while sometimes
necessary is usually undesirable owing to the time con-
sumed. It is also found that undue attention to the draft-
ing of the sketch is very apt to occupy the mind and cause

/•

>!

V

J

omissions of important numerical data. Since recorded
figures and not the size of the field sketch itself must usual-
ly be employed in the subsequent use of the notes, it is im-
portant to review the record 'before leaving the field to detect
omissions or inconsistencies. Making sketches on loose
sheets or in other books and subsequently copying them
into the regular field book is very objectionable practice
and will not be permitted in the class work. Copies of field
notes or sketches are never as trustworthy as the original
record made (luring the progress of the field work. In very
rapid surveys where legibility of the original record must
perhaps suffer somewhat, it is excellent practice to tran-
scribe the notes at once to a neighboring page, thus pre-
serving the original rough notes for future reference. The
original has more weight as evidence, but the neat copy

FIELD NOTES.

Station Value oF Anqle

Amgles or Triangle 5-6-7

^ndlieas.
88''5I'
4.7°4.7'

Mean
88°50'50"
47°47W'
43'r3W
m'00'30"
(D/'FFerence in measurements not to exceed /')

Left Hand Paqe.

Observers, J. Doe & R. l?oe.

With Engineers' TransIt.

lioy./5J9l4, (2 hours). Warm and quiet
Used He/lar& Brightly TrJnsit lioJO.

Riqht Hand Vac\e.

10 GENERAL INSTRUCTIONS.

made before the notes are " cold " is of great help in inter-
preting them.

Numerical Data. — The record of numerical data should
be consistent with the precision of the survey. In obser-
vations of the same class a uniform number of decimal
places should be recorded. When the fraction in a result
is exactly one-half the smallest unit or decimal place to be
observed, record the even unit. Careful attention should
be given to the IcgihUity of numerals. This is a matter in
which the beginner is often very weak. This defect can be
corrected best by giving studious attention and practice to
both the form and vertical alinement of tabulated numerals.

Erasures. — Erasures in the field notes should be avoided.
In case a figure is incorrectly recorded, it should be
crossed out and the correct entry made near by. The neat
cancellation of an item in the notes inspires confidence,
but evidence of an erasure or alteration easts doubt
upon their genuineness. When a set of notes becomes so
confused that erasure seems desirable, it should be tran-
scribed, usually on another page. Rejection of a page of
notes should be indicated by a neat cross mark, and cross
reference should be made between the two places.

Office Copies. — Office copies of field notes will be sub-
mitted promptly, as required. These copies must be actiial
transcripts from the original record contained in the field
note book of the individual submitting the copy. When
office copies are made, a memorandum of the fact should
be entered on the page of the field note book. When so
specified, the office copies will be executed in India ink.

Criticism of Field Notes. — The field notes must be kept
in shape for inspection at any time, and be submitted on
call. All calculations and reductions must be kept up to
date. The points to which chief attention should be di-
rected in the criticism of the field notes are indicated in the
following schedule. The student is expected to criticise his
own notes and submit them ,in as perfect condition as pos-
sible. For simplicity the criticisms will be indicated by
stamping on the note book page the reference letters and
numbers shown in the schedule.

SCHEDULE OF POINTS. 11

SCHEDULE OF POINTS FOR THE CRITICISM OF
FIELD NOTE BOOKS.

A. SUBJECT MATTER.

(1) General:

(a) Descriptive title of problem.

(b) Date.

(c) Weather.

(d) Organization of party.

(e) Equipment used.

(f ) Time devoted to the problem.

(g) Indexing and cro.ss referencing.

(h) Page numbering.

( i ) Title page.

(j) Identification of field note book.

(2) Becord of Data:

(a) Accuracy.

(b) Completeness.

(c) Consistency.

( d) Arrangement.

(e) Originality.

B. EXECUTIDIT.

(1) Lettering:

(a) Style. ("Engineering News")

(b) Size, (a, c, e, i, etc., %o in<^'^ high; b, d, f, g, etc.,
A, B, C, etc., and 1, 2, 3, etc., %o (%) "ich high; t, %o
inch.)

(c) Slant. (In body of notes, "slanting," 1:2.5 right;

(d) Form. ( See Reinhardt's " Freehand Lettering." )

(e) Spacing. (Of letters in words; of numerals; of
words; balancing in column or across page.)

(f) Alinement. (Horizontal ; vertical.)

(g) Permanency. (Use sharp hard pencil with pressure.)

(2) Sketches.

"(a) To be bold, clear and neat.

(b) To be ample in amount.

(c) To be of liberal size.

(d) To be in fair proportion.

( f ) To be made in the field.

12 QENERAIi INSTRUCTIONS.

OFFICE WORK.

Importance of Office Work. — Capable office men are
comparatively rare. Skill in drafting and computing is
within the reach of most men who will devote proper time
and effort to the work. Men who are skillful in both field
and office work have the largest opportunity for advance-
ment.

Calculations. — All calculations and reductions of a per-
manent character must be shown in the field note book in
the specified form. Cross references between field data and
calculations should be shown. Consistency between the
precision of computed results and that of the observed data
should be maintained. Computed results should be verified
habitually, and the verified results indicated by a check
mark. Since most computers are prone to repeat the same
error, it is desirable in checking calculations to employ in-
dependent methods and to follow a different order. A
fruitful source of trouble is in the transcript of data, and
this should be checked first when reviewing doubtful cal-
culations. Skilled computers give much attention to
methodical arrangement, and to contracted methods of
computing and verifying results. Familiarity with the
slide rule and other labor saving devices is important.
(See Chapter X, Methods of Computing.)

Drafting Boom Equipment. — The student is respon-
sible for the proper use and care of drafting room furni-
ture and equipment provided for his use.

Drafting. — The standard of drafting is that indicated in
Reinhardt's " Technic of Mechanical Drafting."

Drafting Boom Decorum. — The decorum of the student
in the drafting room will conform to that observed in first-
class city drafting offices.

CHAPTER II.
THE CHAIN AND TAPE.

METHODS OF FIELD WORK.

Units of IVEeasure. — In the United States the foot is used
by civil engineers in field measurements. Fractions of a
foot are expressed decimally, the nearest 0.1 being taken
in ordinary surveys, and the nearest 0.01 foot (say y^
inch) in more refined work.

In railroad and similar " line " surveys by which a station
stake is set every 100 feet, the unit of measure is really 100
feet instead of the foot. The term " station " was originally
applied only to the actual point indicated by the numbered
stake, but it is now universal practice in this country to
use the word station in referring to either the point or the
100-foot unit distance. A fractional station is called a
" plus " for the reason that a plus sign is used to mark the
decimal point for the 100-foot unit, the common decimal
point being reserved for fractions of a foot. The initial or
starting stake of such a survey is numbered 0.

The 100-foot chain is commonly called the " engineers'
chain " to distinguish it from the 66-foot or lOO-link chain
which is termed the " surveyors' chain " because of its
special value in land surveys involving acreage. The latter
is also called the Gunter chain after its inventor, and is
otherwise known as the four-rod or four-pole chain.
British engineers use the Gunter chain for both line and
land surveys. The " surveyors' " or Gunter chain, while
no longer used in actual surveying, is described in this book
for the reason that the United States rectangular surveys
were made throughout with the 66-foot chain.

In the Spanish-American countries the vara is generally
used in land surveys. The Castilian vara is 32.8748 inches
long, but the state of California has adopted 32.372 inches,
and Texas 331^ inches, as the legal length of the vara.

While the metric system is used exclusively, or in part, in
13

14

THE CHAIN AND TAPE.

each of the several United States government surveys, ex-
cept those for public lands, little or no progress has been
made towards its introduction in other than government
surveys.

Linear Measuring Instruments. — Two general types of
linear measuring devices are used by surveyors, viz., the
common chain and the tape. There are several kinds of
each, according to the length, material, and method of

Fig. 1.

The common chain is made up of a series of links of
wire having loops at the ends and connected by rings so as
to afford flexibility. The engineers' chain is shown in (a),
Fig. 1, the illustration being that of a 50-foot chain, or one-

METHODS OF FIELD WOEK. 15

half the length generally used. The surveyors' or Gunter
chain is shown in (b), Fig. 1. In the common chain the
end graduation is the center of the cross bar of the handle,
and every tenth foot or link is marked by a notched brass
tag. In the 100-foot or 100-link chain the number of points
on the tag indicates the multiple of ten units from the nearer
end, and a circular tag marks the middle of the chain.
The chain is done up hour-glass shape, as shown in the cut.

Chaining pins made of steel wire are used in marking the
end of the chain or tape in the usual process of linear
measurement. A set of pins usually numbers eleven, as
indicated at (c). Fig. 1. The pins are carried on a ring

The flat steel band, shown in (d) and (e), Fig. 1, is the
best form of measuring device for most kinds of work. The
band tape is usually 100 feet long. The end graduations of
the band tape are usually indicated by brass shoiilders,
w^hich "should point in the same direction, as shown in (f),
Fig. 1. The 100-foot band tape is commonly graduated
every foot of its length, and the end foot to every 0.1 foot,
every fifth foot being numbered on a brass sleeve. Brass
rivets are most commonly used in graduating this tape.
The band tape may be rolled up on a special reel, as indi-
cated in (d) and (e), although some engineers dispejise
with the reel and do up the tape in the form of the figure 8
in loops of five feet or so.

The steel tapes shown in (g) and (h) have etched gradu-
ations. This style of tape is commonly graduated to 0.01
foot or yg inch. It is more fragile than the band tape and
is commonly used on inore refined work. The form of the
case shown in (h) has the advantage of allowing the tape
to dry if wound up while damp.

The " metallic " tape (i) , Fig. 1, is a woven linen line hav-
ing fine brass wire in the warp.

The steel tape is superior to the common chain chiefiy
because of the permanency of its length. The smoothness
and lightness of the steel tape are often important advan-
tages, although the latter feature may be a serious draw-
back at times. The tape is both easier to break and more
difiicult to mend than the common chain.

Tapes for measuring base lines with great precision have
recently been made of Invar steel. Invar steel has a very
small coefBcient of expansion. Invar steel tapes are very
expensive.

16 THE CHAIN AND TAPE.

Chaining. — In general, the horizontal distance is chained.
Two persons, called head and rear chainmen, are required.
The usual process is as follows :

The line to be chained is first marked with range poles.
The head chainman casts the chain out to the rear, and
after setting one marking pin at the starting point and
checking up the remaining ten pins on his ring, steps
briskly to the front. The rear chainman allows the chain
to pass through his hands to detect kinks and bent links.
Just before the full length is drawn out, the rear chainman
calls " halt," at which the head chainman turns, shakes out
the chain and straiglitens It on the true line under the
direction of the rear chainman. In order to allow a clear
sight ahead, the front chainman should hold the chain
handle with a pin in his right hand well away from his
body, supporting the right elbow^ on the right knee, if de-
sired. The rear chainman holds the handle in his left hand
approximately at the starting point and motions with his
right to the head chainman, his signals being distinct both
as to direction and amount. Finally, when the straight
and taut chain has been brought practically into the true
line, the rear chainman, slipping the handle behind the pin
at the starting point with his left hand, and steadying the
top of the pin with his right, calls out " stick." The head
chainman at this instant sets his pin in front of the chain
handle and responds " stuck," at which signal and not before
the rear chainman pulls the pin.

Both now proceed, the rear chainman giving the prelim-
inary " halt " signal as he approaches the pin just set by
the head chainman. The chain is lined up, stretched, the
front pin set, and the rear pin pulled on signal, as described
for the first chain length. This process is repeated until
the head chainman has set his tenth pin, when he calls
" out " or " tally," at which the rear chainman walks ahead,
counting his ping as he goes and, if there are ten, transfers
them to the head chainman who also checks them up and
replaces them on his ring. A similar check in the pins may
be made at any time by remembering that the sum, omit-
ting the one in the ground, should be ten. This safeguard
should be taken often to detect loss of pins. The count of
tallies should be carefully kept.

When the end of the line is reached, the rear chainman
units with respect to the brass tags on the chain. The
number of pins in the hand of the rear chainman indicates

METHODS OF FIELD WORK. 17

the number of applications of the chain since the starting
or last tally point. A like method is used in case inter-
mediate points are to be noted along the line.

On sloping ground the horizontal distance may be ob-
tained either by leveling the chain and plumbing down
from the elevated end, or by measuring on the slope and
correcting for the inclination. In ordinary work the for-
mer is preferred, owing to its simplicity. In " breaking
chain " up or down a steep slope, the head chainman first
carries the full chain ahead and places it carefully on the
true line. A plumb bob, range pole or loaded chaining pin
should be used in plumbing the points up or down. The
segments of the chain should be in multiples of ten units,
as a rule, and the breaking points should be " thumbed "
by both chainmen to avoid blunders. Likewise, special cau-
tion is required to avoid confusion in the count of pins dur-
ing this process.

The general method of measuring with the band tape is
much the same as with the common chain. The chief dif-
ference is due to the fact that the handle of the tape extends
beyond the end graduation, so that it is more convenient
for the head chainman to hold the handle in his left hand
and rest his left elbowr on his left knee, setting the pin with
his right hand. Another difEerence is in the method of
estimation, as with the chain, making sure of the feet; then
shifting the tape along one foot, getting. an exact decimal
record of the fraction by means of the end foot graduated
to tenths ; the nearest 0.01 foot is estimated, or in especially

In railroad and similar line surveys, chaining pins are
usually dispensed with and the ends of the chain are indi-
cated by numbered stakes. The stake marked corre-
sponds to the pin at the starting point, and the station
stakes are marked thence according to the number of
100-foot units laid off.

Perpendiculars. — Perpendiculars may be erected and let
fall with the chain or tape by the following methods :

(a) By the 3:4:5 method, shown in (a). Pig. 2, in which
a triangle having sides in the ratio stated, is constructed.

(b) By the chord bisection method, shown in (b), Fig. 3,
in which a line is passed from the bisecting point of the
chord to the center of the circle, or vice versa.

3

18

THE CHAIN AND TAPE.

(c) By the semicircle method, shown in (c). Fig. 2, in
which a semicircle is made to contain the required perpen-
dicular.

The first method corresponds to the use of the triangle
in drafting. Good intersections are essential in the second
and third methods. Eesults may be verified either by using
another process, or by repeating the same method with the
measurements or position reversed, as indicated in (d),
rig. 2.

(^)

^.^5 3

4-

(b)

— ^. —

!d)

\/

(e) ,<

(A

(b)

Cc)

/

t
\ /

V

(d)

le)\

(s) </^

Fig. 2

Fig. 3.

Fig. 4.

In locating a perpendicular from a remote point, the
ratio method shown in (e). Fig. 2, may be used; or a care-
ful trial perpendicular may be erected at a point estimated
by placing the heels squarely on line and swinging the
arms to the front, then proving by precise method.

Parallels. — Parallels may be laid off with the chain in
various ways, a few of the simpler of which are :

(a) By equal distances, as in (a). Fig. 3, in which two
equal distances are laid off, usually at right angles to the
given line.

METHODS OP FIELD WOEK. 19

(b) By similar triangles, as in (b) and (o), Fig. 3. The
ratio may, of course, have any value.

(c) By alternate angles, as in (d). Fig. 3, in which tvi^o
equal angles are laid off in alternation.

The first method is adapted to laying ofE a rectangle, as
in staking out a building, in which case a good check is
found in the equality of the diagonals. Precision of aline-
ment is important, especially where a line is prolonged.

Angles. — Angles may be determined by linear measure-
ments in the following ways :

(a) By the chord method, shown in (a). Fig. 4, in which
the radius is laid off on the two lines forming the angle,
and the chord measured.

(b) The tangent method, shown in (b), Fig. 4, in which
a perpendicular is erected at one end of the radius, and the
length of the perpendicular intercepted by the two lines
measured.

(c) The sine-cosine method, (c), Fig. 4, which is better
suited to constructing than to measuring angles.

The chord method is usually the most satisfactory. The
tangent method may be applied to the bisected angle when
its value approaches a right angle. Measurement of the
supplementary angle affords an excellent check. A 100-foot
with the 50-foot tape. Careful alinement is of the first im-
portance in angular measurements.

It is sometimes necessary to determine angles, at least
approximately, when no tables are at hand. Fair results
may be had on smooth ground by measuring the actual arc
struck off to a radius of 57.3 feet.

For very small angles, the sine, chord, arc and tangent,
(d). Fig. 4, are practically equal. Thus, sin 1° is .017452
and tan 1°, .017455, or either (say) .01745, or 1% per cent.
Also, arc 1' is .000291, or (say) .0003 (three zeros three) ;
and, arc 1" is .00000485, (say) .000005 (five zeros five).

Location of Points. — Points are located in surveying
field practice in the following seven ways.

(a) By rectangular coordinates, that is, by measuring
the perpendicular distance from the required point to a
given line, and the distance thence along the line to a
given point, as in (a). Fig. 5.

(b) By focal coordinates or tie lines, that is, by meas-
uring the distances from the required point to two given
points, as in (b), Fig. 5.

20

THE CHAIN AND TAPE.

(c) By polar coordinates, that is, by measuring the angle
between a given line and a line drawn from any given point
of it to the required point ; and also the length of this
latter line, as in (c). Fig. 5.

(d) By modified polar coordinates, that is, by a distance
from one known point and a direction from another, as in
(d),Fig. 5.

(e) By angular intersection, that is, by measuring the
angles made with a given line by two other lines starting
from given points upon it, and passing through the re-
quired point, as in (e). Fig. 5.

(f) By resection, that is, by measuring the angles made
with each other by three lines of sight passing from the
required point to three points, whose positions are known,
as in (f). Fig. 5.

is) By diagonal intersection, that is, by two lines joining
two pairs of points so as to intersect in the required point,
as in (g). Fig. 5.

</i (g) \ii

In each of these methods, except (f), the point is deter-
mined by the intersection of either two right lines, or two
circles, or a right line and a circle.

Methods (a) and (b) are best suited to chain surveys;
(c) and (d) are used most in the location of railroad
curves; (e) and (f) are employed chiefly in river and ma-
rine surveys for the location of soundings, the latter being
commonly known as the "three-point problem"; the last
method, (g), is much used for "referencing out" transit
points in railroad and similar construction surveys.

Location of Objects. — The location of buildings and
topographic objects usually involves one or more of the
foregoing methods of locating a point.

METHODS OF FIELD WORK.

21

In Fig-. 6, (a), (b), (c), and (d) suggest methods of lo-
cating a simple form, and (e) and (f) illustrate more com-
plex cases.

Tie liine Surveys. — For many purposes tie line surveys,
made with the chaiin or tajDe alone, are very satisfactory.
The skeleton of snch surveys is usually the triangle, the
detail being filled in by the methods just outlined. Much
time may be saved by carefully planning the survey. A
few typical applications of the tie line method are shown
in Fig. 7.

iif;

rh)

/ \/ \< 'v /^erm with Streams

Fig. 6.

Crooked River-

Fig. 7.

JUUL

Ranging in Lines. — The range or flag pole is usually
painted with alternate feet red and white, and the lower
end is shod or spiked. A temporary form of range pole,
called a picket, is sometimes cut from a straight sapling.

In flagging a point, the spike of the pole is placed on the
tack and the pole plumbed by holding it symmetrically be-
tween the tips of the fingers of the two hands, the flagman
being squarely behind the pole.

In hilly or timbered country the two land corners or other
points between which it is desired to range in a line, are
often invisible one from the other. In many cases two in-
termediate points C and D', (a), Fig. 8, may be found, from

22

THE CHAIN AND TAPE.

which the end points B and A, respectively, are visible ; so
that after a few successive lining's in, each by the other,
the true points, C and D, are found.

Otherwise, as shown at (b), Fig-. 8, a random line may
be run from^ A towards B. The trial line is chained and
marked, the perpendicular from B located, and points in-
terpolated on the true line.

If the desired line is occupied by a hedge or other ob-
struction, an auxiliary parallel line may be established in
the adjacent road'or field, after one or two trials, as in (c),
Fig. 8.

A line may be prolonged past an obstacle by rectangular
offsets or by equilateral triangles.

CO B

(c)

' B-

— I-

-i,

3

Fig. 8.

V ^

station ■>
Stake

//ai w/t/>""' ""•
Guard Stake

Fig. 9.

Signals. — There is little occasion for shouting in survey-
ing field work if a proper system of sight signals is used.
Each signal should have but one meaning and that a per-
fectly distinct one. Signals indicating motion should at
once show clearly both the direction and amount of motion
desired. Some of the signals in common use are as follows :

(a) " Eight " or " left," — the arm is extended distinctly in
the desired direction and the motion of the forearm and
hand is graduated to suit the lateral motion required.

(b) " Up " or " down," — the arm is extended laterally and
raised or lowered distinctly with motions to suit the magni-
tude of the movement desired. Some levelers use the left
arm for the " up " signal and the right for " down."

(c) "Plumb the pole (or rod)," — If to the right, that
arm is held vertically with hand extended and the entire
body, arm included, is swung distinctly to the right, or
vice versa.

(d) "All right," — both arms are extended full length
horizontally and waved vertically.

METHODS or FIELD WORK. 23

(e) "Turning point" or "transit point," — the arm is

(f ) " Give line," — the flagman extends both arms upward,
holding the flag pole horizontally, ending with the pole in
its vertical position. If a. precise or tack point is meant,
the signal is made quicker and sharper.

(g) Numerals are usually made by counted vertical swings
with the arm extended laterally. A station number is
given with the right hand and the plus, if any, with the
left ; or a rod reading in like manner. The successive
counts are separated by a momentary pause, emphasized,
if desired, by a slight swing with both hands.

Stakes and Stake Driving. — ^A flat stake is used to
mark the stations in a line survey, and a square stake or
hub to mark transit stations, (a) and (b), Eig. 9. The
station stake is numbered on the rear face, and the hub is
witnessed by a flat guard stake driven slanting 10 inches
or so to the left, Eig. 9. The numerals should be bold and
distinct, and made with keel or waterproof crayon, pressed
into the surface of the wood.

Having located a point approximately vyith the flag pole,
the stake should be driven truly plumb in order that the
final point may fall near the center of its top. In driving
a stake, the axeman should watch for signals. It is better
to draw the stake by a slanting blow than to hammer the
stake over after it is driven. Good stake drivers are scarce.

PROBLEMS WITH THE CHAIN AND TAPE.

General Statement. — Each problem is stated under the

(a) Equipment. — In which are specified the articles and
instruments assigned or required for the proper perform-
ance of the problem. A copy of this manual and of the
regulation field note book, with a hard pencil to keep the
record, form part of the equipment for every problem as-
signed.

(b) Problem. — In which the problem is stated in general
terms. The special assignments will be made by program.

(c) Methods. — In which the methods to be used in the as-
signed work are described more or less in detail. In some
problems alternative methods are suggested, and in others
the student is left to devise his own.

24 THE CHAIN AND TAPE.

PKOBLEM Al. LENGTH OF PACE.

(a) Equipment. — (No instrumental equipment required.)

(b) Problem. — Investig-ate the length of pace as follows:
(1) the natural pace; (2) an assumed pace of 3 feet; and
(3) the effect of speed on the length of the pace.

(c) Methods. — (1) On an assigned course of known length
count the paces while walking at the natural rate. Observe
the nearest 0.1 pace in the fraction at the end of the course.
Secure ten consecutive results, with no rejections, varying
not more than 3 per cent. (3) Repeat (1) for an assumed
3-foot pace. (3) Observe (in duplicate) time and paces for
four or fi\e rates from very slow to very fast, with paces to
nearest 0.1 and time to neare.st second. Record data and
make reductions as in the form.

PROBLEM A3. DISTANCES BY PACING.

(a) Eqiiiiiiiieiit. — (No instrumental equipment required.)

(b) Problem. — Pace the assigned distances.

(c) Methods. — (1) Standardize the pace in duplicate on
measui'ed base. (3) I'ace each line in duplicate, results dif-
fering not more than 3 per cent. Record and reduce as in
form.

PROBLEM A3. AXEMAN AND FLAGMAN PRACTICE.

(a) Equipment. — Flag pole, axe, 4 flat stakes, 1 hub, tacks.

(b) Problem. — Practice the correct routine duties of axe-
man and flagman.

(c) Metliocls. — (1) Number three station stakes to indi-
cate representative cases and drive them properly. (2)
Drive a hub flush with ground and tack it ; number a wit-
ness stake and drive it properly. (3) Arrange program of
signals with partner, separate 1.000 feet or so and practice
same. (4) Signal say flve station numbers to each other
and afterwards compare notes. Make concise record of
the foregoing steps.

PROBLEM A4. RANGE POLE PRACTICE.

(a) Equipment. — 4 flag poles.

(b) Problem. — Given two hubs approximately 1,000 feet
apart, interpolate a flag pole say 100 feet from one hub.

PROBLEMS.

25

^

5spM3,^4,(S Jfrs-J CJeir and Cool ■

"^

lN\

ESTIS

/^TION

OF

LEN6TH OF Pace •

J-Doe, Surveyor- 1

Kind

Races p<

r 400 Ft.

Lengfh

Rem

;rks

EFFECT OF SPEED Or LEN6TH OF PACE- |

oF Pace

Mean

of Pace

Sepm.

■Clear

Kind Paces tn 400 Ff.

Mean

Time

Speed of

Hi

Paces

Paces

Ft.

Smooth

ground'

oF Pace

Obscrv'd Mean

Pace.P

40OFt

Pacing, 5

ttltwsH

J5gl>

mtirt.

eWfnd.

Paces

Paces

Ft-

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714-6

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131-0

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Speed oF Wal

ing, Ft- per Sec>;S* |

(lO-l)

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DiSTA

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BY ?A

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Line

Length

iF Pace

Len(

th oF

-ine

J- Doe, Survey&j

-

No

flirWOft

Ft.

Obarvtd

Mean

Length

Sepl-14,' 14 fulfil

Ts) Clear ^ Coo/'

Paces,

Paces

Ft-

/

i44-0

z

MZ-a
143-0

zsa

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134-0

1 \

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"^

11

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1340

134-0

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i

II

26 THE CHAIN AND TAPE.

remove the distant pole, prolong the line by successive 100-
foot sights and note the error at distant hub. Bepeat
process for 200-foot and 300-foot sights.

(c) Mctlwds. — (1) Set distant flag pole precisely behind
hub and hold spike of pole on tack of near hub ; lying on
ground back of near hub, line in pole 100 feet (paced) dis-
tant ; remove pole from distant hub, and prolong by 100-
foot sights up to distant hub, noting error to nearest 0.01
foot. (2) Eepeat in reverse direction, using 200-foot sights.
(3) Eepeat with 300-foot sights. Avoid all bias. Record
data in suitable form, describing steps concisely.

PROBLEM A5. STANDARDIZING CHAIN OR TAPE.

(a) Equipment. — Chain or tape assigned in any problem
where standard length of chain may be of value.

(b) ProMem. — Determine the length of the assigned
chain or tape by comparison with the official standard
under the conditions of actual use.

(c) Metliods. — In standardizing tape, reproduce the con-
ditions of actual use as regards tension, support, etc., bring
one end graduation of chain or tape to coincide with one
standard mark, and observe fraction at the other end with
a scale. As a general rule, observe one more decimal place
than is taken in the actual chaining.

PROBLEM A6. DISTANCES WITH SURVEYORS' CHAIN.

(a) Equipment. — Surveyors' chain, set of chaining pins, 2
plumb bobs, 2 flag poles (unless instructed otherwise).

(b) Problem. — On an assigned chaining course about one
mile long measure distances with the surveyors' chain to
the nearest 0.1 link, and repeat the measurements in the
opposite direction.

(c) Methods. — (1) Standardize the chain before and after,
as prescribed in A5. (3) Chain along the assigned course,
noting the distances from the starting point to the several
intermediate points and to the end station. Observe frac-
tions to the nearest 0.1 link by estimation. (3) Repeat the
chaining in the opposite direction, noting the distances from
the end point, as before. The difference between the totals
in the two directions should not exceed 1 : 3,000. Retain the
same party organization throughout the problem. Record
the data as in the prescribed form.

PROBLEMS.

27

Line

Chdin

A-B
A-C
A-D

A-B

e-D
B-C
E-B
B-A

Note:

Direction
Cfiained

Befhre

After

B-

DISTjlMCES
Obtrred Dif f- of
Length Total

Ch- Ch.

W-

The
st/hs
the

7■3^7
30-306
eO-3S7
79-g3S

19-473
4AS5I
7M06
73-133

svbsi ^uenf

/ Tea

WITH

Ratio
l:d

^0-OOS

have c 7t3 wL

proble *7?

I-ISS70

crc

(3ee D. a^rsn^

f be i/sed irT3
In t?> 'scuss/pp
-■/sj^n or c bs/mn ^-

CoeF-

C

Lk-

p-oe

Surveyor's CiJai'n

/fff3(f Ch3m/nsj?rJ-l oe^
SepfJS, 'l4-frffourt .
Used &iipferCfr3i'ir\ff~^fO,
haia

Compared Ghain fv/ ^/f

' ''t before and Kiftercf^.

bsfh
Chained 3foi?0 Chan *in0

te^jj7/?m03f

A on ffuard stake,

of W' br/ck w^M

curb J/he

!ll-;thei?ce£'ly

N'brfck side wa.

es to nearest 0-1

marked B, C,

from stsrtmff pi fnfA
Chained same
d/recfion carrying
from Hub ^
fractions of^i/nk

ruie tvas used in

W-^

A B

r/ ^earChalnman, R-Rae,
Ciear^nd Cool-

1 J locker /f^35
off/ciaisfandsrd
'^ chain ing.
Course ''A",
fvifh fack, market^
J, f0C3tetf3tS.-&^
on (freenSt- atf'
•; Urbana^
^lon^ said 5- line of
'kf obsen/^ing distanc-
ik- to tacked hubs
■f the totai distance

being noted-
in therei^rse
total distances

of Oat. 'lews Ave-

Dat'dE,

were estimated- ibcht
standardizing Chain-

PROBLEM A7. DISTANCES WITH THE ENGINEERS'
CHAIN.

(a) Equipment. — Engineers' chain, set of chaining pins, 3
plumb bobs, 2 flag poles (unless instructed otherwise).

(b) Problem. — On an assigned chaining course about one
mile long measure distances with the engineers' chain to
the nearest 0.1 foot, and repeat the measurements in the
opposite direction.

(c) Methods. — (1) Standardize the chain before and after,
as prescribed in A5. (2) Chain along the assigned course,
noting the distances from the starting point to the several
intermediate points and to the end station. Observe frac-
tions to the nearest 0.1 foot by estimation. (3) Repeat the
chaining in the opposite direction, noting the distances from
the end point, as before. The difference between the totals
in the two directions should not exceed 1 : 3,000. Retain
the same party organization throughout the problem. Re-
cord the data as in the form.

28 THE CHAIN AND TAPE.

PROBLEM A8. DISTANCE WITH 100-FOOT STEEL
TAPE.

(a) Equipment. — 100-foot steel band tape with end foot
graduated to tenths, set of chaining- pins, 3 plumb bobs, 2
flag poles (unless instructed otherwise).

(b) Problem. — On an assigned chaining course about one
mile long measure distances with the 100-foot steel band
tape to the nearest 0.01 foot, and repeat the measurements
in the opposite direction.

(c) Methods. — (1) Standardize before and after, as pre-
scribed in A5. (2) Chain along the assigned course, noting
the distances from the starting point to the several inter-
mediate points and to the end station. In observing the
fractions, first determine the foot units, then estimate the
nearest 0.1 foot, then shift the tape along one foot and read
the exact fraction on the end of the tape, estimating the
nearest 0.01 foot. (3) llepeat the mea.surement in the op-
posite direction, noting the distances from the end point, as
before. The diiference between the totals in the two direc-
tions should not exceed 1 : 5,000. Retain the same party
organization. Record data as in the form.

PROBLEM A9. HORIZONTAL DISTANCE ON SLOPE
WITH STEEL TAPE.

(a) Equipment. — 100-foot steel tape with etched gradua-
tions to 0.01 foot, set of chaining pins, 3 plumb bobs, 3 flag
poles, axe, supply of pegs, engineers' level and rod (unless
otherwise instructed).

(b) Problem. — Determine the horizontal distance between
two assigned points on a steep slope, ( 1 ) by direct horizon-
tal measurement, and (3) by measurement on the slope
and reduction to the horizontal.

(c) Methods. — (1) Standardize the tape for each method,
as prescribed in A5, both before and after the day's chain-
ing. (3) In chaining down hill, rear chainniaii lines in flag
pole in hand of head chainman, then holds tape end to tack
on hub ; flagman stands 50 feet or more from line opposite
middle of tape and directs head chainman in leveling front
end, then supports middle point of tape under direction of
tached to tape and using pole as help to steady pull, brings
tension to 13 pounds ; recorder plumbs down front end, and
sets pin slanting sidewise. After checking th? pin, proceed

PROBLEMS.

29

DlST/

NCES

WITI

>

Ensiheer's Chain-

Line

Direction

Obstred

DiFf-of

Ratio

CoeF-

Held Chainman, R-Roe - Rear Chamman, JDk-

Chained

Length

Total

I'd

C

SepMS, '/4-CZ Hours) C/mdyS- Con/-

Ft-

Ft-

Ft

Used WO Ft- Chain ffSS, Locker miS-

Cham

Befire

/iVx/0

Compared chain with official standard

ti

AFfer

WO-IZ

both before and after days chaining

AS

e-

4U0

Chained alonp ciiainlnff course "A",

A-C

tt

ZODZ-Z

beffinninff at hub with tack^ marked

A-D

»'

3Sg7-S

A on^uardstake^ Iff cared 3tS-

A-E

ft

'J

5274-6

\

edge off/' brick waJk on Sreen
St- ate- curt line ofNathewsAve-,

e-D

r

1/86-3

\

llrbana, Hi-; thence f 'iy alon0 said

e-c

ij

il7Z-4

1

5-iine of It- brick walk, observing

e-B

rj

4730-Z

4

distances to nearest O-f ft- to

e-A

r>

SZ74-3

I=/7SSP

0-04

tacked hubs S, C, D and E, the total

1='

n

" e

distances from starting point A

SZ-743

e=

cYlor

^=#

being noted-

(SeeDj

igramy

iJ

Chained same course in tile reverse

Note:

T/ie s

^aye d

?/(? wL

( be ui

ed in

direction, carrying total distances

3 sub

•^e^uer,

tproi

/<sw7 I'r

discas.

from Hub E -

fng ff.

sprecj

^/Of? oi

• chat'i

rng-

fractions of a foot were estimated-

A pocket rate was used to compare
chain with standard*

AS C OS

\

■^ -fi -ft -ft "ft

^

DiSTA

NCES

wit;;

100 Ft Steel Ta

*E Rear Ciiain-J-Doe

Line

DirectiM

111.5er«ii

PiFf-oF

Ratio

CoeF

5epf^^0//4■f^ho^frs^

' Ciear^ moderate-

Cfiainec^

Length

Total

l:d

C-

WOFt-PoeSfeelTap

Ft

Ft.

Ft.

Cemporecf tape w/i

h oFFiciai standard

Tape

Before

100-01

bfffh before and

iFferday^ chaioii^

w

After

iOOOOS

meast/r/ng Fracft

w m'f A engineer scaie

A-B

E-

4S4SS

Chained alon^ CItaiA

f'n0 Course '^^pretri-

A-C

"

zm-79

ffvsiy drained t*

th Si/nterandidO

A-D

It

i99ieg

Ft- CAains, desa

^hedo/T pp- oF

A-E

n

52794S

\

Fieid nofe Book
oi/s distances tt

observing contina-
tiubs 8, C. and

E-P

W-

iZg7-8i

F to nearest 0-

VFt

E-C

n

i^7s-7^

Ciiained coirrse in

reverse direction-

E-B

n

4794-M

*0-O9

Fractions oFa Fo

»/ tvere estimated

E-A

II

SZ7f-S7

/■■jfieo

0-OIZ

to nearest d-di

"5^ on fAe end Foot

L =

II

oFtfre tape wt

fZ-7SS7

E =

cKor

to ten firs oF a

Foot-

(Seel

Ingram,

W"^

-^F

Note:

- The

Tboiie

'ato H

<■// be

fsed

A B C

B

in a

iubse^<

'ent pi
he pn

abiem
cision

in

disc

'ssing

oF

%•% ^

•ft "9

chai

■tin^

^^ ^

30 THE CHAIN AMD TAfJii.

with the next 100 feet. In chaining vp hill, follow same
general method, using plumb bob at rear end. In leveling
the tape the tendency will be to get the down hill end too
low. Chain the line in duplicate, retaining the same or-
ganization. (3) Chain the line again in duplicate, tape
lying on the ground, pull 13 pounds, pins set plumb, frac-
tion direct to nearest 0.01 foot. Set temporary pegs flush
with ground every 100 feet and also at intermediate sudden
changes of slope, for levels. Determine differences of eleva-
tion between successive pegs, unless the leveling data are
supplied to the party. Uecord data and make reductions
and comparisons as in the form.

PEOBLEil AlO. ANGLES OP A TRIANGLE WITH TAPE.

(a) Equipment. — 100-foot steel tape, 50-foot metallic tape,
set of chaining pins, 2 plumb bobs, 2 flag poles, flve-place
tables of trigonometric functions (each member of party
to have tables).

(b) Problem. — Measure the angles of an assigned triangle
with the steel tape and also with the metallic tape, the
error of closure not to exceed 3 minutes.

(c) Methods. — (1) Measure each angle with the steel tape
by both the chord and tangent methods, 100-foot radius,
the difference in the two results not to exceed 2 minutes.
If the angle is near 90°, the tangent method may be applied
to the bisected angle. (2) After securing satisfactory check
on an angle with the steel tape, make a rapid but careful
measurement with the metallic tape, radius 50 feet. The
results may be taken to the nearest half minute. (3) Meas-
ure at least one angle, preferably on smooth ground, by lay-
ing out an arc with radius of 57.3 feet, setting pins every
few feet, and measuring the actual arc. Give close atten-
tion to alinement throughout. Eecord data and make re-
ductions as in the form.

PEOBLEM All. SURVEY OF FIELD WITH STEEL TAPE.

(a) Equipment. — 100-foot steel tape, set of chaining pins,
2 plumb bobs, 4 flag poles, five-place table of functions.

(b) Prohlem. — Make survey of an assigned field with tape,
collecting all data required for plotting the field and calcu-
lating its area by the " perpendicular," " three-side," and
" angle " methods.

PKOBLEMS.

31

Tape I
• 2
I
Z

HOR

ZONTA|L DlS1|AHCE

Direct

Observed
Length

Ft-
}0-3f5
9S-997
76/-4S
761-49

Kisult
Differ
Mean

Horizon tal Mca 5urem< nt'

Mean

Length

Ft-

761-47
MOO-0

By Me
Observed
Length

Ft.
m-007

m-m

761-11
711-7}

-.liO

isurem

Tape /
I

z

Correc^'en for hjdinstAtn

Peducei Horizon
by first

■at^ea:
Hetltail
Differipce tetiveen fie
Hesalts

Mean

Length

Ft-

lOO-OO!

761-10

A14-AI7

Cor- For

Standerd

Ft-

■0-03

nj- on

Cor- for

Standard

Ft

Reduced

Length

Ft.

711-44

Diff-fe)
CoeP-(c)
R3tio(|:d)

e-0-04
c-O-OIS

I:rS3gO

CSee.
ffiagrai^

the

Reduced

Length

Ft

7ii-se

-0-47

76I-M
761-44

t-OS
761-42

Slope

e an

DifF.(E)

CoeP-(C)

R3)io(l--^

C'O-m-
!:3SS90

(See-*
V,

r)
V
'J?
c-MIS'
1:IS130

. Hd- Chain, J-Poe ■ JSh Chain, B-eoe-

,Steep Slope) witi Steel Tape

Recorder, B-F-Keen- j 'lawman, S-W-Sure-
5apf-Zl,''/4- (3Burs) Cieydy; inodarafe-
Vied m-ftfX-Se. efcedfape, tla4l6,

locker Z6, tvit/r sp'ing balance.
1st- Method : Stafidan <'2ed fape (before and
after), supported a f snds and m/dd7e,
pli/mbin^ ends tfotvj ', pn/l iZ poun^^s-
I Chained line in dupll atv, leveling fape 1^
estimation, pull 12. ffs-^ plumbing down
high end, marking f 'ints by chaining pins
leaning sidetvise-
Reduction to the Hbrixontal

2nd Method : Compgn d tape yvifh standard
(before and after), J 'jppor/ed fell lengfh
en ground, pull /? lbs-
Chained line in dt/pli'c Jte, tape supportetf on
ground, pull l^lbs-, ends marked by chain-
ing pins- Prove fen porary pegs every
100 ft- for levels-
Pan levels over line with following results -

c

<i

ft-

Ft-

Ff-

100

IZ-2B

0'0&

100

0-07

JOO

*^'l

16-81

0'08

0-04-

■f-4-S

9 '61

62

i-i-e

z-se

Q-OZ

762

■0-47

I ngo- r
Name
5in-i(l)
ran-(6)
Sin-ire
Tan-rs)
5in-i(S)
Tan-kiS
Sin-id)
Taa-iU)

C
Angle
6
I
S

steei
An^le 6

4N6LE

unctions

Value

MCSO

i-ms

ll-3(96
0-9430
0-6995
0-9796
0-6997
0-9799

_ Wl't^l

Sin-^A*

0-4031
0-3696
0-6999

Assign ng eqva.
tape, fi
47'47'3
4313-3
!l'49'-4

iso'oo'

I OF

Com
Half

?l'4l'S

frmr

£0

4A
Z3'S4'
Zl'41-3
44'ZS'

weigh]
t most

/ ngle

44fl4!i
44'Z4
44'Z5'

tRlAN\$LE
uted
Whole
47'4r
47'47'
43'Z3'
43'Z3'

7f Clesi-ra -

fermA sable

-Meta

A
47'4l'
43'Z3'
SS'30'

1/0 '01'
to the three
7roba3le

«H

/sue

SS'SO'

lie Xy e

Mean

47°47'

43'Zi'

IS'49

I79'S9'

01'

frror \ 03'

r Jselts bf
values are

5-6-8 WITH Tape

Surveyors, J-Doe am ' J?-^oe-
'SeptZe,'l4-(Z Hours) "
VsedPoe lOO-Ft-Sfeel

Clesr and warm'
Tape, 110-36Z. an J

Lufkln 30- ft- Hetall c Tape, lfo-411, lkr-3S-

Though not needed in ^ voblem, noted the
length fi of tapes by standard f 100-01
and 30-01 ft-, resp 'ctfvely.

pleasured each angle by chord method and
checked by tangent method, using radius
eflOtft- m'th steel tape- In measuring
ZS, (nearly 90'J fh, tangent method was
applied fo the bisec ^ed angle • fach
angle was verified before proceeding
to next, a different e of Z' being allow-
ed in each'

After an angle was t ius verified, a rapid
but careFut measui ement was made
with metallic tape, by chord method

Used flagpoles for d. slant and pins for

close targets
Used 3- place
table' J^fg

Arc = 47-!0 Ft'

Rough test of ^6, I Mas =37-3 ft-

32 THE CHAIN AND TAPE.

(c) Methods.— (1) Standardize the tape once. (3) Exam-
ine the field carefully and plan the survey. (3) Measure
the required angles with tape. (4) Locate the perpendicu-
lars. (5) Chain all necessary lines, and also take dis-
tances to feet of perpendiculars. Follow the form.

PROBLEM A13. AEEA OF FIELD P,Y PEEPENDICULAR
METHOD.

(a) Equipment. — Five-place logarithms.

(b) Problem. — Calculate the area of the assigned field by
the perpendicular method, using the data collected in
Problem All.

(c) Methods. — (1) Prepare form for calculations; tran-
scribe data, and carefully verify transcript. (2) Calculate
double areas of the several triangles by contracted multi-
plication, perpendicular method, preserving a consistent
degree of precision. (3) Make the same calculations with
logarithms, as a check. (4) Combine the verified results,
as shown in the form.

PROBLEM A13. AREA OF FIELD BY THREE-SIDE
METHOD.

(a) Equipment. — Five-place logarithms.

(b) Problem. — Calculate the area of the assigned field by
the three-side method, using data collected in All.

(c) Methods. — (1) Prepare form for calcvilation ; tran-
scribe data, and carefully verify transcript. (2) Calculate
the areas of the several triangles by logarithms, three-side
method, preserving proper units in the results. (3) Care-
fully review the calculations, and combine the verified re-
sults, as in the form.

PROBLEM A 14. AREA OF FIELD BY ANGLE METHOD.

(a) Equipment. — Five-place logarithms.

(b) Problem. — Calculate the area of the assigned field by
the " two sides and included angle " method, using the
data collected in All.

(c) Methods. — (1) Prepare form, transcribe data, and
verify copy. (2) Calculate the double areas of the several
triangles by contracted multiplication, angle method, pre-
serving consistent accuracy in results. (3) Make same cal-

PROBLEMS.

33

SUR

'EY OF

Field

A-B-

:-D-E

WITH

Tape- (Data F(ir Area and PlatS

Angle

SiniA

■^^

A ^

ProoF

7, ^-Roe-

ABB

■Z96S

17%'

34'3Z'

eearChamma

n, J-Doe-

eBD

■7/3!

4S'/9-i

90'S9'

\m'io'

5epf-Z5,'14- (

^ /fours) C/oudyJ^Coa/'

DBC

■S347

37'19'S

64'39'

iso'oo'

Used Hoe 100 ft: Vee/ T3peM^361,li:cl!er'^35
5t3/idardlze(f 1 9pe before o/yly-

ABd

■asis

S%'

lO'lZ'

lO'JO

letfallperpemVciilarsAafSbamlBc by

first esllmal in^ poslflons of a, b s/id

Line

Otser/d

Cor- for

iiductd

Cj then ereci

■jn^ j?rac/sff/>er^end/cu!3re

Length

Sfandarel

Length

and shlftlni

'3S required- 5ef /}e0s

Ft-

Ft-

Ft-

at points a.

h and c-

5eptZS

Measi/redanp fsASf, EBD and DBC m'fh

Tape

99-99Z

and cheeked

lymeasurj'n^ snffle be-

Tape

99-SgO

dfjme CB profarfaed)

AB

i}6-8i

-0-07

33e-7e

5ept-?6, '99 (ZM, ars) Prlzz'Ung <f Cold-

BC

4(5-07

-0-09

464-98

Cbalned each

J/ne carefully once-

CD

4S3SZ

-0-10

483-7Z

Skelcb shows }

■educed values ■

DB

uses

-0-lZ

616-53

■■

eA

Z4J-89

-0-OS

Z41-S4

SE

4ZS-93

-0-09

4ZS-84

^^^4".

-"""^^ f

BD

438-70

-0-09

438-61

■-317-'99 ^^7

Ab

190-09

-0-Of

190-aS

£b
Bb
Bb

147-90
30Z-ie
3JS-0S

-0-03
-0-06

-ooe

147-87
302-10
317-99

t \$10

'S3 D

Kfic

Z6i-90

-0-05

20585

J

Triangle
ABE

COMPUTATI IH OF
Ba4e,b

BDE

BCD

I Acre
/9l-f. ■

Line

BE
As

Pe
Bb

CD
Be

43.51 '

Use

dA
shiwn in

/ w^ I

Ft.
4Z5-84

Area

Attitude,:
Ft.

190-fS

616-S3

483-7Z

Mote- To red 'ce s^-if- foaci ^Sfdlv.
43,56 7. 5pea j/mefh >ds are given

6x6'll'tl*l^

= 0-00lKZ95t96Ac

30Z-10

feet'
Ac-''

■he apf llcallo/.

OF

Multipli-
cation

FliLO

4m-t
sesze

2S6

I4SI16
3S69B

133

'J 7 Square

4-3560

one ^ the i lethods

opfios ''te

Logar-
ithms
Z-6MIS
Z-ZIOM

Sept-Z7,14- Compuh r,

A-B-tD-E, Perpendi

Double Area^
5q- Ft.

4-90S49
(81190)

Z-71995
2-410IS

5-pOIO

2-18459
Z-5m9

5-Zt59S
(J1450S)

de by
I elow
chains-
Ac-

Data fromfff>-
Transcript

81190

■■hecked-

(Hesult i t nearest 10 5^- Ft-)

186 ZSO

184 500

Z)45l 940

llTZSMIO_
B )37isi-B&l

lijlnrm

11 1570-631

(Result
5-lllAc-

Contract d Div'n
Used

ZZS9701

znBoo

8170

435&

SB14-

34-85

323

305

Z4

Z£

J-Doe. ^

ULAR Method.

Area- -Lab'

'<7 nearest 0-001 Ac-}

Contracted Mult'n

41191-

4513

Z034

113

15

S-I876

34 THE CHAIN AND TAPE.

culations by logarithms, as a check. (4) Combine the

PROBLEM A15. AEEA OF FIELD FROM PLAT.

(a) Equipment. — Drafting- instruments, paper, etc., pla-
nimeter (as assigned).

(b) Prohlem. — Determine the area of the assigned field
directly from the plat.

(c) Methods. — (1) Make an accurate plat of the field from
the notes secured in All, using a prescribed scale. (2) De-
termine the area of the field by resolving the polygon into
an equivalent triangle. (3) Determine the area from the
plat by the polar planimeter and by one of the following
" home-made " planimeters : " bird shot " planimeter, " jack
knife " planimeter, cross-section paper, parallel strip,
weighing, etc. (4) Prepare on the plat a tabulated com-
parison of results secured by the several methods. (5)
Finish the plat, as required.

PROBLEM A16. SURVEY OF FIELD WITH CURVED
BOUNDARY.

(a) Equipment. — 100-foot tape, 50-foot metallic tape, set
of chaining pins, 2 plumb bobs, 4 fiag poles.

(b) Problem. — Make survey with tape of an assigned tract
having a curved boundary, collecting all data required for
plotting the field and calculating its area.

(c) Methods. — (1) Standardize the tape once to nearest
0.01 foot. (2) Examine the tract carefully and plan the
survey so as to secure a simple laj'out of base lines de-
signed to give short offsets to the curved boundaries. (3)
Locate the perpendiculars, if any, and chain all lines ; on
the curved sides, take offsets so as to secure a definite loca-
tion, and as a riile take equal intervals on the same line.

PROBLEM A17. AREA OF FIELD WITH CURVED
BOUNDARY.

(a) Equipment. — (No instrumental equipment required).

(b) Problem-. — Calculate the area of the assigned field
with curved boundary by " Simpson's one-third rule," using
the data collected in Problem A16.

PROBLEMS.

35

Com

Triangle
ABB

BOB

BCP

Data

PUTAT
Sid
Line

AB=3
SB'-b
BA'C

i)

ON
es
Length

Ft-
336-70
4?S-g4
^41-14

F ARI

ka»b«)

Ft-
50Z-Zi

740-49

(93-00

Tra'nsc

A OF

(S-B)

Ft-
10S-46

30J-SS

ZZS-0S
-!pt cl

Field

(s-b)
Ft-

70-3S

JZ3-90

Z09-94
ecked-

Sept-2S
A-B-C

Ft
Z0O-3S

3J4-05

ZSS-OS

*/4- Compute
D-E, 3 SlDl
Area oF Triangle

r,J-Doe-

■-. Metho

Areas
5<t-Ft-

40030
933S0
9ZI0O

A

V5C5-a)(5-b)(3-c)

Logari+lims

Z -7001 9

Z-ZM9

J-SSZ9S

z-4-jsei

Zl 9-21817

c

(To near-
estlOsq-ft)

-S-J93AC-

III04-44

s =

BD=3
DB=b

SOZ-ZZ

43S-6J
016-S3
425-»t

4-60908

Z-B09SZ
Z-479e3
Z-093Z8
Z- 49 783
Z) 9-94040
4-970Z3

Z-84JIS
Z-2S9Z3
Z-3ZZ/P
Z-4O00Z
Z)9-9Z9W
4-904SS

I48l>-9t

BC=3

CD^b
Db=c

740-49

404-91

'4I3-7Z
431-01

I3!7-3I

From f

093-00

a-

S-3S447

-i-eisog-*-

Z20J9O
-■l- 43 300

0-71338

Triangle
ABB

Part

CoMfUTATipH OP
Value
Ft- or"
330-70
423-14
34'33'

BOB

BCD

AB-a
BB^b
ABB=e

BE'a
BD'b
BBD-e

SC=b

pic=e

425-84
438-61
90°S9'

Are>,

Multip
aSin-C

438-61
4l*-9l
64'39'

OF

ication
abSin-C

16S5S
^02

IflELC

Logar-
itiitns
2-S273Z
2-02920
9-7B3S0

J703JZ

12773

3-^06

ZSB

±

4-9m7
(SI30II)

2-12923

2-64,

9-99994

5-21127

166 750

IS8SS2

23783

1586

357

2-64208
2-66743
9-936113

S-265S4
(114310)

S-3S446
4-63909-

Sepf-22, '14'

A-B-C-D-E, Ahsle

Double Areas
Sq-Ft.

Data From
Transcript

81300

110 750

184 310
2)452 360

pp.
checked-

(Resultto

■-(■r 43500)
(5-I92Ac)

'»-, J-Doe-

Method-

Area^-^absin-C-

nearest lOSij-Ft.)

5-192 Ac-

(Result t J nearest 0-001 Ac-)

36 THE CHAIN AND TAPE.

(c) Methods. — (1) Prepare form for calculation; tran-
scribe data in convenient form for calculation, and carefully
check copy. (2) Calculate the area of the polygon formed
by the base lines, preferably by the perpendicular method.
(3) Calculate the areas of the curved figures by " Simpson's
one-third rule," which is as follows : " Divide the base line
into an errii itumhrr of equal parts and erect ordinates at
the points of division ; then add together the first and last
ordinates, twice the sum of all the other odd ordinates, and
four times the sum of all the even ordinates ; multiply the
sum by one-third of the common distance between ordi-
nates." (The field notes might have been taken with special
reference to the rule, but it is better to take from the notes
the largest cren number of equal segments, assuming the re-
maining portion to be trapezoid or triangle.) (4) Give
signs to the several results by reference to the field sketch,
and combine them algebraically to get the net area, as
shown in the accompanying form.

PROBLEM A18. AKEA OP PTELD WITH CUEVED
BOUNDARY FROM PLAT.

(a) Equipment. — Drafting instruments, paper, etc., pla-
nimeter (as assigned).

(b) Problems. — Determine the area of the field with
curved boundary directly from the plat.

(c) Methods. — (1) Make an accurate plat of the field from
the notes obtained in Al6, tising a prescribed scale. (3)
Determine its area directly from plat by two methods men-
tioned in (3) of A15, other than those used in that problem.
(3) Prepare on the plat a tabulated comparison of the re-
sults by the several methods. (4) Pinish the plat, as re-
quired.

PROBLEM A19. PASSING AN OBSTACLE WITH TAPE.

(a) Equipment. — 100-foot steel tape, set of chaining pins,
plumb bobs, 4 flag poles.

(b) Problem. — Prolong an assigned line through an as-
sumed obstacle by one method and prove by another, finally
checking on a precise point previously established.

(c) Methods. — Given two hubs, A and B, 200 feet apart
prolong line and establish C 200 feet from B : (1) by con-
structing a 200-foot square in one direction; and (2) by lay-

PKOBLEMS.

37

c

Su

RVEY

OF F\

:LD V

ITH

Curved Boundary Line-

Offiiet L-

Dist-

OfFsd-l!'

OffsetL

Dist-

Offset R

HKit)Cl!smjmn,R-K(ie- Re^rChsinimn, J-Vae-

Ft-

ff

Ft-

Ff-

Ft-

Ft-

Oct-2, '14. (3 Hours) Clear and y^arm-

26Z-S

= d

Tspe H136I, locker H^SS = lOO^OI

ll-B

Z4-0

Sketch shows obseri/ed lengths- Final

30-3

too

es

309-1

area resuJt corrected For standard-

39-0

160

300

2-1

39- 1-

IZO

2S0

!S

il-S

go

260

13-2

d

IS- 6

40

240

14-7

^^^lr~>,<'•»^_,

■0

220

lB-0

Line C

CcfoJ.

200
ISO

I4-S
10-0

# %^^

4IS-4

= c

160

2-S

3-S

400

JS4-3

/ .. V.'-'' //

?4-6

iS-4
39-3

360
3Z0
ZSO

7-2
IS-0
■i9-7

MO
120
100

40-7

240

20-8

SO

e

40-3
37-4

200
160

20-2
JS-4

60
40

"^

30-1

no

ja-3

20

»M

<^^ ~~-kj

10 -g

SO
40

LineD

Cdtoe)

/LineA(B09-ih]l,

Line B

= b

I'll toe)

Tepf
Octl't'i

100-01
, Clear

'rWjrm

a

{

^ssdVp)

gesdUp.

\

\

Com >utation of Area of Field

Data for Calculation of Areas

Part

3be
bee
cde

LineB

LineC

LIneP

Chain
TrueA

Triangle, Base' 290-0, Alt ^ 145-3

•• '4IS-'4, <i =261-8

„ » =404-7, " "199-3

^'-"^So„'

I Z 3 4^ S

%\<^.

fM'

■M T;

■M-3

^-e^io'120'- >i ^5^^5«)i ii^?,

•9-1 ftoTTyxxtJi ^z-

(.'

K-6i'?0''l20'->\ _,--■

WO-'OI

■ea = Computed Area "-(H-O-OOOI)
{l+0-00ai)'-(ltll-0ll02) Cnearly)-

Oct-3, 14- Computer, J-Doi^-

WITH Curved Boundary

Indicated Calculations
OafaFrompp- Transcr^t
i[290-C ■-'-'
i(4IS-4
i(404.

\^[(0t9-S)

< +2f2l-l-f37-4t4S-7t32-4)
1 ^■4(|0.8■^3e■l■^40^3t39-5^■24^ii
^^i(9-S>-l8-4)
(^[(0tl3-S;
{ +2(31-8 i-39-O)
\t 4(19-6 ■t39-4 -tSO-S)]
■ i(l3-S'-22-5)
'^[(OtlS-0)
t2(l8■4■^20■S)
\f 4(10-3 +20-2 + 19-7)]
i [20(15-0 +7-2) +(7-2*14-3)]
m[(2-8+8-S)
l+2(l4-8+l4-7)
\+4(IO-0+IS-0+l3-2)J
--i (2-8 "5-7)
^i 1(2-1'- 9-1) + 20(2-l +8-5) J
{98352 -*>
Chain Cor. K2000-0_

tAreas

■.ed-

21068

S6024

40328

(98352 ■
'. 1 2000-0

1487

8

116

II903I
20679

98351

-Areas

II37S
87

6831
152

1961
273

2-?5l^

38 THE CHAIN AND TAPE.

ing off a 200-foot equilateral triangle on the opposite side
using pins to mark points thus established. (3) Prolong
the line by each method to the hub D, 200 feet from C, and
record discrepancies in line. (4) Interpolate a point at G
on true line between B and D, and note errors of prolonga-
tion at G. Record as in the form.

PEOBLEM A20. OBSTRUCTED DISTANCE WITH TAPE.

(a) Equipment. — 100-foot steel tape, set of chaining pins,
2 plumb bobs, 4 flag poles.

(b) Problem. — Determine the distance between two as-
signed points through an assumed obstruction to both vision
and measurement, using two independent methods, and
finally chain the actual distance.

(c) Methods. — (1) Standardize the tape. (2) Determine
the distance between the assigned points by constructing a
line parallel to the given line, and equal or bearing a
known relation to it. (3) Secure a second result by running
a random line from one hub past the other so that a per-
pendicular less than 100 feet long may be let fall, measur-
ing the two sides and calculating the hypothenuse. (4)
After securing two results differing by not more than
1 : 1,000, chain the actual distance. Follow the form.

PROBLEM A21. RUNNING IN CURVE WITH TAPE.

(a) Equipment. — 100-foot steel tape, 50-foot metallic tape,
set of chaining pins, 2 phimb bobs, 3 hubs, 6 flat stakes,
marking crayon, tacks, five-place table of functions.

(b) Prolyicm. — Lay out two lines making an assigned
angle with each other, and connect them with a prescribed
curve by the " chord offset " method.

(c) Methods. — (1) Calculate the radius, R, for the given
degree of curve, D. (2) Calculate the tangent distance, T,
for the given radius, B, and angle of intersection, I. (3)
Calculate the chord offset, d, and tangent offset, t, for the
known radius, R, chord, c and degree, D. (4) At the given
point intersection (P. I.), A, lay off the given angle, /, by
the chord method. (5) Erom the P. I. lay off T along the
two tangent lines and locate point tangent (P. T.) and
point curve (P. G.), setting hubs at P. C. and P. T., with
guard stake at each hub. (6) Run in the curve, by chord
offsets, beginning at P. G. and checking at P. T. Calling P.

PEOBLEMS.

Passing am Obstacle

Oct- 4,*!4 , [2 Hours) Chsr 3f7d W3rm-
Tape ffo'iej, Uckffr /fo-3S, leir^fh ^ WO-OI •
d/vjs/7 thref /ruts, Csef on true //he i>y
transit), B ^Off/'t-frcnr A,3n</ P
400 Ft- beyend 3j. g/f on smostA ^rom<^-
Assumed otst^cle as sJrown j'n sketch,
and then (fgnorfn^ P) passed" ohst^cfe
by ZOO ft- equifaterel triangle to r/^ht
and by t^^ ft- square to left' Resumed
Ii'ne by each method and prolonged to
point D- Used p/ns marked by sJ/ps
of paper to /'nd/cate po/'ntS'
Also interpolated Con BD carefu//y ty

eye-
Results are given ih diagram below »
D

WITH Steel Tape -

Chai'nmen^ J-Pue and J^'^oe-

A (Hub)

Obstructed Distahce

OcfS, */4 fZ Hours) Cloudy a/7d eooJ-
Tape Ho-36/, locker /^o-3S, length \$9-99
Ci'ven tkvo hubs A and B an unknown dis-
tance aparf, on smooth grouncf-
Assumed an ebs/ruction to vi'sfoji and

measurement, asshouv/i /n sketch •
Selected point C visible from A andB, chain-
ed CA and CBf obser\>'inj nearest C-Ifty
and bisected CA at D snd CSatE-
Chained DB- Then ca/cu/ated AB hy

doubiihg fD- ^6o■yx^='^^i'4

^an random line from A as c/ose as
pract/cah/e to obstruct/'ojr so as to reduce
, SF to a minimum • let fait perpendi'cuJar
BF from B on random fine- iieasured
Bf and fA to nearest O'lff- Calculat-
ed hy pothenase AB '
AS' VS0-8^-tSl9-4^ = SZl-0

finally, after securing the above results,
chained the actual distance AB- The
three results are sumarized below

Method-

Obs-Dist. 5td-Cor. Red-Dist-

By similar triangles
By right triangle
By actual measurement

SZl-4
SZhO
SZl-S

-O-l
-O-l
-O-l

SZl-3

SZO-9
5ZI-4

Total range = I tl040 -

WITH Steel Tape .

Chainmen, J-Poe ffnd ^'^off'

pP\^BfHub)

«A (Hub)

40 THE CHAIN AND TAPE.

C. Station 0, establish Station 1 by laying off tangent offset,
t, and chord, c Having one station on the curve, the next
is located by prolonging the chord and forming an isosceles
triangle having the chord offset as a base. Check on the
P. T., noting the discrepancy of distance and line. Also
establish the tangent again by tangent offset and observe
the error of line. Follow the form.

PROBLEM A23. DISCUSSION OF EREOES OF CHAINING

(a) Equipment. — (No instrumental equipment, unless
further data are desired, in which case Problems A6, A7
and A8 may be assigned again).

(b) Prohlein. — Investigate the errors of linear measure-
ment with the several kinds of chains and tape, with the
view to determine practical working tests or coefficients
of precision for actual use.

(c) Methods. — Assume that the conditions in Problems
A6, A7 and A8 are practically coniitant in the same problem,
and that the actual differences between observed lengths
of the several segments when chained in opposite dirc-
tions, represent the normal errors with the particular chain
and chainmen ; then tabulate: (1) the measured lengths of
all possible segments of the chaining course, either from
direct observation or by subtraction; (2) the actual errors
or differences between the two results, giving signs; (3)
the chaining ratios, I: d, and the decimal expressions of the
same to six places; (4) the " coefficients of precision" for
each case, calculated by formula, or more quickly, taken
from the diagram in the chapter on errors of surveying ; (5)
the mean decimal chaining ratio and its equivalent; and
(6) the mean coefficient of precision. Follow the form.

PEOBLEM A23. TESTING (OE ESTABLISHING) AN OF-
FICIAL STANDARD OF LENGTH.

(a) Equipment. — Standard tape (with certified length
given), turnbuckle adjustments with bolts, spring balance,
standard steel rule graduated to 0.01 inch, 2 thermometers,
2 microscopes, strips of wood, a watch.

(b) Problem. — Make a series of ten observations with a
standardized steel tape for the purpose of testing (or estab-
lishing) an official standard of length, observing the near-
est 0.0001 foot. (The Bureau of Standards, Washington,

D. C, will standardize a tape for a small fee.)

PEOBLEMS.

41

Location of Curve

Dcf-e, '/4 ■ f3 tnurs) Clear an J ceo/-
100 ft- Steel Tape f/e-Bl/, IpeJier/fa-JS'/Hi'-ill
ff/fen hub stA and a i/istant Ai/b B, l-a
/ay off a h'ne A-C making an angle I of
so' m'th BApro/onffed, and connect the
tm lines mtli a ^O'ciirve, t/iatis, a
curve having a central angle of ZO '
siiitended fy a W ft- chord, c-
The radii/s was calculated thus : Since the
chord of an arc Is tmce the sine of half

'chord . so ,a-F<, -

Calculated tangent distance thus .' In right
triangle (O-P-C-Fl)

Tan-Ast- - Mad-x tan -y /

=ZS7'9xO-i39lff = Z4l'e

Calculated chord offset d, and tangent off-
set t, thus : By simitar triangles ^: c =
'/''. ''•■%'=^,'34'-97,t'id=l7-^!
(An approx- formula is d-I^D'3S;t-p'l7S)

from A f/hlnt Intersection) laid off Tan-Pist-
(T), locating /hint Curve Cf-C) and Piint
Tangent (P-T-)- Began at PC- and ran

' in curve, asshoivn In sketch' Error of
Closure at P-T- was O^'Z in line and. tl-'l

^ in distance-

WITH Steel Ta

so- 000
34730
ISZ70
I3S91
1378

lie
•A s

'\ -A-if^

tn-9

0-I938-O

ZiOii

164-

?S9

3

2'H-St

Hd Chain, J-Doe-
PE. P'r Chain , S-Koe-
Axeman, B-f-Keen-
flagman, fi-W-Sura-

'i-^/V Chd-Offset-

1/

moo 287-9

IIS2

211

202

9

Line

A-B
B-A
A-C
C-A
A-D
D-A

A-e

f-A
B-C
C-S
B-P
D-S
B-f

e-B

C-D
D-C

c-e
f-c
D-£
E-D

Direction
Chained

E-

m
e-
w-

£■
HI-
E-

w-

E-
W-
E-
W-
E-
W-
E-
W-
E-
W-
E-

Discu

Observed

Length

Ft..

41-f-SS

4l4--(l

zm-79

3991-19
3991-74
S179-4I
K79-S7
ISI9-II
IS19-14
3SII2-II
3Sm3
4794-90
4794-96
1987-90
I987-S9
3Z7S-(9
327S-71
1217-79
I2S7-73
(L-ln

m-ft-
imits)\£,

SSION

Differ-
ence, E
f=t-.

-0-03\

-0-06

-0-OS'

-0.09

-0-03'

-0.02

OF El.RORS

Chaining
Ratio
l-.d

■f.f3<

Hun

;-etT
(Suilf.

Coef' of
Precision

W,ft

/■v

l:i»QO

a-

1:79130

i-.siseo
i-irnms

IU7S3S0

D-miii
i-.niso

i-tmos

1:01790

l:mi90

1:32920

1:41300
<>/■,«:■
■j4^

0-014
0-013
0-00/
0-012
0-OOS
0-003
0-009
0-002

O-OII
0-OIS

0-OOS

E

n

Oct-9, 14 - Computer, J -Doe •

WITH Steel Tape.

Pata from pp- Transcript 0-K-
ABC D E

O- ^ u o

Distances by Subtraction*

S-A 5279-37
e-B 4794-9G
B-A 484-m

A-C 2003-79
A-B

S-C 1519-21

E-B 4794-96
E-B 1217-13

D-B }B07-I3

B-C 327S-72
E-D 1287-83

E-A

5279-57

EC

327S-72

C-A

2003-85

E-B

■4794-96

E-C 3275-72\

C-B

ISI9-24i

A-E 5279-4^

A-B

4£4-S8\

B-E 4794-90\

A-E

S278-4l\

A-C 2003-79

C-E 3279-69

B-A 5279-57
E-D 1217-83

D-A 5991-74

A-D 3991-99
A-B 484-58
B-D 3507-11

A-D 3991-69
A-C 2003-79

C-D 1987-90

A-E 5279-48
A-D 3991-69
D-E 1217-79

D-C 1917-89

Designating Et and W- f4th Column) It Is
seen that the returning results (except
C-D) are greater- This Is explained by
standard tape lengi-ha, vlz-f before
=100-011, after '100-008, l-e- the tape
greater observed lengths- J

42

THE CHAIN AND TAPE.

(c) Methods. — (If a new official standard is being estab-
lished, one standard mark may be made permanent, and the
precise distance taken to an approximate temporary point
on the other bolt, the exact correction being applied after
a sufficient number of results have been obtained. If the
sun is shining, the tape should be protected by a wooden
box or other covering throughout its length. Cloudy days
or night time give best results. The observations should be
made briskly so as to have slight range of temperature.

Ccf/t! '14- Chiicly smf Cool-

Test of 100-Ft- Standard

Selec-f&d c/oi/c/y dsy ivifh s/i'ghf r3nge of
Used Sfanc/srd Tspe Jio-417, msrkeii "US-
3t 62° F- mfh J2- lb- pull, tspe supporte\l,
(-i'ltlt

^^5ffmz<l3spipe Spring Baktite

Program- Arranged "bt/cks3w"3djijsfwenfs,e
strip oF wood- (a) Doe set z^ro
(b) Roe set bslsnce at 12 lbs- Cc)Keen
using Sfarreti steel scale graduatee/ to O-C
reading glass - (d) 5i/re recorded all dsh
thermometers placed one each atSS'snd

■n^0'5tanlarJ-,\,2ero i'Solt in Ifaspipe*
y-, as shofvn in sketch, tape supporfet^ art
'f east standard mark with reading glass.
ob ^eri'ed Fraction at west standard nJark,
'fn-, estimating to nearest 0-001 in ■■ with
and observed time and temperature, ttvo
37'- Released pull between obserirations-

f^rfy:J-Doe,R-lloe, B-F-Keen, S-IV-Sure-

University •

te/npgratune during period oF observations-
^M-Zls;'cert!Fied Iengft7 =S9-99e7 Ft--,
^ coeFFieient oF expansion = 0- 0000061 -

m '/flapo e^fajtJslt

Tornbi/ckle- '

Time

PM-

?:!3

■■28

■-31

es

:39
■Ae
■J3

as

}^:04
•■01

S2"0
52-0
S2-0
S20
£2-0
S20
S2-S
S2-S
B2-0
52-0

Temperature

y*t 67'

ss'o

S3-0
S3-0
Si-0
S2-B
S2S
52-S
S2-0
S2-0
ST-0

Wsan
S2-S
S2-S
S2-S
S2-£
S2-2
S2-2
S2-S
S2-2
S2-0
S2-0

62-H'n

'g.°s

9-S
9-S
9-S
S-8
9-S
9-S
9S
10-0
10-0

Tatnp'Cor

Ft-
0-OOSl
O005S
0-00S8
0-OOSg
0-0060
O-OO60
0-OOSg

0-ooeo

0-OOil
0-0061

Tape
Ft-
99-9909
99-9909
99-9909
99-9909
99-9907
99-9907
999909
99-9901
99-9906
99-9906

West FracHon

0-116

■III

■116
■IIS
■121

■m

■119
■122
■121
■122

0-0091
-0091
■0097
■009!
■0101
■0100
■0099
■0102
■0101
■0102

Standard

Ff-
lOO^OOOB
lOO^OOOl
100-0006
100-0007
100-0008
100-0007
100-0008
100-0009
100-0007
100-0008

Prob-
dfOMl)

/

I

I

I

2

I

Error
d^
/

I

I

I

4

I

'f'-Wl/^='

Mean' 100-0007 2.d^= 3
length oF Standard ^ 100-0007^0-00002 Ft. J

If isolated standard monuments are used, their foundation
should go below frost line, and the monuments should be
located so as to suffer as little as possible from heaving.
If the standard marks are indoors, the conditions are less
difficult to control.)

(1) Arrange "bucksaw" or turnbuckle adjustments, each
held firmly by a bolt dropped into a piece of gaspipe driven
flush with surface of ground, with spring balance and tape
lined up, as shown in sketch in accompanying form ; place
the two thermometers at the one-third points as nearly as
possible under the actual conditions of the tape. (2) With

PROBLEMS. 43

fovir men in party, No. 1 sets end graduation precisely at
one standard mark by means of screw adjustments and mi-
croscope ; No. 2 sets balance at 12 pounds ; No. 3 observes
fraction at other standard mark by means of steel scale
graduated to 0.01 inch, estimating to nearest 0.001 inch
(say 0.0001 foot) by microscope; and No. 4 records all
data, observes time to nearest minute, and temperature to
nearest 0.1 degree. Nos. 1, 2 and 3 should lie flat. Release
the tension between observations. Record and reduce as
in the form.

PROBLEM A24. DETERMINATION OE CONSTANTS OP
A STEEL TAPE.

(a) Equipment. — Steel tape and other articles named in
preceding problem.

(b) Problem. — Determine coefficients of expansion and
stretch of the assigned tape.

(o) Methods. — (See Problem E9.)

PROBLEM A25. MAKING A STANDARD WIRE TAPE.

(a) Equipment — Spring balance, thermometer, etc., as in
A23, and a piece of piano or other suitable steel wire.

(b) Problem. — Make a 100-foot or other standard tape by
graduating the wire with reference to the official standard.

(c) Methods. — (To be devised by the student.)

PROBLEM A26. COMPARISON OP DIFFERENT MAKES
AND TYPES OF CHAINS AND TAPES.

(a) Equipm,ent. — Department equipment and collection of
catalogs of representative instrument makers.

(b) Problem. — Make a critical comparison of the several
types of chains and tapes made by different makers.

(c) Methods. — Study the different catalogs and prepare a
systematic and concise report.

CHAPTER III.
THE COMPASS.

Description. — The magnetic compass consists of a line of
sight attached to a graduated circular box, at the center of
which is a magnetic needle supported on a steel pivot. The
compass box is attached to a tripod or Jacob stafE by a ball
and socket joint, and is leveled by means of the plate levels.
The needle should be strongly magnetized and have an
agate cap to receive the point of the hardened steel pivot.
The dip of the needle is counter-balanced by a small coil of
wire, which can be shifted as desired. The E and W points
are reversed.

In Fig 10 are shown the usual types of niagnetic com-
passes: (a) the vernier compass; (b) the plain compass;

//he of \$1^ 5/0M

Fig. 10. — Types of Magnetic Compasses.
45

46

THE COMPASS.

(c) the vernier pocket compass with folding sights; (d)
the ordinary poclcet compass; (e) the prismatic compass.
Declination of the Needle. — If the needle is allowed to
swing freely, its magnetic axis will come to rest in the
magnetic meridian. The horizontal angle between the mag-
netic meridian and the true meridian at any point is called
the magnetic declination for that point. Imaginary lines
joining points on the earth's surface having the same
declination are called isogonic lines. The isogenic line join-
ing the points of zero declination is called the agonic line.
Fig. 12 is an isogonic chart of the United States. Of the
three agonic lines on the earth's surface, one passes
through Michigan, Ohio, etc.

Declination
5° Wesb 0°Easb 5°

Diaqram oF Secular Variabion oF bhe
Maqnetic Declination in Unibed Sbabes .

6 4Jr,

7 ■■

8 "

9 .■

10 "

11 '•

12 M.
/ RM.
2 "
5 •'

4 >•

5 "

6 >•

V

5^-

<-/^

^^.

i>^

p?^i

::^^^

h

J"*

s

\\ /

^

J _

^

-ss

^

'. — ■

„..^

^

-is=d

L^

-

^

/V

c

k

V^s;

k^

'S

^P l-Ko

^0,^

^'^1

Maqnetic

-^

•~~.

s,

s

"-Sa

^^

^f

- Northern United States

>

^^

6' 5' 4' 3' Z' I' 0' I' t 5' 4' 5' 6'
Fig. 11.

(For additional data see bulletin of Department of Com-
merce, U. S. Coast and Geodetic Survey, entitled " Principal
Facts of the Earth's Magnetism.")

MAGNETIC DECLINATION.

47

Variation of the Declination. — The declination of the
needle is not a constant at any place. The change or
fluctuation is called the variation of the declination. The
variations of the magnetic needle are of several kinds:

48

THE COMPASS.

secular, daily, annual, lunar, and irregular variations aueto
magnetic storms. The most important of these is the
secular variation which is illustrated in the upper diagram
of Fig. 11 for a series of representative points in the United
States. This diagram shows that the extreme range or
swing of the needle is roughly 6° or 7°, and that the period
of time between extreme positions is about a century and a
half. Also that the wave of magnetic influence progresses
across the continent alike in successive cycles. In 1900 the
needle was at its extreme western position at Eastport,
Me., and at its extreme eastern position at San Diego, Cal.
The 3° East isogenic line passed through western Indiana,
and was moving westward at the rate of about 4' per year.
This rate of change was general throughout the central
part of the United States, and is represented by the straight
sections of the curve in the upper diagram of Fig. 11.

The daily variation of the magnetic declination is shown
graphically in the lower part of Fig. 11, the scale being
greatly magnified laterally. It is seen that the needle un-
dergoes each day a vibration similar in a general way to the
grand swing of three centuries or so shown in the upper
diagram. The magnitude of the daily movement in north-
ern United States ranges from 5' in winter to nearly 12'
in summer time. The needle is in its mean daily position
between 10 and 11 a. m. for all seasons. The diagram rep-
resents the normal magnetic day, of which there are per-
haps five or six per month.

Local Attraction. — The pointing of the needle is af-
fected by the close proximity of magnetic substances, such

5_\ ^!

a\

^

PhleLevenhey,/_ J \

(C)

Fig. 13.

USE OF THE COMPASS. 49

as iron ore, wire fences, railroad rails, etc. However, local
attraction does not prevent correct work, provided back
and fore sights are taken withont change of magnetic condi-
tions. It is therefore especially important to avoid disturb-
ances of the needle by the chain, axe, passing vehicles, elec-
tric wires, etc., or by articles on the person of the observer,
such as keys, knife, spectacle frame, wire in the hat rim,
reading glass case, etc. Also the glass cover may become
electrified by friction and attract the needle, in which case
it may be discharged with the moistened finger, or by
breathing on it.

The Vernier. — The vernier is an auxiliary scale used
to read fractional parts of the divisions of the main scale or
limb. Verniers are retrograde or direct, according as the
divisions on the vernier are larger or smaller than those on
the limb. The vernier used on compasses for the setting ofE
of the declination is direct, and is usually of the type shown
in (c) of Fig. 13. In reading a vernier of any kind, blunders
may be avoided by first estimating the fraction by eye be-
fore noting the matched lines on the two scales.

USE OF THE COMPASS.

TJse. — The compass is used: (1)" to determine the bear-
ings of lines ; (2) to measure the angle formed by two lines ;
(3) to retrace old lines. The bearing of a line is the hori-
zontal angle between the line and a meridian through one
end of it. Bearings are measured from the north or south
point 90° each way. The angle between two lines is the
difEerence in their directions as indicated by the bearings.
Having the true bearings of one side of a polygon, the true
bearings of the others may be obtained by algebraic addi-
tion of the angles ; or by using the declination vernier so
as to read the true bearing direct on the fore sights.

Practical Hints. — Point the north end of the compass
box along the line and read the north end of the needle.
Protect the pivot from needless wear by turning the needle
in about the proper direction before releasing it. Always
lift the needle before disturbing the compass. Habitually
the needle by estimation to the nearest five minutes, that
is, to the one-sixth part of one-half degree, which is the
usual subdivision of the compass box. Care should be
taken to avoid parallax in reading the needle.

5

50 THE COMPASS.

Elementary Lines. — The elementary Mnes of the compass,
shown in (a) of Fig. 10, are : (1) the line of sight; (3) the
vertical axis; (3) the plate level lines.

The maker should see: (1) that the needle is strongly-
magnetized; (3) that the magnetic axis corresponds with
the line joining the two ends; (3) that the metal in the
compass box is non-magnetic; (4) that the line of sights
passes through the center of graduation; (5) that the
plates are perpendicular to the vertical axis; (6) that the
zero of the vernier coincides with the line of sights.

The needle may be magnetized with a bar magnet or by
putting it into the magnetic field of a dynamo. The metal
of the compass box may be tested by reading the needle,
then moving the vernier and noting if the needle has moved
the same amount, this process being repeated at intervals
around the full circle.

The Principle of Reversion. — In adjusting surveying
instruments, the presence, direction and amount of the er-
ror are made evident by the method of reversions which
doubles the apparent error. If there is no difEerence after
reversion, there is no error.

Plate Levels. — To make the plane of the plate level lines
perpendieular to the vertical axis. — Level up the instrument
by means of the plate levels and reverse the compass box
in azimuth, that is, turn it through a horizontal angle of
180°. Correct one-half the error, if any, by means of the
adjusting screws at the end of the level tube, and bring the
bubble to the center by the ball and socket joint. The rea-
sons for this process are shown in (a) of Eig. 13.

Sights. — To make the plane of sights normal to the plane
of the plate level lines. — With one sight removed and the
instrument leveled, range in with the remaining sight two
points as far apart vertically as possible, say on the side of
a building. Eeverse in azimuth and bring the bottom of the
sight in range with the lower point ; if the upper point is
then in range, the sight is in adjustment. If not, correct
one-half the error by putting paper under one side, or by
filing oif the other side. Repeat process for the other sight.

The Pivot. — To adjust the pirot to the center of the gradu-
ated eircle. — Set the south end of the needle to read zero,
and read the north end of the needle ; reverse the compass
box in azimuth, repeat the observations, and correct one-
half the difEerence between the two readings of the north

PEOBLEMS.

51

end of the needle by bending the pivot, using the special
wrench for the purpose. Turn the compass box 90° and
repeat. See (b), Fig. 13.

The Needle. — To straighten the needle. — Having adjusted
the pivot, set the north end of the needle to read zero and
bend the needle so that the south end reads zero also. Turn
the compass box and test for other graduations.

PEOBLEMS WITH THE COMPASS.

PEOBLEM Bl.

DECLINATION OP THE MAGNETIC
NEEDLE.

(a) Equipment. — Surveyors' compass, flag pole, reading
glass.

(b) Problem. — At a point on the true meridian determine
the mean magnetic declination with the surveyors' compass.

(c) Methods. — (1) Set the compass over one point and a,
flag pole at another on the true meridian. (3) Lower the
needle and sight at the flag pole carefully with the north
end of the compass box to the front. (3) When the vibra-

DCCLII ATION
Hcedic Mean
Undiaq

nims'e-

IHiS'E-

3 mtuid^as IVesff
thr

most p. vhabJe
jtaf/ffn

OF
Time
PM-
Z'OS
Z=/l
Z-/S
Z:ZZ
Z:Z7
2:31
Z--3S
Z:42
Z:4S
Z.-S4

IHeedl

Mean
P-M-

for da. '/y var^ '9f/ci1

Asstf/tf. 'Tff tfiai fhe m^nef/c

^I't/ejis sre nt\m3/ fqr f/7f

ibf cer.

fy PIffi rvjff of Oa/Iy V, *rf3t/OA
a^detf
'e-aa
tbee

JfJ'Jl'

va/vg o.

fftvei ■ .
fifr thi r part/iff lar Mia/rff~

Z--3C

fo
■Jiff
vJj-

/ esf/m ^t/on

WITH SUBVEVOI^

OcflZ.'m.fZHours)
i/sffef 0ur/ey Ccmpi

reeejtf/y rema^nfi i
Sef- cffmpass on true

Unafi'ff/l
S/ffi^af f/aypo/^ef

a dManoe ofZPfiFf-

neetffe ty

Cdue s/xt/j part

carefuJfy avoi'di't ^ paraJ/ax

magnetic d/sforbi mces

f/me fo nearesf
PJsfvri'et^ neei^Je

pi'vofaatf vei

yi/fien osc//fafm -s

rereatf fJie neeiflff
Cffaf/at/ttt^ fhe prt

utive reatfmpSf

ra/T^e ofaofm

irf'es-f were oht^n7ed-

's Compass-

'Jearanif Coo/-
■sH^Ze-ff/eeaJe
I'zeo^f ant/ ^Vatc/r-
mer/t/jan mff? i/ec-
on n7er/i//an sf"
fo Sm/niftes
'oj?e-/i3/F decree) f
and
Observed
m/nt/te •

iy ///^ft'ny if from
*; t/ren

^r/f 'ed slff/7f/n^;

on/// /en consec-
'lavlng 3 majcfmum
^re f/jan ten min—

62 THE COMPASS.

tions of the needle have ceased, move the vernier by means
of the tangent screw so that the north end of the needle
reads zero, and check the sighting of the compass. (4)
Read the declination on the vernier to the nea,rest minute.
(5) Lift the needle, verify the zero needle reading and the
sighting, read the vernier and record; repeat the process
until ten satisfactory consecutive values of the declination
are obtained. Observe the time of each reading to the near-
est minute. (6) Correct the mean of the ten values for
daily variation by reference to the diagram. Fig. U, using
the mean time. Record and reduce the data as in the form.
( Note that the values in the form were obtained by estimat-
ing the nearest five minutes. Which is better? Try both
if time allows.)

PROBLEM B3. ANGLES OF TRIANGLE WITH COMPASS.

(a) Equipment. — Survej'ors' compass, two flag poles,

(b) rrobtcni. — Measure the angles of a given triangle
with the surveyors' compass.

(c) Methods. — (1) Set the compass over one of the vertices
of the triangle and a flag pole behind each of the other two.
(2) Lower the needle and sight at one of the flag poles care-
fully, with the north end of the box to the front. (3) AVhen
the vibrations have ceased, read the north end of the needle
to the nearest five minutes by estimation. (4) Lift the
needle, verify the sighting and also the reading. (5) Turn
the compass box to the other point and determine the bear-
ing, as before. The required angle is the difference between
the two bearings. (6) Measure the other two angles in
like manner. The error of closure mvist not exceed 5

PROBLEM B3. TRAVERSE OF FIELD WITH COMPASS.

(a) Equipment. — Surveyors' compass, 2 flag poles, engi-
neers' chain, set of chaining pins.

(b) I'rohiem. — Determine the bearings of the sides of an
assigned field with the surveyors' compass and measure the
lengths of the sides with an engineers' chain.

(c) Mcthod-i. — (1) Set the compass over one of the corners
of the fielfl which is free from local attraction, and set off
the declination with the vernier. (2) Take back sight on
the last point to the left and fore sight to the next point

PROBLEMS.

53

A

Statfon
S
8
6

NSLI5
Line

S-6
S-g
S-5

e-6

6-g
6-S

OF Tl
ObMrvEd
Bearing
5-g3'js')V

hsWe

V49'm
M9°M'£

ilAMSL

Needle
Angle

77'3S'

S4'4S'

47'4S'

z 5-6

-8

WITH Surveyors

Observerfi, R-Roe

0cf/3//4-fZ//our£^

Used hurley Comf

£sch bearing w&

(fup/j'cafe, the /

turbed 3/?d tJn

(P/screpency not i

Compass ■

fed/e being d/'S"

JSD'HS'

9 ejtceed S m/nufes^

\

^x — r
X

*.5

a

/

>,

7RAVE

RSE • FlEU

A-B-C

D-E

WITH Compass At

D Chain-

station

Line

Observed! Inten'or

Distance

Observers : J- Doe &

'K-^ae-

Bearing

Angle

Bearing

Ft-

0ct-J6, '14. f 3 Naurs)

Clears Windy

A

A-E

5-6s'sm

03'15'

Used Si/r/eyCompa

^s, locker if^Z4-

A-S

i-3Z'4S'f-

iJ2'45i

33e-£

era wiief7 poinfing

B

B-A

mi'fsh

/SdW

trae fiar/h hysej

fin0 off declination

B-C

V43'/Si'

54i'K'B

4e4-e

m'fJj vernier an o

iciinai-iar7 arc oF

C

C-B

f43'^f'n

SS'fS'

Jf3'36'F-

C-D

ss/'j5'n

w'si'n

4n-3

th a- E/?d af{ffjnpaff&

D

D-C

fs/is'e

m'ss'

. ,

toward tiie fanv

ird station and

p-e

vzr^m

m'm

6J6-0

Heedls-

£

E-P

iZZ^S'i

S7'S0'

£-A

IfSO'jf^

mWe

241.6

N A

S4S'PS'

E<t^ I \

H

See C3

'eiflafm

aflat

'ft/des

wddep

irfi/res

JV 5 \

on

pp.

\

i 5

See a

la/latie

n qF ai

•ea on

pf,.

Al/owabie error of

V "^

closure ™ ^/O^'

Cliained each line

once w/fh Fngineers*

chain -lengi-fi of 1

'Aa/n^/aa-as/^f-

\

J

54 THE COMPASS.

to the right, following the methods used in Problem B3.
(3) Repeat this process for the remaining corners of the
polygon taken in succession to the right. (4) Chain the
sides of the field to the nearest 0.1 foot by estimation. (5)
Compare the chain with standard. (6) From the observed
bearings compute the interior angles of the field, and the
true bearings of the sides. The angular error of closure
must not exceed 10 minutes for a five-sided field. Record
and reduce data as in the form.

PROBLEM B4. AREA OP FIELD WITH COMPASS.

(a) Equipment. — Five-place logarithms.

(b) ProWem. — Compute the area of the assigned field by
means of latitudes and departures.

Laiiiude Cprojection on merid'ran)
]pfl>?rti/^efrjgg = Oisfance "Cosine Bearing ■

'^"^^■'ho/ A N Depdrfure (pnyech'ononEandW line)
~ Disiance '^ Sine Bearing ■
\Mendian Disfance of a point is itb
E distance Eor Wofan assumed
reference meridian ■

' Meridian d/sfance of a line is fhe

Compass % Merid-Disf of ifs middle poinh

(c) Methods. — (1) Prepare forms for calculations; tran-
scribe data, and carefully verify copy. (2) Compute lati-
tudes and departures by contracted multiplication, preserv-
ing results to the nearest 0.1 foot. (3) Make the same cal-
culations by logarithms, as a check. (4) Determine the ac-
tual linear error of closure. (5) Determine the permissible
error of closure (see chapter on errors of surveying). (6)
If consistent, distribute the errors in proportion to the sev-
eral latitudes and departures, respectively, repeating the
justed latitudes and departures, and verify transcript. (8)
Calculate the meridian distances of the several stations and
lines. (9) Calculate the latitude coordinates. (10) Calcu-
late the partial trapezoidal areas by multiplying the merid-
ian distances of the lines by the respective latitudes, pre-
serving consistent accuracy, and observing algebraic signs.
(11) Determine the area by taking the algebraic sum of the
partial areas. Reduce to acres, and correct for standard.

PROBLEMS.

55

ComiIass T(iaver4e

Observed
Distance

Line

Bearing

AB

CD

DE

ff/inW

£A H-eaWf. T4I-S

Ft-

3}e-s

464-6

4S3-3

OF

Compdtation
Multipli ■
cation
(Lat-Oisi

HTlesyi 616-0

Distribution oF Error
Line Lat. Dep<

AB . -- • '-

BC

CO

ne

-h^h -^^

Field

Logar-
ithms
xCos-Bg]
2-52\$9i
9-\$248l

U5ISQ
(2S5-0I)
2-B6708
9-86355

4-£3Ji

e/e-0

123Z
370

\-B-C-D
oF Lati
Computed
Latitude

Ft.
S-ZS3.0

S-330.3

2-S3KI
(33\$3i)
2-6S4Z2
9-ISZ4S

zisin

(6S-6S}
2-7S9SS

i-7S64}

(nasi)

Z-3!3I0
■M3«

M71S3
Erro

tudes'

/■ 6S-6

S-6.

s- o-t

- oF C
£rror

fO-S'

Oct-n, '/4 Compt/fer, J-Poe-

P3t3 rrom pp- Jrsnscripf 0-K-

E Latitudes and Departures-

Latitude

Ft-

s-^g^-g

Multipli-
cation
[Deji-^Dist

5- 61-6

itni-i

lf-M-4

tl.6)C-S
3-190-S

Computation of Departures_

Logar-
itnme
KSinBs)
Z-5Z191
9-733IS

2-?eoie
(m-M)

2-66701
■S34tl

S5

Z31-S8

J4SO
217

Computed
Departure

Ft-
E-|g^■0

2-50IS4

(3ms)

2-11422
9-39S57

2-17979
(471-40)
2-7!9Sg
9-S7SI4

2-3(472
(231-59)
2-3g3IO
9-93934

2-32244

m47!-4

W-231-6

£-70S-S
»7IO-0

W- OS

Departure

Fh
E-lgZ-0

-0-2

W-47g-2

W-23/-S

e-7e9-7
0709-7

(See Distrain)

Pu-misasUe £rnr= -^S,

1/1 nnn -^ * ^''

Ocf- 77,14 ■ Compufer, J-Ooe.
lljtj frompp- Tr3nscr/pf 0-K-

C-D-E, Compass Traverse.

56 THE COMPASS.

Follow the form. (13) Make plat of field, using total rect-
angular coordinates, and checking by polar planimeter.

PROBLEM B5. ADJUSTMENT OF THE COMPASS.

(a) Equipment. — Surveyors' compass, adjusting pin, small
screw driver.

(b) Prolilem. — Make the necessary tests and adjustments
of the surveyors' compass.

(c) Methods. — Observe the following program: (1) test
the magnetism of the needle; (2) test the metal of the
compass box; (3) test and adjust the plate levels; (4) test
the sights; (5) test the pivot; (6) test the needle.

PROBLEM B6. COMPARISON OF DIFFERENT MAKES
AND TYPES OP COMPASSES.

(a) Equipment. — Department equipment, catalogs of rep-
resentative makers of compasses.

(b) Prohlem. — Make a critical comparison of the several
types of compasses.

(c) Methods. — Examine the department equipment and
study the several catalogs carefully, noting the character-
istic features, prices, etc. The following items, at least,
should be included in the tabulated report : name of instru-
ment, length of needle, length of alidade, vernier, tripod,
weight, price, etc.

CHAPTER IV.
THE LEVEL.

Description. — The engineers' level consists of a line of
sight attached to a bubble vial and a vertical axis. Two
types of level, the wye and dumpy, Fig. 14, are used by engi-
neers. In the former the telescope rests in Y-shaped sup-
ports, from which it may be removed. In the dumpy level
the telescope is fixed. The dumpy is a favorite with IJritish

Engineers' Wye Level.

Fig. 14.

Dumpy Level.

Fig. 15. — Types of Levels.

57

58 THE LEVEL.

and the wye level with American engineers. (The dumpy
level with erecting eye-piece has been adopted as standard
by the Division of Valuation, Interstate Commerce Com-
mission.) The two types differ chiefly in the methods of
adjustment. A third type, not shown in the cuts, is called
the level of precision because of its use solely for work of
extreme refinement.

In Fig. 15 are shown: (a) an architects' or builders' level
of the wye type; (b) a road builders' level of the dumpy
type; (c) a reconnaissance level with a decimal scale for
reading horizontal distances direct; (d) a water level some-
times used in locating contours; (e) a Locke hand level;
(f) a clinometer; (g) a binocular hand level.

THE TELESCOPE.

Principles. — The telescope used in the engineers' level
and transit, shown in section in Figs. 16 and 23, consists
of an objective or ohject glass which collects the light and
forms an image in the plane of the cross-hairs, and an ocular
or eyepiece which magnifies the image and cross-hairs. The
cross-hairs are thus at the common focus of the oujective
and eyepiece. The principle of this type of telescope, both
optically and mechanically, may be illustrated by the photo-
graphic camera if cross lines be ruled on the ground glass
focusing plate and a microscope be used in viewing the
image formed by the lens. Telescopes of the above class are
called measuring telescopes, while those of the opera glass
type are termed seeing telescopes. The latter have no real
image formed between the object glass and eyepiece.

Line of Colliniation. — The telescope of the level or tran-
sit may be represented by a line, called the line of collima-
tion, which joins the optical center of the objective and the
intersection of the cross-hairs. The optical center is a point
such that a ray of light passing through it emerges from
the lens parallel to its original direction. The line of coUi-
mation is independent of the eyepiece.

Objective. — The objective is a double convex or plano-
convex lens. In all good telescopes the objective is com-
pound, that is, made up of two lenses, with the view to cor-
rect two serious optical defects to which a simple lens is
subject. These defects are called chromatic aberration and
spherical aberration.

Chromatic aberration is the separation, by the objective,
of white light into its component colors. A lens which is

Tangent Line of level Tube

Optical Center
: oF Objective

Intersection of
Cross Hairs^f

(a)

ObjeclCkss
(forms imaqe in plane
oFeross-hairs)

Vertical Axisf.
Clip^....,^ ..'rising

Tangent to Bubble

/Azimuth 5crew5 1^

I eye I Bar-
(b)

i''^l?imj5 fgual -'■""<{

Line oF 'WoliimationMxis oFW&derJ

Bottom \ElementoFFin(f3 i

Tangent Y ~VtoBubbIe "T l

W Eyepiece'r
(MagniFiesimaqe
and cross-hairs)

-""Vertical Axis
Clip^-:.^jrl?inff

WyeMs
'^Altitude Screirs
footScrem

^;_Bj.q^J

^«i-

L>L

(C) \

I

,_ True Line of Collimalion

True Level Line from Target
i- — Length oFBack Sight egaa/j
True Level Line Through

' Bottom\FkmentoF things jf ^

Tanaeni:V ~\"to Bubble 1 f

(d) j

SI

_ TrueJJnejf Collima2ion_ l^

toTargetiSase oFCone)

^■fo Length of foresight -—
S\ Top oF Peg.

(e) Correct Levels by Equal Sights.

^ True Line oFCollimation

hi He t hod.

1 True Line oFCollimation.

True Level line fnd-^
^indfleiho'd.

60 THE LEVEL.

free from this defect is called achromatic. A telescope is
tested for the chromatic defect by focusing on a bright ob-
ject, such as a piece of paper with the sun shining on it,
and noting the colors on the edge of the object and es-
pecially at the edge of the field of view as the focus is
slightly deranged. Yellow and purple are the characteris-
tic colors indicating good qualities in the lens.

Spherical aberration is a defect which prevail? to a serious
extent in a simple lens having spherical surfaces. It is due
to a difference in the focal distance for different concentric
or annular spaces of the objective, so that the plane of focus
for rays passing through the outer edges of the lens is dif-
ferent from that of the middle portion. A telescope is
tested for this defect by focusing on a well defined object,
such as a printed page, with the raj's of light cut off alter-
nately from the middle and the edge of the lens. This is
best done by means of a circular piece of paper with a
small round hole in it.

As a rule, the object glass in good levels and transits con-
sists of a double convex lens of crown glass fitted to a con-
cavo-convex or a plano-concave lens of flint glass, the
former to the front. The defects described above are
avoided through the different dispersive and refractive
powers of the two kinds of glass, and by grinding the sur-
faces of the two lenses to the proper curvatures.

Eyepiece. — As in the camera, the image formed by the
objective is inverted, so that if a simple microscope be used
as an eyepiece, the observer sees objects inverted. Such
an eyepiece is commonly used on the dumpy level, as shown
in rig. 14. This form of eyepiece consists of two plano-
convex lenses with their convex sides facing each other.
The form of eyepiece most used in American instruments is
the erecting eyepiece in which two plano-convex lenses re-
place each of the two in the simpler form. The erecting
eyepiece is much longer than the simple one, as may be
seen at a glance in Fig. 14. While the simple eyepiece
causes a little confusion at first, owing to the inversion of
objects, it is much siiperior to the erecting eyepiece in the
matter of clearness and illumination.

The chief inherent defect in the eyepiece is a lade of
flatness of the field. A single lens usually causes a distor-
tion or curving of straight lines in the image, especially to-
wards the edge of the field. A telescope is tested for this
defect by observing a series of parallel right lines, prefer-

THE TELESCOPE. 61

ably a series of concentric squares, which fill the entire
field of view.

In the best achromatic eyepieces, one or more of the sep-
arate lenses may be compounded, the curvatures being sucli
as to eliminate the color defect and give rectilinear qualities
to the lens or combination of lenses.

Definition. — The definition of a telescope depends upon
the finish and also the accuracy of the grinding of the
curved surfaces of the lenses. It may be tested by reading
the time on a watch or a finely printed page at some dis-
tance from the instrument.

Illumination. — Illumination and definition are apt to
be confused. Poor definition causes indefinite details, while
poor illumination causes faintness in the image. The latter
may be tested about dusk, or in a room which can be grad-
ually darkened, and can be best appreciated if two tele-
scopes of different illuminating qualities be compared.

Aperture of Objective. — The aperture or effective di-
ameter of the objective is determined by moving the end of
a pencil slowly into the field and noting the point where it
first appears to the eye when held say 8 or 10 inches back
from the eyepiece. The process should be repeated in the
reverse order. The annular space is deducted from the
actual diameter to obtain the real aperture.

Size of Field. — The field of the telescope is determined by
noting the angle between the extreme rays of light which
enter the effective aperture of the objective. With the tran-
sit telescope, the limiting points may be marked on the side
of a building and the angle measured directly with the
plates ; or with either level or transit the angle may be cal-
culated from the measured spread in a given distance. For
simplicity, a distance of 57.3 feet may be taken, and the re-
sult reduced to minutes.

Magnifying Power. — The magnifying power of a tele-
scope is expressed in diameters, or as the multiplication of
linear dimension. It is determined most readily by making
an observation with both eyes open, one looking through
the telescope and the other by natural vision. The com-
parison may be made by means of a leveling rod, or the
courses of brick or weather-boarding on the side of a house
may be used in like manner.

Parallax. — Parallax is the apparent movement of the
cross-hairs on the object with a slight movement of the ej'e,
and is due to imperfect focusing of the eyepiece on the
cross-hairs before focusing the objective. The eyepiece

62

THE LEVEL.

should be focused tritli the eye normal, the cross-hairs being
illuminated by holding the note book page or other white
object a few inches in front of the objective.

(/) (2)

(5) f4)
(b)

Fig. 17.

Cross-Hairs. — The cross-hairs are attached to a ring or
reticule ■n'hich is held by two pairs of capstan headed
screws. The hairs usually consist of spider lines, although
some makers use platinum wires for the purpose. To re-
move the reticule the eyepiece is taken out, one pair of
screws is removed and a sharpened stick is inserted in a
screw hole. The best spider lines are obtained from the
spider's e.^^ nest.

In Fig, 17, (a) shows the usual arrangement of the cross-
hair ring and the method of attaching the hairs ; (b) shows
the number and positions of hairs used, (1) being the most
common, (2) the form for stadia work with the transit and
also for estimating the lengths of sights with the level, (3)
a form used by some makers with the level, and (4) a style
found in English levels ; (c) shows the e^^ pod or case of
the large brown spider (about half size) which yields the
best lines for engineering instruments; (d) illustrates a
convenient vest pocket outfit for replacing cross-hairs in
the field, consisting of a supply of spider lines and some
adhesive paper (bank note repair paper) each in a capsule
or tin tube, and several sharpened sticks for stretching the
hairs. Cross-hairs stretched in this manner may last indefi-
nitely, or they may be fastened on permanently with shel-
lac at the first opportunity.

THE BUBBLE VIAL.

Principle. — The spirit level consists of a sealed glass
tube nearly filled ^^■ith ether or other liquid, and bent or
ground so that the action of gravity on the liquid may indi-

THE BUBBLE VIAL.

63

cate a level line by means of the bubble. The delicacy of the
buble depends upon the radius of the curvature in a verti-
cal plane, the greater the radius the more delicate the level.
Thus, for example, a perfectly straight tube could not be
used as a level.

Curvature of Bubble Vials. — Good bubble vials are now-
made by grinding or polishing the interior surface of a se-
lected glass tvibe by revolution, as indicated in exaggerated
form at (a) Pig. 18. As a general rule, only one side of
the vial is actually used, it being customary to encase it in

Tophnqent_ Line_ _
J Axh ofLevelTabe_

i\ b \\

(9> \

i r-4d

tfsecfienk)

Fig. 18.

a brass tube having a slot or race on one side. However,
both sides of the vial may be utilized, as in (b) and (c),
Fig. 18, which show the reversion level adapted to the tran-
sit and wye level, respectively. Bubble vials of several sizes
are shown in (d), Fig. 18. It was formerly customary to
grind out only a portion of the upper side of the glass tube,
as shown at (e). The cheap vial, consisting merely of a
bent tube, used mostly in carpenters' and masons' levels, is

64 THE LEVEL.

shown at (f) ; and a method of increasing the precision of
the bent tube by tilting it is indicated at (g), Fig. 18.

Delicacy. — The delicacy of the bubble vial is designated
either by the radius, usually in feet, or by the central angle
in seconds corresponding to one division or one inch of the
bubble scale. Two methods are employed to determine the
delicacy of level vials, (1) by the optical method, as at (h),
Fig. 18, where the radius is calculated from an observed tar-
get movement at a given distance for an observed bubble
movement, the two triangles being similar; and (2) by the
level tester, as at (i), by means of which the angular move-
ment of the bubble. The engineer usually employs the radial
designation, while the maker expresses the delicacy in an-
gular units. As shown at (h) and (i),Pig. 18, the radius in
feet is equal to 17,189 divided by seconds per inch of bubble.

Bubble Line. — The relations of the bubble to the other
parts of the instrument are best understood by representing
the vial by a line. This line may be either the axis of the
surface of revolution in (a). Fig. 18, or to provide for either
of the three forms of vial shown, it may be taken as the
tangent line at the middle or top point. This tangent line
will be meant hereafter in referring to the bubble line.

LEVELING EODS.

Types. — There are two classes or types of leveling rods ;
(1) target rods, having, a sliding target which is brought
into the line of sight by signals from the leveler ; and (2)
the leveler.

In Fig. 19, (a) is the Philadelphia rod ; (b) the New York
rod; and (c) the Boston rod. The first is either a target
or self-reading rod ; the second is a target rod, but may be
read from the instrument when the rod is " short " ; the
Boston rod is strictly a target rod. The Philadelphia rod is
perhaps the favorite for most purposes, and the Boston rod
is used least. A folding self-reading rod is shown at (d).
Fig. 19 ; (e) is a woven pocket device which may be tacked
to a strip of wood and used as a leveling rod; (f) is a rail-
road contouring rod with an adjustable base ; (g) is a plain
rod graduated to feet, for use with the water level.

Targets. — The targets shown on the Philadelphia and
Xew York rods, (a) and (b). Fig. 19, are called quadrant
targets. That on the Boston rod, (c), is a modified form of

USE OF THE LEVEL.

65

^

2

6.
4

do

6,
±

.2.

4
6.

PC

4

a

pzi

4
6^

lS,

4

6.

. .a.

4

6

a

r/)

D

Pig. 19.

the diamond target. A special form, called the corner tar-
get, is bent to fit two sides of the rod to assist in plumb-
ing it, and another target has two parallel planes for
the same purpose. A detachable rod level is shown at (h).
The target on rod (b), with the zero of the vernier 0.09 foot
below the center of the target, frequently causes blunders.

USE OF THE LEVEL.

Use. — The engineers' level is used: (1) to determine dif-
ferences of elevation; (2) to make profile surveys; (3) to
locate contours; (4) to establish grade lines; (5) to cross
section; (6) to run lines.

66 THE LEVEL.

Differential Leveling. — Differential leveling consists of
finding the difference of elevation between two or more
points. In the simplest case the difference of elevation be-
tween two points may be found from a single setting of
the level, the leveling rod being used to determine the
vertical distance from the plane of the instrument to each
of the two points, and the difference between the rod read-
ings taken. When the distance between the two points is
too great, either vertically or horizontally, or both, to ad-
mit of this simple process, two or more settings of the level
are taken so as to secure a connected series of rod read-
ings, the algebraic sum of which gives the desired differ-
ence of elevation. This difference may be expressed either
by the numerical result of the algebraic sum of the rod
readings, or by assuming an elevation for the beginning
point and calculating the elevation of the closing point by
means of the observed rod readings.

A haelc sight is a rod reading taken to determine the height
of the instrument. A fore sight is a rod reading taken to de-
termine the height of a point. A hench mark is a point se-
lected or established for permanent reference in leveling
operations. A turning point is a temporary reference point
used in moving the instrument ahead to a new setting. The
same point is often both a turning point and bench mark.
The datum is the plane or surface of reference from which
the elevations are reckoned ; it may be sea level, or an arbi-
trary local datum. A level line is a line parallel to the sur-
face of a smooth body of water. A horizontal line is
tangent to a level line at any point. The curvature varies
as the square of the distance from the point of tangeney,
and is 0.001 foot in 304 feet, or 8 inches in one mile.

In Fig. 19, (i) shows a metal and also a wooden peg com-
monly used for turning points. Several forms of bench
marks are shown in Fig. 19 ; ( j) is a mark on the corner
of a stone water-table ; (k) a rivet leaded into a hole
drilled in a stone slab ; (1) a railroad spike driven into a
wooden post or telegraph pole ; (m) a projection cut on the
root of a tree, preferably with a spike driven vertically into
the top of the bench, and usually with a blaze above
marked " B. M. No. — ." All bench marks and also turning
points should be clearly described in the notes.

Fig. 19a shows the essential details of differential level-
ing. In practice the calculations are made mentally.

Two chief essentials in correct differential leveling are :
(1) that the 'bu'bl)le lie in exactly the same position (usu-

USE OF THE LEVEL.

67

ally the middle) on hoth hack and fore sight; and (2) that
the length of hack sight and fore sight, horizontally, shall
be balanced. It is seen at (e), Fig. 16, that with the bubble
always in the middle, the line of collimation generates a
horizontal plane when in perfect adjustment, but a cone
with axis vertical when out of adjustment; so that in tak-
ing equal distances in the opposite directions, the base of
the cone is used, this base being parallel to the true colli-

ejf.l.

level Line from B.M.Iio.I

5ta.

B.5.

H,l.

F5.

Elev.

Di5t.

Calculations

Description of 6.(1.5 and 05.

B.ni.

mo.oo

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100.00 mi
t4U B.i.

CilylbtmmBo/i.H.mciin,
mtertible, litM.BankBld'q

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-1.16 F.5.

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103.26

340

10320 01
H37.B.5.
107.63 /^Z

Peq, KEcanJ. 6reen'3 lot.

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104.01

1300

104.08 ez

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105.93 H3
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300

103.59 mi

n.W.bolt, (nickeJ)H3terplaa

+10.70

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(Bal3ifcedB.5.andF.5 0/st.

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+ -i.i3

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0,ecJtedF/er.iyI(B.5.,F5.IJ

Fig. 19a. — Details of Differential Leveling.

mation plane. In the best leveling practice the instrument
is adjusted as perfectly as possible and then used so that
the residual errors balance each other.

The three common styles of leveling rods may be read to
0.001 foot by vernier or by estimation on a scale to 0.005
foot. However, for most kinds of leveling, it is an absurd
refinement to read the rod closer than 0.01 foot, especially
with the usual maximum length of sight of 350 to 400 feet,
and with the more or less sluggish bubbles supplied in the
general run of leveling instruments. Furthermore, the
horizontal hair usually covers 0.01 foot or so of the target
at the maximum length of sight, that is, the target can
move that amount without being noticed by the observer.

68 THE LEVEL.

Profile Leveling. — Profile leveling consists of finding
the relative elevations of a series of representative points
along a surveyed line, for the purpose of constructing a pro-
file or vertical section. The skeleton of profile leveling, that
is, the precise bench marks and turning points with the
successive heights of instrument, is identical with differen-
tial leveling, already described. Having determined the
height of instrument by taking a back sight on a bench
mark of known or assumed, elevation, rod readings are
taken at proper intervals along the measured and staked
line. These readings are fore sights, but they are usually
termed intermediate siplits to distinguish them from the
more precise rod readings taken on turning points and
bench marks. On railroad surveys intermediate sights are
taken usually to the nearest 0.1 foot on the ground ; but in
other cases, such as tile and sewer surveys, intermediates
are often read to the nearest 0.01 foot on small pegs driven
beside the station stakes flush with the surface of the
ground. In railroad work, the benches, turning points,
and intermediates of special importance are commonly read
to 0.01 foot, although some engineers persist in the ques-
tionable practice of taking the nearest 0.001. In drainage
surveys the nearest 0.01 foot is usually taken on bench
marks, although more carefully than on the intermediate
peg points, and the nearest 0.1 foot is read on ground points.

The errors of profile leveling are balanced on turning
points by equal back and fore sights, as in differential lev-
eling. If the instrument is seriously out of adjustment, an
error is made in the case of odd bench marks with unbal-
anced sights, and also on all intermediate sights. However,
the error is usually unimportant when ground readings are
taken to the nearest 0.1 foot. In important leveling, such
as canal and drainage work, it is customary to run a line of
check levels to prove benches, before construction begins.

The profile is plotted to an exaggerated scale vertically
on a special paper, called profile paper. Three kinds, known
as plates A, B and C, are in general use. The most common
is plate A, which is ruled in ^4"iiich squares with a further
subdivision to %o inch vertically. In railroad profiles the
scales most used are 400 feet to the inch horizontally and
20 feet vertically. A still greater exaggeration is generally
used in drainage profiles.

Reciprocal Leveling. — The application of differential
leveling to the determination of the difference of elevation
between two bench marks separated by a wide river or gorge

USE OF THE LEVEL. 69

is termed reciprocal leveling. A setting of the level is
taken on each side of the river, and the mean of the two re-
sults is taken. The necessary unbalancing of distances in
one setting is balanced \ip in the other. Each back or fore
sight should be the mean of a series of careful observations.
In best practice, simultaneous readings are taken with two
levels. .

Contour Leveling. — Contour leveling is an application
of the methods of profile leveling to the location of contour
lines, that is, lines having the same elevation. Two methods
are employed: either (1) actually establishing points on
the adopted contour planes on the ground and then locat-
ing these points; or (2) taking random elevations at rep-
resentative points and interpolating the contour lines from
the plotted data. The latter is the more common. Tlie
chief ptirpose of contour leveling is to make a contour map,
and the process is essentially a part of topographic survey-
ing, where it will be more fully considered.

ally the concluding part of profile leveling. After making
the profile, the grade line is established by stretching a fine
thread through the ruling points, taking into account the
controlling conditions, such as maximum gradient or earth-
work quantities on a railroad profile, the carrying capacity
or the scour in the case of a ditch, etc. After laying the
grade line on the profile, notes are made of the data, and
the actual grade line is established. Two methods are used :
(1) the height of instrument is determined as usual, and
stakes are driven at measured intervals with their tops to
match calculated rod readings; and (2) a limited number
of ruling points are established by the first method or
otherwise, and the remaining stakes are " shot in " by con-
structing a line parallel to the ruling line used. The latter
is more rapid, since a constant rod reading is used ; how-
ever, the method is unreliable unless the foresight be
checked frequently on a fixed target.

Cross-Sectioning. — Cross-sectioning consists of staking
out the limits of the transverse section of an excavation or
embankment for the purpose of construction, and usually
includes the collection of data for the calculation of the
quantities. This may be done either with the engineers'
level, rod and tape line, or with special rods called cross-
section rods. The notes are taken as rectangular coordi-
nates, usually with reference to the center of the finished

70 THE LEVEL.

roadbed. The slope stakes are set where the side slope
lines pierce the surface of the ground.

Running Lines. — Lines are sometimes run with the en-
gineers' level, provision being made in most good levels for
the attachment of a plumb bob. A line may be prolonged
by sighting in two points ahead. A clamp and tangent
movement are necessary. Some builders' levels have a
needle and also a roughly divided horizontal circle for use
in staking out buildings.

Practical Hints. — The following practical suggestions
apply more or less directly to all kinds of leveling, and
also in a general sense to transit work.

Speed. — Cultivate the habit of briskness in all the de-
tails of the work. While undue haste lowers the standard
of the results, an effort should be made to gain speed
steadily without sacrificing precision. Gain time for the
more important details by moving rapidly from point to
point. On rapid surveys both leveler and rodman often
move in a trot. Neither rodman nor leveler should delay
the other needlessly.

Care of Instruments. — Do not carry the level on the shoul-
der in climbing fences. Clamp the telescope slightly when
hanging down Keep the tripod legs at the proper tight-
ness, and avoid looseness in the tripod shoes. Avoid undue
exposure to the elements, and guard the level from injury.
Do not leave the instrument standing on the tripod in a
room over night.

Setting Up — In choosing a place to set the level up, con-
sider visibility and elevation of back point and probable
fore sight. Set up with plates about level. On side-hill
ground place one leg up hill. In general, place two tripod
shoes parallel to the general line of the levels.

Leveling Up. — A pair of foot screws should be shifted to
the general direction of the back or fore sight before level-
ing up. Set the foot screws up just to a snug bearing and
no tighter. If either pair of screws binds, loosen the other
pair a little The bubble moves with the left thumb. Level
up more precisely in the direction of the sight than trans-
verse to it, but do not neglect the latter. Inspect the bubble
squarely to avoid parallax, and also to prevent such blun-
ders as reading the bubble iive spaces off center.

Observations. — Adjust the eyepiece for parallax with the
eye unstrained. It is much easier on the eye to observe
with both eyes open. Read at the intersection of the cross-
hairs, since the horizontal hair may be inclined. Set the

USE OF THE LEVEL. 71

target approximately, check the bubble, and repeat the proc-
ess several times before approving the sight. Be certain
that the bubble is exactly in the middle at the instant of
approving the target. If the level has horizontal stadia
lines, beware of reading the wrong hair (the reticule may be
rotated one-quarter so as to have the extra hairs vertical,
or a filament may be attached to the middle horizontal hair
to assist in identifying it) . Avoid disturbance of the tripod
by stepping about the instrument. Assist the rodman in
plumbing the rod. Let signals be perfectly definite both as
to direction and amount, using the left hand for " up " and
the right for " down," or vice versa.

The leveler can work much more intelligently if he knows
the space covered on the rod by one division of the bubble
scale at the maximum length of sight, and also the space
on the rod hidden by the cross hair.

and then use it as though it were out of adjustment.

Balancing Sights. — Balance the length of back sight and
fore sight, and record the approximate distances. The dis-
tances in the two directions may be made equal roughly by
equality of focus, but it is better on careful work to pace
the distances or determine them by means of the stadia
lines in the level. If necessary to unbalance the sights,
they should be balanced up at the first opportunity, and in
general they should be in balance when closing on import-
ant benches. When leveling up or down steep slopes, fol-
low a zigzag course to avoid short sights. Take no sights
longer than 350 or 400 feet.

Leveling Rod. — The rod should be carefully plumbed, to
accomplish which the rodman should stand squarely behind
the rod and support it symmetrically between the tips of
the extended fingers of the two hands. In precise work
wave the rod to and fro towards the observer and take
the minimum reading of the target. With " short " rods
avoid the somewhat common blunder of 0.09 foot when the
vernier slot is below the center of the target. With " long "
rods, see that the target has not slipped from its true set-
and avoid blunders of 1 foot, 0.1 foot, etc. Careless rodinen
sometimes invert the rod. Each rod reading on turning'
points and bench marks should, when practicable, be read
independently by both rodman and leveler.

Bench Marks and Turning Points. — Wooden pegs or other
substantial points shoiild be used to turn the instrument

72 THE LEVEL.

on. Select bench marks with reference to ease of identifica-
tion, the balancing of sights, freedom from disturbance, etc.
As a rule, each bench mark should be used as a turning
point so that the final closure of the circuit may prove the
bench. Mark the benches and turning points and describe
them in the notes so plainly that a stranger may readily
find them. Green rodmen sometimes hammer at turning
point pegs with the rod. When leveling near a still body
of water, its surface may be used to save time and check
the work.

Record and Calculations. — Describe bench marks and turn-
ing points clearly. It is good practice to apply algebraic
signs to the back and fore sight rod readings. The eleva-
tions should be calculated as fast as the rod readings are
taken, and calculations on turning points should be made
independently by leveler and rodman, and results compared
at each point. The rodman may keep turning point notes
in the form of a single column. The calculations should be
further verified by adding up the columns of back sights
and fore sights for each circuit, or page, or day's work, and
the algebraic sum of the two compared with the difference
between the initial and last calculated elevation.

Error of Closure. — A circuit of levels run with a good
level by careful men, observing all the foregoing pre-
cautions, should check within 0.05 foot into the square root
of the length of the circuit in miles (equivalent to 0.007 foot
into the square root of the length of the circuit in 100-foot
stations). In closing a circuit, the error should be care-
fully determined, as above indicated, and the value of the
coefficient of precision found. (See discussion of errors of
leveling and precision diagrams in Chapter IX, Errors of
Surveying.)

Elementary Lines. — The principal elementary lines of
the wje level, as shown in Fig. 16, are: (1) the line of col-
limation ; (2) the bubble line; (3) the vertical axis. For
(4) the axis of the rings; (5) the bottom element of the
rings. The following relations should exist between these
lines ; (a) the line of collimation and bubble line should be
parallel ; (b) the bubble line should be perpendicular to the
vertical axis. The first of these relations involves two
steps, viz., (1) to make the bubble line parallel to the bot-

torn element of the rings, and (2) to make the line of col-
limation coincide with the axis of the rings. The other
relation involves the wye adjustment, and is similar to the
plate level adjustment described in the chapter on the com-
pass.

Bubble. — To make the 'bubble line parallel to the bottom
element of the rings. — Two steps are involved, (a) to place
the bubble line in the same plane with the bottom element,
and (b) to make the two lines parallel.

Azimuth Screws. — To make the bubble line in the same
plane with the bottom element of the rings. — Clamp the
level over a pair of foot screws, loosen the wye clips, and
level up ; rotate the telescope through a small angle, and
if the bubble mov^s away from the middle, bring it back
by means of the aximuth adjusting screws. Test by rotat-
ing in the opposite direction. Leave the screws snug.

Altitude Screws. — To make the bubble line and the bottom
element of the rings parallel. — Jlake the element level with
the foot screws and bring the bubble to the middle by
means of the altitude adjusting screws. The element is
made level by the method of reversions as follows : With
the level clamped over a pair of foot screws, as above, lift
the clips and level up precisely ; cautiously lift the tele-
scope out of the wyes, turn it end for end, and very gently
replace it in the wyes ; if the bubble moves, bring it half
way back by means of the foot scretvs. Before disturbing
adjusting screws make several reversals, and conclude the
adjustment with screws snug. This end for end reversal
is similar to that made with the carpenter's level, the
straight edge of the level corresponding to the element of
the rings. The lines involved are shown in Fig. 16.

Line of CoUimation. — To make the line of collimation co-
incide with the axis of the rings. — Loosen clips, sight on a
point, say a nail head or the level target, more distant than
the longest sight used in leveling; rotate the telescope half
way and note the movement of the hair, if any. The line
of collimation generates a cone, the axis of which is that
of the rings, and the apex of which is at the optical center
of the objective. Correct one-half the observed error by
means of the capstan headed screws which hold the cross-
tersection of the cross-hairs remains fixed on the same
point when reversed by rotation with reference to either
hair. The adjustment should be concluded with the screws
at a snug bearing.

74 THE LEVEL.

After collimating the instrument for a long distance, the
adjustment should be checked for a short distance, say 50
or 100 feet, so as to test the motion of the optical center
of the objective.

Bings. — The theory of the wye level demands perfect
equality of the rings, that is, the parallelism of the axis and
element, as in (c), Fig 16. Should the rings be unequal,
either from poor workmanship or uneven wear in service,
they form a cone instead of a cylinder, and the axis is not
parallel to the element, as in (d), Fig. 16. Under the latter
conditions, the principle of the wye level fails, and an in-
dependent test is demanded. This is known as the two-peg
test, the details of which are shown in (e) and (f). Fig. 16,
and described in the adjustments of the dumpy level. If,
after making the wye level adjustments above described,
the two-peg test shows that the line of collimation and
bubble line are not parallel, the rings are probably unequal
and the instrument should thereafter be adjusted as a
dumpy level. However, hasty conclusions should be guarded
against.

In case the instrument has a reversion level, shown at
(c), Fig. 18, the equality of the rings may be tested by
first adjusting the top tangent line of the bubble vial par-
allel to the bottom element of the rings, and then after ro-
tating the telescope half way round in the wyes, compare
the bottom (now above) tangent line of the vial with the
top (now below) element of the rings, all by the end for
end reversion. However, the exact parallelism of the top
and bottom tangent lines of the reversion level should first
be proven by the two-peg method.

Wyes. — To make bttihle line perpendicular to the vertical
axis. — Make the vertical axis vertical and bring the bubble
to the middle by means of the wye nuts. The vertical axis
is made vertical by reversion thus : With clips pinned, level
up ; reverse over the same pair of screws, and bring the
bubble half way back with the foot screws. When adjusted,
the bubble will remain in the middle during a complete rev-
olution. This adjustment is identical in principle with the
plate level adjustment of the compass and transit, illus-
the adjustment of the bubble line parallel to the element
of the rings. The wye adjustment is a convenience, not
a necessity.

Centering the Eyepiece. — After collimating the level,
the cross-hairs should appear in the center of the field.

The eyepiece is centered by moving its ring held by four
screws. This adjustment is desirable, but not essential.

Elementary Lines. — The principal elementary lines of
the dumpy level are identical vvith those of the wye level
(1) the line of coUimation; (2) the bubble line; (3) the
vertical axis. As in the wye level, the bubble line should be
(1) perpendicular to the vertical axis, and (2) parallel to
the line of coUimation. However, owing to the difference
in the construction of the two types of instrument, the
auxiliary elementary lines are not recognized in the dumpy
level. The transit with its attached level is identical in
principle with the dumpy level.

Bubble. — To make the iuhhle line perpendicular to the
vertical axis. — Make the vertical axis vertical ty the method
of reversions, and adjust the Jtuhhle to the middle. This
adjustment is identical in principle with the plate level
adjustment, shown in (a). Fig. 13. The bubble should re-
main in the middle through a complete revolution.

Line of CoUimation. — To make the line of coUimation
parallel to the iuiile line. — Construct a level line, and ad-
just the cross-hairs to agree with it. The level line is de-
termined either by using the surface of a pond of water, or
by driving two pegs at equal distances in opposite directions
from the instrument, and taking careful rod readings on
them with the bubble precisely in the middle, as shown at
(e). Fig. 16. For simplicity, the two pegs may be driven to
the same level, or two spikes may be driven at the same
level in the sides of two fence posts, say 400 feet apart.
Otherwise, determine the precise difference of elevation, as
indicated in (e). Fig. 16. Then set the level almost over
one of the pegs, level up, and as in the first method of (f).
Fig. 16, set the target of the leveling rod at the line of col-
limation, as indicated by the center of the object glass or
eyepiece (this can be done more precisely than most levels
will set the target at 400 feet distance) ; now with the rod
on the other peg, sight at the target (shifted to allow for
the difference if the two pegs are not on the same level) ;
adjust the cross-hair to the level line so constructed. If
preferred, the second method shown in (f). Fig. 16, mgy be
used ; the level is set back of one peg, rod readings are
taken on both pegs, allowance made for the difference in
level of the two pegs, if any, the inclination of the line of

76 THE LEVEL.

collimation determined, correction made for the small
triangle from the level to the first peg, and finally the level
line constructed by means of the calculated rod readings.
The second method is simplified and made practically
equivalent to the first by setting the level at minimum
focusing distance from the first peg. The small corrective
triangle is thus practically eliminated. Strictly speaking
the rod readings should be corrected for the earth's curva-
ture (0.001 foot in about 200 feet, or say 0.004 foot in 400
feet distance). However, the effect of curvature is reduced
by atmospheric refraction ; and with errors of observation,
sluggishness of bubble, etc., to contend with, the curvature
correction should be ignored, especially when the rod is
read to the nearest 0.01 foot.

(The foregoing process is known as the "two-peg adjust-
ment." Although exceedingly simple, this adjustment is
commonly regarded as a " bug-bear " by many American
engineers. But for it, the dumpy level would have the ex-
tended use in this country which it merits. It is said that
" the wye level is easy to adjust and usually needs adjust-
ment." Many good levelers employ the " two-peg test " to
prove the wye level adjustments. Time may be saved by
dumpy level are very stable.)

Uprights. — In some dumpy levels the uprights which
connect the telescope with the level bar are adjustable,
similar to the wyes of the wye level. This adjustment is
designed to bring the bubble line perpendicular to the ver-
tical axis in case the bubble is first adjusted parallel to the
line of collimation. However, the best order is that already
described, viz., first adjust the bubble line perpendicular
to the vertical axis, and then the line of collimation par-
allel to the bubble line, in which case the adjustable up-
rights are unnecessary.

PROBLEMS WITH THE LEVEL.

PROBLEM CI. DIFFERENTIAL LEVFILING WITH THE
HAND LEVEL (OR WATER LEVEL).

(a) Eqvipwent. — Hand level (or water level), rod gradu-
ated to feet.

(b) ProMem. — Run an assigned level circuit with the
hand level (or water level), observing the nearest 0.1 foot
by estimation, and closing baclt on the starting point.

PKOBLEMS. 7Y

(c) Methods. — (1) Determine the correct position of the
bubble of the hand level by sighting along a water table,
or sill course of a building, or by the principles of the two-
peg test. (If the water level is used, fill the tube so as to
have a good exposure of the colored water in the glass up-
rights.) (2) Take sights of 100 feet or so (paced), estimat-
ing the rod reading to the nearest 0.1 foot; balance back
and fore sights ; assume the elevation of the starting point,
and keep the notes in a single column by addition and sub-
traction, as in the 7th column. Fig. 19a. (3) Check back
on the first point. Determine coefficient of precision. (The
error of closure in feet should not exceed 0.5 Vdistance in

miles.)

PROBLEM C3. DIFFERENTIAL LEVELING WITH EN-
GINEERS' LEVEL (OR TRA^'SIT WITH ATTACHED
LEVEL).

(a) Equipment. — Engineers' level (or transit with at-
tached level), leveling rod, hatchet, pegs, spikes.

(b) Problem. — Run the assigned level circuit, observing
the nearest 0.01 foot, and closing back on the initial point.

(c) Methods. — Follow the practical suggestions given at
the conclusion of the " Use of the Level," giving special at-
tention to the following points: (1) eliminate parallax of
the eyepiece; (2) balance back and fore sight distances;
(3) have the bubble precisely in the middle at the instant
of sighting ; (4) both rodman and leveler read each rod and
also make the calculations independently; (5) calculate ele-
vations as rapidly as rod readings are obtained; (6) plumb
the rod; (7) avoid blunders; (8) determine coefficient of
precision; (9) no sights longer than 350 or 400 feet. Fol-
low the first form shown to begin with, — ^the other after
several circuits have been run.

PROBLEM C3. PROFILE LEVELING FOR A DRAIN.

(a) Equipment.— ^ngmeers' leveling instrument, leveling
rod, 100-foot steel tape, stakes, pegs, axe.

(b) Problem. — Make a survey, plat and profile, with esti-
mate of cuts and quantities for a drain under assigned con-
ditions.

78

THE LEVEL.

(c) Methods. — (1) Examine the ground, determine the
head and outlet of the drain, and select the general route.
(3) Stake out the line, set stakes every 50 feet, or oftener
if required to get a good profile, and drive a ground peg
flush, say 2 feet to the right (or left) of each stake ; record
data for mapping the line. (3) Starting with the assigned
datum or bench mark, run levels over the line of the pro-
posed drain, observing the nearest 0.01 foot both on turning
points and ground pegs, the former somewhat more care-
fully ; take rough ground levels, as required, to the nearest
0.1 foot; locate and determine the depth of intersecting
drains or pipe lines, or other objects which may influence
the grade line of the drain, and secure full data for placing
the same on the profile ; observe due care with the back and

Pig. 19b.

fore sights, as in differential leveling, and conclude the
leveling work with a line of check levels back to the initial
bench mark ; a permanent bench mark should be established
at each end of the drain, and if the length is considerable,
at one or more intermediate points as well. (4) Make plat
and profile of the drain line ; lay the grade line, taking into
account all ruling points ; calculate the cuts, both to the
nearest 0.01 foot, and also to the nearest 14"ii'ch; mark the
latter on the stakes for the information of the ditcher,
using waterproof keel and plain numerals ; make estimate
of the quantity of drain pipe, and of the cost of the job.
Follow the form and the profile in Eig. 19b.

PROBLEMS.

79

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PEOBLEMS. 81

(a) Equipment. — Eng^ineers' leveling instrument, leveling
rod, 100-foot steel tape, stakes, axe.

(b) ProMem. — Bun levels over a short section of line
staked out after the manner of railroad surveys, for the
purpose of constructing a profile.

(c) Methods. — Follow the general process outlined in the
preceding problem, taking rod readings to the nearest 0.01
foot on turning points and bench marks, and also on im-
portant profiling points, when consistent ; but take ground
rod readings only to the nearest 0.1 foot. In calculating
elevations, preserve the same degree of exactness in the re-
sult as observed in the rod reading, that is, when the rod
readings are taken to the nearest 0.1 foot, use only the
nearest 0.1 foot in the height of instrument to determine
the elevations. When a hub or station stake is to be used
as a turning point, the notes should show the ground rod
and elevation to the nearest 0.1 foot on the line preceding
the precise turning point record. Bench marks should be
selected with reference to their freedom from disturbance
during construction, and they should be located not more
than 1500 or 2000 feet apart along the line. Check levels by
the same parties should not differ more than 0.05 foot into
the square root of the length of circuit in miles. Back and
fore sights should be balanced, and no sight longer than
350 or 400 feet should be taken. In order to secure a repre-
sentative profile, ground rods should be taken not only at
every station stake, but also at every important change of
slope between station points. Pluses may be determined
either by pacing, or when short, by means of the leveling
rod. The rodman should keep a record of the turning
points. The notes should be checked and the other safe-
guards taken, as outlined in the practical hints under the
" Use of the Level." Bottoms of deep gullies may be taken
by means of the hand level, or with the engineers' level
tised like the hand level; or a "long" rod of 17 feet or
more may be obtained by holding the 12-foot rod 5 feet or
more from the ground.

The profile is best plotted by having another person read
off the data. The horizontal scale on railroad profiles is
usually 400 feet to the inch and the vertical scale 20 feet to
the inch. Gradients are expressed to the nearest 0.01 per

82

THE LEVEL.

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PKOBLEMS. 83

cent. It is usual to give the alinement notes and prominent
topography, as shown in Fig. 19c.

(The complete series of steps involved in railroad and
similar leveling for location and construction purposes is :
(1) setting the station stakes ; (2) running the levels ; (3)
making the profile; (4) laying the grade line on profile;
(5) calculating vertical curves; (6) cross-sectioning for
earthwork; (7) calculating earthwork quantities; (8) set-

PEOBLEM C5. VERTICAL CURVE.

(a) Equipment. — Drafting instruments, profile paper.

(b) ProMem. — Connect two grade lines by a parabolic
curve, as assigned.

(e) Methods. — (1) Plot the given grade lines, station
numbers, etc., on a sheet of profile paper. (2) Pind the
grade angle, i. e. the algebraic difference of the two rates
of grade. (3) Determine the length of the vertical curve by
of grade per station (notice the analogy to simple circular
curves). (4) Calculate the apex correction. (5) Determine
the corrections at the several stations or fractional stations
(as assigned), and tabulate the stations and elevations.
(6) Plot the vertical curve from the data so determined,
as in Fig. 19d. (7) Also compute and plot the same curve
by the method of chord gradients.

PROBLEM C6. ESTABLISHING A GRADE LINE.

(a) Equipment. — Leveling instrument, leveling rod, flag
pole, 100-foot steel tape, stakes, axe.

(b) Problem. — Establish an assigned grade line, (1) by
measured distances and calculate rod readings, and (2) by
" shooting in " the same line, for comparison..

(c) Methods. — (1) Stake oflE the distance between ruling
points, and drive stakes to the required grade, or if desir-
able, parallel to it, by dividing up the fall in proportion to
the distance. (2) Set the level over one ruling point and
determine the height from the point to the line of collima-
tion by means of the leveling rod ; set the flag pole behind
the other ruling point and establish a target, consisting of a
rubber band holding a strip of paper wrapped about the

84

THE LEVEL.

Vertfcd/ Curve.
■(J) ^ x Tangent Correct fans y.

COMPARISOM OF RESULTS

Elevation

By Tanijer?

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Sbabion.

Tanqenb

Curve

Chord 6ra

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Tanqent .

Correction.

Elevation.

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Elevation.

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Fb.

Fb.

Percent.

Percent.

Fb.

84

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Fig. 19d.

pole at a height equal to the rod reading ; having thus con-
structed a line parallel to the desired grade line, direct the
telescope on the fore sight target, and with the same rod
reading, " shoot in " the same stakes. Make careful record
of data and comparative results.

PEOBLEMS. 85

PROBLEM C7. SETTING SLOPE STAKES.

(a) Equipment. — Leveling instrument, self-reading level-
ing rod, 50-foot metallic tape, stakes, axe, marking crayon.
(Or, instead of levelling instrument and rod, use special
cross-sectioning rods, if assigned.)

(b) Prohlcm. — Set slope stakes for the construction of a

(c) Methods. — (Follow the methods described in Chap-
Sectioning.")

PROBLEM C8. CALCLTjATION OF QUANTITIES.

(a) Equipment. — (No' instrumental equipment imless pla-
nimeter is assigned.)

(b) ProMem.. — Compute the quantity of earthwork for
an assigned set of cross-section notes.

(c) Methods. — (1) Transcribe the notes and carefully
verify the copy. (2) Calculate the sectional area for each
station and intermediate in the notes, and prove the re-
sults. ( 3 ) Calculate the volume by the " average end area "
method, results to nearest 0.1 cubic yard, and check the
same. (4) If so instructed, plot the notes on cross-section
paper and determine the areas by means of the planimeter
as a check. Record the results.

PROBLEM C9. STAKING OUT A BORROW PIT.

(a) Equipm,ent. — Engineers' level or transit with at-
tached bubble, leveling rod tape, stakes, axe.

(b) Problem. — Stake out a borrow pit and take notes re-
quired for calculation of earthwork quantities.

(c) Methods. — (1) Select a base line, preferably outside
the limits of the proposed borrow pit, set substantial station
stakes say 50 or 100 feet apart along this base ; designate
these stakes A, B, C, etc. (2) Establish auxiliary refer-
ence lines by erecting perpendiculars to the base line at the
several stakes, driving temporary stakes for pegs at suit-
able distances on these lines. (3) Establish a permanent
bench mark and run levels, as in profile leveling, along
lines starting at A, B, C, etc., noting elevations both at
pegs and at marked intermediate changes of slope. (4) In

86

THE LEVEL.

case actual construction is undertaken, repeat the levels
along- the same auxiliary lines from time to timie and cal-
culate the quantities. (5) Eecord complete data.

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Usid SaFf ^ Barker Dumpy Levg/, locker JfSJS-
Chained down, center of street, lining in
with transit poles, tskjn0 Jeye/s en roi/t^

PROBLEM CIO. LEVELS FOE PEOFILE AND QUANTI-
TIES FOR PAVING A STREET.

(a) Equipment. — Level, level rod, 3 flag poles, 100-foot
steel tape, chaining pins, 50-foot metallic tape, hubs, axe.

(b) Prohlem. — Take level rod readings on the center line,
right and left curb lines, right and left sidewalk lines, and
right and left property lines to determine profiles and quan-
tities for paving street. Plot profiles on Plate A profile
paper to a scale of 100 feet to 1 inch horizontal and 10 feet
to 1 inch vertical. Estimate the quantities of cut and fill,
and paving materials.

(c) Methods. — (1) Locate the center line of the street and
set flag poles on line about 400 feet apart by ranging in
vsfith the eye. (3) Drive a hub at one end of the street and
call this point station zero. (3) Run a line of differential
levels from the Standard B. M. to the zero end of the line.

PROBLEMS.

87

J-Poe, leveler. X-Koe. Xodmsn.

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Fig. 19e. — Contour Plat and Device for the Kapid Inter-
polation of Contours.

88 THE LEVEL.

Eead the rod to 0.01 foot. (4) Bead the level rod to 0.1
foot on the ground at center hub. (5) Measure the dis-
tance out to the right curb line, right sidewalk and right
property lines with the metallic tape and read the rod to
0.1 foot on the ground at station zero. (5) Measure the
distance out to the center line to station 1. (8) Measure
to the right and left from the chaining pin the required
distances with the metallic tape and take rod readings as at
station zero. (9) Repeat the process at each station and at
abrupt changes intermediate. (10) Check the level circuit.
(11) Make profile on Plate A paper, scales 100 feet to the
inch horizontal and 10 feet vertical, indicating the several
lines by conventional lines or colors. (12) Lay grade line
as directed. (13) Show plat at bottom of profile. (14)
Plot sections to a scale of 20 feet to the inch and determine
areas. (15) Compute quantities of earthwork, paving, etc.

Pig. 19f.

*80*85-Z Ed'

PROBLEM Cll. CONTOUE LEVELING.

(a) Equipment. — Engineers' leveling instrument, leveling
rod, 100-foot steel tape, stakes, axe.

(b) Prohlem. — Make a rapid contour survey of an as-
signed tract of ground with the level and chain.

(c) Methods. — (1) Examine the tract and plan the system
of reference lines for locating the points at which levels
are to be taken ; if the ground is comparatively regular, a
simple subdivision into squares of 100 feet may suffice ; but
if much broken, special lines along gullies and ridges
should be included in the survey plan. (2) Stake off the
tract according to the plan, and make a record of the same.
(3) Starting from an assigned bench, determine the eleva-
tions of the ground at the various stakes and at such other

PROBLEMS. 89

points as may be required to give a correct basis for accu-
rate contouring'. (4) Plot the data, and interpolate con-
tours at a specified interval, employing both numerical cal-
culations and geometrical methods, Pig. 19e. (5) Finish
the plat, as required.

PROBLEM C12. USE OP CONTOUR MAP.

(a) Equipment. — Contour map, drafting instruments, etc.

(b) Prohle'M. — From the f;iven contour map : (1) construct
profiles on the assigned lines; (3) project a line of specified
grade through assigned points on the contour map ; make
profile, lay grade line and estimate earthwork quantities
approximately; (3) calculate the earthwork quantities
from the map for given grade planes and limitations of
area. (The third step may, perhaps, best be taken with a
different map from the first two.)

(c) Methods. — (1) Use profile paper for the profiles. (3)
To project the line on the map, set the dividers at the
horizontal distance in which the specified gradient will sur-
mount the vertical interval between successive contour
planes, Pig. 19f ; then beginning at a specified point, locate
points on the successive contour lines up or down on the given
gradient, as required ; sketch in the route roughly, and pro-
ject a series of connected curved and tangent lines approxi-
mating to it ; construct a profile along the new line ; lay

.the required grade line on the profile, and estimate approxi-
mate earthwork quantities for specified dimensions and
slopes of roadbed. (3) By means of end area method cal-
culate the earthwork quantities required to establish the
specified grade planes on the designated contoured area.

PROBLEM C13. RECIPROCAL LEVELING.

(a) Equipment. — Engineers' level, 3 leveling rods.

(b) Problem. — Determine the difference of elevation be-
tween two bench marks on opposite sides of a river (or
wide ravine) by reciprocal leveling.

(c) Methods. — (1) Set the level up so that a rod reading
may be taken on both benches at one setting. Station a
rodman at each bench. (3) Take a back sight consisting
of a series of say 5 or 10 careful oonsecvitive rod readings.
(3) Without delay take a like series of readings for a fore-
sight. (4) Set the instrument on the opposite side of the

8

90

THE LEVEL.

river or ravine and repeat the above process. (5) Deter-
mine a difference of elevation by taking the difference be-
tween the mean back sight and fore sight for each setting,
and finally take the mean of the two results. Observe rigid
care in all details of the problem.

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PKOBLEM C14. TEST OF DELICACY OF BUBBLE VIAL.

(a) Equipment. — Engineers' leveling instrument, leveling
rod, tape, level tester.

(b) Prohlem. — Determine the radius of curvature of the
assigned bubble vial. (1) by means of the optical test, and
(3) by the level tester.

(c) Methods. — (1) Measure off a base line say 100 feet
long, set level at one end and hold rod on a peg driven at
the other end ; note the target movement corresponding to a
given bubble movement, both in the same linear unit ; cal-
culate the radius by the method shown at (h). Fig. 18. (3)
Set the level tester on a solid base and place the instru-
ment on it, a? jjDd.icated at (i), Fig. 18; by means of the

PROBLEMS. 91

micrometer head and known relations of the level tester,
determine the angular equivalent in seconds for one divi-
sion and also one inch movement of the bubble, from which
calculate the radius of curvature of the vial in feet. Fol-
low the form.

PROBLEM C15. COMPARISON OF LEVEL TELESCOPES.

(a) Equipment. — Five (or other specified number) engi-
neers' levels (both wye and dumpy), leveling rod, metallic
tape.

(b) Prohlem. — ^Malce a critical examination and compari-
son of the telescopes of the assigned instruments.

(c) Methods. — Carefully read the discussion of the tele-
scope in the text. Then compare the telescopes with refer-
ence to : (1) magnifying power ; (2) chromatic aberration ;
(3) spherical aberration ; (4) definition; (5) illumination;
(6) flatness of fields; (7) angular width of field; (8) effec-
tive aperture of objective. Make tabulated record of com-
parisons, giving in separate columns; (a) locker number;
(b) kind of level; (c) name of maker; (d) magnifying
power, and so on for the other points examined.

PROBLEM C16. TESTS OF THE WYE LEVEL.

(a) Equipment. — ^Wye level, leveling rod, tape.

(b) Problem. — Test the essential relations and adjust-
ments of the wye level.

(c) Methods. — Carefully note the construction of the as-
signed level and the positions of the elementary lines. Then
following the methods outlined in the text, test the fol-
screws) : (1) The bubble, both as to the azimuth and alti-
tude movements ; find the position of the bubble when par-
allel to the element of the rings. (2) The line of collima-
tion ; its deviation from the axis in 400 feet. (3) The wyes ;
finding the position of the bubble when the vertical axis is
vertical. Keep a neat and systematic tabulated record of
observed numerical data, with explanation of the several

92 THE LEVEL.

PEOBLEM C17. ADJUSTMENT OF THE WYE LEVEL.

(a) Equipment. — Wye level (reserved expressly for ad-
justment), leveling rod, tape, adjusting pin.

(b) ProMem. — Make the full series of adjustments 'of the
wye level.

(c) Methods. — Follow the methods detailed in the text
according to the following program: (1) Adjust the bubble
line (a) into the same plane with the bottom element of
the rings, and (b) parallel to that element. (3) Adjust the
line of collimation to coincide with the axis of the rings,
first on a long distance ; and then, to test the object glass
slide, try it for a short distance ; if necessary, shift the
reticule in rotation to make the horizontal hair horizontal,
and also center the eyepiece. (3) Adjust the bubble line
perpendicular to the vertical axis by means of the wye
nuts. (4) Test the rings of the wye level by the two-peg
test ; if the level has a reversion bubble, first test the paral-
lelism of the top and bottom tangent lines, and then test
the rings. Keep a, clear and systematic record. In each
case, state (a) the desired relation, (b) the test, and (c)

PEOBLEM C18. SKETCHING THE WYE LEVEL.

(a) Equipment. — Wye level.

(b) Pro6?e)».— Make a first-class freehand sketch of the
assigned wye level.

(c) MetlKidt^'. — The sketch should be correct in proportion
and clear in detail. The essential parts should be desig-
nated in neat and draftsmanlike form, and the elementary
lines clearly indicated.

PEOBLEM C19. TESTS OF THE DUMPY LEVEL.

(a) Equipment. — Dumpj' level, leveling rod, tape.

(b) Prohlem. — Test the essential relations and adjust-
ments of the dumpy level.

(c) Methods. — Carefully note the construction of the as-
signed level and the position of the elementary lines. Then,
following the methods outlined in the text, test the follow-
ing adjustments: (1) the bubble line, whether perpendicu-
lar to the vertical axis ; and if not, what is the angular
inclination of the vertical axis when the bubble is in the

PROBLEMS. 93

middle? (3) The line of collimation, whether parallel to
the bubble line. Record the errors and observations sys-
tematically.

PROBLEM C30. ADJUSTMENT OP THE DUMPY LEVEL.

(a) Equipment. — Dumpy level (reserved expressly for ad-
justment), leveling- rod, tape, peg-s, axe, adjusting pin.

(b) Pro6?c»).— Make the essential adjustments of the as-
sig-ned dumpy level.

(c) Methods. — (1) Adjust the bubble line perpendicular
to the vertical axis. (2) Adjust the line of collimation par-
allel to the bubble line by the two-peg method. In describ-
ing the adjustments, the record should state (a) the desired
relation, (b) the test, and (c) the adjustment.

PROBLEM C21. SKETCHING THE DUMPY LEVEL.

(See Problem C18.)

PROBLEM C33. STRETCHING CROSS-HAIRS.

(a) Equipment. — Engineers' level or transit (or cross-
hair reticule), pocket cross-hair outfit, reading glass.

(b) ProMem. — Renew the cross-hairs in a level or transit
instrument by a method applicable to field use.

(c) Methods. — (If instrument is provided, follow the
complete program outlined below ; otherwise, merely stretch
the lines on the reticule and test same.) (1) Remove the
eyepiece, carefully preserving the screws from loss. (2)
Remove one pair of the capstan headed reticule screws ;
turn the ring edgewise and insert a sharpened stick in the
exposed screw hole, take out the other two screws and re-
move reticule from telescope tube. (3) Clean the cross-hair
graduations, and support the reticule on a sharpened stick,
or (if a transit) place it on the object glass with a piece of
paper interposed to protect the lens. (4) Select from the
capsule (see (d), Eig. 17) two spider lines 3 inches or more
long, and fasten a stick to either end of each hair by means
of glue from the adhesive paper. ( 5 ) Put the hairs in place,
(with the bits of wood hanging loose), shifting them as
desired with a pin point or knife blade. (6) Apply a bit of
the moistened adhesive paper to the reticule over each hair.

94

THE LEVEL.

and after a few minutes cut or break the sticks loose. (7)
Test the hairs by blowing- on them full force. (8) If they
stand this test, replace the reticule, and adjust the instru-
ment. Make a record of the process.

PROBLEM C33. ERROR OF SETTING A LEVEL TARGET.

(a) Equipment. — Engineers' leveling instrument, leveling
rod (preferably a New York or Boston rod), tape, pegs.

(b) Protlem. — Determine the probable error of setting
the level target at distances of 100 and 300 feet (or such
other distances as may be assigned).

>(7W7<? lew/. y^
BosCon/fod.Lr.lZ.

i 'loudy, cool, breezy.
' .Metallic Tape.

Error

Disbance 100 feet

" Ft.

i.l69

i.m

3.170
i.l69

i./7l

i

Ft.
0.000
.001
.001
.001
.000
.001
.002
.00!
.000
.001

P/lsn Me3fi=f 5u/n-l

Frob.frror Single Ods.
[=0.67]/id.':0.0007i

' 1/7'/

Apprax.Pwb. imgkfmr
Ob3.=0.8SF=0.00068

Freli.Errtfr /jfHean

l,/,-fy=o.ooon

ofSettims
Distance 300 Feet.

Ft.
mi

mi

1837

<JS7

113J

d

Ft.
0.006
.002
.001
.000
.003

/I i.S37

/lean Me3/i=F 5unj=l

tfO.OOW

£i lsppmx.1 = 0. 0017

E„ = 0.0005

Ltveler, R.Roe.
Hodman, J- Doe.
Level Target.

t1ov.l,l9l4,(Z hours).
Distances mUSO-f

vM fup.

5et//75trurrjent in s

ol'nOOFt.a/7dd/-i>v
Placed pair offa/t s

of pegs and le, '

snug.
Focused eyepj'eceon

Fuiiy,hep/nf eye i
5et target ten time

careFulJy veriFyio

bubble each time be
Peiernr/ined magnlF

hy comparing 77.1/, .

with one eye a/j, '

ifitliothereye.

28diameters.

K=hD=MlxlOi
t 0.0/13
Pidm.hor.h3ir,h=

lelteredplace^measured
anddr/JV ^peg. Same at 500 Ft.

•reivs on general line
7,leamgscremjust

=0.OO00ZH.

crosshairs very care-
normal condition.
at each distance,
'Fyin y the posiiio/? oFche
ne be 'ore apprm'n/i sight,
^gnit 'Jng power or telescope
0.1/, . on rod natural size
'dw^gnlFied l]y telescope
roL idMag.Poivertolie

'atore oFdudb/e
=/4S.'s
r. ff.ff/xD.S
'■" 400
0.1^024/n. rod-.

(c) Methods. — (1) Determine the magnifying power of
the telescope. .(2) Determine the radius of curvature of
the level vial by the field method. (3) Determine the space
on the rod covered by the diameter of the hair. (4) Drive
a peg at 100 feet from the level, level up, and secure ten sat-
isfactory consecutive rod readings with rod held truly plumb
on the peg ; shift the target several inches between read-

PEOBLEMS. 95

ings, and reset without bias ; reject no readings ; watch the
bubble closely, but work briskly. (4) Repeat the series at
300 feet. (5) Determine for each distance the mean rod,
the probable error of a single reading, and of the mean, as
indicated in the form.

PROBLEM C24. MAKING A LEVELING ROD.

(a) Equipment. — Piece of straight dressed clear white
pine of proper dimensions, steel tape graduated to 0.01 foot,
carpenter's tri-square, paint, etc.

(b) Proilem. — Make a self-reading leveling rod.

(c) Methods. — (To be devised by the student. See Fig. 27

PROBLEM C25. COMPARISON OE DIEEERENT MAKES
AND TYPES OF ENGINEERS' LEVELS.

(a) Equipment. — Department equipment, catalogs of rep-
resentative engineering instrument makers.

(b) Problem. — Make a critical comparison of the several
types and makes of engineers' levels.

(c) Methods. — Examine the department equipment and
stiidy the several catalogs carefully, noting the usual and
special features, prices, etc., and prepare a systematic sum-
mary or digest of the same. Prepare brief specifications
for a leveling instrument, and also suggest the preferred
make.

CHAPTER V.
THE TRANSIT.

Description. — The engineers' transit consists of an ali-
clade, carrying the line of sight, attached to an inner verti-
cal spindle (or upper motion) which turns in an outer an-
nular spindle (or lower motion). The latter carries the
horizontal graduated circle or limb, and is supported by the
needle with its graduated circle, and the vernier ; it may be
revolved while the graduated limb remains stationary. The
horizontal limb is graduated to degrees and half degrees
and sometimes to twenty minutes, and is numbered prefer-
ably from zero to 360° in both directions.

The complete transit differs from the plain transit, Fig.
20, in having a vertical arc and level bubble attached to
the telescope.

Complete Transit,

Plain Transit.

Fig. 30.
97

998

THE TRANSIT.

(h)

USE OF THE TRANSIT. 99

In Fig. 21 are shown: (a) the English theodolite; (b)
the shifting plates and foot screws of a transit ; (c) the
Saegmuller solar attachment to the transit; (d) the gra-
dienter; (e) tripods ;•(£) reflectors; (g) reading glass ; (h)
flagpoles; (i) plumb bobs; (j) the Brunton pocket transit.

The Vernier. — The vernier is an auxiliary scale used to
The least count of a direct vernier is found by dividing the
value of one division of the limb by the number of divisions
on the vernier. With a limb graduated to half degrees and
a direct vernier reading to single minutes 30 divisions on
the vernier cover 29 divisions on the limb.

In reading a direct vernier observe the following rule :
Bead from the zero of the limb to the zero of the vernier,
then along on the vernier until coincident lines are found.

In setting the vernier to a given reading, as for example
a zero reading for measuring an angle, the tangent move-
ment should be given a quick short motion to secure the
last reflnement, since a slow movement is not noticed by

In Pig. 23, (c) is a vernier reading to single minutes, (d)
to half minutes (30"), and (e) to thirds of minutes (20").
The slant in the numerals on the limb corresponds with
that on the vernier.

USE OP THE TRANSIT.

Use. — The complete transit is used: (1) to prolong lines;
(2) to measure horizontal angles; (3) to measure vertical
angles; (4) to run levels ; (5) to establish grade lines. The
plain transit is conflned to the flrst two uses, unless it has
a vertical clamp and tangent movement, when it may be
used to " shoot in " grade lines.

Prolongation of Lines. — If the instrument is in adjust-
ment a line can be prolonged by sighting at the rear sta-
tion and reversing the telescope in altitude. It is, however,
not safe to depend on the adjustments of the transit, and
important lines should always be prolonged by the method
of " double sights," as given in Problem D2. Lines may be
prolonged with the plates by sighting at the rear station
azimuth and locating stations ahead with the A vernier
of the instrument.

100 THE TRANSIT.

Measurement of Horizontal Angles. — Horizontal angles
are measured as described in Problem Dl. If greater ac-
curacy is required, angles may be measured by series or
by repetition.

By Series. — In measuring an angle by series all the
angles around the point are read to the right, both verniers
being read to eliminate eccentricity. The instrument is
then reversed in altitude and azimuth and all the angles
checked by sighting back gn the first point in each case.
These observations constitute one " set." The vernier is
shifted between sets 360° divided by the number of sets.
The arithmetical mean of the observed values is taken as
the true value.

By Repetition. — Angles are measured by repetition as
described in Problem D13. This method is especially suited
to the accurate measurement of angles with an ordinary
transit, and is to be preferred to the series method, which is
a favorite where precise instruments are used. In the repe-
tition method all the instrumental errors are eliminated
and the error of reading is very much reduced. It is doubt-
ful if it is ever consistent to make more than 5 or 6 repe-
titions.

Azimuth.. — The azimuth of a line is the horizontal angle
which it makes with a line of reference through one of its
ends, the angles being measured to the right from 0° to
360°, as in (f) Fig. 23. It is usual to assume that the true
meridian is the line of reference, the south point being
taken as zero in common surveying.

Deflection. — The deflection of a line is the angle that it
makes with the preceding line produced, and is called de-
flection right or left depending upon whether the angle is
on the right or left side of the line produced, as in (h).
Fig. 23.

Vertical Angles. — Vertical angles are referred to the
horizon determined by the plane of the level under the
telescope, and are angles of depression or elevation relative
to that plane. In measuring vertical angles the instrument
should be leveled by means of the level under the telescope
and correction should be made for index error of the ver-
nier. With a transit having a complete vertical circle, the
true vertical angle may be obtained by measuring the
angle with the telescope normal and reversed and taking
the mean.

Traversing. — A traverse is a series of lines whose

USE OF THE TEx\NSIT. 101

lengths and relative directions are known. Traverses are
used in determining' areas, locating highways, railroads, etc.

Azimuth Traverse. — In an azimuth traverse the azimuths
of the lines are determined, nsiially passing around the
field to the right. In orienting the transit at any station
the A vernier is set to read the azimuth of the preceding
cotirse, the telescope is reversed, directed towards the pre-
ceding station and the lower motion clamped ; the telescope
is then reversed in altitude. The reading of the A vernier
with telescope normal will then give the azimuth of any line
sighted on. If there is any error in collimation the transit
may be oriented by sighting back ^vith the A vernier read-
ing the back azimuth of the preceding course. In a closed
traverse the last front azimuth should agree with the first
back azimuth. The azimuth traverse is especially adapted
changed to bearings, if desired.

Deflection Traverse. — In a deflection traverse the de-
flection of each line is determined, usually passing around
the fleld to the right. To avoid discrepancies due to error
in collimation, the transit may be oriented by sighting at
the preceding station with the A vernier set at 180°, the
telescope being in its normal position, and the lower mo-
tion clamped. The reading of the A vernier will then give
the deflection of any line sighted on.

Compass Bearings. — Compass bearings should always
be read on an extended traverse as a check against such
errors as using the wrong motion or an erroneous reading
of the vernier. To guard against errors due to local attrac-
tion, back and front bearing's should always be read, and
the angle thus determined compared with the transit angle.

Leveling ■with the Transit. — The transit with an at-
tached level is the complete equivalent for the engineers'
level. The instrument is leveled up with the plate levels
first, after which the position of the attached bubble is con-
trolled by means of the vertical tangent movement.

transit either by means of known distances and calculated
rod readings, or by " shooting in " a parallel line by means
of the inclined telescope, as described under the use of the
engineers' level. For the latter purpose the transit is
rather more convenient than the level.

Setting up the Transit. — To set the transit over a point,
spread the legs so that they will make an angle of about
30°, place them symmetrically about the point with two legs

102 THE TRANSIT.

down hill. Bring one plate level parallel to two of the legs,
force these legs firmly into the ground and bring the plumb
bob over the point and the plates approximately level with
the third leg, changing the position of the plumb bob with
a radial motion and leveling the plates with a circular mo-
tion of the leg. Finish the centering with the shifting
plates. In leveling up, the bubbles mo^'e with the left
thumb. Use care to bring the foot screws to a proper
bearing.

Parallax. — Before beginning the observations the eye-
piece should be carefully focused on the cross-hairs so as to
prevent parallax.

Back Sight With Transit. — Ahrays check the bacTc sight
icfore moving the transit to see that the instrument has not
been disturbed or that a wrong motion has not been used.

Instrumental Errors. — The transit should be kept in as
perfect adjustment as possible, and should be used habit-
ually as though it were out of adjustment, that is, so that
the instrumental errors will balance. No opportunity
should be lost to test adjustments.

Elementary Lines. — Fig. 22 shows the elementary lines
of the transit, viz., (1) line of coUimation ; (2) horizontal
axis; (3) vertical axis; (4) plate level lines; (5) attached
level lines. These lines should have the following relations :
(a) the plate levels should be perpendicular to the vertical
axis ; (b) the line of collimation should be perpendicular to
the horizontal axis; (c) the horizontal axis should be per-
pendicular to the vertical axis; (d) the attached level line
should be parallel to the line of collimation. The following
additional relations should exist : (e) the vertical axes of
the upper and lower motions should be coincident; (f) the
optical center of the objective should be projected in the
line of collimation ; (g) the center of the graduated circle
should be the center of rotation, i. e., there should be no
eccentricity.

Plate Levels. — To make the plate levels perpendicular to
the vertical axis. — Make the vertical axis vertical and ad-
just the bubbles to the middle of their race. The vertical
axis is made vertical by leveling up, reversing in azimuth,
and if the bubbles move, bring them half way back with
the foot screws. The adjustment is the same as for the
compass, and the reasons are shown in (a). Fig. 13.

103

After adjusting the plate levels with reference to say the
upper motion, test them with the lower motion to prove
the coincidence of the vertical axes.

Op tied I Center
(^^ oFObiective,

fntersectior? oF
Cro55-Hair5

line ofCollimatioi^'':.,

cc:i<!

Fig. 22.

Line of Collimation. — To make the line of oolUmation
perpendicular to the horizontal axis. — Construct a straight
line and adjust the vertical hair so that the instrument will
reverse in altitude on it The straight line may be estab-
lished either by prolongation beyond a point in front, or

104 THE TEANSIT.

preferably by the methods of double sighting', described in
Problem 1)2. One-fourth the apparent error is corrected in
second case as shown in (a), Fig. 23. In deciding which
way to move the hair, notice that the optical center is the
fulcrum. The transit should be cbllimated first for equal
back and fore sights, say 100 feet or so, and then checked
for a distant point in one direction and perhaps 50 feet in
the other, so as to test the motion of the optical center of
the objective. The points should all be as definite as pos-
sible. Chaining pins may be used, or V-marks may be made
on the side of a stake driven securely. Each altitude re-
versal should be checked back and forth to make sure of
the prolongations, and the telescope should be handled very
carefblly. If the cross-hair reticule is removed from the
instrument or should be much disturbed, the foregoing ad-
tical by sighting on a plumb line, such as the corner of a
building, or by noting whether the hair continuously covers
the same point as the telescope is moved in altitude ; the

Horizontal Axis. — To make the hori::ontal axis perpen-
dicular to the vertical axis. — Adjust the horiontal axis so
that the line of collimation icill foUoio a pluml) line. An
actual plumb line may be used ; or preferably a vertical line
may be constructed by first sighting on a high point, then
depressing the telescope and marking a low point ; then re-
versing in altitude and azimuth (turning the horizontal axis
end for end), sighting at the high point again and marking
u, second low point beside the first one. The mean of the
two low points is vertically beneath the upper one. The
transverse plate level is especially important in this process.
One end of the horizontal axis is changed, as in (b),
Fig. 23.

Attached Level. — To malie the attached level and the line
of collimation parallel to each other. — Construct a Icrel line
and adjust the instrument to agree with it. The level line
may be obtained either by using the surface of a still body
of water, as of a pond, or it may be constructed by equal
back and fore sights, as indicated in (e), Fig. 16. Either
the horizontal hair may be changed to bring the line of
collimation parallel to the bubble line, or vice versa. The
method is the same as used for the dumpy level.

If the bubble vial is a reversion level, as shown at (b).
Fig. 18, the adjustment is much simpler. However, the

105

two-peg test should be applied at least once to the rever-
sion level to prove the parallelism of the top and bottom
tangent lines of the bubble vial.

^itr

"i-i— 4^'

:hf Position

True Position

^^-Znd Position

(d)

I ^ '0 'i '"

2,0 15 \0 5

10
iSO

seo

"fo

M

N

(9)

w-so°

*^

d

^

90°E-

Fig. 23.

Vertical Arc. — After the last preceding adjustment, the
when the bubble is at the center of the tube. Bring the
bubble to the center and shift the vernier to read zero. If
the vernier is fixed, an index correction may be applied to
all vertical angles ; or the bubble may be made to agree
with the vernier and the horizontal hair then adjusted by
the two-peg method.

Eccentricity. — Eead the two verniers at intervals around
the circle ; if the verniers have changed the same amount in

106 THE TRANSIT.

each case the circle is well centered. If the two verniers
have not changed the same amount, the mean of the angles
passed over by the verniers is the actual angle through
which the instrument has turned. The error cannot be ad-
justed.

Centering the Eyepiece. — If the intersection of the
cross-hairs is not in the center of the field of view, move
the inner ring of the eyepiece slide by means of the screws
which hold it.

PROBLEMS WITH THE TRANSIT.
PROBLEM Dl. ANGLES OF A TRIANGLE WITH TRANSIT.

(a) Equipment. — Transit, 2 flag poles, reading-glass.

(b) Problem. — Measure the angles of a given triangle
with the transit.

(c) Methods. — (1) Set the transit over one of the vertices
of the triangle and plumb a transit pole over each of the
other two. (2) Set the A vernier to read zero, sight at the
left hand point approximately, clamp the lower motion and
make an exact bisection with the lower tangent movement.
(3) Unclamp the upper motion, sight at the right hand
point approximately and make an exact bisection with the
upper tangent movement. (4) Read the A vernier to the
nearest single minute. This reading is the angle sought.
(5) With the A vernier set to read zero repeat the measure-
ment, sighting first at the right hand station and then at
the left. The recorded value of the angle is to be the mean
of these two determinations which must not differ by more
than one minute. (6) Measure the other angles in like
manner. The error of closure must not exceed one minute.

PROBLEM D3. PROLONGATION OF A LINE WITH
TRANSIT.

(a) Equipment. — Transit, 3 flag poles, axe, 6 hubs, 6 flat
stakes, tacks.

(b) Problem. — Prolong a 300-foot base line successively
with the transit by the method of " double sights " about
1500 feet, and check on a hub previously established.

(c) Methods. — (1) Drive two hubs, A and F, about 1500
feet apart. (2) Set the transit over tack in hub A, sight at

PROBLEMS.

107

f

Station

AtfdLES

Valu«

l9tM«3S<

SS'SO'

47'47'
43''Z3'

(Biff 7rance hsftvea/f me3si(rffmen.

»xcee> ^ J')
ffrrkr liot . v exe^d /O

At 9I1

OP Tr
F

Si'Sl'

47%7'
43'Z3'

ANGLI: 5-«i-8

Mean

Observers, J-Doe cF

WITH Engineers'

Nov-]B,imj(Z hears).
Used Mellar^ Brightly
The Jst- measuremeni

on Sta-g y^ifh the

phfes clamped di

on Sfj-S with the

re3dw0 the
The

sigi}tin0 on St3-
Used trans/f poies

them very

mentS'
Sketch siiows

f plafi s
f second measure, ffenf

Transit-

Warm and ijuiet.
Transit No- 10-
'ower motion^ the
xero} then sf^ht/n^
i^per matio/j, and

^nd then on Sta- 8-
targets f piumbtn^
careA 'i/y over the monu-

obsej '^ed angles-

DOUBLE SIGHTINGS-^
Pbolonsation of Line--

i^JSh^/Sh

I

'Setupaf-B "double
sightedbF- /few
tsck F is'o-OI left-
of anginal f^ck-
(Alhweble error is
t'oiymSigM/ngs
For 3aO's/0hts)

I ^"'X syhted" toE.
^,'iK^a)fSef up 3fC, "double

a \

a ^

I

e/e'

Did'

A/

'Set up sfB," double
sighted to C/ ss
FoUoiVsjfSee^ote-*-)
(g) Back sighted on A
(b)Plungedto c'

^(c) Rotated to A
Cd) Plunged to c"
fe) Bisected c'c" to
' l^ locffte tack C-

{'Set up fft A; sighted
on Flag 3t Ff dnve hub
Bi removed Flag F-
.(Brvxe MsA 3ndF

Obsermsl-'"" "-'^-''f-C^hrs) Cool.doudy^
Vl-goe UsedK-S-F- Transit NS4-

WITH Ensineers' Transit
-Interpolation of Point.

\Biseeted pp' St P- Set

•K ■<: ^ •*,

k,«.||l

■■^5S
1-^ ^^

up 3tA and checked Pf
error, 0-02 to right-
[Reversed in azimuth ?

„e

K

C4)

m

shlFted transit so it ^ .-
would again plunge *i /
exactly on A and B ^ /
Drove hub p" to bob- \i ,
'Set up and shIFted P\0-I1
transit laterally ^.

until it would \

plunge exactly ^.

on A and B-(/eelfoh:)-^ \
Drove hub p* to plumb ii;b-^\ ^

{'Set Flags on tacks ah \ \

A and B, and determined
point P- by lining in two ri
poles- successively by eye-(See p-dS-J
. Drove temporary peg-
(Drove hubs A and B about 61X>' apart,
(l)\ assumed to have hill between them^
I both visible From desired hub P-

\
\

ohi\p"

'i

HOTE. Watched plate leveis cioseiyj

especially transverse bubble' ^J

108 THE TRANSIT.

flag pole plumbed over tack in hub F, drive hub B about
300 feet from the transit and locate a tack in line very
carefully. Eemove the flag pole from hub F. (3) Set the
transit over hub B, back sight on hub A and clamp the ver-
tical axis. (4) Reverse the telescope, drive hub C at a dis-
tance of about 300 feet and mark line very carefully with a
pencil. (5) Reverse the transit in azimuth, sight on hub A;
reverse the telescope and locate a second point on hub C.
Drive a tack midway between these two points. (6) Set the
transit over the mean point on hub C, back sight on hub
B, prolong 300 feet and set hub D by double sights. (7) Set
over hub D, back sight on hub C, prolong, 300 feet and set
hub E, as before. (8) Finally prolong from hub E, with
back sight on D, and establish mean tack at terminal hub
/''. Record the collimation errors at G, D, E, and the final
error at F. Follow the form.

PROBLEM D3. INTERSECTION OP LINES BY TRANSIT.

(a) Equipment. — Transit, 3 flag poles, plumb bob string,
axe, 6 hubs, 6 flat stakes, tacks, marking crayon.

(b) Proileni. — Determine the intersection of the bisect-
ing lines of two angles of a triangle and check by bisect-
ing the third angle.

(c) Methods. — (1) Drive and tack three hubs so as to
form a triangle approximately equilateral and having sides
about 400 feet long ; properly witness the hubs with guard
stakes. (2) Set the transit over one of the vertices of the
triangle, and measure the angle as in Problem Dl. (3) Set
two hubs on the bisecting line, about 6 feet apart, so that
the point of intersection of the bisecting lines will come
between them, and mark the line by stretching a string be-
tween the hubs. Check by measuring each half angle inde-
pendently. (4) Set the transit over one of the other ver-
tices of the triangle, measure the angle and determine the
bisecting line as at the first point. (5) Drive a hub at the
intersection of the two bisecting lines and mark the exact
point with a tack ; check by measuring each half angle in-
dependently. (6) Set the transit over the third vertex and
determine the angular and linear error of intersection. (7)
As a final check measure the angles around the point of in-
tersection of the bisectors. The angular error of closure of
any triangle should not exceed one minute. Follow the
form.

PROBLEMS.

109

static

/
3
Z

Station

v_

Whole
Angle

ez'is'

73'm'
44'4S'

An
I-Q-i

i-o-z

Z-0-}

Alliiwsli: ?

INTEP
L'HalF
Angle

ii'n'jo'
zz'zzin'

SE

RHa:F
Angle
3/WX'
}6'Mk'
ZZ'ziW

ilhwah.

Chick

iiz'zi'm

IZS'Sl'lO

izMn

m't.

er/vr

CT ON OlF LiH
■or
Distance

•dJ'-O

Er
Angle

■ OJ-'O

:S

0-C3f1

WITH TRAN

/t<ivIS,19/4,(ZHo
Used KSie trsr,

chaining locker
Sef rrspsif over

setoff i Z3-;-

and " b" on Imc
^ef over A3 J

offi/./-3^f;_

hefween "

J/uii el-
se f overAZ;^eB

checked fnf^.
5ef overAV,snd

SIT.

^J'Coe-
s) CJear Jr Cool-
it; Locker tf^^, snd

a i; jTjeasi/red Z 3-/-Z-J
?andseffiu^s '5"
aiioi/t 6 'apart'

l-3r-i;set
Jtretcfied string
\<nd "f'and /ocated

measured Zi

sured angles and
'. 'ect/on •
measured angles-

t? ^^ nV *"-

•fip

■i

^gp.

K-hs>

PEOBLEM D4. KEFEEENCING OUT A POINT.

(a) Equipment. — Transit, 2 flag poles, 100-foot steel tape,
axe, 6 hubs, 6 flat stakes, marking crayon, tacks.

(b) Problem. — Eeference out a point with a transit and
tape.

110 THE TKANSIT.

(c) Methods. — (1) Drive two hubs about 500 feet apart
and mark them with guard stakes. (2) Set the transit
over one of the hubs and reference it out as shown in the
diagram. All hubs should be driven flush with the ground,
and the exact points should be marked by means of tacks
driven into the tops of the hubs. Record in proper form.

PROBLEM D5. TEIANGULATION ACROSS RIVER.

(a) Equipment. — Transit, 3 flag poles, 100-foot steel tape,
axe, 4 hubs, 4 flat stakes, tacks.

(b) Proilem. — Determine the distance across an imag-
inary river by triangulating with the transit arid check by
direct measurement.

Simpfe and Rapid Methods oFTrianquIation.

l\\\\\\\ WWW!

///7g oF Survey Prolonged Across Fiver. .1

-m

:^\\\v ..*-^"'*

AB=;rT^^j=BC-Coseo5°U''=BCxl0.0l'(BCxH>hl^§Jr^x0.l)
Sin544 il'l'

'Rule oFTen'.' (DWithtrsnsitatA, line in liubstBon opposite side of river.
(ilTurnoFFangle 5°44'3ndwitlioneendoFtapeheldatB locate C by
swinqinq on arc under direction oF transttman; IF the Front Flaqrnan be provid-
ed with a metallic tape , he may locate C alone by hooking the ring oF the tape
on 3 projecting tack in hul? B.

The desired distance ABmaybe
taken roughly as ten times the meas -
ured distance BC. For greaterexact-
nessj add 0. 1 Foot For each 100 Foot
unit in the distance ABbs Found by
the simple " rale oF ten "juststated.

Leveling ^

Instrument

(c) Methods. — (To be devised by the student. Use this
and the next problem to learn the relative merits of several
good methods. The " rule of ten " method in the sketch be-
low is very rapid and also quite accurate.)

PROBLEM D6. PASSING OBSTACLE WITH TRANSIT.

Ca) Equipment. — Transit, 100-feet steel tape, 2 flag poles,
axe, hubs, flat stakes, tacks.

PROBLEMS.

Ill

Tri.insuution 'Vcros ; a E iver

station

B
C

Distance
Ft-

Ill-tS

UjB-

D-B-C
S-C-D

Co leulatii n

I Cxtar SO

lo0-B-'D=lc}-

' - - 2.

III

s-.>

3-D='i<-

Chalped dii tance
Dift vrence

Per. tiiessb.

AnjJe
Value

0°3O

B-D

'0- ,
ltf09+Itt-CIS39l

SH-ffff* Ug- ti nSOJi ■

Z.r993

CM
M/-06

J.W-t

1 tsulf-

isd
fj:d

Sff'Jl
■Zli/0

/■f-

/tl-9e

mis

Ft

ft-

Transit

» R-Rae-

i) CaJdiilCItar-
lacker H^8;

■ert{233-
B.sethabefD

Li ckt

WITH ENSINEERS'
Observers - J- Dot
Mciv-Z7,'r4 -fZJfi
Used f3uth

and Chai'mng .
Kth transit ove.\

by "Method cF Double Sights',

tvifh A asa ba
Set /fab ate,.

care,
Checked -

chaining 8'D-
length cF Tape, il9-9SFt- ffisrrved

distances ract rded'

^9°iO'(Complemait)

■ksight.

iB-CwIth
, andmeashred /. B<-D-
computet' distance by

Imaginary .

r

station

A

e

D
A-D

A

B

C

D

A-D

A
P
S

A-e

Pass

Distance
Ft.
"E,u

m-t>o
ZOO-OS

101-03
199-93

"Ri^

ZO-00

ZOO-00

loose

200- OS

izo-00
izo-00

ZOi-SS
Z39-0S

MS /N 08 5TACIE

An]g

lateral
N-A-F
A-e-D
e-D-H

B^f 'ectii n

if-A-F

a-F-6

F-S-H

le

Value
Triang
60W
bO'OO'
S9°S9'

Error
Pish Ft.
Meth

:iht Andle Of fs|[t Metljod

ff-A-B

A-S-C

B-C-D

C-B-H

90 '00'
90'00'

eo'oi'

Met

s'oo'

lO'OO'

s'oo'

of Closure
Line Ft.
od

-0-07

tO-OS

lod"

0-09R-

0-tOL-

0-03R-

WITH' EHSIHEERS'
Observers :J-Dat
Hovl7,i9/4,
Used Soriey
chaining
Wth tlie'transit
end a, in tbe line
and prolonged tl
lateral Triangle
Angle OFFset
"DtFlection

f/V
t,

Transit

S R-Roe-
CZifo vrs) iVarm ff cloudy-
~, and.
HS3Z-
<erif, set hubs at A
MM- Set transit at A
e line MA by the "Fiji-
Method" the "'
lethod" and the

Tram !t. Locker If^S, ,
7 Lockt r

Me'-hod-

g Length qF tape
•=100-01 Ft.
Observed measure-
ments recorded-

ir-90'

112 THE TRANSIT.

(b) Prohlcm. — Prolong a line beyond, an imaginary ob-
stacle by three methods and check by direct measurement.

(c) Methods. — (To be devised by the student.)

PEOBLEM D7. TEAVEPtSE OF FIELD WITH TEANSIT.

(a) Equipment.- — Transit, 2 flag poles, 100-foot steel tape.

(b) Problem. — Determine the deflections of the sides of
an assigned field with the transit, check angles by observing
the magnetic bearings, and measure the lengths of the
sides with a steel tape.

(c) Methods. — (1) Set the transit over one corner of the
field, set the A vernier to read 180°, and sight at a flag
pole plumbed over the point to the left with the telescope
normal. Eead and record the magnetic bearing. (3) Keep
the telescope normal and sight at the next point to the right.
The reading of the A vernier will be the deflection of the
second line. (3) Eead and record the magnetic bearing
and compare the transit and magnetic deflections. (4) Ee-
peat this process for the remaining corners of the polygon
taken in succession to the right. Deflections will be based
on duplicate readings agreeing within one minute. (5)
Jleasure the sides to the nearest 0.01 foot with the tape.
Compare the tape ^^ith the standard at the beginning and
conclusion of the chaining. (6) From the observed deflec-
tions determine the bearings of the field assuming one side
as a true meridian. The angular error of closure must not
exceed one minute. Eecord and reduce data as in the pre-
scribed form. Should a side of the field be obstructed, use
one or more auxiliary points (see (c) of D8).

(Most engineers prefer "plunge reversals" to the above
method of " plate reversals." To avoid the collimation error
involved in a single plunge reversal, the principles of
" double sights " must be used and the mean angle taken.
To save time, some engineers try to keep the transit always
in first-class adjustment, so as to omit one altitude reversal
in the " plunge " method, and some turn the transit " end
for end" (reverse in azimuth) every setting or so.)

PEOBLEM D8. AEEA OF FIELD WITH TEANSIT.

(a) Equipment. — Five-place table of logarithms.

(b) Problem.— Coimpnie the area of the assigned field by
means of latitudes and departures.

PEOBLEMS.

113

Trai

ERSE (

F FlELl

A-B-

:-D-E

WITH

:nsineers Transit, D

EFLECTioN Method

Stat

on

Distance

Drfltdim

Majneiic

Chcclt

CslculaM

Oi7servsrs • J-Po&i-

?JS-Mae ■

Insf.

Obj.

Ft-

Angle

Bearing

Angle

Bearing

tfoviO-lS/4- CZff<:rrs)WarmS-Nisfy. \

A

e

Keo'ssi

VsedKevffeJ3^esi

er Trans/ f J lacker *^

s

iSS-OD

MZUsi

0'4i>il

^■^M

AssuiTied thatA-S

was 3 frue jj7er/(/Zan^

B

A

lO'/l'L

5-33'ilSi

C3refi/7/y checker

f eachsngle^

c

4(4-9!

S43iS'i

joi^l

S-/t!'/3'E

Bahile dcwrr on 3. t sights- I

C

B

B

4S3-7t

IZ4i3'x

543^^'B
SIJ'JB'H/

lU'sSR

IflSln't

£ei7fffh of Tspe -
^ fiec/aced measure.

JOO^CIFf-
77enfs recordacf-

D

C

7f'llfji

s^f/'Ssk

A//cwab7e error o

"closure = J'-

e

116-Sl

mi'zm

7e't^/i

lfll7U3'£

L

e

D
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Z42-14

Slili'lt

HZl^s'lli

SZ'33'R

ss('3;'e

i

E

i2

A

r

— r

i70'J3'

370'JS'

.*•''

jfn'

(Cluck)

IfW

1
/
1

\

30W

iS9'3S'

Calcula

ion oF

Bearin

gs— '

'

\

R

AS

swWe

P-E

HII)'43'e

\

Y

B

JO'li'L

e

/??*>

\

\

B-C

SD'B'e

e-A

iH'3l'E

A

\

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im'si'r

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W'JJ'R

(Check,

)

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. \

D-e-

M'4}'e

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V

/^

l1ov-ZI,}9I4 C0777pufer, J -Doe. ^\
Data frompp- Transcr/pt 0-JC- 1

TRAt

SIT t

RAVES

SE, Fl

:LD A

B-C-D-I

Latitudes and Departure?,-

Line

Eefced

COMPl

TATlOt

OF L/

fit

UDE3

COWh

TATIOt

OF DE

PAR

rUEES

Bearmg

Distance

Multipli-

Lo^ar-

Computed

Lat.

Multipli-

Logar-
ithms

Computed

Dep.

cation

itlims

Latitude

Cor.

Latitude

cation

)eparfure

Cor.

)eparture

Ft.

Ft.

Ft-

Ft-

Ft.

Ft.

Ft.

AB

S-OCV

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Perm'

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ure
55 ft.
6 Ft.

Line
AB

ec

CO

2
2
1

Cor

-/O

-/a

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5-107-K

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S-807-2Z

Line
AS
BC
CO

I
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5- 0-40

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tA

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II

y

114 THE TRANSIT.

(c) Methods. — (Follow the instructions in the correspond-
ing- problem with the compass, Problem B4, preserving the
same degree of precision in the computed latitudes and de-
partures as in the field measurements. In case auxiliary
stations are used on an obstructed side of the field, calcu-
late the latitudes and departures of the polygon actually
traversed in the field, and then to find the area drop the
false cornej-s in calculating the meridian distance and the
latitude of the real side of the field.)

PROBLEM B9. STAKING OUT A BUILDING.

(a) Equipment. — Transit, 100-foot steel tape, 2 flag poles,
axe, hubs, tacks, plan of building

(b) Problem. — On an assigned plot of ground stake out
the assigned building.

(c) Methods. — (1) Orient one side of the enclosing rect-
■ angle with reference to a true meridian or a street line.

(2) Locate and check up the corners of the rectangle by set-
ting over each corner in turn, passing around to the right,
back-sighting on the corner to the left, turning off 90°
and locating the corner to the right. (3) Locate the corners
of the building by setting stakes on the side lines of the
building produced, using the rectangle as a base line. (4)
Check all stakes by additional measurements. The rect-
angle should close to the nearest minute, the linear error
should not exceed 1 : 50,000. Follow the form.

PROBLEM DIO. HEIGHT OF TOWER WITH TRANSIT.

(a) Equipment. — Complete transit, 2 flag poles, leveling
rod, 100-foot steel tape, axe, hubs, tacks.

(b) Problem. — Determine the height of an assigned tower
with the transit and steel tape.

(c) Methods. — (1) Set the transit over a hub located a
little further from the base than the height of the tower.

(2) Level the instrument very carefully with the attached
level and determine the index error of the vertical circle.

(3) Bring the bubble of the attached level to the center
and read a level rod held on the base of the tower (4)
Sight at the top of the tower, read the vertical angle,- cor-
rect for index error and record. (5) Reverse the telescope
and locate a second point at least as far from the flrst as
the height of the tower, check by " double sights." (6) Set

PROBLEMS.

115

ass

ass rmeef 3. r ^fanc ard fo

locfff 'd

esfi
A

oF th.
hub

h / obsei vah'on

That a nstn

Set
Set-
Sat

The

5tai:ims bUT _

ofta^^m'OOffh,

's buflo
4 to Fi.

<3 trm ■

as

^snsit
md
y7 line

J ran sit

Bnd sei

9/1 Hne

ransit

checke\f

9n0le

For }/n^

rema/r,

locfftec '

checks

c ted 3 t heckeo recta. 7^/e

J vJtoi^s

over hiBA ahdset hub B

tei nporar v hub 0) £et

1170*

rhssffe

B{UtLDI^ S
tape
-the

merle Tan

Pc fan's

'I'ghte t.
tem/iarary
hnd di's fance

Fo r disti. nee

h lbs

f3Z

der of-

with
y recthng/e ASCD-

the
-ffFers

being
chnsfuc-

,ano

• th

then
-ou^h

/fub
Measm 'edaffc Vsfanct s fmce-
over hMbBjS. ghted 9tA
fempc rary h ib C) s ft hu

a/? 8
bubD
error

Wire
7ce tc the

and
For

r

Surveyors, -^o^ JJoe , F Ki'ch&rd J^off, ^

WITH Ehgineers' Transit-

//ov. t!, im'. (Shiar. ). Cool and cJesr.
Used 6urlay Tram if, Locker No- 6 and

locker /fo-30-
Hubs sre set- en Jme S' from corners-

r

station

A
B

(I)
(i)
(3)
W

Subs.

Heisiit of

Vertical D|-D;
^ngle Ft-
?0'I6' ISO-00

latio

^/fufi'n^
Hz

M •

Tow

F-5-
(Levels)
■f-SO-

eot-fi-
■ Coi--N\-

(h,

Cct-
16-4-7

90-Z9

of Hg^ht

Calc

H= H,
H,'H,

in CO i rid sub '■rscfin ^ (2) From
DrDi • -

Hz

■■Hz(Cof-M-Coi-

Cit-M-Cit-lf-

150 i-t '-^SO-i gZ) Col-ZO'

zo'ie

. (2)
■t/)-Ch,
Cof-h

Cot-4l '
l-Z+3-

'n^r4)

(I),

■hz)C(it-M

'14'

Observers, J-Doe^ ^-^oe- ^

WITH Ehsimeers Transit.

Mov-2g,/M,fZ hours). tVsrm S? Cloudy.

Used 6ur/ey Transit , locker No-S, and
Clieining locker iio. 35.

5et transit over A and measured t/je
vertical angle //, lisvin^ First determin-
ed tile indeji error oF vertical circie-

Read fevei rod on Base oF tower, (hi)

Set 8 in-line mfh A and top oF tower
and measured D/-Dz as base line.

Set transit oyer B and Found itan^'iig.

Length oFtape = S9-^t Ft.

deduced measurements recorded-

i<-/j»j,. l>,-i>t-^

116

THE TRANSIT.

the transit over the second hub, sight at the top of the
level rod on the base of the tower as before. Each angle
low the form.

PROBLEM Dll. SURVEY OP LINE SHAFTING.

(a) Equipment. — Engineers' transit with attached bubble,
leveling rod (or instead of these engineers' instruments, a
16-foot metal-bovmd straight-edge with an adjustable bubble
of say 20-foot radius, a long braided fishing line, and 3 long
metal' suspenders made exactly alike, from which to sus-
pend straight-edge from line of shafting), 2 good plumb
bobs, 50-foot etched steel tape, copper tacks, hatchet.

(b) Problem. — Make a survey of a line of shafting in a
machine shop, and establish a true alinement for it, both
vertically and transversely.

Eesuryey oF North Line ShdFtinq, F/etal Shop.

leveJs. :■■■?

Line
Hangers

wmmmm4m

(c) Methods. — (1) Establish a reference line for lateral
deviations and carefully mark the same. (2) Select a suit-
able permanent bench mark to which the levels may be re-
ferred. (3) Determine the horizontal distance from the
vertical reference plane to the line shafting at selected
points, say at each hanger. (4) Determine the elevations
of the same points by the methods of profile leveling. (5)
Plot the data as suggested in the diagram. (6) Note the
ruling points and permissible change both laterally and
vertically at each hanger, and record the data. (7) Lay
grade lines, and prepare data to shift the line shafting tp a
ti'se position. (8) Make complete record of results.

PROBLEMS.

117

PEOBLEM D12. SUEVEY OP EACE TEACK.

Outfit for transit party (instrument
ire, say No. 20, spring balance, ther-

race track, as in-

(a) Equipment.
assig-ned, a long
mometer, etc.).

(b) Problem. — Make the survey for a
structed.

(c) Methods. — (1) Standardize steel tape, noting temper-
ature and pull. (3) Make a careful examination of the tract
of land with a view to secure the best location for the race

Requlah'on One-Mile and Half-Mile Trottinq Tracks.

I Grand Stand \

The standard distance kmeasured ona line 3 Feet From t/?e
hub-board. The inner edge of tiie trsck is thus 2Tr-3=i8.85feeb
shorter than the standard distance. The trac/c is banlted 'on
curves Fron?l:iZtoi:i5, and, to provide drair7aqe, shouid be sioped
one Foot on the straight stretches. The ends of curves are some -
times Flattened.

track as regards visibility, drainage, economy of construc-
tion and maintenance, etc. (3) After fixing the ruling
points, establish the principal axis of the track by locating
the centers of the two semi-circles and the intersections of
the axis with the curves ; also establish the ends of the
curves, preferably on the true measured line (3 feet from
the hub plank for a sulky track, and 18 inches from the
inner edge for a bicycle track). (4) Eun in each quadrant,

118

THE TEANSIT.

either by the deflection angle method, or, if trees or other
obstructions do not prevent, by using the wire as a radius
with observed pull ; set points 16 feet apart unless in-
structed otherwise. (5) After locating the true line, check
up the total distance very carefully. (6) Make plat and
complete record of survey.

PROBLEM D13. ANGLES OE TRIANGLE BY REPETITION.

(a) Equipment. — Transit, reading glass, 3 chaining pins,
2 tripods with plumb bobs (if necessary).

(b) Probletn. — Measure the angles of a prescribed tri-
angle with transit by repetition.

/'

Observers : John Doe t

■ Richard Kos- \

At

ISLES

IF Tri

ANSLE

5-6

-8

BY Repetition. Bi

■ffifBerger Transit''9-\

station

BubDirec

Object

Vern-A-

Vern-B-

Mean

Difference

Angle

Mean Angle

Remarks

ble

tion

Ae

hm

msh

AS

mh'u

o'mW

osW

HovSD,').

tfZJfours) .

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ss'si'zt

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lip

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AS

fWdn'

mVis

' OO'M"

AS

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' 47'0(l'

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73S'S6'4I>"

47'47'Zll"

47'4:'lg"

SKeps-

AS

a

R

AB

oWss''

7!th'm

' rn'os"

J>ecI,'9S-

''3//mrs) I

y^rm^iiv

ref.

AS

43'IZ'ZII

mirit

'Z2V

4i'ZZ'Z0"

5/ngfc-

2ieViv

36i7'0

■ sm'

zis'sz'do"

43°ZZ'Z4"

S£<ips-

u

L

AS

m'M'Ki

' o'mW

M'W'

A6

m'pit

'4}1f2l'

' Z2'ai:

43°ZZ'Zl>°

S/ngJe

it'siki

iW'SI^

' Sl'40'

ZJ6'Sl'4l!"

43'ZZ'Z^''

43'ZZ'ZZ"

SXrps-

A3

D

R

AS

i^'tsii!''

rn'mit

me^

Ae

n'mii'

m'siit

' so'zs'

H'SI!'ZI>''

Single

l4'lZ'0

'ze4V4i

' jz'4e>

444'JZ'4(>''

Sf'Sl'SZ'

SReps-

u

L

Ae

mVit'

t'ltio'

u'mi'

AS

m'si'a

'sfy'zti

' Sl'Zl'

s/'so'zo"

Single

Z64'/l'Z<l

'M'li'ze

' Il'ZH'

444'/Z'Zll"
^rrormfi

gs'so'ze'i

7 exceecfIS

J/'SO'JO"

seeps-

ISP'W'IO"

\

)

(c) Methods. — (1) Set the transit over one of the vertices
of the triangle and set chaining pins in the tops of the mon-
uments at the other two. (2) Set the A vernier to read
zero. (3) Sight at the left hand station with the bubble
down, and clamp the lower motion. (4) Unclamp the upper
motion, sight at the right hand station, read both verniers
and record. (5) Unclamp the lower motion, sight at the

PEOBLEMS. 119

left hand station, and check the verniers to see that they
have not moved. (6) Unclamp the upper motion and sight
at the right hand station but do not read verniers. Repeat
until five repetitions of the angle are secured, and read
both verniers to eliminate errors of eccentricity. (7) Di-
vide the arithmetical mean of the two vernier readings by
five and compare with the value obtained by single measure-
ment. (8) Reverse the instrument in altitude, and set the
A vernier to read zero. (9) Sight at the right hand station
with the bubble up, and clamp the lower motion. (10) Un-
clamp the vipper motion, sight at the left hand station, read
both verniers and record. (11) Unclamp the lower motion,
sight at the right hand station, and check the verniers to
see that they have not moved. (12) Unclamp the upper
motion and sight at the left hand station, but do not read
the verniers. Repeat until five repetitions of the angle are
secured, and read both verniers to eliminate errors of ec-
centricity. (13) Divide the mean of the two vernier read-
ings by five and compare with the value obtained by single
measurement. (14) Take the mean of the two sets as the
most probable value. (15) Measure the other angles in the
same manner-. The angular error of closure should not

PROBLEM D14. DETERMINATION OP TRUE MERIDIAN
BY OBSERVATION ON POLARIS AT ELONGATION.

(a) Equipment. — Complete transit, reading glass, hub, 2
flat stakes, board 2"x 4"x 3', 4 8d nails, axe, 2 lanterns,
good w^atch set and regulated to keep railroad time.

(b) Proilem. — Determine a true meridian by an observa-
tion on Polaris at elongation.

(c) Methods. — (1) Calculate the time of elongation of
Polaris, and regulate and set a good reliable watch to keep
railroad time (mean solar time). Calculate the time of
elongation of Polaris from Table II.

Set the transit over a hub about 40 minutes before the time
of elongation. Level the instrument very carefully, and
set the vernier of the vertical circle to read the latitude of
the place. (2) Focus the objective on a bright star; sight
at Polaris which will be found by following the pointers of
the Great Dipper, at an elevation equal to the latitude of
the place. (3) With a reflector or a piece of white paper re-
flect light into the telescope so that the cross-hairs and the

120

THE TRANSIT.

imag-e of Polaris will be visible at the same time. (4) De-
press the telescope and establish a, target at a distance of
about 500 feet ; place the plank on the ground and nail it
firmly to flat stakes, driving one at each end. (5) Level up
again and follow Polaris with the telescope by means of the
tangent movement ; at elongation it will appear to traverse
the vertical hair for several minutes. (6) Depress the tele-

Oet

Inb-

Up

Obs-

Z 30
Z 4-0
2 4S

Azfrni
Observed
I'ZSk

I°!9-SI
J'29-9

'^RMiN^TioN t)F True Mepidiah

th of q'olaris
Corect'n

o'a'o
o'o'i
o'o's
oVo

Hssn
{All iwable

l'2S4
J'ZB-O
f30-4
J'30-S
l'Z9-B

Error
/limuth
-0'7
-O'l
+l-'3i
i-l'S

I.'O)

CalcuUtion of Railroad Ti'ma of Elongation'
UtJtuds 40°06', LmS'tude sns' .
Astron-Tlms U-C-Pchns,Pec/,l9/S S SO.B
KeducNon ForSdiys/s 3x3-94"'' - !3-6
AstmfTimsU-C-Fi>hri3,Dec-7,19/S S ZJ-Z
Correction far Psi'lroad Tfme - 7-0

g-RTfmeV-C-PaJsnXDec.7,l9/S S ZD-Z
kaducf-ion for Western flongatfon + 5 55'0

Calculation of Azimutli of Polaris at Elon^'n
Az!miifhPt>raris,Btons'f'n,Jsn-J,Wi l'Z9'9
CorrecHon far Pec- T, 1913 - 0-8

A:iimutj7 Polaris, flong't'n, Pec^T, 1913 l'Z9't
Time of Elongation of Polaris'
For Western flongsfion scfd S''SS'"- fa
time U-C- Polaris • far Eastern flon^ation
subtract S''SS'"- from time U-C-

Observers, J-Oae ^ R-Roe-

BY Ob's on Polaris at Elongatioh-

Pec.7,l9ISfZ Hours), CIe.sr 3nc^ warm-
duffSBer^er Transit lia-S, ZLanferns,
Inbs, Zflatstalces,plank lS"t4''Z', 4-
Sd nails, axe, watch settalceep Railroad
time-
Set transit over hvi> 3t i!40A'i^; si^iited
at Polaris, depressed file teiescape and
estabjisiied target about 500 ft- fn>m
instrament- The pianic was placed af-
r/^iit 3n0ies to Jine and fjai/ed to
3 Stake driven <af each end-
Made first observation at western elsfiption.
Reversed instrument in slfifude and

azijnutti between Znd 3nd ^rd readih^s-
Beduced observations

Z,3S-4bythe
following formula.'-
corr-"=0-CISBt'
where ;^= time
from elongation
in minutes; the
correction be-
ing seconds of arc-
(farl3tifude40°,
30 min- from
elongation') -

J^:^J'oiaris
\ "^P-Pole

Polaris at
Upper Culmination

scope, sight at a pencil held on the target and mark the
point very carefully. (7) As a check make three observa-
tions within half an hour after elongation, noting the time
of sighting on the star. Eeverse the instrument in altitude
and azimuth after the first check observation. (8) Reduce
the check observations to observations at elongation by the
following rule : Multiply the square of the time since
elongation in minutes by 0.058, and the product will be the
correction to the azimuth of Polaris in seconds of arc, for
latitude 40°. (9) The next morning lay off the azimuth of
Polaris for each observation to the east or west, depending
upon whether the observation was made at western or east-

PEOBLEMS. 121

ern elong-ation. (10) Check the observed meridian with the
standard meridian. The error of the mean of the four ob-
servations should not exceed one minute. Eecord and re-
duce the data as in the form.

PROBLEM D15. DETERMINATION OP TRUE MERIDIAN
BY OBSERVATION ON POLARIS AT ANY TIME.

(a) Equipment. — The same as in Problem D14.

(b) ProMcm. — Determine a true meridian by observing
Polaris at any time.

(c) Methods. — Make the observations as described in
Problem D14, noting- tlie time of observation to the near-
est minute, and reversing the instrument in altitude and
azimuth between the 3rd and 4th observations. The transit
should be leveled up very carefully with the attached
bubble, particular attention being- given to the horizontal
plate level at right angles to the line of sight. (3) Reduce
the observations by means of the tables.

A star comes to the meridian 4 minutes (nearly) earlier
each day than it did the preceding day. The sideral day is
therefore shorter than tlie solar day, the time from upper
culmination to upper culmination being 23 hours 56.1 min-
utes mean solai time. The time from Upper Culmination to
Lower Culmination is 11 hours 58 minutes.

Astronomical time, or Local Mean Solar time, is the time
that would be kept by the mean sun and is obtained from
utes for each degree of longitude that the place of obser-
vation is east or west of the Standard Meridian. The As-
tronomical day begins at noon of the civil day of the same
date, and is reckoned from zero to 24 hours.

The Hour Angle of Polaris is found by subtracting the
correct Local Mean Solar time of Upper Culmination,
Table II, from the Local Mean Solar time of observation.

The Time Argument used in entering Table IV is the
Hour Angle of Polaris, or 23 hours 56.1 minutes minus the
Hour Angle of Polaris. Table IV is used as follows : Find
the " hours and minutes " of the time argument in the left
hand column of either page of Table IV. On the horizontal
line with the "time before or after upper culmination"
(time argument), the azimuth of Polaris for a declination
of Polaris of 88° 51' will be found in the column under the
given latitude. The correction to the azimuth for each
10

122

THE TRANSIT.
TABLE I.

Azimuth of Polabis at Elongation foe Any Ybab Betweew
1915 AND 1924.

Latitude

1915

1916

1917

1918

1919

1920

1921

1922

1923

1924

o

o /

o /

o /

o /

/

o /

o /

/

O 1

o /

30

I 19.6

119.2

1 18.8

118.5

1 18.1

1 17.8

1 17.4

1 17.0

116.7

116.4

31

20.4

20.0

19.7

19.3

18.9

18.6

18.2

17.9

17.5

17.2

32

21.2

20.9

20.5

20.1

19.8

19.4

19.1

18.7

18.3

18.0

33

22.1

21.8

21.4

21.0

20.7

20.3

19.9

19.6

19.2

18.8

34

23.1

22.7

22.4

22.0

21.6

21.2

20.9

20.5

20.1

19.8

35

124.1

123.7

123.3

123.0

122.6

122.2

121.8

121.5

121.1

120.7

36

25.2

24.8

24.4

24.0

23.6

23-3

22.9

22.5

22.1

21.7

37

26.3

25.9

25.3

25.1

24.7

24.3

24.0

23.6

23.2

22.8

38

27-4

27.0

26.6

26.2

25.9

25.5

25.1

24.7

24.3

23.9

39

28.6

28.2

27.8

27.5

27.1

26.7

26.3

25.8

25.5

25.1

40

1299

129.5

129.1

128.7

128.3

127.9

127.5

127.1

126.7

126.3

41

31.3

30.9

30.4

30.0

29.6

29.1

28.8

28.4

28.0

27.6

4-2

32.7

32.3

31.9

31.5

31.0

30.6

30.2

29^

29.4

29.0

43

34.2

33.8

33.4

32.9

32.5

32.1

31.8

31.2

30.8

30.4

44

35.8

35.3

34.9

34.5

34.1

33.6

33.2

32.8

32.4

31.9

45

137.4

137.0

136.6

136.1

135.7

135.3

134.8

134.4

134.0

133.5

46

39.2

38.7

38.3

37.8

37.4

37.0

365

36.1

35.6

35.2

47

41.0

40.6

40.1

39.7

39.2

38.8

38.3

37.9

37.4

37.0

48

43.0

42.5

42.0

41.6

41.1

40.7

40.2

39.8

39.3

38.8

49

45.0

44.5

44.1

43.6

43.1

42.7

42.2

41.7

41.3

40.8

50

147.2

146.7

146.2

145.7

145.3

144.8

144.3

143.8

143.4

142.9

Correction For Above Table

The above table waa computed with the mean declination of

*olaris

for each year. A more accurate result will be had by applying

to the

tabular values the following corrections, which depend on the difference 1

of the mean and the apparent place of the star. The deduced azimut

hwill,

in general

be correct within 0'.3.

For

Correction

For

Correction

mi
Jan

ddle of

in minutes

middle of

in minutes

uary . . ,

-0.5

July

+0.2

February . ,

-0.4

August. . . .

+0.1

March

-0 3

September .
October. .

—0.1

April

0.0

—0 4

May

+0.1

November .'

-o'e

June

+0.2

December . .

-0.8

minute of change in Declination of Polaris are given in the
last two columns on each page. The changes for latitudes
between 30° and 40° and between 40° and 50° may be in-
terpolated. The Declination of Polaris at any date may be

PROBLEMS.

123

TABLE II.

Local Mean (Astronomical) Time of the Culmination

AND Elongation of Polaris in the Year 1915.

(Computed for latitude 40° and longitude 90° or 6" west of

Greenwich.)

Date

East

Upper

West

Lower

elongation

culmination

elongation

culmination

1915

h m

h

m

h m

h m

January 1 . . .

B1.7

6 46.9

12 42.1

18 44.9

15. ..

23 52.5

5 51.6

11 46.8

17 49.6

February 1 . .

22 45.3

4 44.5

10 39.7

16 42.5

15..

21 50.1

3 49.2

9 44.4

15 47 2

March 1

20 54.8

2 54.0

8 49.2

14 52.0

15 ....

19 59.6

1 58.8

7 54.0

13 56.8

April 1

18 62.7

Bl.9

6 47.1

12 49.9

15

17 57.7

23 52.9

5 52.0

11 54.8

May 1

16 54.8 .

22 50.0

4 49.2

10 52.0

15

15 59.9

21 55.1

3 54.2

9 57.0

June 1

14 53.3

20 48.5

2 47.6

8 50.4

15

13 58.5

19 53.7

1 52.8

7 55.6

July 1

12 55.9

18 51.1

B0.2

6 53.0

15

12 01.1

17 56.3

23 51.5

5 58.2

August 1 . . . .

10 54.5

16 49.7

22 44.9

4 51.7

15

9 59.8

15 55.0

21 50.2

3 56 9

September ] .

8 53.2

14 48.4

20 43.6

2 50.3

15.

7 58.3

13 53.5

19 48.7

1 55.4

October 1 . . .

6 55.5

12 50.7

18 45.9

B2.7

15...

6 00.6

11 55.8

17 51.0

23 53.8

November 1 .

4 53.7

10 48.9

16 44.1

22 46.9

15.

3 58.6

9 53.8

15 49.0

21 Sl.8

December 1 .

2 55.6

8 50.8

14 46

20 48.8

15 .

2 00.4

7 55.6

13 50.8

19 53.6

Correctio

n For Years

After 1915

1.6 up to Mar

chl

1922

1916-^ subtract

2.3 on and aft

er

1923

1

March 1

1924

March 1

1917 subtract

0.7

after March 1

0.9

1925

2.5

1926

4.0 up to Mar

chl

1927

1920 -(subtract

0.1 on and aft

er

1928

Warch 1

1

March 1

after March 1

1.6

found from Table III. For example the azimuth of Polaris
with a time argument of 9 hours and 15 minutes in latitude
40°, on April 21, 1915, was as follows : From Table III the
declination of Polaris on April 21 was very closely 88°
51.25'. From Table IV for declination 88° 51' the azimuth
of Polaris was 58.65' ; the correction for 0.25' was 0.83 X
0.25=0.31', and the azimuth was 58.65' — 0.21 =58.44'. If
the exact time argument is not found in the table, the azi-
muth may be found with sufBcient accuracy by direct inter-

124

THE TRANSIT.

polation. Azimuths for latitudes between values given in
Table IV may be found by direct interpolation. The
nearest whole degree of latitude is usually suflficiently ac-

TABLE III.
Declination of Polaris fob 1915.

Date

Declination

Date

Declination

Jan. 1

88° 51'.54

July 1 ....

88° 51'.05

15

51.58

15. ...

5105

Feb. 1

51.57

August 1 . ...

61.09

IS

51.54

15

51.13

March 1

51.49

Sept. 1

51.21

15

51.43

15

51.28

April 1

51.35

Oct. 1

51.38

15

51.27

15

51.47

May 1

51.20

Nov. 1. ...

51.58

15

51.14

15 ...

51.67

June 1

51.08

Dec. 1

51.75

15

51.06

15

51.81

To obtain t

le declination for th

e corresponding time for years after

n. for each year to tl

le corresponding declination for 1915.

ETERMINAriON C

Up

RR-Timc
Obs-

h. m.
8 Z6

8 41
I 56

9 II
9 IS
S Z(

Mean So-
lar Time
oF Obs-
h- m-
S 3i
S 48
3 03
S IS
S25
S3i

fTrui Mer

Hour An-
gle oF
Polaris
h- m.
9 00
B 15
S iO
9 4i
S5Z
JO 00

Aiimufh
of Pol's
W-oFN-

1

384
33.9
49.Z
46.5
UI

DIAH

Error

oF

Obs-

a I

'J-0
0-0

H-0

tZ-0

= i-0-3

AlhwabJe ^rrc/\= J-0
Calculation
latjJude 40^06' Lanpiti/de 88''J3'W-
Mean Solar Time Upper Culmination ^
Afronom'- Time U-C-F0l3r/s,J\pr-I5, '/S^ Z3 :5i.9
J?educfjon fo Apr- Z/,'= 394X3 = - I9-7

A strommical (local Mean Solar)

Tjme l/-C-p0/3rjs,Apr.Z'; ^ ^3;'33.Z

Jfot/r Angle of foJarfs =Mean Solar T/me
of Oi>s--f-Z4^ff(^'"~Mean SoJarT/Jne,
U-C-foJ3rJs CZ3^Z6^J
T//7?e armament = /four An^/e of P<?/3r/s

(ore3^3£-I^-JfourAff^/eJ
Asfro/Tom/cal Time (local Mean SoJarJ
J3 reckoned from /? o'clock Jfoo/? Qn
\fhe CM/ I>ay of f/ie same cfafs--

Obs- on Polaris at any Time .

Apr/IZI, J915 (2/fours) Clear 3? Warm.

Buff 3? Ber^dr Transit If ^9j Z lanterns,
hubs, Zf/at stakes, p/ank3e'x4'x 21
4~i(f-n3iJs, watch set to keep ^all"

Set transit aver hub at S'lS P-ff;
sighted gt Polar/s, depressed teles-*
cope and established target SOO ft'
from Instrument, the plank was
placed at right angles to line and
nailed to a stake driven at each end-

Set vertical hair on star, noted
time, depressed the telescope
and marked line en target with
pencil-

Apr 22, 19/5 (2 Ifrs^
f educed obsfra-
-tlons using Azi-
muth Tables'

laid off Azimuth
fa the fast and
measured angle
with the True
tferld/an-

Polaris at Time
oF Observation

Ir

^<ppir

PROBLEMS.

125

curate. The time used in making observations should be
correct to the nearest minute, if accuracy is desired.

Table II was compiled from " Ephemeris for the Sun and
Polaris and Tables of Azimuth of Polaris for the year
Land Office. Tables I, II and IV were compiled from " Prin-
S. Coast and Geodetic Survey, 1914.

The observations should be made as near elongation as
possible, for the reason that Polaris is moving most rapidly
in azimuth near culmination and errors in observing the
time and using the table are then a maximum.

With careful work the range of 6 reduced observations
should in no case exceed 1' of are. Eecord the data and
make the calculations as in the form.

r 1 K^"^--

Zenith-, rDecIiiiation+liefractM

/'

./

\

Horizon \
,-

/

(b) /-- S!^ /f"

Fig. 23a.

PROBLEM D16. DETERMINATION OP TRUE MERIDIAN
WITH SOLAR TRANSIT.

(a) Equipmen t. — Complete transit with solar attachment,
reading glass, solar ephemeris, axe, hubs, tacks.

(b) Prohlem. — Determine a true meridian with a. solar
transit.

(c) Methods. — (There are various forms of solar attach-
ments, to transits, among which are the Saegmuller, (a),
and the Buff and Berger, (b). Fig. 23a; the former is the
best known. The theory of all solar attachments in gen-
eral use is the same, and is as follows: In order to bring

TABLE IV. Azimuths of Polabis at Any Hour Angle.

Hour

Angle

before or

after
upper cul-
mination

Azimuths of Polaris computed for Declination

88° 51'

Azimuths given in minutes

Correction for

1' increase in

declination of

Polaris

Lat.
30°

Lat.
32°

Lat.
34°

Lat.
36°

Lat.
38°

Lat.
40°

Lat.
30°

Lat.
40°

h

m
15
30
45

05!28
10.52
15.73

05'38
10.75
16.07

05'52
11.02
16.45

05'67
11.30
16.88

05^82
11.62
17.35

06^00
11.95
17.87

-0'08
-0.15
-0.23

-o'os

-0.17
-0.27

1
1

1

1

00
15
30
45

20.85
25.90
30.82
35.62

21.32
26.47
31.50
36.40

21.83
27.10
32.25
37.27

22.40
27.80
33.08
38.23

23.02
28.57
34.00
39.28

23.70
29.43
35.02
40.45

-0.30
-0.38
-0.45
-0.62

-0.35
-0.43
-0.52
-0.60

2
2
2
2

00
15
30
45

40.25
44.70
48.95
53.00

41.13
45.68
50.03
54.17

42.12
46.77
51.22
55.45

43.20
47.97
52.53
56.87

44.38
49.30
53.98
58.42

45.70
60.75
55.58
60.15

-0.58
-0.65
-0.72
-0.77

-0.67
-0.75
-0.82
-0.88

3
3
3
3

CO
15
30
45

56.80
60.37
63.67
66.68

58.05
61.68
65.05
68.13

59.42
63.13
66.58
69.73

60.93
64.75
68.27
71.50

62.62
66.52
70.13
73.45

64.47
68.48
72.20
75.60

-0.83
-0.88
-0.93
-0.97

-0.95
-1.00
-1.05
-1.10

4
4
4
4

00
15
30
45

69.40
71.82
73.93
75.73

70.90
73.38
75.53
77.35

72.57
75.10
77.30
79.15

74.40
76.98
79.23
81.13

76.43
79.08
81.38
83.33

78.67
81.38
83.75
85.75

-1.02
-1.05
-1.07
-1.10

-1.15
-1.20
-1.23
-1.25

5
5
5
S

00
15
30
45

77.18
78.32
79.12
79.57

78.85
79.98
80.80
81.25

80.67
81.83
82.67
83.12

82.68
83.88
84,72
85.18

84.92
86.13
86.98
87.45

87.38
88.63
89.50
89.97

-1.13
-1.15
-1.15
-115

-1.27
-1.28
-1.30
-1.30

6
6
6
6

00
15
30
45

79.68
79.43
78.87
77 97

81.37
81.12
80.53
79.60

83.22
82.97
82.37
81.42

85.28
85.02
84.40
83.40

87 55
87.28
86.63
85.62

90 07
89.77
89.10
88.05

-1.17
-1.15
-1.13
-1.12

-1.30
-1.30
-1.28
-127

7
7
7

7

00
15
30
45

76.73
75.17
73.28
71.10

78.33
76.73
74.82
72.57

80.10
78.47
76.50
74.20

82.07
80.38
78.35
76.00

84.23
82.50
80.42
77.98

86.62
84.83
82 68
80.18

-1.10
-1.08
-1.07
-103

-125
-1.22
-1.20
-1.15

8
8
8
8

00
15
30
45

68.60
65.82
62.77
59.45

70.02
67.18
64.07
60.67

71.60
68.68
65.48
62.02

73.33
70.35
67.07
63.52

75 23
72.18
68.82
65 17

77.35
74.20
70.73
66.98

-1.00
-0.95
-0.90
-0 85

-1.10
-1.07
-1.02
-0.97

9
9
9
9

00
15
30
45

55.88
52.08
48.07
43.85

57.03
53.15
49.05
44.73

58.30
54.33
50.13
45.73

59.72
55.63
51.33
46.82

61.23
57.07
52.65
48 02

62.95
58.65
54.12
49.35

-0.80
-0.75
-0.70
-0.63

-0.90
-0.83
-0.77
-0.70

10
10
10
10

00
15
30
45

39.45
34.88
30.17
25.33

40.25
35.58
30.78
25.85

41.13
36.37
31.47
26.42

42.12
37.23
32.20
27.05

43.20
38.20
33.03
27.73

44.40
39.25
33.95
28.50

-0.57
-0.50
-0.43
-0.37

-0.63
-0.57
-0.48
-0.40

11
11
11

11

00
15
30
45

20.40
15.37
10.28
05.15

20.82
15.68
10.48
05.25

21.27
16.03
10.72
05.37

21.77
16.40
10.97
05.50

22.33
16.83
11 25
05.63

22.95
17 28
11.57
06.78

-0.30
-0.22
-0.15
-0.07

-0.33
-0.25
-0.17
-0.08

Elongation

Azimuth
Hour Angle

l''l9'68
b m s
5 57 21

l°2l'37 l''23'23
b m s 'h m s
5 57 08,5 56 54

1°25'30
h m s
5 56 39

1°27;57
h m s
5 56 24

1°30'08
h m s
5 56 08

-l'l5

s
+2

-l'.30

s
+3

TABLE IV. Azimuths of Polaeis at Ant Houb Angle.

Hour

Angle

before or

after
upper cul-
mination

Azimuths of Polaria computed for declination

88° 51'

Azimuths given in minutes

Correction for

1' increase in

declination of

Polaris

Lat.
40°

Lat.
42°

Lat.
44°

Lat.
46°

Lat.
48°

Lat.
50°

Lat.
40°

Lat.
60°

h

m
15
30
45

06^00
11.95
17.87

06.18
12,35
18.45

06!40
12.77
19.08

06!63
13.23
19.78

06'90
13,77
20.57

07^20
14.35
21.45

-o!o8

-0.17
-0.27

-o!io

-0.22
-0 32

1
1

1
1

00
15
30
45

23.70
29.43
35.02
40.45

24.47
30.37
36.13
41.75

25.30
31.42
37.38
43.18

26.23
32.57
38.77
44.77

27.28
33.87
40,30
46.55

28.45
35.30
42.02
48.52

-0,35
-0.43
-0,52
-0,60

-0.42
-0.53
-0.63
-0.72

2
2
2
2

00
15
30
45

45.70
50.75
55.58
60.15

4717
52.37
57.35
62.07

48.78
54.17
59.30
64.18

50.58
66.15
61.48
66.53

52,58
58.37
63.90
69.15

54.82
60.85
66.62
72.07

-0,67
-0.76
-0,82
-0,88

-0,82
-0.90
-0.98
-1.07

3
3
3
3

00
15
30
45

64.47
68.48
72.20
75.60

66.50
70.65
74.48
77 98

68.77
73.05
77.00
80.62

71.28
75,72
79.82
83.55

74.08
78.68
82.93
86.82

77.20
82.00
86,42
90,45

-0,95
-1.00
-1.06
-1.10

-1.13
-1.20
-1.27
-1.33

4
4
4
4

00
15
30

45

78.67
81.38
83.75
85.75

81.13
83.93
86.37
88.43

83.88
86,77
89.27
91.38

86.90
89.90
92.50
94.68

90.30
93.40
96.10
98,33

94,08

97,28

100.07

102,42

-1,16
-1,20
-1.23
-1.25

-1.38
-1.43
-1.47
-1.60

5
5
5
5

00
15
30

45

87.38
88.63
89.50
89.97

90.10
91.38
92.27
92.75

93.10
94,42
95.33
95.83

96.45
97.80
98.73
99.23

100,17
101,57
102.52
103.03

104,32
105,76
106,73
107.27

-1.27
-1.28
-1.30
-1.30

-1.62
-1.63
-1.55
-1.57

6
6
6
6

00
15
30
45

90.07
89.77
89.10
88.05

92.83
92.53
91.83
90.73

95.92
95 58
94.85
93.72

99.32
98.97
98.20
97.02

103.10
102.73
101.93
100 68

107.32
106.93
106.07
104.77

-1.30
-1.30
-1.28
-1.27

-1.56
-1.66
-1.53
-1.52

7
7
7
7

00
15
30
45

86.62
84.83
82.68
80.18

89.25
87.40
85.18
82.60

92.18
90.27
87.97
85.28

95 42
93.42
91.03
88.27

99.02
96.93
94.45
91.57

103.03
100.85
98.27
95.25

-125
-122
-1.20
-1.15

-1.48
-1.45
-1.42
-1.37

8
8
8
8

00
15
30

45

77.35
74.20
70.73
66.98

79.68
76.43
72.87
68.98

82,27
78.90
75.20
71.20

85.13
81.65
77.82
73.67

88.32
84.68
80.72
76.40

91.85
88.08
83.93
79.45

-1.10
-1.07
-1.02
-0.97

-1.32
-1.27
-1.20
-L13

9
9
9
9

00
15
30
45

62.95
58.65
54.12
49.35

64.83
60.40
55.73
50.82

66,92
62.33
57.52
52.45

69.22
64.48
59.48
54.25

71.78
66.87
61.68
56.25

74.63
69.53
64.13

58.48

-0.90
-0.83
-0.77
-0.70

-1.07
-0.98
-0.92
-0.83

10
10
10
10

00
15
30
45

44.40
39.25
33.95
28.50

45.72
40.42
34.95
29.35

47.17
41.70
36,07
30.28

48.78
43.13
37.30
31.32

50.58
44.72
38,67
32.47

52.58
46.48
40.20
33.75

-0.63
-0.57
-0,48
-0.40

-0.76
-0.67
-0.57
-0.48

11
11
11
11

00
15
30
45

22.95
17.28
11.57
05.78

23.62
17.80
11.90
05.97

24.37
18.37
12.28
06,15

25.20
19,00
12.70
06,37

26.13
19.70
13.17
06.60

27.16
20.47
13,72
06.85

-0.33
-0.25
-0,17
-0.08

-0.38
-0.30
-0.20
-0.10

Elongation

Azimuth
Hour Angle

1 30.08
h m s
5 56 08

1°32'85
h m s
5 55 51

1°35'93
h m s
5 55 33

1°39!33
h m s
5 55 14

l°43!l3
h m s
5 64 53

1°47!36
h m s
5 54 31

-l'.30

s
+3

-166

s
+3

127

128

THE TRANSIT.

the image of the sun into the center of the solar telescope
when the line of collimation of the solar telescope makes an
angle with the line of collimation of the main telescope
equal to the sun's declination corrected for refraction, and
the line of collimation of the main telescope is elevated at
an angle equal to the co-latitude of the place of observa-
tion, it is rigidly necessary that the line of collimation of
the main telescope lie in a true meridian as shown in (b),
Fig. 23a.

The elementary lines of a solar attachment are : (1) The
polar axis; (2) the line of collimation of the solar tele-
scope; (3) the attached level line. These lines should have

Fig. 23b.

the following relations: (1) The polar axis should be per-
pendicular to the line of collimation of the solar telescope
and the horizontal axis of the main telescope; (2) the line
of collimation of the solar telescope and the attached level
line should be parallel. The methods of making these ad-
justments are obvious.

The declination of the sun (see Pig. 23b for explanation
of astronomical terms) for the place of observation is
found by adding, algebraically, the hourly change multi-
plied by the number of hours since Greenwich mean noon
(6 A. M., 90th Meridian) to the declination of the sun, as
given in the solar ephemeris for Greenwich mean noon for
the given date. The setting (apparent declination) is found

PROBLEMS.

129

by taking the algebraic sum of the refraction correction and
the declination of the sun obtained as above. The refrac-
tion is always plus ; the declination is plus when the sun
is north and minus when south of the celestial equator ;
and the hourly chang-e in declination is plus when the sun
is moving north and minus when moving south.

The " Pocket Solar Ephemeris and Eefraction Tables for
Use with Saegmuller's Solar Attachment," is given in
"Handbook for; Engineers" by George N. Saegmuller, pub-
lished by Bausch & Lomb Optical Co., Eochester, N. Y. An
" Ephemeris of the Sun and Polaris, and Tables of Azimuths
of Polaris " is published by the General Land Office for each
year. This Ephemeris may be obtained by addressing the
Department of Interior, General Land Office, Washington,
D. C, or may be purchased at a price of 5 cents per copy
from the Government Printing Office, Washington, D. C.
The true' local mean solar time should always be iised, and
or subtracting four minutes for each degree that the place
of observation is east or west of the standard meridian.
The mean refraction of the sun for different altitudes is
given in Table V.)

TABLE V.
Mean Eefraction of the Sun.

Babometeb 30 Inches, Temperatuke 50° Eahe.

(Eefraction makes observed altitude too large.)

Altitude,

Refraction,

Altitude,

Refraction,

Altitude,

Refraction,

Degrees

Minutes

Degrees

Minutes

Degrees

Minutes

10

5.10

24

2.02

50

0.70

12

4 25

26

183

55

0.58

14

3.62

28

1.67

60

0.48

16

3 17

30

1.53

65

0.38

18

2 80

35

1.25

70

0.30

20

2 48

40

1.03

80

0.13

22

2.22

45

0.85

90

0.00

(1) Calculate the apparent declination (setting) of the
sun for several different times, varying by 15 minutes, be-
tween 8 and 10 o'clock A. M. and 2 and 4 o'clock P. M. (2)
Set the transit over the hub, level up very carefully with
the attached bubble, and very carefully adjust the main
transit and solar attachment. Determine the index error
of the vertical circle, and either correct it or apply it to all
vertical angles with its proper sign. (3) Level the transit

130

THE TEANSIT.

Determinatioh

Time
oF ObS'

sho'

9:011
3: IS
9--30
PM-
2=30
Z:4S
3 ■■OP
3:15
3:30

OF True Meiibiah

Declina-
tion
f0'S4'g
fIS'U'S

ta'ss'z

tl9'SB'3

fJ3'}li
Hi'Sl'l
HS'SS'Z
H9'S8'4
WSS'S

Rtfrac-

Cor.
tO-7
tO-'l
tO'6
tO'S
f-O'S

■/■O'S
K>'6
fO-'»
fO-'7

Setting

fff'SF3
Hl'SSi
tJ9'SS-'?
m'iS'7

f/9'SS'6
W'JS-'S

H3'i9-'Z

Azi-

riufh
tZ'll'
ZZ'IZ'

Z2'ja'
zz'ji

ZZ'JI'

zz'ts'
z?'/o'
zz'os'

ZZ'll'
ZZ'P9'

ZZ'JO'i

zzWi.

o'm

True Atiffiufh of Line
MIoivabJe error 01'- frrorA

Calculation of Setting fApf-Bedil 8!'30'"AM-
LetltUife 40'(!6'tl; langifude SS'JS'W-
App-Pecliffafion ffreenivlch Mean /foon.
(etllT^ AM- ltere),l1syZt>,J90J, fJS'Si'S
Correction for Z''Z3'"=Z-4i<)-S}, * Ol-'S
tecJinat-Jon ofJunstf--3l>AM- = i-l9'£4'-S
Xefracl-hn Cor^-3''30'ie/iremi'n-+ 0-7
Apii-Dccl-3tS:3Ci\M- - f/S'SS^S

Apparent- Pec/- (^ett/n^) for f/je other
times was eaJci/fsfeet in JiMe manner-

WITH Solar Transit-

Observers, J- Doe if B- Roe-
MayZO,190/. C4 ffrs-) C/ear i^ warm-
SuffifSergerTrans/f/i^S, ty/fh Saeg-
muiler SffJarAttacIimentjItubs, ajie^
fphemeris CMsndbmk for Enqineers,
By 6eo./1.S3egr/7Uller, Bausch&Lomb
OpticdICo. Rochester,/1.Y.)
Tested Transitand SolarAttacbment
and found both in perfect- adjustment.
Set tran sit over hubj JeveJedup very
carefui/y yvit/i ianf bt/bbJej found
Index frror of Vert- Circle ='Zerff-
Setoff -i9-°SS'S (-App-Pecl-) on Vertical
circie-and Jeve/ed solar tet escape by
means of its attached bubble-
Set off t49'S4' Cfo-lafJ on Vertical

circle-TeJescopff pointed S-bo/tr times-
Set A vernierat zero and sighted af

5ta-3 with Jower rnotion-
Unclamped upper motion, moved transif-
on vertical axis andsoiar on its
polar axis, and brought image of sun
into center of solar <af dijff A- fl-
Mean Solar Time-(ff-S-Tijne=P-P-timei-7-?)
were determined- (S-A-M- ^ S P-pf-J I

very carefully with the attached bubble. Bring the line of
collimation of the main telescope and the line of collima-
tion of the solar telescope parallel by sighting on a distant
point, and point the main telescope south. (4) Set off the
apparent declination (setting) with opposite sign on the
vertical circle, i. e., dip the telescope when the declination is
plus (north), and elevate the telescope when the declina-
tion is minus (south). (5) Level the solar telescope by
means of its attached bubble. (6) Set off a plus vertical
angle on the vertical circle equal to the co-latitude of the
place. (7) Set the A vernier at zero and sight at a point on
the true meridian. (8) Unclamp the upper motion, turn
the main telescope about its upper motion and the solar
telescope about its polar axis until the image of the sun is
brought to the center of the cross lines in the solar tele-
scope at the time for which the declination was computed,
and clamp the upper motion. The line of collimation of the
main telescope will then be in the meridian. (9) Kead the
horizontal plates. The reading will be the azimuth of the
line first sighted on. (10) Eepeat, using the setting corre-
sponding to the time of observation, until ten values are ob-
tained. If possible make five determinations in the A. M.,

PROBLEMS.

131

and five in the P. M., about the same time from noon. The
mean of these observations will eliminate instrumental
errors. The most favorable time for making observations
with a solar transit is from 8 to 10 A. M. and from 3 to 4 P.
M. (11) Determine the true azimuth of the given line. The
error of the determination of the meridian should not ex-
ceed one minute. Eecord as in the form.

Solar Observation-

Tele.scope

Vertical Circle

Dates Time

On Mark.

On Sun

Dlreci- °r
Rev'd -6

ISO'SO'OO"
O'SO'OO"

I00°4I'P0"
?80'4T30"

4S°59'00"
46°0T00"

May 15,1301
S/S-M-nemSalar
Time

Mean

iso'so'oa"

IOO'44'I5"

4e'03'00"

Computation

Oeclination at 6reenwich Hoon, 6AM-St'd Tima SOtf Md- = IS°4S'S6-l" N-
Hourly Change =35-g"- Change FerShrs >= 3 x-iS-S" = l'47-4" N-

Dedinafion at 9 A-M- = /f'47'43-SV.

Average Vertical Angle liy Otservetfort 4e'o3'PO"
(^rrecffon for PeFr3cf:or7 00'36' '

True Altifi/da

Latitude of Obseri^atoryj U- of I-
Station JOO'tl-

Latitude of Stafian

4e'0Z'04"
40' 06' 00"

40' 06' 00"

Co, J- P2S=. }/^'"i S- xSin-dS'PoleDist^
Cos-^PZS- \l 5A7. CoAlt-r. Sin- Ca-Laf-

where S=Pole Plot. + Co-Alt--t Co-laf-

Pole Dist- = 7/" 12' 16"

Co-Alt = 43'STS6''

Co-Lat- = 49° £4' 00"

S = ies°04'ie"

is = SZ'32'06"

Pole Diet- = 7J'IZ'IG"

^5- Pole Dist- = ll'J9'S0"

Log-Sln-82'3Z'0e'= 3-99630

Log- Sin- II'I9'S0"= 9-29313

Co-lag- Sin- 43°S7'S6"= 0-ISSSO

Co-log- Sin- 49'S4'00"=' 0-11638

2)l9-S6441 -20
Log-Cos-iPZS = 3-78224

iPZS= 32 '4-5 'IS"
, PIS = IOS'26'36"
Azimuth of Sun from the North
Angle between Sun and Mark
Observed Azimuth from Harth Station to Mark
True Azimuth from Iforth Sfatj'on to Mark-
Error

PROBLEM D17. DETERMHSTATION OF TRUE MERIDIAN
BY DIRECT OBSERVATION ON THE SUN.

(a) Equipment. — Complete transit, reading glass, hub,
axe, colored eyepiece or colored shade to fit over objective,
good watch set to keep standard time, solar ephemeris.

132 THE TRANSIT.

(b) Problem. — Determine a true meridian by a direct ob-
servation on the svin with a transit.

(c) Methods. — (1) Set the transit over a hub and level up
very carefully with the attached bubble. (3) Test the ad-
justments of the transit very carefully, and determine the
index error of the vertical circle. (3) Sight on a horizontal
mark and read the horizontal plates. (4) Sight at the sun
directly, by the aid of the colored eyepiece or colored glass
shade, and bring his image tangent to the horizontal and
vertical wires. (5) Read vertical circle and horizontal
plates. (6) Reverse the telescope and make a second ob-
servation the same as the first except that the sun should
be in the opposite quarter of the field of view. (7) The
mean of the vertical and horizontal circle readings will
give the apparent altitude and plate reading of the sun's
center. (8) Observe the standard time of the observation
and reduci to mean solar time by adding or subtracting 4
minutes for each degree that the place of observation and
reduce to mean solar time by adding or subtracting 4
minutes for each degree that the place of observation is
.>ast or west of the standard meridian. (9) Calculate the
angle PZS in the P Z S triangle as shown in the accom-
panying form. Refraction makes the sun appear too high
and it should therefore be subtracted. (10) Determine the
azimuth of the line from the hub to the mark and check
the observed azimuth. (The data for this problem may be
obtained from Saegmuller's " Solar Ephemeris and Refrac-
tion Tables,'' or from the " Ephemeris of the Sun and
Polaris, and Tables of Azimuths of Polaris," by the General
Land Office, mentioned in Problem D16. Mean refraction
of the sun for different altitudes is given in Table V.) (11)
Where considerable accuracy is desired, make a second ob-
servation when the sun is about the same distance on the
opposite side of the meridian. The error of the determina-
tion should not exceed 1 minute.

PROBLEM D18. COMPARISON OF TRANSIT TELESCOPES.

(a) Equipment. — Eive engineers' transits.

(b) Prohlem. — Make a critical comparison of the tele-
scopes of five engineers' transits.

(c) Methods. — Follow the methods outlined in the com-
parison of level telescopes.

PROBLEMS. 133

PROBLEM D19. TEST OF A TRANSIT.

(a) Equipment. — Transit, reading glass, leveling rod,
chaining pins, foot rule.

(b) Proilem. — Test the following adjustments of an as-
signed transit: (1) Test the graduation for eccentricity.
(2) Test the plate levels to see if they are perpendicular to
the vertical axis. (3) Test the line of collimation to see if
it is perpendicular to the horizontal axis. (4) Test the
horizontal axis to see if it is perpendicular to the vertical
axis. (5) Test the level under the telescope to see if the
tangent to the tube at the center is parallel to the line of
collimation. (6) Test the vertical circle to see if the
vernier reads zero when the line of sight is horizontal.

(c) Methods. — Make the tests as described in the first
part of this chapter but do not make any of the adjust-
ments or tamper with any of the parts of the instrument.
Check each test. Make a careful record of the methods and
errors, including a statement of the manner of doing cor-
rect work with each adjustment out.

PROBLEM D20. ADJUSTMENT OF A TRANSIT.

(a) Equipment. — Transit, reading glass, leveling rod,
chaining pins, adjusting pin, small screw driver.

(c) Methods. — Make the following tests and adjustments
of an assigned transit that has been thrown out of adjust-
ment by the instructor: (1) Test the graduation for eccen-
tricity. (2) Adjust the plate levels perpendicular to the
vertical axis. (3) Adjust the line of collimation perpendicu-
lar to the horizontal axis. (4) Adjust the horizontal axis
perpendicular to the vertical axis. (5) Adjust the level
nnder the telescope parallel to the line of collimation. (6)
Adjust the zero of the vertical circle to read zero when the
line of sight is horizontal. (7) Center the eyepiece.

(c) Methods. — Make the tests and adjustments as de-
scribed in the first part of this chapter. Use extreme care
in manipulating the screws and if any of the parts stick
or work harshly, call the instructor's attention before pro-
ceeding. Repeat the tests and adjustments. Make a care-
ful record of methods and errors.

PROBLEM D21. SKETCHING A TRANSIT.

(a) Equipment. — Engineers' transit.

(b) Prohletn. — Make a first-class sketch of an engineers'
transit.

(c) Methods.- — (See similar problem with the level.)

134

THE TRANSIT.

PROBLEM D33. ERROR OF SETTING FLAG POLE WITH
TRANSIT.

(a) Equipment. — Transit, iron flag pole, flat stake l"x
2"x 15", foot rule.

(b) Prohlem. — Determine the probable error of setting a
flag pole with the transit at a distance of 300 feet. Repeat
for 600 feet.

^

Observers, J-Doe c

R^Roe- ^

Err

OR OF

SET-

IHG I

UA6

'OLE

WITH ENSINEER

>' Transit-

)Ht3nce

No.oF

Distance

d

d^

Dec-6,m4.(2hiiun

1 Cool and Quiet-

Ft-

StHlnj

In-

In-

Used Suff S Seiyar

Transit, LocJcer/io^^i

300

/

I-/S

0-JS

0'03Z4

f/afsfake,/'"?'

^lS"and iron flagpole-

Z

J-3S

■02

■0004

Sighted 3 f- /ran F/ag

poie set- on stake

3

/■}0

■06

■0036

laced on ground <?/"

*

I-S3

■17

■0219

ab.out 300 ft- fn

m ti?e Tr,ynsiff 3nd

S

I-3Z

■04

■0016

clamped bot-h pj.

^tes; then measured

6

I-3S

■02

■0004

The distance in i

iches from <? ling

7

JZS

■07

■0049

dr^wn across t

he beared'

e

1-46

■10

■0100-

With tct/i plates cl.

'mped, lined in the

9

M6

■10

■0100

rod 10 times in a

'if the flagman not-

w

Nean

1-30

■06

■0036

-td^

ing the distance
The pole tvas sh.

fram the line-
'f ted each time-

1-36

O-OSSS

Repeated test For

00 ft-

600

1

I-J4

0^25

0^0623

Probable Error for

300 Ft-

?
3

1-56
1-14

■J7
■2S

■0219
■062S

e,-c.usi^,^

H74?l'-^ =0-105 in

5

1-76

■)7
■37

■02S9
■136S

--#=^

^i- = 0-032 in.' 0-0027 ft-

6

hSS

■16

■0236

£m (Angle) =

.,„- 0^^ ,:i.

7

1-23

■/6

■0256

Probable trror Fo

■ 600 Ft.

S

HO
I-5S

■23
■16

■OSH
■0256

£,' 0-6745 f^

■ = 0-247 !n-

JO
Mean

I-6S

•26

■0676

=Zd^.

^'"- no -

0-O7Sin. = e-OO6Sft-

1-39

0^5472

^-—J

Bm (Angte}A

tan-">-Zf=2rz

600 J

(c) Methods. — (1) Set the transit up and sight at the flag
pole plumbed near the middle of the stake at a distance of
about 300 feet. (2) Measure the distance from the point of
the flag pole to a mark on the stake. (3) Keep the vertical
axis clamped, and move the pole to one side. (4) Set the
pole with the transit, and measure the distance from the
first line. (5) Repeat until at least ten consecutive satis-
factory results are obtained. (6) Compute the probable
error of a single observation and of the mean of all the
observations (see chapter on errors of surveying), and re-
duce the mean error to its angular value. (7) Repeat
for 600 feet. Determine distances by pacing. Follow the
form.

PROBLEMS. 135

PROBLEM D23. REPORT ON DIFFERENT MAKES AND
TYPES OP TRANSITS.

(a) Equipment. — Department equipment, catalogs of the
principal makers of engineers' transits.

(b) Prolileni. — Make a critical comparison of the several
types of transits made by the different makers.

(c) Methods. — (See similar problem with the level.)

CHAPTER VI.
TOPOGRAPHIC SURVEYING.

Topographic Map. — A topographic map is one which
shows with practical accuracy all the drainage, culture, and
relief features that the scale of the map will permit. These
features may be grouped under three heads as follows :
(1) the culture, or features constructed by man, as cities,
villages, roads; (2) the hypsography, or relief of surface
forms, as hills, valleys, plains; (3) the hydrography, or
water features, as ponds, streams, lakes. The culture is
usually represented by conventional symbols. The surface
forms are shown by contours (lines of equal height), (a).
Fig. 24, or hachures, (b), Fig. 24. The -water features are
shown by soundings, conventional signs for bars, etc.

Fig. 24.

Topographic maps may be divided into two classes de-
pending upon the scale of the map. Small scale topographic
maps are made by the U. S. Coast and Geodetic Survey and
the U. S. Geological Survey, and are drawn to a scale of
1 : 62,500, 1 : 125,000 or 1 : 250,000 with corresponding contour
intervals of 5 to 50, 10 to 100, and 200 to 250 feet. These
maps show the streams, highways, railroads, canals, etc., in

1.W

138 TOPOGRAPHIC SURVEYING.

outline but do not show any features of a temporary char-
acter. For topographic symbols, see Chapter XI.

Large scale topographic maps are drawn to a scale of 400
feet to 1 inch ( 1 . 4800) , or greater, with contour intervals
from 1 to 10 feet depending upon whether the ground is iiat
or hilly Roads, streets, dwellings, streams, etc., are drawn
to scale. Features too small to be properly represented
when drawn to scale are drawn out of proportion to the
scale of the map.

Topographic Survey. — The object of a topographic sur-
vey is the production of a topographic map, and hence
neither time nor money should be wastefully expended in
obtaining field data more refined than the needs of the map-
ping demand. A topographic survey may be divided into
three parts: (1) the reconnaissance; (2) the skeleton of
the survey; (3) filling in the details.

Reconnaissance. — The reconnaissance is a rapid prelim-
inary survey to determine the best methods to use in mak-
ing the survey and the location of the principal points of
control. A careful reconnaissance enables the topographer
to choose methods that are certain to result in a better map
and a distinct saving of time.

Skeleton. — There are three general methods of locating
the skeleton of a topographic survey: (1) tie line survey
with chain only, (2) traverse method with transit or com-
pass; (3) triangulation system, (f), Fig. 30. The first
method is used for the survey of small tracts. The second
method, in which the distances are measured with the
veys. The third method, in which ■ triangulation stations
are connected with each other and with a carefully meas-
ured base line and base of verification, is used on surveys
for small scale maps and on detailed or special surveys,
such as surveys of cities and reservoir sites.

Filling in Details. — There are three general methods em-
ployed for filling in the details : ( 1 ) with transit or compass
and chain; (2) with transit and stadia; (3) with plane
Mississippi and Missouri River Commissions. The plane
table and stadia are used by the TJ. S. Coast and Geodetic
and the U. S. Geological Surveys.

Topographic City Survey. — A topographic city survey is
one of the best examples of a survey for a large scale map.
It is usually based on a system of triangulation executed
with precision and connected with carefully measured base

lines. The details of the survey are usually taken up in the
following- order: (1) reconnaissance and location of trian-
gulation stations ; (2) measurement of base line and base of
verification; (3) measurement of angles by repetition ; (4)
establishment of bench marks by running duplicate levels ;
(5) adjustment of angles of triangulation system; (6) com-
putation of sides, azimuths and coordinates; (7) filling in
details, usually with transit and stadia; (8) plotting of
triangulation and other important points on the map by
rectangular coordinates; (9) plotting the details and com-
pleting the map. The instructions given on the succeeding
pages are for a survey of this type.

Hydrographic Survey. — Hydrographic surveying is di-
vided into river and marine. The first includes the location
of bars and obstructions to navigation, and the determina-
tion of the areas of cross-section, the amount of sediment
carried, etc. The second includes the making of soundings,
location of bars, ledges, buoys, etc. The depth of the water
is determined by making soundings with a lead or rod,
and the velocity is gaged by means of fioats or a current
meter, (d). Fig. 31.

Soundings are located: (1) by two angles read simulta-
neously from both ends of a line on the shore, (f). Fig. 31;
(2) by keeping the boat in line with two flags on shore, and
determining the position on the line by means of an angle
read on the shore, or by a time interval ; ( 3 ) by intersecting
ranges, (g). Fig. 31 ; (4) by stretching a rope or wire across,
the stream; (5) by measuring with a sextant in the boat
at the instant that the sounding is taken two angles to three
known points on the shore, (c). Fig. 31 ; the point is located
by solving the three point problem graphically with the
three arm protractor, (e). Fig. 31 ; (6) by locating the posi-
tion of the boat at the instant that the soundings are taken
with transit and stadia. The first three methods are used
on small river or lake surveys. The fourth method is used
where soundings are taken at frequent intervals. The fifth
method has been used almost exclusively in locating sound-
ings in harbors, lakes, and large rivers. The sixth method
is rapidly coming into general use and promises to be the
favorite method.

Description. — The stadia is a device for measuring dis-
stadia-hairs, shown in (g) , Fig. 27, are carried on the same

140

TOPOGEAPHIC SURVEYING.

reticule as the cross-hairs and are placed equidistant from
the horizontal hair. The stadia-hairs are sometimes placed
ever, considered better practice by most engineers to have
the stadia-hairs fixed and use an interval factor, rather
than try to space the hairs to suit a rod or to graduate
a rod to suit an interval factor.

type. In Fig. 27, (a) and (b) are the kind used on the U.
S. Coast Survey; (c) on the U. S. Lake Survey; (d) and
(c) by the U. S. Engineers. A target for marking on the
rod the height of the horizontal axis of the transit above
the station occupied is shown in (f), Fig. 27.

Theory of the Stadia. — In Fig. 25, by the principles of
optics, rays of light passing from points A and B on the
rod through the objective so as to emerge parallel and pass
through the stadia-hairs a and 6, respectively, must inter-

Fig. 25.

sect at the principal focal point (J in front of the objective ;
therefore the rod intercept, s is proportional to the dis-
tance, g from the principal focal point in front of the ob-
jective.

Stadia Formula For Horizontal Line of Sight and Ver-
tical Rod. — In Fig. 25, from similar triangles we have

From which

:: i : f

g^ rS = k. S

(1)

(2)

and

D = k. s + (c -f f )

(3)

Stadia Formula For Inclined Line of Sight and Vertical
Rod. — In Fig. 26 we have

and
but

also

BD=iAE. cosa (approx.) (4)

D =k. s. cos a + (c + f ) (5)

H ^ D. cos a

k. s. cos2 a + (c + f ) cos a (7)

= k. s — k. s. sin2 a -(- (c -|- f ) cos a (8)

V = D. sin a (9)

= k. s. sin a. cos a +(c + f) sin a (10)

= l,^k. s. sin 2 a+(c+f) sin a (11)

Use of the Stadia. — The transit is set up over a station
of known elevation and with a given direction or azimuth
to another visible station ; the height of the line of coUima-
tion above the top of the station Is determined either by-
holding the rod beside the instrument and setting the
target, or preferably by graduating one leg of the tripod
and using the plumb bob ; then with the transit oriented on
a given line, " shots " are taken to representative points,
and record made of the rod intercept, vertical angle and azi-
muth. In reading the intercept the middle hair is first set
roughly on the target, then one stadia-hair is set at the
nearest foot-mark on the rod and the intercept read with
the other stadia-hair, after which the precise vertical angle
is taken, and tUe azimutli is read,

142

TOPOGRAPHIC SURVEYING.

Beducing' the UTotes. — The notes may be reduced by
means of tables, diagrams, or a special slide rule. The
slide rule is the most rapid. There are several forms of
stadia slide rule that are very accurate and are convenient
for field use.

<

i
<

4
i
<
4

<

(3J

X

X

(b)

>
>

CCJ (d)

Fisr. 27.

M

M

(6)

r ^

(f)

THE PLANE TABLE.

Description.^The plane table consists of an alidade, car-
rying a line of sight and a ruler with a fiducial edge. The
alidade is free to move on a drawing board mounted on a
tripod. The drawing board is leveled by means of plate
levels. The line of sightf should make a fijced horizontal
angle with the fiducial edge of the ruler. The complete
plane table is a transit in which the horizontal limb has
been replaced by a drawing board.

There are three general types of plane tables: (1) the
Coast Survey plane table, (a). Fig. 28; (2) the Johnson
planB table, (b), Fig. 28; (3) the Gannet plane table, (d),
Fig. 39.

TTse of the Plane Table. — In making a survey with the
plane table the angles are measured graphically and the

THE PLANE TABLE.

143

lines and points are plotted in the field. The principal

methods of making a survey with a plane table are: (1)

radiation; (3) traversing; (3) intersection; (4) resection.

Radiation. — In this method a convenient point on the

Complete Plane Tables.
Fig. 38.

paper is set over a selected point in the field, and the table
clamped. The line of sight is then directed towards each
point to be located in turn and a line is drawn along the

Eg. 39.

144

TOPOGKAPHIC SUKVEYING.

fiducial edge of the ruler. The distances, which may be de-
termined by measuring with chain, tape or stadia, are
plotted to a convenient scale, (a). Fig. 30.

Traversing. — This method is practically the same as
traversing with a transit, (b). Fig. 30. Care should be used
in orienting the plane table to get the point on the paper
over the corresponding point on the ground as nearly as
the character of the work requires.

C

m--:

^D

A

E

f3)

'K'

-y^R

I

r^j

3 --^bl

Kg. 30.

THE PLANE TABLE. 145

Intersection. — In this method the points are located by
intersecting lines drawn from the ends of a measured base
line, (c), Kg. 30.

Resection. — In the resection method the plane table is set
up at a random point and oriented with respect to either
three or two given points, which gives rise to two methods
known respectively as the three-point and two-point prob-
lems.

Three Point Problem. — Where three points are located on
the map and are visible but inaccessible, the plane table is
oriented by solving the " three point problem." There are
several solutions, the best known of which are: (1) the
mechanical solution; (3) the Coast Survey solution; (3)
Bessel's solution; (4) algebraic solution. The problem is
indeterminate if a circle can be passed through the four
points.

In the mechanical solution the two angles subtended by
the three points are plotted graphically on a piece of trac-
ing paper, and the point is located by placing the tracing
paper over the plotted points.

In Bessell's solution, (d), Fig. 30, a, 6, c are three points
on the map corresponding to the three points, A, B, C on
the ground, and D is the random point at the instrument
whose location, d, it is desired to find on the map. Con-
struct the angle 1 with vertex at point c as follows : Sight
along the line ca at the point A, and clamp the vertical axis.
Then center the alidade on c and sight at B by moving the
alidade, and draw a line along the edge of the ruler. Con-
struct the angle 3 with vertex at a in the same manner. The
line joining 6 and e will pass through the point d required.
Orient the board by sighting at B with the line of sight
along the line e 6, and locate d by resection.

Two Point Prohlem. — To orient the board when only two
points are plotted, proceed as follows : Select a fourth
point, C, that is visible, and with these two points as the
ends of a base line, (e). Fig. 30, laid off to a convenient
scale, locate two points a' and 6' on the map by intersec-
tion. The error of orienting the board will be the angle
between the lines o-6 and a'-h'. The table can now be
oriented and the desired point located on the board by re-
section.

(1) the plate levels; (3) the line of collimation; (3) the
horizontal axis; (4) the attached level. These adjustments
are practically the same as those for the transit.
11

146

TOPOGRAPHIC SUEVEYING.
THE SEXTANT.

Description. — The sextant consists of an arc of 60°,
with each half degree numbered as a, whole degree, (a),
Fig. 31, combined with mirrors so arranged that angles can
be measured to 120°.

Boat Boaf

Boaf

Boat

(9)

%^.

^>C-.viC<-.-

Fig. 31.

THE SEXTANT. 147

Theory. — The principle upon which the sextant is con-
structed is that if a ray of light is reflected successively be-
tween two plane mirrors, the angle between the first and
last direction of the ray is twice the angle of the mirrors.

In (b), Fig. 31, the angles of incidence and reflection
are equal,

i = r and i' :^r', and

E = (i-|-r) _(i' + r')=2(r-r')

C = (90° — i') _ (90° — r) = (r — r')

and therefore E ^ 2 C

but C = angle CIC, by geometry, since the

mirrors are parallel for a zero reading.

TTse of the Sextant. — To measure an angle between two
objects with a sextant, bring its plane into the plane of
the two objects ; sight at the fainter object with the tele-
scope and bring the two images into coincidence. The
reading is the angle sought. The angle will not be the true
horizontal angle between the objects unless the objects are
in the same level with the observer. Since the true vertex
of the measured angle shifts for different angles, the sex-
tant should not be used for measuring small angles be-
tween objects near at hand.

Adjustments, Index Glass. — To make the index glass, 1,
perpendicular to the plane of the limb, bring the vernier to
about the middle of the arc and examine the arc and its
image in the index glass. If the glass is perpendicular to
the plane of the limb, the image of the reflected and direct
portions will form a continuous curve. Adjust the glass by
means of the screws at the base.

Horizon Glass. — To make the horizon glass, H, parallel
to the index glass, I, for a zero reading. With the vernier
set to read zero, sight at a star and note if the two images
are in exact coincidence. If not, adjust the horizon glass
until they are. If the horizon glass cannot be adjusted,
bring the images into coincidence by moving the arm and
must be applied with its proper sign to all the angles
measured.

Line of Collimation. — To make the line of collimation
parallel to the limb. Place the sextant on a plane surface

148 TOPOGRAPHIC SURVEYING.

and sight at a point about 20 feet away. Place two objects
of equal height on the extreme ends of the limb, and note
whether both lines of sight are parallel. If not, adjust the
telescope by means of the screws in the ring that carries it.

PROBLEMS IN TOPOGRAPHIC SURVEYING.
PROBLEM El. DETERMINATION OP STADIA CON-
STANTS OF TRANSIT WITH FIXED STADIA-HAIRS.

(a) Equipment. — Complete transit, stadia rod, steel tape,
set chaining pins, foot rule.

(b) Prohlem. — Determine the stadia constants c, f and Ic
for an assigned transit.

(c) Methods. — (1) Set up the transit and set ten chaining
pins in line about 100 feet apart on level ground. (2)
Plumb the stadia rod by the side of the first pin. (3) Set the
lower hair on an even foot or half foot mark keeping the
Record the intercept. (5) Read the intercept on the rod at
the remaining pins. (6) Measure the distance from the
center of the transit to each pin with the steel tape. (7)
Focus the objective on a distant object, measure /' (the dis-
tance from the plane of the cross-hairs to the center of the
objective), and c (the distance from the center of the ob-
jective to the center of the instrument). (8) Calculate the
value of the stadia ratio, /r, for each distance by substitut-
ing in the fundamental stadia formula. (9) Take the arith-
metical mean of the ten determinations as the true value.
(10) Compute the probable error of a single observation
and of the mean of all the observations. The interval factor
should be determined by the instrument man under the con-
ditions of actual work. The determination should be
checked at frequent intervals during the progress of the
field work. Follow the prescribed form.

(a) Equipment. — (No instrumental equipment required.)

(b) ProTjlem. — Compute a stadia reduction table giving
the horizontal distances from a point in front of the objec-
tive equal to the principal focal distance for the stadia in-
tervals from 0.01 feet to 10 feet, for the transit used in
Problem El.

PROBLEMS.

149

DlTERMIHATIO )

Ho. S
Ft.
IS/

• 2-70

• 3-SS
: 4-)S

■' S-61

' 6-Sll.

! 7-90

1 ill

ieM-71

D

Ft.
m-4I
Z6S-4I>
iS5-3Z

4gZ-S0

sse-io
e4i-ss

7S6-93

■.7m t

' n-'

m

mz4

Z67-S3
iS4-15
399-7Z
4gMi
SS5-J3
e4Z-4I
71S-76
m-Z3

m}-S4

= 0-6

= 0-e

0-47 f't-

0-70

1-17

OP

k

Ft.
39-lPZ
Sg-96
91-0Z
93-71
S9-J/
99-ZO
91S4
9S-47
9S-9!
99Ze

i

S9-1/4

ft-

0-0 I
O-OS

0-n

0-3Z
0-07

o-je
o-z:!

M3
0-J3
0-ZZ

0(!M4
(/■0064
11.0144
0-J0Z4
M049

o-ozse

0-04C0
0-IS45
M163
II-0484

0-4443
0-lBff

H-OSfr

Cot|sTAHTS - Fixed Iairs-

Obseri'ers, J-Poe ■ F ^-Rae-

DecJ4, '14-CZHmn ■) Cau/ iS C/ni/dy-

Used Buffs Berffs r 7r3nsifj Lacker J2,
and Chaining Loc. -.er f{^3S'

Ssf JO chaining /jfns m fins alra^f/ff^f/-
apart on leveJ t around'

Wjfh felescopff af 'fr,5i7sjf J7ear/y
level and defff. •/Pined Intercepf
"s"3Teach pm 4 'Seff/jj^ Joiver
Jiair an a fotffar ha'/f fafft mark

Measured d/sfanci from cenfer af
frans/f fo eacJ ' p/n tvifb sfeeS
tape fa nearas/- d-07 ft-

ohject defermin 'd c apd jf By
measi/r//?^ d/sA 'nee frojn center
of objecfj'ye fc center of the
horJzonfaJ sx/i - -and tfiep/ane
of the cross-wires respectively-

i/etermined tl7e different vaiues

of ic by sui>sfj
formuJa D=icS

^uij/?ffij7 file
■tctf-

Station
Imt. Obj.
A
F
3

A
C

B
D

C
E

D
F

E
A

frroi •
Allows

AXIMITH TiAVERSE WITH

hmv^

I6°B'
227'I6'

47°I6'
O'Oi'

ISOW
6'I4'

ISe°J4'

ze9°4e'
no'oi'

= o'or

bte erro.

M33-
6e3rin9

/l-4-'00%
S-aVM-

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hyAzimuti? reversals-

150

TOPOGRAPHIC SURVEYING.

(c) Methods. — (1) Prepare form for calculation. (3)
Compute the horizontal distances by substituting the dif-
ferent values of s in the stadia formula. Compute D' for
values of s varying from 0.01 foot to 0.1 foot varying by
0.01 foot ; from 0.1 foot to 1 foot varying by 0.1 foot ; and
from 1 foot to 10 feet varying by 1 foot.

(c+F)= 1-20 Feet-

k=ll5-,75

D=kS + (c+F) = D'+Cc+F) 1

Distance
D'=kS

Distance
D'=k.5

»3

[)ist^nce
D'=k.5

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(To use the table, take the sum of the values of D' cor-
responding to the units, tenths and hundredths of s as given
in the table. To the value of D' thus obtained add c plus /.)

PROBLEM E3.

AZIMUTH TRAVERSE WITH TRANSIT

(a) Equipment. — Complete transit, stadia rod, steel
pocket tape.

(b) Problem. — Make a traverse of the perimeter of an
assigned field with a transit and stadia.

(c) Metlwds. — (1) Set the transit over one corner of the
field and set the A vernier to read the back azimuth of the
preceding course. (2) Sight at a stadia rod held edgewise on
the last station to the left with the telescope normal, and
clamp the lower motion. (3) Read the intercept on the rod
to the nearest 0.01 foot. (4) Sight at the target set at height
of first station and read the vertical angle to the nearest
minute. (The observer should measure the height of the
horizontal axis above the station with the steel pocket tape,
or one tripod leg may be graduated and the instrument
height determined by swinging the plumb bob out against

PROBLEMS. 161

tHe leg.) (5) Unclamp the upper motion, sight at the next
station to the right and clamp the upper motion. (6) Read
the A vernier, (this will be the azimuth of the course) . (7)
Read the intercept on the rod. (8) Measure the vertical
angle by sighting at the target set at the height of the hori-
zontal axis as before. (9) Set the transit over the next
station to the right and determine the intercepts and ver-
tical angles as at the first station. (10) Determine the
stadia intercepts and vertical angles at the remaining sta-
tions, passing around the field to the right. (11) Reduce
the intercepts to horizontal distances before recording.
(12) Compute the vertical differences in elevation using
mean distances and vertical angles. (13) Compute latitudes
and departures to the nearest foot using a traverse diagram
or traverse table. FoUow^ form B4. (14) Compute the per-
missible error of closure of the traverse by means of Baker's
formula (see Chapter IX "Errors of Surveying") ; using
" a " equals one minute times square root of number of
sides, and " 6 " equal 1 : 500. If consistent, distribute the
errors in proportion to the several latitudes and departures,
respectively. (15) Compute the area by means of latitudes
and departures, and reduce to acres.

PROBLEM E4. SURVEY OP FIELD WITH PLANE TABLE

(a) Equipment. — Plane table, stadia rod, 2 flag poles,
engineers' divided scale, drawing paper, 6H pencil.

(b) Problem. — ^Make a survey of an assigned field by

(c) Methods. — (1) Set the plane table up at some conven-
ient point in the field and select a point on the drawing
board that will allow the entire field to be plotted on the
paper. (2) Sight at one of the stations with the ruler cen-
tered on the point on the paper. (3) Draw a line along the
fiducial edge of the ruler towards the point. (4) Measure
the distance to the point with the stadia. (5) Lay ofE the
distance on the paper to the prescribed scale. (6) Locate
the remaining points in the same manner. (7) Complete
the map in pencil. The map should have a neat title, scale,
meridian, etc. (8) Trace the map on tracing linen. (9)
Compute the area by the perpendicular method, scaling the
dimensions from the map.

152 TOPOGRAPHIC SURVEYING.

PROBLEM E5. SURVEY OF A FIELD WITH PLANE
TABLE BY TRAVERSING.

(a) Equipment. — Plane table, stadia rod, 2 flag poles,
engineers' divided scale, drawing paper, 6H pencil.

(b) Prohlem. — Make a survey of an assigned field by tra-
versing with the plane table.

(c) Methods. — Follow the same general methods as those
given for traversing with the transit. Adjust the plane
table before beginning the problem. Complete the map and
compute the area as in Problem E4.

PROBLEM E6. SURVEY OF FIELB WITH PLANE TABLE
BY INTERSECTION.

(a) Equipment. — Plane table, 3 flag poles, engineers' di-
vided scale, drawing paper, 6H pencil.

(b) Prohlem. — Make a survey of an assigned field with
the plane table by intersection.

(c) Methods. — (1) Select and measure a base line having
both ends visible from all the stations in the field. (3) Set
the plane table over one end of the base line, sight at the
other end of the base line and at each one of the stations
of the field. (3) Se't the plane table over the other end of
the base line, orient the instrument by sighting at the
station first occupied and sight at all the stations in the
field. (4) Complete map and compute area as in E4.

PROBLEM E7. THREE POINT PROBLEM WITH PLANE
TABLE.

(a) Equipment. — Plane table, 2 flag poles, engineers' di-
vided scale, 6H pencil.

(b) Problem. — Having three points plotted on the map,
required to locate a fourth point on the map by solving
the " three point problem " with the plane table.

(e) Methods.— (1) Use Bessell's solution. (2) Check by
using the mechanical solution.

PROBLEM E8. ANGLES OP TRIANGLES WITH SEXTANT.

(a) Equipment. — Sextant, 3 flag poles.

(b) Problem. — Measure the angles of an assigned tri-
angle with the sextant.

(c) Methods.— (To determine index error, sight at a d.is-

PEOBLEMS.

153

<

Angle

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station

Sextant

Observed

Index

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Mean

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Error

Angle

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taut object and bring the direct and reflected images into
coincidence. The reading of the vernier will give the index
error, which, with proper sign, must be applied to all angles
measured.) (1) Set the flag poles behind the monuments
at two of the vertices of the triangle and stand on the
monument at the third. (2) Hold the plane of the sextant
horizontal, siglit at one flag pole directly with the tele-
scope and bring the image of the other flag pole into coin-
cidence by moving the arm. (3) Kead the vernier, and cor-
rect the angle for index error. (4) Repeat the measure-
ment with the sextant inverted. Take the mean of the two
readings, which should not differ more than 2', as the true
value of the angle. (5) Measure the other angles in the
same manner. The error of closure should not exceed 3'.
Eecord the data in the form.

PEOBLEM E9. DETERMINATION OE COEFFICIENTS
OF A TAPE.

(a) Equipment. — Steel tape, spring balance, 3 thermom-
eters, steel rule, 3 stout stakes, axe, 3 pieces sheet zinc 3 by
3 inches.
12

154 TOPOGRAPHIC SURVEYING.

(b) Problem. — Determine the coefficients of expansion,
stretch, and sag of an assigned tape. Make three deter-
minations of each, and take the arithmetical mean as the
true value.

(Standard Tapes. — In laying ofE a standard or measuring
a base line where a high degree of precision is required it
is important that all measurements be referred to the same
standard. The Bureau of Standards, Washington, D. C,
will compare a tape with the government standard for a
small fee. The tape tested is certified to be of a given
length for a given temperature and pull. Por example the
standard tape marked " U. S. W. & M. 215 " used in laying
off the 100-ft. standard in Problem A23, was certified to be
99.9967 feet long at a temperature of 62° P. and a pull of
12 pounds, when tested on a plane surface. The coefficient
of expansion of this tape was 0.0000061 per degree P. Tapes
for measuring base lines with great precision have recently

(c) Methods. — (1) Correction for Expansion. — Measure
the length of the tape on a plane surface at two different
temperatures but with a constant pull determined by a
spring balance. Then substitute the lengths, { and L, and
temperatures, t and T, in the formula

l — L = e{t — T)l

where e is the coefficient of expansion. Repeat the test
and obtain three values of the coefficient e. As large a
range of temperatures as possible should be secured. Take
the arithmetical mean of the three determinations as the
true value.

(2) Correction for Stretch. — Measure the length of the
tape on a plane surface with two different pulls but at a
constant temperature. Determine the pull with a spring
balance. Then substitute the lengths, I and L, and the pulls
/) and P, in the formula

l — L = s{p~P)l

where s is the coefficient of stretch. Repeat the test and
obtain three values of the coefficient s. The pulls should
range from 10 to 40 pounds. Take the arithmetical mean
of the three determinations as the true value.

(3) Correction for Sag. — Remove the handles from the
tape and determine its weight very carefully. Divide the
weight by the length to obtain the weight per foot, w.

PROBLEMS. 155

Drive two stout hubs a little less than 100 feet apart and
fasten a piece of sheet zinc with a line ruled at right angles
to the line on the top of each stake. With a pull of 10
pounds, as determined by the spring balance, measure the
distance between the stakes. Calculate the correction for
sag by substituting the lengths, I and L, pull p, and weight
per foot w, in the formula.

'--'i

(t-T

Repeat the measurements using a pull of 20 and 30 pounds,
respectively. Add the corrections for sag to each measure-
ment and compare the results. The temperature should re-
main constant during the tests. To remove the possibility
of an error due to temperature, observe the temperature at
the time of each observation and correct the observed
length for e pansion before substituting in the formula.

Eeport the methods, data, computations and results on a
suitable form.

PEOBLEM ElO. MEASUREMENT OF BASE LINE.

(a) Equipment. — Standard tape, transit or level, stakes
(number and size to be specified by instructor), axe, spring
balance, 2 thermometers, lath stakes, 8-d nails, steel rule,
pieces sheet zinc 2 by 2 inches.

(b) Problem. — Measure an assigned base line with a stan-
dard tape.

(c) Methods. — (1) Set the transit over one end of the base
line, sight at the other end and determine the difference
in elevation and grade. (2) Drive stout square stakes to
grade, by " shooting " them in with the instrument in true
line, a little less than a full tape length apart. The top
of the lowest stake should not be less than 6 inches above
the ground. (3) Fasten a piece of sheet zinc, with a fine
line ruled at right angles to the direction of the base line,
on the top of each stake. (4) Drive lath stakes in line
about 20 feet apart. (5) Drive an 8-d nail through each
lath stake at grade to support the tape. (6) Measure from
stake to stake, the men working as follows : No. 1 plumbs
up from the rear monument or holds the zero on the raark
on the rear stake ; No. 2 takes the spring balance and puts
a pull of 16 pounds on the tape ; No. 3 reads the tape and
measures the fraction of a tenth with a steel rule to 0.001

156 TOPOGRAPHIC SUEVEYING.

feet ; No. 4 records the reading of the tape and reads the
two thermometers placed at the quarter points of the tape.
(7) Obtain at least three determinations of the length of
the base line. (8) Correct each measurement of the base
for standard, expansion, sag, stretch, and slope (see prob-
lein on coefficients of a tape). The three measurements
should not differ more than 1 : 100,000. Report methods,
computations and results on a suitable form.

PROBLEM Ell. CALCULATION OF TRIANGULATION
SYSTEM.

(a) Equipment. — Seven-place table of logarithms.

(b) Problem. — Adjust and calculate an assigned triangu-
lation system and plot the skeleton.

(c) Methods. — Observe the following program: (1) pre-
pare forms for calculations and transcribe data; (2) adjust
the angles of the triangulation system (see chapter on er-
rors of surveying) ; (3) calculate the front and back azi-
muths of each line; (4) beginning with the base line com-
pute the sides, to the nearest 0.001 foot; (5) calculate the
latitudes and departures to the nearest 0.001 foot (6) cal-
culate the coordinates of the triangulation stations to the
nearest 0.001 foot. In computing the coordinates of the
stations take the mean of the values found by taking the
different routes from the base line as the true value. (7)
Plot the skeleton of the triangulation system to the pre-
scribed scale by means of the coordinates of the points.
Check by lengths of sides. Use a steel straight edge.

PROBLEM E13. SKETCHING TOPOGRAPHY.

(a) Equipment. — Small drawing board or plane table,
plat of assigned field, 4H pencil.

(b) Problem. — Sketch in the roads, walks, buildings and
five-foot contours on the plat of the assigned field by eye
having given the elevations of the ruling points.

(c) Methods. — (1) Transfer from the level notes to the
plat the elevations of the ruling points of the field. (3)
Locate the roads, buildings, etc., on the map as nearly as
possible in their relative positions (the topographers' esti-
mate of distance should be frequently checked by pacing).
(3) Estimate the slopes and locate the contour points be-
tween the points of known elevation. (4) Join these points
by smooth curved lines. (5) Finish the map in pencil, put-

PROBLEMS.

157

ting on a neat title, the scale of the map and a meridian.
(6) Compare the finished map with a contour map fur-
nished by the instructor, i

PEOBLEM E13, FILLING IN DETAILS WITH TRANSIT

(a) Equipment. — Complete transit, 2 stadia rods, pocket
tape.

(b) Problem. — Locate the topographic details of an as-
signed area with the transit and stadia.

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(c) Methods. — (1) Set transit up over assigned triangu-
lation or other point. (2) Orient instrument, i. e., set
plates to given azimuth and sight at given back sight. (3)
Measure height of axis above station hub with tape or by
graduations on tripod leg, and set target to correspond.
(4) Take shot on given back sight and reduce results as a
check before proceeding. (The program for each shot is:
(a) set middle hair roughly on target, then set one stadia

158

TOPOGRAPHIC SURVEYING.

hair on nearest foot-mark and read intercept ; (b) set
middle hair precisely on target and signal rodman " all
Take side shots to representative points, keeping in mind
the scale of the proposed map. Select points according to
a systematic plan, following along ridges, gullies, etc. Con-
tour points should be taken with reference to change of

slope. (6) Reduce and plot the notes, and interpolate the
contours, as in the accompanying diagram. (This topo-
raphy sheet should be carefully preserved for use in Prob-
lem E15.) (7) After completing the survey at the assigned
taking both fore and back sights. (8) Lose no opportunity
to take check sights at other triangulation stations, tra-
verse points, etc.

PROBLEMS.

159

PROBLEM E14. PILLING IN DETAILS WITH PLANE

(a) Equipment. — Complete plane table (preferably with
prismatic eyepiece), 2 stadia rods, engineers' divided scale,
drawing paper, 6H pencil, pocket tape.

(b) Problem. — Locate the topographic details of an as-
signed area with the plane table and stadia.

(c) Methods. — Follow the same methods as in Problem
E13 except that the notes are to be plotted on the drawing
paper in place of being recorded m the field book. Mark
the points by number and write the elevation of each point
imder the number in the form of a fraction. Locate the
contour points by interpolation on the map and connect the
points by smooth curves. Complete the map in pencil and
make a tracing if required.

PROBLEM E15. TOPOGRAPHIC SURVEY.

(a) Equipment. — Complete transit, 2 stadia rods, stakes,
hubs, spring balance, pocket tape, stadia slide rule, seven-
place logarithm table, (extra tripods, stadia reduction
table, stadia reduction diagrams, etc., as required).

(b) Problem. — Make a complete topographic survey of an
assigned area and make a topographic map.

(c) Methods. — (1) Make a reconnaissance and locate the
triangulation stations. Care should be used to select the
triangulation stations so that the sights will be clear and
the triangles well formed. A system composed of quad-
rilaterals or more complicated figures will give more con-
ditions and checks than a simple string of triangles. A
system composed of simple triangles is sufBcient for this
survey. (2) Mark the triangulation stations with gas pipe

160 TOPOGRAPHIC SURVEYING.

monuments about 4 feet long, the exact point being marked
by a hole drilled in a bolt screwed into a cap on the top of
the gas pipe. (3) Measure the base line and base of veri-
fication as described in Problem ElO. (4) Measure the
angles by repetition as described in Problem D13. (5) Cal-
culate the skeleton as described in Problem Ell. (6) Es-
tablish permanent bench marks and determine their eleva-
tions and the elevation of the stations of the triangulation
system by running duplicate levels with the engineers' level,
reading the rod to 001 foot. (7) Fill in the details with
or both, as described in Problems E13 and E14. (8) Com-
plete the map in pencil on manila paper, and after it has
been approved by the instructor trace it on tracing linen.
The title, meridian, scale, lettering and border should re-
ceive careful attention.

CHAPTER VII.
LAND SURVEYING.

Kinds of Surveys. — Surveys of land are of two kinds :
(a) original surveys ; (b) resurveys.

Original Surveys. — An original survey is made for the
purpose of establishing monuments, corners, lines, bound-
aries, dividing land, etc. The survey of a townsite and the
government survey of a section are examples of original
surveys.

Resurveys. — A resurvey is made for the purpose of iden-
tifying and locating corners, monuments, lines and bound-
aries that have been previously established. The resurvey
of a city block, or a survey to relocate a section corner are
examples of resurveys.

Functions of a Surveyor. — In an original survey it is
the function of the surveyor to make a perfect survey, es-
tablish permanent monuments and true markings, and
make a correct record of his work in the form of field
notes and a plat.

In a resurvey it is the function of the surveyor to find
where the monuments, courses, lines and boundaries orig-
inally were, and not where they ought to have been. Fail-
ing in this it is his business to reestablish them as nearly
as possible in the place they were originally placed. No
reestablished monument, no matter how carefully relocated,
will have the same weight as the original monument if the
latter can be found. In making resurveys the surveyor has
no ofHcial power to decide disputed points. He can act only
as an expert witness. If the interested parties do not agree
to accept his decision the question must be settled in the
courts.

Also see Problem F6, " Eesurvey of a City Block."

Responsibility of the Stirveyor for the Correctness of
His Survey. — An engineer in the discharge of his profes-
sional duties requiring an exercise of judgment can be held
liable only for failure to exercise reasonable care and skill,
or :^or negligence or fraud. A surveyor is liable not only

161

162 LAND SURVEYING.

for negligence or fraud but for want of skill. A surveyor
agrees to not only do his work carefully, honestly, dili-
gently, but skillfully as well. The precision required in
making any particular survey in order to satisfy the re-
quirement for skill will depend upon the conditions ; greater
accuracy being required for making a survey of an ex-
pensive city lot than for a survey of a farm. Surveying is
a trade and the precision required in any particular case
to show proper skill is a matter to be decided by the court
after evidence has been submitted.

Ownership of Surveyors' Notes. — Survey notes, data,
maps, plats and records obtained by a surveyor while in
the employ of a city, state, railroad or other corporation,
or of a consulting or independent engineer belong to the
employer. A city engineer or a county surveyor has no
ownership rights in the notes, data, maps, plats and records
which he prepares or obtains, or are prepared or obtained
by him or by his assistants, in the exercise of the duties of
his olBce as city engineer or county surve3'or. Survey notes,
data, maps, plats and records obtained by a consulting or
independent engineer in preparing a report or plans for a
client, belong to the consulting or independent engineer.
The client, whether it be an individual, city, state, or cor-
poration, is entitled only to the finished report or plans,
and is not entitled to the notes and data used in the prep-
aration of the report or the plans.

Bules for Besurveys. — The following rules may be
safely observed in making resurveys.

(1) The description of boundaries in a deed are to be
taken as most strongly against the grantor.

(2) A deed is to be construed so as to make it effectual
rather than void.

(3) The certain parts of a description are to prevail over
the uncertain.

(4) A conveyance by metes and bounds will convey all
the land included within.

(5) Monuments determine boundaries and transfer all
the land included.

(6) When a survey and a map disagree the survey pre-
vails.

(7) Marked lines and courses control courses and dis-
tances.

(8) The usual order of calls in a deed is; natural ob-
jects, artificial objects, coiirse, distance, quantity.

(9) A long established fence line is better evidence of

SYSTEM OF PUBLIC LAND SURVEYS. 163

actual boundaries than any survey made after the monu-
ments of th^ original survey have disappeared.

(10) A resurvey made after the monuments have disap-
peared is to determine where the monuments were and not
where they should have been.

(11) All distances measiired between known monuments
are to be pro rata or proportional distances.

If the above rules do not cover the ease in question spe-
cial court decisions on that particular point should be con-
sulted.

THE UNITED STATES EECTANGULAE SYSTEM OE
PUBLIC LAND SUEVEYS.

Historical. — The United States rectangular system of
subdividing lands was adopted by congress May 20, 1785.
The first public land surveys were made in the eastern part
of the present state of Ohio under the direction of Capt.
Thomas Hutchins,* Geographer of the United States, and
were known as the " Seven Eanges." The townships were
six miles square, and were laid out in ranges extending
northward from the Ohio river ; the townships were num-
bered from south to north, the ranges from east to west.
In these initial surveys only the exterior lines of the town-
ships were run, but mile corners were established on the
township lines, and sections one mile square were marked
on the plat and numbered from 1 to 36, commencing with
section 1 in the southeast corner and running from south
to north in each tier to 36 in the northwest section.

The act of congress approved May 18, 1796, provided for
the appointment of a surveyor general and changed the law
relating to the surveys of public lands. Under this law the
townships were subdivided into sections by running paral-
lel lines two miles apart each way and setting a corner at
the end of each mile. This law also provided that the sec-
tions be numbered beginning with section 1 in the north-
east corner of the township, thence west and east alter-
nately to 36 in the southeast corner. This is the method
of numbering still in use, shown in Figs. 33 and 34.

* The earliest published reference to the rectangular sys-
tem of land surveys is found in an appendix to " Bouquet's
March," published in Philadelphia, 1764. Hutchins was en-
gineer with this expedition to the forks of the Muskingum
river, and wrote the appendix. (See reprint by Eobt.
Clarke, Cincinnati.)

164

LAND SURVEYING.

The act of congress approvefl May 10, 1800, required that
townships be subdivided by running parallel lines through
the same from east to west and from south to north at a
distance of one mile from each other. Section corners and
half section corners on the lines riinning from east to west
were required to be set. The excess or deficiency was to be
thrown into the north and west tiers of sections in the
townships.

Initial
Point

Standard P/ji

First Standard Parallel North.

T4N-,
R IE.

T3N-,
RIE-

T-2N-,1

IN-.
IE.

T4N-.
R2E-

T3N.,
R2E-,

_ B|

r
T-?N-, ■
R£E-

T- 1 N-,
R.£E.

T-4N-,
R3E-

I T-
R-

T-3N-,
R-3E. ,

T-?N-,
R-3E-

4N-,
4E.

3N-.
4E-

At-

±

T- I N-,
R-3E-

i T-

1K-.
4-E.

Base Line
rig. 33.

The act of congress approved February 11, 1805, required
that interior section lines be run every mile ; that corners
be established every half mile on both township and sec-
tion lines ; that discrepancies be thrown on the north and
west sides of the township. This act of congress further
provided " that all corners marked in the original surveys
shall be established as the proper corners of sections, or
subdivisions of sections ; and that corners of half and
quarter sections not marked shall be placed as nearly as
possible ' equidistant ' from those two corners which stand
on the same line. The boundary lines actually run and
marked shall be established as the proper boundary lines
of the sections or subdivisions for which they were intended ;
and the length of such lines as returned by the surveyor
shall be held and considered as the true length thereof, and

SYSTEM OF PUBLIC LAND SUEVEYS. 165

the boundary lines which shall not have been actually run
and marked as aforesaid shall be ascertained by running'
straight lines from the established corners to the opposite
corresponding- corners." Under this law, which is still the
established rule of procedure, each reported distance be-
tween established monuments is an independnt unit of
measure.

The revised instructions issued in 1855 required that the
sections be subdivided as shown in Fig. 33. The full lines
representing " true " lines, are parallel to the east exterior
line of the township, and the dotted lines, representing
" random " lines, close on corners previously established.
The order of the survey of the interior section lines is in-
dicated by the small niimerals. Double corners on the
north and west township lines, which were common in the
earlier surveys, were thus avoided in the revised practice.

Laws Inconsistent. — It is obviously impossible to pre-
serve a true rectangular system on a spherical surface, ow-
ing to the convergency of meridians.* To harmonize the
methods of making surveys, the General Land Office has
issued instructions for the survey of public lands from
time to time.

DETAILS OF SURVEY.— The details of the survey are
taken up in the following order: (1) selection of initial
points; (3) establishment of the base line; (3) establish-
ment of the principal meridian; (4) running standard par-
allels; (5) running the guide meridians; (6) running the
township exteriors; (7) subdividing the township; (8)
meandering lakes, rivers, streams, etc. See Figs. 33 and 33.

Initial Points. — Initial points from which to start the
survey are established whenever necessary under special
instructions prescribed by the Commissioner of the General
Land Office.

Base Line. — The base line is extended east and west
from the initial point on a parallel of latitude. The proper
township, section and quarter comers are established and
meander corners at the intersection of the line with all
meanderable streams, lakes, or bayous. Two sets of chain-

* The angular convergency, a, of two meridians is m. sin L,
where m is the angular difEerence of longitude of meridians
and L is the mean latitude of the two positions. The linear
convergency, c, for a length, t, is t. sin a. Latitude 40°,
the difference between the north and south sides of a town-
ship is 0.60 chains.

166

LAND SURVEYING.

men are employed and the mean of the two measurements
is taken as the true value. When the transit is used, the
base line — which is a small circle parallel to the equator —
is run by making offsets from a tangent or secant line, the
direction of the line being frequently checked by an obser-
vation of Polaris.

t

6 5^

i^zzlL

Random K

1 ♦

1 . '

r'43-^

1 *!l *

1 ^

Random)^
SS-:^

Randon?!^

9 ^
4/ — >-

Random^
30-^

Random^

"A

Random
8 >-

" ^
-<—S5

k" ^

" h

" h

15 si,

28— >-

■' A
J7—>-

13

6 >-

1 " ^
•^49--^

" h

37—^
" A

" \

" \
24—^

15— >■

ff

z- — ^

-<--47

^ " h

" A

" A

55 1

" h

35 1
1

rr

36

Fig. 33.

Principal Meridian. — The principal meridian is extended
either north or south, or in both directions from the initial
point on a true meridian. The same precautions are ob-
served as in the measurement of the base line.

Standard Parallels. — Standard parallels, which are also
called correction lines, are extended east and west from the
principal meridian, at intervals of 24 miles north and south
of the base line. They are surveyed like the base line.

Guide Meridians. — Guide meridians are extended north
from the base line, and standard parallels, at intervals of
24 miles east and west from the principal meridian, in the

SYSTEM OF PUBLIC LAND SURVEYS.

167

manner prescribed for running the principal meridian.
When existing conditions require that guide meridians shall
be run south from the base or correction lines, they are
initiated at properly established closing corners on such
lines.

Township Exteriors. — The township exteriors in a tract
24 miles square, bounded by standard lines, are surveyed
successively through the block, beginning with the south-

TowTUhip yo. 5 yorth, Ran&e Nil. 9 Weat, of a Principat Meridian
Eatt

I'.l "l«

s '"tsef. 7

*!&,< . M__. i

iiBiMj, \jj , .iBo:ao]; tjA t iii|ni; t^ « ijsoiooij ij^ « iiaotooi. ijj t iiw^ii i

^

5-1*

67

^

3-fIO

'M St!p. St__
f^~63 ijH' ti

! Weat

--^k'-

I JVest

oo

West

West

g Sec It
M "ft •) '

W fc-

West

a;
West

00

ei

80|00
g SecllO

S 6^0

[ West

"mJoo
I

3 "*io"

Scd f 7__g

g e^

80)00

sojoo
Sec 25

P See

TFe^t

00

/A
I

iS West In-iT^
^ ' Kr«( Standard Parailei

Sees I £ce. £ I Sec. 4 | £ec. 5

«<0 3
West

« g
» ^3

To
fi-c

3 rdo~ s

Sec. 8 I Asc. r I

Th« abovo plot represents a tfteoreticeU toton~hip tUth perfect subd-'.visfcne,
eonliffwrus to the tiorHi side of a Stimd^ard .Parallel; ,fn atsumtC I
ieiS'jr.. ^ Lm jUiide IQOOOO' W. of Or. Aria£S0ai.J6 A.'

Fig. 34.

western township. The meridional boundaries are run first
from south to north on true meridians with permanent cor-
ners at lawful distances ; the latitudinal boundaries are run
from east to west on random or trial lines and corrected
back on true lines. Allowance for the convergency of

168

LAND SURVEYING.

Township Subdivisions. — A true meridian is established
at the southeast corner of the township and the east and
south boundaries of section 36 are retraced. Then begin-
ning' at the corner to sections 35 and 36 on the southern
boundary, a line is run north parallel to the township line,
corners are established at a distance of 40 and 80 chains ;
from the last named corner a random line is run eastward,
parallel to the south boundary line of section 36, to its
intersection with the east boundary of the township. A
temporary corner is set at a distance of 40 chains, and a
permanent corner is afterwards established midway be-

Tti'04

to-ooi

40-00

— o

o
o
o

1

1

■--t

1
1

1

o
o
o

o
o
o

o

4

lzo-10

ro-oo.

40-00

1

rcj

Zl-O0\ ZO-OO'. 40-00 s'

I

-^ — -

f ^Z\-OV,Z0-0Oi 40-00

f<^J

Fig. 35.

tween the two permanent corners. The other corners are
located in a similar manner, as shown in Pig. 33. The lines
closing on the north and west boundary lines of the town-
tablished. A theoretical township with perfect subdivisions
is shown in Fig. 34.

Meandering. — Navigable rivers and other streams hav-
ing a width of three chains and upwards are meandered on
both banks, at the ordinary high water line by taking the
general course and distances of their sinuosities. The

SYSTEM OF PUBLIC LAND SUEVEYS.

169

meanders of all lakes, navigable bayous, and deep ponds of
the area of twenty-five acres and upwards are surveyed as
directed for navigable streams. Meander corners are estab-
lished where meander lines cross base lines, township lines,
or section lines. ,

Subdivision of Sections. — In Kg. 35, (a) gives the sub-
division of an interior section, (b) of section.2 on the north
side, (c) of section 7 in the west tier, and (d) of section 6
in the northwest corner.

Pig. 36.

Description of Land. — Land is described in the rectan-
gular system by giving its location in a civil township ; for
example, in Kg. 36, the northeast quarter, containing
160 acres, would be described as: N E 14, Sec. 8, T 19 N,
R 9 E, 3 P. M. The ten acre lot indicated in the northwest
quarter would be described as: S E %, N W ^, N W %,
Sec 8, T 19 N, R 9 E, 3 P. M.

Corners. — The corner monuments may be as follows :
(a) stone with pits and earthen mound; (b) stone with
mound of stone ; (c) stone with bearing trees ; (e) post in
mound of earth; (f) post in mound of stone ; (g) post with,
bearing trees ; (h) simple mount of earth or stone ; (i) tree
without bearing trees ; (j) tree with bearing trees ; (k) rock
in place, etc. The trees on line are required to be blazed.
The size, markings and proper corners to be used in any
particular case and all other details are given in the

170 LAND SUEVEYING.

" Manual of Surveying Instructions for the Survey of Pub-
lic Lands of the United States," issued by the General Land
Office, Washington, D. C.

The last edition of the " Manual of Surveying Instruc-
tions for the Survey of Public Lands " was issued in 1902
and may be obtained from the Superintendent of Docu-
ments, Government Printing Office, Washington, D. C, price
75 cents per copy. A new edition of the Manual is prom-
ised for 1915. The circular on the " Restoration of Lost
and Obliterated Corners " mentioned in the next paragraph
gives instructions for malting resurveys, and may be ob-
tained free by addressing the Department of Interior, Gen-
eral Land Office, Washington, D. C.

Bestoration of Lost or Obliterated Corners.* — "An ob-
literated corner is one where no visible evidence remains
of the work of the original surveyor in establishing it. Its
location maj', however, have been preserved beyond all
question by acts of landowners, and by the memory of
those who knew and recollect the true position of the
original monument. In such cases it is not a lost corner.

" A lost corner is one whose position can not be deter-
mined beyond reasonable doubt, either from original marks
or reliable external evidence."

General Bales. — The following rules are derived from a
brief synopsis of congressional legislation relating to sur-
veys.

" (1) The boundaries of the public lands established and
returned by the duly appointed government surveyors, when
approved by the surveyor general and accepted by the gov-
ernment, are unchangeable.

" (2) The original township, section, and quarter-section
corners established by the government surveyors must
stand as the true corners which they were intended to rep-
resent, whether the corners be in place or not.

" (3) Quarter-quarter corners not established by the gov-
ernment surveyors shall be placed on the straight line
joining the section and quarter-section corners and mid-
way between them, except on the last half mile of section
lines closing on the north and west boundaries of the
townships, or on other lines between fractional sections.

" (4) All subdivisional lines of a section running between
corners established in the original survey of a township

* Circular on the " Restoration of Lost and Obliterated
Corners and Subdivision of sections," Department of In-
terior, General Land Office, Washington, D. C.

SYSTEM OF PUBLIC LAND SURVEYS. 171

must be straight lines, rtmning from the proper comer in
one section line to its corresponding corner in the opposite
section line.

" (5) That in a fractional section where no opposite cor-
responding corner has been or can be established, any re-
quired subdivision line of such section must be run from the
proper original corner in the boundary line due east and
west, or north and south, as the case may be, to the water
course, Indian reservation, or other boundary of such sec-
tion, with due parallelism to section lines."

" From the foregoing it will be plain that extinct cor-
ners of the government surveys must be restored to their
original locations, whenever it is possible to do so ; and
hence resort should always be first had to the marks of the
survey in the field. The locus of the missing corner should
be first identified on the ground by the aid of the mound,
pits, line trees, bearing trees, etc., described in the field
notes of the original survey.

" The identification of mounds, pits, buried memorials,
witness trees, or other permanent objects noted in the field
notes of survey, affords the best means of relocating the
missing corner in its original position. If this can not be
done, clear and convincing testimony of citizens as to the
place it originally occupied should be taken, if such can be
obtained. In any event, whether the locus of the corner be
fixed by the one means or the other, such locus should
always be tested and confirmed by measurements to known
corners. No definite rule can be laid down as to what shall
be sufficient evidence in such cases, and much must be left
to the skill, fidelity, and good judgment of the surveyor in
the performance of his work.

" Actions or decisions by county surveyors which may re-
sult in changes of boundaries of tracts of land and involve
questions of ownership in connection therewith, are sub-
ject to review by the local courts in proceedings instituted
in accordance with the local statutes governing such
matters."

The pamphlet also contains much additional informa-
tion of value.

liOcations of Principal Meridians. — Principal meridians
have been established as the needs of the surveys war-
ranted. There are twenty-four principal meridians in all,
the locations of which are given in the " Manual of In-
structions," mentioned above.

172

LAND SURVEYING.

Abridging Field Notes. — The government surveyors use
the method of abridging field notes shovpn in Fig. 38. Cor-
ners in the township boundary are referred to by letter;
interior section corners are referred to by giving the num-
bers of the sections meeting at the corner ; interior quarter
section corners are referred to by giving the number on the
section lines produced.

OfFeEdDcCbBaA

h -

M

m

6-

7-

I

\6

^19--^

— F-

I

-31-

16

'--5-

8--

-^-9

's

-^-16-

-f

1^

---II—-

-4--

4-

zi-

t

\

I

'-11-

I

-5--'

-10-

I
13

-"-15-

?-■

-"-1

II-

— li-

|3

-^-3'4-

13

■14-

12

-26-

I

-+-

^-12-
1/

^-15-

I

"2,5-^

;/

-36-

1/

1^

y

N n o P p Q cj R

Fig. 38.

r 5

SURVEYS BY METES AND BOUNDS.

That portion of the United States settled before the adop-
tion of the rectangular system was surveyed by the method
of metes and bounds. For the most part these surveys were
very irregular and often involved complex and conflicting
conditions. The entire eastern portion of the United States,
and the state of Kentucky, were surveyed in this manner,

PROBLEMS.

173

and further examples are found in tlie French, surveys in
the states of IMichigan, Indiana, Illinois, Missouri, Louisiana,
etc., and the Spanish surveys of Texas, California, etc. The
general principles underlying the questions of ownership,
priority of survey, the restoration of lost corners, etc., are
identical whatever the system of survey used,

PEOBLEMS IN LAND SURVEYING.
PROBLEM Fl. INVESTIGATION OF A LAND CORNER.

(a) Equipment. — Digging outfit, tape, etc., as required.

(b) ProMem. — Collect complete evidence relative to an as-
signed land corner, and after giving due w^eight to the same,
laake a decision as to the true corner,

(c) Methods. — (1) Make careful examination of the offi-
cial field notes and records pertaining to the land corner in
question and make extracts from the same for further ref-
erence. (3) Seek oral evidence from those acquainted with
the history of the corner. (3) Make a survey of fence lines
and other physical evidence, such as witness trees or their
stumps, etc., near the corner under investigation. (4) Make

? ^sAv /

INVESTISATIOH OF S-W- CORNER.,

On'gingl UnifeiJ States Field Notes, on fl/e
the S-W' Cor-, Sec-S, T-!9N.,R'9E; 3 P-M-_
Jocated on the Prairis remote from
other three corners of the sect ton-

On 0ct-tB,/g96, Col-S-T-Susey, when
mvest/gat/on, stated that &bout 1850.
ifvas then County Surveyor, was calfed
the time mentioned the section lines
fence' CohSusey says that hfs Fathe.
surveyor near the fence corner evi
the ar/0/ha/ U' 5- Survey comer- I^r-
spot and found the decayed jooi'nt
marked the true posit/on of the ^Po
or more previous to Campbells resurve^
the boulder which was set in place t
section come/} and that this monuir.
pisced iy 3 much iar^er stone when
iines'

This stone stood /S^orso shove the ievel
it was carefully towered by the Stree.
Cify Engineer of Urban a- Resurveys
that its present posit/on is fdenticai

Conclusion- in view of Coi- Buseys

other credible soi/rceSf and the enttn •
character, it is conduced that the
recognized is the true S'W- corner of

Ca. npbej

J-.Ooe. Survsyori >

SECTIOH 8,T-19K.,R-9E-,3D:P-M-

at Courfffousa &f Urbana, Hi-, describe
as "^Post in Mound" the corner being
the heavyxtimber which surrounds the
' for /nfo/:ai& tio/7 about the corner under

when he vwa^ a boy, i^r- Campbeii, who
on to re-est36. 'ish the SW- Cor, Sec-S- At
near the cojux, " were occupied by rail

fa pioneer setj Ver) pointed out to the

?s of a motrnd which he believed marked
'c/if the j9sarvi yor dug cere fully at the

<7 sassafr^^s sft jJce which unquestiorrably
•t in Plound **esti biished some TS years
'- CohBusey sfai es that he himself carried
/ the County Surv eyor to pepetuate the
'nt was not cfistur bed until it was re-
''he roads wj^re op\ ''ned up on the section

' Commission sr undei ' the direction of the
since thet stone was /owered^ indicate
with that previous -to the change •
'le statenoe m« with fh*. ' corroborat/on from
abscence o,P conflicting evidence of any
ihonument > tow and fof ft, any years so
Sections, 7'i\$H,,!i-\$£^:\$l P'M-

• ^ : , ^ A >

174

LAND SURVEYING.

careful examination of the site of the corner with the dig-
ging outfit ; the digging should be done cautiously so as to
avoid disturbance of existing stakes or other monuments.
(5) If more than one monument be found, make due record
of their character and positions, and make further inquiry-
respecting them. (6) If no monument of any sort be found
at first, continue the search diligently and do not give up
finding the true corner as long as there is a remote chance
of locating it. In any event, avoid wanton disturbance of
any object or evidence that may have a bearing on the
same. Keep a clear and concise record.

PEOBLEM E3. PERPETUATION OF A LAND CORNER.

(a) Equipment. — Digging outfit, a large boulder or other
permanent monument, cold chisel, hatchet, plumib bob,
string, stakes.

(b) Problem. — Replace a temporary land corner by a per-
manent monument.

(c) Methods. — (1) Uncover the identified temporary mon-
ument and carefully determine the true point with consist-

Ar

Survey of SeoH,T-2S-,R10W.

C^/nmB/iced ef the 5B. cor. oF .^ec-/4:
fcr the ccr. which //ujh f/^aftsr says
//aeSf vnqbesfionedf as fhs cor- for otf
mafije, Suis- d/sm.,S-4a'W., 77 Iks-
J sef up a fall flag on the cor- and
temporary stakes every JO chi ■
^ sec- cor. lasf: i
Intersected the W-lfee oFSec-14; 4Z
correctpoint, Il-t6'£;l04 Iks
bearing tree of- (/-\$• \$t/rveyj havl
piece of steel T rail ^S /hs- long
locust 16 ins- diam; 5-ZS'Mf
iarroakIS " " ,N-7S'E.

CHAINS

40-00
eO-24

4!K

60-1\$

C-Rowland . ^ear w 5-fom'/)gs,f/ajm3n.

FOR. J-R- Comings ahd H- Rowland-

Fi ^und apiece of strap railroad iron driven
knows to have Ireen kept in the sama
■30 years' Ftsrkecl:

d/st.

disf.

then ran W- or} random,var.Z°l5'e;setting
in fine-

Ran thence F- on corrected line

Found cedar stake J ft- belowsurface

Ho other evidence oF cor- to lie

top of the stake For^ sec- cor.^

Planted granite toulder ZOi^Kxi

cor., in true line hot ween qr-

maple, IZins- diam., S-IS^F.

ittrroak,l6 " it ll'34''F.

'ks' S- of the cor. Found rotten stake aF
Fj om stump oF wh- oakj 24 ins- diam.f
^g surveyor^ mark distinct on it' Seta
For cor. Marked:
, lie Iks- disf.
ISZ /« »»

(lO!30A-M)
at single sight will} transitfFrom con to for. l^rZ'^ST.
oF road crossing and Zz Iks- 5 of line •
. -ound' Put a piece oF T rail Z4" long on
SBIks-SoFS. rail oF tt-e-k-B. Ifo tree near-
ins, f with cross + mark For ^ quan^ec.
Poland sec- cor. and marked:
' SS iks-dist.
IIS n It .

PKOBLEMS. 175

cut exactness. (2) Keference out the point by driving' two
pairs of stakes with strings stretched so as intersect
squarely over the corner. (3) After carefully checking the
referencing, dig out the old monument to a depth suiKcient
to receive the boulder and permit its top to set several
inches beneath the natural surface if located in a road or
where disturbance is probable. (4) Cut a plain cross mark
on the top of the stone, and set it in place in the hole,
packing the earth about it, testing the position of the
mark by means of the reference stakes and strings and
plumb bob ; finally leave the boulder set firmly in the cor-
rect position. (5) Make reference measurements to suitable
permanent points such as marks on curbing, gas pipes, wit-
ness trees, etc., selected with respect to good intersections,
and make reliable record of the witness notes after check-
ing the same. (Other forms of permanent monuments are :
gas pipe ; fish plate ; section of T-rail ; farm tile or vitrified
pipe filled with cement mortar ;. post hole filled with mor-
tar ; special solid monument burned like farm tile ; special
casting similar to a gas main valve box, with hole in top
to receive flag pole ; etc.)

PROBLEM F3. REESTABLISHING A QUARTER-SECTION
CORNER.

(a) Equipment — Transit party outfit, digging tools, etc.

(b) Problem. — Reestablish a quarter-section corner that
has been obliterated or lost.

(c) Methods. — (1) Collect and record all the available
evidence which may assist in the discovery and identifica-
tion of the corner. Examine the field notes of the original
survey, the surveyors' plat book and the county atlas on file
at the court house, and make diligent inquiry for credible
and competent information, either written or oral as to the
location of the corner. (3) Make a careful search for the
monument. Trace all the lines of the original survey, pay-
ing particular attention to bearing and sight trees. Dig in
all the places indicated by the different lines and give up
the search only after you have exhausted every possible
clue. (3) If the corner cannot be found, reestablish it, giv-
ing due weight to all the evidence. The surveyor should
remember that the corner should be reestablished where it
originally was and not where it ought to be. After having
located a stake at the supposed location of the original
monument, reference it out and renew the search. (4)

176 LAND SURVEYING.

After the monument has been relocated, mark it in a per-
manent manner as indicated in Problem F3, by a stone
with a cross cut in its top or with a gas pipe well driven
into the ground. Reference it out to at least two perma-
nent objects selected with a view to securing a first class
intersection. Make a careful record and preserve con-
sistent accuracy in the work.

PROBLEM r4. REESTABLISHING A SECTION CORNER.

(a) Equipment. — Transit party outfit, digging tools, etc.

(b) Problem. — Reestablish an obliterated or lost section
corner.

(c) Methods. — Follow the various methods described in
Problem P3, giving special attention to the search for the
original corner ; upon failing to find trace of it, run out
lines with reference to the section, quarter, and quarter-
quarter corners in the four directions, with linear measure-
ments from the same and finally reach the most consistent
decision with reference to such survey lines, ownership
lines, fences, hedges, road centers, etc. (A fruitful cause
of disturbance of section and other corners is careless use
ciently below the surface of the road.)

PROBLEM F5. EESURVEY OF A SECTION.

(a) Equipment. — Transit party outfit, digging tools, etc.

(b) Problem. — Make a resurvey of an assigned section.

(c) Methods. — (1) Make extracts from the field notes of
the original survey and of all resurveys on file at the court
house, and other notes that may be of value. Make dili-
gent inquiry among the property owners for evidence as to
the location of corners. (2) Retrace the lines, recording
the location of old fences, timber markings and other evi-
dences as to prior recognition of lines and corners. Use
consistent accuracy. Record the original notes as given in
the forms. Record the field notes in narrative style using
the designation of corners as given in the resurvey plat in
the form. Make a plat of the section in the manner pre-
scribed by state law for a resurvey.

PROBLEMS.

177

/^

iHVESTIGATIO^ OF lAND CORHERS
■COLLECTION OF EVIDENCE

Extracts from Surveyors Plat Book
Nov-Sf IS97, Found in the County Recorder's
offi'ce fff UrBsna, J//., the '^Surveyors Plat Sook"
containing plats offownsfiips showing exist-
ing monaments and st/bdivisfons oF sections
made by the County Surveyor, with cerf/F/'-^ -
cafes oF various resurveyS' /fade ff?eFoffow~
/ng exfrscfs refating to Sec-8 , T'J9fii^-9F;
^HD- P'M' :-

(From P-/S6)
"l>sc-S,Jg7\$, Surveyed at the reijuesf of
F-Adams tlie east fine oFSec-B- Beginning
atasf^ne prev/ousfy planted stJff cor' a/'
said sectioHf and running thence S--to S'B'
Cor' oFsame, wf)ere f Found a stone previous-
ly set by Jofin Tfirasher and lewis Sommers,
divided tfie distance pro rata ^ndsef Cor*
sf/ffCar-oFS-f^: oFsame.'*

• (Signed) T/?os-S-Xyfe

Co' Surveyor.
(From p' IS?)
"Apr- if, IS84' Surveyed by reguesF oP
5-T-Susay the W- fines oF Sees- 8 and S •■
Seginnfng survey at S'W- Cffr- Sec- S wfyere

Surveyor, J-Poe-
OF 5EC-8,T.19N.,F-
Apr.?5,IS99-
oF Resurveys oF Ch

a sfonq ispfanfedand
running thence ff- to
fiW-Cor-5ec-B, Found
an excess oF40 Ilis.,
corrected back, came
on fo^a stone planted
by lewis Sommers at
^Sec-Can on fine be-
fweei? Sees- Sand 6 •
iafso pfanteda stone
atSec.Cor'(S'e-7-S)
ihg witnesses to tfie
burr oak, fS "dfam-
bearing ft- \$0;^"^,
lOZiiks-jafsoa Wfi-
Oak, f4"di3m., bear-
ing ff-SSi^fSgfks.
fafso set a stone aF
them Cor. oFtbeSlV^
oFtfie5Wi,a'FSec.£*
(Signed) Tffss-B'Kyfe
Co-Surveyor'

smpaign County-

(portion oF Plat o/7
p-fSS, strewing exist-
ing monuments-)

9E.,5rd. P-M-

Stona

5fone
' 'Stcine

T"

stone
Stone

Stone

Stone
N

8

j^

r

Surveyor, J-Doe- ^

INVESTISATIOH OF LANB CORNERS

F Sec.8,T-I9N.,R-<

iE.,3rdPM-

COLLECTION OF EVIDENCE (Contmued)

Aor. 2S,

I8S9.

Extracts prom FieM Motes of Origin;

>"

Hov-^,1897, found in the CotinfyTressunsrs

(5^c.6)

' (Sec

£) ^

(Sec-4)

Office It Urbsns, III-, tlie Pl^ Book contain -
in} Plots sn(/ Abstracts of Field Hotss of

'l^^ ''"

00

Y'*^

\\

s

'-'■*■ H

Original United States Survey of Champaign

Coonty, and made the following extracts

v\

relating to Sec S, TISfi.,ll-SB, 3eo PM- .—
DESCRIPTIONS OF ORISIHAL CORNERS (P-30)

(Sec-7)

1 Se

Xs

/(Sec-9)

s %e4

\^^

Corners

WitnCK

Trees

Inches

Courses

Lints

)Mi5i«tipn

K

3d

Diameter

they Bear

Distant

%

^\^>=

5cc-Cor5-

^k

4,S,S,S

p5A
XBOak
[W-Oak
\w-Oak

24
14

S-S8°e-
N-64'W
M-U'E-

is

230

(5ec-ll),

798

m

mi ,

(Secie)

Y^ (S"-

^

S,l,7,8

ZO

H-16%

zn

DESCRIPTIONS OF "OBJE

CTS OK THE LINE5"(P-75)

WVt

Pastil.

Mound

DESIGNATION

DiSTAKCES

PE5CRIPTI0K

(fiWali
\S-IVol.

lit

24-

n-sz'B-

44

Chs- Lies

i,%ieji

'Ut

Z4-

S-IO'W-

42

ZS-00

iSecdr-

slang the channel of the

HhYatS

[am
eim

IZ

s-ei'w-

?l>

same 13 chs- then leaving

S

N-78'B-

30

it running B'ly.

T'X«S

[W-Oak
\w-ll3k

6

MSB'S-

Z3

SO- 19

Ash IZ'diam.

6

S-Wf-

20

e - S'i7

Z4-50

SmkSlks-rs-X-f'ly-

R-CnS

[Ash

\eim

n

s 7'e-

IB

3\$-00

fnfered timber bs-HS5.

8

H-IO-E-

13

£■ ' S'S

4- 00

gntemit fimber Is-lf-fS.

\^«B'S

Pastil

Mound

16-SO

S/aokeoiks-ii- S'ly-J

13

178

LAND SURVEYlIsra.

RESURVfiY OF 5E<;-t7,TltN-,R.l6W-,3D,

CHAINS

Se^an ■?/ 7' found sfske inpUee snd both
bearing frets sfsndfng- Planted stoji£
ZS"^\$"* 6'; marked-^ for oer*

Thence 1} on random, var-Z^O'f-f setting
temp, stakes every 10 etis-

Intersected sec t/he T£Iks.W-af£.

At S found rotten stake at correct point,
5-tS'W; eSlks- from stamp ofwti- oak,
hearfng free of l/S-Survey • Prcve
stake for con and put broken
.earthenware and glass around ff'
ftkd. wh-oak^/Z^dfam, tfS6^e.j
i4Ztks; diso wh- oak /S'd/am-j
if-S^'m, 63 Iks-

from S ran B- on random, setting temp-
stakes every 10 chs-

Intersected sec- line 12 Iks- It- of Z-

At Z Found earthen post in correct-
position snd bearing trees of
resurvey standing-

Thence W- on corrected line.

Set stake on true line-
on neKtpage")

to-n

39-9Z

0-98
(Cont

V_

/■£■ Wilson, Pear tf ^-W-Smifh, Flagman.

PM- FOR. THE Estate of Johh W. Smith.

JuIyiZ, '^Z- Cloudy with showers-

RESURVEY REFERENCE PLAT*

e

a

F

'' /•

b

h

c

9

14 J /3 3

<

Resurvey. Sec-H, Smith

Estate (comtihued)

CHAINS

)M6

(Line S-Z cont'd} At to set stake with
stones around it and marked :

pine, IZ"diam.,ti46'W., 79 Iks.
redo3k,Z4'di3m.,5:I\$^°W; 7ZIks.

?3-}4-

Set stake on true line-

from 10 ran S- on random, vanZ'tS^B-,
and set temp- stakes at Z0gnd40 chs-

Then went fo &• found post 3nd 6eant\£

trees of resurvey sfandinq.
Ran thence Wen random, var-Z'ZO'f.

Z0-OZ

Intersected random line from N- 6Jks.
S- of temp- stake -

4C-IS

Intersected random ^ line Slks-t^of
temp- stske.

gtl-04-

Intersected sec- line 10 Iks- S- ofS-
Cor- post dug out in road- Set Iron ptorr
besm for con, 5-79'y/., 76 Iks.,
from bearing tree of U-S-Sur^^y.
Thence B- on corrected line-

!9-\$3

At intersectton ofquarhsr ii>?ej set-

^_

post

1 1 1 1 1 II 1 1 1^

PROBLEMS. 179

PROBLEM F6. RESURVEY OP A CITY BLOCK.

(a) Equipment. — Transit, 100-foot steel tape, chaining'
pins, axe, hubs, stakes, 4 pieces one-inch gas pipe 2 feet
long, notes of previous surveys, etc.

(b) Problem. — Make a resurvey of an assigned city block.

(c) Methods. — (1) Procure full notes of all the surveys
and resurveys of the assigned block from the records at the
court house and from any other source available. (2) Make
a resurvey of the block, using the notes, and drive hubs for
temporary corners. (3) Compute the latitudes and depar-
tures of the courses, and if consistent balance the survey.
(4) If the corners of the block as located are consistent
with the existing property and street lines, drive gas pipes
as permanent corners. (5) Subdivide the block into lots as
shown in the notes. (6) Make a plat of the block on manila
paper to the prescribed scale, showing block and lot lines,
distances and angles obtained in making the survey, the
names of the owners of the property and the names of the
streets. Prepare a surveyors' certificate as provided by law.
Trace the map if required. (The accuracy attained should
be based on the valuation and other local conditions. Be-
fore beginning the survey use every possible care to find
the corners with reference to which the original survey was
made. When lots are sold by number, the excess or de-
ficiency should be divided pro rata. However, when lot lines
have been long acquiesced in, it is doubtful if the courts
will uphold the surveyor in interfering with the ancient
lines of ownership. It then becomes necessary either to
make a compromise survey that will be satisfactory to the
owners, or to make a survey that is strictly according to
the letter of the law, and submit the map and certificate to
the courts for settlement. The surveyor should remember
that he is simply an expert witness and that he had no final
judicial powers.)

PROBLEM F7. RESURVEY BY METES AND BOUNDS.

(a) Equipment. — Transit party outfit, digging tools, etc.

(b) Prohlem. — Make a resurvey of an assigned tract
whose original survey was made by metes and bounds.

(c) Methods. — (1) Collect full notes and data relating to
the monuments, magnetic bearings, magnetic variation,
date of survey, lengths of lines, etc. (2) Make a careful
investigation of the lines and corners on the ground and

180 LAND SURVEYING.

make notes of any evidence there found. (3) Locate and
identify witli certainty as many as possible of the original
monuments ; where double or contested corners exist, locate
each definitely for further reference ; if corners are gen-
erally lacking or doubtful, concentrate attention on at least
two which give most promise of definite relocation, and re-
establish these corners as carefully as possible. (4) Having
at least two corners, retrace by random line the perimeter
of the tract, according to the original description, begin-
ning at one and closing on the other corner ; set temporary
corner stakes at the several points ; note the linear and an-
gular error of closure of the random traverse on the last
monument. (5) Calculate the latitudes- and departures of
the random survey, and determine the angular and linear
relations between the random and the original survey ; also
fix the position of the several random stakes relative to
the supposed true positions of the respective corners. (6)
Set stakes in the true positions, as calculated, reference
them out, and renew the search for the original monu-
ments. (7) Finally reestablish each corner in the most
consistent position, put permanent corners in place, and
take witness notes for each, making comiplete notes of the

PEOBLEM F8. PAETTTION OF LAND.

(a) Equipment. — Transit party and digging outfits, etc.

(b) ProMetn. — Make a partition of an assigned tract of
land in accordance with instructions.

(c) Methods. — (1) Make the necessary resurveys of the
assigned tract, Identifying original monuments, and rees-
tablishing lost corners as required. (2) JViake a plat of the
partition. (3) Subdivide the land and set permanent cor-
ners ; carefully establish witnesses to the corners and se-
cure witness notes. (4) Prepare and file plat and descrip-
tion as required by law.

PEOBLEM FO. DESIGN AND SUEVEY OF A TOWN SITE

(a) Equipment. — Equipment for topographic survey for
both field and office.

(b) Problem. — Make a preliminary topographic survey
of the proposed town site (or addition), design the plat,
and make the surveys for blocks, lots, etc.

PROBLEMS.

181

Resurvey of "Mission Rid6e"

Consulted Cot/nty Records snd con Firmed
Following Meander Notes fvr cenfer
line oF highway ss descnbed in J-W-Msrt/n^s
deed fo J-D-Clsrk-

"H-eZ'B; 14-ch.; Il-43i% 8ch.; N-S'lV., 12 ch.;
l1-7Zi'£;ll!-2Sch-; S-!2'W; e-43ch."

Descriph'on referred fo sfones ef hegin--
ning and ending points-

Fai'nd First- stone projecting above road,
but could not locate last corner.

Began at First monument ani^ ran on
random according to meander notes,
with Z'n'E' as magnetic declination-
Drove temporary stake at each deFlect/on
point and made careful search For monu-
ments- Found no corners at infermediefz
points, but identiFied marked boulder"
as true corner at closing point SZ links
due west of last sf alee oF r-andom •

Made careFul calculation oF notes For
shiFting over From random to true
corners- (See plat opposite and cal-
culations on next pair oF pa^es-)

J-Doe, Surveyor- Mar- 10, 191S'

TransFerred corners sccording fo
calculations and renetved searct?
For original -monuments, keeping
close watch For decayed stakes,
but without success -

Set stone at each true corner.

Sta
('iandoi 1
A

1
A'
B'
C
D'
£''
F'

Hate.

CALClfLATIOfIS

Dist-

Ch.
14-00

8-00
l?-00

loss

e-43

Line)

H-ezio'e.

H-43'il'B

ns'm'w-
H-nii'i-
s-iik'if.

al Sun
irms ol

H-ei'u'e-

U-42'4t'B.
lt-S'4Zk
H-ll'4t'B-
S-II'IS'W-

The abi
0-ilch
AF anc
needle

ey in
Reau
13-SO

7-lS
11-8S
10-10

e-34

ve solu
atFF
AF'

Lat-
ch-
H-6-S7
tl-S-gO
H-11-9S
11-3-08
S-e-Z9

R^SURV^Y
Dep.

I1-Z7-40

S e-zs

lt-Zl-11

ifvey)
11-6-33
N-S-7S
11-11-77
11-2 IB

S- e-ii

H-Z7-3i
S- 6-?Z

H-il-ll

\on J3
Is due
d englt
corrections-

Ch
B-IMS
E- S-Sl
W- MS
B- 978
W-1-34

E-?7-lS
W- B-31

E-mO
E- S3S
Hi- 1-17
E- 9-0!
W- 1-14-

E-n-os

W- Ml

B-24-B4

\
. -<.

I ased

to dlFf^rence
HAF

OF

Tot- Lat-
ch- (N)

6-S7
li-ij
24-32
27-40
21-11

-/
/

e-es

lZ-41
24-18
27-ii
21-1!

/

:/

Data transcribed From pp- Copy OK-

Tot-Dep

Ch-(E)

12-il
17-87
16-82
26-60
2S-26

12-10
17-4S
16-28
ZS-ll
24.64

/

N

kt?i

f 24.64'—

Notss For Shifting from
Random to True Coi;pers

Lat-

Dep-

Lks-

Lks-

SB'

I1-6

W-26

CC'

H-4

W-42

DD'

S14

W-S4

£E'

5- 7

W-72

FF'

W-62

Dist-
Lks-

26-7
42-2
SS-'l
72-i
62-0

Bearing

H-77'^Om
H-84'3fk

s-isifm

S-84'27'U-

w.

sumption that the error oF closure oF

> <F both chain and needle • Distances

' were calculated, giving chain and

182 LAND SURVEYING.

(c) Methods. — (1) Make a careful resurvey of the entire
tract. Eeferenee the existing monuments and carefully re-
locate all missing corners. (3) After the monuments have
been carefully located, reroeasure the distances and angles
very carefully. Before beginning the chaining, a standard
should be established as described in Problem A23. (3)
Fill in the topographic details with the transit and stadia,
unless directed otherwise, using consistent accuracy. (4)
Make a complete topographic map of the tract. (5) Design
the townsite and sketch it in on the map. The questions of
surface drainage, sewerage, possible overflow, street grad-
ients, principal thoroughfares, diagonal streets, alleys, etc.,
should be carefully considered. The streets should be of
ample width, and be laid out with reference to ease of
should face desirable streets and the cross streets in the
residence district should not be too numerous. The prin-
cipal thoroughfare should pass through the business por-
tion and have minimum gradients. The system of sewer-
age and (^rainage should be worked out roughly before the
design is completed. Much expensive construction can be
avoided by using care in designing the town site. (6)
Make preliminary profiles of all the streets on Plate A
profile paper to the prescribed scale. (7) Carefully locate
the block and other important corners and mark them by
permanent monuments of stone, gas pipe, tiling, etc. (8)
Subdivide the blocks into lots and mark the lot corners by
means of gas pipes or hubs. (9) After the streets have been
located carefully, take levels on the same, make profiles,
and lay grade lines for all streets, sidewalks, and improve-
ments.

Use accuracy consistent with the value of the property
throughout the problem. Make a careful record of the
notes. Complete the maps and profiles.

CHAPTER VIII.

Classification. — For the purpose of class instruction,
railroad surveying will be discussed under the following
heads: (1) curve practice, (2) reconnaissance, (3) prelim-
inary survey, (4) location survey, (5) construction, (6)
maintenance.

Curve practice is designed to give the student familiarity
with the methods of running curves so that the location
survey may be made without needless delay. It consists of
a series of typical problems covering the usual range of
conditions found in such surveys.

The reconnaissance is a rapid prelim'nary examination
of a district or area for the purpose of selecting ruling
points to control the general routes of the preliminary sur-
vey lines. The distances are paced or scaled from a map;
elevations are determined by means of the barometer or
hand level.

The preJiminary survey is designed to obtain information
and to obtain it rapidly, as a guide in making the location
survey. A rapid deflection angle traverse is run, following
the general route of the proposed line, but keeping in clear
ground as far as may be to gain time ; levels are run, topog-
raphy including contours taken, the map made, and one or
more location lines projected on the map.

The location survey fixes the exact lines, including the
curves, preparatory to building the proposed railroad. Some
engineers prefer to run one or more trial location lines, but
it is best practice to locate the line as projected on a re-
liable contour map.

Construction surveys are made for the purpose of fixing
the roadbed limits and other constructive details, and esti-
mating earthwork and other quantities.

Maintenance surveys and resurveys are made after the
line is built, for ballasting, yard construction or other pur-
pose.

183

Field Organization of Class.— In order to carry out the
foreg-oing steps, the following- field parties are required:
(a) transit party, (b) leveling party, (c) topography party,
(d) land-line party, (e) cross-sectioning party, (1) bridge
and masonry party, (g) resurvey party.

General Bequirements. — Each party should work with
snap and vigor and accomplish the best results practicable,
both as to quality and quantity. To this end each member
of the party should not only be careful, exact, and rapid in
the discharge of his own duties, but avoid interfering with
the work of others, such as obstructing the view of the
transitman. In order to give each student practice in all
the positions, the posts will be shifted daily, progressing to
the higher positions in the party. The student should not
underrate his practice in the subordinate positions, nor fail
to make proper use of his more responsible duties. The
usual decorum of field parties will be observed.

TRANSIT PARTY.— It is the duty of the transit party
to establish the traverse line upon which to base the levels
and topography. The student transit party will consist of
the following members : (1) chief of party, (2) transitman,
(3) head chainman, (4) rear chainman, (5) stakeman, (6)
axeman, (7) front flagman, (8) rear flagman. The duties
and equipment of the respective members are stated below.

Chief of Party. — (Party list, map of line, 50-foot metallic
tape, railroad curve text book.) The chief of party is re-
sponsible for the general progress and quality of the work.
It is his duty to direct the survey ; see that each man does
his work properly and with sufficient accuracy and de-
spatch ; check the transitman's work when necessary ; keep
the transit notes if the transitman is pushed ; and make
himself generally useful. He should be thoroughly ac-
quainted, before going to the field, with the situation and
with the data applicable to the work of the day. In requir-
ing subordinate members of the party to perform their
work properly, he should carefully preserve the dignity of
his own position. Should there be no chief, these duties
will be shared by the transitman and head chainman under
the former's directions.

The transitman runs the transit, keeps the notes, and in
the absence of the chief, directs the work of the party. He
should do careful and exact as well as rapid work, since the

TRANSIT PARTY. 186

progress and character of the siirvey are usually controlled
chiefly by the skill of the transitman.

In leveling up, keep the lower parallel plate about level.
Avoid undue tightness of foot screws. In setting the ver-
nier to zero, use a quick converging motion with the tangent
sit has lost motion, learn which way to get the slack on thcj
tangent screws. As a rule, use the lower motion by prefer-
ence. Habitually back sight to the rear with telescope re-
versed, then plunge the telescope on prolongation and read
the deflection right or left. If practicable, base the cal-
culated bearings on a true meridian ; otherwise, allow for
the magnetic declination at a station which seems to be free
from local attraction and thus obtain a reference meridian.
Check all deflection angles by needle reading, both as to
amount and direction. Lack of proper adjustment is no
excuse for error. Always prolong a tangent line by double
sightings. Also check deflection angles from time to time,
by double sightings. Check on back sight before finally
approving any precise point ; likewise never fail to con-
clude the observations at each transit station by checking
on the back sight. In such check it is usually best to sight
back precisely on the point and then note whether the ver-
nier has the proper reading. Assist the flagman in plumb-
ing the pole, and always sight as near the bottom of the
pole as possible. The transitman should admonish the
chainmen, etc., to keep clear of the line.

On preliminary surveys, usually let the rear chainman
line in the head chainman by eye, at least for short
stretches. Do not hesitate to offset or zig-zag more or less
along open ground to gain time. A rapid method for pass-
ing through heavy timber is to zig-zag on slight deflection
angles right and left, tabulate the lengths in stations and
deflections in minutes, and the products of the two in sep-
arate columns on the right hand page. The original line is
regained by making the algebraic sum of. the products zero,
and the original direction is resumed by turning ofE a de-
flection which balances the deflection angle columns.

On location, each stake should be lined in carefully by
transit. Small obstructions, such as trees, may be passed
by parallel lines, using offsets of one foot or so at two hubs
a few stations apart ; the line is resumed in like manner.
Where plate readings are used in rectangular or other off-
set methods, no sights shorter than 50 feet should be used.
The equilateral triangle one station or more on a side is
14

often used. Obstructions on curves may usually be passed
readily with the aid of tables of long chords and mid-or-
dinates.

Curve index-readings should be calculated as though the
entire curve were to be run in from the P. C. ; starting with
the index-reading of P. C. always equal to zero, check the
calculations by noting that the index of M. C. is 14 I' ^""^ °^
P. T. is 1/^ I. In using the notes, remember that with the
transit at any point whatever on the curve, the following
rules apply: (1) When pointing to any station, the ver-
station; and (2) when pointing on tangent at any station,
station. As a rule, the best program in curve location is:
Having P. I. located, (1) measure I and assume D ; (3) cal-
culate T and E; (3) establish P. T. by chaining ofE T on
front tangent ; (4) establish M. C. by laying ofE E on bisect-
ing line ; (5) locate P. C. by interpolating hub at calculated
station number on back tangent; (6) move transit to P. C.
and foresight on P. I. ; (7) calculate curve notes (if not al-
ready done) ; (8) check sight on P. T. and M. C. and if sat-
isfactory; (9) run in curve, checking for distance and angle
on M. C. and P. T., moving transit ahead if desirable or
necessary; (10) set up at P. T. and resume front tangent.
One minute is the limit of allowable error in any curve.
Mistakes in calculations or in measurements of angles will
be counted serious errors. On final location the curves will
be spiraled. After the line is located, reference out P. C,
P. T., and other important hub points by two intersecting
lines and take careful notes of the same (see method (g),
Fig. 5, Chapter II).

The transit notes should be reliable, complete, neat and
distinct. Each entry should have but one reasonable mean-
ing and that the correct one. Record station numbers from
the bottom upwards, usually with ten stations per page.
Repeat the last station at the bottom of the next page.
Allow two lines per station so as to provide for sketching at
200 feet to the inch. On the middle line of the right hand
page mark each station with a dot and number every fifth
station which should also be enclosed in a circle. The
transit notes should include sketches of prominent land
and street lines, stream crossings and other prominent
topographic details, with pluses shown in the sketch. The
notes should include date, weather, organization of party,
etc. An appropriate title page giving name of survey, date

TRANSIT PAETY.

187

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86

85

84

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of commencement and completion, etc., should be prepared.
The notes will be kept in the prescribed form. The field
notes are to be returned at the close of the day's work.
All estimated data should be noted as such.

Completeness and neatness of notes and records, facility
and accuracy in handling- the instrument, and promptness
in advancing the progress of the survey will count in the
estimate of the work of the transitman.

Head Chainman. — (Flag pole.) The progress of the
chaining depends chiefly on the activity of the head chain-
man. After setting a stake he should move off briskly (pre-
ferably at a trot) and be prepared for the " halt " signal as
he approaches the next station. When the full chain length
is pulled out, the head chainman turns, holding the flag pole
in one hand and the chain handle in the other, and sets the
pole in line by signal from the rear chainman or transit-
man. Much time can be saved in this process if the head
chainman habitually walks about on line and if he sights
back over the two stakes last set. If on curve location, he
should line himself in on the prolongation of the preceding
station chord, and then offset by. pacing or with flag pole
a distance in feet equal to 1% times the degree of the
curv& ; the calculation is made mentally and the pole can
usually be set within a few inches of the correct position
by the time a speedy transitman has the deflection angle
set off. Having the line established, the pole is shifted to
the correct distance, and the stake is driven plumb in the
hole made by the flag pole spike. If the survey is a rapid
stant the stake is started at the proper point, although in
a more careful preliminary the chainmen check the dis-
tance to the driven stake. On location surveys it is custo-
mary for the chainmen to wait until the stake is driven
and mark the exact distance on the top of the stake with
the axe blade, and the exact line of signal from the transit-
man. In this process the head chainman should keep in
mind the convenience of the transitman, and in case the
line is being run to a front flag, the chainman should be
careful to clear the liMe frequently to allow check sights
ahead. In breaking chain on steep slopes the full length
of chain should usually be pulled out ahead and the chain
thumbed at the breaking points so as to avoid blunders ; a
plumb bob or flag pole should be iised in the process. In
passing over fences it often saves time to drive a 10-d nail,
with " butterfly " attached, in the top plank to serve as a

TRANSIT PAETY. 189

check back sight from the next transit point. The chain-
men should carefully avoid obstructing the transitman's
view, to which end they should walk on the outside when
locating curves.

Bear Chainman. — (100-foot chain or tape, chaining pins
(if allowed) , figuring pad or note book.) As the rear chain-
man approaches the stake just set, he calls out " halt " and
holds the end of the chain approximately over the stake,
quickly lines in the flag pole in the hand of the head chain-
man (or the pole is lined in by the transitman), the precise
distance is given, and the chainmen move on briskly. As a
rule, pluses should be read by the rear chainman, the front
end being held at the point to be determined. Fractions
will usually be taken to the nearest 0.1 foot, although 0.01
foot may at times be properly noted. It is the duty of the
rear chainman to keep a record of pluses and topographic
details when the transitman is not at hand. This record
may be kept on a figuring pad and the memoranda handed
at the first opportunity to the transitman, who transfers
the data to his book and carefully preserves the slips for
future reference. It is usually better, however, to keep
the auxiliary notes in a memorandum book instead of on
the loose slips. The chainmen should carefully avoid dis-
turbing the transit legs.

The responsibility for correct numbering of the station
stakes rests chiefly on the rear chainman. It is his duty
to remember the number of the previous station so as to
catch blunders on the part of the stakeman. As he reaches
the stake just driven, he mentally verifies its number and
repeats it distinctly for the guidance of the stakeman in
marking the stake to be driven ; the stakeman responds by
calling the new number, and each repeats his number as
a check before final approval. The rear chainman then
charges his mind with the numbers and checks the newly
set stake on reaching it. In case of dovibt he returns to
the preceding stake and notes its numljer.

Stakeman. — (Sack of flat and hub stakes, marking
crayon, handaxe.) The stakeman with his supply of flat
and hub stakes in a sack, should keep up with the head
chainman and be standing, with stake and marking keel
in hand, ready to number the new station stake on hearing
the rear chainman call out the preceding station number ;
the numbering is repeated, as already explained, before the
.stake is driven. Chaining pins are not used, but their
equivalent in checking tallies may be had by numbering the

stakes ahead and tieing them up in sets of ten. By num-
bering stakes at slack moments the stakeman gains time
to assist the axeman in clearing the line, etc. However,
special care should be taken to avoid omissions and dupli-
cates. The stakeman should finish numbering the stake
and hand it to the axeman by the time the head chainman
has fixed the exact station point. The stakes should be
numbered in a bold and legible manner, the keel being
pressed into the wood for permanency. The number should
read from the top of the stake downward. Stakes on an
offsetted line should be so marked as 4'L or 3'R, beneath
the station number. When survey lines are lettered, the
serial letter should precede the station number. Guard
stakes for P. I., P. C, P. T., reference points (R. P.), etc.,
should be clearly marked. The stakeman should assist the
axeman in clearing the line and should drive stakes when
the axeman is delayed. He should carefully avoid obstruct-
ing the transitman's view. The stakeman is under the di-

Axeman. — (Axe, tacks, (and if so instructed) an extra
sack of stakes with marking keel.) It is the duty of the
axeman to drive stakes, remove underbrush from the line,
clear an ample space about the transit station, etc. He is
expressly warned, however, in student field practice, not to
hack or cut trees or damage other property in any way,
and in general, not to trespass on the rights of owners of
premises entered in the progress of the survey.

The flat station stakes are driven firmly crosswise to the
line with the numbered face to the rear. Hubs are driven
witnessed by a flat guard stake driven 10 inches or so to the
left, the marked face slanting towards the hub, as shown
in Fig. 9, Chapter II. The axeman receives the marked
stake from the stakeman and drives it plumb at the point
marked by the spike of the flag pole. On location or careful
preliminary surveys when the stakes are being lined in
by transit, the axeman should stand on one side when driv-
ing and keep a lookout for signals from the transitman.
In shifting the stake as signaled he should use combined
driving and drawing blows with the axe. When the precise
point comes much to one side of the top of the hub, an-
other hub should be driven alongside and the first one
driven out of sight before the tack is set. The axeman
should move ahead briskly and avoid delay to the chaining.
The stakeman should, when necessary, drive the stake with

LEVEL PAETY. 191

the spare handaxe. When the field force is scant, one
man may serve in both capacities. The axeman is under
the direct charge of the head chainman.

Front Flagman. — (Flag pole, small supply of hubs and
guard stakes in stake sack, handaxe, a few 10-d nails.) It
is the duty of the front flagman to establish hub points
ahead of the chaining party under the direction of the chief
and transitman. In selecting transit stations he should
keep in mind visibility and length of both fore sight and
back sight, and to this end, points should be taken on ridge
lines and where underbrush, etc., is least in the way. The
practice of planting the flag pole behind the hub may be
warranted occasionally, as for example, when the field
party is shorthanded, but never when the regular flagman
is not specially detailed for other duties. The front flag-
man should keep close watch on the transitman and should
habitually stand with the spike of the flag pole on the tack
head and plumb the pole by standing squarely behind it
and supporting it between the tips of the fingers of the two
hands. Should the front flagman be flagging for an inter-
polated point depending on a foresight which his pole would
conceal, he should clear the line for a check sight by lean-
ing the pole to one side. When crossing fences he should,
when convenient, establish check sights on the top plank
by driving a spike and attaching a " butterfly "

Bear Flagman. — (Flag pole, hatchet, slips of paper.)
The rear flagman gives back sight on the preceding transit
station. The details of his duties are much the same as
those of the front flagman. It is an excellent plan for him
to cut a straight sapling or limb and plant it exactly be-
hind the hub when signaled ahead. This picket pole is
made more visible by splitting the top and inserting a slip
of paper, to make a " butterfly." A series of such pickets
on a long tangent line often afEords a flne check on the
work when an elevated transit point is reached.

LEVEL PARTY.— It is the purpose- of the level party to
secure data concerning the elevations of the points along
the line so that an accurate proflle may be made and the
grade line established. The leveling party should be on the
alert to detect errors in the work of the transit party, such
as omitted or duplicated stations, etc. The party consists of
two members: (1) leveler, (2) rodman. In very brushy
country an axeman may be added, but this is usually un-
necessary if the line cleared by the transit party is fol-
lowed.

192

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Leveler. — (Level, adjusting pin, level note book.) The
leveler should follow the most approved methods described
under the head of differential and profile leveling' in Chap-
ter IV. The nearest 0.01 foot should be observed on turn-
ing points and bench mark rod readings and elevations and
on occasional inaportant profile points. The fore sight rod
readings on ground profile points are to be taken only to
the nearest 0.1 foot and the nearest 0.1 foot in the height of
instrument is to be used in calculating the elevation. (Be-
ginners sometimes calculate elevations to 0.01 foot when
the rod readings are taken only to the nearest 0.1 foot.)
The leveler should be rax^id with his level as well as with
figures. He should calculate elevations as fast as the rod
readings are taken and should systematically cheek up the
turning point and instrument heights as the work proceeds.
As results are verified the same should be indicated by check
marks. Each page of notes should be checked by summing
up turning point back and fore sight rod readings, and com-
paring -their difference with the difference between the first
and last elevations or instrument heights, as the case may
be, on the page. Follow the prescribed form. As far as

LEVEL PARTY. 193

possible, bench, marks should be cheeked by including them
in the circuit as turning points. Balance back and fore
sight distances on turning points. Permanent bench marks
should be established at least every 1500 feet, and located
in places at once convenient and free from disturbance
during construction. Later levels should check within
0.05 foot into the square root of the length of circuit in
miles. When a discrepancy is found, a line of check levels
must be run to fix responsibility for the error. In cross-
ing streams, secure high water elevations, with dates, es-
pecially of extraordinary floods, also low water level. In
crossing highways obtain elevations each side for some
distance with a view to avoid grade crossings. In going up
or down steep slopes, gain all the vertical distance possible
each setting, and follow a zig-zag course. The bottom of
deep gullies may be determined by hand level. Assist the
rodman in plumbing the rod, and on turning points and
benches have the rod gently swung in a vertical plane to
and from the instrument and take the minimum reading.
The self-reading rod is to be preferred. Many levelers use
the Philadelphia rod without target. If the target is used
on turning points, the leveler should check the rod read-
ing when practicable.

Completeness, correctness and neatness of notes and rec-
ords, and facility and accuracy in handling the level will
be given chief weight in fixing the merit of the leveler's
work. The level notes are to be returned at the end of the
day's work.

Biodman. — (Leveling rod, peg book, hatchet, turning
point pegs, spikes, keel.) The rodman holds the rod at
station stakes and at such plus points as may be required
to make a representative profile. It is his duty to identify
each station point and be on the lookout for duplicated or
omitted stations. To this end he should habitually pace in
each station, especially in grass or underbrush, and call out
or signal the station number to the leveler. Should a blun-
der in station numbering appear, he should positively con-
firm the fact by retracing several stations, and then carry
the corrected stationing ahead. The rod should be held
truly plumb, which is best done by standing squarely be-
hind the rod and supporting it with the tips of the fingers
of both hands. On turning points, the rod should be waved
gently in a vertical plane to and from the instrument. The
rodman should pay special attention to placing the target
right for long rods and examine it to note if it has slipped

before reading the rod. Errors of 1 foot, 0.1 foot, etc.,
should be carefully guarded against. Turning points should
be selected with special reference to their solidity, and care
should be taken not to disturb them. Station pegs and
hubs are often used for turning points ; when so used, the
precise fore sight to 0.01 foot should follow the usual ground
rod reading to the nearest 0.1 foot. The rodman should use
good judgment in selecting bench marks, locating them out
of reach of probable disturbance during construction and
describing them so as to be easily found. He should be ac-
tive and do his best to keep close up with the transit party.
The rodman should keep a peg book for recording turning
points and instrument heights, and check his computations
independently and compare results with the leveler.

TOPOGKAPHY PARTY.— It is the purpose of the
topography party to secure full data for mapping contours,
property lines, buildings, roads, streams, and other import-
ant topographic details. The width of territory to be em-
braced in the survey depends on local conditions ; in places
it may be as much as one-fourth or one-half mile from the
line, although it is usually better to run alternate lines when
the distance to be included becomes so great. The topog-
raphy party often consists of only two men, but a party
of four is much more efficient. Sometimes no regular topog-
raphy party is provided, but after running a few miles of
line ahead, the transit and level parties are formed into
several parties to bring the topography up to the end of the
preliminary line. For student practice the topography
party will consist of four members: (1) topographer, (2)
assistant topographer, (3) topography rodman, (4) tape-
man.

Topographer.- — (Topography board, topography sheet (or
several sheets), hard pencil, compasses, eraser, etc.) The
topography sheet should be prepared before going to the
field, showing the alinement and other data needed from
the transit notes, and elevations of all stations and pluses
from the level notes. Cross-section paper is to be preferred.
The center line may be plotted to one side of the center
line of the sheet, when the topography is to be taken far-
ther in one direction than the other. In order to secure
full details, the scale of the field plat may well be double
(or even more) that of the finished map. The topography
sheet should show local conditions, such as gravel banks,
rock ledges, etc., suitable for ballast or other constructive
use ; out-croppings of rock or other material which may

TOPOGKAPHY PAKTl. 195

affect the classification of the graduation; character of
substrata at sites of bridge or other masonry work ; springs,
wells, streams, etc., suitable for water supply ; approximate
flood levels and other data relating to waterways or surface
drainage ; location of streams, especially with reference to
desirable crossings, freedom from probable change of chan-
nel, etc. ; location of highways including elevations some
distance either way with special reference to avoiding
in view ; character and condition of crops and other farm
improvements, names of owners, etc., — in short, any and all
information that is at all likely to be of service in mapping
the route, projecting the location, during construction, etc.
In locating a group of buildings some distance from the
line, fix the principal one by tie lines, by intersection or
polar coordinates, and the others by measurement and
sketch from it. Locate buildings near the line by rectangu-
lar offsets, or by intersections of the principal outlines
with the survey line. Contours are located by means of
the hand level used by the assistant topographer. The con-
tour interval should be five feet ordinarily, but niay be in-
creased to ten or more feet on very steep slopes. The con-
tour data should be selected with special reference to
ridge and gully lines (see problem and plat on contour level-
ing. Chapter IV). Ordinarily hand level lines may be run
out at right angles ; angling lines along gulches and ridges
may be located by estimation, pocket compass or tie lines.
The plat is made by the topographer from data collected by
the other members of the party. A common fault with the
beginner in such work is the omission from the plat of im-
portant numerical data, such as station numbers of land-
line crossings, etc., owing to an undue attention to the
minute details of the drafting work. A good topography
record with contour notes on the left hand page and field
sketch showing all numerical data on the right, is shown
in the accompanying form.

Assistant Topographer. — (Hand level, pocket compass,
topography note book.) It is the duty of the assistant
topographer to collect data for the use of the topographer
in making the plat. He uses the hand level, notes station
numbers, distances, bearings, etc., and makes such record
of the same as may be required to fit local conditions. In
contouring, a special rod with adjustable base (see Fig. 19,
Chapter IV.), if available, may be used; otherwise, an or-
dinary flag pole with alternate feet red and white is em-

196

EAILEOAB SURVEYING.

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ployed. BeginniBg with the known profile elevation, as ex-
tracted from the leveler's record, even five-foot contours are
located, as a rule, nominally every 200 to 500 feet at right
angles to the line, except as ruling ridges or gullies may
suggest other directions. His record should be ample and
legible, and include data and information which may not
properly be placed on the plat. All estimated elevations,
distances or dimensions should be noted as such. The as-
sistant topographer works under the direction of the topog-
rapher, but is expected to take the initiative in the collec-
tion of data so as to permit his superior to devote proper
attention to the field plat.

Topography Bodman. — (Topography rod with adjust-
able base (see (f). Fig. 19, Chapter IV.) or flag pole,
hatchet.) It is the duty of the rodman to hold the topog-
raphy rod as directed by the assistant topographer. He
should be active and continually on the alert for informa-
tion or data which the record book or sheet should contain.
The rodman holds the zero end of the tape in measuring
the distances. He should acquire skill in pacing on rough
as well as smooth ground, and when sufficiently exact es-

OFFICE WOKK. 197

pecially on ground remote from the surveyed line, lie should
gain time by pacing in the distances to contour lines.

Tapeman. — (Metallic (or band) tape, set of chaining
pins, flag pole.) It is the duty of the tapeman to deter-
mine distances with the help of the rodman. He should
be vigilant in checking up tallies, reading fractions, level-
ing the tape, breaking chain, plumbing down ends, etc.,
and should never be the cause of needless delay in the
work. When required, he should measure angles, take tie
lines, etc., with the tape.

OFFICE WOBK.— The office work of each student in-
cludes : (1) reconnaissance map, profile and report; (2)
map showing preliminary lines with topography and pro-
jected location lines; (3) preliminary profile with grade
lines, approximate estimate of quantities, etc.; (4) final lo-
cation map (traced from preliminary map) ; (5) location
profile; (6) copies of field notes; (7) cross-section notes
and estimate of graduation quantities; (8) estimate of
cost of constrution ; (9) monthly estimates, progress pro-
file, haul, prismoidal and curvature corrections, vouchers,
etc., final estimate.

B>econnaissance Report. — The reconnaissance map show-
ing the area examined will be based upon such maps of the
route as may be available. It should show the several
ruling points and general routes selected for actual survey.
The profile should be based upon barometric or hand level
observations and distances scaled from the map or deter-
mined roughly by pacing or otherwise on the ground. The
report should refer to the map and profile and state the
general scheme, the several ruling considerations or condi-
tions, the details of the examination, a rough comparison
of the several alternative routes, and a final summary
and conclusion with definite recommendations. The report
should be made in accordance with best usage as to form,
composition, etc.

(Considering the limited point of view of the beginner,
the reconnaissance reports may not be required until the
actual surveys are well along. In such case, however, the
student is not to draw data from sources other than those
above outlined.)

Preliminary Hap. — The mapping should be the best
product of the student's skill as a draftsman, and should
conform closely to the department standards, which are
based upon best current usage of leading American rail-
roads. Unless otherwise instructed, the preliminary map

198

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OFFICE WOEK. 199

will be made on eggshell or paragon paper. There are
three ways to plot the skeleton of the preliminary survey :
(1) by laying ofE each successive deflection angle and dis-
tance from the preceding line; (2) by laying ofE the suc-
cessive calculated courses and distances from a precisely
drawn meridian or other reference line; and (3) by rect-
angular coordinates. The first method should not be used,
since cumulative errors are probable. The second is rapid
and free from serious objection ; if preferred, a modified
base line may be assumed and the calculated bearings
transferred to the same ; the angles may be laid ofE by
means of scale and table of natural trigonometric functions
from a precisely drawn base line and then transferred, as
required, by parallel ruler or triangle ; this method is used
most in practice. The third method is the most exact, and
will be used by the student unless the second is specified.
It involves the calculation of a plotting sheet, as shown in
the accompanying form. The axis is usually a meridian
line, but any line may be taken and the courses changed
to suit. In making the plotting table, the data, calculated
bearings, distances, etc., should be carefully checked through
to the last point in the skeleton before the plotting is be-
gun. Only one axis should be plotted, preferably the one
having greater totals, so as to give short perpendiculars.
Starting from the origin, 1000-foot points are pricked in
along the axis to the specified scale, and marked 0, 10, 20,
etc. ; the totals are interpolated on the axis and lettered ;
exact perpendiculars about the right length are erected ;
the second point is established by scaling the perpendicu-
lar and the line is checked back on the preceding point ; if
correct, the stations are pricked in and every fifth station
and deflection points are enclosed in a small circle and
neatly numbered ; the next course is so located and checked
back by length of hypothenuse, the stations fixed and num-
bered, and so on to the end of the line ; the courses should
be taken in their order and none passed without checking
satisfactorily. After the skeleton is completed, the topog-
raphic details are penciled in, and the map finished and
inked. The title, border, meridian (both true and mag-
netic), etc., should be first-class in quality and in keeping
with the rest of the map. Crude or careless lettering or
other details of the map will cause its rejection. The title
of the map, profile, etc., should be given in brief on the
outside of the sheet or roll at each end.

Preliminary Profile. — Use Plate A profile paper in mak-
ing the profiles. The level notes should first be carefully
verified and then one person should read off while another
plots the data. A hard pencil, 6H or 7H, sharpened to a
long needle point should be used. The stations are first
numbered along the bottom from left to right (or the re-
verse, as prescribed) ; leaving six inches or so at the left for
a title, and beginning at a prominent line with station 0,
every tenth station is so numbered. The notes are exam-
ined for lowest and highest elevation and a prominent line
is assumed as an even 50 or 100-foot value relative to the
datum. The horizontal scale is 400 feet and the vertical
scale 20 feet to the inch. Points should be plotted no
heavier than necessary, since the surface of profile paper
will not permit much erasing. The surface line should be
traced in close up to the plotted points, owing to the
danger of overlooking abrupt breaks such as streams,
ditches, etc. Pluses should be fixed by estimation. The
surface line when completed should be inked with a ruling
pen used freehand ; the weight of the line should be about
the average of the ruled lines on the profile paper. (A
special profiling" or contouring pen is much used for this
intersection, elevations and rates of grade in red ; water
levels, and data relative to same in blue ; surface line, sta-
tion numerals, etc., in black ; the alinement, important land-
lines, streams, etc., should be shown at the bottom of the
profile in black. The grade line should be laid nominally
with a view to balance the cut and fill quantities, but this
should be varied to suit local conditions, such as drainage,
the elimination of grade crossings, classification of ma-
terials, etc. The maximum gradients, the rate of compen-
sation for curvature, etc., will be made to suit the specified
conditions. The compensation for curvature will be al-
lowed for on the preliminary profile by dropijing the grade
intersection elevations and rates of grade will be given to
the nearest 0.01 foot.

Approximate Estimates. — Rapid estimates of earthwork
quantities may be made direct from the profile either
by reference to a table of level sections, or preferably by
means of an earthwork scale. Estimates made in this way
from the profile of a careful preliminary survey, often do
not vary more than five per cent from the final construction
quantities.

OFFICE WORK. 201

Iiocation Map. — The location map may be traced from
the preliminary map and should include the topography
and such details as usually appear in the iinal record map
of the located line. Contoiir lines may be traced in cad-
mium yellow to insure satisfactory blue printing.

Location Profile. — The location profile should be exe-
cuted according to the standard specimen, and should in-
clude estimates of earthwork as determined from the ac-
tual cross-section notes, and quantities of other construc-
tion materials. Curvature compensation will be shown on
the location profile by reduced maximum gradients. Verti-
cal curves will be calculated at a rate of change not to ex-
ceed 0.05 foot per station, except at summits where it may
be 0.10 foot or more per station. It should be prepared as
the final record profile. Approximate profiles of projected
lines, determined from the contour map, with rough esti-
mates of quantities will also be prepared, as specified.

Office Copies of Notes. — The complete level and transit
notes, and topography notes as assigned, must be copied
in the individual books by each student. These copies will
be in pencil (or ink if so specified) and will be executed in
a faithful and draftsmanlike manner according to the de-
partment standards of lettering, etc.

Estimates of Quantities. — The cross-section notes will
be copied and the quantities of excavation and embankment
calculated, as assigned. The cross-sectional areas will be
calculated arithmetically and checked, especially on rough
ground, by means of planimeter. The quantities will be
calculated by average end areas, by tables, and by diagrams,
so as to afford ample practice for the student in all the cur-
rent methods. The estimate will also include all the other
materials of construction.

Bstlmate of Cost. — Each student will make a detailed
summary of the quantities, fix prices, and estimate the
probable total cost of the work, or of the assigned section.
The prescribed form will be followed. The prices should
be based on local conditions as far as possible.

Construction Estimates. — Monthly estimates, estimates
of haul, borrow^ pit estimates, classification, prismoidal and
curvature corrections, progress profile, vouchers, force ac-
count, etc., and final estimate will be prepared by each
student in accordance with prescribed forms and standards.

Right of Way Records. — Each student will be assigned
a share of work in the preparation of right of way deeds
and record maps. The following forms (from the " Engi-

neering Rules and Instructions," Northern Pacific R. R.)
will be used as models in preparing right of way descrip-
tions.

(Through government subdivisions) : " A strip, piece or
parcel of land one hundred feet in width, situated in the
northwest quarter of the northwest quarter of section ten,
in township two north, range one west (S. 10, T. 2 N., R.
1 W.), Madison county, Montana, and having for its bound-
aries two lines that are parallel with and equidistant from
the center line of the railroad of the Railway Com-
pany, as the same is now located (and constructed). For a
more particular description, reference may be had to the
plat drawn upon and made a part of this deed."

(Lots in platted tracts) : "Lot seven (7), block six (6),
in Smith's addition to Helena, Lewis and Clark county,
Montana, according- to the recorded plat thereof."

CROSS-SECTIONING PARTY.— It is the duty of the
cross-sectioning party to set slope stakes for the proposed
roadbed and to secure data for the calculation of earth-
work quantities. The data should first be transcribed from
the location level notes and profile into the cross-section
book, including station numbers, surface and grade eleva-
tions, rates of grade, bench mark record, etc. In order to
avoid confusion in relation to directions right and left, the
station numbers should run up the page, and plenty of
space left for pluses in the notes, especially on rough
ground. As shown in the form, the left hand page should
be used for data and the other for the cross-section notes.
The organization and equipment of the cross-sectioning
party when using the engineers' level is: (1) recorder
(note book), (3) leveler (engineer's level), (3) rodman
(self-reading leveling rod, 50-foot tape), (4) axemen (axe,
sack of flat stakes, marking keel). The usual routine is:
(1) Determine height of instrument by back sight on iden-
tified bench or turning point. (When a bench mark is re-
mote and an original turning point can not be found, it may
suffice in an emergency to check on the ground at several
stations to the nearest 0.1 foot and use the mean height of
instrument. Such places .should be verified later.) (2)
Having the height of instrument, check the original eleva-
rod and checking off the elevation if it does not difl'er more
than 0.1 foot or so ; in case of a new plus, take a rod read-
ing and record the elevation. (3) Determine the "grade
rod " for the station by subtracting the height of Instru-

CEOSS-SECTIUJN IWU

203

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Notes
R

sm%i

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J6-0

Renrar-kft

(i-levei section //i cuf')
(Level secf-f9n In cvf)

{2 leveJ sect/on in Fill)
{Levd sect/on in fill)
li-and stringer, 8r.0J8-
(Toe of Dump)
Bridge its IS ]jtS*34
6,J4sp3/?3 \{B7+£0

S-end sMngeri Sr-IKIS
Ditch 2-'4'-4'T- 33'-
(3 level section In Fill)
(Srsde point- right)
(Srsde point leFt)
(3 level section In ct/f)
(level section in cut)
(4 level section in cut)

(S level section in cut)
Cijk,tl',li:I, Fillsll',lhl-J

Typical Cases

L^vel Sections.

-\{760)

Riqht. Center. Lefi^

5iac-HiJI5ectioi}. FndofFillatTrestle.
H.D.

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204

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OrossSection at Station iZ6^

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Cross-Section at Station J25

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CEOSS-SECTIONING 205

ment from the grade elevation ; then note that cut or fill at
any point of the cross-section is equal to surface rod minus
grade rod (counting rods as minus when downward from
the plane of the level dnd those upward as plus, this rule
gives results always plus for cut and minus for fill, which
agrees with the conception that cross-section notes are
rectangular coordinates of the sectional area referred to
the center of the finished roadbed as an origin) . (4) If the
ground is level transversely, that is, does not vary more
than 0.1 foot or so within the limits of the proposed grad-
ing, then the distance from the center out to each side
slope stake is half width of roadbed plus center cut or fill
times rate of side slope; (thus for 20-foot roadbed, side
slopes 1 to 1, and a cut of 18.6 feet, the distance out to slope
stake on a level section would be 28.6 feet, or with a slope
of 11.^ to 1, the distance out would be 10 plus 1% times 18.6,
or 37.9 feet. Calculations of this sort should be done men-
tally in an instant). (5) On three-level ground estimate
the rise or fall of the surface from the center to about
where the side slope stake should come, and add the same
to, or subtract it from the center cut or fill, as the case
may be ; compute the distance out to the point where the
side slope line would pierce the ground surface and test
the same with tape, rod and level by the foregoing rule for
cut or fill ; continue to construct points on the side slope
line until the common point is found. (6) The axeman
marks " S. S." (slope stake) on one side of the stake with
the cut or fill to the nearest 0.1 foot (as C 6.8 or F 10.2)
and the station number on the other side ; the stake is
driven slanting towards or away from the center line ac-
cording as it is cut or fill. (7) On five-level ground or, in
general, on ground involving any number of points or
angles in the section, the cut or fill is taken at each break.
(8) Should there appear to be danger of land slips, the
cross-sectioning should be carried well beyond the limits
of the slope stake points. (9) The cross-section notes are
recorded as in the accompanying form, expressing the co-
ordinates of each point in the form of a fraction, and dis-
tinguishing the slope stake points by enclosure in a circle.
(10) Having completed the cross-sectioning^ at the station,
the same program is followed at the next point, first check-
ing the elevation obtained in the original location levels ;
the grade rod should be determined as before by subtract-
ing the height of instrument from the grade elevation, and
then checked by applying to the preceding grade rod th?

rise or fall of grade from, the preceding point. (H)
Cross-sections should be taken as a general rule at every
station and at such intermediate points as will insure a
reliable measurement of the earthwork quantities. It is
not necessarily the lowest and highest points that are re-
quired, but those points which, when joined by straight
lines, will give the contents as nearly as possible equal to
the true volume ; if the " average end areas " method is to
be used in calculating the quantities, sections should be
taken every 50 feet when the difference of center height is
as much as 5 feet ; as a rule, slope stakes need not be set
at cross-seclions taken between stations. (12) "Grade
point" stakes (marked 0.0), should be set where the center
line and each edge of the roadbed pierce the ground ; and
also in side-hill sections in both cut and fill, where the road-
bed plane cuts the ground line ; if the width of road-
bed is different in cut and fill, the greater half-width is
commonly used in locating the side grade point ; in the
simplest case a contour line is perpendicular to the center
line and the three grade points are at the same cross-sec-
tion, forming two wedges ; in the more usual case the con-
tour line is diagonal, and the three grade points are not
in the same section, so that two pyramids are formed ;
if the station numbers of the two side grade points differ
by only a few feet, it is usual to simplify the record by
taking the notes as for a wedge at the station number of
are set in their true positions ; as a rule, a complete cross-
section is taken at each grade point. (13) In cross-section-
ing for the end of an embankment at a wooden trestle the
end slope is made the same as the side slope, and the end
and side planes are joined by conical quadrants ; the dis-
tance between " heads of dump " (H. D.) is usually 10 feet
(5 feet at each end) less than the total length of stringers;
a complete cross-section is taken at the " head of dump,"
and the "toe of dump" (T. D.) on each edge of the end
slope is located and recorded ; on level ground the volume
of the wedge-like solid so formed is found by dividing it
into a triangular prism and two right conical quadrants ;
on ground sloping transversely the end of dumip is made up
of a middle prismoid and two conical quadrants, each of
the latter being generated by a variable triangle revolved
plane at " head of dump."

The calculations in the foregoing method of cross-section-

207

ing may be simplified by preparing a table of distances out
for the standard roadbed widths and slopes, or by using a
special tape having the zero graduation at a distance from
the end equal to the half-width of roadbed, and the re-
maining graduations modified to suit the side slope ratio.
The calculations may be further simplified by using a, spe-
cial rod having an endless sliding tape graduation. The
student will be given practice with these labor saving
devices after he has first acquired familiarity with the
principles of cross-sectioning without these aids.

Cross-sectioning with rods alone is done in much the
same manner as that described above. Two rods are used.
The usual length of the rods is ten feet, and each is gradu-
ated to tenths and has a bubble vial in one or both ends.
The slope stake point is determined by leveling out from
the ground at the center stake with reference to the center
cut or fill, each rod being held alternately level and plumb.
Other points in the cross-section, as well as grade points,
etc., are determined in tEe same manner. The notes are
kept as in the other method. On very rough ground, the
rod method is usually the more rapid. Some engineers
cross-section on rough ground by taking the elevation of
each point and plotting the notes on cross-section paper,
then using the planimeter to determine the areas. Borrow
pits are often cross-sectioned by taking elevations at the
intersections of two series of parallel lines forming squares.

Laud-Line Party. — It is the duty of the right of way
party to secure data for the preparation of right of way
deeds. The party should consist of at least four: (1) re-
corder, (2) transitman, (3) head chainman, (4) rear chain-
man, (the chainmen also to serve as axemen and flagmen
as required). Their equipment is the usual one of a transit
party for such work. The party should secure ties with
all section and other laud lines whenever crossed. The
notes should show station numbers and angles of intersec-
tion and distance along land line to the nearest identified
land corner and also to important fences. As a rule, make
the intersection by running through from one corner to the
other. Where the line passes through a town, tie the cen-
ter line to the plats, block lines, monuments, etc. Secure
any records and make tracings of any plats, etc., at the
recorder's office, that may be of service in preparing deeds.

Bridge and Masonry Party. — The bridge and masonry
survey party will determine drainage areas for culverts and
other waterways, prospect for foundations, and stake out

trestles, masonry work, etc. The usual organization will
be four men : (1) recorder (in charge), (2) transitman or
leveler, (3) chainman, rodman, flagman, etc., (4) chainman,
axeman, flagman, etc., as the work assigned may demand.
Besurvey Party. — The resurvey party will be assigned
to such duties as the resurvey of yards, the collection of
data for crossings frogs, running centers on old track, in-
cluding spiraling, etc. It will usually be a, party of four.

PROBLEM Gl. ADJUSTMENTS OF LEVEL AND TRANSIT.

(a) Equipment. — Engineers' level and transit, adjusting
pin.

(b) Problem. — Test the essential adjustments of the as-
signed instruments and correct any discrepancies found.

(c) Methods. — This problem is designed to freshen the
student's knowledg'e of the adjustments of the instruments,
as well as to place the equipment in condition for accurate
direction of the instructor. The student should attempt to
be speedy as well as accurate in testing and making the

PROBLEM G2. USE OF FIELD EQUIPilENT.

(a) Equipment. — Complete equipment for railroad transit
and level party, as specified in foregoing pages.

(b) Prohlem. — Practice the detailed duties of each posi-
tion in the transit and level party.

(c) Methods. — This problem is designed as a "breaking
in " exercise preparatory to engaging in the regular field
work qf railroad location. With the manual in hand the
duties of each position will be studied and practiced in
turn.

For example, each student will go through the following
exercise with the transit as briskly as possible: (1) set
transit over tack in hub, (2) level up, (3) set plate to zero,
(4) reverse telescope and sight on back flag, (5) release
needle, (6) phmge telescope, (7) read and record needle on
back line prolonged, \8) sight at front flag pole, (9) read
and record deflection angle right or left, (10) read and
record needle on front line, (11) lift needle, (13) plunge
telescope and check on back flag, (13) calculate needle

PKOBLEMS.

209

angle and compare with plate reading, and if checked,
shoulder transit; now repeat entire process at the same
hub, more briskly than at first, if practicable, avoiding ref-
erence to preceding record until the full series of steps is
completed.

Problem 2. Calculation* of Curve Elemen+s.

DM'n'):sl337.6^ \^ (b) By Tails I'C.

.,TandE.

, tn'go.

tan 3efo8.'s= o-seoes
•exsec jd'o8.'s=0. /Sffjff

Wf7'= eo°.S833-t-
tl'n'= 4? 2833 +

{Results to 0.0/-fti)

Msthod.

I4.07SS
776.71

109. IS

D'lff.

776.77
209.17

Indicated WorK-

Calculations.

Leng th of Curve , i. ,
I — SO' 17'
*-■" ■*V7'

'"' = ^^. = (etszS^

eo'.2»33 _,^7Z^

f6) =

2T7\^c n)r4.0739

g-gy a*.

10^7
to 38

laoo

1799
loio

Z330

ea!S33 )*.SB333

Tangent Pi'stanee . T.
(at 7-= n tan-kZ

— '337.e sx a.s8»se

=<^ff.77)

lb) r=

776.71
cH.

Titecfie) = 33as.a
^Ko'/si = 33saj_
r, fea'n'j = 33s 7. /s )^se33(S)

2998.33 776.77

3Batz o,k.

2.9983
2899
2370

776.77

776.7/

O.06

Di-f^ due to approK.
basis of method Cb),

JS9
300

External Distance , £.
<a} E-RexSec-tl

= i33-i6S x o. isese

=(20915)

(b) E=

— £ii

1337. 6S
6363 t.O

1 33 re
eesa

803
40

S9S.9S
1iZS33

209.IS

; c.H.

£,f60Wj= B9S..f.

El (6a'lff) =S96S

E, (eo'l7'J= 89S.9S ) 4.2S33&

836 67 209.17
209.17 3928 „K

O.02 - Jl

30

Dl-ff. due -to 3o

epproxw basis of method fbj.

Let the student prepare a similar numbered program for
each of the other positions and practice the same systemat-
ically. This series of exercises may profitably occupy two
or more assignments, since the speed and quality of the
actual surveys to follow are certain to be much enhanced.

15

PEOBLEM G3. PEELIMINAEY FIELD CUEVE PEAC-
TICE.

(a) Equipment. — Transit party equipment, as prescribed
in instructions.

(b) Problem. — Eun out the assigned practice curves in
the field, with the prescribed organization and conditions.

(c) Methods. — The preliminary curve practice is designed
to give the student a practical knowledge of the principles
of railroad curves and the routine methods iised in location
surveys. The several positions in the field party will be
filled in succession, and each student is expected to respond
heartily to the spirit of the practice, whatever his assigned
duties. Each member of the party should engage in the
calculations as far as practicable. The report of the field
work should state the precision of linear and angular
checks. The field practice will be based in part on the
indoor curve problems.

PEOBLEM G4. CUEVE PEOBLEMS.

(a) Equipment. — Drafting instruments, paper, etc.

(b) Prohlcm. — Solve the assigned problems in railroad
curves and submit results in a neat and draftsmanlike
form.

(c) Methods. — (1) Draw a plain figure to the largest con-
venient scale. (2) State problem and present data in a
concise and systematic manner. (3) Show the separate
steps clearly ; first state formulas in general terms, then
substitute values and give results ; as a rule, .show actual
calculations adjacent to the indicated work ; habitually
verify results by an independent process ; use common sense
checks and contracted methods of calculation ; in general,
make full use of the opportunity to gain skill as a com-
puter. (As a rule, the nearest 0.1 foot only is required in
field measurements on curve location, but it is excellent
practice, especially for the beginner, to preserve the nearest
0.01 foot in the calculations.)

CHAPTER IX.
ERRORS OF SURVEYING.

Errors. — Errors of observations are of three kinds, viz.,
(1) mistakes; (2) systematic errors ; (3) accidental errors.
Systematic errors include all errors for whicli corrections
can be made, as erroneous length of standard, errors of
adjustment, refraction, etc. Accidental errors are those
which still remain after mistakes and systematic errors
have been eliminated from the results.

It has been found from experience that accidental errors
are not distributed at random but follow mathematical
laws. These laws are fundamental in the Theory of Least
Squares and are : ( 1 ) small errors are more frequent than
large ones; (2) positive and negative errors are equally
numerous ; ( 3 ) very large errors do not occur.

Arithmetical Mean. — The most probable value of a quan-
tity obtained by direct measurements is the arithmetical
mean of all the determinations where the observations are
of equal weight, or is the weighted mean where the obser-
vations are of unequal weight.

Precision of Observations. — In the adjustment of obser-
vations it is often necessary to combine results of different
degrees of precision or weight. It is also desirable to have
some means of comparing observations so that the com-
puter may know what degree of confidence to place in the
results. The quantity commonly used for comparing the
precision of observations is the probable error.

Probable Error. — The probable error is such a quantity
that it is an even wager that the number of errors greater
is the same as the number of errors less than the probable
error. It is also the limit within which the probaloility is
one-half that the truth will fall. For example, if 4.63 +
0.12 is the mean of a number of observations, the true value
is as likely to be between 4.51 and 4.75 as it is to be some
value greater or less.

Probable error is also useful in finding the relative
weights that should be given different sets of observations,
as it has been found that the weights of observations vary
inversely as the squares of their probable errors.

211

212 EREORS IN SUEX'EYING.

Formulas:

Let i\ = probable error of a single observation.

Bm=: probable error of the mean of all the observa-
tions.

H = the number of observations.

d = the diiference between any observation and the
mean of all the observations.

2 = symbol signifying sum of.
Then from the Theory of Least Squares

^. = 0.6745^^^ (1)

i?,„ = 0.6745X|^ (2)

l/ n
The probable error of the weighted or general mean is

(3)

' \(7i-i)::

,y (4)

where S p =: summation of the weights.

The probable error of a quantity with a weight p is equal
to E„ divided by the square root of p.

The probable error of Z, where Z = Si + St.,, and R„ r,.
and ;■, are the probable errors of Z, e^ and z.,, respectively,
is

iJ^2 = ri2 + r/ (5)

The probable error of Z, where Z z^ a. z '\%

Iii' = a'-r' (6)

The probable error of Z, where Z = z^. Z2 is

-Ri' = 2i'-'-2' + z.^-V (7)

This would be the probable error of the area of a rect-
angle where r^ and r.. are the probable errors of the sides ~i
and «2, respectively.

Example. — As an example of the application of these
formulas consider the two following series of measurements
of an angle given in Table I. The first set was taken with
a transit reading to 10 seconds, the second with a transit

PEOBABLE ERROR.
TABLE I

213

FIIIST TRANSIT.

SECOND TRANSIT.

No.

Angle.

d

6?

No.

Angle.

d

d^

o / //

/ //

i

34 55 35

2

4

1

34 56 15

39

1521

2

35

2

4

2

55 30

6

36

3

20

13

169

3

54 30

66

4356

4

05

28

784

4

55 15

21

441

5

56 15

42

1764

5

56 00

24

576

6

55 40

7

49

6

55 45

9

81

7

10

23

529

7

55 30

6

36

8

30

3

9

, 8

55 30

6

36

9

50

17

289

9

56 00

24

576

10

30

3

9

10

55 45

9

81

Mean 34° 55' 33"

Sd^ =

= 3610

Mean 34° 55' 36"

Sd2 = 7740

t4".3

E^=

E„ =

=«™V^.

)
0~ =

= <>-^^^^V9x'lO-^«"-3

The weights of these mean values vary inversely as the
squares of the probable errors, or in this ease the weights

are as — ^ to r-^ or as 13 to 5. The most probable value

4.0 D.o

of the angle measured with the two transits will be the
weighted mean.

Z= 34° 55' +

33X12" + 36X5"
17

= 34° 55' 33". 9
The probable error of this result from (5) since

Substituting r^'^i^-r^ we have

iJ, = ± 4. "3 VTI = ± 3".6.

214 ERRORS IN SURVEYING.

Eor other examples in the use of probable error see prob-
able error of measuring a base line, probable error of set-
ting a level target, probable error of setting a flag pole.

Angle Measurement. — The measurement of an angle re-
quires two pointings and two readings. If r^ and r., are the
probable errors of reading and pointing, respectively ; the
probable error of the measurement of an angle will from
(5) be

If i\ is the probable error of a single reading

If the value of an angle is determined by n separate meas-
urements the probable error due to reading will be

nV2

If the value of an angle is determined by measuring the
angle n times by repetition the probable error due to read-
ing will be

ni/2

It will thus be seen that the probable error due to reading
is very much reduced by measuring an angle by the method
of repetition. The errors of pointing, etc., however, make
it doubtful whether it is ever advantageous to make n ex-
ceed 5 or 6 with an engineers' transit.

Angle Adjustment. — When the three angles of a triangle
have been measured with equal care they should be adjusted
by applying one-third of the error as a correction to each
angle.

When the interior angles of a polygon having n sides
have been measured with equal care they should be adjusteJ
by applying oiic-iith of the error as a correction to each
angle.

When n — 1 angles and their sum angle at a point have
been measured with equal care they should be adjusted by
applying one-nth part of the error as a correction to each
angle.

In a quadrilateral the triie values of the angles fulfil the
following geometrical conditions : (1) the sum of the angles
of each triangle is equal to 180° plus the spherical excess

TESTS OF PEECISION. 215

(the spherical excess in seconds of arc is equal approxi-
mately to the area in square miles divided by 78) ; (2) the
computed length of any side when obtained from any other
side through two independent sets of triangles is the same
in both cases.

When the angles of a quadrilateral have been measured,
errors are certain to be present and the corrections that
satisfy one of these conditions will not satisfy the other.
The most probable values of the corrections to the angles
are then determined by the Theory of Least Squares.

TESTS OF PRECISION.

Practical Tests. — In careful surveying where blunders
are eliminated and the systematic and accidental errors are
small and under control, it is found that the magnitude of
the errors increases in close accord with the foregoing
rational basis, tliat is, as the square root of the number of
observations. The following practical tests of precision are
based on this truth.

Linear Errors. — Cumulative or systematic errors usually
increase directly as the length of the line chained, while
compensating or accidental errors vary about as the square
root of the length. While both kinds of errors afEect all
linear measurements, the former chiefly control the results
of crude and the latter of accurate chaining. It is thus
fairly consistent to express the precision of chaining in
crude work in terms of the simple ratio of the length ; but
as the chaining becomes more and more exact, the varia-
tion of the differences between duplicate measurements
approximates more and more closely to the law of square
roots.

Coefficients of precision derived from the latter relation
may be based on either 100-foot units or foot units in the
distance chained, as preferred. The former basis is used in
the chaining diagram while the latter is found in the last
paragraph of the explanatory matter on the second page
referring to the precision of traverse surveys.

The diagram of chaining errors shows chaining ratios by
right lines radiating from the origin, and the law of square
roots by means of parabolas. The coefficient of precision
for a given observed difference between duplicate chainings
is determined by inspection from the diagram, interpolat-
ing between curves if an additional decimal place is desired
in the result. In actual practice a pair of careful chain-

216 ERRORS IN SURVEY J JNU.

men may determine the coefficient corresponding to a given
degree of oare, and then vise this value either in testing
their duplicate results, or in estimating the probable uncer-
tainty of the lengths chained.

For accurate chaining with the steel tape, duplicate
measurements reduced for temperature, etc., or made under
sensibly identical conditions, should not diifer more than
0.05 foot into the square root of the distance in 100-foot
iniits. Careful work with the common chain- (estimating
fractions to 0.1 foot) should not differ more than 0.1 foot
into the square root of the distance in 100-foot units.

Angular Errors. — In measuring deflection angles by alti-
tude reversals, as in railroad traversing, there is, of course
a cumulative discrepancy due to the collimation error, but
generally speaking, careful angular measurements with
good instruments are subject only to compensating or ac-
cidental errors. Under the latter conditions the magnitude
of the error of closure in a series of angles, either in a
root of the number of angles. This relation is indicated
graphically in the diagram of angular errors.

In measuring angles with a transit reading to the nearest
minute, the compensating uncertainty of a single reading is
probably somewhat under 0.5 minute per angle, or about
one minute for the closure of a triangle. If a reading glass
be used and the vernier reads to the nearest half minute,
the uncertainty is still further reduced.

Again, in estimating the needle reading of a compass to
the nearest 5 minutes (one-sixth part of a half-degree), the
uncertainty of reading alone is perhaps 3 minutes, although
this is increased by other conditions such as sluggishness
of needle, etc., probably causing an uncertainty of as much
as 5 minutes per angle, which later limit would produce an
error of closure of a triangle of say 10 minutes, and of a,
five-sided polygon of perhaps the same amount. (See dia-
gram.)

Traversing Errors. — The errors of traversing are made
lip of the combined errors of linear and angular measure-
ments. If the error of closure as determined from the lati-
tudes and departures is large, the work should be scanned
closely to detect blunders such as the substitution of sine
for cosine, errors of 100 feet in chaining, misplacing deci-
mal point, etc. After establishing the consistency of the
residvial errors, they should be distributed either in propor-
tion to the lengths of the several courses, as in the more

TESTS OF PRECISION.

217

THE PRECISION OF CHAINING.

10 10 ^0 40

Lcn^h of Line Chained, l, in tOO'

THE PRECISION OF ANGULAR MEASUREMENTS.

"0 5 10

Number of An^Us in PoIy^^*^ °^

IS
Series, W.

to

£5

16

21S

ERIiOES IN SURVEYING.

THE PRECISION OF TRAVERSE SURVEYS.

The error of cfosure of a traverse /'s usually expressed as the
ratio of the calculated linear error tt> the length of the perimeter of the
fie/ol or polygon. The following table shows the h'mits prescribed by
various author/ ties

PrescHbed Limits For C/osure Of Traverses

Authority.

Conditions.

Limits.

Gillespie, (lassj.

"Suri^eying,' p. 149.

Compass Surveys.

1:300 to i:iooo

A/sop. (I8S7).

Compass Surveys.

I.SOO

"Surveying" p. 199.

Transit Surveys.

i.iooo to risoo

Davi'es. (/S70>.

"Surveying" p. 137.

Farm Surveys-

i:soo to I.IOOO

Jordan. 0877).

German Gov't Surveys.

"Handbuch der

/:400

Vermessungs-

Prussian Instructions.

1:333 to l-.IOOO

kunde;' Vol.1, p.a96.

Stviss Gov't Surveys.

Ordinary Country.

1:400 to 1:800

Mountainous Country,

i:S67 to I: S3 3

Hodgnian. OS8SJ.

"Surveying" p. 119.

Compass Surveys.

1:300 to 1:1000

Johrjson. 0886).

Farm Surveys.

i:300

"Sur veyi'ng" p. 301.

City Surveys.

1:1000 to ItSODO

Baker. * (1888).

"Engineers' Surveying

/ns trum ents" p. S3.

(See Foottiote).

(See Footnote).

Carhart. 0888).

"Surveying' p. ISI.

Ordinary Farm Surveys.

i:Soo

Level Ground.

1:1000

Rougit Ground.

1:200 to l:3O0

Average Transit Surveys.

i:i200

Wood.

(See Footnote).

(See Footnote).

(Roanoke, Va., 1692).

_' Precise Traverses wit/A
Repeated /Ingles. J

1:10 000

(Baltimore, Md-, 1394)

1:15 000 -^.04 Ft.

Raymond. (/396J.

"Surveying," p. 144.

Ordinary Farm Surveys.

nsoo

Good Farm Surveys.

1:2000

Baker derives the fortnu/a E.

= -/]

where

' d^ ~^ /2 000 000
E IS the permissible /inear error of c/osure, P the /erjgth of the
perimeter, I'd the ratio of the chaining error, and a the angular
error of closure in minutes. A thorough te^t of this formula under
a wide range of conditions proves if to be trustworthy'

However, the use of a chaining rcrtio^ /:d, presumably of fixe'd
value for the same chainmen, does not accord tv^th th& resu/ts of
experience in careful ivarHj for it is found that the differences
between duplicate chainings yary about as the square foot of the
iength of fine.

On the fo/low/n^ poge a sftnpfifred fhrmufa }s oisr^amed by as-
suming the more cot7sistent re/a/ion Just stated for fhe chaitving
errors. The resu/ts are about fhe^ame as thos^ obtained yv^ith
Batter's formuta^ and the fbmt of the express iOf> is icfejrticaf
tvith that used by iVood in the &t/titr?ore Surrey.

TESTS OF PRECISION.

219

THE PRECISION OF TRAVERSE SURVEYS.

The reasonable or perm/ssibte error of closure of a traverse
Survey may he determined by the formula derived Leiotv, provided
the errors of ff'e/d tvorft are under oorttrol and their magn/ttida
is ftnotn/n, at /east apfsroxrmarely.
Let P= length of perimeter.

L= calculated error of latitudes.
D~ calculated error of departures.

E^ actual or calculared linear error of cfoSurG offravcr^c
c = coefficient of precision of chaitring.
C = linear error of closure due to chai/ilng errors.
a= angular error of closure in miriutes.
A •= //near error of closure due fo angular errors.
Ef^ permissible or reasonable linear error of closure cfue fo
errors of chaining and angle.
In the triang/e of error the hypothenuse is y^="v/-*+D".
In Dtagram A oe/otv lvalues of Eg may he read close enough for
most cases. Diagram A may also serve as a crude grap/iical rrav~
erse table, and blunders in r/ye fie/d v^r/f may be /ocated by ir.

/n careful chaining by men of some training, the error Marie's about
as the sguare root of the distance, ff^c be the compensating error
for the unit d/sfance, f/rei? C= cifp ,

The angu/ar error of closure in careful surveys prt^ai>/y occurs
arrrong the sides in proportion to t/?eir /engths. Assuming this To be
the case, the resulting linear error is A — aP.arc !=> .OOOSaP.

In good worM the errors are snjalf in amount and egual/y
If able to be plus and minds. Hence, the probable error of c/osarc
due t-o the tirvo causes, i.e. thi> reasonali/e or pern?issib/e //near er-
ror of closure is Ep=l/A'-*-C' —^/'.OOff'SaePJ'-t-c^P

This formula may be much simp/if led by completing the sguarc
and dropping rhe negative tern? under the radical, whence vvirh
sufficient exactness, there resu/ts the genera/ formti/a
Ep^.0003af*-^ I700c^ s • • • -fl)
The very exact standard, P-^/SOOO-*:ad-ft.,used of Baltitporc,
(see table_, preceding page), may be obtained from (O by tnatdng tt
somewhat less than y- minute, and cs.oosft., these va/uas oeing
chnsistent wit/^ the fie/d vnorH of that survey.

The va/ue of c may be def-ermihed for the given ehoin/nen, or
The chaining term of (I) may be taMen as fol/otvs:~ for heat tvarf^
(c^oos-ft.), .OSft} for dverage worH (c^.OIOft.),,Zft.; for fair
worH CcK.O'SJ, ,•? ft.' and for poor nvorH (ci^.OZO), .8 ft. /n care*'
ful traverse Surveys the angle ternf a/one affords a rigid test, so that
formula (B) maybe used except vrhen a='0. Diagrcing 3 gives f£J
for the genera/ run of traverse prob/err^s.

Ep=.0003aP=.^sPg. f£,

A. Actual Error.

0* S; 10' 15* 20" IV 30" 35'

1

ifi i^

^^p

1

1

1

B. Permissible Error.

Sse Formula (2)

rpgro

8 9 10

Error of Deporture, O.

tDOO SOW 3000 4000 5000 6000 7000 6000 9000 now
Length of Perimeter. /? Feet (or LinKsJ

220

EREORS IN SURVEYING.

THE PRECISION OF LEVEL CIRCUITS.
(For Good Average Practice.)

when the length of the level circuit is known in lOO-ft stations,
or when merely the number of settings of the Instrument and the approx-
imate average distarjce covered per setting are hnown, the following
modlficatiofjs of the preceding lest are valuable.

Let £= maximum permissible error of closure of level circuit.
M = length of level circuit it) miles.

L= lOD-ft. stations.

L'~ approximate average tdisfartce covered per setting

of the instrument in WO-ff: staflotis.
5 = number of instrumental settings in the circuit

f^or ^ood average worH with the engmeers' level
E = 0.05ft?fM

from which E = 0.007 fhl/L

and E = 0.007 fffES

Substituting for 100 -ft. average sights, L'=8, E = O.OISS ft.VJ

. 350— ■ - L'=7, E=O0lBZft.TlS

• 300-- • ■ 11=6, E= 0.0163 fi.iS

■ SSO- ■ ■ L=S, E=0.0IS4ft.l/S

For a very rapid approximate check under ordirtary conditions, it may

be assumed that E^O.OlftYS. A graphical representation of these

formulas is given belorv.

Permissible Error of Closure of Level Circuits
For Careful WorK with a Good Engineers' Level.

Length of Circuit Given In Miles (Upper Curye); Or in
the Number of Insfromental Settings fMialc/le Group of
Curves); or in 100-Foot Units (Lower Carre in Diagram^.

Length of Level Circuit, M, Miles.
5 10 15 20

035

0.30

iJo.JS

1 0.20

S0.I5

J 0.00

30

40

«

10

Length of

EO 30
Level Circuit, L,

50

i

:

; : :; ::

-M M ;;;::;-;;

:

:

:: :^

::: :

:

:^

\

:

1 ^lili j ;[ iiMj|:j

y

1

1

::

llllllmlllllliraaairfiliTtiJITfflTlilUI^

%

&

ilMIIMtHi 1 111 1

'.

::

;:g

II

lHjiLUiliJIll liWfi|Hr*Ki

-U-U-

■ ■

i: ;Ji ■ ■■■■■■■■ fflB

1

\ ; : :::-

0.35

0.25

40 50 60 70 80 90 100

100-Foot 5tation5; or Number of Level Seftinq5,5-

TESTS OF PRECISION.

221

THE PRECISION OF LEVEL CIRCUITS.

The precision of spirit leveling is expressed by the formula

Error of Closure =s Constant 1/ Length of Circuit

In the fallonlnj summary of practice in representative surveys of
The United States^ E is the majrimum limit of error of closure of a
level circuit having a length of K kilometers or M miles.

Precision of Leveling in Representative Surveys.

MAXIMUM PERMISSIBLfi ERROR OF CLOSURE,
Metric Unifi British Units.

Coefficient to Coefficient to nearest

nearest mm. O.OOIft. OiOlft.

E=3mm?/K'=0.0ISftiM =0.om.'iM
E= imm?/si<= 0.018 ft.iM\
Mississippi Piver Commission. (Ml). E= imm'SER-= 0.018 ft.T/M V= O.oiftiM
Mississippi Kiver Com'nlBefore 1890. E= 5mm:>flf = O.OSI ff.W)
United States Coast Survey. E= Smm^lZK = 0.0^9 ff.l/M -O.OiftM

United States Lake Survey E=IOmm?[K = 0.012^.^^ =O.O^ft.iM

Vnlted States Geological Survey. E= O.OSO ft.T/M = 0.05 ft.iM

A simple practical test of the degree of precision attained in spirit
leveling is found In the last column of the above table. This graduated
scale of precision is given below graphically for distances to ten miles.

NAME OF SURVEY.

Chicaijo Sanitary District.
Missouri River Commission.

Precision Diagram for Level Circuits.

I 2 3 4

Length (f Level Circuit M, Miles>

222 EEEORS IN SUEVEYING.

common usage, or in the proportion of the respective lati-
tudes and departures, as would seem to be more consistent.
If the several courses have not been surveyed with like
precision, weights should be assigned in distributing the
errors. Absurd refinement should be avoided in making
the distribution of errors.

Leveling Errors. — Perhaps in no phase of surveying
measurements is it more clearly established that accidental
errors follow the law of square roots than in careful level-
ing. The precision diagrams are based on best current
usage.

CHAPTER X.
METHODS OF COMPUTING.

Introduction. — To no one is the ability to make calcula-
tions accurately and rapidly of more value than to the engi-
neer. Many fail to appreciate the value of rapid methods
of calculation, and have no conception of the amount of
time that can be saved by the skillful use of arithmetic,
logarithms, reckoning tables and computing machines.

In the field the engineer has to depend upon the ordinary
methods of arithmetic, or a table of logarithms for his
results. The use of these aids should therefore receive
special attention, for the engineer cannot afford to lose the
time of his assistants while he makes unnecessary or ex-
tended computations.

In the ofBce tables of squares, reckoning tables, slide
rules and computing machines can be used in many cases
with profit.

Consistent Accuracy. — It is safe to say that at least one-
third of the time expended in making computations is
wasted in trying to attain a higher degree of precision than
the nature of the work requires.

In making arithmetical computations where decimals are
involved it is a common practice to carry the result out to
its farthest limit and then drop a few figures at random.

In using logarithms time and labor are lost by using
tables that are more extensive than the data will warrant.
The relative amount of work In using four, five', six and
seven-place tables is about as 1, 2, 3 and 4. Besides the
extra labor involved, the computer has u, result that is
liable to give him an erroneous idea of the accuracy of his
work.

In making computations, in general, calculate the result
to one more place than it is desired to retain.

If several numbers are multiplied or divided, a given
percentage of error in any one of them will produce the
same percentage of error in the result.

223

224 METHODS OP COMPUTING.

In taking the mean of a series of quantities it is consist-
ent to retain one more place than is retained in the quan-
tities themselves.

In direct multiplication or division retain four places of
significant figures in every factor for an accuracy of about
one per cent ; retain five places of significant figures in
every factor for an accuracy of about one-tenth of one per
cent.

LOGAEITHMIC CALCULATIONS.

Iiogarithm Tables. — Logarithm tables contain the deci-
mal part of the logarithm called the mantissa, the integral
part called the characteristic is supplied by the computer.

Four-place tables give the mantissa to four decimal
places of numbers from 1 to 999, and by interpolation give
the mantissa of numbers from 1 to 9,999. Four-place log-
arithms should be used where four significant figures are
sufficient, and should not be xised where an accuracy
greater than one-half of one per cent is required.

Five-place tables give the mantissa to five decimal places
of numbers from 1 to 9,999, and by interpolation give the
mantissa of numbers from 1 to 99,999. Five-place loga-
rithms should be used where five significant figures are
sufficient, and should not be used where an accuracy greater
than one-twentieth of one per cent is required. Five-place
tables are sufficiently accurate for most engineering work.

Six-place tables give the mantissa to six decimal places
of numbers from 1 to 9,999, and by interpolation give the
mantissa of numbers from 1 to 99,999, the same as the five-
place tables. Six-place tables give practically no gain in
precision over fi.ve-place tables since the same numbers of
significant figures are given in both tables, and in addition
the labor of using a six- instead of a five-place table is
about as 3 to 2, due to interpolation with larger diffier-
ences. For the above reasons five-place tables have been
selected for use in this book as being the most suitable
tables for use in surveying.

Seven-place tables give the mantissa to seven decimal
places of numbers from 1 to 99,999, and by interpolation
of numbers from 1 to 999,999. Seven place tables are
rarely needed in engineering work, except in triangulation
work where the angles are measured by repetition.

ARITH^iIETICAL CALCULATIONS. 225

AEITHMETICAL CALCULATIONS.

Requirements. — To become a rapid computer the follow-
ing requirements are essential :

(1) A good memory for retaining certain standard num.-
bers for reference.

(3) The power of performing the ordinary simple arith-
metical operations of multiplication, division, etc., on num-
bers with facility, quickness and accuracy.

(3) The power of registration, i. e., of keeping a string
of numbers in the mind and working accurately upon them.

(4) The power of devising instantly the best method of
performing a complicated problem as regards facility,
quickness and certainty.

It is obvious that all do not have the ability to become
rapid computers, but even these can become fairly skillful
by constant practice and perseverance. The ordinary pro-
cesses of arithmetic should be performed with numbers in
all possible positions. No more figures should be put down
than necessary, and all operations should be performed
mentally whenever possible. In the mental part the results
should alone be stated, much time being lost by repeating
each separate figure.

Checks. — In order to check his work the computer should
keep the following well known properties of numbers well
fixed in his mind :

(1). The sum or difference of two even or of two odd
numbers is even.

(3) The sum or difference of an even and odd number is
odd.

(3) The product of two even numbers is even.

(4) The product of two odd numbers is odd.

(5) The product of an even number and an odd number
is even.

(6) Checking results by the familiar operation of east-
ing out the 9's depends upon the following properties of
numbers :

(a) A number divided by 9 leaves the same remainder
as the sum of the digits divided by 9. For example :

4384 -H 9 = 487 -|- 1

(4-t-3H-8-l-4)^9 = 3-Fl

(7)) The excess of 9's in the product equals the excess of
9's in the product of the excesses of the factors.

226 ilETHODS OF COMPUTING.

473,295 Excess = 3
4,235 Excess = 5
15 Excess = 6

2,004,404,325 Excess =

Check

(e) The excess of 9's in the dividend equals the excess
of 9's in the product of the excesses in tlie di%'isor and quo-
tient, plxis the excess in the remainder :

56)2443 Excess in divisor ^2

43 -)- 35 Excess in quotient = 7

Excess in remainder := 8
Excess in (2 X 7 + 8) =41
Excess in dividend —4j-^'^eck

(7) Results should be checked by taking aliquot parts
wherever possible, and by performing the operations in
inverse order or performing inverse operations. Computa-
tions performed by means of logarithms should be checked
by making the computations roughly by means of arith-
metic. Tlie prohahility of error should be recognized and
precaution fallen, to verify results.

ADBITIOUr. — Since the eye is accustomed to pass from
left to right time can be saved, where the cohimns are not
too long, by adding in the same way. The device of in-
creasing or diminishing the numbers to make them mul-
tiples of ten and then subtracting or adding to the result
is very convenient, especially where several columns are

Ex. 1. — 96

47 143
212 69
32

87 331
49

380

The mental work in detail is as follows :
100 + 47 = 147 ; 147 — 4 = 143 ; 143 + 70 =: 213 ; 213 — 1 ^
212; 212 + 30 + 90 = 332; 332 — 1 = 331; 331 + 50 = 381;
381 — 1=:380.

Expert accountants use the method of adding columns
in groups of 10, 20, 30, etc., small figures, indicating the
mimlier of the group, being placed along the column at in-
tervals depending upon the computer. This method is well

MULTIPLICATION. 227

to be called away from his work. The progress of the
work being then shown by the number of the group, plus
the excess.

MULTIPLICATIOUr. — In order to make the best use of
the methods given, the computer should have perfect com-
mand of the multiplication table as far as 20 at least.

(1) When the tens differ by unity and the sum of the
units equals 10, numbers may be multiplied by the follow-
ing rule : Prom the squares of the tens of the larger number
subtract the square of the units of the larger number.
For the numbers may be represented by (a -\- i) and
(a — 6), and the product will be (a + 6) {a — 6)^o^ — 6^

E.T. i.— (93 X87)=90= — 3= =8,100 — 9 = 8,091.

(3) The product of composite numbers is best obtained
mentally by resolving them into their factors and taking
the products of the factors.

ESB. 2.— 26 X 36 = 9 X 13X 8 — 936.

Ex.3.— 48 X24=(24)^X 3 = 1,152.

Multiples of 10. — To multiply by some number which is
a factor of 10 or some multiple of 10, for example: Multi-
ply

CIO"
A by B, where B = — —
a
Annex n ciphers to A, multiply by C and divide by d.

Ex. i.— 4,324 X 625 = 4,334 ^ =(4,324,000 X 5)-H 8

= 3,702,500.

Ex. 2.-7,924 X 25 = 792,400 H- 4 = 198,100.

Squaring Small Numbers. — Numbers may be squared
mentally by the following rule : Add to or subtract from
one factor enough to make its units figure zero. Subtract
from or add to the other factor tne same amount. Multiply
together this sum and difEerence, and to the product add
the square of the amount by which the factors were in-
creased or diminished.

Proof.— a^ — B^=(a-f6)(a— 6)

a= = (a + 6)((i — 6)+6'.

Ex. i.— (76) = = (73X80) + 4- = 5,776.

228 METHODS OF COMPUTING.

Ex. 2.— (137) = = (124 X 130) + 3^ = 16,139.

Ex. S.— ( 61/i) ^ = ( 6 X 6%) + (1/4) ^ = 39%e-
Ex. J,.— (61^)^ = (6 X 7) + (1^)^ = 421/4.
Ex.5.— (7.5)^ = (7x8) + (-5)' = 56.25.

It will be seen that the process is very simple where the
units place Is 5.

(3) Having- the square of any number the square of the
number next higher is obtained by the following rule : To
the known square add the number and the next higher and
the result will be the square of the next higher number.

Ex.6.— (25)^=635. (26)^ = 635 + 35 + 36 = 676.

(3) A very close approximation to the square of a quan-
tity which is very near unity is obtained by adding algebra-
ically two times the difference between the quantity and
unity to the quantity.

Proof. — (1 + 6)''= 1 + 36 + 6^ = 1 + 26, (approximate).

Ex. 7.— (1.05) = = 1 + 2(1.05 — 1)=1+ 10=110.

Ex. 8.— (.94)^=1 — 2(1 — .94)=1 — .12= 88.

E.r. 9.— (2.034) = = 2=(1 + 2 X .017)= 4(1.034)= 4.136.

Cross-Multiplication. — This consists in taking the prod-
uct of each digit in the multiplicand by each digit in the
multiplier and taking the sums, products of the same de-
nomination being determined thus : units X units gives
units ; tens X units and units X tens gives tens ; units X
hundreds, tens X tens and hundreds X units give hundreds
being put down.

Ex. i.— (2,347) = = 5,508,409 the final result being all that
it is necessary to write down. The mental work is as
follows, the figures in heavy t pe being figures in the prod-
uct ; 7X7 = 49; 4 + 2(7X4)=60; 6 + 2(7X3) + 4= =
64; 6 + 3(3 X 7)+3(3 X 4)=58; 5 + 3(2 X 4) + 3= = 30;
3 + 2(3 X 2)= 15; 1 + 3==5.

Ex. 2. — The product of any two numbers may be found
in the same manner.

9,433
3,583

24,362,856

CEOSS-MULTIPLICATION. 229

The mental work is as follows :3X2 = 6;3X3 + 8X2
:=:25; 3 + 3X4 + 8X3 + 5X2 = 48; 4 + 3X9 + 8X4
+ 5X3 + 2X3 = 82; 8 + 8X9 + 5X4 + 3X3 = 106;
10 + 5X9 + 3X4 = 63; 6 + 2X9 = 34.

Ear.. 3. — The process of cross-multiplication may be sim-
plified as follows : Eequired to multiply 4,338 by 736 ; write
the multiplier on a slip of paper in inverse order and place
it below the multiplicand with the left hand figure below
the units place of the multiplicand thus :

IMultiply together the figures in the same vertical column
6 X 8 ^ 48 ; set down the 8 and carry the 4 ; then move the
slip one space to the left thus :

4,338

I ^37"!
8

Multiplying together the figures in the same vertical col-
umns and taking the sum, 4 + 6X2 + 3 X8 = 40; set
down the and carry the 4 ; then move the slip one space
to the left, multiplying together the figures in the same
vertical columns, adding, etc., we will finally have the work
standing thus :

4,338
I 637 I

3,185,408
Removing the slip we have

4,328
736

3,185,408

The multiplier may be written on the bottom of a sheet
in inverse order and placed above the multiplicand instead
as above described. The work, however, is very much
simplified by simply writing the multiplier in inverse order
without using the slip :

4,328

637

3,185,408

230 ilETHODS OF GOAli'UTlJNCi.

The mental work being as follows : 6X8^ 48; 4 + 6X
3 + 3X8 = 40; 4 + 6X3 + 3X3 + 7X8 =84 ; 8 + 6 X
4+3X3+7X3 = 55 ;5 + 3X4 + 7X3 =38 ; 3 + 7 X 4
= 31. It will be seen that this device removes most of the
mental strain, there being no cross-products.

CONTBACTED MULTIPLICATION.— In multiplying
decimals, when the product is required to a few places of
decimals, the work may be shortened as follows : Kequired
a product correct to the nth decimal place. Write the multi-
plier with its figures in reverse order, its units place under
the nth decimal place of the multiplicand. Multiply the
multiplicand by the figures in the multiplier, beginning
with the right hand figure ; rejecting those figures in the
multiplicand which are to the right of the figure used as a
multiplier, increasing each product by as many units as
would have been carried from the rejected part of the mul-
tiplicand, taking the nearest unit in each case ; place the
right hand figure of each partial product in the same col-
umn, and add as in common multiplication.

In most cases it is best to compute one more place than
required. The following examples illustrate the process :

Ex. 1. — The radius of a circle is 420.17 ft. What is its
semicircumference to nearest 0.01 ft.? (vr^S. 14159265.)

In the work below the partial products in the contracted
multiplication are seen to correspond to the partials of the
common method, taken in reverse order, the part to the
right of the vertical line being rejected. The contracted
multiplication is carried one more place than required. A
dot is j)laced above each figure when it is rejected from the
multiplicand.

4 2 0.1 7 O 4 3 0.1 7

5 6 2 9 5 1 4 1.3 S.1 4 1 5 9 3

!«0510 112 6051

42017 37 8153

16807 210|0 85

4 2 4 2 017

210 16 8 6 8

3 8 4 2 17

1 126051 I

1 3 2 0.0 O 3 1 3 2 0.0 3|1 3 8 1

Ex. 2. — The observed length of a line is 2231.63 ft. with
a tape having a length of 100.018 ft. Required the reduced
length of the line to the nearest 0.01 ft.

CONTKACTED DIVISION. 231

Noting that each foot of the tape = 1.00018 ft.

2 2 3 1.6 3 2 2 3 1.6 3

8 1 0.1 1.0 1 8

223163 1785304

22 223163
18 - 223163000

2 2 3 2.0 3 2 2 3 2.0 3|1 6 9 3 4

Ex. 3. — Same observed length with a tape 99.982 ft. long.
Required the reduced length.

Each foot of the tape = 0.99983 =(1 — 0.00018) ft.

2 2 3 1.6 3
8 10 0.0-

22

18

— 0.4

2 3 3 1.6 3

0.9 9 8 3

4 4 6 3 2 6

1785304

200S467

2008467

2008467

2 2 3 1.2 3

223 1.2 283066

Ex. Jt. — To compare contracted multiplication with log-
arithmic work, calculate 861.3 ft. X sin 17° 19' to the
nearest 0.1 ft.

log. 8 6 1.3 = 2.9 3 5 1 5

log. sin 17° 19' = 9.4 7 3 7 1
log. (2 5 6.4) =2.4 8 8 6

2 5 6.4

CONTBACTED DIVISION.— If the quotient is desired
correct to the nth decimal place, the following method may
be used : Find one-half of the desired figures in the quotient
in the usual way and do not bring down a figure for the
last remainder. Drop a figure from the right of the divisor
and find another figure in the quotient. Then without
bringing down any more figiires continue to discard figures
from the divisor until the required places are obtained.

Ex 1. — Divide 443.9425 by 24.311 to nearest hundredth.
There will be four figures in the quotient, so we will find

8 6 1.3
5 6 7 9 2.0

1723

776

60

5

232 METHODS OF COMPUTING.

the first two in the ordinary way. A dot is placed over
each figure in the divisor when it is rejected.

2 4.3 2 ) 4 4 3.9 4 2 5 ( 1 8.2 5
2432
20074
10456

618

486

132
122

10

Divisor Near Unity. — '\A'hen the divisor is near unity a
very close approximation is given by the method shown in
the following problems :

EJ!. i.— , „„^. ,, = 5(1 — .003554)= 5 X .996746 = 4.98373
1.003204

correct to within one unit in the fifth place.

E^- 2.— -^=7(1+(1 — .9982))=7 X 1.0018 = 7.0126
correct to the last place.

CONTBACTED SQTTAIIE ROOT. — A result correct to a
required number of decimal places may be found by a
process similar to the method employed for contracted divi-
sion.

Ex. 1. — Required the square root of 12,598.87325 correct
to thousandths. We see by inspection that the root will
contain six figures. Find in the ordinary way the first
three figures. Form a new trial divisor in the usual way,

1 2 5 » S.S 7 3 2 5 ( 1 1 2.2 4 5

1

21)35
21

222 ) 498
444

224)548
448
100

89

11

11

CONTRACTED SQUARE ROOT. 233

and bring down only one figure for the dividend in place of
two. Eind the remaining figures by contracted division.

The last figure brought down is not increased whatever it
may be followed by, since the contracted process tends to
make the result a little too large. This method may be ap-
plied to the extraction of cube roots, where it saves much
work in finding long trial divisors.

Square Koot of Small Numbers. — The approximate
square roots of small numbers may be found by means of
the following rule : Divide the given number by the number
whose square is nearest the given number. The arith-
metical mean of the quotient and divisor will be the ap-
proximate square root of the number. The nearer the
number is to a perfect square the less the error. For
example,

Ex. i.— V~35=(35/g -I- 6) -=- 3 = 5.93.

Ex. 2.— V~8=(% + 3)-=-3 = 3.83.

Ex 3.— V"^ =(7% -1-9)-:- 2 = 8.89.

Ex. 4.— V128=(12%i + ll)-=-3n=11.31.

Square B.oot by Subtraction. — ^While it possesses no
points of merit in this connection, it would not be proper to
pass the subject of square root without presenting the novel
method of extracting square roots used with the Thomas
Computing machine. The method depends upon the rela-
tion existing between odd numbers and squares in the sys-
tem of numbers having a radix ten. If we sum up the odd
numbers, beginning at 1, we will observe the following
relation :

1 = 1=; 1-1- 3 = 4 = 3=; 1-1- 3 -I- 5 = 9 = 3^; 1 -1-3 -f-S-f- 7
= 16 = 4", etc. It will be seen that the square root of the
sum in each case is the number of the group.

The method of extracting square roots is as follows :
Point off in periods of two figures each. Subtract from
the left hand period the odd numbers in order, beginning
at unity, until a remainder is obtained less than the next
odd number. Write for the first figure in the root the
number which represents the number of subtractions made.
Double the root already found and annex unity. Subtract
as before, using for subtrahends the successive odd num-
bers, the root figure being the number of subtractions

234 METHODS OF COMPUTING.

Ex. 1. — Extract the square root of 53,824.

r. 3834(232
_1

4

3 2 Hiibtractinns.

41)138
41

97
43

54

4 5 3 subtractions.

401)924
461

463

4 6 3 ... 2 subtractions.

RECKONING TABLES. — Tables for use in computing
are so numerous and well known that it would be useless
to try to refer to them by name. Two valuable tables for
obtaining products of numbers — which are well known in
Germany, but comparatively unknown in this country — are,
'■ Crelle's Eechentafeln," which gives the products of num-
bers of three significant figures by three significant figures
to 999 by 999 ; and " Zimmerman's Eechentafeln," which
gives the products of numbers of two places of significant
figures by numbers of three significant figures to 100 by
999. .

COMPUTING MACHINES.— In Fig. 40, (a) is a Kutt-
ner reckoning machine ; (b) a Thomas computing machine ;
(c) a Fuller slide rule; (d) a Thacher slide rule; (e) an
ordinary slide rule; (f) a Colby Stadia slide rule; (g) a
Colby sewer slide rule; (h) a Grant calciilating machine;
(i) a full circle protractor; (j) a Crozet protractor; (k) a
protractor tee square ; (1) a polar planimeter ; (m) a " jack
knife "' planimeter ; (n) a pantagraph ; (o) a, section liner ;
(p) a spherical planimeter.

In using the " jack knife " planimeter, the point is placed
at the center of gravity, and the knife edge is placed on a
line passing through the center of gravity of the figure.
The point is then made to traverse the perimeter of the
figure to be measured ; passing out to the perimeter and
returning to the center of gravity of the figure on the same
line. The distance from the final position of the knife edge
to the line through the center of gravity, multiplied by the

COMPUTING INSTRUMENTS.

ra)

235

236 METHODS OF COMPUTING.

length of the arm of the planimeter will give the area of
the figure. The arm of the planimeter is usually made ten
inches long and the distance measured in inches.

The correct area may be obtained by means of the hatchet
planimeter, without using the center of gravity of the
figure, as follows: (1) Draw a tangent to the figure. (2)
Trace the figure with the point starting with the hatchet on
the tangent and the point at the point of tangency. (3)
Trace the figure as before except that the point is to move
around in the opposite direction. (4) The arithmetical
mean of the two areas will be the true area. That this
method is correct can be easily proved by the student.

The other machines are described in the instructions ac-
companying them when purchased.

CHAPTER XI.
TOPOGRAPHIC DRAWING AND LETTERING.

LETTERING. — A magnified scale is used in the first six
plates to giFB familiarity with form of letter and numeral,
and also to produce freedom of hand motion. The six
plates should first be made with a soft pencil sharpened
to a needle point, and afterward with pen and india ink. In
Plate 7 the height of letter is that prescribed in Chapter I.
This standard size is not only well adapted to field notes
and general drafting, but is economical of execution.

The student should train the eye and acquire a " swing " of
the hand by industrious practice in such exercises as the fol-
lowing: (1) Pass a line freehand through two points; first
sketch in the line roughly by a free swing of the forearm ;
then partially erase and retrace ; finally test result with
ruler. (2) Pass a circular arc through three points free-
hand; follow sketch method just described and, after per-
fecting the arc, sketch in the chords and locate the center
freehand; test result mechanically. (3) Inscribe a circle
in a square. (4) Inscribe an ellipse in a rectangle. (5)
Inscribe an ellipse in an oblique parallelogram. (In the
last three exercises give particular attention to points and
lines of tangency and axes of symmetry.) After making
the line or figure satisfactorily with pencil, it should be
executed freehand in India ink.

Practice should include spacing of letters and words, and
for this purpose it is suggested that the student use the
" specifications for a good engineer " following the preface.

The student should not be content until he can make
letters freehand so well that a close inspection is required
to determine that they were not made mechanically.

Freehand Titles. — Good freehand titles suffice for most
drawings. In a good title consistent emphasis is given to
the several parts, and the title as a whole accords with the
purpose and character of the drawing. Elaborate and or-
namental titles have a limited application, and should not
be attempted at all unless the draftsman has special skill

237

238 TOPOGltiU'HIC DKAWING AND LETTEEING.

FKEE HAND LETTEIUNG.

239

240 TOPOGKAPHIC DRAWING AND LETTERING.

FKEE HAND LETTERING. 241

3^:

iiiiiiii
lllllll

iiiiiiii

iliiiliB

iliUliil

17

242 TOPOGEAPHIC DEAWING AND LETTERING.

s

i

-5^1

m

m

||

li^i

I

I

I

w

I

i^^

H

m

m

m

m

FKEE HAND LETTERING. 243

244 TOPOGRAPHIC DEAWING AJSfD LETTERING.

/

/

/

/

7

/

1

/

/

c [ 3 /

1

1

7 -'

1

/

/

TECzztmr

II H

1

Ml

)()

P

^

■i

/

//

7 7

1

/

IM^UtEJl'Bi

IK

/

fHI

10

P

K

'i

/

^

X

^^ ^^

/

/

TELfTEMlL

IK

//

Ml

i()

^t^

H

•i

//

WY

/

/

_

^

cLinU'E.^'niii

j-kt

f

fr,i

ITt

77

7

I'

""f

r

77

V

7

^ 1

1

/

]_

/

zAaH^^TfE^T-

T't

t

ih(

777

77

?7

-J-'

T

t

TT"

'/

/

J

whTLU-EJEirELTi

frK

7

-rr);/

) 1)

77

f?/

r

77

t

W^

r

""

-L d d

T^

/

y

.(

t^jit'Bznt

()\

/

f-

u

^

^

1

H

9

/;

rV

/

/

/

I

JZZlfEET^R

()\

f

/'

^4

6^

IH

,y

^

/

/

-

-

!

SBLDIEEHI

,IK

\

Mr

\{)

P

(^

K

S

1

11

V

SEEDIEEII

IK

\

Ml

\{)

P

Q

i

S

1

w

y

1

5:3i:dei5hi

IK

\

Ml

10

F

(^

{

s

1

Y

7

— -1-

—

h: icdEtqin

T"^

T

THT

1

TT

n

r

"S-

T:

TT

T

—

,r

1

nJJnBsJiqSi

--\

T

TTl

1

TT

n

r

"=T

t

TT

■^r

—

\

zzi'icdErqiin:

T"t^

T

m 1

n

T7"

"n

rr"

^

t

W

V

—

i

1

;

115251122

()<\

\

'(

'A

s

b

»

^

h

135255155

\\h

\

'( '

^4

b

f)

(

H

4

h

T

115255125

{)h

1

'( '

4

b

h

/

«

<)

.

DRAWING PENS.

245

in sueli work. In designing titles, whether freehand or
mechanical, skill in sketching in the outlines, guide lines,
axes of symmetry, etc., is of much importance. On the
following pages are a few examples of good titles.

W

Ei

1

^

I

:s

a

s

z

DBA WING PENS. — The following pens, arranged in
order of fineness, will give sufficient variety for ordinary
work.

Gillott's 170, very fine, for very small lettering.

Gillott's 303, extra fine, for small lettering.

Gillott's 404, fine, for small lettering.

Hunt 21, medium, for ordinary lettering.

Hunt 513, Shot Point, for ordinary lettering.

Leonardt 510, E. E. Ball Point, for large lettering and
titles.

Hunt 513, Round Point, for large lettering and titles.

Leonardt 516, E. E. Ball Point, for large lettering and
titles.

Leonardt 516 E., Ball Point, for very large lettering and
titles.

Payzant Pens, K. & E. Co., Nos. 6, 5, 4, 3, 2, 1, for titles.

The following rules should be observed in making letters
on drawings free hand.

Use the quill in inking the pen.

Never dip the pen in the ink bottle.

Keep the pen clean.

Ink must not be allowed to dry on the pen and spread the
points.

Before rising a new pen moisten the points and wipe it
dry to insure a free flow of ink.

TOPOGRAPHIC SYMBOLS.— The standard symbols for
topographic drawings adopted by the American Railway
Engineering Association are given on pages 248 to 351.

246 TOPOGRAPHIC DEAWING AND LETTERING.

Right-of-way Map

liEwYoRK AMD Denver R.R.

Shahion 551+55 to Station 54Z+75

Scale lin.=400 Ft. January 3, 1915

Of Fice 0? ChieF Engineer

Right°of=Way Map

NewYork and Denver RR

station 351+55 to Station 511+10
Scale 1 in =400 ft. January 30,1915

Office oF Chief Engineer

•(oPOGRAPH/c Map

OFTHE

Surveyed by the

Class in Topographic Surveying

First Semester I9I4-I?

Scale lin= 500 Ft.

MAP TITLES DRAWING AND LETTEKING. 247

Right-of-WayMap
flEWYORKAMDDEflVERR.R.

Station 55k55 to Station 54^+75

ScaielinrMFL January3J9i5

' Office of Chief fnqineer

RmihitofWay Map
New YORK& Denver KK

St a tion 33U55 to Station 511 f 10
Scatelin-rdOOft. January 1,1915

Office of Chief Engineer

lojjographic Map

teucrsifg of (Hofora^o

¥ir\$f0'tmtshr 1914-15
JcaU iMrSQOfJt.

248 TOPOGRAPHIC DRAWING AND LETTERING.

HVDROfeRAPHY.

Stream

Springs and Sinks

Lakes and Ponds

Falls and Rapids

Water Line

Marsh

Canals
Ditches

Contour System

Sand

Cliffs

Cut

Embanhment

Top of Slope
Bottom of Slope

Name

Relief.

tuiiuiiijiiiiiiKliiuir:

uuuimuiuiiiiiifiiii.

TOPOGRAPHIC SYMBOLS. 249

-■^Railways (Topographical Maps.)

Steam t — i — i — t-n — i — i — i — i — i — i

Electric i i i i i i i i i i i

Street Railways mimi »■ i

« Railway Tracks (Track Maps.)

Railway Track or Old Track to Remain —

Old Track to he Taken up rz^-jin^-.^-.^-.^z

Proposed Tracks —

Proposed (Future) Tracks ~-:rz^z^rz^rL:n^rz

_ . -^ , Color o ther than Reef or

Foreign Tracks ~

■^ dloc/i with Initials of Road

Alinement rj"^^^^*^^^ I
12" ■ Left- )

4'C.R. Z'C.L

Boundary and Survey Lines.

( Political Divisions -, State, County Bethel T wp.-w.v ne Co.,Mich.

1 or Township Lines. pS^Tw7^TOcl!>d"

J Government Surveys, Base, Meridian, sec i6.t. i zn.,r. i e..5"' pm^

"l Township,6ection or Harbor Line 5eoi3.T iiTT^ifir^p'M

Street, Block or other Property Line

Survey Lines -^ 4^^

-' Location

oenrer l ineb if Monumented, Show Location

and Proper Symbol

Company Property Line

_ State Kind and Height

Fence (on Street Line ) ' ■ ' ■ '-

_ , - -, , , State Hind and Height

Fence (on Company Property Line) ■■ ' ■ •_--.-.

5H For Railway Trach and Yard Stvdiei Use

Single or Double Linei^

18

250 TOPOGRAPHIC DRAWING AND LETTERING.

City

!□□□□□!

Village

■ Jr 1

• ir>

City Lim

its

k^;^f^/^;^3;^3i

Fire Limits

\Ica}A/,/\£^Aa7\.£ZU.

Section

Corner

17 1 16

20 1 11
Section Center —-.^q.^-

Triangulafion Station or Transit Point A
Bench l^arl< B.M.Xl23H

Stone Monument u

Iron /Monument ■

Ml

SCELLANEOUS.

Pole Wire Lines
Railway Tunnel

Dimension Lines

'ndicaieNcofWim ^Ownership

-^ — f — r

=^

^^

~Blas,U

True and
Magnetic l^eridian

I Graphic
'■^ CO let/

50' 100

TOPOGRAPHIC SYMBOLS. 251

Culverts, Sewers, etc.
Masonry Arch or Flat Top Culvert \v.V.'.Z'.'~S~~"^

„. .,, I n V- , ^^ . JSrafe Kind and Lengihjjnd,

Pipe or Wood Boj! Culvert or Dram \:-~:r---r::.:.":\

'^ ' Kind af Walls, If any.) '

Catch Basin D'

C.B.

Manhole -p-

M. M

Sump Qsump

Water Supply and Pipe Lines.

Give /f-v

Water Tank o.STS^C^"'-^-

OiveSite

Water Column o 1

Track Pan u-uo-uoaxxaj

Company Water Pipe 'Give'Siie

Other Water Pipe ->-*->-->-->->-»-->->->-*->

^, r Oive Size

Steam or Oas - —

Give Size

Compreaed Air <■-■• ■■ •' '■-' ' •

Highways and Crossings.

Trails

Street and Public Road Crossings - // //'

Bridges.

Girder ^

Truss ^—

Trestle . )— 1— ' i

SURVEYING MANUAL

PAKT II

FIELB AND OFFICE TABLES FOB TTSE IN
SURVEYING.

BY

WILLIAM D. PENCE

AND

MILO S. KETCHUM

Table 1. Logarithms of Numbers.

Table 2. Logarithmic Functions of Angles.

Table 3. Natural Functions of Angles.

Table 4. Squares, Cubes, Square Roots, Cube Roots and
Circles.

Table 5. Trigonometric Functions.

Explanation of Tables.

The authors wish to thank the J. B. Lippincott Company
for the use of Tables 1 and 2 taken from Suplee's " Five
Place Logarithms," and Table 3 taken from Suplee's " Me-
chanical Engineers' Reference Book " ; and the McGraw-
Hill Book Company for the use of Tables 4 and 5, taken
from Harger and Bonney's " Highway Engineers' Hand-
book."

All of the above tables are fully protected by copyright.

253

254

LOGARITHMS OF NUMBERS.

Table 1.

N

um.

100 to

139.

Log

. 000 to

145.

N

L

0,

1

2

3

4

5

6

7

8 9

P. P.

100

00

000

043

087

130

173

217

260

303

346 389

44

43

101

432

475

518

661

604

647

689

732

776 817

1
2

4.4
8.8

4.3
8.6

102

860

903

945

988 *030

*072 *1]5 *157 *199 *242

103

01

284

326

368

410

462

494

536

578

620 662

3

13.2

12.9

104

703

745

787

828

870

912

953

995 *086 *078

4
5

17.6
22.0

17.2
21.5

105

02

119

160

202

243

284

325

366

407

449 490

6
7

8

26.4
30.8
85.2

25.8
30.1
34.4

106

531

572

612

663

694

735

776

816

857 898

107

938

979 *019 *060 *100

*141 *181 *222 *262 *302

9

89.6

38.7

108

03

342

383

423

463

503

543

583

623

663 703

42

41

109

743

782

822

862

902

941

981 *021 *060*100

110

04

139

179

218

258

297

336

376

416

454 493

1

2

4.2
8.4

4.1
8.2

111

532

571

610

660

689

727

766

805

844 883

3

12.6

12.3

112

922

961

999 *038 *077

*115 *154 *192 *231 *269

4
5
6

16.8
21.0
26.2

16.4
20.5
24.6

113

05

308

346

385

423

461

500

538

576

614 652

114

06

690
070

729
108

767
145

806
183

843
221

881
258

918
296

956
333

994 *032
371 408

7
8
9

29.4
33.6
37.8

28.7
32.8
36.9

116

446

483

521

558

595

633

670

707

744 781

Jf\ tn

117

819

856

893

930

967

*004 *041 *078 *115 *151

4U

ov

118

07

188

225

262

298

835

372

408

415

482 518

1

4.0

3.9

119

555

591

628

664

700

787

773

809

846 882

2
3

8.0.
12.0

7.8
117

120

121

08

918
279

954
314

990 *027 *063
350 386 422

*099 *135
468 493

*171 *207 *243
529 565 600

4
5
6

16.0
20.0
24.0

16.6
19.5
23 4

122

636

672

707

743

778

814

849

884

920 955

7

28.0

27.3

123

991 *026 *061 *096 *132

*167 *202 *237 *272 *307

8

32.0
36.0

31.2

124

09

342

377

412

447

482

517

662

587

621 656

9

86.1

125

691

726

760

795

830

864

899

934

968 *003

38

37

126

10

037

072

106

140

176

209

243

278

812 346

1

3.8

3.7

127

380

415

449

483

617

551

585

619

653 687

2

7.6

7.4

128

721

755

789

823

857

890

924

968

992 *025

3
4

11.4
15.2

11.1
14.8

129
130

11

059
391

093
428

126
461

160
494

193
528

227
561

261
594

294
628

827 361
661 694

5
6

7

19.0
22.8
26.6

18.5
22.2
25.9

131

727

760

793

826

860

893

926

959

992 *024

8

30.4

29.6

132

12

057

090

123

166

189

222

254

287

320 352

9

34.2

33.3

133

385

418

450

483

516

548

581

613

646 678

36

35

134

710

743

775

808

840

872

905

937

969 *001

1

3 6

3.5

135
136

13

033
354

066
386

098

418

130
450

162

481

194
613

226
545

268
577

290 822
609 640

2
3
4

7!2
10.8
14 4

7!o
10.5
14

137

672

704

735

767

799

830

862

893

925 956

5

18.0

17.5

138

988 *019 *051 *082 *1U

*146 *176 *208 *239 »270

6

21.6

21.0

139

14

301

333

364

396

426

467

489

520

551 582

7
8

25.2
28.8

24.5
28.0

140

613

644

675

706

737

768

799

829

860 891

9

32.4

31.5

N

L

1

2

3

4

5

6

7

8 9

P. P.

Table 1.

LOGAEITHMS OF NUMBERS.

255

Num. 140 to 179. Log. 146 to 255.

N

L

1

2

3

4

5

6

7 8 9

P. P.

140

14 613

644

675

706

737

768

799

829 860 891

34

33

141

922

953

983 *014 *045 1

*076 *106 *137 *168 *198

1

2

3.4
6.8

3.3
6.6

142

15 229

259

290

320

351

381

412

442 473 503

143

534

564

594

625

655

685

715

746 776 806

3

10.2

9.9

144

836

866

897

927

957

987 *017 *047 *077 *107'

4
6

13.6
17.0

13.2
16.5

145

16 137

167

197

227

256

286

316

346 376 406

6

7
8

20.4
23.8
27.2

19.8
23.1
26.4

146

4S5

465

495

524

554

684

613

643 673 702

147

732

761

791

820

860

879

909

938 967 997

9

30.6

29.7

118

17 026

056

085

114

143

173

202

231 260 289

32

31

149

319

348

377

406

435

464

493

522 551 580

150

609

638

667

696

725

754

782

811 840 869

1
2

3.2
6.4

3.1
6.2

151

898

926

955

984 *013

*041 *070 *099 *127 *156

3

9.6

9.3

152

18 184

213

241

270

298

327

355

384 412 441

4
5
6

12.8
16.0
19.2

12.4
15.5
18.6

153

469

498

526

554

583

611

639

667 696 724

154

752

780

808

837

865

893

921

949 977 *005

7
8

22.4
26.6

21.7
24.8

155

19 033

061

089

U7

145

173

201

229 267 285

9

28.8

27.9

156

312

340

368

396

424

451

479

507 535 562

30

29

157

590

618

645

673

700

728

758

783 811 838

158

866

893

921

948

976

*003 *030 *058 *085 *112

1

3.0

2.9

159

20 140

167

194

222

249

276

303

330 358 385

2
3

6.0
9.0

5.8
8.7

160

412

439

466

493

520

548

675

602 629 656

4
5
6

12.0
15.0
18.0

11.6
14.5

161

683

710

737

763

790

817

844

871 898 926

17^4

162

952

978 *005 *032 *059

*086 *112 *139 *165 *192

7

21.0

20.3

163

21 219

245

272

299

325

352

378

405 431 458

8
9

24.0
27.0

23.2
26.1

164

484

511

537

564

590

617

643

669 696 722

165

748

775

801

827

854

880

906

932 958 986

28

27

166

22 Oil

037

063

089

115

141

167

194 220 246

1

2.8

2.7

167

272

298

324

350

376

401

427

453 479 505

2
3

5.6
8 4

5.4
8 1

168

531

557

583

608

634

660

686

712 737 763

4

ll!2

io!8

169

789

814

840

.866

891

917

943

968 994 *019

5
6

14.0
16.8

13.5
16.2

170

23 045

070

096

121

147

172

198

223 249 274

7

19.6

18.9

171
172

300
553

325
578

350
603

376
629

401
654

426
679

452
704

477 502 528
729 754 779

8
9

22.4
25.2

21.6
24.3

173

805

830

865

880

905

930

955

980 *005 *030

26

25

174

24 055

080

105

130

155

180

204

229 254 279

1

2.6

2.5

175

304

329

353

378

403

428

462

477 602 527

2
3

5.2
7.8

5.0
7 5

176

551

576

601

625

650

674

699

724 748 773

4

io!4

io!o

177

797

822

846

871

895

920

944

969 993 *018

5

13.0

12.5

178

25 042

066

091

115

139

164

188

212 237 261.

6

7

15.6
18 2

15.0
17.5

179

285

310

334

358

382

406

431

455 479 503

8

20.8

20.0

180

527

551

575

600

624

648

672

696 720 744

9

23.4

22.5

N

L

1

2

3

4

5

6

7 8 9 1 P. P.

256

LOGAKTTHMS OF Nx^T^rp.ERS.

Table 1.

Num.

180 to 219.

Log.

255 to 342.

N

L

1

2

3

4

S

6

7 8 9

P. P.

ISO

25 527

551

575

600

624

648

672

696 720 744

24

181

768

792

816

840

864

888

912

935 959 983

1

2.4 ■

182

26 007

031

055

079

102

126

150

174 198 221

2

4.8

183

245

269

293

316

340

364

387

411 435 458

3

7.2

184

482

505

529

553

576

600

623

647 670 694

4
5

9.6
12.0

185

717

741

764

788

811

834

858

881 905 928

6

7

14.4
16 8

186

951

975

988 *021 *045

*068 *091 *114 *138 *161

8

19 2

187

27 184

207

231

254

277

300

323

346 370 393

9

21.6

188

410

439

462

485

508

531

554

57Z, 600 623

23

189

646

669

692

715

738

761

784

807 830 852

IPO

875

898

921

944

967

989 *012 *035 *058 *081

1
2

4.6

191

28 103

126

149

171

194

217

240

262 285 307

3

6.9

192

330

353

375

398

421

443

466

488 511 533

4
5

9.2
11.5

193

556

578

601

6'23

616

668

691

713 735 758

6

13!8

194

780

803

825

847

870

892

914

937 959 981

7
8

16.1
18.4

195

29 003

026

048

070

092

115

137

169 181 203

9

20.7

196

226

248

270

292

314

336

358

380 403 425

22

197

447

469

491

513

535

557

579

601 623 045

198

667

688

710

732

7.51

776

798

820 842 863

1

2.2

199

885

907

929

951

973

994 *016 *038 *060 *081

2
3

4.4
6.6

200

30 103

125

146

168

190

211

233

265 276 298

4
5

8.8
11.0

201

320

811

363

384

406

428

449

471 49? 514

6

13^2

202

535

557

578

600

621

643

664

685 707 728

7

15.4

203

750

771

792

814

835

856

878

899 920 942

8
g

17.6
19.8

204

963

984 *006 *027 *048

*069 *091 *112 *133 *154

205

31 175

197

218

239

260

281

302

323 346 366

Zl

206

387

408

429

450

471

492

613

534 555 576

1

2.1

207

597

618

639

660

681

702

723

744 765 785

2
3

4.2
6.3

208

806

827

848

869

890

911

931

952 973 994

4

8!*

209

32 015

035

056

077

098

118

139

160 181 201

6
6

10.5
12.6

210

222

243

263

2K4

305

325

346

366 387 408

7

14.7

211

428

449

469

490

510

531

552

572 593 613

8
9

16.8
18.9

212

634

664

675

695

715

736

756

777 797 818

213

838

858

879

899

919

940

960

980 *001 *021

20

19

214

33 041

062

082

102

122

143

163

183 203 224

1

2.0

1.9

215

24-1

264

284

304

325

345

365

385 405 425

2
3
4

4.0
6.0
8.0

3.8
5.7
7.6

216

445

465

486

506

626

546

566

586 606 626

217

646

666

686

706

726

746

766

786 806 826

5 1

0.0

9.5

218

846

866

885.

905

925

945

965

985 *005 *025

6 1

7 1

8 1

2.0
4.0
6.0

11.4
13 3
15.2

219

34 044

064

084

104

124

143

163

183 203 223

220

212

262

282

301

321

341

361

380 400 420

9 1

8.0

17.1

N

L

1

2

3

4

S

6

7 8 9

P. P.

Table 1.

LOGARITHMS OF JNUMBiiKa.

vn

Num. 220 to 259. Log. 342 to 414.

N

L

1

2

3

4

5 6 7 8 9

P. P.

220

34 242

262

282

301

321

341 361 380 400 420

221

439

459

479

498

518

537 657 677 596 616

20

222

635

655

674

694

713

733 753 772 792 811

1

2.0

223

830

850

869

889

908

928 947 967 986 *005

2

4.0

224

35 025

044

061

083

102

122 141 160 180 199

3
4

6.0
8.0

225

218

238

^7

276

295

315 334 353 372 392

5
6

10.0
12

226

411

430

449

468

488

607 526 545 564 583

7

iiio

227

603

622

■641

660

679

698 717 736 755 774

8

16.0

228

793

813

832

851

870

889 908 927 946 985

9

18.0

229

984 *003 *021 *04D *D59

*078 *097 *116 *135 *154

230

36 173

192

211

229

248

267 286 305 324 342

19

231

361

380

399

418

436

455 474 493 611 530

232

519

568

586

605

624

642 661 680 698 717

1

1.9

233

736

754

773

791

810

829 847 866 884 903

2
3
4

3.8
5.7
7.6

234

922

940

959

977

996

*014 *033 *051 *070 *088

235

37 107

125

144

162

181

199 218 236 254 273

5
6

9.5
11.4

236

291

310

328

316

365

383 401 420 438 457

7

13.3

237

475

493

511

530

548

566 585 603 621 639

8
9

15.2
17.1

238

658

676

694

712

731

749 767 786 803 822

239

840

858

876

894

912

931 949 967 985 *003

240

38 021

039

057

076

093

112 130 148 166 184

241

202

220

238

266

274

292 310 328 346 364

•
18

242

382

399

417

435

453

471 489 507 525 543

243

561

578

596

614

632

650 668 686 703 721

1

1.8

214

739

757

775

792

810

828 846 863 881 899

2
3

3.6
5.4

245

917

934

952

970

987

*005 *023 *041 *058 *076

1

5

7.2
9

246

39 094

111

129

146

161

182 199 217 235 262

6

io!8

247

270

287

305

322

340

358 375 393 410 428

7

12.6

248

445

463

480

498

515

533 650 668 685 602

8
9

14,4

Ifl 9.

249

620

637

655

672

690

707 724 742 759 777

250

794

811

829

846

863

881 898 915 933 960

251

967

985 *002 *019 *037

*064 *071 *088 *106 *123

262

40 140

157

175

192

209

226 243 261 278 295

253

312

329

346

361

381

398 415 432 449 466

17

254

483

600

518

535

652

569 586 603 620 637

1

1.7

255

654

671

688

705

722

739 756 773 790 807

2
3

3,4
5.1

266

824

841

858

875

892

909 926 943 960 976

4

6.8

257

993 *010 *027 *044 *061

*078 *095 *111 *128 *145

5
6

7

8.5
10.2
11.9

258

41 162

179

196

212

229

246 263 280 296 313

259
260

330
497

347
514

363
531

380
547

397
564

414 430 447 464 481
581 597 614 631 647

8
9

13.6
15.3

N

L

1

2

3

4

S 6 7 8 9

P. P.

258

LOGARITHMS OP NUMBEliS.

Table 1.

Num

260 to 299.

Log. 414 to 476.

N

L

1

2

3

4

5 6 7 8 9

P. P.

260

41 497

514

531

547

564

581 697 614 031 047

261

604

681

697

714

731

747 764 780 797 814

262

830

847

863

880

896

913 929 946 963 979

263

996 *012 *029 *045 *002

*078 *095 *111 *127 144

264

42 100

177

193

210

226

243 259 275 292 308

17

205

325

341

367

374

390

406 423 439 455 472

1 1 "7

260

488

504

521

537

653

570 580 602 619 635

2

3.4

207

651

667

684

700

716

732 749 766 781 797

3

5.1

208

813

830

846

862

878

894 911 927 943 959

4
6
6

6.8
8.5
10.2

269

975

991 *008 *024 *040

*056 *072 *088 *1(M *120

270

43 136

152

169

185

201

217 233 249 265 281

7
8

11.9
13.6

271

297

313

329

345

361

377 393 409 425 441

9

15.3

272

457

473

489

505

521

637 553 509 584 600

273

616

632

648

664

680

696 712 727 743 769

274

775

791

807

823

838

854 870 886 902 917

275

933

949

965

981

996

*012 *028 *044 *059 *075

270

44 091

107

122

138

154

170 185 201 217 232

16

277

248

264

279

295

311

326 342 358 373 389

278

404

420

436

451

467

483 498 514 529 545

1

1.6

279

560

576

592

607

623

638 654 669 685 700

2
3

3.2
4.8

280

7}6

731

747

762

778

793 809 824 840 855

4
5
6

6.4
8.0
9.6

281

871

886

902

917

932

948 963 979 994 *010

282

43 025

040

056

071

086

102 117 133 148 163

7

11.2

283

179

194

209

225

240

255 271 286 301 317

8
9

12.8
14.4

284

332

347

362

378

393

408 423 439 454 469

285

484

500

515

530

545

561 576 591 606 621

280

637

652

667

682

697

712 728 743 758 773

287

788

803

818

834

849

864 879 894 909 924

288

939

954

969

984 *000

*015 *030 *045 *060 *075

289

40 090

105

120

135

150

166 180 195 210 226

15

290

240

255

270

286

300

315 330 345 359 374

1

1.5

291

389

404

419

434

419

464 479 494 609 523

2
3
4

3.0
4.5
6.0

292

538

553

508

683

598

613 627 642 667 672

293

687

702

716

731

716

761 776 790 805 820

5

7.5

294

835

850

864

879

894

909 923 938 953 967

6

7

9.0
10.6

295

982

997 *012 *020 *041

*066 *070 *086 *100 *114

8
9

12.0
13.5

296

47 129

144

159

173

188

202 217 232 246 261

297

276

290

305

319

334

349 363 378 392 407

298

422

430

451

465

480

494 609 624 538 553

299

567

582

596

611

625

640 654 669 683 698

300

712

727

741

756

770

784 799 813 828 842

N

L

1

2

3

4

S 6 7 8 9

P. P.

Table 1.

LOGARITHMS OF NUMBERS.

259

Num. 300 to 339. Log. 477 to 531.

1

8

P. P.

47 712 727 741 756 770
857 871 885 900 914

48 001 015 029 044 058
144 159 173 187 202
287 302 316 330 344

430 444 458 473 487

572 586 601 615 629

714 728 742 756 770

855 869 883 897 911

996 *010 *024 *038 *052

49 136 150 164 178 192
276 290 304 318 332
415 429 443 457 471
•554 568 582 596 610
693 707 721 734 748

831 845 859 872 886

969 98? 996 *010 *024

50 106 120 133 147 161

243 256 270 284 297
379 393 406 420 433

515 529 542 556 569

651 664 678 691 705

786 799 813 826 840

920 934 947 961 974

51 055 068 081 095 108

188 202 216 228 242

322 335 348 362 375

455 468 481 495 508

587 601 614 627 640

720 733 746 759 772

851 865 878 891 904

983 996 *009 *022 *035

52 114 127 140 153 166

244 257 270 284 297
375 388 401 414 427

504 617 530 543 556

634 647 660 673 686

763 776 789 802 815

892 905 917 930 943

53 020 033 046 058 071

148 161 173 186 199

784 799 813 828 842
929 943 958 972 986
073 087 101 116 130
216 230 244 259 273
359 373 387 401 416

501 515 530 544 558

643 657 671 686 700

785 799 813 827 841
926 940 964 968 982

*066 *080 *094 *108 *122

206 220 234 248 262

346 360 374 388 402

486 499 513 527 541

624 638 651 665 679

762 776 790 803 817

900 914 927 941 965

*037 *051 *066 *079 *092

174 188 202 215 229

311 326 338 362 365

447 461 474 488 501

583 596 610 623 637

718 732 746 759 772

853 866 880 893 907

987 *001 *014 *028 *041

121 136 148 162 175

255 268 282 295 308

388 402 415 428 441

521 534 548 561 574

654 667 680 693 706

786 799 812 825 838

917 930 943 957 970

*048 *a61 *075 *088 *101

179 192 205 218 231

310 323 336 349 362

440 463 466 479 492

569 582 595 608 621

699 711 724 737 750
827 840 853 866 879
956 969 982 994 *007
084 097 110 122 135

212 224 237 250 263

P. P.

260

LOGARITHMS OF NUMBERS.

Table l.

Num

340 to 379.

Log

531

to

579.

N

L

1

2

3

4

5

6

7

8 9

p. p.

340

63 148

161

173

186

199

212

224

237

250 203

311

275

2XS

301

314

326

339

352

364

377 390

312

403

415

428

441

453

400

479

491

504 517

313

529

542

555

667

580

593

605

618

631 643

344

656

668

081

694

706

719

732

744

757 769

13

345

782

794

807

820

832

845

857

870

882 895

1
2

1.3
2.6

340

908

920

933

945

958

970

983

995 *008 *020

347

54 033

046

058

070

083

095

108

120

133 145

3

3.9

348

158

170

183

195

208

220

233

245

258 270

4
5

5.2
6 5

349

283

295

307

320

332

345

357

370

:«2 394

7^8

350

407

419

432

444

456

469

481

494

506 518

7
8

9.1
10.4

361

531

543

555

568

580

593

605

617

630 042

9

11.7

352

654

667

679

691

704

716

728

741

753 765

353

777

790

802

814

827

839

851

804

876 888

354

900

913

925

937

949

902

974

986

998 *011

365

55 023

036

047

060

072

084

096

108

121 133

356

145

157

169

182

194

206

218

230

242 255

12

367

267

279

291

303

315

328

340

352

364 376

358

388

400

413

425

437

449

401

473

485 497

1

1.2

359

509

522

534

546

558

570

582

594

606 618

2
3

2.4
3.0

360

630

642

654

666

678

691

703

715

727 739

4
5
6

4.8
6.0
7.2

361

761

763

775

787

799

811

823

835

847 859

362

871

883

895

907

919

931

943

955

967 979

7

8.4

363

991 *003 *015 *027 *038

*050 *062

*074 *080 *098

8
9

9.6
10.8

3G4

56 110

122

134

146

158

170

182

194

205 217

365

229

241

253

265

277

289

301

312

324 336

366

348

360

372

384

396

407

419

431

443 465

367

467

478

490

502

514

526

538

549

601 673

368

585

597

608

620

632

044

056

667

679 691

369

703

714

726

738

750

761

773

785

797 808

11

370

820

832

844

855

807

879

891

902

914 920

1

1.1

371

937

949

961

972

984

990 *008 *019 *031 *043

2
3

2.2
3.3
4.4

372

57 054

066

078

089

101

113

124

130

148 159

4

373

171

183

194

200

217

229

241

252

264 276

6

5.5

374

287

299

310

322

334

345

367

308

380 392

6

7

0.0

7.7

375

403

415

426

438

449

401

473

484

196 507

8
9

8.8
9.9

376

519

530

512

553

5(,5

570

,688

000

Oil 623

377

634

646

057

669

080

692

703

715

726 738

378

749

761

772

784

795

807

818

830

841 852

379

864

875

887

898

910

921

933

944

955 967

380

978

990 »001 *013 *0a4

*035 *047 *058 TO70 *081

N

L

i

2

3

4

S

6

7

8 9

P. P.

Table 1.

LOGARITHMS OF NUMBERS.

2111

Num.

380 to 419.

Log.

579 to 623.

N

L

1

2

3

4

S

6

7

8 9

P. P.

380

67 978

990 *001 *013 *024

*035 *047 *058 *070 *081

381

58 092

104

115

127

188

149

161

173

184 196

382

206

218

229

240

252

263

274

286

297 309.

383

320

331

343

354

365

377

388

399

410 422

384

433

444

456

467

478

490

501

612

524 535

1 1

385

546

557

569

580

591

602

614

625

636 647

386

659

670

681

692

704

715

726

737

749 760

1 ■* ■*

2

1.1
2.2

387

771

782

794

805

816

827

838

860

861 872

3

3.3

388

883

894

906

917

928

939

950

961

973 984

4
6
6

4.4
6.5
6.6

389

995 *006 *017 *028 *040

*051 *062 *073 *084 *095

390

59 106

118

129

140

151

162

173

184

196 207

7
8

7.7
8.8

391

218

229

240

251

262

273

284

295

306 318

9

9.9

392

329

340

351

362

,373

384

395

406

417 428

393

439

450

461

472

483

494

606

517

628 539

394

550

561

572

583

594

605

616

627

638 649

395

660

671

682

693

704

715

726

787

748 759

396

770

780

791

802

813

824

835

846

857 868

10

397

879

890

901

912

923

934

945

956

966 977

398

988

999 *010 *021 *032

*043 *054 *066 *076 *086

1

1.0

399

60 097

108

119

130

141

152

163

173

184 195

2
3

2.0
3.0

400

206

217

228

239

249

260

271

282

293 304

4
6
6

4.0
6.0
6.0

401

314

325

336

347

358

369

379

390.

401 412

402

423

433

444

455

466

477

487

498

809 520

7

7.0

403

531

541

552

563

574

584

595

606

617 627

8
9

8.0
9.0

404

638

649

660

670

681

692

703

713

724 735

405

746

756

767

778

788

799

810

821

831 842

406

853

863

874

885

895

906

917

927

938 949

407

959

970

981

991 *002

*013 *023 *034 *045 *055

408

61 066

077

087

098

109

119

130

140

151 162

409

172

183

194

204

215

225

236

247

257 268

410

278

289

300

310

321

331

342

352

363 374

411

384

395

405

416

426

437

448

458

469 479

412

490

500

511

521

532

512

563

563

574 584

413

595

606

616

627

637

648

658

669

679 690

414

700

711

721

731

742

752

763

773

784 794

415

805

815

826

836

847

857

868

878

888 899

416

909

920

930

941

951

962

972

982

993 *003

417

62 014

024

034

045

055

066

076

086

097 107

418

118

128

138

149

169

170

180

190

201 211

419

221

232

242

262

263

273

284

294

304 315

420

325

335

346

356

366

377

387

397

408 418

N

L

1

2

3

4

S

6

7

8 9

P. P.

262

LOGARITHMS OF NUMBERS.

Table 1.

Num

. 420 to 459.

Log

. 623 to 662.

N

L

1

2

3

4

S

6 7 8 9

P. P.

420

62 325

335

346

356

366

377

387 397 408 418

421

428

439

449

459

469

480

490 500 511 521

422

531

542

552

562

572

583

593 603 613 624

423

634

644

655

665

675

685

696 706 716 726

424

737

747

757

767

778

788

798 808 818 829

425

839

849

859

870

880

890

900 910 921 931

426

941

951

961

972

982

992 *002 *012 *022 *033

427

63 043

053

063

073

083

094

104 114 124 134

428

144

155

165

175

185

195

205 215 225 236

10

429

246

256

266

276

286

296

306 317 327 337

430

347

357

367

377

387

397

407 417 428 438

1 ' '
2

1.0
2.0

431

448

458

468

478

488

498

508 518 528 538

3

3.0

432

548

558

568

579

589

599

609 619 629 639

4
5
6

4.0
5.0
6.0

433

649

659

669

679

689

699

709 719 729 739

434
435

749
849

759
859

769
869

779
879

789
889

799
899

809 819 829 839
909 919 929 939

7
8
9

7.0
8.0
9.0

436

949

959

969

979

988

998 *008 *018 *028 *038

437

64 048

058

068

078

088

098

108 118 128 137

438

147

157

167

177

187

197

207 217 227 237

439

246

256

266

276

286

296

306 316 326 335

440

345

355

365

375

385

395

404 414 424 434

441

444

454

464

473

483

498

503 513 523 532

442

512

552

562

572

582

591

601 611 621 631

443

640

650

660

670

680

689

699 709 719 729

444

738

748

758

768

777

787

797 807 816 826

445

836

846

856

865

875

885

895 904 914 924

9

446

933

943

953

963

972

982

992 *002 *011 *021

1

0.9

447

65 031

040

050

060

070

079

089 099 108 118

2
3

4

1.8
2.7
3.6

448

128

137

147

157

167

176

186 196 205 215

449
4S0

225
321

234
331

244
341

254
350

263
360

273
369

283 292 302 312
379 389 398 408

5
6

7

4.5
6.4
6.3

451

418

427

437

447

456

466

475 485 495 504

8
9

7.2
8.1

452

514

523

533

543

552

562

571 581 591 600

453

610

619

629

639

648

.658

667 677 686 696

454

706

715

725

734

744

753

763 772 782 792

455

801

811

820

830

839

849

868 868 877 887

456

896

906

916

925

935

944

954 963 973 982

457

992 *001 *011 *020 *030

*039 *049 *058 *068 *077

458

66 087

096

106

115

124

134

143 153 162 172

459

181

191

200

210

219

229

238 247 257 266

460

276

285

295

304

314

323

332 342 351 361

N

L

i

2

3

4

S

6 7 8 9

P. P.

Table 1.

LOGARITHMS OF NUMBERS.

263

Num

460 to 499.

Log.

662 to 698.

N

L

1

2

3

4

5

6

7 8 9

P. P.

460

66 276

285

295

304

314

323

332

342 351 361

461

370

380

389

398

408

417

427

436 445 455

462

464

474

483

492

502

511

521

530 539 549

463

558

567

577

586

596

605

614

624 633 642

464

652

661

671

680

689

699

708

717 727 736

465

745

755

764

773

783

792

801

811 820 829

466

839

848

857

867

876

885

894

904 913 922

467

932

941

950

960

969

978

987

997 *006 *015

468

67 025

034

043

052

062

071

080

089 099 108

10

469

117

127

136

145

154

164

173

182 191 201

470

210

219

228

237

247

256

265

274 284 293

1
2

1.0
2.0

471

302

311

321

330

339

348

357

367 376 385

3

3.0

472

394

403

413

422

431

440

449

459 468 477

4
5
6

4.0
5.0
6.0

473

486

495

504

514

523

532

511

550 560 569

474
475

578
669

587
679

596
688

605
697

614

706

624
715

633

724

642 651 660
733 742 752

7
8
9

7.0
8.0
9.0

476

761

770

779

788

797

806

815

825 834 843

477

852

861

870

879

888

897

906

916 925 934

478

943

952

961

970

979

988

997 *006 *015 *024

479

68 034

043

052

061

070

079

088

097 106 115

480

124

133

142

151

160

169

178

187 196 205

481

215

224

233

242

251

260

269

278 287 296

482

305

314

323

332

341

350

359

368 377 386

483

395

404

413

422

431

440

449

458 467 476

484

485

494

502

511

520

529

538

547 556 565

485

574

683

592

601

610

619

628

637 646 655

9

486

664

673

681

690

699

708

717

726 735 744

1

0.9

487

753

762

771

780

789

797

806

815 824 833

2
3

4

1.8
2.7
3.6

488

842

851

860

869

878

886

895

904 913 922

489
490

931

69 020
-7108

940
028

949
037

958
046

966
055

975
064

984
073

993 *002 *011
082 090 099

5
6

7

4.5
5.4
6.3

491

117

126

135

144

152

162

170 179 188

8
9

7.2
8.1

492

197

205

214

223

232

241

249

268 267 276

493

285

294

302

311

320

329

338

346 366 364

494

373

381

390

399

408

417

425

434 443 452

495

461

469

478

487

496

504

513

522 531 539

496

548

557

566

574

583

592

601

609 618 627

497

636

644

653

662

671

679

688

697 705 714

498

723

732

740

749

758

767

775

784 793 801

'

499

810

819

827

836

845

854

862

871 880 888

SCO

897

906

914

923

932

940

949

958 966 975

N

L

1

2

3

4

5

6

7 8 9

P. P.

264

LOGARITHMS OF NUMBERS.

Table 1.

Num

. 500 to 539.

Lo^

. 698 to 732.

N

L

1

2

3

4

5

6

7

8 9

P. P.

soo

69 897

906

914

922

932

940

949

958

966 975

501

984

992 *001 *010 *018

*027 *036 *044 *053 *062

502

70 070

079

088

096

105

114

122

131

140 148

503

157

165

174

183

191

200

209

217

226 234

504

243

252

260

269

278

286

295

303

312 321

505

329

338

346

355

361

372

381

389

398 406

506

415

424

432

441

449

458

467

475

484 492

507

501

509

518

526

535

544

552

561

569 578

508

^586

595

603

612

621

629

638

646

655 663

9

509

' 672

680

689

697

706

714

723

731

740 749

510

757

766

774

783

791

800

808

817

825 834

1

2

0.9
1.8

511

842

851

859

868

876

885

893

902

910 919

3

2.7

512

927

935

944

952

961

969

978

986

995 *003

4
5
6

3.6
4.5
5.4

513

71 012

020

029

037

046

054

063

071

079 088

511
515

096

181

105
189

113
198

122
206

130

214

139
223

147
231

155
240

164 172
248 257

7
8
9

6.3
7.2
8.1

516

265

273

282

290

299

307

315

324

332 341

517

349

357

366

374

383

391

399

408

416 4'25

518

433

441

450

458

466

475

483

492

500 508

519

517

525

533

542

550

559

567

575

584 592

520

600

609

617

625

634

642

650

659

667 675

521

684

692

700

709

717

725

734

742

750 759

522

767

775

784

792

800

809

817

825

834 842

623

850

858

867

875

883

892

900

908

917 925

521

933

941

950

958

966

975

983

991

999 *008

525

72 016

0'24

032

041

049

057

066

074

082 090

8

526

099

107

115

123

132

140

148

156

165 173

1

0.8

527

181

189

198

206

214

222

230

239

247 265

2
3
4

1.6
2.4
3.2

528

263

272

280

288

296

304

313

321

329 337

529
530

346
428

351
436

362

444

370
452

378
460

387
469

395

477

403

485

411 419
'493 501

5
6
7

4.0
4.8
6.6

531

509

518

526

534

542

550

558

567

575 583

8
9

6.4
7.2

532

591

599

607

616

624

632

640

648

656 665

533

673

681

689

697

705

713

722

730

738 746

534

754

762

770

779

787

795

803

811

819 827

535

835

843

852

860

868

876

884

892

900 908

536

916

925

933

941

949

957

965

973

981 989

537

997 *006 *0U *D22 *030

*038 *046 *054

062 *070

538

73 078

086

094

102

HI

119

127

135

143 151

539

159

167

175

183

191

199

207

215

223 231

540

239

247

255

263

272

280

288

296

304 312

N

L

1

2

3

4

5

6

7

8 9

P. P.

Table 1.

LOGARITHMS OF NUMBEKS.

265

Num. 540 to 579. Log. 732 to 763.

N

L

1

2

3

4

5 6 7 8 9

P.

P.

540

73 239

247

255

263

272

280 288 296 304 312

541

^20

328

336

344

352

360 368 376 384 392

542

400

408

416

424

432

440 448 456 464 472

543

480

488

496

504

512

520 528 536 544 552

544

560

568

576

584

592

600 608 616 624 632

545

640

648

656

664

672

679 687 695 703 711

546

719

727

785

743

751

759 767 775 783 791

547

799

807

815

823

830

838 846 854 862 870

548

878

886

894

902

910

918 926 933 941 949

8

-549

957

965

973

981

989

997 *005 *013 *020 *028

550

74 036

044

052

060

068

076 084 092 099 107

1
2

0.8
1.6

551

115

123

131

139

147

155 162 170 178 186

3

2.4

552

194

202

210

218

225

233 241 249 257 265

4
5
6

3.2
4.0
4.8

553

273

280

288

296

304

312 320 327 335 343

554
555

351
429

359
437

367
445

374
453

382
461

390 398 406 414 421
468 476 484, 492 600

7
8
9

5.6
6.4
7.2

556

507

515

523

531

539

547 554 562 570 578

557

586

593

601

609

617

024 632 610 648 656

558

663

671

679

687

695

702 710 718 726 733

559

741

749

757

764

772

780 788 796 803 811

560

819

827

834

842

850

858 865 873 881 889

561

896

904

912

920

927

935 943 950 958 966

562

974

981

989

997 *005

*012 *020 *028 *035 *043

563

75 051

059

066

074

082

089 097 105 113 120

564

128

136

143

151

159

166 174 182 189 197

565

205

213

220

228

236

243 251 259 266 274

7

566

282

289

297

305

312

320 328 335 343 351

1

0.7

567

358

366

374

381

389

397 404 412 420 427

2
3
4

1.4
2.1
2.8

568

435

442

450

458

465

473 481 488 496 504

569

511

519

526

534

542

549 557 565 572 580

5
6

7

3.5
4.2
4,9

570

587

595

603

610

618

626 633 641 648 656

571

664

671

679

686

694

702 709 717 724 732

8
9

5.6
6.3

572

740

747

755

762

770

778 785 793 800 808

573

815

823

831

838

846

853 861 868 876 884

574

891

899

906

914

921

929 937 944 952 959

575

967

974

982

989

997

*005 •012 *020 *027 *035

576

76 042

050

057

065

072

080 087 095 103 110

577

-118

125

133

140

148

155 163 170 178 185

578

193

200

208

215

223

230 238 245 253 260

579

268

275

283

290

298

305 313 320 328 335

580

343

350

358

365

373

380 388 395 403 410

N

L

J

2

3

4

5 6 7 8 9

P.

P.

266

liOGAEITHMS OP NUMBERS.

Table 1.

Num. 580 to 619. Log;. 763 to 792.

N

L

1

2

3

4

S

6

7

8 9

P. P.

580

76 343

360

358

365

373

380

388

395

403 410

8

581

418

425

433

440

448

455

462

470

477 485

1
2

0.8
1.5

582

492

500

507

515

523

530

537

545

552 659

583

567

574

582

589

697

604

612

619

626 634

3

2.4

684

641

649

666

664

671

678

686

693

701 708

4
5

3.2
4

585

716

723

730

738

745

753

760

768

775 782

6

7
8

4.8
5.6
6.4

686

790

797

805

812

819

827

834

842

849 856

587

864

871

879

886

893

901

908

916

923 930

9

7.2

688

938

945

963

960

967

975

982

989

997 *004

589

77 012

019

026

034

041

048

056

063

070 078

590

085

093

100

107

115

122

129

137

144 151

591

159

166

173

181

188

196

203

210

217 226

592

232

240

247

264

262

269

276

283

291 298

593

305

313

320

327

335

342

349

367

364 371

594

379

386

393

401

408

415

422

430

437 444

595

452

469

466

474

481

488

495

503

610 517

596

525

532

539

646

654

561

568

576

583 590

697

697

605

612

619

627

634

641

648

656 663

7

598

670

677

685

692

699

706

714

721

728 735

599

743

760

767

76i

772

779

786

793

801 808

1

2

0.7
1.4

600

815

822

830

837

844

861

859

866

873 880

3

4
5

2!l
2.8
3.6

601

887

895

902

909

916

924

931

988

946 952

602

960

967

974

981

988

996 *003 *010 *017 *025

6

4.2

603

78 032

039

046

053

061

068

075

082

089 097

7

4.9

604

104

111

118

125

132

140

147

154

161 168

8
9

5.6
6.3

606

176

183

190

197

204

211

219

226

233 240

606

247

254

262

269

276

283

290

297

305 312

607

319

326

333

340

347

355

362

369

376 383

608

390

398

405

412

419

,426

433

440

447 456

609

462

469

476

483

490

497

504

512

519 626

610

633

540

647

554

561

669

576

583

690 597

611

604

611

618

626

633

640

647

654

661 668

612

675

682

689

696

704

711

718

725

732 739

613

746

753

760

767

774

781

789

796

802 810

614

817

824

831

838

845

852

859

866

873 880

015

888

895

902

909

916

923

930

937

944 951

616

958

965

972

979

986

993 *000 *007 *014 *021

617

79 029

036

043

060

057

064

071

078

085 092

618

099

106

113

120

127

134

141

148

155 162

619

169

176

183

190

197

204

211

218

226 232

620

239

246

253

260

267

274

281

288

295 302

N

L

1

2

3

4

5

6

7

8 9

P. P.

Table 1.

LOGAEITHMS OP NUMBEKS. .

267

Num. 620 to 659. Log;. 792 to 819.

N

L

1

2

3

4

5

6

7

8

9

P. P.

620

79 239

246

253

260

267

274

281

288

295

302

621

309

316

323

330

337

344

351

358

365

372

622

379

386

393

400

407

414

421

428

435

442

623

449

456

463

470

477

484

491

498

505

511

624

518

525

532

539

546

553

560

567

574

581

625

588

595

602

609

616

623

630

637

644

660

626

657

664

671

678

685

692

699

706

713

720

627

727

734

741

748

754

761

768

775

782

789

628

796

803

810

817

824

831

837

844

851

858

629

865

872

879

886

893

900

906

913

920

927

630

934

941

948

955

962

969

975

982

989

996

631

80 003

010

017

024

030

037

044

051

058

065

632

072

079

085

092

099

106

113

120

127

134

633

140

147

154

161

168

175

182

188

195

202

634

209

216

223

229

236

243

250

257

264

271

635

277

284

291

298

305

312

318

325

332

339

636

346

353

359

366

373

380

387

393

400

407

7

637

414

421

428

434

441

448

455

462

468

475

638

482

489

496

502

509

516

523

530

536

543

1

0.7

639

550

557

564

570

577

584

591

598

604

611

2
3

1.4
2.1

640

618

625

632

638

645

652

659

665

672

679

4
5
6

2.8
3.5
4.2

641

686

693

699

706

713

720

726

733

740

747

642

754

760

767

774

781

787

794

801

808

814

7

4.9

643

821

828

835

841

848

855

862

868

875

882

8
9

5.6
6.3

644

889

895

902

909

916

922

929

936

943

949

645

956

963

969

976

983

990

996 *003 *010 *017

646

81 023

030

037

043

050

057

064

070

077

084

647

090

097

104

111

117

124

131

137

144

151

648

158

164

171

178

184

191

198

204

211

218

649

224

231

238

245

251

258

265

271

278

285

650

291

298

305

311

318

325

331

338

345

351

651

358

365

371

378

385

391

398

405

411

418

652

425

431

438

445

451

458

465

471

478

485

653

491

498

505

511

518

525

531

538

544

551

654

558

564

571

578^584

591

598

604

611

617

655

624

631

637

644

651

657

664

671

677

684

656

690

697

704

710

717

723

730

737

743

750

657

757

763

770

776

783

790

796

803

809

816

658

823

829

836

842

849

856

862

869

875

882

659

889

895

902

908

915

921

928

935

941

948

660

954

961

968

974

981

987

994 *000 *007 *014

N

L

I

2

3

4

5

6

7

8

9

P. P.

268

LOGARITHMS OP NUMHilJltS.

^um

. 660 to 699.

Log

. 819 to 845.

N

L

1

2

3

4

5

6

7

8

9

P. P.

660

81 954

961

968

974

981

987

994 *000 *007 *014

7

661

82 020

027

033

040

046

053

060

066

073

079

662

086

092

099

105

112

119

125

132

138

146

1

u. /

1.4

663

151

158

164

171

178

184

191

197

204

210

3

2.1

664

217

223

230

236

243

249

256

263

269

276

4
6

2.8
3.5

665

282

289

295

302

308

315

321

328

334

311

6
7
8

4.2
4.9
5.6

666

347

354

360

367

373

380

387

393

400

406

667

413

419

426

432

439

445

452

458

465

471

9

6.3

668

478

484

491

497

504

510

517

523

530

536

669

543

549

556

562

569

575

682

588

596

601

670

607

614

620

627

633

640

646

653

659

666

671

672

679

685

692

698

705

711

718

724

730

672

737

743

750

756

763

769

776

782

78?

795

673

802

808

814

821

827

834

840

847

853'

860

674

866

872

879

886

892

898

905

911

918

924

675

930

937

943

950

956

963

969

975

982

988

676

995 *001 *008 *014>

*020

*027 *033 *040 *046 *052

677

83 059

065

072

078

085^

091

097

104

110

117

6

678

123

129

136

142

149

155

161

168

174

181

679

187

193

200

206

213

219

225

232

238

215

1 ; 0.6

2l 1.2

680

251

257

264

270

276

283

289

296

302

308

3 1.8

4 2.4

5 3.0

681

315

321

327

334

340

347

353

369

366

372

682

378

385

391

398

404

4le> 417

423

429

436

6 3.6

683

442

448

455

461

467

474'

480

487

493

499

7 4.2

8 ! 4.8

9 1 5.4

684

506

512

518

625

531

637

.544

650

556

563

685

569

675

582

588

694

601

607

613

620

626

686

632

639

645

651

658

664

670

677

683

689

687

696

702

708

715

721

727

734

740

746

753

688

769

765

771

778

7Si

790

797

803

809

816

689

822

828

835

841

847

853

860

866

872

879

690

885

891

897

904

910

916

923

929

935

942

691

948

954

960

967

973

979

985

992

998 *004

692

84 Oil

017

023

029

036

042

048

055

061

067

693

073

080

086

092

098

105

111

117

123

130

694

136

142

148

155

161

167

173

180

186

192

695

198

205

211

217

223

230

236

242

218

255

696

261

267

273

280

286

292

298

305

311

317

697

323

330

336

342

348

354

361

367

373

379

698

386

392

398

404

410

417

423

429

435

442

699

448

451

460

466

473

479

485

491

497

504

700

510

516

522

528

636

541

647

653

559

566
9

N

L

1

2

3

4

S

6

7

8

P. P.

Table 1.

LOGARITHMS OF NUMBERS.

269

Num. 700 to 739. Log. 845 to 869.

85

510 516 522 528 535

572 578 584 590 597

634 640 6^16 652 658

696 702 708 714 720

757 763 770 776 782

819 825 831 837 814

880 887 893 899 905

942 948 954 960 967

003 009 016 022 028

065 071 077 083 089

126 132 138 144 150

187 193 199 205 211

248 254 260 266 272

309 315 321 327 333

370 376 382 388 394

431 437 443 449 455

491 497 503 509 516

552 558 664 570 576

612 618 625 631 637

673 679 685 691 697

733 739 745 751 757

794 800 806 812 818

854 860 866 872 878

914 920 926 932 938

974 980 986 992 998

86 034 040 046 052 058

094 100 106 112 118

153 159 165 171 177

213 219 225 231 237

273 279 285 291 297

332 338 344 350 356

392 398 404 410 415

451 457 463 469 475

510 516 522 528 534

570 576 581 587 593

629 635 611 646 652

688 694 700 705 711

747 753 759 764 770

806 812 817 823 829

864 870 876 882 888

923 929 935 941 947

L 1 2 3 4

541 547 553 559 666

603 609 615 621 628

665 671 677 6S3 689

726 733 739 715 751

788 794 800 807 813

860 856 862 868 874

911 917 924 930 936

973 979 985 991 997

034 040 046 052 058

095 101 107 114 120

166 163 169 175 ISl

217 224 230 236 242

278 285 291 297 303

339 345 362 358 364

400 406 412 418 425

461 467 473 479 485

522 528 634 540 546

582 588 594 600 600

643 649 655 661 667

703 709 715 721 727

763 769 775 781 788

824 830 836 842 848

884 890 896 902 908

944 950 956 962 968

*004 *010 *016 *022 *028

064 070 076 082 088

124 130 136 141 147

183 189 195 201 207

243 249 255 261 267

303 308 314 320 326

362 368 374 380 386

421 427 433 439 445

481 487 493 499 504

540 546 562 558 564

599 605 611 617 623

658 664 670 676 682

717 723 729 735 741

776 782 788 794 800

835 841 847 853 859

894 900 906 911 917

953 968 964 970 976

P. P.

0.6
1.2
1.8
2.4
3.0
3.6
4.2
4.8

P. P.

270

LOGARITHMS OP NUMBERS.

Table 1.

Num

. 740 to 779.

Log

. 869 to 892

N

L

(

2

3

4

S

6

7

8

9

P. P.

740

8ti 923

929

935

941

947

953

958

964

970

976

741

982

988

994

999 *005

*011 *017 *023 *029 *035

742

87 040

046

052

058

064

070

075

081

087

093

743

099

105

111

116

122

128

134

140

146

151

744

157

163

169

175

181

186

192

198

204

210

745

216

221

227

233

239

245

251

256

262

268

746

274

280

286

291

297

303

309

315

320

326

747

332

338

344

349

855

361

367

373

379

384

748

390

396

402

408

413

419

425

431

437

442

749

448

454

460

466

471

477

483

489

495

500

7S0

506

512

518

523

529

535

541

547

552

568

751

564

570

576

581

687

593

599

604

610

616

752

622

628

633

639

645

651

656

662

668

674

753

679

685

691

697

703

708

714

720

726

731

754

737

743

749

754

760

766

772

777

783

789

755

795

800

806

812

818

823

829

835

841

846

756

852

858

864

869

876

881

887

892

898

904

757

910

915

921

927

933

938

944

960

965

961

6

758

967

973

978

984

990

996 *001 *007 *013 *018

759

88 024

030

036

041

047

053

058

064

070

076

1 " '^
2

u.o

1.2

760

081

087

093

098

104

110

116

121

127

133

3

4

1.8

2.4

761

138

144

150

156

161

167

173

178

184

190

5

3.0

762

195

201

207

213

218

224

230

235

241

247

6
7
8

3.6
4.2
4.8

763

252

268

264

270

275

281

287

292

298

304

764

309

315

321

326

332

338

343

349

356

360

9

5.4

765

366

372

377

383

389

395

400

406

412

417

766

423

429

434

440

446

451

457

463

468

474

767

480

485

491

497

502

508

513

519

625

530

768

536

542

547

553

559

564

570

576

581

687

769

593

598

604

610

615

621

627

632

638

643

770

649

655

660

666

672

677

683

689

694

700

771

705

711

717

722

728

734

739

745

750

756

772

762

767

773

779

784

790

795

801

807

812

773

818

824

829

835

840

846

852

857

863

868

774

874

880

885

891

897

902

908

913

919

925

775

930

936

941

947

953

958

964

969

975

981

776

986

992

997 *003 *009

*014 *020 *025 *031 *037

777

89 042

048

053

059

064

070

076

081

087

092

778

098

104

109

115

120

126

131

137

143

148

779

154

159

165

170

176

182

187

193

198

204

780

209

215

221

226

232

237

243

248

254

260

N

L

1

2

3

4

5

6

7

8

9

P. P.

Table 1.

LOGAEITHMS OF NUMBERS.

271

Num

780 to 819.

Log

892 to 913.

N

L

1

2

3

4

S

6

7

8

9

P. P.

780

89 209

215

221

226

232

237

243

248

254

260

781

265

271

276

282

287

293

298

304

310

315

782

321

326

332

337

343

348

354

360

365

371

783

376

382

887

393

398

404

409

415

421

426

784

432

437

443

448

454

459

465

470

476

481

785

487

492

498

504

509

515

620

526

531

537

786

542

548

553

559

564

570

575

581

586

592

787

597

603

609

614

620

625

631

636

642

647

788

653

658

664

669

675

680

686

691

697

702

789

708

713

719

724

730

735

741

746

752

757

790

763

768

774

779

785

790

796

801

807

812

791

818

823

829

834

840

845

851

856

862

867

792

873

878

883

889

894

900

905

911

916

922

793

927

933

938

944

949

955

960

966

971

977

794

982

988

993

998 *004

*009 *015 *020 *026 *031

795

90 037

042

048

053

059

064

069

075

080

086

796

091

097

102

108

113

119

124

129

135

140

797

146

151

157

162

168

173

179

184

189

195

5

798

200

206

211

217

222

227

233

238

244

249

1
2

0.5
1.0

799

255

260

266

271

276

282

287

293

298

304

800

309

314

320

325

331

336

342

347

352

358

3

4

1.5
2.0

801

363

369

374

380

385

390

396

401

407

412

5

2.5

802

417

423

428

434

439

445

450

455

461

466

6
7
8

3.0
3.5
4.0

803

472

477

482

488

493

499

504

509

515

520

804

526

531

536

542

547

553

558

563

569

574

9

4.5

805

580

585

590

596

601

607

612

617

623

628

806

634

639

644

650

655

660

666

671

677

682

807

687

693

698

703

709

714

720

725

730

736

808

741

747

752

757

763

768

773

779

784

789

809

795

800

806

811

816

822

827

832

838

843

810

849

854

859

865

870

875

881

886

891

897

811

902

907

913

918

924

929

934

940

945

950

812

956

961

966

972

977

982

988

993

998 *004

813

91 009

014

020

025

030

036

041

046

052

057

814

062

068

073

078

084

089

094

100

105

110

815

116

121

126

132

137

142

148

153

158

164

816

169

174

180

185

190

196

201

206

212

217

817

222

228

233

238

243

249

254

259

265

270

818

275

281

286

291

297

302

307

312

318

323

819

328

334

339

344

350

355

360

365

371

376

820

381

387

392

397

403

408

413

418

424

429

N

L

1

2

3

4

S

6

7

8

9

P. P.

272

LOGARITHMS OF NUMBERS.

Table 1.

Num. 820 to 859. Log. 91-3 to 934.

8

P. P.

93

381 387 392 397 403

431 440 445 450 455

487 4'.I2 498 503 508

540 545 531 556 561

593 598 603 609 614

645 651 656 661 666

698 703 709 714 719

751 756 761 766 772

803 808 814 819 824

855 861 866 871 876

908 913 918 924 929

960 965 971 976 981

012 018 023 028 033

065 070 075 080 085

117 122 127 132 137

169 174 179 184 189

221 226 231 236 241

273 278 283 288 293

324 330 335 340 345

376 381 387 392 397

428 433 438 443 449

480 485 490 495 500

531 536 542 547 552

583 588 593 598 603

634 639 645 650 655

686 691 696 701 706

737 742 747 752 758

788 793 799 804 809

840 845 850 855 860

891 896 901 906 911

942 947 952 957 962

993 998 *003 *008 *013

044 049 054 059 064

095 100 105 110 115

146 151 156 161 166

197 202 207 212 217

247 252 258 263 268

298 303 308 313 318

349 354 359 364 369

399 404 409 414 420

450 455 460 465 470

408 413 418 424 429

461 466 471 477 482

514 519 524 529 535

566 572 577 582 587

619 624 630 635 640

672 677 6S2 687 693

724 730 735 740 745

777 782 787 793 798

829 834 840 845 850

882 887 892 897 903

934 939 944 950 955

986 991 997 *002 *007

038 044 049 054 059

091 096 101 106 111

143 148 153 158 163

195 200 205 210 215

247 252 257 262 267

298 304 309 314 319

350 355 361 366 371

402 407 412 418 423

454 459 464 469 474

505 511 516 521 526

557 562 567 572 578

609 614 619 624 629

660 665 670 675 681

711 716 722 727 732

763 768 773 778 783

814 819 824 829 834

865 870 875 881 886

916 921 927 932 937

967 973 978 983 988

*018 *024 *029 *034 *039

069 075 080 085 090

120 125 131 136 141

171 176 181 186 192

222 227 232 237 242

273 278 283 288 293

323 328 334 339 344

374 379 384 389 394

425 430 435 440 445

475 480 485 490 495

1

0.5

2

1.0

3

1.5

4

2.0

5

2.5

6

3.0

7

3.5

8

4.0

9

4.5

P. p.

Table 1.

LOGAEITHMS 0¥ NUMBERS.

273

Num. 860 to 899. Log. 934 to 954.

N

L

1

2

3

4

5

6

7

8

9

P. P.

860

93 450

455

460

465

470

475

480

485

490

495

861

500

505

510

515

520

526

531

536

541

546

862

551

556

561

566

571

576

581

586

591

696

863

601

606

611

616

621

626

631

636

641

646

864

651

656

661

666

671

676

682

687

692

697

865

702

707

712

717

722

727

732

737

742

747

866

752

757

762

767

772

777

782

787

792

797

867

802

807

812

817

822

827

832

837

842

847

868

852

857

862

867

872

877

882

887

892

897

869

902

907

912

917

922

927

932

937

942

947

870

952

957

962

967

972

977

982

987

992

997

871

94 002

007

012

017

022

027

032

037

042

047

872

052

057

062

067

072

077

082

086

091

096

873

101

106

111

116

121

126

131

136

141

146

874

151

156

161

166

171

176

181

186

191

196

875

201

206

211

216

221

226

231

236

240

245

876

250

255

260

265

270

275

280

285

290

295

877

300

305

310

315

320

325

330

335

340

345

S

878

349

354

359

364

369

374

379

384

389

394

1
2

0.5
1.0

879

399

404

409

414

419

424

429

433

438

443

880

448

453

458

463

468

473

478

483

488

493

3

4

1.5
2.0

881

498

503

507

512

517

522

527

532

537

542

5

2.5

882

547

552

557

562

567

671

576

581

586

591

6

7
8

3.0
3.5
4.0

883

596

601

606

611

616

621

626

630

635

640

884

645

650

655

660

665

670

675

680

685

689

9

4.6

885

694

699

704

709

714

719

724

729

734

738

886

743

748

753

758

763

768

773

778

783

787

887

792

797

802

807

812

817

822

827

832

836

888

841

846

851

856

861

866

871

876

880

885

889

890

895

900

905

910

915

919

924

929

934

890

939

944

949

954

959

963

968

973

978

983

891

988

993

998 *002 *007

*012 *017 *022 *027 *032

892

95 036

041

046

051

066

061

066

071

075

080

893

085

090

095

100

105

109

114

119

124

129

894

134

139

143

148

153

158

163

168

173

177

895

182

187

192

197

202

207

211

216

221

226

896

231

236

240

245

260

255

260

265

270

274

897

279

284

289'

294

299

303

308

313

318

323

898

328

332

337

342

347

352

357

361

366

371

899

376

381

386

390

395

400

405

410

415

419

900

424

429

434

439

444

448

463

458

463

468

N

L

1

2

3

4

5

6

7

8

9

P. P.

19

274

LOGARITHMS OF NUMBERS.

Table 1.

Num

900 to 939.

Log

954 to 973.

N

L

i

2

3

4

5

6

7

8

9

P. P.

900

95 424

429

434

439

444

448

463

458

463

468

901

472

477

4S2

487

492

497

501

506

511

616

902

521

5'26

530

535

540

645

650

654

669

564

903'

569

571

578

583

588

693

698

602

607

612

904

617

622

626

631

636

641

646

650

655

660

905

665

670

674

679

684

689

694

698

703

708

906

713

718

722

727

732

737

742

746

751

756

907

761

766

770

775

780

785

789

794

799

801

908

809

813

818

823

828

832

837

842

847

852

909

856

861

866

871

875

880

886

890

896

899

910

904

909

914

918

923

928

933

938

942

947

911

952

957

961

966

971

976

980

985

990

995

912

999 *004 *009 *014 *019

*023 *028 *033 *038 *042

913

96 047

052

057

061

066

071

076

080

085

090

914

095

099

104

109

114

118

123

128

133

137

915

142

147

152

156

161

166

171

175

180

185

916

190

194

199

204

209

213

218

223

227

232

917

237

242

246

251

256

261

265

270

275

280

S

918

284

289

294

298

303

308

313

317

322

327

1
2

0.5
1.0

919

332

336

341

346

350-

355

360

365

369

374

920

379

384

388

393

398

402

407

412

417

421

3
4

1.5
2.0

921

426

431

43ft

440

445

450

454

459

464

468

5

2.5

922

473

478

483

487

492

497

601

506

511

615

6
7
8

3.0
3.5
4.0

923

520

525

530

534

539

544

548

553

658

562

924

667

572

677

681

586

591

695

600

606

609

9

4.5

925

614

619

624

628

633

638

612

647

652

656

926

661

666

670

675

680

686

689

694

699

703

927

708

713

717

722

727

731

736

741

745

750

928

765

759

764

769

774

778

783

788

792

797

929

802

806

811

816

820

825

830

834

839

844

930

848

853

858

862

867

872

876

881

886

890

931

896

900

904

909

914

918

923

928

932

937

932

942

946

951

956

960

965

970

974

979

981

933

988

993

997 *002 *007

*011 *016 *021 *026 *030

934

97 035

039

044

049

053

058

063

067

072

077

935

081

086

090

095

100

104

109

114

118

123

936

128

132

137

142

146

151

165

160

165

169

937

174

179

183

188

192

197

202

206

211

216

938

220

225

230

234

239

243

248

263

257

262

939

267

271

276

280

285

290

294

299

304

308

940

313

Sl7

322

327

331

336

340

345

850

354

N

L

1

2

3

4

S

6

7

8

9

P.

P.

Table 1.

LOGARITHMS OF NUMBERS.

275

Num. 940 to 979. Log. 973 to 991.

P. P.

313 317 322 327 331

359 364 368 373 377

405 410 414 419 424

451 456 460 465 470

497 502 506 511 516

548 548 552 557 562

589 594 598 603 607

635 640 644 649 653

681 685 690 695 699

727 731 736 740 745

772 777 782 786 791

818 823 827 832 836

864 868 873 877 882

909 914 918 923 928

955 959 964 968 973

000 005 009 014 019

046 050 055 059 064

091 096 100 105 109

137 141 146 150 155

182 186 191 195 200

227 232 236 241 245

272 277 281 286 290

318 322 327 331 336

363 367 372 376 381

408 412 417 421 426

453 457 462 466 471

498 502 507 511 516

543 547 552 556 561

588 592 597 601 605

632 637 641 646 650

677 682 686 691 695

722 726 731 735 740

767 771 776 780 784

811 816 820 825 829

856 860 865 869 874

900 905 909 914 918

945 949 954 958 963

989 994 998 *003 *007

034 038 043 mj 052

078 083 087 092 096

123 127 131 136 140

336 340 345 350 354

382 387 391 396 400

428 433 437 442 447

474 479 483 488 493

520 525 529 534 539

566 571 575 580 585

612 617 621 626 630

658 663 667 672 676

704 708 713 717 722

749 754 759 763 768

795 800 804 809 813

841 845 850 855 859

886 891 896 900 905

932 937 941 946 950

978 982 987 991 996

023 028 032 037 041

068 073 078 082 087

114 118 123 127 132

159 164 168 173 177

204 209 214 218 223

250 254 259 263 268

295 299 304 308 313

340 345 349 354 358

385 390 394 399 403

430 435 439 444 448

475 480 484 489 493

520 525 529 534 538

565 570 574 579 583

610 614 619 623 628

655 659 664 668 673

700 704 709 713 717

744 749 753 758 762

789 793 798 ■ 802 807

834 838 843 847 851

878 883 887 892 896

923 927 932 936 941

967 972 976 981 985

*012 *016 *021 *025 *029

056 061 065 069 074

100 105 109 114 118

145 149 154 158 162

P. P.

276

LOGARITHMS OP NUMBERS.

Table 1.

N

um.

980 to 1

000.

Log

99

to 999.

N

L

1

2

3

4

5

6

7

8

9

P. P.

980

99 123

127

131

136

140

145

149

154

158

162

981

167

171

176

180

185

189

193

198

202

207

982

211

216

220

224

229

233

2:B8

242

247

251

983

255

260

261

269

273

277

282

286

291

295

984

300

304

308

313

317

322

326

330

335

339

985

344

348

352

357

361

366

370

374

379

383

986

388

392

396

401

405

410

414

419

423

427

987

432

436

411

445

449

454

458

463

467

471

988

476

480

481

489

493

498

602

506

611

516

989

520

624

528

533

537

642

646

550

555

569

4

990

564

568

572

577

581

685

590

594

599

603

1

0.4

991
992

607
651

612
656

616
660

621
664

625
669

629
673

634
677

638
682

642
686

647
691

2
3
4

0.8
1.2
1.6

993

695

699

704

708

712

717

721

726

730

734

6

2.0

994

739

743

747

752

756

760

765

769

774

778

6

7

2.4
2.8

995

782

787

791

795

800

804

808

813

817

822

8
9

3.2
3.6

995

826

830

835

839

843

848

862

856

861

865

997

870

874

878

883

887

891

896

900

904

909

998

913

917

922

926

930

935

939

944

948

952

999

957

961

965

970

974

978

983

987

991

996

1000

000 000

043

087

130

174

217

260

304

347

391

N

L

1

2

3

4

S

6

7

8

9

P. P.

Logarithms of Im

portant Numbers.

Number.

Logarithm.

TT

== 3.141 .593

0.497 150

i^

= 4.188 790

0.622 089

i^

= 0..523 599

1.718 999

1

TT

= 0.318 310

1.502 850

TT-

= 9.869 604

0.994 300

1

TT-

= 0.101 321

1.005 700

)n

« 1.772 4M

0.248 675

\n

= 0.564 190

T.751 425

r";

=- 1.464 592

0.165 717

fn

= 0.682 784

1.834 283

»/6

A'1

= 1.240 701

0.093 667

Table 2. LOGAEITHMIC ANGULAR FUNCTIONS.

277

0°

Logarithms.

179°

M.

Sine.

Cosecant.

Tangent.

Cotangent.

Secant.

Cosine.

M.

Inf. Neg.

Infinite.

Inf. Neg.

Infinite.

10.00000

10.00000

60

1

6.46373

13.53627

6.46373

13.53627

00000

00000

59

2

76476

23524

76476

23524

00000

00000

58

3

94085

05915

94085

05915

00000

00000

57

4

7.06579

12.93421

7.06579

12.93421

00000

00000

56

5

7.16270

12.83730

7.16270

12.83730

10.00000

10.00000

55

6

24188

75812

24188

75812

00000

OuOOO

54

7

30882

69118

30882

69118

00000

00000

53

8

36682

63318

36682

63.318

00000

00000

52

9

41797

58203

41797

58203

00000

00000

51

10

7.46373

12.53627

7.46373

12.53627

10.00000

10.00000

50

11

50512

4948S

50512

49488

00000

00000

49

12

54291

45709

54291

45709

■ 00000

00000

48

13

57767

42233

57767

42233

00000

00000

47

14

60985

39015

60986

39014

00000

00000

46

15

7.63982

12.36018

7.63982

12.36018

10.00000

10.00000

45

16

66784

33216

66785

33215

00000

00000

44

17

69417

30583

69418

30582

00001

9.99999

43

18

71900

28100

71900

28100

00001

99999

42

19

74248

25752

74248

25752

00001

99999

41

20

7.76475

12.23525

7.76476

12.23524

10.00001

9.99999

40

21

78594

21406

78595

21405

00001

99999

39

22

80615

19385

80615

19385

00001

99999

38

23

82545

17455

82546

17454

00001

99999

37

24

81393

15607

84394

15606

00001

99999

36

25

7.86166

12.13834

7.86167

12.13833

10.00001

9.99999

35

26

87870

12130

87871

• 12129

00001.

99999

34

27

89509

10491

89510

10490

00001

99999

33

28

91088

08912

91089

08911

00001

99999

32

29

92612

07388

92613

07387

00002

99998

31

30

7.94084

12.05916

7.94086

12.05914

10.00002

9.99998

30

31

95508

04192

95510

04490 .

00002

99998

29

32

96887

03113

96889

03111

00002

99998

28

33

98223

01777

98225

01775

00002

99998

27

34

99520

00480

99522

00478

00002

99998

26

35

8.00779

11.99221

8.00781

11.99219

10.00002

9.99998

25

36

02002

97998

02004

97996

00002

99998

24

37

03192

96808

03194

96806

00003

99997

23

38

04350

95650

04353

95647

00003

99997

22

39

05478

94522

05481

94519

00003

99997

21

40

8.06578

11.93422

8.06581

11.93419

10.00003

9.99997

20

41

07650

92350

07653

92347

00003

99997

19

42

08696

91304

08700

91300

00003

99997

18

43

09718

90282

09722

90278

00003

99997

17

44

10717

89283

10720

89280

00004

99996

16

45

8.11693

11.88307

8.11696

11.88304

10.00004

9.99996

15

40

12647

87353

12651

87349

00004

99996

14

47

13581

86419

13585

86415

00004

99996

13

48

14495

85505

14500

85500

00004

99996

12

49

15.391

84609

15395

84605

00004

99996

11

50

8.16268

11.83732

8.16273

11.83727

10.00005

9.99995

10

61

17128

82872

17133

82867

00005

99995

9

52

17971

82029

17976

82024

00005

99995

8

53

18798

81202

18804

81196

00005

99995

7

54

19610

80390

19616

80384

00005

99995

6

55

8.20407

11.79593

8.20413

11,79587

10.00006

9.99994

5

66

21189

78811

21195

78805

00006

99994

4

57

21958

78042

21964

78036

00006

99994

3

58

22713

77287

22720

77280

00006

99994

2

59

23456

76544

23462

76538

00006

99994

1

60

24186

75814

24192

75808

00007

99993

M.

Cosine.

Secant.

Cotangent

Tangent.

Cosecant.

Sine.

M.

90°

89°

278 LOGARITHMIC ANGULAR FUNCTIONS. TaUe 2.

1°

Logarithms.

178=

M.

Sine.

(!osi_'<"int.

Tangent.

Cotani^ent.

Secant.
10.00007

Co.sine.
9.99993

M.

8.24186

11.76814

8.24192

11.75808

60

1

24903

75097

24910

75090

00007

99993

59

2

25609

74391

25616

74384

00007

99993

58

3

26304

73696

26312

73688

00007

99993

57

4

26988

73012

26996

73004

00008

99992

56

5

8.27661

11.72339

8.27669

11.72331

10.00008

9.99992

55

6

28324

71676

28332

71668

00008

99992

54

7

28977

71023

28986

71014

00008

99992

53

8

29621

70379

29629

70371

00008

99992

52

9

30265

69745

30263

69737

00009

99991

51

10

8.30879

11.69121

8.30888

11.69112

10.00009

9.99991

50

11

31495

68506

31605

68498

00009

99991

49

12

32103

67897

32112

67888

00010

99990

48

13

32702

67298

32711

67289

00010

'99990

47

14

33292

66708

33302

66698

00010

99990

46

15

8.33875

11.66125

8.33886

11.66114

10.00010

9.99990

45

16

34450

65550

34461

65839

00011

99989

44

17

36018

64982

. 36029

64971

00011

99989

43

18

35678

64422

35590

64410

00011

99989

42

19

36131

63869

36143

63857

00011

99989

41

20

8.36678

11.63322

8.36689

11.63311

10.00012

9.99988

40

21

37217

62783

37229

62771

00012

99988

39

22

37760

62250

37762

62238

00012

99988

38

23

38276

61724

38289

61711

00013

99987

37

24

38796

61204

38809

61191

00013

99987

36

25

8.39310

11.60690

8.39323

11.60677

10.00013

9.99987

35

26

39818

60182

39832

60168

00014

99986

84

27

40320

59680

40334

59666

00014

99986

33

28

40816

59184

40830

69170

00014

99986

32

29

41307

58693

41321

58679

00015

99985

31

30

8.41792

11.58208

8.41807

11.58193

10.00015

9.99985

30

31

42272

57728

. 42287

67713

00015

99985

29

32

42746

57254

42762

57238

00016

99984

28

33

43216

56784

43232

56768

00016

99984

27

34

43680

56320

43696

56304

00016

99984

26

35

8.44139

11.55861

8.4415G

11.55844

10.00017

9.99983

25

36

44594

65406

44611

56389

00017

99983

24

37

45044

54956

45061

54939

00017

99983

23

38

48489

54511

46507

54493

00018

99982

22

39

45930

54070

46948

54052

00018

99982

21

40

8.46366

11.53634

8.46385

11.83616

10.00018

9.99982

20

41

46799

53201

46817

63183

00019

99981

19

42

47226

52774

47245

52755

00019

99981

18

43

47660

52350

47669

52331

00019

99981

17

44

48069

51931

48089

51911

00020

99980

16

45

8.48486

11.61515

8.48505

11.51495

10.00020

9.99980

15

46

48896

51104

48917

51083

00021

99979

14

47

49304

60696

49326

80675

00021

99979

13

48

49708

60292

49729

80271

00021

99979

12

49

60108

49892

50130

49870

00022

99978

11

50

8.60604

11.49496

8.60627

11.49473

10.00022

9.99978

10

51

50897

49103

50920

49080

00023

99977

9

62

51287

48713

51310

48690

00023

99977

8

63

51673 i

48327

51696

48304

00023

99977 -

7

64

52055

47945

52079

47921'

00024

99976

6

55

8.62434

11.47566

8.52459

11.47541

10.00024

9.99976

5

56

52810

47190

6'2835

47165

00025

99975

4

67

53183

46817

63208

46792

00026

99976

3

68

53552

46448

63578

46422

00026

99974

2

59

53919

46081

53945

46056

00026

99974

1

60

54282

46718

64308

46692

00026

99974

M.

Cosine.

Secant.

Cotangent.

Tangent

Cosecant.

Sine.

M.

91°

88="

Table 2. LOGARITHMIC ANGULAR FUNCTIONS. 279

2=

Logarithms.

77°

M.

Sine.

Cosecant.

Tangent.

Cotangent.

Recant.

Cosine.

M.

8.54282

11.45718

8.54308

11.46692

10.00026

9.99974

60

1

54642

45358

64669

45331

00027

99973

59

2

54999

45001

55027

44973

00027

99973

58

3

55354

44646

65382

44618

00028

99972

57

4

55705

44295

65734

44266

00028

99972

56

5

8.56054

11.43946

8.56083

11.43917

10.00029

9.99971

55

6

56400

43600

56429

43571

00029

99971

64

7

56743

43257

56773

43227

00030

99970

53

8

57084

42916

57114

42886

00030

99970

52

9

57421

42579

67462

42548

00031

99969

61

10

8.57757

11.42243

8.57788

11.42212

10.00031

9.99969

60

11

58089

41911

58121

41879

00032

99968

49

12

58419

41581

6S451

41549

00032

99968

48

13

58747

41253

68779

41221

00033

99967

47

14

59072

40928

59105

40895

00033

99967

46

15

8.59395

11.40605

8.59428

11.40572

10.00033

9.99967

45

16

59715

40285

59749

40251

00034

99966

44

17

60033

39967

60068

3993,2

00034

99966

43

18

60349

39651

60384

39616

00035

99965

42

19

60662

39338

60698

39302

00036

99964

41

20

8.60973

11.39027

8.61009

11.38991

10.00036

9.99964

40

21

61282

38718

61319

38681

00037

99963

39

22

61589

38411

61626

38374

00037

99963

38

23

61894

38106

61931

38069

00038

99962

37

24

62196

37804

62234

37766

00038

99962

36

25

8.62497

11.37503

8.62535

11.37466

10.00039

9.99961

35

26

62795

37205

62834

37166

00039

99961

34

27

63091

36909

63131

36869

00040

99960

33

28

63385

36615

63426

36574

00040

99960

32

29

63678

36322

63718

36282

00041

99959

31

30

8.63968

11.36032

8.64009

11.35991

10.00041

9.99959

30

31

64256

35744

64298

36702

00042

99958

29

32

64543

35457

64685

35416

00042

99958

28

33

64827

36173

64870

351.S0

00043

99957

27

34

65110

34890

65154

34846

00044

99956

26

35

8.66391

11.34609

8.65435

11.34565

10.00044

9.99966

25

36

65670

34330

65715

34286

00045

99955

24

37

65947

34053

66993

34007

00045

99955

23

38

66223

33777

66269

33731

00046

99954

22

39

66497

33603

66543

33467

00046

99954

21

40

8.66769

11.33231

8.66816

11.33184

10.00047

9.99953

20

41

67039

32961

67087

32913

00048

99952

19

42

67308

32692

67356

32644

00048

999.52

18

43

67575

32426

67624

32376

00049

99951

17

44

67841

32159

67890

32110

00049

99951

16

45

8.68104

11.31896

8.68154

11.31846

10.00050

9.99960

15

46

68367

31633

68417

31583

00051

99949

14

47

08627

31373

68678

31322

00051

99949

13

48

68886

31114

68938

31062

00062

99948

12

49

69144

30856

69196

30804

00062

99948

11

50

8.69400

11.30600

8.69453

11.30547

10.00053

9.99947

10

51

69654

30346

69708

30292

00054

99946

9

52

69907

30093

69962

30038

00054

99946

8

53

70159

29841

70214

29786

00065

99945

1

54

70409

29591

70466

29636

00056

99944

6

55

8.70658

11.29342

8.70714

11.29286

10.00056

9.99944

5

56

70905

29095

70962

29038

00057

99943

4

57

71151

28849

71208

28792

00068

99942

3

58

71395

28605

71453

28647

00058

99942

2

59

71638

28362

71697

28303

00059

99941

1

60

71880

28120

71940

28060

ooOeo

99940

M.

Cosine.

Secant.

Cotangent,

Tangent.

Cosecant.

Sine.

M.

87°

280

LOGARTTHMTC ANGULAR FUNCTIONS. Table 2.

3°

Logarithms.

176°

M.

Sine.

Cosecant.

Tangent.

Cotangent

Secant.

Ctirtine.

M.

8.71880

11.28120

8.71940

11,28060

10.00060

9.99940

60

1

72120

27880

72181

27819

00060

99940

59

2

72359

27641

72420

27580

00061

99939

58

3

72597

27403

72659

27341

00062

99938

57

4

72834

27166

72896

27104

00062

99938

66

5

8.73069

11.26931

8,73132

11,26868

10.00063

9.99937

55

6

73303

26697

73366

2B634

00064

99936

54

7

73635

26465

73600

26400

00064

99936

53

8

73767

26233

73832

26168

00066

99935

52

9

73997

26003

74063

25937

00066

99934

51

10

8.74226

11.25774

8,74292

11.25708

10.00066

9.99934

50

11

74454

25546

74521

25479

00067

99933

49

12

74680

2,5320

74748

25252

00068

99932

48

13

74906

26094

74974

26026

00068

99932

47

14

75130

24870

76199

24801

00069

99931

46

15

8.75353

11.24647

8.75423

11.24577

10,00070

9.99930

45

15

75576

24426

75645

24355

00071

99929

44

17

76796

24206

75867

24133

00071

99929

43

18

76015

23985

76087

23913

00072

99928

42

19

76234

23766

76306

23694

00073

99927

41

20

8.76451

11.23549

8.76626

11,23475

10.00074

9,99926

40

21

76667

23333

76742

23258

00074

99926

39

22

76883

23117

76958

23042

00075

99925

38

23

77097

22903

77173

22827

00076

99924

37

24

77310

22690

77387

22613

00077

99923

36

25

8.77522

11.22478

8.77600

11.22400

10,00077

9,99923

36

26

77733

22267

77811

22189

00078

99922

34

27

77943

22057

78022

21978

00079

99921

33

28

78152

21848

78232

21768

00080

99920

32

29

78360

21640

78441

215,59

00080

99920

31

30

8.78568

11,21432

8.7,8649

11,21351

10.00081

9,99919

30

31

78774

21226

78855

21145

00082

99918

29

32

78979

21021

79061

20939

00083

99917

28

33

79183

20817

79266

20734

00083

99917

27

34

79386

20614

79470

20530

00084

99916

26

35

8.79588

11.20412

8.79673

11,20327

10.00085

9,99915

25

36

79789

20211

79875

20125

00086

99914

24

37

79990

20010

80076

19924

00087

99913

23

38

80189

19811

80277

19723

00087

99913

22

39

80388

19612

80476

19524

00088

99912

21

40

8.80585

11.19415

8,80674

11.19326

10,00089

9.99911

20

41

80782

19218

80872

19128

00090

99910

19

42

80978

19022

81068

18932

00091

99909

18

43

81173

18827

81264

18736

00091

99909

17

44

81367

18633

81459

18641

00092

99908

16

45

8.81660

11.18440

8,81653

11,18347

10,00093

9.99907

16

46

81752

18248

81846

18154

00094

99906

14

47

81944

18056

82038

17962

00095

99905

13

48

82134

17866

82230

17770

00096

99904

12

49

82324

17676

82420

17580

00096

99904

11

60

8.82613

11.17487

8.82610

11.17390

10.00097

9.99903

10

61

82701

17299

82799

17201

00098

99902

9

52

82888

17112

82987

17013

00099

99901

8

53

83075

16925

83176

16825

00100

99900

7

54

83261

16739

83361

16639

00101

99899

6

56

8.83446

11.16554

8.83647

11,16453

10.00102

9.99898

5

56

83630

16370

83732

16268

00102

99898

4

57

83813

16187

83916

16084

00103

99897

3

58

83996

16004

84100

16900

00104

99896

2

59

84177

1.5823

84282

16718

00105

99895

1

60

84358

15642

84464

15636

00106

99894

mTI'

Cosine.

Secant.

Cotangent.

Tangent,

Cosecant.

Sine.

M.

86"

■Table 2. LOGARITHMIC ANGULAR FUNCTIONS.

281

4°

Logarithms.

175°

M.

Sine.

Cosecant.

Tangent.

Cotangent.

Secant.

Coeine.

M.

8.84358

11.15642

8.84464

11.15536

10.00106

9.99894

60

1

84539

15461

84646

15S54

00107

99893

59

2

84718

15282

84826

15174

00108

99892

58

3

84897

15103

86006

14994

00109

99891

57

4

85075

14925

85185

14815

00109

99891

56

5

8.85262

11.14748

8.85363

11.14637

10.00110

9.99890

55

6

85429

14571

86540

14460

00111

99889

54

7

85605

14395

85717

14283

00112

99888

63

8

85780

.., 14220

85893

14107

00113

99887

52

9

85955

■ ' 14045

86069

13931

00114

99886

51

10

8.86128

11.13872

8.86243

11.13757

10.00115

9.99885

50

H

86301

13699

86417

13583

00116

99884

49

12

86474

13526

86591

13409

00117

99883

48

13

86645

13355

86763

13237

00118

99882

47

14

86816

13184

86935

13065

00119

99881

46

15

8.86987

11.13013

8.87106

11.12894

10.00120

9.99880

45

16

87156

12844

87277

12723

00121

99879

44

17

87325

12675

87447

12553

00121

99879

43

18

87494

12506

87616

12384

00122

. 99878

42

19

87661

12339

87785

12215

00123

99877

41

20

8.87829

11.12171

8.87953

11.12047

10.00124

9.99876

40

21

87995

12005

88120

11880

00125

99875

39

22

88161

11839

88287

11713

00126

99874

33

23

88326

11674

88453

11547

00127

99873

37

24

88490

11510

88618

11382

00128

99872

36

25

8.88654

11.11346

8.88783

11.11217

10.00129

9.99871

35

26

88817

11183

88948

11052

00130

99870

34

27

88980

11020

89111

10889

00131

99869

33

28

89142

10858

89274

10726

00132

99868

32

29

89304

10696

89437

10563

00133

99867

31

30

8.89464

11.10536

8.89598

11.10402

10.00134

9.99866

30

31

89625

10375

89760

10240

00135

99865

29

32

89784

10216

89920

10080

00136

99864

28

33

89943

10057

90080

09920

00137

99863

27

34

90102

09898

90240

09760

00138

99862

26

35

8.90260

11.09740

8.90399

11.09601

10.00139

9.99861

25

36

90417

09583

90557

09443

00140

99860

24

37

90574

09426

90715

09285

00141

99859

23

38

90730

09270

90872

09128

00142

99858

22

39

90885

09115

91029

08971

00143

99857

21

40

8.91040

11.08960

8.91185

11.08815

10.00144

9.99856

20

41

91195

08805

91340

08660

00145

99855

19

42

91349

08651

91495

08505

00146

99854

18

43

91502

08498

91650

08350

00147

99853

17

44

91655

08345

91803

08197

00148

99852

16

45

8.91807

11.08193

8.91957

11.08043

10,00149

9.99861

15

46

91959

08041

92110

07890

00150

99850

14

47

98110

07890

92262

07738

00152

99848

13

48

92261

07739

92414

07586

00153

99847

12

49

92411

07589

92565

07435

00164

99846

11

50

8.92561

11.07439

8.92716

11.07284

10.00155

9.99845

10

•51

92710

07290

92866

07134

■ 00156

99844

9

52

' 92859

07141

93016

06984

00157

99843

8

53

93007

06993

93165

06835

00158

99842

7

5J

93154

06846

93313

06687

00159

99841

6

55

8.93301

11.06699

8.93462

11.06538

10.00160

9.99840

5

56

93448

06552

93609

06391

00161

99839

4

57

93594

06406

93756

06244

00162

99838

3

58

93740

06260

93903

06097

00163

99837

2

59

93885

06115

94049

05951

00164

99836

1

60

94030

05970

94195

05805

00166

99834

M.

Cosine.

Secant.

Cotangent.

Tangent.

Cosecant.

Sine.

M.

20

282 LOGARITHMIC ANGULAK FUNCTIONS. Table 3.

5°

Logarithms.

74°

M.

Sine.

Cosecant.

Tangent.

Cotangent,

Scran t.

Cosine.

M.

8.94030

11.06970

8.94195

11.05805

10.00166

9.99834

60

1

94174

06826

94340

05660

00167

99833

69

2

94317

05683

94486

06516

00168

99832

58

3

944G1

05539

94630

06370

00169

99831

57

4

94603

05397

94773

05227

00170

99830

56

5

8.94746

11.06254

8.94917

11.06083

10.00171

9.99829

66

6

94887

06113

95060

04940

00172

99828

64

7

96029

04971

96202

04798

00173

99827

63

8

95170

04830

95344

04656

00175

99825

52

9

9.5310

04690

95486

04614

00176

99824

51

10

8.95450

11.04650

8.95627

11.04373

10.00177

9.99823

50

11

95589

04411

95767

04233

00178

99822

49

12

95728

04272

95908

04092

00179

99821

48

13

95867

04133

96047

03953

00180

99820

47

14

96005

03995

96187

03813

00181

99819

46

16

8.96143

11.03857

8.96326

11.03676

10.00183

9.99817

45

16

96280

03720

96464

03536

00184

99816

44

17

96417

03583

96602

03398

00185

99815

43

18

965.53 .

03447

96739

03261

00186

99814

42

19

96689

03311

9(i»77

03123

00187

99813

41

20

8.96825

11.03175

8.97013

11.02987

10.00188

9.99812

40

21

96960

03040

97150

02850

00190

99810

39

22

97095

02905

97285

02715

00191

99809

38

23

97229

02771

97421

02679

00192

99808

37

24

97363

02637

97556

02444

00193

99807

36

25

8.97496

11.0'2501

8.97691

11.02309

10.00194

9.99806

35

26

97629

02371

97825

02175

00196

99804

34

27

97762

02238

97959

02041

00197

99803

33

28

97894

02106

98092

01908

00198

99802

32

29

98026

01974

98225

01775

00199

9^801

31

30

8.98157

11.01843

8.98358

11.01642

10.00200

9.99800

30

31

98288

01712

98490

01510

00202

99798

29

32

98419

01681

98622

01378

00203

99797

28

33

98649

01451

98753

01247

00204

99796

27

34

98679

01321

98884

01116

00205

99795

26

35

8.98808

11.01192

8.99015

11.00985

10.00207

9.99793

25

36

98937

01063

99145

00855

00208

99792

24

37

99066

00934

99275

00726

00209

99791

23

38

99194

00806

99405

00596

00210

99790

22

39

99322

00678

99534

00466

00212

99788

21

40

8.99450

11.00550

8.99662

11.00338

10.00213

9.99787

20

41

99577

00423

99791

00209

00214

99786

19

42

99704

00296

99919

00081

00215

99785

18

43

99830

00170

9.00046

10.99954

00217

99783

17

44

99966

00044

00174

99826

00218

99782

16

45

9.00082

10.99918

9.00301

10.99699

10.00219

9.99781

15

46

00207

99793

00427

99573

00220

99780

14

47

00332

99668

00553

99447

00222

99778

13

48

00456

99544

00679

99321

00223

99777

12

49

00681

99419

00805

99196

00224

99776

11

50

9.00704

10.99296

9.00930

10.99070

10.00225

9.99775

10

51

00828

99172

01055

98945

00227

99773

9

62

00951

99049

01179

98821

00228

99772

8

53

01074

98926

01303

98697

00229

99771

7

54

01196

98804

01427

98673

00231

99769

6

55

9.01318

10.98682

9.01650

10.984.50

10.00232

9.99768

5

56

01440

98560

01673

98327

00233

99767

4

57

01661

98439

01796

98204

00236

99765

3

68

01682

98318

01918

98082

00236

99764

2

69

01803

98197

02040

97960

00237

99763

1

60

01923

98077

02162

97838

00239

99761

M.

Cosine.

Secant.

Cotangent.

Tangent.

Cosecant.

Sine.

M.

84°

Table 2. LOGAEITHMIC ANGULAR FUNCTIONS.

283

6°

Logarithms.

173°

M.

Sine.

Cosecant.

Tangent.

Cotangent.

Secant.

Cosine.

M.

9.01923

10.98077

9.02162

10.97838

10.00239

9.99761

60

1

02043

97957

02283

97717

00240

99760

59

2

02163

97837

02404

97596

00241

99759

58

3

02283

97717

02525

97475

00243

99757

57

4

02402

97598

02645

97355

00244

99756

56

5

9.02520

10.97480

9.02766

10.97234

10.00248

9.99755

55

6

02639

97361

02885

97115

00247

99753

54

7

02757

97243

03005

96995

00248

99752

53

8

02874

97126

03124

96876

00249

99751

52

9

02992

97008

03242

96758

00261

99749

51

10

9.03109

10.96891

9.03361

10.96639

10.00252

9.99748

50

U

03226

96774

03479

96621

00253

99747

49

12

03342

96668

03597

96403

00285

99745

48

13

03458

96542

03714

96286

00266

99744

47

14

03674

96426

03832

96168

00268

99742

46

15

9.03690

10.96310

9.03948

10.96052

10.00259

9.99741

45

16

03805

96198

04065

95935

00260

99740

44

17

03920

96080

04181

95819

00262

99738

43

18

04034

95966

04297

96703

00263

99737

42

19

04149

95861

04413

95587

00264

99736

41

20

9.04262

10.95738

9.04528

10.95472

10.00266

9.99734

40

21

04376

96624

04643

95357

00267

99733

39

22

04490

9.5510

04768

95242

00269

99731

38

23

04603

95397

04873

95127

00270

99730

37

24

04715

9.5286

04987

95013

00272

99728

36

25

9.04828

10.95172

9.05101

10.94899

10.00273

9.99727

35

26

04940

9.5060

05214

94786

00274

99726

31

27

05052

94948

05328

94672

0)270

99724

33

28

05164

94836

05441

94559

00277

99723

32

29

05275

94725

05653 .

94447

00279

99721

31

30

9.05386

10.94614

9.05666

10.94334

10.00280

9.99720

30

31

05497

94503

05778

91222

00282

99718

29

32

05607

94393

05890

94110

00283

99717

28

33

05717

94283

06002

93998

00284

99716

27

34

05827

94173

06113

93887

00286

99714

26

35

9.05937

10.94063

9.06224

10.93776

10.00287

9.99713

25

36

06046

93954

06335

93665

00289

99711

24

37

06155

93845

06445

93565

00290

99710

23

38

06264

93736

06566

93144

00292

99708

22

39

06372

93628

06666

93334

00293

99707

21

40

9.06481

10.93519

9.06775

10.93225

10.00295

9.99705

20

41

06589

93411

06885

93115

00296

99704

19

42

06696

93304

06994

93006

00298

99702

18

43

06804

93196

07103

92897

00299

99701

17

44

06911

93089

07211

92789

00301

99699

16

45

9.07018

10.92982

9.07320

10.92680

10.00302

9.99698 •

15

46

07124

92876

07428

92572

00304

99696

14

47

07231

92769

07536

92464

00305

99695

13

48

07337

92663

07643

92357

00307

99693

12

49

07442

92568

07751

92249

00308

99692

11

50

9.07548

10.92452

9.07858

10.92142

10.00310

9.99690

10

51

07653

92347

07964

92036

00311

99689

9

52

07768

92242

08071

91929

00313

99687

8

53

07863

92137

08177

91823

00314

99686

7

54

07968

92032

08283

91717

00316

99684

6

55

9.08072

10.91928

9.08389

10.91611

10.00317

9.99683

5

66

08176

91821

08495

91505

00319

99681

4

57

08280

91720

08600

91400

00320

99680

3

58

08383

91617

08705

91295

00322

99678

2

59

08486

91514

08810

91190

00323

99677

1

60

08689

91411

08914

91086

00325

99675

M.

Cosine.

Secant.

C^5 tangent.

Tangent.

Cosecant.

Sine.

M.

83°

284 LOGARITHMIC ANGULAR FUNCTIONS. Tables.

7°

Logarithms.

72°

M.

Sine.
9.0S5S9

Cosocant.

Tangent.

Cotangent.

Recant.

Cosine.

M.

10.91411

9.08914

10.91086

10.00325

9.99675

60

1

08692

91308

09019

90981

00326

99674

69

2

08795

91205

09123

90877

00.328

99672

58

3

08897

91103

09227

90773

00330

99670

57

4

08999

91001

09330

90670

00331

99669

,56

6

9.09101

10.90899

9.09434

10.90666

10.00333

9.99667

,55

e

09202

90798

09537

90463

00334

99666

54

7

09304

90696

09610

90360

00336

99664

53

8

09105

90595

09742

90258

00337

99663

52

9

09506

90494

09845

90155

00339

99661

51

10

9.09606

10.90394

9.09947

10.900.53

10.00341

9.99659

■50

H

09707

90293

10049

89961

00342

99658

49

12

09807

90193

10150

89850

00344

99656

48

13

09907

90093

10252

89748

00345

99655

47

1-1

10006

89994

10353

89647

00347

99663

46

15

9.10106

10.89894

9.10464

10.89,546

10.00349

9.99651

45

16

10205

89795

10565

89445

00350

99650

44

17

10304

89696

10656

89344

003.52

99648

43

18

10402

89598

10756

89244

00353

99647

42

19

10501

89499

108.56

89141

00355

99646

41

20

9.10.599

10.89401

9.10966

10.89044

10.00.357

9.99643

40

21

10697

89303

11066

88944

00368

99642

39

22

10795

89205

11155

88845

00360

99640

38

23

10893

89107

11254

88746

00362

99638

37

24

10990

89010

11353

88647

00363

99637

36

25

9A1087

10.88913

9.114.52

10.88548

10.00365

9.99635

35

26

11184

88816

11.561

88449

00367

99633

34

27

11281

88719

11649

88351

00368

99632

33

28

11377

88623

11747

88253

00370

99630

32

29

11474

88526

11845

88155

00371

99629

31

30

9.11570

10.88430

9.11943

10.88067

10.00373

9.99627

30

31

11666

88334

12040

87960

00375

99625

29

32

11761

88239

12138

87862

00376

99624

28

33

11857

88143

12235

87765

00378

99622

27

34

11952

88048

12332

87668

00380

99620

26

35

9.12047

10.87953

9.12428

10.87572

10.00382

9.99618

25

36

12142

87858

1'2525

87475

00383

99617

24

37

12236

87764

12621

87379

00386

99615

23

38

12331

87669

12717

87283

00387

99613

22

39

12425

87575

12813

87187

00388

99612

21

40

9.12619

10.87481

9.12909

10.87091

10.00390

9.99610

20

41

12612

.S73SM

13004

86996

00392

99608

19

42

12706

87294

13099

86901

00393

99607

18

43

12799

87201

13194

86806

00395

99605

17

44

12892

87108

13289

86711

00397

99603

16

45

9.12985

10.87015

9.13384

10.86616

10.00399

9.99601

15

46

13078

86922

13478

86522

00400

99600

14

47

13171

86829

l;5573

86127

00402

99598

13

48

13263

86737

13667

86333

00404

99596

12

49

13355

86645

13761

86239

00405

99595

11

60

9.13447

10.86553

9.13864

10.86146

10.00407

9.99593

10

51

13539

86461

13948

86052

00409

99.591

9

52

13630

86370

11041

8.5959

00411

99.589

8

53

13722

86278

141.34

85866

00412

99.588

7

54

13813

86187

14227

85773

00414

996S6

6

55

9.13904

10.86096

9.14320

10.a5680

10.00416

9.99684

5

56

13994

86006

14412

85588

00418

99,-82

4

57

14085

85915

14504

85496

00419

99581

3

58

14175

85825

14597

86403

00421

99579

2

69

14266

8.5734

14688

8.5312

00423

99577

1

60

14356

85644

14780

85220

00425

99575

sr.

Cosine.

Secant.

Cotangent

Tangent.

Cosecant.

Sine.

M.

Table 2. LOGARITHMIC ANGULAR FUNCTIONS.

285

8°

Logarithms.

171°

M.

Sine.

Cosecant.

Tangent.

Cotangent.

Secant.

Co.siue.

M.

9.14356

10.85644

9.14780

10.86220

10.00425

9.99675

60

1

14445

85655

14872

85128

00426

99574

59

2

14535

85465

14963

85037

00428

99572

68

3

14624

85376

16054

849-16

00430

99570

,57

4

14714

85286

15145

84866

00432

99568

56

5

9.14803

10.85197

9.16236

10.84764

10.004.34

9.99566

55

6

14891

85109

16327

84673

00435

99565

64

7

14980

85020

15417

84583

00437

99563

53

8

15069

84931

16508

84492

00439

99561

52

9

15157

84813

1.5698

84402

00441

99559

51

10

9.15245

10.84765

9.15688

10.84312

10.00443

9.99657

50

11

15333

84667

15777

84223

00444

99556

49

12

15421

84579

15867

84133

00446

99554

48

13

15608

84492

15966

84044

00448

99552

47

14

15596

84404

16046

83954

004,50

99550

46

15

9.15683

10.84317

9.16136

10.83865

10.00462

9.99548

45

16

15770

84230

16224

83776

004.51

99546

44

17

15857

84143

16312

83688

00465

99545

43

18

15944

84056

16401

83599

00457

99513

42

19

16030

83970

16489

83611

00459

99641

41

20

9.16116

10.83884

9.16577

10.83423

10.00461

9.99539

40

21

16203

83797

16665

83336

00463

99537

39

22

16289

83711

16753

83217

00465

99536

38

23

16374

83626

16841

83169

00467

99633

37

24

16460

83640

16928

83072

00468

99632

36

25

9.16545

10.83466

9.17016

10.82984

10.00470

9.99530

35

26

16631

83369

17103

82897

C0472

99528

34

27

16716

83284

17190

82810

00474

99526

33

28

16801

83199

17277

82723

00476

99524

32

29

16886

83114

17.363

82637

00478

99522

31

30

9.16970

10.83030

9.17450

10.82550

10.00480

9.99520

30

31

17055

82945

17536

82464

00482

99518

29

32

17139

82861

17622

82378

00483

99617

28

33

17223

82777

17708

82292

00485

99515

27

3-t

17307

82693

17794

822C6

00487

99613

26

35

9.17391

10.82609

9.17880

10.82120

10.00489

9.99511

25

36

17474

82626

17965

82035

00491

99509

24

a^7

17558

82442

18051

81949

00493

99507

23

^8

17641

82359

18136

81864

00495

99505

22

39

17724

82276

18221

81779

00497

99503

21

40

9.17807

10.82193

9.18306

10.81694

10.00499

9.99501

20

41

17890

82110

18391

81609

00501

99499

19

42

17973

82027

18475

81525

00503

99497

18

43

18065

81945

18560

81440

00605

99495

17

44

18137

81863

186U

J1356

00506

99494

16

45

9.18220

10.81780

9.18728

10,81272

10.00.508

9.99492

15

46

18302

81698

18812

81188

00510

99490

14

47

18383

81617

18896

81104

00512

99488

13

48

18465

81535

18979

81021

00614

99486

12

49

18547

81453

19063

80937

00516

99484

11

50

9.18628

10.81372

9.19146

10.80851

10.00518

9.99482

10

51

18709

81291

19229

80771

00520

99480

9

52

18790

81210

19312

80688

00522

99478

8

53

18871

81129

19395

80605

00524

99476

7

54

18952

81048

19478

80622

00526

99474

6

55

9.19033

10.80967

9.19661

10.80439

10.00628

9.99472

5

56

19113

80887

19643

803,57

00630

99470

4

57

19193

80807

19725

80276

00532

99468

3

58

19273

80727

19807

80193

00534

99466

2

59

19353

80647

19889

80111

. 00.536

99464

1

60

19433

80567

19971

80029

00538

99462

M.

Cosine.

Secant.

Cotangent.

Tangent.

Cosecant.

Sine.

M.

81°

^86

LOGARITHMIC ANGULAR FUNCTIONS. Table 3.

9°

Logarithms.

170°

M.

Sine.

Cosecant.

Tangent.

Cotangent

Secant.

Cosine.

M.

9.19433

10.80507

9.19971

10.80029

10.00538

9.99462

60

1

19513

804S7

20053

79947

00540

99460

59

2

19592

80408

20134

79866

00542

99468

58

3

19672

80328

20216

79784

00544

99456

67

i

19751

80249

20297

79703

00546

994.54

56

5

9.19830

10.80170

9.20378

10.79622

10.00548

9.99452

65

6

19909

80091

20459

79541

00550

99450

54

7

19988

80012

20540

79460

C0552

99448

63

8

20067

79933

20621

79379

00554

99446

52

9

20145

79855

20701

79299

00666

99444

51

10

9.20223

10.79777

9.20782

10.79218

10.00558

9.99442

50

11

20302

79698

20862

79138

00560

99440

49

12

20380

79620

20912

79058

00562

99438

48

13

20458

79542

21022

78978

00564

99436

47

14

20535

79465

21102

78898

00666

99434

46

15

9.20613

10.79,387

9.21182

10.78818

10.00568

9.99432

45

16

20691

79309

21261

78739

00571

99429

44

17

20768

79232

21341

78659

00573

99427

43

18

20845

79155

21420

78580

00575

99425

42

19

20922

79078

21499

78501

00677

99423

41

20

9.20999

10.79001

9.21578

10.78422

10.00579

9.99421

40

21

21076

78924

21657

78343

00581

99419

39

22

21153

78847

21736

78264

00583

99417

38

23

21229

78771

21814

78186

00585

99415

37

24

21306

78694

21893

78107

00587

99413

36

25

9.21382

10.78618

9.21971

10.78029

10.00589

9.99411

35

26

21458

78642

22049

77951

00691

99409

34

27

21534

78466

22127

77873

00593

99407

33

28

21610

78390

22205

77795

00596

99404

32

29

21685

78315

22283

77717

00598

99402

31

30

9.21761

10.78239

9.22361

10.77639

10.00600

9.99400

30

31

21836

78164

22438

77562

00602

99398

29

32

21912

78088

22516

77484

00604

99396

28

33

21987

78013

22593

77407

00600

99394

27

34

22062

77938

22670

77330

00608

99392

26

35

9.22137

10.77863

9.22747

10.77253

10.00610

9.99390

25

36

22211

77789

22824

77176

00612

99388

24

37

22286

77714

22901

77099

00615

99385

23

38

22361

77639

22977

77023

00617

99383

39

22435

77665

23054

76946

00619

99381

21

40

9.22509

10.77491

9.23130

10.76870

10.00621

9.99379

20

41

22583

77417

23206

76794

00623

99377

19

42

22667

77343

23283

76717

00625

99375

18

43

22731

77269

23359

76641

00628

99372

17

44

22805

77195

23435

76565

00630

99370

16

45

9.22878

10.77122

9.23510

10.76490

10.00632

9.99368

15

46

22952

77048

23586

76414

00634

99366

14

47

23025

76975

23661

76339

00636

99364

13

48

23098

76902

23737

76263

00638

99362

12

49

23171

76829

23812

76188

00641

99359

11

50

9.23244

10.76756

9.23887

10.76113

10.00643

9.99357

10

51

23317

76683

23902

76038

00645

99355

9

52

23390

76610

24037

75963

00647

99353

8

53

23462

76538

24112

75888

00649

99361

7

54

23535

76465

2J1S0

75814

00652

99348

6

55

9.23607

10.76393

9.24261

10.7.5739

10.00654

9.99346

5

56

23679

76321

24335

75665

00656

99344

4

57

23752

76248

24410

75590

00658

99342

3

58

23823

76177

•2US-1

75516

00660

99340

2

59

23895

76105

24 .WS

75442

00663

99337

1

CO

23967

76833

24632

75368

00665

99335

M.

Cosine.

Seciint.

Cotangent.

Tangent.

Cosecant.

Sine.

M.

pp.

80°

Table 2. LOGARITHMIC ANGULAR FUNCTIONS.

287

10°

Logarithms.

J 69°

M,

Sine.

Cosecant.

Tangent.
9.24632

Cotangent.
10.75368

Secant.

Cosine.

M.

9.23967

10.76033

10.0066.5

9.99336

60

1

24039

75961

24706

75294

00667

99333

69

2

24110

76890

24779

76221

00669

99331

58

3

21181

75819

24853

75147

00672

99328

57

4

24253

75747

24926

76074

00674

99326

56

5

9.24324

10.75676

9.26000

10.76000

10.00676

9.99324

56

6

24395

75605

25073

74927

00678

99322

54

7

24466

7.5534

25146

74854

00681

99319

53

8

21536

75464

25219

74781

00683

99317

52

9

24607

75393

25292

74708

00685

99315

51

10

9.24677

10.75323

9.26305

10.74635

10.00687

9.99313

50

11

24748

76252

25437

74563

00690

99310

49

12

24818

76182

25510

74490

00692

99308

48

13

21888

76112

25682

74418

00694

99306

47

14

24958

75042

25665

74345

00696

99304

46

15

9.25028

10.74972

9.25727

10.74273

10.00699

9.99301

46

16

25098

74902

25799

74201

00701

99299

44

17

25168

74832

26871

74129

00703

99297

43

18

25237

74763

26943

74057

00706

99294

42

19

25307

74693

26016

73985

00708

99292

41

20

9.25376

10.74624

9.26086

10.73914

10.00710

9.99290

40

21

25445

74566

26168

73842

00712

99288

39

22

25514

74486

26229

73771

00715

99285

38

23

25583

74417

26301

73699

00717

99283

37

24

25652

74348

26372

73628

00719

99281

36

25

9.25721

10.74279

9.26443

10.73557

10.00722

9.99278

35

26

25790

74210

26514

73486

00724

99276

34

27

26858

74142

26585

73115

00726

99274

33

28

25927

74073

26655

73345

00729

99271

32

29

25995

74006

26726

73274

00731

99269

31

30

9.26063

10.73937

9.26797

10.73203

10.00733

9.99267

30

31

26131

73869

26867

731.33

00736

99264

29

32

26199

73801

26937

73063

00738

99262

28

33

26267

73733

27008

72992

00740

99260

27

34

26335

73666

27078

72922

00743

99257

26

35

9.26403

10.73597

9.27148

10.72852

10.00745

9.99255

25

36

26470

73530

27218

72782

00748

99252

24

37

26638

73462

27288

72712

00760

99250

23

38

26605

73395

27357

72643

00762

99248

22

39

26672

73328

27427

72573

00755

99245

21

40

9.26739

10.73261

9.27496

10.72504

10.00757

9.99243

20

41

26806

73194

27566

72434

00759

99241

19

42

26873

73127

27635

72365

00762

99238

18

43

26940

73060

27704

72296

00764

99236

17

44

27007

72998

27773

72227

00767

99233

16

45

9.27073

10.72927

9.27842

10.72168

10.00769

9.99231

15

46

27140

72860

27911

72089

00771

99229

14

47

27206

72794

27980

72020

00774

99226

13

48

27273

72727

28049

71951

00776

99224

12

49

27339

72661

28117

71883

00779

99221

11

50

9.27405

10.72596

9.28186

10.71814

10.00781

9.99219

10

51

27471

72529

28254

71746

00783

99217

9

52

27537

72463

28323

71677

00786

99214

8

53

27602

72398

28391

71609

00788

99212

7

64

27668

72332

28459

71641

00791

99209

6

55

9.27734

10.72266

9.28527

10.71473

10.00793

9.99207

5

56

27799

72201

28595

71405

00796

99204

4

57

27864

72136

28662

71338

00798

99202

3

58

27930

72070

28730

71270

00800

99200

2

59

27995

72005

28798

71202

00803

99197

1

60

28060

71940

28865

71135

00806

99195

M.

Cosine.

Secant.

Cotangent.

Tangent.

Cosecant.

Sine.

M.

100°

79°

288 LOGARITHMIC ANGULAR FUNCTIONS. Table 3.

11°

Logarithms.

68°

M.

Sine.

Cosecant.

Tangent.

Cotangent.

Secant.

Cosine.

M.

9.28060

10.71940

9.28865

10.711.36

10.00806

9.99195

60

1

28125

71875

28933

71067

00808

99192

69

2

28190

71810

29000

71000

00810

99190

58

3

28254

71746

29067

70933

00813

99187

57

4

28319

71681

29134

70866

00816

99185

56

5

9.28384

10.71616

9.29201

10.70799

10.00818

9.99182

55

6

28448

71652

29268

70732

00820

99180

64

7

28512

71488

29335

70665

00823

99177

53

8

28677

71423

29402

70598

00826

99175

52

9

28641

71359

29468

70532

00828

99172

51

10

9.28705

10.71295

9.29536

10.70405

10.00830

9.99170

60

n

28769

71231

29601

70399

00833

99167

49

12

28833

71167

29668

70332

00835

99165

48

13

28896

71104

29734

70266

00838

99162

47

14

28960

71040

29800

70200

00840

99160

46

15

9.29024

10.70976

9.29866

10.70134

10.00843

9.99157

45

16

29087

70913

29932

70068

00845

99156

44

17

29150

70850

29998

70002

00848

99152

43

18

29214

70786

30064

69936

00850

99160

42

19

• 29277

70723

30130

69870

00853

99147

41

20

9.29340

10.70660

9.30196

10.69805

10.00856

9.99145

40

21

29403

70597

30261

69739

00858

99142

39

22

29466

70534

30326

69674

00860

99140

38

23

29529

70471

30391

69609

00863

99137

37

24

29591

70409

30457

69643

00865

99135

36

25

9.29654

10.70346

9.30522

10.69478

10.00868

9.99132

35

26

29716

70284

30587

6941.3

00870

99130

34

27

29779

70221

30&52

69348

00873

99127

33

28

29841

70169

30717

69283

00876

99124

32

29

29903

70097

30782

69218

00878

99122

31

80

9.29966

10.70034

9.30846

10.69154

10.00881

9.99119

30

31

30028

69972

30911

69089

00883

99117

29

32

30090

69910

30976

69026

00886

99114

28

33

30151

69849

31040

68960

00888

99112

27

34

30213

69787

31104

68896

00891

99109

28

35

9.30275

10.6972.5

9.31168

10.68832

10.00894

9.99106

25

36

30336

69664

31233

68767

00896

99104

24

37

30398

69602

31297

68703

00899

99101

23

38

30459

69541

31361

68639

00901

99099

22

39

30621

69479

31425

68675

00904

99096

21

40

9.30582

10.69418

9.31489

10.68511

10.00907

9.99093

20

41

30643

69357

31552

68448

00909

99091

19

42

30704

69296

31616

CS3M4

00912

99088

18

43

30766

69235

31679

68321

00914

99086

17

44

30826

69174

31743

68267

00917

99083

16

45

9.30887

10.69113

9.31806

10.68194

10.00920

9.99080

15

46

30947

69053

31870

68130

00922

99078

14

47

31008

68992

31933

68067

00925

99076

13

48

31068

68932

31996

68004

00928

99072

12

49

31129

68871

32059

679J1

00930

99070

11

50

9.31189

10.68811

9.32122

10.67878

10.00933

9.99067

10

51

31250

68750

32185

67815

00936

99064

9

52

31310

68690

32248

67752

00938

99062

8

53

31370

68630

32311

67689

00941

99059

7

54

31430

68570

32373

67627

00944

99056

6

65

9.31490

10.68510

9.32436

10.67564

10.00946

9.99064

5

56

31649

68451

32498

67502

00949

99061

4

57

31609

68391

32561

67439

00952

99048

3

58

31669

68331

32623

67377

00954

99046

2

59

31728

68272

32685

67315

00957

99043

1

60

31788

68212

32747

67253

00960

99040

M.

Coeiiie.

Secant.

Cotangent,

Tangent.

Cosecant.

Sine.

M.

10i°

78°

Table 2. LOGARITHMIC ANGULAR FUNCTIONS. 289

12°

Logarithms.

167°

M-.

Sine.

CofMicant.

Tangent.

Cotangent

Secant.

Cosine.

M.

9.31788

10.68212

9.32747

10.67253

10.00960

9.99040

60

1

31847

68153

32810

67190

00962

99038

59

2

31907

68093

32872

67128

00965

99035

58

3

31966

68034

32933

67067

00968

99032

57

4

32025

67975

32995

67005

00970

99030

56

5

9.32084

10.67916

9.33057

10.66943

10.00973

9.99027

56

6

32143

67857

83119

66881

00976

99024

54

7

32202

67798

33180

66820

00978

99022

53

8

32261

67739

33242

66758

00981

99019

52

9

32319

67681

38303

66697

00984

99016

51

10

9.32378

10.67622

9.33365

10.66635

10.00987

9.99013

50

11

32437

67563

83426

66574

00989

99011

49

12

32495

67505

33487

66513

00992

99008

48

13

32553

67447

33548

66452

00995

99005

47

14

32612

67388

33609

66391

00998

99002

46

15

9.32670

10.67330

9.33670

10.66330

10.01000

9.99000

45

16

32728

67272

33731

66269

01003

98997

44

17

32786

67214

33792

66208

01006

98994

43

18

32844

67156

33853

66147

01009

98991

42

19

32902

67098

33913

66087

01011

98989

41

20

9.32960

10.67040

9.33974

10.66026

10.01014

9.98986

40

21

83018

66982

34034

65966

01017

98983

89

22

33075

66925

84095

65905

01020

98980

38

23

33133

66867

341,55

65845

01022

98978

37

24

33190

66810

34215

65785

01025

98975

86

25

9.33248

10.667.52

9.34276

10.65724

10.01028

9.98972

35

26

33305

66695

34336

65664

01031

98969

34

27

33362

66638

34.396

65604

01083

98967

38

28

33420

66580

34456

65544

01036

98964

32

29

33477

66523

34516

65484

01039

98961

31

3D

9.33534

10.66466

9.34576

10.65424

10.01042

9.98958

30

31

33.591

66409

34635

65365

01045

98955

29

32

33647

66353

34695

65305

01047

98953

28

33

33704

66296

34755

6.5245

01060

98950

27

34

33761

66239

34814

65186

01063

98947

26

35

9.33818

10.66182

9.34874

10.65126

10.01056

9.98944

25

36

33874

66126

34933

65067

01059

98941

24

37

339.31

66069

34992

66008

01062

98938

23

38

33987

66013

35051

64949

01064

98936

22

39

34043

65957

35111

64889

01067

98933

21

40

9.34100

10.65900

9.35170

10.64830

10.01070

9.98980

20

41

34156

65844

35229

64771

01073

98927

19

42

34212

65788

35288

64712

01076

98924

18

43

34268

65732

35347

64663

01079

98921

17

44

34324

6.5876

35405

64696

01081

98919

16

45

9.34380

10.65620

9.35464

10.64536

10.01084

9.98916

15

46

34436

65564.

35523

64477

01087

98913

14

47

34491

65509

35581

64419

0109U

98910

13

48

34547

65453

,35640

64360

01093

98907

12

49

34602

65398

3.5698

64302

01096

98904

11

50

9.34658

10.65342

9.35757

10.64243

10.01099

9.98901

10

51

34713

65287

35815

64185

01102

98898

9

52

34769

65231

35873

64127

01104

98896

8

53

34824

66176

35931

64069

01107

98893

7

54

34879

65121

35989

64011

OHIO

98890

6

55

9.34934

10.65066

9.36047

10.63953

10.01113

9.98887

5

56

34989

65011

36105

63895

01116

98884

4

57

35044

64956

36163

63837

01119

98881

3

58

35099

64901

36221

63779

01122

98878

2

59

351.54

64846

36279

63721

01125

98875

1

60

35209

64791

36336

63664

01128

98872

M.

Cosine.

Secant.

Cotangent

Tangent.

Cosecant.

Sine.

M.

102°

77°

290 LOGARITHMIC ANGULAR FUNCTIONS. Table 3.

13°

Logarithms.

166°

M,

Sine.

Cosecant.

Tangent.

Cotangent,

Secant.

Cosine.

M,

9.35209

10.64791

9.36336

10.63664

10.01128

9.98872

60

1

35263

64737

36394

63606

01131

98869

69

2

35318

64682

36452

63548

01133

98867

68

3

35373

64627

36509

63491

01136

98864

57

4

36427

64573

36566

63434

01139

98861

66

5

9.36481

10.64519

9.36624

10.63376

10.01142

9.98858

55

6

36536

64464

36681

63319

01115

98855

54

7

36590

61410

36738

63262

01148

98852

63

8

35644

04356

36795

63205

01161

98849

52

9

35698

61302

36852

63148

01154

98846

51

10

9.35752

10.64248

9.36909

10.63091

10,01157

9.98843

50

11

35806

64194

36966

63034

01160

98840

49

12

35860

64140

37023

62977

01163

98837

48

13

35914

64086

.37080

62920

01166

98834

47

14

35968

64032

37137

62863

01169

98831

46

15

9.36022

10.63978

9.37193

10.62807

10,01172

9.98828

45

16

36075

03925

37250

62750

01175

98825

44

17

30129

63871

37306

62694

01178

98822

43

18

36182

63818

37363

62637

01181

98819

42

19

36236

63764

37419

62581

01184

98816

41

20

9.36289

10.63711

9.37476

10.6'2521

10.01187

9.98813

40

21

36342

63658

37532

62468

01190

'98810

39

22

36395

63606

37688

62412

01193

98807

38

23

36449

63551

37644

62356

01196

98804

37

24

36502

63498

37700

62300

01199

98801

36

26

9.36555

10.63445

9.37766

10.62214

10.01202

9.98798

36

26

36608

63392

37812

62188

01205

98795

34

27

36660

63340

37868

62132

01208

98792

33

28

36713

63287

37924

62076

01211

98789

32

29

36766

63234

.37980

62020

01214

98786

31

30

9.36819

10.63181

9.38035

10.61965

10.01217

9.98783

3D

81

36871

63129

38091

61909

01220

98780

29

32

36924

63076

38147

61853

01223

98777

28

33

36976

63024

38202

61798

01226

98774

27

34

37028

62972

38257

61743

01229

98771

26

35

9.37081

10.62919

9.38313

10.61687

10.01232

9.98768

25

36

37133

62867

38368

01632

01235

98765

24

37

37185

62815

38423

61677

01238

98762

23

38

37237

62763

38479

61521

01241

98759

22

39

37289

62711

38534

61466

01244

98756

21

40

9.37341

10.62659

9.38589

10.61411

10.01247

?, 98753

■20

41

37393

62607

38644

61356

01250

98760

19

42

37445

6-2555

38699

61301

01254

98746

18

43

37497

6'2603

38754

61246

01257

98743

17

44

37549

6'2451

38808

61192

01260

98740

16

45

9,37600

10.62400

9.38863

10.61137

10.01263

9,98737

15

46

37652

62348

38918

61082

01266

98734

14

47

37703

62297

38972

61028

01269

98731

13

48

37755

62245

39027

60973

01272

98728

12

49

37806

62194

39082

60918

01275

98725

11

60

9.37858

10.62142

9.39136

10,60864

10.01278

9,98722

10

51

37909

62091

39190

60810

01281

98719

9

62

37960

62040

39245

607.55

01285

98715

8

63

38011

61989

39299

60701

01288

98712

7

54

38062

61938

39353

60647

01291

98709

6

55

9.38113

10.61887

9.39407

10.60593

10.01294

9,98706

5

56

38164

61836

39461

60.539

01297

98703

4

57

38215

61785

39516

60485

01300

98700

3

58

38266

61734

39569

60431

01303

98697

2

69

38317

61683

39623

60377

01306

98694

1

60

38368

61632

39677

60323

01310

98690

mT

CoBiue.

Secant.

Cotangent.

Tangent.

Cosecant.

Sine,

M.

103°

76°

Table 2. LOGARITHMIC ANGULAR FUNCTIONS. 291

J4°

Logarithms.

165°

M.

Sine.

Cosecant.

Tangent.

Cotangent,

Secant.

Cosine.

M.

9.38368

10.61632

9.39677

10.60323

10.01310

9.98690

60

1

38418

61582

39731

60269

01313

98687

59

2

38469

01531

39785

60215

01316

98684

58

3

38519

61481

39838

60162

01319

■98681

57

4

38570

61430

39892

60108

01322

98678

56

5

9.38620

10.61380

9.39945

10.60055

10.01325

9.98675

55

6

38670

61330

39999

60001

01329

98671

54

7

38721

61279

40052

59948

01332

98668

53

8

38771

61229

40106

59894

01335

98665

52

9

38821

61179

40159

59841

01338

98662

51

10

9.;^8871

10.61129

9.40212

10.59788

10.01341

9.98659

50

11

38921

61079

40266

59734

01344

98656

49

12

38971

61029

40319

59681

01348

98652

48

13

39021

60979

40372

59628

01351

98649

47

H

39071

60929

40425

59575

01354

98646

46

15

9.39121

10.60879

9.40478

10.59522

10.01357

9.98643

45

16

39170

60830

40531

59469

01360

98640

44

17

39220

60780

40584

59416

01864

98636

43

18

39270

60730

40636

59364

01367

986.33

42

19

39319

60681

40689

59311

01370

98630

41

20

9.39369

10.60631

9.40742

10.59258

10.01373

9.98627

40

21

39118

60582

40795

59205

01377

98623

39

22

39467

60533

40847

59163

01380

98620

38

23

39517

60483

40900

59100

01383

98617

37

24

39566

60434

40952

59048

01386

98614

36

25

9.39615

10.60385

9.41005

10.58995

10.01390

9.98610

35

26

39664

60336

41057

58943

01393

98607

34

27

39713

60287

41109

58891

01396

98604

33

28

39762

60238

41161

58839

01399

98601

32

29

39811

60189

41214

58786

01403

98597

31

30

9.39860

10.60140

9.41266

10.58734

10.01406

9.98594

30

31

39909

60091

41318

58682

01409

98591

29

32

39958

60042

41370

58630

01412

98588

28

33

40006

59994

41422

58578

01416

98584

27

34

40055

59945

41474

58526

01419

98581

26

35

9.40103

10.59897

9.41526

10.58474

10.01422

9.98578

25

36

40152

59848

41578

58422

01426

98574

24

37

40200

59800

41629

58371

01429

98571

23

38

40249

59751

41681

58319

01432

98568

22

39

40297

59703

41733

58267

01435

98565

21

40

9.40346

10.59654

9.41784

10.58216

10.01439

9.98561

20

41

40394

59606

41836

58164

01442

98558

19

42

40442

59658

41887

58113

01445

98555

18

43

40490

59510

41939

58061

01449

98551

17

44

40538

59462

41990

58010

01452

98548

16

45

9.40586

10.59414

9.42041

10.57959

10.01455

9.98545

15

46

40634

59366

42093

57907

01459

98541

14

47

40682

59318

42144

57856

01462

98538

13

48

40730

59270

42195

57805*

01465

98536

12

49

40778

59222

42246

57754

01469

98531

11

50

9.40825

10.59175

9.42297

10.57703

10.01472

9.98528

10

51

40873

59127

42348

57652

01475

98525

9

52

40921

59079

42399

57601

01479

98521 ■•

8

53

40968

59032

42450

57550

01482

98518

7

54

41016

58984

42501

57499

01485

98515

6

55

9.41063

10.58937

9.42552

10.57448

10.01489

9.98511

5

56

41111

58889

42603

67397

01492

98508

4

57

41158

58842

42653

57347

01495

98505

3

58

41205

58795

42704

57296

01499

98501

2

59

41252

58748

42755

57245

01502

98498

1

60

41300

58700

42805

57195

01506

98494

M.

Cosine.

Secant.

Cotangent.

Tangent.

Cosecant.

Sine.

M.

104°

7S°

292 LOGARITHMIC ANGULAR FUNCTIONS. Table 2.

IS':

Logarithms.

J 64°

M.

Sine.

Cosecant.

Tangent.

Cotangent

1 Secant.

Cosine.

M.

9.41300

10.58700

9.42805

10.57195

10.01506

9.98494

60

1

41347

58653

42855

57144

01509

98491

59

2

41394

58606

42906

57094

01512

98488

58

3

41441

58559

42957

67043

01516

98484

57

4

41488

58512

43007

56993

01619

98481

56

6

9.41535

10.58465

9.43057

10.56943

10.01523

9.98477

55

6

41,=.82

.584 1,S

43108

56892

01526

98474

.54

7

41(iiS

.58372

43158

56842

01529

98471

63

8

JlC.To

58325

43208

56792

01533

98467

52

9

41722

58278

43268

56742

01536

98464

51

10

9.417I-.S

10.58232

9.43308

10.56692

10.01540

9.98460

60

11

41815

.58185

43.3.58

56642

01543

98457

49

12

41861

58139

43408

66592

01547

98463

48

13

41908

.58092

43458

56542

01650

98450

47

14

419.54

58046

43508

56492

01553

98447

46

15

9.42001

10.57999

9.43.558

10.56442

10.01557

9.98443

45

16

42047

57953

43607

56393

01560

98440

44

17

42093

57907

43657

56343

01564

98436

43

18

42140

57860

43707

56293

01667

98433

42

19

42186

57814

43756

66244

01571

98429

41

20

9.42232

10..57768

9.43806

10.56194

10.01674

9.98426

40

21

42278

57722

43855

66145

01578

98422

39

22

42324

57676

43906

56095

01581

98419

38

23

42370

57630

43964

56046

01586

98415

37

24

42416

57584

44004

55996

01688

98412

36

25

9.42461

10.57539

9.44063

'10.56947

10.01691

9.98409

35

26

42507

57493

44102

,55898

01.595

98405

34

27

42553

57447

44151

55849

01598

98402

33

28

42599

67401

44201

65799

01602

98398

32

29

42644

57356

44250

55760

01605

98396

31

30

9.42690

10.57310

9.44299

10..55701

10.01609

9.98391

30

31

42735

57265

44348

55652

01612

98388

29

32

42781

57219

44397

66603

01616

98384

28

33

42826

57174

44446

.55654

01619

98381

27

34

42872

57128

44495

56505

01623

98377

26

35

9.42917

10.57083

9.44.544

10.56456

10.01627

9.98373

25

3G

42962

57038

44592

55408

01630

98370

24

37

43008

56992

44641

66359

01634

98366

23

38

43053

56947

44690

55310

016.37

98363

22

39

43098

56902

44738

6.5262

01641

98369

21

40

9.43143

10.56857

9.44787

10.55213

10.01644

9.98356

20

41

43188

56812

44836

55164

01648

98352

19

42

43233

56767

44884

55116

01661

98349

18

43

43278

56722

44933

55067

01655

98345

17

44

43323

56677

44981

55019

01658

98342

16

45

9.43367

10.56033

9.45029

10.54971

10.01662

9.98338

15

46

43412

56.588

45078

54922

01666

98334

14

47

43457

66543

45126

54874

01669

98831

13

48

43502

66498

46174

54826

01673

98327

12

49

43546

564,54

45222

54778

01676

98324

11

60

9.43591

10.56409

9.46271

10.54729

10.01680

9.98320

10

51

43635

66365

45319

54681

01683

98317

9

52

43680

56320

45367

54633

01687

98313

8

53

43724

56276

45415

54.585

01691

98309

7

64

43769

56231

45463

54537

01694

98306

6

55

9.43813

10.56187

9.45511

10.,54489

10.01698

9.98302

5

56

43857

56143

45559

64441

01701

98299

4

57

43901

56099

45606

54394

01705

98295

3

58

43946

56054

45654

54346

01709

98291

2

59

43990

56010

4.5702

54298

01712

98288

1

60

44034

55966

45750

54250

01716

98284

M.

Cosine.

Secant.

Cotangent.

Tangent.

Cosecant.

Sine.

M.

105°

74°

Table 2. LOGARITHMIC ANGULAR FUNCTIONS.

293

16°

Logarithms.

163°

M.

Sine.

Cosecant.

Tangent.

Cotangent.!

Secant.

Ci>.''Ule.

M.

9.44034

10.56966

9.45750

10.54250

10.01716

9.98284

60

1

44078

55922

45797

54203

01719

982S1

59

2

44122

55878

45845

541.55

01723

98277

68

3

44166

55834

45892

54108

01727

98273

57

4

44210

55790

46940

54060

01730

98270

56

5

9.44253

10.55747

9.45987

10.54013

10.01734

9.98266

55

6

44297

55703

46035

53965

01738

98262

54

7

44341

55659

46082

53918

01741

98269

53

8

44385

55615

46130

53870

01745

982.65

52

9

44428

55572

46177

53823

0]74'9

98251

51

10

9.44472

10.55528

9.46224

10.53776

10.01752

9.98248

50

11

44516

55484

46271

53729

01756

98244

49

12

44559

55441

46319

53681

01760

98240

48

13

44602

55398

46366

63634

01763

98237

47

14

44646

55354

46413

5.3587

01767

98233

46

15

9.44689

10.55311

9.46460

10.5.3540

10.01771

9.98229

45

16

44733

55267

46507

63493

01774

98226

44

17

44776

55224

46554

53446

01778

98222

43

18

44819

55181

46601

53399

01782

98218

42

19

44862

55138

46648

53352

01785

98215

41

20

9.44905

10.55095

9.46694

10.53306

10.01789

9.98211

40

21

44948

55052

46741

53259

01793

98207

39

2^^

44992

55008

46788

63212

01796

98204

38

23

45035

54965

46835

63165

01800

98200

37

24

45077

54923

46881

53119

01804

98196

36

25

9.45120

10.54880

9.46928

10.53072

10.01808

9.98192

35

26

45163

54837

46975

53025

01811

98189

34

27

45206

54794

47021

52979

01815

98185

33

28

45249

54751

47068

62932

01819

98181

32

29

45292

54708

47114

52886

01823

98177

31

30

9.45334

10.54666

9.47160

10.52840

10.01826

9.98174

30

31

45377

54623

47207

52793

01830

98170

29

32

45419

54581

47263

52747

01834

98166

28

33

45462

54538

47299

52701

01838

98162

27

34

45504

54496

47346

52654

01841

98159

26

35

9.45547

10.54453

9.47392

10.52608

10.01845

9.98165

25

36

45589

54411

47438

52562

01849

98151

24

37

45632

54368

47484

52516

01853

98147

23

38

45674

54326

47530

52470

01856

98144

22

39

45716

54284

47576

62424

01860

98140

21

40

9.45758

10.54242

9.47622

10.52378

10.01864

9.98136

20

41

45801

54199

47668

52332

01868

98132

19

42

45843

54157

47714

52286

01871

98129

18

43

45885

64115

47760

62240

01875

98125

17

44

45927

54073

47806

52194

01879

98121

16

45

9.45969

10.54031

9.47852

10:52148

10.01883

9.98117

15

46

46011

53989

47897

52103

01887

98113

U

47

46053

53947

47943

52057

01890

98110

13

48

46095

53905

47989

52011

01894

98106

12

49

46136

53864

48035 ■

51965

03898

98102

11

50

9.46178

10.53822

9.48080

10.51920

10.01902

9.98098

10

51

46220

53780

48126

61874

01906

98094

9

52

46262

53738

48171

51829

01910

98090

8

53

46303

53697

48217

51783

01913

98087

7

54

46345

53655

48262

61738

01917

98083

6

55

9.46386

10.53614

9.48307

10.51693

10.01921

9.98079

5

56

46428

53672

48353

51647

01925

98075

4

57

46469

53531

48398

51602

01929

98071

3

58

46511

53489

48443

51557

01933

98067

2

59

46552

53448

48489

51511

01937

98063

1

60

46594

53406

48534

51466

01940

98060

M.

Cosine.

Secant.

Cotangent.

Tangent.

Cosecant.

Sine.

M.

106°

73°

294

LOGARITHMIC ANGULAR FUNCTIONS. Table 2.

Logarithms.

162°

M.

Sine.

Coaecant..

Tangent.

Cotangent.

Secant.

Cosine.

M.

9.46594

10.53406

9.48534

10.51466

10.01940

9.98060

60

1

46635

53365

48579

51421

01944

98056

59

2

46676

53.324

48624

51376

01948

98052

68

3

46717

53283

48669

61331

01952

98048

57

4

46758

53242

48714

61286

01956

98044

56

5

9.41K00

10.53200

9.48759

10.51241

10.01960

9.98040

55

6

iimi

53159

48804

51196

01964

98036

54

7

46,H82

53118

48849

51151

01968

98032

63

8

46923

53077

48894

51106

01971

98029

52

9

46964

53036

48939

51061

01975

98025

51

10

9.47005

10.52995

9.48984

10.61016

10.01979

9.98021

50

11

47045

52955

49029

60971

01983

98017

49

12

47086

52914

49073

50927

01987

98013

48

13

47127

52873

49118

50882

01991

98009

47

14

47168

5'2832

49163

60,837

01996

98005

46

15

9.47209

10.52791

9.49207

10.60793

10.01999

9.98001

45

16

47249

52751

49252

50748

02003

97997

44

17

47290

52710

49296

50704

02007

97993

43

18

47330

52670

49341

50659

02011

97989

42

19

47371

52629

49385

60615

02014

97986

41

20

9.47411

10.52589

9.49430

10.50570

10.02018

9.97982

40

21

47452

52548

49474

50626

02022

97978

39

22

47492

52508

49.519

60481

02026

97974

38

23

47533

52467

49563

50437

02030

97970

37

24

47573

52427

49607

50393

02034

97966

36

25

9.47613

10.52387

9.49652

10..50348

10.02038

9.97962

35

26

47654

62346

49696

60304

02042

97958

34

27

47694

52306

49740

50260

02046

97964

33

28

47734

52266

49784

50216

02050

97960

32

29

47774

52226

49828

50172

02054

97946

31

30

9.47814

10.52186

9.49872

10..50128

10.02058

9.97942

30

31

47854

52146

49916

50084

02062

97938

29

32

47894

52106

49960

50040

02066

97934

28

33

47934

52066

50004

49996

02070

97930

27

34

47974

52026

50048

49952

02074

97926

26

35

9.48014

10.51986

9.50092

10.49908

10.02078

9.97922

25

36

48054

51946

50136

49864

02082

97918

24

37

48094

51906

50180

49820

02086

97914

23

38

48133

51867

50223

49777

02090

97910

22

39

48173

51827

50267

49733

02094

97906

21

40

9.48213

10.51787

9.50311

10.49689

10.02098

9.97902

20

41

48252

51748

50355

49645

02102

97898

19

42

48292

51708

50398

49602

02106

97894

18

43

48332

51668

50442

49558

02110

97890

17

44

48371

51629

50485

49515

02114

97886

16

45

9.48411

10.51589

9.50529

10.49471

10.02118

9.97882

15

46

48450

51550

,50572

49428

02122

97878

14

47

48490

51510

50616

49384

02126

97874

13

48

48529

51471

60659

49341

02130

97870

12

49

48568

51432

50703

49297

02134

97866

11

50

9.48607

10.51393

9.60746

10.49254

10.02139

9.97861

10

51

48647

51353

60789

49211

02148

97857

9

52

48686

51314

60833

49167

02147

97853

8

53

48725

51275

50876

49124

02151

97849

7

54

48764

51236

50919

49081

02155

97845

6

55

9.48803

10.51197

9.50962

10.49038

10.021,59

9.97841

5

56

48842

51158

51006

48995

02163

97837

4

57

48881

51119

,51048

48962

02167

97833

3

58

48920

51080

51092

48908

02171

97829

2

59

48959

51041

51136

48865

02175

97826

1

60

48938

51002

51178

48822

02179

97821

M.

Cosine.

Secant.

Cotangent.

Tangent.

Cosecant.

Sine.

M.

107°

Table 2. LOGARITHMIC ANGULAR FUNCTIONS.

295

18°

Logarithms.

161°

M.

Sine.

CoBGcant.

Tangent.

Cotangent.

Secant.

Cosine.

M.

9.48998

10.51002

9.51178

10.48822

10.02179

9.97821

60

1

49037

50963

51221

48779

02183

97817

59

2

49076

50924

51264

48736

02188

97812

68

3

49115

60885

51306

48694

02192

97808

57

4

49153

50847

51349

48651

02196

• 97804

56

5

9.49192

10.50808

9.51392

10.48608

10.02200

9.97800

56

6

49231

50769

51435

48565

02204

97796

54

7

49269

50731

51478

48522

02208

97792

53

8

49308

50692

51620

48480

02212

97788

52

9

49347

50653

51563

48437

02216

97784

51

10

9.49385

10..50615

9.51606

10.48394

10.02221

9.97779

50

11

49424

50576

51648

48352

02226

97775

49

12

49462

50538

61691

48309

02229

97771

48

13

49500

50500

51734

48266

02233

97767

47

14

49539

50461

61776

48224

02237

97763

46

15

9.49577

10.50423

9.51819

10.48181

10.02241

9.97769

45

16

49615

50385

51861

48139

02246

97754

44

17

49654

50346

61903

48097

02250

97750

43

18

49692

60308

61946

48064

02254

97746

42

19

49730

50270

51988

48012

02258

97742

41

20

9.49768

10.60232

9.62031

10.47969

10.02262

9.97738

40

21

49806

50194

62073

47927

02266

97734

39

22

49844

50156

52115

47886

02271

97729

38

23

49882

50118

52157

47843

02275

97725

37

24

49920

50080

62200

47800

02279

97721

36

25

9.49958

10.50042

9.52242

10.47758

10.02283

9.97717

35

26

49996

50004

52284

47716

02287

97713

34

27

50034

49966

52326

47674

02292

97708

33

28

50072

49928

52368

47632

02296

97704

32

29

50110

49890

52410

47590

02300

97700

31

30

9.50148

10.49852

9.52452

10.47548

10.02304

9.97696

30

31

50185

49815

52494

47506

02309

97691

29

32

50223

49777

52536

47464

02313

97687

28

83

50261

49739

52678

47422

02317

97683

27

34

50298

49702

52620

47380

02321

97679

26

35

9.50336

10.49664

9.62661

10.47339

10.02326

9.97674

25

86

50374

49626

52703

47297

02330

97670

24

37

50411

49589

52745

47255

02334

97666

23

38

50449

49551

52787

47213

02338

97662

22

39

50486

49514

62829

47171

02343

97657

21

40

9.50523

10.49477

9.62870

10.47130

10.02347

9.97663

20

41

50561

49439

52912

47088

02351

97649

19

42

50598

49402

52953

47047

02355

97645

18

43

50635

49365

52995

47005

02:360

97640

17

44

50673

49327

53037

46963

02364

97636

16

45

9.50710

10.49290

9.53078

10.46922

10.02368

9,97632

16

46

50747

49253

53120

46880

02372

97628

14

47

50784

49216

53161

46839

02377

97623

IS

48

50821

49179

63202

46798

02381

97619

12

49

50858

49142

53244

46756

02385

97616

11

50

9.50896

10.49104

9.53285

10.46715

10.02390

9.97610

10

51

50933

49067

53327

46673

02394

97606

9

52

50970

49030

53368

46632

02398

97602

8

53

51007

48993

53409

46591

02403

97697

7

54

51043

48957

53450

46560

02407

97593

6

55

9.51080

10.48920

9.63492

10.46508

10.02411

9.97689

5

56

51117

48883

53633

46467

02416

97684

4

57

51154

48846

53574

46426

02420

97580

3

58

51191

48809

53615

46385

02424

97576

2

59

51227

48773

53666

46344

02429

97571

1

60

51264

48736

53697

46303

02433

97567

M.

Cosine.

Secant.

Cotangent.

Tangent. |

Cosecant.

Sine.

M.

71°

296

LOGAEITHMIC ANGULAR FUNCTIONS. Table 2.

19°

Logar

thms.

Si'cant.
10.024:53

60°

M.

Sine.

CnSCCilllt. i

10.48730

Tiingent.
9.,53697

Cotangent.!
10.46303

Cosine.
9,97567

M.

9.51264

60

1

51301

48699

53738

46262

024:17

97563

59

2

51338

48602

53779

46221

024 12

97558

53

3

51374

48626

53820

46180

0214(1

97584

57

i

51411

18,689

5;3861

46139

02 160

97,550

56

5

9.51447

10.486,63

9..63902

10.46098

10.024:66

9.97545

55

6

51484

4861li

53943

46057

02459

97,641

54

7

51620

484S0

53984

46016

02464

97536

53

8

.^1557

4.S4-13

.64025

4.5975

02468

97632

52

9

51593

48407

54065

45935

01^172

97528

51

10

9..51(i29

10.48371

9.64106

10.45894

10.02477

9.97523

50

11

51666

483: M

54147

4,5853

02481

97519

49

12

51702

48298

,64187

45813

02485

97515

48

13

51738

482C.2

,54228

45772

02490

97510

47

14

51774

48226

,64269

45731

02494

97506

46

15

9..51S11

10.48189

9.,>4309

10.45691

10.02499

9.97501

45

16

51847

48163

54350

45650

0'2503

97497

44

17

51883

48117

54390

45610

02508

97492

43

18

51919

48U81

64431

45569

02512

97488

42

19

51955

48046

,54471

4.5529

02516

97484

41

20

9.61991

10.48009

9.64612

10.4,648S

10.0'2521

9.97479

40

21

.62027

47973

54552

4:6418

02626

97475

39

22

62063

47937

54593

4.6407

02630

97470

38

23

.62099

47901

54633

46:167

02634

97466

37

24

,62136

47865

,54673

46:127

02,639

97461

36

25

9..62171

10.47829

9,54714

10. 1.6286

10.02,643

9.974,57

35

26

."v>2()7

17793

64764

4.6246

02647

974.63

34

27

.62242

477,68

64794

462011

0'2552

97448

33

28

.62278

47722

64H36

45166

02556

97444

32

29

,62314

4768i;

,61876

45126

02.561

97439

31

30

9.62360

10.476,60

9..14916

10.4.6086

10.0'2565

9.97435

30

31

52385

47615

54966

46046

02570

97430

29

32

52421

47579

54996

4.6006

0'2574

974-26

28

33

52456

47,644

. .65035

44966

02579

97421

27

34

52492

47508

55075

44926

02583

97417

26

35

9..62527

10.47473

9.,65115

10.44886

10.02588

9.97412

25

36

.626Ci:!

47437

551,65

44846

02592

97408

24

37

52598

47402

55195

44,S06

0-2697

97403

23

38

52634

47366

,65235

44766

02601

97399

22

39

52669

47331

55275

44725

02606

97394

21

40

9.52705

10.47295

9.,65315

10.446,S6

10.02610

9.97390

20

41

52740

47260

65355

44645

02615

97385

19

42

52775

47225

55395

44605

02619

97381

18

43

52811

47189

55434

44566

02624

97376

17

44

62846

47154

56474

44526

02628

97372

16

45

9.52881

10.47119

9.,55,514

10.44486

10.02633

9.97367

15

46

52916

47084

56554

44446

02637

97363

14

47

52951

47049

55593

44407

02642

97358

13

48

52986

47014

55633

44:l{;7

02647

97353

12

49

63021

46979

,66673

44:127

02651

97,349

11

50

9..63056

10.46944

9,55712

]0.442,S8

10.02656

9.97344

10

51

53092

46908

55752

44248

02660

97340

9

52

53126

46874

,55791

44209

02665

97335

8

53

53161

46839

55831

44169

02669

97331

7

54

53196

46804

,65870

44130

02674

97:326

6

55

9.53231

10.46769

9.56910

10.44090

10.02678

9.97322

5

56

,53266

46734

5.6949

440.51

02683

97317

4

57

53301

46699

55989

44011

02688

97312

3

58

,53336

46664

56028

43972

02692

97308

2

59

53370

46630

56067

43933

02697

97303

i

60

53405

46,595

56107

43893

02701

97299

M.

Cosine.

Set-ant.

Colaiipent.

Tangent.

Cosecant.

Sine,

M.

109°

70°

Table 2. LOGAKITHMIC ANGULAR FUNCTIONS.

297

20°

Log:arithins.

159°

M.

Sine.

Cosecant.

Tangent.

Cotangent

Secant.

Cosine.

M.

9.53405

10.46595

9.56107

10.43893

10.02701

9.97299

00

1

53440

46560

56146

43854

02706

97294

59

2

53475

46525

56186

43815

02711

97289

58

3

53509

46491

56224

43776

02716

97285

57

4

53544

46456

66264

43736

02720

97'280

56

5

9.53578

10.46422

9.56303

10.43697

10.02724

9.97276

55

6

53613

46387

56342

43658

02729

97271

54

7

53647

46353

56381

43619

02734

97266

53

8

53682

46318

56420

43580

02738

97262

62

9

53716

46284

66159

43541

02743

97257

51

10

9.53751

10.46249

9.56498

10.43502

10.02748

9.97252

50

11

53785

46215

66537

43463

02762

97248

49

12

53819

46181

56676

43424

02757

97'243

48

13

53854

46146

56615

43385

02762

97238

47

14

53888

46112

66654

43346

02766

97234

46

15

9.53922

10.46078

9.56693

10.43307

10.02771

9.97229

45

16

53957

46043

56732

43268

02776

97224

44

17

53991

46009

66771

43229

02780

97220

43

18

54026

45975

66810

43190

02785

97216

42

19

.54059

45941

56849

43151

02790

97210

41

20

9.54093

10.45907

9.56887

10.43113

10.02794

9.97206

40

21

54127

45873

56926

43074

02799

97201

39

22

54161

45839

56965

43035

02804

97196

38

23

54195

45805

57004

42996

02808

97192

37

24

54229

45771

57042

42958

02813

97187

36

25

9.54263

10.45737

9.67081

10.42919

10.02818

9.97182

85

26

54297

43703

67120

42880

02822

97178

34

27

54331

45669

57168

42842

02827

97173

33

28

51365

4.5635

57197

42803

02832

97168

32

29

54399

45601

67236

42765

02837

97163

31

30

9.54433

10.45567

9.57274

10.42726

10.02841

9.97159

30

31

54466

45534

57312

42688

02846

97164

29

32

54500

45500

57361

42649

02851

97149

28

33

54534

45466

57389

42611

0'2855

97145

27

34

54567

4.W33

57428

42572

02860

97140

26

35

9.54601

10.45399

9.57466

10.42584

10.02865

9.97136

25

36

54635

45365

57504

42496

02870

97130

24

37

54668

45332

57543

42457

02874

97126

23

38

54702

45298

57581

42419

02879

97121

22

39

54735

45265

57619

42381

02884

97116

21

40

9.54769

10.45231

9.57658

10.42342

10.02889

9.97111

20

41

54802

45198

57696

42304

02893

97107

19

42

54836

45164

57734

42266

02898

97102

18

43

54869

45131

67772

42228

02903

97097

17

44

54903

45097

57810

42190

02908

97092

16

45

9.54936

10.45064

9.67849

10.42151

10.02913

9.97087

15

46

54969

45031

67887

42113

02917

97083

14

47

55003

44997

57925

42075

02922

97078

13

48

55036

44964

57963

42037

02927

97073

12

49

55069

44931

58001

41999

02932

97068

11

50

9.55102

10.44898

9.58039

10.41961

10.02937

9.97063

10

51

55136

44864

58077

41923

02941

97059

9

52

55169

44831

68115

41885

02946

97064

8

53

55202

44798

58153

41847

02961

97049

7

54

55235

44765

68191

41809

02966

97044

6

55

9.55268

10.44732

9.58229

10.41771

10.02961

9.97039

5

56

55301

44699

58267

41733

02965

97035

4

57

55334

44666

58304

41696

02970

97030

3

58

55367

44633

58342

41658

02975

97026

2

59

55400

44600

58380

41620

02980

97020

1

60

55433

44567

58418

41582

02985

97015

31.

CosJDe.

Secant.

Cotangent.

Tangent.

Cosecant.

Sine.

M.

110°

69°

298 LOGARITHMIC ANGULAR FUNCTIONS. Table 2.

21°

Logarithms.

158°

M.

Sine.

CoHecunt.

Tangent.

Cotangent.

Secant.

Cosine.

M.

9.55433

10.44567

9.58418

10.41682

10.02985

9.97015

60

1

55466

44534

58456

41645

02990

97010

59

2

55499

44501

58493

41607

02995

97005

58

3

55532

44468

58531

41469

02999

97001

57

4

65564

44436

68569

41431

03004

96996

56

5

9.65697

10.44403

9.68606

10.41394

10.03009

9.96991

.55

6

65630

44370

58644

41356

03014

96986

64

7

55663

44337

58681

41319

03019

96981

53

8

55695

44306

58719

41281

03024

96976

52

9

55728

44272

68767

41243

03029

96971

61

10

9.55761

10.44239

9.58794

10.41206

10.03034

9.96966

50

11

65793

44207

58832

41168

03038

96962

49

12

65826

44174

58869

41131

03043

96967

48

13

55858

44142

68907

41093

03048

96952

47

14

65891

44109

58944

41056

03053

96947

46

15

9.56923

10.44077

9..58981

10.41019

10.03068

9.96942

46

16

66956

44044

69019

40981

03063

96937

44

17

56988

44012

.59056

40944

03068

96932

43

18

56021

43979

69094

40906

03073

96927

42

19

56053

43947

59131

40869

03078

96922

41

20

9.56086

10.43915

9.59168

10.40832

10.03083

9.96917

40

21

56118

43882

69206

40795

03088

96912

39

22

66160

43850

59243

40757

03093

96907

38

23

56182

43818

59280

40720

03097

96903

37

24

56215

43785

69317

40683

03102

96898

36

25

9.56247

10.43753

9.69354

10.40646

10.03107

9.96893

85

26

56279

43721

59391

40609

03112

96888

34

27

66311

43689

59429

40571

03117

90883

33

2«

66343

43657

59466

40534

03122

96878

32

29

56375

43626

59603

40497

03127

96873

31

30

9.56408

10.43592

9..595-10

10.40460

10.03132

9.96868

30

31

56440

43660

59577

40423

03137

96863

29

32

56472

43528

69614

40386

03142

96858

'28

33

66604

4;i490

69651

40349

03147

96858

27

34

66536

4:3464

69688

40312

03152

96848

26

35

9.56568

10.43432

9.59725

10.40276

10.03157

9.96843

25

36

56599

43401

69762

40238

03162

96838

24

37

56631

43369

59799

40201

03167

96833

23

38

66663

43337

69836

40165

03172

96828

22

39

66695

43305

59872

40128

03177

96823

21

40

9.66727

10.43273

9.59909

10.40091

10.03182

9.96818

20

41

,56759

43241

59946

40054

03187

96813

19

42

56790

43210

59983

40017

03192

96808

18

43

56822

43178

60019

39981

03197

96803

17

44

56864

43146

60056

39944

03202

96798

16

45

9.66886

10.43114

9.60093

10.39907

10.03207

9.96793

15

46

56917

43083

60130

39870

03212

96788

14

47

56949

43051

60166

39834

03217

96783

13

48

56980

43020

60203

39797

03222

96778

12

49

67012

42988

60240

39760

03228

96772

11

50

9.67044

10.42966

9.60276

10.39724

10.03233

9.96767

10

51

67076

42925

60313

39687

03238

96762

9

52

57107

4'2893

60349

396.51

03243

96757

8

53

67138

42862

60386

39614

03248

96762

7

54

57169

4'2831

60422

39578

03253

96747

6

65

9.57201

10.42799

9.60469

10.39541

10.03258

9.96742

5

56

57232

42768

60495

39505

03263

96737

4

57

67264

42736

60532

39468

03268

96732

3

58

67296

42706

60568

39432

03273

96727

2

59

67326

42674

60605

39396

03278

96722

1

60

57368

42642

60641

39369

03283

96717

M.

Cosine.

Secant.

Cotangent.

Tangent.

Cosecant.

Sine.

M.

111°

68°

Table 2. LOGARITHMIC ANGULAR FUNCTIONS. 299

22°

Logarithms.

IS?"^

M.

Sine.

Cosecant.

Tangent.

Cotangent.

Secant,

Cosine,

M.

9.57358

10.42642

9.60641

10.39359

10,03283

9.96717

60

1

57389

42611

60677

39323

03289

96711

59

2

57420

42580

60714

39286

03294

96706

88

3

57451

42549

60750

39250

03299

96701

57

4

57482

42518

60786

39214

03304

96696

56

6

9,57514

10.42486

9.60823

10.39177

10,03309

9,96691

55

6

57545

4'2455

60859

39141

03314

96686

54

7

57576

42424

60895

39105

03319

96681

53

8

57607

42393

60931

39069

08324

96676

52

9

57638

42362

60967

39033

03380

96670

61

10

9.57669

10.42331

9.61004

10.38996

10,03335

9,96665

50

11

57700

42300

61040

38960

03340

96660

49

12

57731

42269

61076

38924

03345

96655

48

13

57762

42238

61112

38888

03350

96650

47

14

57793

42207

61148

38852

03355

96645

46

15

9.57824

10.42176

9.61184

10.38816

10,03360

9,96640

45

16

57855

42145

61220

38780

03366

96634

44

17

57885

42115

61266

38744

03371

96629

43

18

57916

42084

61292

38708

03376

96624

42

19

57947

42053

61328

38672

03381

96619

41

20

9.57978

10.42022

9.61364

10.38636

10,03386

9,96614

40

21

58008

41992

61400

38600

03392

96608

39

22

58039

41961

61436

38564

03397

96603

38

23

58070

41930

61472

38528

03402

96598

37

24

58101

41899

61508

38492

03407

96593

36

25

9.58131

10.41869

9.61544

10.38456

10,03412

9,96588

35

26

58162

41838

61579

38421

03418

96582

34

27

58192

41808

61615

38385

03423

96577

33

28

.58223

41777

61651

38349

03428

96572

32

29

58253

41747

61687

38313

03433

96667

31

30

9.58284

10.41716

9.61722

10.38278

10,03438

9.96562

30

31

58314

41686

61758

38242

03444

96556

29

32

58345

41655

61794

38206

03449

96551

28

33

58375

41625

61830

38170

03454

96546

27

34

58406

41594

61865

38136

03159

96541

26

35

9.58436

10.41564

9.61901

10.38099

10,03465

9.96536

26

36

58467

41533

61936

38064

03470

96530

24

37

58497

41503

61972

38028

03475

96525

23

38

58527

41473

62008

37992

03480

96520

22

39

58557

41443

62043

37957

03486

96514

21

40

9.58588

10.41412

9.62079

10.37921

10,03491

9.96509

20

41

58618

41382

62114

37886

03496

96504

19

42

58648

41352

62150

37850

03502

96498

18

43

58678

41322

62185

37815

03507

96493

17

44

58709

41291

62221

37779

03512

96488

16

45

9.58739

10.41261

9.62266

10.37744

10,03517

9.96483

15

46

58769

41231

62292

37708

03523

96477

14

47

58799

41201

62327

37673

03528

96472

13

48

58829

41171

62362

37638

03533

96467

12

49

58859

41141

62398

37602

03539

96461

11

50

9.58889

10.41111

9.62433

10.37567

10,03544

9.96466

10

51

58919

41081

62468

37532

03549

96451

9

52

58949

41051

62604

37496

03555

96445

8

53

58979

41021

62539

37461

03560

96440

7

54

59009

40991

62574

37426

03565

96435

6

55

9.59039

10.40961

9.62609

10.37391

10,03571

9.96429

5

56

59069

40931

62645

37355

03576

96424

4

57

59098

40902

62680

37320

03581

96419

3

58

59128

40872

62715

37285

03587

96413

2

59

59158

40842

62750

37250

03592

96408

1

60

59188

40812

62785

37215

03597

96403

M.

Cosine.

Secant.

Cotangent.

Tangent,

Cosecant,

Sine.

M.

J 12°

67°

300

LOGAEITHMIC ANGULAE FUNCTIONS. Table 2.

23°

Logarithms.

1S6°

M.

Sine.

CuStTilllt.

10.40812

Tangent.

Cotangent

Secant.

Cosine.

M.

9.59188

9.62785

10.37215

10.03597

9.96403

60

1

59218

40782

628'20

37180

03603

96397

59

2

59247

40753

62855

87145

03608

96392

58

3

59277

40723

62890

37110

03613

96387

57

4

59307

40693

62926

37074

03619

96381

56

5

y..'.933ll

10.40664

9.62961

10.37039

10.03624

9.96376

66

6

59361!

40634

62996

37004

03630

96370

64

7

.■.93911

40604

63031

36969

03636

96365

53

8

59-l-i'i

40575

63066

369.34

03640

96360

52

9

59455

40M6

63101

36899

03646

96364

51

10

»..59484

10.40516

9.63135

10.36865

10.03651

9.96349

50

11

59514

40486

63170

36830

03657

96343

49

12

59M3

40457

63205

36795

03662

96338

48

13

59573

40427

63240

36760

03667

96333

47

1-1

69602

40398

63275

30725

03673

96327

46

15

9.59632

10.40368

9.63310

10.30690

10.03678

9.96322

45

16

59661

40339

63345

36655

03684

96316

44

17

59690

40310

6.3379

36621

03689

96311

43

18

59720

40280

63414

36686

03695

96305

42

19

59749

40261

63449

36551

03700

96300

41

20

9.5977S

10.40222

9.63484

10.36516

10.03706

9.96294

40

21

59h08

40192

63619

36481

03711

96289

39

22

59837

40163

63653

36447

03716

96284

38

23

5986(1

40134

63588

36412

03722

96278

37

24

59895

40105

63623

36377

03727

96273

36

25

9.59924

10.40076

9.63657

10.36343

10.03733

9.96267

35

26

59964

40046

63692

36308

03738

96262

34

27

59983

40017

63726

36274

03744

96256

33

28

60012

39988

63761

36239

03749

96261

32

29

60041

39959

63796

36204

03755

96246

31

30

9.60070

10.39930

9.63830

10.36170

10.03760

9.96240

30

31

60090

39901

63865

36135

03766

96234

29

32

60128

39872

63899

36101

03771

96229

28

33

60157

39843

63934

36066

03777

96223

27

34

60186

39814

63968

36032

03782

96218

26

35

9.60215

10.39785

9.64003

10.36997

10.03788

9.96212

25

36

(10244

39766

64037

35963

03793

96207

24

37

60273

39727

64072

35928

03799

96201

23

38

60302

39698

64106

85894

03804

96196

22

39

60331

39669

64140

35860

03810

96190

21

40

9,60359

10.39641

9.64175

10.35825

10.03815

9.96185

20

41

60388

39612

64209

36791

03821

96179

19

42

60417

39583

64243

36757

03826

96174

18

43

60446

396.54

64278

35722

03832

96168

17

44

60474

39626

64312

35688

03838

96162

16

46

9.60503

10.39497

9.64346

10.35654

10.03843

9.96157

15

46

60532

39468

64381

36619

03849

96151

14

47

60581

39439

64416

35585

o:«54

96146

13

48

60589

39411

64449

35551

03860

96140

12

49

60618

39382

64483

35517

03865

96136

11

50

9.60646

10.39354

9.64517

10.35483

10.03871

9.96129

10

51

60675

393'26

64.552

35448

03877

96123

9

52

60704

39296

64586

35414

03882

96118

8

53

60732

392 J8

64620

35380

03888

96112

7

64

60761

39239

64664

35346

o;«93

96107

6

65

9.60789

10.39L'l

9.64688

10.36312

10.03899

9.96101

5

56

60818

39182

64722

36278

03905

96096

4

57

60846

39154

64756

35244

03910

96090

3

58

60875

39125

64790

35210

03916

96084

2

59

60903

39097

64824

35176

03921

96079

1

60

60931

33069

P4858

35142

03927

96073

M.

Cosine.

Secant.

Cotangent.

Tangent.

Cosenint.

Sine.

M.

U3°

66°

Table 3. LOGARITHMIC ANGULAR FUNCTIONS. 301

24°

Logarithms.

1SS°

M.

Sine.

Cosecant.

Tangent.

Cotangent.

Secant.

Cosine.

M.

9.60931

10.39069

9.64858

10.35142

10.03927

9.96073

60

1

60960

39040

64892

35108

03933

96067

69

2

60988

39012

64926

35074

03938

96062

58

3

61016

38984

64960

35040

03944

96056

67

4

61045

38955

64994

35006

03950

96050

56

5

9.61073

10.38927

9.65028

10.34972

10.03955

9.96045

56

6

61101

38899

65062

34938

03961

96039

54

7

61129

38871

65096

34904

03966

96034

63

8

61188

38842

66130

34870

03972

96028

62

9

61186

38814

65164

34836

03978

96022

51

10

9.61214

10.38786

9.65197

10.34803

10.03983

9.96017

50

11

61242

38768

G5231

34769

03989

960H

49

12

61270

38730

66265

34735

03995

96005

48

13

61298

38702

65299

34701

04000

96000

47

14

61326

38674

65333

34667

04006

95994

46

15

9.61364

10.38646

9.66366

10.34634

10.04012

9.95988

46

16

61382

38618

65400

34600

04018

95982

44

17

61411

38589

65434

34566

04023

95977

43

18

61438

38562

65467

34533

04029

95971

42

19

61466

38684

65601

34499

04035

95965

41

20

9.61494

10.38606

9.65636

10.34466

10.04040

9.95960

40

21

61522

38478

65568

S4432

04046

96954

39

22

61560

38460

66602

34398

04052

95948

38

23

61578

38422

65636

34364

04058

96942

37

24

61606

38394

65669

34331

04063

96937

36

25

9.61634

10.38366

9.65703

10.34297

10.04069

9.95931

35

26

61662

38338

66736

34264

04075

95925

34

27

61689

38311

66770

34230

04080

95920

33

28

61717

38283

65803

34197

04086

95914

32

29

61745

38255

66837

34163

04092

95908

31

30

9.61773

10.38227

9.66870

10.34130

10.04098

9.95902

30

31

61800

38200

66904

34096

04103

96897

29

32

61828

38172

65937

34063

04109

96891

28

33

61856

38144

65971

34029

04115

96886

27

34

61883

38117

66004

33996

04121

95879

26

35

9.61911

10.38089

9.66038

10.33962

10.04127

9.95873

25

36

61939

38061

66071

33929

04132

96868

24

37

61966

38034

66104

33896

04138

96862

23

38

61994

38006

66138

33862

04144

95856

22

39

62021

37979

66171

33829

04150

95850

21

40

9.62049

10.37951

9.66204

10.33796

10.04156

9.95844

20

41

62076

37924

66238

33762

04161

95839

19

42

62104

37896

66271

33729

04167

96833

18

43

62131

37869

66304

33696

04173

96827

17

44

62169

37841

66337

33663

04179

95821

16

45

9.62186

10.37814

9.66371

10.33629

10.04185

9.95815

15

46

62214

37786

66404

33596

04190

95810

14

47

62241

37769

66437

33563

04196

95804

13

48

62268

37732

66470

33630

04202

95798

12

49

62296

37704

66503

33497

04208

95792

11

50

9.62323

10.37677

9.66637

10.33463

10.04214

9.95786

10

51

62360

37650

66570

33430

04220

96780

9

52

62377

37623

66603

33397

04226

96776

8

63

62405

37595

66636

33364

04231

95769

7

54

62432

37668

66669

33331

04237

95763

6

65

9.62469

10.37541

9.66702

10.33298

10.04243

9.95767

6

56

62486

37514

66735

33266

04249

95751

4

57

62613

37487

66768

33232

04255

96746

3

58

62541

37459

66801

33199

04261

96739

2

69

6'2668

37432

66834

33166

04267

9.5733

1

60

62596

37405

66867

33133

04272

95728

M.

Cosiue.

Secant.

Cotangent.

Tangent.

Cosecant.

Sine.

M.

114°

302

LOGARITHMIC ANGULAE FUNCTIONS. Table 2.

25°

Logarithms.

154°

M.

Sine.

Cosecant.

Tangent.

Cotangent.

Secant.

Cosine.

M.

9.62595

10.37405

9.66867

10.33138

10.04272

9.95728

60

1

62622

37378

66900

33100

04-278

95722

69

2

62649

37351

66938

83067

04284

95716

58

3

62676

37324

66966

33034

04290

95710

57

4

6-2703

37297

66999

38001

04296

95704

56

5

9.62730

10.37270

9.67032

10.32968

10.04302

9.96698

55

6

62757

37243

67065

32985

04308

95692

54

7

62784

87216

67098

32902

04314

95686

53

8

62811

37189

671.81

82869

04320

95680

52

9

62838

37162

67163

32837

04326

96674

51

10

9.62865

10.37135

9.67196

10.32804

10.04332

9.95668

50

11

62892

37108

67229

32771

04337

95663

49

12

62918

37082

67262

3-2788

04343

98657

48

13

62945

.37055

67295

32705

M349

95661

47

14

62972

37028

67827

8-2673

04355

95645

46

15

9.62999

10.37001

9.67360

10.32640

10.04361

9.95639

45

16

68026

36974

67393

32607

04367

95633

44

17

63052

36948

67426

32574

04378

95627

43

18

63079

36921

67458

3-2542

04379

956-21

42

19

63106

36894

67491

32509

04385

95615

41

20

9.63138

10.36867

9.67524

10.32476

10.04391

9.95609

40

21

63159

36841

67,556

32444

04397

95603

39

22

63)86

86814.

67689

32411

04403

96597

38

23

68213

86787

67622

3'2378

04409

95.591

37

24

68239

36761

67654

32346

04415

95585

36

25

9.68266

10.36734

9.67687

10.32313

10.04421

9.95579

35

26

68292

^ 36708

67719

8-2281

04427

96578

34

27

63319

A 36681

67752

32-248

04433

96567

33

28

63345

36655

67785

32215

04439

96661

82

29

63372

36628

67817

82183

04445

95565

31

30

9.68398

10.36602

9.67850

10.32150

10.04451

9.95549

30

81

68426

36575

67882

3'2118

04457

95543

29

32

63451

36.549

67915

32085

04463

96537

28

33

63478

36522

67947

32053

04469

96531

•27

34

63504

36496

67980

32020

04476

955-25

26

35

9.63531

10..36469

9.68012

10.81988

10.04481

9.95519

25

36

63557

36448

68044

81956

04487

95518

24

37

63588

86417

68077

81928

04493

95507

23

88

63610

36890

68109

81891

04500

96500

22-

39

63636

36864

68142

31858

04506

95494

21

40

9.63662

10.86838

9.68174

10.31826

10.04512

9.9.5488

20

41

63689

86811

68206

31794

04518

9.5482

19

42

63715

36285

68-289

31761

04524

95476

18

43

63741

86259

68-271

31729

04530

95470

17

44

63767

86233

68303

31697

04536

96464

16

45

9.63794

10.86206

9.68386

10.31664

10.04542

9.95458

15

46

63820

86180

68368

81682

04548

95462

14

47

63846

86154

68400

81600

04554

95446

13

48

63872

86128

68432

81568

04560

95440

12

49

63898

86102

68465

31535

04566

96484

11

50

9.63924

10.36076

9.68497

10.31503

10.04573

9.96427

10

51

63960

36050

685-29

31471

' 04579

95421

9

52

63976

36024

68.561

81439

04585

95415

8

53

64002

85998

68598

81407

04591

95409

7

54

64028

35972

68626

31374

04597

95403

6

55

9.64054

10.85946

9.68658

10.31342

10.04603

9.95397

5

56

64080

85920

68690

81310

04609

95391

4

57

64106

35894

68722

31-278

04616

95384

3

58

64132

35868

68754

31246

04622

95378

2

59

64158

35842

68786

31214

04628

95372

i

60

64184

35816

68818

31182

04634

95366

M.

Cosine.

Secant.

Cotangent.

Tangent.

Cosecant.

Sine.

M.

115°

Table 2. LOGARITHMIC ANGULAK FUNCTIONS.

303

26°

Logarithms.

153°

M.

Sine.

Cosecant.

Tangent.

Cotangent,

Secant,

Cosine.

M,

9.64184

10.35816

9.68818

10.31182

10.04634

9.95366

60

1

64210

35790

68850

311.50

04640

95360

59

2

64236

35764

68882

31118

04646

95354

58

3

64262

35738

68914

■ 81086

04652

95348

57

4

64288

35712

68946

31054

046,59

95341

56

5

9.64313

10.35687

9.68978

10.31022

10,04665

9.9.5335

55

6

64339

35661

69010

30990

04671

95329

54

7

64365

35635

69042

.30958

04677

95323

53

8

64391

35609

69074

30926

04683

9.5317

.52

9

64417

35583

69106

,30894

04690

95310

51

10

9.64442

10.35558

9.69138

10.30862

10,04696

9.95.304

50

11

64468

35532

69170

30830

04702

95298

49

12

64494

35506

69202

30798

04708

95292

48

13

64519

35481

69234

30766

04714

95286

47

14

61545

35455

69266

30734

04721

95279

46

15

9.64571

10.35429

9.69298

10.30702

10.04727

9.9.5273

45

16

64596

35404

69329

30671

04733

95267

44

17

64622

35378

69361

30639

047.39

95261

43

18

61647

35353

69393

30607

04746

95254

42

19

64673

35327

69125

30.575

04752

95248

41

20

9.64698

10.35302

9.69457

10.30.543

10.047.58

9.95242

40

21

64724

35276

69488

30512

04764

95236

39

22

64749

35251

69520

30480

04771

95229

38

23

64775

35225

69552

30448

04777

95223

37

24

64800

35200

69584

80416

04783

95217

36

25

9.64826

10.35174

9.69615

10.30385

10,04789

9.95211

35

26

64851

35149

69647

30353

04796

95204

34

27

64877

35123

69679

30321

04802

95198

33

28

64902

35098

69710

30290

04808

95192

32

29

64927

35073

69742

30258

04815

95185

31

30

9.64953

10.35047

9.69774

10.30226

10,04821

9.95179

30

31

64978

35022

69805

30195

04827

95173

29

32

65003

34997

69837

30163

04833

95167

28

33

65029

34971

69868

30132

04840

95160

27

34

65054

34946

69900

30100

04846

95154

26

35

9.65079

10.34921

9,69932

10.30068

10,04852

9,95148

25

36

65104

34896

69963

30037

04859

95141

24

37-

- '>6.5130

34870

69995

30005

01865

95135

23

38

65155

34845

70026

29974

04871

95129

22

39

65180

34820

70058

29942

04878

95122

21

40

9.65205

10.34795

9.70089 •

10.29911

10,04884

9,95116

20

41

65230

34770

70121

29879

04890

95110

19

42

65255

34745

70152

29848

04897

95103

18

43

65281

34719

70184

29816

04903

9.5097

17

44

65306

34694

70215

29785

04910

95090

16

45

9.65331

10.34669

9.70247

10,29753

10,04916

9,95084

15

46

65356

34644

70278

29722

04922

95078

14

47

65381

34619

70309

29691

04929

95071

13

48

65406

34594

70341

29659

04935

95065

12

49

65431

34569

70372

29628

04941

95059

11

50

9.65456

10.34544

9.70404

10.29596

10,04948

9,95052

10

51

65481

34519

70435

29565

049.54

95046

9

52

65506

34494

70466

29.534

04961

95039

8

53

65531

34469

70498

29502

04967

9.5033

7

54

65556

34444

70529

29471

04973

95027

6

55

9.65580

10.34420

9,70.560

10,2944r

10.0498f^

9.95020

5

56

65605

34395

70592

29408

04986

95014

4

57

65630

34370

70623

29377

04993

95007

3

58

65665

34345

70654

29346

04999

95001

2

59

65680

34320

70685

29315

05005

94995

1

60

65705

34295

70717

29283

05012

94988

M.

Cosine.

Secant.

Cotangent

Tangent,

Cosecant.

Sine,

M,

116°

63°

304

LOGAETTHMTC ANGULAR FUNCTIONS. Table 2.

27°

Logarithms.

52°

M,

Sine.

Cosecant.

Tangent.

Cotangent,

Secant.

Cosine.

M.

9.65705

10.34295

9.70717

10,29283

10.05012

9.94988

60

1

65729

34271

70748

29252

06018

94982

59

2

65754

34246

70779

29221

0,5026

94975

58

3

65779

34221

70810

29190

05031

94969

57

4

6.M14

34196

70841

29159

05038

94962

66

5

9,li5SiH

10.34172

9.70873

10.29127

10,05044

9.94956

55

6

li.iSS;!

31147

70904

29096

05051

94949

54

7

65878

34122

70935

29065

05057

94943

53

8

65902

34098

70966

29034

05064

94936

52

9

65927

34073

70997

29003

06070

94930

51

10

9.65952

10.34048

9.71028

10.28972

10,06077

9.91923

50

11

65976

34024

71059

28941

05083

94917

49

12

66001

33999

71090

28910

05089

94911

48

13

66025

33975

71121

28879

05096

94904

47

14

660.10

33950

71153

28847

05102

94898

46

15

9.66075

10.33925

9.71184

10.28816

10,05109

9.94891

45

16

66099

33901

71215

28785

05115

94885

44

17

66124

33876

71216

28754

05122

94878

43

18

66148

33852

71277

28723

05129

94871

42

19

66173

33827

71308

28692

05135

94865

41

20

9.66197

10.33803

9.71339

10.28(i61

10.06142

9.94868

40

21

66221

33779

71370

28630

05148

94852

39

22

66216

33754

71401

28599

05155

94845

38

23

66270

33730

71431

28569

05161

94839

37

24

66295

33705

71462

2,S53,H

05168

94832

36

26

9.66319

10.33681

9.71193

10.28507

10.05174

9.94826

35

26

66343

33657

71524

28476

05181

94819

34

27

66368

33632

715.55

28146

05187

94813

33

28

66392

33608

71586

28414

05194

94806

32

29

66416

S3584

71617

28383

06201

94799

31

30

9.66441

10.335.59

9.71648

10.28352

10.05207

9.94793

30

31

66465

33536

71679

28321

05214

94786

29

32

66489

33511

71709

28291

05220

94780

28

33

66513

33487

71740

28260

05227

94773

27

34

66537

33463

71771

28229

062:B

94767

26

35

9.66.562

10.33438

9.71802

10.28198

10.06240

9.94760

25

36

66586

33414

71833

28167

06247

94753

24

37

66610

33390

71863

28137

05253

94747

23

38

66634

33366

71894

28106

05260

94740

22

39

66658

3,3342

71926

28075

05266

94734

21

40

9.66682

10.33318

9.71955

10.2.><0J5

10.05273

9.94727

20

41

66706

33294

71986

28014

05280

94720

19

42

66731

33269

72017

27983

05286

94714

18

43

66765

33245

72048

27952

05293

94707

17

44

66779

33221

72078

27922

05300

94700

16

45

9.66S03

10.33197

9.72109

10.27891

10.05306

9.94694

15

46

66827

33173

72140

27860

06313

94687

14

47

66861

33149

72170

27830

05320

94680

13

48

66875

331'25

72201

27799

0,5326

94674

12

49

66899

33101

72231

27769

0,5333

94667

11

50

9.66922

10.33078

9.72262

10,27738

10.05340

9.94660

10

51

66946

33054

72293

27707

06346

94654

9

52

66970

33030

72323

27677

05353

94647

8

53

66994

33006

723,54

27646

06360

94640

7

54

67018

32982

72384

27616

05366

94634

6

55

9.67042

10.32958

9.72415

10.27,585

10.05373

9.94627

5

56

67066

32934

72445

275,55

05380

94620

4

57

67090

32910

72476

27524

08386

94614

3

58

67113

32887

725Uli

27191

06393

94607

2

59

67137

32863

72537

27463

0.5400

94600

1

60

67161

32839

72567

27433

06407

91593

M.

Cosine.

Secant.

Cotangent,

Tangent.

Cosecant.

Sine.

M.

Table 3. LOGARITHMIC ANGULAR FUNCTIONS.

305

28°

Logarithms.

1S1°

M.

Sine.

Cosecant.

Tangent.

Cotangent.

Secant.

Cosine.

M.

9.67161

10.32839

9.72567

10.27433

10.05407

9.94593

60

1

67185

32815

72598

27402

05413

94587

59

2

67208

32792

72628

27372

05420

94580

58

3

67232

32768

72659

27341

05427

91573

57

4

67256

32744

72689

27311

06433

94567

56

5

9.67280

10.32720

9.72720

10.27280

10.06440

9.94860

55

6

67303

32697

72760

27260

05447

94553

54

7

67327

32673

72780

27220

05454

94516

63

8

67350

32650

72811

27189

05460

94640

52

9

67374

32626

72841

27169

05467

94533

51

10

9.67398

10.32602

9.72872

10.27128

10.05474

9.94526

60

11

67421

32679

72902

27098

06481

94519

49

12

67445

32655

72932

27068

05487

94513

48

13

67468

32532

72963

27037

05494

94506

47

14

67492

32508

72993

27007

05601

94499

46

15

9.67515

10.32486

9.73023

10.26977

10.05508

9.94492

45

16

67539

32461

73054

26946

06515

94485

41

17

67562

32438

73084

26916

05521

94479

43

18

67586

32414

73114

26886

06528

94472

42

19

67609

32391

73144

26866

05535

94465

41

20

9.67633

10.32367

9.73175

10.26826

10.05642

9.94468

40

21

67666

32344

73205

26795

06549

94451

39

22

67680

32320

73235

26765

06556

94446

38

23

67703

32297

73265

26735

05562

94438

37

24

67726

32274

73295

26705

05669

94431

36

25

9.67750

10.32250

9.73326

10.26674

10.05576

9.94424

35

26

67773

32227

73356

26644

05583

94417

34

27

67796

32204

73386

26614

06590

94410

33

28

67820

32180

73416

26584

05596

94404

32

29

67843

32157

73446

26554

05603

94397

31

30

9.67866

10.32134

9.73476

10.26524

10.05610

9.94390

30

31

67890

32110

73507

26493

05617

94383

29

32

67913

32087

73637

26463

05624

94376

28

33

67936

32064

73567

26433

05631

94369

27

34

67959

32041

73597

26403

05638

94362

26

35

9.67982

10.32018

9.73627

10.26373

10.0O646

9.9ii355

26

36

68006

31994

73657

26343

05651

94349

24

37

68029

31971

73687

26313

0o658

9*342

23

38

68052

31948

73717

26283

05666

94335

22

39

68075

31925

73747

26263

0o672

9*328

21

40

9.68098

10.31902

9.73777

10.26223

10.0O679

9.94321

20

41

68121

31879

73807

26193

O0686

94314

19

42

68144

31856

73837

26163

0o693

94307

18

43

68167

31833

73867

26133

06700

94300

17

44

68190

31810

73897

26103

06707

94293

16

45

9.68213

10.31787

9.73927

10.26073

10.0D714

9.94286

15

46

68237

31763

73957

26043

05721

94279

14

47

68260

31740

73987

26013

05727

94273

13

48

68283

31717

74017

25983

05734

94266

12

49

68305

31695

74047

25953

06741

94259

11

50

9.68328

10.31672

9.74077

10.26923

10.06748

9.94252

10

51

68351

31649

74107

26893

05755

94246

9

52

68374

31626

74137

25863

05762

94238

8

53

68397

31603

74166

26834

05769

94231

7

54

68420

31580

74196

26804

05776

94224

6

55

9.68443

10.31.557

9.74226

10.25774

10.06783

9.94217

5

56

68466

31534

74256

25744

05790

94210

4

57

68489

31511

74286

25714

05797

94203

3

58

68512

31488

74316

25684

05804

94196

2

59

68534

31466

74345

26666

05811

94189

1

60

68557

31443

74375

26626

05818

94182

M.

Cosine.

Secant.

Cotangent.

Tangent.

Cosecant.

Sine.

M.

118°

21

61°

306

LOGARITHMIC ANGULAR FUNCTIONS. Table 3.

29°

Logarithms.

150°

M.

Sine.

("'itKLTailt.

Tangent.

Cotangent.

Secant.

CoBJne.

M.

9.68557

10.31443

9.74375

10.25625

10.0,5818

9.94182

60

1

e.'^.wo

31420

74406

2.5595

05825

94175

59

2

fisco:)

31397

74435

25565

05832

94168

58

3

6S625

:!i:i75

7-1465

25535

05839

94161

57

4

fism.s

3l:l.'-i2

74494

2.5506

0.5846

94154

56

5

9.liSli71

10.:n;)29

9.74524

10.25476

10.05863

9.94147

55

6

(■„Sli94

;-ii;;o«

74554

25446

05860

94140

54

7

twn;

31284

74683

25417

0.5867

94133

53

8

(;s7n9

31261

74613

26387

05874

94126

52

9

(iS7l'i2

31238

74643

25357

05881

94119

61

10

O.CWSl

10.31216

9.74673

10.25327

10.05888

9.94112

50

11

6.SS07

31193

74702

2.5298

05895

94105

49

12

C,SX29

31171

74732

25268

06902

94098

48

13

iisK-a

31148

74762

25238

05910

94090

47

14

6S,S7.')

31125

74791

26209

05917

94083

46

15

9.6S.S97

10.31103

9.74821

10.25179

10.05924

9.94076

45

16

|;,S920

31080

74851

25149

0,6931

94069

44

17

68942

310.68

74880

25120

05938

94062

43

18

68965

31035

74910

25090

0,5945

94055

42

19

68987

31013

74939

25061

05952

94048

41

20

9.69010

10.30990

9.74969

10.25031

10.05959

9.94041

40

21

69032

30968

74998

25002

05966

94034

39

22

69055

30945

75028

24972

05973

94027

38

23

69077

30923

750.58

24942

06980

94020

37

24

69100

30900

76087

24913

06988

94012

36

25

9.69122

10.30878

9.75117

10.24883

10.05996

9.94005

35

26

69144

308.56

75146

248,54

06002

93998

34

27

69167

30833

75176

24824

06009

93991

33

28

69189

30811

75205

24796

06016

93984

32

29

69212

30788

75235

24765

06023

93977

31

30

9.69234

10.30766

9.75264

10.24736

10.06030

9.93970

30

31

69266

30744

75294

24706

06037

93963

29

32

69279

30721

75323

24677

06045

93955

28

33

69301

30699

75353

24647

06052

93948

27

34

69323

30677

7,5382

24618

06059

93941

26

35

9.69345

10.30655

9.754n

10.24589

10.06066

9.93934

25

36

69368

30632

75441

24.559

06073

93927

24

37

69390

30610

75470

24530

06080

93920

23

88

69412

30588

75500

24500

06088

93912

22

39

69434

30566

75529

24471

06095

93905

21

40

9.69466

10.30.544

9.75558

10.24442

10.06102

9.93898

20

41

69479

30521

75588

24412

06109

93891

19

42

69501

30499

75617

24383

06116

93884

18

43

69523

30477

75647

24:!63

00124

93876

17

44

69545

30455

75676

24324

06131

93869

16

45

9.69567

10.30433

9.75705

10.24295

10.06138

9.9.3862

15

46

69.589

30411

75735

24265

06145

93855

14

47

69611

30389

75764

24236

08153

93847

13

48

69633

30367

75793

24207

06160

93840

12

49

69655

30345

75822

24178

06167

93833

11

50

9.69677

10.30323

9.75862

10.24148

10.06174

9.93826

10

51

69699

30301

76881

24119

06181

93819

9

52

C9721

30279

76910

24090

06189

93811

8

53

69743

30257

76939

24061

06196

93804

7

54

69705

30236

76969

24031

06203

93797

6

55

9.69787

10.30213

9.76998

10.24002

10.06211

9.93789

5

56

69809

30191

76027

23973

06218

93782,

4

57

69831

30169

76056

23944

06225

93776

3

58

69S53

30147

76086

23914

06232

93768

2

59

69875

30125

76116

23,SS5

06240

93760

1

60

C9897

30103

76144

23856

06247

93753

M.

t'l>MJH\

Secant.

Cotangent.

Tansent.

1 Cosecant.

Sine.

M.

119°

60°

Table 2. LOGAKITHMIC ANGULAR FUNCTIONS.

307

30°

Logarithms.

49°

M.

Sine.

Cosecant.

Tangent.

Cotangent,

Secant.

Cosine.

M.

9.69897

10.30103

9.76144

10.23856

10.06247

9.93763

60

1

B9919

30081

76173

23827

06254

93746

59

2

69941

30059

76202

23798

06262

93738

58

3

69963

30037

76231

23769

06269

93731

67

4

69984

30016

76261

23739

. 06276

93724

66

5

9.70006

10.29994

9.76290

10.23710

10.06283

9.93717

55

6

70028

29972

76319

23681

06291

93709

64

7

70050

29950

76348

23652

06298

93702

53

8

70072

29928

76377

23623

06305

93695

52

9

70093

29907

76406

23594

06313

93687

61

10

9.70115

10.29885

9.76435

10.23565

10.06320

9.93680

60

11

70137

29863

76464

23536

06327

93673

49

12

70159

29841

76493

23507

06335

93665

48

13

70180

29820

76522

23478

06342

93668

47

14

70202

29798

76551

23149

06350

93650

46

15

9.70224

10.29776

9.76580

10.23420

10.06357

9.93643

45

16

70245

29755

76609

23391

06364

93636

44

17

70267

29733

76639

23361

06372

93628

43

18

70288

29712

76668

23332

06379

93621

42

19

70310

29690

76697

23303

06386

93614

41

20

9.70332

10.29668

9.76725

10.23275

10.06394

9.93606

40

21

70353

29647

76754

23246

06401

93599

39

22

70375

29625

76783

23217

06409

93591

38

23

70396

29604

76812

23188

06416

93584

37

24

70418

29582

76841

2.3159

06423

93577

36

25

9.70439

10.29561

9.76870

10.23130

10.06431

9.93669

35

26

70461

29539

76899

23101

06438

93562

34

27

70482

29518

76928

23072

06446

93554

33

28

70504

29496

76957 ■

23043

06453

93547

32

29

70525

29475

76986

23014

06461

93539

31

30

9.70547

10.29453

9.77015

10.22985

10.06468

9.93532

30

31

70568

29432

77044

22956

06475

93525

29

32

70590

29410

77073

22927

06483

93517

28

33

70611

29389

77101

22899

06490

93510

27

34

70633

29367

77130

22870

06498

93502

26

35

9.70654

10.29346

9.77159

10.22841

10.06505

9.93495

25

36

70675

29325

77188

22812

06513

93487

24

37

70697

29303

77217

22783

06520

93480

23

38

70718

29282

77246

22764

06528

93172

22

39

70739

29261

77274

22726

06535

93465

21

40

9.70761

10.29239

9.77303

10.22697

10.06543

9.93467

20

41

70782

29218

77332

22668

06550

93450

19

42

70803

29197

77361

22639

06558

93442

18

43

70824

29176

77390

22610

06565

93435

17

44

70846

29154

77418

22582

06573

93427

16

45

9.70867

10.29133

9.77447

10.22553

10.06680

9.93420

15

46

70888

29112

77476

22524

06588

93412

14

47

70909

29091

77505

22495

06695

93405

13

48

70931

29069

77533

22467

06603

93397

12

49

70952

29048

77562

22438

06610

93390

11

50

9.70973

10.29027

9.77591

10.22409

10.06618

9.93382

10

51

70994

29006

77619

22381

06625

93375

9

52

71015

28985

77648

22352

06633

93367

8

53

71036

28964

77677

22323

06640

93360

7

54

71058

28942

77706

22294

06648

93352

6

55

9.71079

10.28921

9.77734

10.22266

10.06656

9.93,344

5

56

71100

28900

77763

22237

06663

93337

4

57

71121

28879

77791

22209

06671

93329

3

68

71142

28858

77820

22180

06678

93322

2

59

71163

28837

77849

22161

06686

93314

1

60

71184

28816

77877

22123

06693

93307

M.

Cosine.

Secant.

Cotangent.

Tangent.

Cosecant.

Sine.

M.

120°

59°

308

LOGAEITHMIC ANGULAR FUNCTIONS. Table 2.

31°

Logarithms.

J48°

M.

Sine.

Cosecant.

Tangent.

Cotangent.

Secant.

Cosine.

M.

9.71184

10.28816

9.77877

10.22123

10.06693

9.93307

60

1

71205

28795

77906

22094

06701

93299

69

2

71226

28774

77935

22065

06709

93291

58

3

71247

28753

77963

22037

06716

93284

57

4

71268

28732

77992

22008

06724

93276

66

5

9.71289

10.28711

9.78020

10.21980

10.06731

9.93269

55

6

71310

28690

78049

21951

06739

93261

54

7

71331

28669

78077

21923

06747

93253

53

8

71352

28648

78106

21894

06754

93246

52

9

71373

28627

78135

21865

06762

93238

51

10

9.71393

10.28607

9.78163

10.21837

10.06770

9.93230

50

11

71414

28586

78192

21808

06777

93223

49

12

71435

28565

78220

21780

06785

93216

48

13

71456

28544

78249

21751

06793

93207

47

14

71477

28523

78277

21723

06800

93200

46

15

9.71498

10.28502

9.78306

10.21694

10.06808

9.93192

46

16

71519

28481

78334

21666

06816

93184

44

17

71539

28461

78363

21637

06823

93177

43

18

71560

28440

78391

21609

06831

93169

42

19

71581

28419

78419

21581

06839

93161

41

20

9.71602

10.28398

9.78448

10.21562

10.06846

9.93154

40

21

71622

28378

78476

21524

06854

93146

39

22

71643

28367

78505

21495

06862

93138

38

28

71664

28336

78533

21467

06869

93131

37

24

71685

28315

78562

21438

06877

93123

36

25

9.71705

10.28295

9.78590

10.21410

10.06885

9.93115

35

26

71726

28274

78618

21382

06892

93108

34

27

71747

28253

78647

21353

06900

93100

33

28

71767

28233

78675

21325

06908

93092

32

29

71788

28212

78704

21296

06916

93084

31

30

9.71809

10.28191

9.78732

10.21268

10.06923

9.93077

30

31

71829

28171

78760

21240

06931

93069

29

32

71860

28150

78789

21211

06939

93061

28

33

71870

28130

78817

21183

06947

93063

27

34

71891

28109

78845

21165

06954

93046

26

35

9.71911

10.28089

9.78874

10.21126

10.06962

9.93038

25

36

71932

28068

78902

21098

06970

93030

24

37

71952

28048

78930

21070

06978

93022

23

38

71973

28027

78959

21041

06986

93014

22

39

71994

28006

78987

21013

06993

93007

21

40

9.72014

10.27986

9.79015

10.20985

10.07001

9.92999

20

41

72034

27966

79043

20967

07009

92991

19

42

72065

27945

79072

20928

07017

92983

18

43

72075

27925

79100

20900

07024

92976

17

44

72096

27904

79128

20872

07032

92968

16

45

9.72116

10.27884

9.79156

10.20844

10.07040

9.92960

15

46

72137

27863

79185

20815

07018

92952

14

47

72157

27843

79213

20787

07066

92944

13

48

72177

27823

79241

20769

07064

92936

12

49

72198

27802

79269

20731

07071

92929

11

50

9.72218

10.27782

9.79297

10.20703

10.07079

9.92921

10

51

72238

27762

79326

20674

07087

92913

9

52

72259

27741

79354

20646

07095

92906

8

53

72279

27721

79382

20618

07103

92897

7

54

72299

27701

79410

20690

07111

9'2889

6

65

9.72320

10.27680

9.79438

10.20562

10.07119

9.92881

5

56

72340

27660

79466

20534

07126

92874

4

57

72360

27640

79495

20505

07134

92866

3

58

72381

27619

79523

20477

07142

92858

2

59

72401

27599

79551

20449

07150

92850

1

60

72421

27579

79579

20421

07158

92842

M.

Cosine.

Secant.

Cotangent.

Tangent.

Cosecant.

Sine.

M.

121°

58°

Table 3. LOGARITHMIC ANGULAR FUNCTIONS.

309

32°

Logarithms.

147°

M.

Sine.

Cosecant.

Tangent.

Cotangent.

Secant.

Cosine.

M.

9.72421

10.27579

9.79579

10.20421

10.07158

9.92842

60

1

72441

27559

79607

20393

07166

92834

59

2

72461

27539

79635

20365

07174

92826

58

3

72482

27518

79663

20337

07182

92818

57

4

72502

27498

79691

20309

07190

92810

56

5

9.72522

10.27478

9.79719

10.20281

10.07197

9.92803

56

6

72642

27458

79747

20253

07205

92795

54

7

72562

27438

79776

20224

07213

92787

53

8

72582

27418

79804

20196

07221

92779

52

9

72602

27398

79832

20168

07229

92771

51

10

9.72622

10.27378

9.79860

10.20140

10.07237

9.92763

50

11

72643

27357

79888

20112

07245

92765

49

12

72663

27337

79916

20084

07253

92747

48

13

72683

27317

79944

20056

07261

92739

47

14

72703

27297

79972

20028

07269

92731

46

15

9.72723

10.27277

9.80000

10.20000

10.07277

9.92723

45

16

72743

27257

80028

19972

07285

92715

44

17

72763

27237

80056

19944

07293

92707

43

18

72783

27217

80084

19916

07301

92699

42

19

72803

27197

80112

19888

07309

92691

41

20

9.72823

10.27177

9.80140

10.19860

10.07317

9.92683

40

21

72843

27157

80168

19832

07325

92675

39

22

72863

27137

80195

19805

07333

92667

38

23

72883

27117

80223

19777

07341

92659

37

24

72902

27098

80251

19749

07349

92661

36

25

9.72922

10.27078

9.80279

10.19721

10.07357

9.92643

35

26

72942

2'7058

80307

19693

07365

92635

34

27

72962

27038

80335

19665

07373

92627

33

28

72982

27018

80363

19637

07381

92619

32

29

73002

20998

80391

19609

07389

92611

31

30

9.73022

10.26978

9.80419

10.19581

10.07397

9.92603

30

31

73041

26959

80447

19553

07405

92595

29

32

73061

26939

80474

19526

07413

92587

28

83

73081

26919

80502

19498

07421

92579

27

84

73101

26899

80530

19470

07429

92571

26

35

9.73121

10.26879

9.80558

10.19442

10.07437

9.92563

25

36

73140

26860

80586

19414

07445

92555

24

37

73160

26840

80614

19386

07454

92546

23

88

73180

• 26820

80642

19358

07462

92538

22

39

73200

26800

80669

19331

07470

92530

21

40

9.73219

10.26781

9.80697

10.19303

10,07478

9.92622

20

41

73239

26761

80725

19275

07486

92614

19

42

73259

26741

80753

19247

07494

92506

18

43

73278

26722

80781

19219

07502

92498

17

44

7329S

26702

80808

19192

07510

92490

16

45

9.73318

10.26682

9.80836

10.19164

10.07518

9.92482

15

46

73337

26663

80864

19136

07527

92473

14

47

73357

26643

80892

19108

07535

92465

13

48

73377

26623

80919

19081

07543

92467

12

49

73396

26604

80947

19053

07551

92449

11

50

9.73416

10.26584

9.80975

10.19025

10.07559

9.92441

10

51

73435

26565

81003

18997

07567

92433

9

52

73455

26545

81030

18970

07575

92425

8

53

73474

26526

81058

18942

07584

92416

7

54

73494

26506

81086

18914

07592

92408

6

55

9.73513

10.26487

9.81113

10.18887

10.07600

9.92400

5

56

73533

26167

81141

18859

07608

92392

4

57

73652

26448

81169

18831

07616

92384

3

58

73572

26428

81196

18804

07624

92376

2

59

73591

26409

81224

18776

07633

92367

1

60

73611

26389

81252

18748

07641

92359

M.

CosinG.

Secant.

Cotangent.

Tangent.

CoBecant.

Sine.

M.

122°

57°

110

LOGARITHMIC ANGULAR FUNCTIONS. Table 2.

3°

Logarithms.

146°

9.

Sine.

Cosecant.

Tangent.

Cotangent,

Secant.

Cosine.

M.

9.73611

10.26389

9.81252

10.18748

10.07641

9.92359

60

1

73630

26370

81279

18721

07649

92351

59

2

73650

26360

81307

18693

07657

92343

58

3

73069

26331

813.35

18665

07665

92335

57

4

73689

26311

81362

18638

07674

92326

56

5

9.73708

10.26292

9.81390

10.18610

10.07682

9.92:318

55

6

73727

26273

81418

ia5S2

07690

9-2310

54

7

73747

26253

81445

18556

07698

92302

53

8

73766

26234

81473

18627

07707

92293

52

9

73785

26215

81500

18500

07715

92285

51

10

9.73805

10.26196

9.81628

10.18472

10.07723

9.922/7

50

LI

73824

20176

81656

18444

07731

92269

49

L2

73843

26167

81683

18417

07740

922b0

48

13

73863

26137

81611

18389

07748

92262

47

14

73882

26118

81638

18362

07756

92244

46

15

9.73901

10.26099

9.81666

10.18334

10.07765

9.92236

45

16

73921

26079

81693

18307

07773

92227

44

17

73940

26060

81721

18279

07781

92219

43

18

73959

26041

S1748

18262

07789

92211

42

19

73978

26022

81776

18224

07798

92202

41

20

9.73997

10.26003

9.81803

10.18197

10.07806

9.92194

40

21

74017

25983

81831

18169

07814

92186

39

22

74036

25964

81868

18142

07823

92177

38

23

74055

25945

81886

18114

07831

92169

37

24

74074

25926

81913

18087

07839

92161

36

25

9.74093

10.25907

9.81941

10.18059

10,07848

9.92152

35

26

74113

25887

81968

18032

07856

92144

34

27

74132

25868

81996

18004

07864

92136

33

28

74161

25849

82023

17977

07873

92127

32

29

74170

25830

82051

17949

07881

92119

31

3D

9.74189

10.25811

9.82078

10.17922

10.07889

9.92111

30

31

74208

25792

82106

17894

07898

92102

29

32

74227

25773

82133

17867

07906

92094

28

33

74246

25754

82161

17839

07914

92086

27

34

74265

25735

82188

17812

07923

92077

26

35

9.74284

10.25716

9.82216

10.17785

10.07931

9.92069

25

36

74303

25697

82243

17757

07940

92060

24

37

74322

26678

82270

17730

07948

92052

23

38

74341

25659

82298

17702

07956

92044

22

39

74360

26640

82325

17676

07965

92035

21

10

9.74379

10.26621

9.82352

10.17648

10.07973

9.92027

20

41

74398

25602

82380

17620

07982

92018

19

12

74417

25583

82407

17593

07990

92010

18

13

74436

25564

82436

17566

07998

92002

17

14

74455

25645

82462

17538

08007

91993

16

15

9.74474

10.25526

9.82489

10.17511

10.08015

9.91985

15

16

74493

25607

82517

17483

08024

91976

14

17

74512

25488

82544

17456

08032

91968

13

18

74531

26469

82671

17429

08041

91969

12

19

74649

25451

82699

17401

08049

91951

11

50

9.74568

10.25432

9.82626

10.17374

10.08068

9.91942

10

51

74587

26413

82653

17347

08066

91934

9

52

74606

25394

82681

17319

08076

91925

8

53

74625

25375

82708

17292

08083

91917

7

54

74644

26356

82735

17266

08092

91908

6

55

9.74662

10.25338

9.82762

10.17238

10.08100

9.91900

6

56

74681

25319

82790

17210

08109

91891

4

57

74700

25300

82817

17183

08117

91883

3

58

74719

25281

82844

17156

08126

91874

2

59

74737

25263

82871

17129

08134

91866

1

60

74756

26244

82899

17101

08143

91857

M.

Cosine.

Secant.

Cotan{?ent.

Tangent.

Cosecant.

Sine.

M:

23°

Table 2. LOGARITHMIC ANGULAR FUNCTIONS.

311

54°

Logarithms.

145°

M.

Sine.

Cosecant.

Tangent.

Cotangent.

Secant,

Cosine.

M.

9.74756

10.25244

9.82899

10.17101

10.08143

9.91857

60

1

74775

25225

82926

17074

08151

91849

59

2

74794

25206

82953

17047

08160

91840

58

3

74812

25188

82980

17020

08168

91832

57

4

74831

25169

83008

16992

08177

91823

66

5

9.74850

10.25150

9.83035

10.16965

10.08185

9.91815

55

6

74868

25132

83062

16938

08194

91806

54

7

74887

25113

83089

16911

08202

91798

53

8

74906

25094

83117

16883

08211

91789

52

9

74924

25076

83144

16856

08219

91781

51

10

9.74943

10.25057

9.83171

10.16829

10.08228

9.91772

50

11

74961

25039

83198

16802

08237

91763

49

12

74980

25020

83225

16775

08245

91755

48

13

74999

25001

83252

16748

08254

91746

47

14

75017

24983

83280

16720

08262

91738

46

15

9.75036

10.24964

9.83307

10.16693

10.08271

9.91729

45

16

75054

24946

83334

16666

08280

91720

44

17

75073

24927

83361

16639

08288

91712 ■

43

18

75091

24909

83388

16612

08297

91703

42

19

75110

24890

83415

16585

08305

91695

41

20

9.75128

10.24872

9.83442

10.16558

10.08314

9.91686

40

21

75147

24853

83470

16530

08323

91677

39

22

75165

24835

83497

16503

08331

91669

38

23

75184

21816

83524

16476

08340

91660

37

24

75202

24798

83551

16449

08349

91651

36

25

9.75221

10.24779

9.83578

10.16422

10.08357

9.91643

35

26

75239

24761

83605

16395

08366

91634

34

27

75258

24742

83632

16368

08375

91625

33

28

75276

21724

83659

16341

08383

91617

32

29

75294

24706

83686

16314

08392

91608

31

30

9.75313

10.24687

9.83713

10.16287

10.08401

9.91599

30

31

75331

24669

83740

16260

08409

91691

29

32

75350

24650

83768

16232

08418

91682

28

33

75368

24632

83795

16205

08427

91573

27

34

75386

24614

83822

16178

08435

91665

26

85

9.75405

10.21595

9.83849

10.16151

10.08444

9.91656

25

36

75423

24577

83876

16124

08453

91547

24

37

75441

24559

83903

16097

08462

91538

23

38

75459

24541

83930

16070

08470

91530

22

39

75178

24522

83957

16043

08479

91521

21

40

9.75496

10.24504

9.83984

10.16016

10.08488

9.91612

20

41

75514

24486

84011

15989

08496

91604

19

42

75533

24467

84038

15962

08505

91495

18

43

75551

24449

84065

15935

08514

91486

17

44

75569

24431

84092

15908

08523

91477

16

45

9.75587

10.21413

9.84119

10.15881

10.08531

9.91469

15

46

75605

24395

84146

15854

08540

91460

14

47

75624

24376

84173

15827

08549

91451

13

48

75642

24358

84200

15800

08558

91442

12

49

75660

24340

84227

15773

08567

91433

11

50

9.75678

10.24322

9.84254

10.15746

10.08575

9.91425

10

51

75696

24304

84280

15720

08584

91416

9

52

75714

24286

84307

15693

08593

91407

8

53

75733

24267

84334

15666

08602

91398

7

54

75751

24249

84361

15639

08611

91389

6

55

9.75769

10.24231

9.84388

10.15612

10.08619

9.91381

5

56

75787

24213

84415

15585

08628

91372

4

57

75805

24195

84442

15568

08637

91363

3

5S

75823

24177

84469

15531

08646

91354

2

59

76841

24159

84496

15504

■08655

91345

1

60

75859

24141

84523

15477

08664

91336

M.

Cosine.

Secant.

Cotangent

Tangent.

Cosecant.

Sine.

M.

124°

J12 LOGARITHMIC ANGULAR FUNCTIONS. Table 2.

(5°

Logarithms.

144°

M.

Sine.

Cosecant.

Tangent.

Cotangent.

Secant.

Cosine.

M.

9.75859

10.24141

9.84523

10.15477

10.08664

9.91336

60

1

75877

24123

84650

15460

08672

91328

59

2

75895

21105

84576

15424

08681

91319

58

3

75913

24087

84603

15397

08690

91310

57

4

75931

24069

84630

15370

08699

91301

56

5

9.75949

10.24051

9.84657

10.15343

10.08708

9.91292

65

6

75967

24033

84684

15316

08717

91283

54

7

75985

24015

84711

15289

08726

91274

53,

8

76003

23997

84738

15262

08734

91266

52

9

76021

23979

84764

15236

08743

91257

51

10

9.76039

10.23961

9.84791

10.15209

10.08752

9.91248

50

11

76057

23943

84818

15182

08761

91239

49

n

76076. .

23925

84845

15155

08770

91230

48

13

76093

23907

84872

15128

08779

91221

47

14

76111

23889

84899

15101

08788

91212

46

15

9.76129

10.23871

9.8-1925

10.15075

10.08797

9.91203

.45

16

76146

23854

84952

15048

08806

91194

44

17

76164

23836

84979

16021

08815

91185

43

18

76182

23818

86006

14994

08824

91176

42

19

76200

23800

86033

14967

08833

91167

41

20

9.76218

10.23782

9.85059

10.14941

10.08842

9.91158

40

21

76236

23764

85086

14914

08851

91149

39

22

76253

23747

85113

14887

08859

91141

38

23

76271

23729

8.5140

14860

08868

91132

37

24

76289

23711

85166

14834

08877

911^^

36

25

9.76307

10.23693

9.85193

10.14807

10.08886

9.91114

35

26

76324

23676

85220

14780

08895

91105

34

27

76342

23658

85247

14753

08904

91096

33

28

76360

23640

85273

14727

08913

91087

32

29

76378

23622

85300

14700

08922

91078

31

)0

9.76395

10.23606

9.86327

10.14673

10.08931

9.91069

30

Jl

76413

23587

85364

14646

08940

91060

29

!2

76431

2:3569

86380

14620

08949

91051

28

i3

76448

23652

86407

14693

08958

91042

27

!4

76466

23534

85434

14566

08967

91033

26

!5

9.76484

10.23516

9.85460

10.14540

10.08977

9.91023

25

56

76501

23499

85487

14513

08986

91014

24

57

76519

23481

86514

14486

08995

91005

23

!8

76537

23463

85540

14460

09004

90996

22

i9

76554

23446

85567

14433

09013

90987

21

10

9.76672

10.23428

9.85594

10.14406

10.09022

9.90978

20

11

76690

23410

86620

14380

09031

90969

19

12

76607

23393

85647

14.363

09040

90960

18

13

76625

23375

85674

14326

09049

90961

17

14

76642

23368

85700

14300

09058

90942

16

15

9.76660

10.23340

9.85727

10.14273

10.09067

9.90933

15

16

76677

23323

86754

14246

09076

90924

14

17

76695

23305

86780

14220

09085

90915

13

18

76712

23288

85807

14193

09094

90906

12

19

767.30

23270

85834

14166

09104

90896

11

lO

9.76747

10.23253

9.85860

10.14140

10.09113

9.90887

10

p1

76765

23235

85887

14113

09122

90878

9

i2

76782

23218

85913

14087

09131

90869

8

.3

76800

23200

85940

14060

09140

90860

7

.4

76817

23183

86967

14033

09149

90851

6

.5

9.76835

10.23165

9.85993

10.14007

10.09158

9.90842

5

i6

76852

23148

86020

13980

09168

90832

4

i7

76870

23130

86046

13954

09177

90823

3

iS

76887

23113

86073

13927

09186

90814

2

i9

76904

23096

86100

13900

09195

90805

1

iO

76922

23078

86126

13874

09204

90796

I.

Cosine.

Secjlnt.

Cofcmgent.

Tangent.

Cosecant.

Sine.

M.

25°

Table 2. LOGAEIJHMIC ANGULAR FUNCTIONS. 313

36°

Logarithms.

143°

M.

Sine.

Cosecant.

Tangent.

Cotangent.

Secant.

Cosine.

M.

9.76922

10.23078

9.86126

10.13874

10.09204

9.90796

60

1

76939

23061

86153

13847

09213

90787

59

2

76957

23043

86179

13821

09223

90777

58

3

76974

23026

86206

13794

09232

90768

67

4

76991

23009

86232

13768

09241

90769

66

5

9.77009

10.22991

9.86259

10.13741

10.09250

9.90750

55

6

77026

22974

86285

13716

09269

90741

54

7

77043

22957

86312

13688

09269

90731

63

8

77061

22939

86338

13662

09278

90722

52

9

77078

22922

86365

13635

09287

90713

51

10

9.77095

10.22905

9.86392

10.13608

10.09296

9.90704

50

11

77112

22888

86418

13,582

09306

90694

49

12

77130

22870

86445

13555

09315

90685

48

13

77147

22853

86471

13529

09324

90676

47

14

77164

22836

86498

13502

09333

90667

46

16

9.77181

10.22819

9.86524

10.13476

10.09343

9.90667

45

16

77199

22801

86551

13449

09352

90648

44

17

77216

22784

86577

13423

09361

90639

43

18

77233

22767

86603

13397

09370

90630

42

19

77250

22750

86630

13370

09380

90620

41

20

9.77268

10.22732

9.86656

10.13344

10.09389

9.90611

40

21

77285

22715

86683

13317

09398

90602

39

22

77302

22698

86709

13291

09408

90592

38

23

77319

22681

86736

13264

09417

90683

37

24

77336

22664

86762

13238

09426

90574

36

25

9.77353

10.22647

9.86789

10.13211

10.09435

9.90665

35

26

77370

22630

86815

13185

09445

90556

34

27

77387

22613

86842

13168

09464

90546

33

28

77405

22595

86868

13132

09463

90537

32

29

77422

22578

86894

13106

09473

90627

31

30

9.77439

10.22561

9.86921

10.13079

10.09482

9.90618

30

31

77456

22544

86947

13053

09491

90509

29

32

77473

22527

86974

13026

09501

90499

28

33

77490

22510

87000

13000

09610

90490

27

34

77507

22493

87027

12973

09520

90480

26

35

9.77524

10.22476

9.87053

10.12947

10.09529

9.90471

25

36

77541

22459

87079

12921

09538

90462

24

37

77558

22442

87106

12894

09548

90452

23

38

77575

22425

87132

12868

09557

90443

22

39

77592

22408

87158

12842

09666

90434

21

40

9.77609

10.22391

9.87185

10.12815

10.09676

9.90424

20

41

77626

22374

87211

12789

09685

90415

19

42

77643

22357

87238

12762

09595

90405

18

43

77660

22340

87264

12736

09604

90396

17

44

77677

22323

87290

12710

09614

90386

16

45

9.77694

10.22306

9.87317

10.12683

10.09623

9.90377

15

46

77711

22289

87343

12657

09632

90368

14

47

77728

22272

87369

12631

09642

90358

13

48

77744

22-256

87396

12604

09651

90349

12

49

77761

22239

87422

12678

09661

90339

11

50

9.77778

10.22222

9.87448

10.12652

10.09670

9.90330

10

51

77795

22205

87475

12525

09680

90320

9

52

77812

22188

87501

12499

09689

90311

8

53

77829

22171

87527

12473

09699

90301

7

54

77846

22154

87564

12446

09708

90292

6

55

9.77862

10.22138

9.87680

10.12420

10.09718

9.90282

5

56

77879

22121

87606

12394

09727

90273

4

57

77896

22104-

87633

12367

09737

90263

3

58

77913

22087

87669

12341

09746

90254

2

59

77930

22070

87685

12315

09766

90244

1

60

77946

22054

87711

12289

09765

90235

M.

Cosine.

Secant.

Cotangent.

Tangent.

Cosecant.

Sine.

M.

126°

22

53°

514

LOGARITHMIC ANGULAR FUNCTIONS. Table 3.

Logarithms.

142°

M.

Sine.

Cosecant.

Tangent.

Cotangent,

Secant.

Cosine.

M.

9.77946

10.22054

9.87711

10.12289

10.09765

9.90235

60

1

77963

22037

87738

12262

09776

902'2,5

59

2

77980

22020

87764

12236

09784

90216

58

3

77997

22003

87790

12210

09794

90200

57

4

78013

21987

87817

12183

09803

90197

66

6

9.78030

10.21970

9.87843

10.12167

10.09813

9.90187

55

6

78047

21953

87869

12131

09822

90178

54

7

78063

21937

87895

12106

09832

90168

53

8

78080 .

21920

87922

12078

09841

90169

52

9

78097

21903

87948

12052

09851

90149

61

10

9.78113

10.21887

9.87974

10.12026

10.09861

9.90139

50

11

78130

21870

88000

12000

09870

90130

49

12

78147

21853

88027

11973

09880

90120

48

13

78163

21837

88053

11947

09889

90111

47

14

78180

21820

88079

11921

09899

90101

46

15

9.78197

10.21803

9.88105

10.11896

10.09909

9.90091

45

16

78213

21787

88131

11869

09918

90082

44

17

78230

21770

88158

11842

09928

90072

43

18

78246

21764

88184

11816

09937

90063

42

19

7«263

21737

88210

11790

09947

90063

41

20

9.78280

10.21720

9.88236

10.11764

10.09957

9.90043

40

21

78296

21704

88262

11738

09966

90034

39

22

78313

21687

88289

11711

09976

90024

38

23

78329

21671

88315

11686

09986

90014

37

24

78346

21654

88341

11659

09995

90005

36

25

9.78362

10.21638

9.88367

10.11633

10.10005

9.89995

36

26

78379

21621

88393

11607

10015

89985

34

27

78395

21606

88420

11680

10024

89976

33

28

78412

21588

88446

11664

10034

89966

32

29

78428

21572

88172

11628

10044

89956

31

30

9.78445

10.21555

9.88498

10.11.602

10.10063

9.89947

30

31

78461

21539

88524

11476

10063

89937

29

32

78478

21522

88660

11460

10073

89927

28

33

78494

21506

88677

11423

10082

89918

27

34

78510

21490

88603

11397

10092

89908

26

35

9.78527

10.21473

9.88629

10.11371

10.10102

9.89898

26

36

78543

21457

88655

11346

10112

89888

24

37

78560

21440

88681

11319

10121

89879

23

38

78576

21424

88707

11293

10131

89869

22

39

78592

21408

88733

11267

10141

89859

21

40

9.78609

10.21391

9.88759

10.11241

10.10151

9.89849

20

11

78625

21375

88780

11214

10160

89840

19

12

78642

21358

88812

11188

10170

89830

18

13

78658

21342

88838

11162

10180

89820

17

14

78674

21326

88864

11136

10190

89810

16

16

9.78691

10.21309

9.88890

10-11110

10.10199

9.89801

16

16

78707

21293

88916

11084

10209

89791

14

17

78723

21277

88942

11058

10219

89781

13

18

78739

21261

88968

11032

10229

89771

12

19

78756

21244

88994

11006

10239

89761

11

)0

9.78772

10.21228

9.89020

10.10980

10.10248

9.89752

10

)1

78788

21212

89046

10954

10258

89742

9

)2

78805

21195

89073

10927

10268

89732

8

)3

78821

21179

89099

10901

10278

89722

7

)4

78837

21163

89125

10875

10288

89712

6

)5

9.78853

10.21147

9.891.51

10.10849

10.10298

9.89702

5

)6

78869

21131

89177

10823

10307

89693

4

i7

78886

21114

89203

10797

10317

89683

3

i8

78902

21098

89229

10771

■ 10327

89673

2

.9

78918

21082

89255

10746

10337

89663

1

;o

78934

21066
StTunt.

89281

10719

10347

89653

I.

Cosine.

Cotangent.

Tangent. |

Cosecant.

Sine.

M.

27°

52°

Table 2. LOGARITHMIC ANGULAR FUNCTIONS.

315

38°

Logarithms.

141°

M.

Sine.

CoBecaut.

Tangent.

Cotangent.

Secant.

Cosine.

M.

9.78934

10.21066

9.89281

10.10719

10.10347

9.89653

60

1

78950

21050

89307

10693

10357

89643

59

2

78967

21033

89333

10667

10367

89633

58

3

78983

21017

89359

10641

10376

89624

57

4

78999

21001

89385

10615

10386

89614

56

5

9.79015

10.20985

9.89411

10.10589

10.10396

9.89604

56

6

79031

20969

89437

10563

10406

89594

54

7

79017

20953

89463

10537

10416

89584

53

8

79063

20937

89489

10511

10426

89574

52

9

79079

20921

89515

10485

10436

89564

51

10

9.79095

10.20905

9.89541

10.10459

10.10446

9.89554

50

U

79111

20889

89567

10433

10466

89544

49

12

79128

20872

89593

10407

10466

89534

48

13

79144

20856

89619

10381

10476

89624

47

14

79160

20840

89515

10365

10486

89614

46

15

9.79176

10.20824

9.89671

10.10329

10.10496

9.89504

45

16

79192

20808

89697

10303

10505

89495

44

17

79208

20792

89723

10277

10515

89486

43

18

79224

20776

89749

10251

105'2.5

89476

42

19

79240

20760

89775

10225

10536

89465

41

20

9.79256

10.20744

9.89801

10.10199

10.10546

9.89455

40

21

79272

20728

89827

10173

10655

89445

39

22

79288

20712

89853

10147

10665

89435

38

23

79304

20696

89879

10121

10575

89426

37

24

79319

20681

89905

10095

10585

89416

36

25

9.79335

10.20665

9.89931

10.10069

10.10595

9.89406

35

26

79351

20649

89957

10043

10606

89395

34

27

79367

20633

89983

10017

10616

89385

33

28

79383

20617

90009

09991

10625

89375

32

29

79399

20601

90035

09965

10636

89364

31

30

9.79415

10.20585

9.90061

10.09939

10.10646

9.89354

30

31

79431

20569

90086

09914

10656

89344

29

32

79447

20553

90112

09888

10666

89334

28

33

79463

20537

90138

09862

10676

89324

27

34

79478

20522

90164

09836

10686

89314

26

35

9.79494

10.20506

9.90190

10.09810

10.10696

9.89304

25

36

79510

20490

90216

09784

10706

89294

24

»

79526

20474

90242

09758

10716

89284

23

38

79542

20458

90268

09732

10726

89274

22

39

79558

20442

90294

09706

10736

89264

21

■40

9.79573

10.20427

9.90320

10.09680

10.10746

9.89254

20

41

79589

20111

90346

09664

10756

89244

19

42

79605

20395

90371

09629

10767

89233

18

43

79621

20379

90397

09603

10777

89223

17

44

79636

20364

90423

09577

10787

89213

16

45

9.79652

10.20348

9.90449

10.09551

10.10797

9.89203

16

46

79668

20332

90475

09526

10807

89193

14

47

79684

20316

90501

09499

10817

89183

13

48

79699

20301

90527

09473

10827

89173

12

49

79715

20285

90553

09447

10838

89162

11

50

9.79731

10.20269

9.90578

10.09422

10.10848

9.89152

10

51

79746

20254

90604

09396

10858

89142

9

52

79762

20238

90630

09370

10868

89132

8

53

79778

20222

90656

09344

10878

89122

7

64

79793

20207

90682

09318

10888

89112

6

55

9.79809

10.20191

9.90708

10.09292

10.10899

9.89101

5

56

79825

20175

90784

09266

10909

89091

4

57

79840

20160

90759

09241

10919

89081

3

58

79856

20144

90785

09215

10929

89071

2

59

79872

20128

90811

09189

10940'

89060

1

60

79887

20113

90837

09163

10950

89060

M.

CoBioe.

Secant.

Cotangent.

Tangent,

Cosecant.

Sine. ,

M.

128°

51°

;16 LOGARITHMIC ANGULAR FUNCTIONS. Table 3.

9°

Logar

thms.

140°

H.

Sine.

Cosecant.

Tangent.

Cotangent.

Secant.
10.10950

Cosine.

M.

9.79887

10.20113

9.90837

10.09163

9.89050

60

1

79903

20097

90863

09137

10960

89040

59

2

79918

20082

90889

09111

10970

89030

58

3

79934

20066

90914

09086

10980

89020

67

4

79950

20050

90940

09060

10991

89009

56

5

9.79965

10.20035

9.90966

10.09034

10.11001

9.88999

55

6

79981

20019

90992

09008

11011

88989

64

7

79995

20004

91018

08982

11022

88978

53

8

80012

19988

91043

08957

110.32

88968

52

9

80027

19973

91069

08931

11042

889.58

51

10

9.80043

10.19957

9.91095

10.08905

10.110.52

9.88948

.60

11

80058

19942

91121

08879

11063

88937

49

12

80074

19926

91147

08853

11073

88927

48

13

80089

19911

91172

08828

11083

88917

47

14

80105

19895

91198

08802

11094

88906

46

15

9.80120

10.19880

9.91224

10.08776

10.11104

9.88896

45

Ifi

80136

19864

91250

08750

11114

88886

44

17

80151

19849

91276

08724

11125

88875

43

18

80166

19834

91301

08699

11135

88865

42

19

80182

19818

91327

08673

11145

88855

41

20

9.80197

10.19803

9.91353

10.08647

10.111.56

9.88844

40

21

80213

19787

91.379

08621

11166

88834

39

22

80228

19772

91404

08696

11176

88824

38

23

80244

19756

91430

08570

11187

88813

37

24

80259

19741

91456

08.544

11197

88803

36

25

9.80274

10.19726

9.91482

10.08518

10.11207

9.88793

35

26

80290

19710

91507

08493

11218

88782

34

27

80305

19695

91533

08467

11228

88772

33

28

80320

19680

91559

08441

11239

88761

32

29

80336

19664

91585

08416

11249

88761

31

30

9.80351

10.19649

9.91610

10.08390

10.11259

9.88741

30

31

80366

19634

91636

08364

11270

88730

29

32

80382

19618

91662

08338

11280

88720

28

33

80397

19603

91688

08312

11291

88709

27

34

80412

19588

91713

08287

11301

88699

26

35

9.80428

10.19572

9.91739

10.08261

10.11312

9.88688

25

36

80443

19.157

91765

08235

11322

88678

24

37

80458

19.V12

91791

08209

11332

88668

23

38

80473

19527

91816

08184

11343

88657

22

39

80489

19511

91842

08158

11353

88647

21

40

9.80504

10.19496

9.91868

10.08132

10.11364

9.88636

20

41

80519

19481

91893

08107

11374

88626

19

42

80534

19466

91919

08081

11385

88616

18

43

80550

19450

91945

08056

11395

88605

17

44

80565

19435

91971

08029

11406

88594

16

45

9.80580

10.19420

9.91996

10.08004

10.11416

9.88584

15

46

80595

19405

92022

07978

11427

88573

14

47

80610

19390

92048

07952

11437

88563

13

48

80626

19375

92073

07927

11448

88552

12

49

80641

19359

92099

07901

114.58

88542

11

50

9.80656

10.19344

9.92126

10.07875

10.11469

9.88531

10

61

80671

19329

92150

07850

11479

88521

9

52

80686

19314

92176

07824

11490

88510

8

53

80701

19299

92202

07798

11501

88499

7

54

80716

19284

92227

07773

11.511

88489

6

55

9.80731

10.19269

9.92253

10.07747

10.11522

9.88478

5

56

80746

19254

92279

07721

11532

88468

4

57

80762

19238

92304

07696

11543

88467

3

58

80777

19223

92330

07670

11553

88447

2

59

80792

19208

92356

07644

11564

&84a6

1

60

80807

19193

92381

07619

11575

88426

M.

Cosine.

Secant.

Cotangent.

Tangent.

Cosecant.

Sine.

M.

29°

50°

Table 2. LOGARITHMIC ANGULAE FUNCTIONS. 317

40°

Logarithms.

39°

M.

Sine.

Cosecant.

Tangent.

Cotangent.

Secant.

Cosine.

M.

9.80807

10.19193

9.92381

10.07619

10.11575

9.88425

60

1

80822

19178

92407

07593

11585

88415

59

2

80837

19163

92433

07667

11596

88404

58

3

80a62

19148

92158

07542

11606

88394

57

4

80867

19133

92484

07516

11617

88383

66

5

9.80882

10.19118

9.92510

10.07490

10.11628

9.88372

66

6

80897

19103

92635

07466

11638

88362

54

7

80912

19088

92561

07439

11649

88361

53

8

80927

19073

92687

07413

11660

88340

62

9

80942

19058

92612

07388

11670

88330

51

10

9.80957

10.19043

9.92638

10.07362

10.11681

9.88319

50

11

80972

19028

92663

07337

11692

88308

49

12

80987

19013

92689

07311

11702

88298

48

13

81002

18998

92716

07285

11713

88287

47

14

81017

18983

92740

07260

11724

88276

46

15

9.81032

10.18968

9.92766

10.07234

10.11734

9.88266

45

16

81047

18953

92792

07208

11745

88266

44

17

81061

18939

92817

07183

11766

88244

43

18

81076

18924

92843

07167

11766

88234

42

19

81091

18909

92868

07132

11777

88223

41

20

9.81106

10.18894

9.92894

10.07106

10.11788

9.88212

40

21

81121

18879

92920

07080

11799

88201

39

22

81136

18864

92945

07065

11809

88191

38

23

81161

18849

92971

07029

11820

88180

37

24

81166

18834

92996

07004

11831

88169

36

25

9.81180

10.18820

9.93022

10.06978

10.11842

9.88158

36

26

81195

18805

98048

06952

11852

88148

34

27

81210

18790

93073

06927

11863

88137

33

28

81226

18775

93099

06901

11874

88126

32

29

81240

18760

93124

06876

11886

88115

31

30

9.81254

10.18746

9.93160

10.06850

10.11896

9.88106

30

31

81269

18731

93175

06825

11906

88094

29

32

81284

18716

93201

06799

11917

88083

28

33

81299

18701

93227

06773

11928

88072

27

34

81314

. 18686

93262

06748

11939

88061

26

35

9.81328

10.18672

9.93278

10.06722

10.11949

9.88061

26

36

81343

18657

93303

06697

11960

88040

24

37

81358

18642

93329

06671

11971

88029

23

38

81372

18628

93354

06646

11982

88018

22

39

81387

18613

93380

06620

11993

88007

21

40

9.81402

10.18598

9.93406

10.06594

10.12004

9.87996

20

41

81417

18583

93481

06569

12015

87986

19

42

81431

18569

93467

06543

12025

87976

18

43

81446

18554

93482

06618

12036

87964

17

44

81461

18639

93508

06492

12047

87963

16

46

9.81475

10.18526

9.93633

10.06467

10.12068

9.87942

16

46

81490

18610

93559

06441

12069

87931

14

47

81506

18495

93584

06416

12080

87920

13

48

81519

18481

93610

06390

12091

87909

12

49

81534

18466

93636

06364

12102

87898

11

50

9.81549

10.18451

9.93661

10.06339

10.12113

9.87887

10

51

81563

18437

93687

06313

12123

87877

9

52

81578

18422

93712

06288

12134

87866

8

53

81692

18408

93738

06262

12145

87866

7

54

81607

18393

93763

06237

12166

87844

6

55

9.81622

10.18378

9.93789

10.06211

10.12167

9.87833

6

66

81636

18364

93814

06186

12178

87822

4

57

81661

18349

93840

06160

12189

87811

3

58

81665

18335

93865

06135

12200

87800

2

59

81680

18320

93891

06109

12211

87789

1

60

81694

18306

93916

06084

12222

87778

M.

Cosine.

Secant.

Cotangent.

Tangent.

Cosecant.

Sine.

M.

130°

49°

!18 LOGARITHMIC ANGULAR FUNCTIONS. Table 2.

i°

Logarithms.

38°

M.

Sine.

Cosecant.

Tangent.

Cotangent.

■Secant,

Cosine,

M.

9.81694

10.18306

9.93916

10.06084

10,12222

9,87778

60

1

81709

18291

93942

06058

12233

87767

59

2

81723

18277

93967

06033

12244

87766

58

3

81738

18262

93993

06007

12255

87746

S'

4

81752

18248

94018

05982

12266

87734

56

5

9.81767

10.18233

9.94044

10.0,5966

10,12277

9,87723

65

6

81781

18219

94069

05931

12288

87712

54

-7

S1796

18204

94095

05906

12299

87701

53

8

81810

18190

94120

06880

12310

87690

52

9

8182.5

18175

94146

osse-i

12321

87679

51

10

9.81839

10.18161

9.94171

10.06829

10,12332

9.87608

50

11

81854

18146

94197

05803

12343

87657

49

12

81868

18132

94222

05778

12354

87640

48

13

81882

18118

94248

05752

12365

87635

47

14

81897

18103

94273

06727

12376

87624

46

15

9.81911

10.18089

9.94299

10,05701

10,12387

9,87613

45

16

81926

18074

94324

05676

12399

87601

44

17

81940 V

18060

94350

06650

12-110

87590

43

18

81955

18045

94375

0,5625

12421

87579

42

19

81969

18031

94401

05699

12432

87668

41

20

9.81983

10.18017

9.94426

10,0,5574

10,12443

9,87667

40

21

81998

18002

94462

05.548

124.54

87546

39

22

82012

17988

94477

05523

12465

87535

38

23

82026

17974

94503

05497

12476

87624

37

24

82041

17959

94.528

0.5472

12487

87513

36

25

9.82055

10.17945

9.94,564

10,05446

10.12499

9,87501

35

26

82069

17931

94579

05421

12.510

87490

34

27

82084

17916

94604

05396

12521

87479

33

28

82098

17902

94630

05370

12532

87468

32

29

82112

17888

94G55

05345

12513

874,57

31

30

9,82126

10.17874

9.94681

10,05319

10.12.554

9,87446

30

SI

82141

17859

94706

05294

125ri6

87434

29

32

82155

17845

94732

0.5268

12577

87423

28

33

82169

17831

94757

05'243

12588

87412

27

34

S21S1

17816

94783

0,5217

12599

87401

26

36

9.K219X

10.17802

9.94808

10,05192

10,12610

9.87390

25

36

.S2212

17788

94834

05166

12622

87378

24

37

82226

17774

94859

05141

121 ;33

87367

23

38

82240

17760

94884

06116

12644

87356

22

39

82255

17745

94910

05090

126.55

87346

21

10

9.82269

10.17731

9.94935

10,06065

10,12666

9.87334

20

41

82283

17717

94961

0,5039

12678

87322

19

42

82297

17703

94986

05014

12689

87311

18

43

82311

17689

95012

04988

12700

87300

17

44

82326

17674

95037

04963

12712

87288

16

45

9.8'2340

10.17660

9.95062

10,04938

10,12723

9-87277

15

le

82354

17646

96088

04912

12734

87266

14

17

82368

17632

95113

04887

12745

87255

13

18

82382

17618

95139

04861

12757

87243

12

19

82396

17604

95164

04836

12768

87232

11

50

9.82410

10.17690

9,95190

10,04810

10,12779

9,87221

10

51

82424

17576

95216

04786

12791

87209

9

52

8'2439

17561

95240

04760

12802

87198

8

53

82453

17647

95266

04734

12813

87187

7

54

82467

17533

95291

04709

12825

87175

6

55

9.82481

10.17519

9.95317

10,04683

10,12836

9,87164

5

56

82495

17505

9.5342

04658

12847

87153

4

57

8'2509

17491

95368

04632

12859

87141

3

58

82523

17477

95393

04607

12870'

87130

2

59

82637

17463

95418

04.582

12881

87119

1

30

82651

17449

95444

04556

12893

87107

H.

Coaine.

Secant.

Cotangent.

Tangent,

Cosecant,

Sine,

M,

31°

48°

ile 2. LOGAEITHMIC ANGULAR FUNCTIONS.

319

Logarithms.

137°

Sine.

Cosecant.

Tangent.

Cotangent,

Secant.

Cosine.

M,

9.82551

10.17449

9.95444

10.04556

10.12893

9.87107

60

82565

17435

95469

04531

12904

87096

89

82579

17421

95495

04505

12915

87085

58

82593

17407

95520

04480

12927

87073

57

82607

17393

95545

04455

12938

87062

56

9.82621

10.17379

9.95571

10.04429

10.12950

9 87050

55

82635

17365

95596

04404

12961

87039

54

82649

17351

95622

04378

12972

87028

53

82663

17337

95647

04353

12984

87016

52

82677

17323

95672

04328

12995

87005

51

9.82691

10.17309

9.95698

10.04302

10.13007

9.86993

50

82705

17295

95723

04277

13018

86982

49

82719

17281

95748

04252

13030

86970

48

82733

17267

95774

04226

13041

86959

47

82747

17253

95799

04201

13053

86947

46

9.82761

10.17239

9.95826

10.04175

10.13064

9.86936

45

82775

17225

95850

04150

13076

86924

44

82788

17212

95875

04125

13087

86913

43

82802

17198

95901

01099

13098

86902

42

82816

17184

95926

04074

13110

86890

41

9.82830

10.17170

9.95952

10.04048

10.13121

9.86879

40

82844

17166

95977

04023

13133

86867

39

82858

17142

96002

03998

13145

86855

38

82872

17128

96028

03972

13156

86844

37

82885

17115

96053

03947

13168

86832

36

9.82899

10.17101

9.96078

10.03922

10.13179

9.86821

35

82913

17087

96104

03896

13191

86809

34

82927

17073

96129

03871

13202

86798

S3

82941

17059

96155

03845

13214

86786

32

82955

17045

96180

03820

13225

86775

31

9.82968

10.17032

9.96205

10.03795

10.13237

9.86763

30

82982

17018

96231

03769

13248

86752

29

82996

17004

96256

03744

13260

86740

28

83010

16990

96281

03719

13272

86728

27

83023

16977

96307

03693

13283

86717

26

9.83037

10.16963

9.96332

10.03668

10.13295

9.86705

25

83051

16949

96357

03643

13306

86694

24

83065

16935

96383

03617

13318

86682

23

83078

16922

96408

03592

13330

86670

22

83092

16908

96433

03567

13341

86669

21

9.83106

10.16894

9.96459

10.03541

10.13353

9.86647

20

83120

16880

96484

03516

13365

86635

19

83133

16867

96510

03490

13376

86624

18

83147

16853

96535

03465

13388

86612 -,

17

83161

16839

96560

03440

13400

86600

16

9.83174

10.16826

9.96586

10.08414

10.13411

9.86589

15

83188

16812

96611

03389

13423

86577

14

83202

16798

96636

03364

13435

86665

13

83215

16785

96662

03338

13446

86554

12

83229

16771

96687

03313

13458

86542

11

9.83242

10.16758

9.96712

10.03288

10.13470

9.86530

10

83256

16744

98738

03262

13482

86518

9

83270

16730

96763

03237

13493

86507

8

83283

16717

96788

03212

13505

86495

7

83297

16703

96814

03186

13517

86483

6

9.83310

10.16690

9.96839

10.03161

10.13528

9.86472

5

83324

16676

96864

031.36

13.540

86460

4

83338

16662

96890

03110

13552

86448

3

83351

16649

96915

03085

13564

86436

2

83365

16635

96940

03060

13575

86425

1

83378

16622

96966

03034

13587

86413

Cosine.

Secant.

Cotangent.

Tangent.

Cosecant.

Sine.

M.

47°

320 LOGARITHMIC ANGULAR FUNCTIONS. Table 3.

43°

Logarithms.

1

36°

M.

Sine.

Cosecant.

Tangent.

Cotangent.

Secant.

Cosine.

M.

9.83378

10.16622

9.96966

10.03031

10.13587

9.86413

60

1

83392

16608

96991

03009

13599

86401

59

2

83405

16595

97016

02981

13611

86389

58

3

83419

16581

97012

02958

13623

86377

57

4

83432

16568

97067

02933

13634

86366

50

5

9.83446

10.16551

9.97092

10.02908

10.13646

9.86354

.56

6

83459

16511

97118

02882

13658

86342

54

7

83173

16527

97143

02857

13670

86330

53

8

83486

16511

97168

02832

13682

86318

52

9

83500

16500

97193

02807

13694

85306

51

10

9.83513

10.16187

9.97219

10.02781

10.13705

9.86295

50

11

83527

16173

97211

02756

13717

86283

49

12

83510

16160

97269

02731

13729

86271

48

13

83554

16116

97295

02705

13741

86259

47

U

83567

16133

97320

02680

13753

86247

16

15

9.83581

10.16119

9.97315

10.02655

10.13765

9.86235

45

16

83591

16406

97371

02629

13777

86223

14

17

83608

16392

97896

02601

13789

86211

43

18

83621

16379

97121

02679

13800

86200

42

19

83634

16366

97117

02553

13812

86188

41

20

9.83648

10.16352

9.97172

10.02528

10.13824

9.86176

40

21

83661

16339

97197

02503

13836

86164

39

22

83674

16320

97523

02177

13848

86162

38

23

83688

16312

97518

02452

13860

86140

37

24

83701

16299

97573

02127

13872

86128

36

25

9.83715

10.16285

9.97598

10.02102

10.13884

9.86116

35

26

83728

16272

97621

02376

13896

86101

34

27

83741

16259

97649

02351

13908

86092

33

28

83755

16245

97674

02326

13920

86080

32

29

83768

16232

97700

02300

13932

86068

31

30

9.83781

10.16219

9.97725

10.02275

10.13944

9.86056

30

31

83795

16205

97750

02250

13966

86044

29

32

83808

16192

97776

02221

13968

86032

28

33

83821

16179

97801

02199

13980

86020

27

34

83831

16166

97826

02171

13992

86008

26

35

9.83848

10.16152

9.97851

10.02119

10.14004

9.86996

25

36

83861

16139

97877

02123

14016

85984

24

37

83874

16126

97902

02098

14028

85972

23

38

83887

16113

97927

02073

14040

86960

22

39

83901

16099

97953

02047

11052

85948

21

40

9.83914

10.16086

9.97978

10.02022

10.14064

9.85936

20

41

83927

16073

98003

01997

14076

85924

19

42

83940

16060

98029

01971

14088

85912

18

43

83951

16046

98054

01946

14100

85900

17

44

83967

16033

98079

01921

14112

85888

16

45

9.83980

10.16020

9.98101

10.01896

10.14124

9.85876

16

46

83993

16007

98130

01870

11136

85864

14

47

84006

15994

98155

01815

11119

85851

13

48

84020

15980

98180

01820

11161

85839

12

49

84033

15967

98206

01794

11173

85827

11

50

9.84046

10.15954

9.98231

10.01769

10.11185

9.85815

10

51

84059

15911

98256

01744

11197

85803

9

52

84072

15928

98281

01719

11209

85791

8

53

84086

15915

98307

01693

11221

85779

7

54

81098

15902

98332

01668

14234

85766

6

55

9.81112

10.15888

9.98357

10.01643

10.14246

9.85754

5

56

81125

15875

98383

01617

14268

85742

4

57

81138

15862

98108

01592

14270

86730

3

58

81151

15819

98133

01567

14'282

85718

2

59

81164

15836

9S4.')S

01542

14294

85706

1

60

81177

15823

984 Si
Cotan^^ent.

01516

14307

85693

M.

CoBine.

Secant.

Tangent.

Cosecant.

Sine.

M.

Table 2. LOGARITHMIC ANGULAR FUNCTIONS.

321

44°

Logarithms.

135°

M.

■ Sine.

Cosecant.

Tangent.

Cotangent.

Secant.

Cosine.

M.

9.84177

10.15823

9.98484

10.01616

10.14307

9.85693

60

1

81190

35810

98609

01491

14319

85681

59

2

84203

15797

98534

01466

14331

85669

58

3

84216

1,5784

98560

01440

14343

85667

67

4

84229

15771

98585

01415

14356

85645

56

5

9.84242

10.15758

9.98610

10.01390

10.14368

9.85632

56

6

84255

15745

98635

01365

14380

85620

54

7

84269

15731

98661

01339

t 14392

8.5608

53

8

84282

15718

98686

01314

14404

85696

62

9

84295

15705

98711

01289

14417

85583

61

10

9.84308

10.15692

9.98737

10.01263

10.14429

9.85571

50

11

84321

15679

98762

01238

14441

86559

49

12

84334

15666

98787

01213

14463

86547

48

13

84347

15653

98812

01188

14466

85534

47

14

84360

15640

98838

01162

14478

86522

46

15

9.84373

10.15627

9.98863

10.01137

10.14490

9.85510

46

16

84385

15615

98888

01112

14503

85497

44

17

84398

15602

9S913

01087

14515

86485

43

18

84411

15589

98939

01061

14527

85473

42

19

84424

15576

98964

01036

145-10

85460

41

20

9.84437

10.15563

9.98989

10.01011

10.14652

9.85448

40

21

84450

15550

99016

00985

14564

85436

39

22

84463

16537

99010

00960

14577

85423

38

23

84476

16524

99065

00935

14589

85411

37

24

84489

16511

99090

00910

14601

85399

36

25

9.84502

10.15498

9.99116

10.00884

10.14614

9.85386

35

26

84515

15485

99141

00859

14626

85374

34

27

84528

15472

99166

00834

14639

85361

33

28

84540

16460

99191

00809

14651

85349

32

29

84553

16447

99217

00783

14663

86337

31

3D

9.84566

10.16434

9.99'242

10.00768

10.14676

9.86324

30

31

&lo79

15421

99267

00733

14688

86312

29

32

84592

15408

99293

00707

14701

85299

28

33

84605

15395

99318

00682

14713

86287

27

34

84618

15382

99343

00667

14726

85274

26

35

9.84630

10.15370

9.99368

10.00632

10.14738

9.85262

26

36

84643

15357

99394

00606

14750

85250

24

37

84656

15344

99419

00581

14763

85237

23

38

84669

15331

99444

00656

14776

85225

22

39

84682

15318

99469

00531

14788

86212

21

40

9.84694

10.15306

9.99495

10.00505

10.14800

9.86200

20

41

84707

16293

99620

00480

14813

85187

19

42

84720

16280

99545

00155

14825

85175

18

43

84733

15267

99570

00430

14838

86162

17

44

84745

1.5255

99696

00404

14850

86150

-16

45

9.84758

10.15242

9.99621

10.00379

10.14863

9.86137

16

46

84771

16229

99646

00354

14875

86125

14

47

84784

16216

99672

00328

14888

85112

13

48

84796

15204

99697

00303

14900

86100

12

49

84809

1.5191

99722

00278

14913

85087

11

50

9.84822

10.15178

9.99747

10.00263

10.14926

9.85074

10

61

84835

15165

99773

00227

14938

85062

9

52

84847

15153

99798

00202

14951

85049

8

53

84860

15140

99823

00177

14963

86037

7

54

84873

16127

99848

00152

14976

86024

6

55

9.84885

10.15116

9.99874

10.00126

10.14988

9.85012

5

56

84898

15102

99899

00101

15001

84999

4

57

84911

15089

99924

00076

16014

84986

3

68

84923

15077

99949

00051

16026

84974

2

59

84936

15064

99975

00025

15039

84961

1

60

84949

16051

10.00000

00000

15051

84949

M.

Cosine.

Secant.

Cotangent.

Tangent.

Cosecant.

Sine.

M.

134°

122

NATURAL FUNCTIONS.

Table 3.

Natural Trigonometrical Functions.

179°

rl.

Sine.

Vrs. cos.

Cosec'nt

Tang.

Co tang.

Secant.

Vrs. ein.

Cosine,

M.

.00000

1.0000

Infinite,

.00000

Infinite.

1.0000

.00000

1,0000

60

1

. 0029

.99971

3437,7

. 0029

3437,7

.0000

. 0000

,0000

59

2

. 0058

. 9942

1718.9

. 0058

1718.9

.0000

. 0000

.0000

58

3

. 0087

. 9913

1145.9

. 0087

1146.9

.0000

. 0000

.0000

57

4

. 0116

. 9884

859.44

. 0116

869,44

.0000

. 0000

.0000

66

5

.00145

.99854

687.55

.00145

687.55

1,0000

.00000

1.0000

55

6

. 0174

. 9826

572.96

. 0174

572.96

.0000

. 0000

.0000

54

7

. 0204

. 9796

491.11

. 0204

491.11

.0000

. 0000

.0000

53

8

. 0233

. 9767

429.72

. 0233

429.72

.0000

. 0000

.0000

62

9

. 0262

. 9738

381.97

. 0262

381.97

.0000

. 0000

.0000

51

.00291

.99709

313.77

.00291

343.77

1.0000

.00000

.99999

50

1

. 0320

. 9680

312.52

. 0320

312.52

.0000

. 0000

. 9999

49

2

. 0349

. 9651

286,48

. 0349

286.48

.0000

. 0001

. 9999

48

3

. 0378

. 9622

64,14

. 0378

64.44

.0000

. 0001

. 9999

47

4

. 0407

. 9593

45.55

. 0107

45.65

.0000

. 0001

. 9999

46

5

.00436

.99564

229.18

.00436

229,18

1,0000

.00001

.99999

45

6

. 0465

. 9534

14.86

. 0465

11,86

.0000

. 0001

. 9999

44

7

. 0194

. 9505

02.22

. 0494

02,22

.0000

. 0001

. 9999

43

8

. 0524

. 9476

190.99

. 0524

190.98

.0000

. 0001

. 9999

42

9

. 0553

. 9447

80.93

. 0553

80.93

.0000

. 0001

. 9998

41

.00582

.99418

171.89

.00582

171.88

1.0000

.00002

.99998

40

1

. OGll

. 9389

63.70

. 0611

63.70

.0000

. 0002

. 9998

39

2

. 0640

. 9360

56.26

. 0640

56.26

.0000

. 0002

. 9998

38

3

. 0669

. 9331

49.47

. 0669

49.46

.0000

. 0002

. 9998

37

i

. 0098

. 9302

43.24

. 0698

43.24

.0000

. 0002

. 9997

36

5

.00727

.99273

137.51

.00727

137,51

1.0000

.00003

.99997

36

6

. 0756

. 9244

32.22

. 0756

32,22

.0000

. 0003

. 9997

34

7

. 0785

. 9215

27.32

. 0785

27,32

.0000

. 0003

. 9997

33

8

. 0814

. 9185

22.78

. 0814

22.77

.0000

. 0003

. 9997

32

9

. 0843

. 9156

18.54

. 0844

18.64

.0000

. 0003

. 9996

31

.00873

.99127

114,59

.00873

114.59

1.0000

.00004

.99996

30

1

. 0902

. 9098

10,90

. 0902

10,89

.0000

. 0004

. 9996

29

2

. 0931

. 9069

07,43

. 0931

07,43

.0000

. 0004

. 9996

28

3

. 0960

. 9040

04.17

. 0960

04,17

.0000

. 0005

. 9995

27

■1

. 0989

. 9011

01,11

. 0989

01.11

.0000

. 0005

. 9995

26

5

.01018

.98982

98.223

.01018

98.218

1.0000

.00005

.99995

25

6

. 1047

. 8953

5.495

. 1047

6.489

.0000

. 0005

. 9994

24

7

. 1076

. 8924

2.914

. 1076

2.908

.0000

. 0006

. 9994

23

8

. 1105

. 8895

0.469

. 1105

0.463

.0001

. 0006

. 9994

22

9

. 1134

. 8865

88,149

. 1134

88.143

.0001

. 0006

. 9993

21

.01163

.98836

85.946

.01164

85.940

1,0001

.00007

.99993

20

1

. 1193

. 8807

3.849

. 1193

3.843

.0001

. 0007

. 9993

19

2

. 1222

. 8778

1.853

. 1222

1.847

.0001

. 0007

. 9992

18

3

. 1251

. 8749

79.950

. 1261

79.943

.0001

. 0008

. 9992

17

i

. 1280

. 8720

8.133

. 1280

8.126

.0001

. 0008

. 9992

16

5

.01309

.98691

76.396

.01309

76.390

1.0001

.00008

.99991

15

6

. 1338

. 8662

4.736

. 1338

4.729

.0001

. 0009

. 9991

14

7

. 1367

. 8633

3.146

. 1367

3,139

.0001

. 0009

. 9991

13

8

. 1396

. sr,04

1.622

. 1396

1.615

.0001

. 0010

. 9990

12

9

. 1425

. S575

0.160

. 1125

0.153

.0001

. 0010

. 9990

11

.01454

.9X546

68.757

.01454

68.750

1.0001

.00010

.99989

10

1

. 1183

. 8.)16

7.409

. 1184

7,102

.0001

. 0011

. 9989

9

2

. 1512

. S4N7

6.113

. 1513

6,105

.0001

. 0011

. 9988

8

3

. 1512

. 8458

4,866

. 1542

4,858

.0001

. 0012

. 9988

7

i

. 1571

. 8429

3.664

. 1571

3,657

.0001

. 0012

. 9988

6

5

.01600

.98400

62.507

.01600

62,499

1.0001

.00013

.99987

5

6

. 1629

. 8371

1,891

. 1629

1,383

.0001

. 0013

. 9987

4

7

. 1658

. 8342

0,314

. 1668

0,306

.0001

. 0014

. 9987

3

3

. 1687

. 8313

59.274

. 1687

59,266

.0001

. 0014

. 9986

2

9

. 1716

. 8284

8.270

. 1716

8,261

.0001

. 0015

. 9985

1

. 1745

. 8265

7.299

. 1745

7,290

.0001

. 0015

. 9985

I.

Cosine.

Vrs. sin.

Secant.

Cotang.

Tang,

Cosec'nt

Vrs. COS.

Sine.

M.

Table 3.

NATURAL FUNCTIONS.

323

1°

Natural Trigonometrical Functions.

178°

M.

Sine.

Vrs. COS.

Cosec'nt

Tang.

Cotang.

Secant,

Vra. sin.

Cosine.

M.

.01745

.98255

57.299

.01745

57.290

1.0001

.00015

.99985

60

1

. 1774

. 8226

6.359

. 1775

6.350

.0001

. 0016

. 9984

59

2

. 1803

. 8196

5.450

. 1804

5.441

.0001

. 0016

. 9984

58

3

. 1832

. 8167

4.570

. 1833

4.561

.0002

. 0017

. 9983

67

i

. 1861

. 8138

3.718

. 1862

3.708

.0002

. 0017

. 9983

56

5

.01891

.98109

52.891

.01891

52.882

1.0002

.00018

.99982

55

6

. 1920

. 8080

2.090

. 1920

2.081

.0002

. 0018

. 9981

54

7

. 1949

. 8051

1.313

. 1949

1.303

.0002

. 0019

. 9981

63

8

. 1978

. 8022

0.568

. 1978

0.548

.0002

. 0019

. 9980

62

9

. 2007

. 7993

49.826

. 2007

49.816

.0002

. 0020

. 9980

51

10

.02036

.97964

49.114

.02036

49.104

1.0002

.00021

.99979

60

11

. 2065

. 7935

8.422

. 2066

8.412

.0002

. 0021

. 9979

49

12

. 2094

. 7906

.7.750

. 2095

7.739

.0002

. 0022

. 9978

48

13

. 2123

. 7877

7.096

. 2124

7.085

.0002

. 0022

. 9977

47

14

. 2152

. 7847

6.460

. 2163

6.449

.0002

, .,0023

. 9977

46

15

.02181

.97818

46.840

.02182

45.829

1.0002

.00024

.99976

45

16

. 2210

. 7789

5.237

. 2211

5.226

.0002

. 0024

. 9975

44

17

. 2240

. 7760

4.650

. 2240

4.638

.0002

. 0026

. 9975

43

18

. 2269

. 7731

4.077

. 2269

4.066

.0002

. 0026

. 9974

42

19

. 2298

. 7702

3.520

. 2298

3.608

.0003

. 0026

. 9974

41

20

.02327

.97673

42.976

.02327

42.964

1.0003

.00027

.99973

40

21

. 2356

. 7644

2.445

. 2367

2.433

.0003

. 0028

. 9972

39

22

. 2385

. 7615

1.928

. 2386

1.916

.0003

. 0028

. 9971

38

23

. 2414

. 7586

1.423

. 2415

1.410

.0003

. 0029

. 9971

.37

24

. 2443

. 7557

0.930

. 2444

0.917

.0003

. 0030

. 9970

36

25

.02472

.97528

40.448

.02473

40.436

1.0003

.00030

.99969

35

26

. 2501

. 7499

39.978

. 2502

39.966

.0003

. 0031

. 9969

34

27

. 2530

. 7469

9.518

. 2531

9.506

.0003

. 0032

. 9968

33

28

. 2559

. 7440

9.069

. 2560

9.057

.0003

. 0033

. 9967

32

29

. 2589

. 7411

8.631

. 2589

8.618

.0003

. 0033

. 9966

31

30

.02618

.97382

38.201

.02618

38.188

1.0003

.00034

.99966

30

31

. 2647

. 7353

7.782

. 2648

7.769

.0003

. 0036

.. 9966

29

32

. 2676

. 7324

7.371

. 2677

7.358

.0003

. 0036

. 9964

28

33

. 2705

. 7295

6.969

. 2706

6.966

.0004

. 0036

. 9963

27

34

. 2734

. 7266

6.676

. 2736

6.663

.0004

. 0037

9963

26

35

.02763

.97237

36.191

.02764

36.177

1.0004

.00038

.99962

25

36

. 2792

. 7208

5.814

. 2793

5.800

.0004

. 0039

. 9961

24

37

. 2821

. 7179

5.445

. 2822

5.431

.0004

. 0040

9960

23

38

. 2850

. 7150

5.084

. 2851

5.069

.0004

. 0041

. 9959

22

39

. 2879

. 7121

4.729

. 2880

4.715

.0004

. 0041

. 9958

21

40

.02908

.97091

34.382

.02910

34.368

1.0004

.00042

.99958

20

41

. 2937

. 7062

4.042

. 2939

4.027

.0004

. 0043

. 9957

19

42

. 2967

. 7033

3.708

. 2968

3.693

.0004

. 0044

. 9966

18

43

. 2996

. 7004

3.381

. 2997

3.366

.0004

. 0046

. 9955

17

44

. 3025

. 6975

3.060

. 3026

3.046

.0004

. 0046

. 9954

16

45

.03054

.96946

32.746

.03055

32.730

1.0005

.00046

.99963

15

46

. 3083

. 6917

2.437

. 3084

2.421

.0005

. 0047

. 9962

14

47

. 3112

. 6888

2.134

. 3113

2.118

.0005

. 0048

. 9961

13

48

. 3141

. 6869

1.836

. 3143

1.820

.0005

. 0049

. 9951

12

49

. 3170

. 6830

1.544

. 3172

1.528

.0005

. 0050

. 9950

11

50

.03199

.96801

31.267

.03201

31.241

1.0005

.00051

.99949

10

61

. 3228

. 6772

• 0.976

. 3230

0.960

.0005

. 0052

. 9948

9

52

. 3267

. 6743

0.699

. 3259

0.683

.0005

. 0053

. 9947

8

53

. 3286

. 6713

0.428

. 3288

0.411

.0005

. 0054

. 9946

7

54

. 3315

. 6684

0.161

. 3317

0.145

.0005

. 0065

. 9945

6

55

.03344

.96665

29.899

.03346

29.882

1.0005

.00056

.99944

5

56

. 3374

. 6626

9.641

. 3375

9.624

.0006

. 0057

. 9943

4

57

. 3403

. 6597

9.388

. 3405

9.371

.0006

. 0058

. 9942

3

58

. 3432

. 6668

9.139

. 3434

9.122

.0006

. 0069

. 9941

2

69

. 3461

. 6539

8.894

. 3463

8.877

.0006

. 0060

. 9940

1

60

. 3490

. 6510

8.664

. 3492

8.636

.0006

. 0061

. 9939

M.

Cosine.

Vrs. sin.

Secant.

Cotong.

Tang.

Cosec'nt

Vrs. cos.

Sine.

M.

91°

88°

324

NATURAL FUNCTIONS.

Table 3.

2°

Natural Trigonometrical

Functions.

177°

M.

Sine.

Vrs. COS.

Cosec'nt

Tang.

Cotang.

Secant.

Vra. sin.

Cosine.

M.

.03490

.96510

28.654

.03492

28.636

1.0006

.00061

.99939

60

1

. 3519

. 6481

8.417

. 3521

8.399

.0006

. 0062

. 9938

69

2

. 3548

. 6452

8.184

. 3550

8.166

.0006

. 0063

. 9937

58

3

. 3577

. 6423

7.955

. 3579

7.937

.0006

. 0064

. 9936

57

4

. 3606

. 6394

7.730

. 8608

7.712

.0006

. 0065

. 9935

56

5

.03635

.96365

27.508

.03638

27.490

1.0007

.00066

.99934

55

C

. 3664

. 6336

7.290

. 3667

7.271

.0007

. 0067

. 9933

54

7

. 3693

. 6306

7.075

. 3696

7.066

.0007

. 0068

. 9932

53

8

.3722

. 6277

6.864

. 3725

6.845

.0007

. 0069

. 9931

52

9

. 3751

. 6248

6.655

. 3754

6.637

.0007

. 0070

. 9930

51

10

.03781

.96219

26.450

.03783

26.432

1.0007

.00071

.99928

50

11

. 3810

. 6190

6.249

. 3812

6.230

.0007

. 0073

. 9927

49

12

. 3839

. 6161

6.050

. 3842

6.031

.0007

. 0074

. 9926

48

13

. 3868

. 6132

5.354

. 3871

5.835

.0007

. 0075

. 9925

47

14

. 3897

. 6103

5.661

. 3900 ■

6.642

.0008

. 0076

. 9924

46

15

.03926

.96074

25.471

.03929

25.452

1.0008

.00077

.99923

45

16

. 3955

. 6045

6.284

. 3968

5.264

.0008

. 0078

. 9922

44

17

. 3984

. 6016

8.100

. 3987

5.080

.0008

. 0079

. 9921

43

18

. 4013

. 5987

4.918

. 4016

4.898

.0008

. 0080

. 9919

42

19

. 4042

. 5968

4.739

. 4046

4.718

.0008

. 0082

. 9918

41

20

.04071

.95929

24.562

.04075

24.642

1.0008

.00083

.99917

40

21

. 4100

. 5900

4.388

. 4104

4.367

.0008

. 0084

. 9916

39

22

. 4129

. 5870

4.216

. 4133

4.196

.0008

. 0085

. 9915

38

23

. 4158

. 5841

4.047

. 4162

4.026

.0009

. 0086

. 9913

37

24

. 4187

. 5812

3.880

. 4191

3.859

.0009

. 0088

. 9912

36

25

.04217

.95783

23.716

.04220

23.694

1.0009

.00089

.99911

35

26

. 4246

. 5754

3.553

. 4249

3.532

.0009

. 0090

. 9910

34

27

. 4275

. 5725

3.393

. 4279

3.372

.0009

. 0091

. 9908

33

28

. 4304

. 5696

3.235

. 4308

3.214

.0009

. 0093

. 9907

32

29

. 4333

. 5667

3.079

. 4337

3.068

.0009

. 0094

. 9906

31

30

.04362

.95638

22.925

.04366

22.904

1.0009

.00095

.99905

30

31

. 4391

. 5609

2.774

. 4395

2.752

.0010

. 0096

. 9903

29

32

. 4420

. 6580

2.624

. 4424

2.602

.0010

. 0098

. 9902

28

33

. 4449

. 5551

2.476

. 4453

2.454

.0010

. 0099

. 9901

27

34

. 4478

. 5622

2.330

. 4483

3ie08

.0010

. 0100

. 9900

26

35

.04507

.95493

22.186

.04512

22.164

1.0010

.00102

.99898

25

30

. 4536

. 5464

2.044

. 4541

2.022

.0010

. 0103

. 9897

24

37

. 4565

. 5435

1.904

. 4570

1.881

.0010

. 0104

. 9896

23

38

. 4594

. 5405

1.765

. 4599

1.742

.0010

. 0106

. 9894

22

39

. 4623

. 6376

1.629

. 4628

1.606

.0011

. 0107

. 9893

21

40

.04652

.96347

21.494

.04657

21.470

1.0011

.00108

.99892

20

41-

. 4681

. 6318

1.360

. 4687

1.337

.0011

. 0110

. 9890

19

42

. 4711

. 5289

1.228

. 4716

1.205

.0011

. 0111

. 9889

18

43

. 4740

. 5260

1.098

. 4745

1.075

.0011

. 0112

. 9888

17

44

. 4769

. 5231

0.970

. 4774

0.946

.0011

. 0114

. 98S6

16

45

.04798

.95202

20.843

.04803

20.819

1.0011

.00115

.99885

15

46

. 4827

. 5173

0.717

. 4832

0.693

.0012

. 0116

. 9883

14

47

. 4856

. 5144

0.593

. 4862

0.569

.0012

. 0118

. 9882

13

48

. 4885

. 5115

0.471

. 4891

0.446

.0012

. 0119

. 9881

12

49

. 4914

. 6086

0.350

. 4920

0.325

.0012

. 0121

. 9879

11

50

.04943

.96057

20.230

.04949

20.205

1.0012

.00122

.99878

10

51

. 4972

. 6028

0.112

. 4978

0.087

.0012

. 0124

. 9876

9

62

. 5001

. 4999

19.995

. 5007

19.970

.0012

. 0125

. 9875

8

53

5030

4970

9.880

. 5037

9.854

.0013

. 0127

. 9873

7

54

. 6059

. 4941

9.766

. 5066

9.740

.0013

. 0128

. 9872

G

65

.05088

.94912

19.653

.05095

19.627

1.0013

.00129

.99870

5

66

. 5117

. 4883

9.541

. 5124

9.515

.0013

. 0131

. 9869

4

57

. 5146

. 4853

9.431

. 6153

9.405

.0013

. 0132

. 9867

3

58

. 5175

. 4824

9.322

. 6182

9.296

.0013

. 0134

. 9866

2

59

. 5204

. 4795

9.214

. 6212

9.188

.0013

. 0135

. 9864

1

60

. 5234

. 4766

9.107

. 5241

9.081

.0014

. 0137

. 9863

M.

Cosine,

Vrs. sin.

Secant.

Co tang.

Tang.

Cosec'nt

AVs. cos.

Sine.

M.

92°

87°

lies.

NATUEAL FUNCTIUJNS.

325

Natural Trigonometrical P|fnctions.

176°

Sine.

Yra. COB.

Cosec'nt

Tang.

Co tang.

Secant.

Vrs. sin.

Cosine.

M.

.05234

.94766

19.107

.05241

19.081

1.0014

.00137

.99863

60

. 5263

. 4737

9.002

. 5270

8.975

.0014

. 0138

. 9861

59

. 6292

. 4708

8.897

. 5299

8.871

.0014

. 0140

. 9860

58

. 5321

. 4679

8.794

. 5328

8.768

.0014

. 0142

. 9868

57

. 5350

. 4650

8.692

. 5357

8.665

.0014

. 0143

. 9857

56

.05379

.94621

18.591

.05387

18.564

1.0014

.00145

.99865

55

. 5408

. 4592

8.491

. 5416

8.464

.0016

. 0146

. 9854

64

. 5437

. 4563

8.393

. 5445

8.365

.0015

. 0148

. 9852

53

. 5466

. 4534

8.295

. 5474

8.268

.0016

. 0149

. 9850

62

. 5495

. 4505

8.198

. 5503

8.171

.0015

. 0151

. 9849

51

.05524

.94476

18.103

.05532

18.075

1.0015

.00153

.99847

50

. 5553

. 4447

8.008

. 5562

7.980

.0015

. 0154

. 9846

49

. 5582

. 4418

7.914

. 5591

7.886

.0016

. 0156

. 9844

48

. 5611

. 4389

7.821

. 5620

7.793

.0016

. 0157

. 9842

47

. 5640

. 4360

7.730

. 5649

7.701

.0016

. 0159

. 9841

46

.05669

.94331

17.639

.05678

17.610

1.0016

.00161

.99839

45

. 5698

. 4302

7.549

. 5707

7.520

.0016

. 0162

. 9837

44

. 5727

. 4273

7.460

. 5737

7.431

.0016

. 0164

. 9836

43

. 5756

.4244

7.372

. 5766

7.343

.0017

. 0166

. 9834

42

. 6785

. 4214

7.285

. 5795

7.256

.0017

. 0167

. 9832

41

.05814

.94185

17.198

.05824

17.169

1.0017

.00169

.99831

40

. 5843

. 4156

7.113

. 5853

7.084

.0017

. 0171

. 9829

39

. 4127

7.028

. 5883

6.999

.0017

. 0172

. 9827

38

5^02

. 4098

6.944

. 5912

6.915

.001?

. 0174

. 9826

37

! 5931

. 4069

6.861

. 5941

6.832

.0018

. 0176

. 9824

36

.05960

.94040

16.779

.05970

16.750

1.0018

.00178

.99822

35

. 5989

. 4011

6.698

. 5999

6.668

.0018

. 0179

. 9820

34

. 6018

. 3982

6.617

. 6029

6.587

.0018

. 0181

. 9819

33

. 6047

. 3953

6.538

. 6053

6.507

.0018

. 0183

. 9817

32

. 6076

. 3924

6.459

. 6087

6.428

.0018

. 0186

. 9815

31

.06105

.93895

16.380

.06116

16.350

1.0019

.00186

.99813

30

. 6134

. 3866

6.303

. 6145

6.272

.0019

. 0188

. 9812

29

. 6163

. 3837

6.226

. 6175

6.195

.0019

. 0190

. 9810

28

. 6192

. 3808

6.150

. 6204

6.119

.0019

. 0192

. 9808

27

. 6221

. 3777

6.075

. 6233

6.043

.0019

. 0194

. 9806

26

.06250

.93750

16.000

.06262

15.969

1.0019

.00196

.99804

25

. 6279

. 3721

5.926

. 6291

6.894

.0020

. 0197

. 9803

24

. 6308

. 3692

5.853

. 6321

6.821

.0020

. 0199

. 9801

23

. 6337

. 3663

5.780

. 6350

6.748

.0020

. 0201

. 9799

22

. 6366

. 3634

5.708

. 6379

5.676

.0020

. 0203

. 9797

21

.06395

.93605

15.637

.06408

15.605

1.0020

.00205

.99795

20

. 6424

. 3576

5.566

. 6437

5.534

.0021

. 0206

. 9793

19

. 6453

. 3547

5.496

. 6467

6.464

.0021

. 0208

. 9791

18

. 6482

. 3518

5.427

. 6496

5.394

.0021

. 0210

. 9790

17

. 6511

. 3489

5.358

. 6525

5.325

.0021

. 0212

. 9788

16

.06540

.93460

15.290

.06554

16.267

1.0021

.00214

.99786

15

. 6569

. 3431

5.222

. 6583

5.189

.0022

. 0216

. 9784

14

. 6598

. 3402

5.155

. 6613

5.122

.0022

. 0218

. 9782

13

. 6627

. 3373

5.089

. 6642

5.066

.0022

. 0220

. 9780

12

. 6656

. 3343

5.023

. 6671

4.990

.0022

. 0222

. 9778

11

.06685

.93314

14.958

.06700

14.924

1.0022

.00224

.99776

10

. 6714

. 3285

4.893

. 6730

4.860

.0023

. 0226

. 9774

9

. 6743

. 3256

4.829

. 6759

4.795

.0023

. 0228

. 9772

8

. 6772

. 3227

4.765

. 6788

4.732

.0023

. 0230

. 9770

7

. 6801

. 3198

4.702

. 6817

4.668

.0023

. 0231

. 9768

6

.06830

.93169

14.640

.06846

14.606

1.0023

.00233

.99766

5

. 6859

. 3140

4.578

. 6876

4.644

.0024

. 0235

. 9764

4

. 6888

. 3111

4.517

. 6905

4.482

.0024

. 0237

. 9762

3

. 6918

.3082

4.456

. 6934

4.421

.0024

. 0239

. 9760

2

. 6947

. 3053

4.395

. 6963

4.361

.0024

. 0241

. 9758

1

. 6976

. 3024

4.335

. 6993

4.301

.0024

. 0243

. 9766

Cosine.

Vrs. sin.

Secant.

Cotang.

Tang.

CoBBC'nt

Vrs. cos.

Sine.

86°

326

NATURAL FUNCTIONS.

Table 3.

4°

Natural Trigonometrical Functions.

175°

M.

Sine.

Vrs. C08.

Cosec'nt

Tang.

Cotang.

Secant.

Vrs. sin.

Cosine.

M.

.06976

.93024

14.335

.06993

14.301

1.0024

.00243

.99756

60

1

. 7005

. 2995

4.276

. 7022

4.241

.00'25

. 0246

. 9754

59

2

. 7034

. 2966

4.217

. 7051

4.182

.0025

. 02'18

. 9752

58

3

. 7053

. 2937

4.159

. 7080

4.123

.0026

. 0250

. 9750

57

4

. 7092

. 2908

4.101

. 7110

4.065

.0025

. 0252

. 9748

66

5

.07121

.92879

14.043

.07139

14.008

1.0025

.00254

.99746

65

6

. 7150

. 2850

3.986

. 7168

3.961

.0026

. 0256

. 9744

54

7

. 7179

. 2821

3.930

. 7197

3.894

.0026

. 0268

. 9742

53

8

. 7208

. 2792

3.874

. 7226

3.838

.0026

. 0260

. 9740

,52

9

. 7237

. 2763

3.818

. 7256

3.782

.0026

. 0262

. 9738

61

10

.07266

.92734

13.763

.07285

13.727

1.0026

.00264

.99736

50

11

. 7295

. 2705

3.708

. 7314

3.672

.0027

. 0266

. 9733

49

12

. 7324

. 2676

3.654

. 7343

3.617

.0027

. 0268

. 9731

48

13

. 7353

. 2647

3.600

. 7373

3.563

.0027

. 0271

. 9729

47

l-l

. 7382

. 2618

3.547

. 7402

3.510

.0027

. 0273

. 9727

46

15

.07411

.92589

13.494

.07431

13.457

1.0027

.00276

.99725

45

16

. 7440

. 2560

3.441

. 7460

3.404

.0028

. 0277

. 9723

44

17

. 7469

. 2.531

8.389

. 7490

3.351

.0028

. 0279

. 9721

43

18

. 7498

. 2502

3.337

. 7.519

3.299

.0028

. 0281

. 9718

42

19

. 7527

. 2473

3.286

. 7648

3.248

.0028

. 0284

. 9716

41

20

.07556

.92444

13.235

.07.577

13.197

1.0029

.00286

.99714

40

21

. 7585

. 2415

3.184

. 7607

3.146

.0029

. 0288

. 9712

39

22

. 7614

. 2386

8.134

. 7636

3.096

.0029

. 0290

. 9710

38

23

. 7643

. 2357

3.084

. 7665

3.046

.0029

. 0292

. 9707

37

24

7672

. 2328

3.034

. 7694

2.996

.0029

. 0295

. 9705

36

2.>)

.07701

.92299

12.985

.07724

12.947

1.0030

.00297

.99703

35

26

7730

. 2270

2.937

. 7763

2.898

.0030

. 0299

. 9701

34

27

. 7759

. 2241

2.888

. 7782

2.849

.0030

. 0301

. 9698

33

28

. 7788

. 2212

2.840

. 7812

2.801

.0030

. 0304

. 9696

32

29

7817

. 2183

2.793

. 7841

2.764

.0031

. 0306

. 9694

31

30

.07846

.92154

12.745

.07870

12.706

1.0031

.00308

.99692

30

31

. 7875

. 2125

2.698

. 7899

2.659

.0031

. 0310

. 9689

29

32

. 7904

. 2096

2.052

. 7929

2.612

.0031

. 0313

. 9687

28

83

. 7933

. 2067

2.006

. 7968

2.566

.0032

. 0315

. 9685

27

34

. 7962

. 2038

2.560

. 7987

2.520

.0032

. 0317

. 9682

26

35

.07991

.92009

12.614

.08016

12.474

1.0032

.00320

.99680

25

36

. 802O

. 1980

2.469

. 8046

2.429

.0032

. 0322

. 9678

24

37

. 8049

. 1951

2.424

. 8075

2.384

.0032

. 0324

. 9675

23

38

. 8078

. 1922

2.379

. 8104

2.339

.0033

. 0327

. 9673

22

39

. 8107

. 1893

2.335

. 8134

2.295

.0033

. 0329

. 9671

21

40

.08136

.91864

12.291'

.08163

12.250

1.0033

.00331

.99668

20

41

. 8165

. 1835

2.248

. 8192

2.207

.0033

. 0334

. 9666

19

42

. 8194

. 1806

2.204

. 8221

2.163

.0034

. 0336

. 9664

18

43

. 8223

. 1777

2.161

. 8251

2.120

.0034

. 0339

. 9661

17

44

. 8282

. 1748

2.118

. 8280

2.077

.0034

. 0341

. 9659

16

45

.08281

.91719

12.076

.08309

12.035

1.0034

.00343

.99656

15

46

. 8310

. 1690

2.034

. 8339

1.992

.0035

. 0346

. 9654

14

47

. 8339

. 1661

1.992

. 8368

1.950

.0035

. 0348

. 9652

13

48

. 8368

. 1632

1.960

. 8397

1.909

.0035

. 0351

. 9649

12

49

. 8397

. 1603

1.909

. 8426

1.867

.0035

. 0353

. 9647

H

50

.08426

.91574

11.868

.08466

11.826

1.0036

.00356

.99644

10

51

. 8455

1545

1.828

. 8485

1.785

.0036

. 0358

. 9642

9

52

. 8484

. 1516

1.787

. 85i4

1.746

.0036

. 0360

. 9639

8

53

. 8513

. 1487

1.747

. 8544

1.704

.0036

. 0363

. 9637

7

54

. 8542

1468

1.707

. 8.573

1.664

.0037

. 0365

. 9634

6

55

.08571

.91429

11.668

.08602

11.625

1.0037

.00368

.99632

5

66

. 8600

. 1400

1.628

. 8632

1.685

.0037

. 0370

. 9629

4

67

. 8629

1371

1.589

. 8661

1.546

.0037

. 0373

. 9627

3

58

. 8658

1342

1.560

. 8690

1.507

.0038

. 0376

. 9624

2

69

. 8687

. 1313

1.512

. 8719

1.468

.0038

. 0378

. 9622

1

60

. 8715

. 1284

1474

. 8719

1.430

.0038

. 0380

. 9619

M.

CuBilin.

Vrs. sin.

Secant.

Cotang.

Tang.

Cosec'nt

Vra. COS.

Sine.

M.

40

85°

Table 3.

NATURAL FUNCTIONS.

327

s°

Naturcl Trigonometrical Functions,

174°

M,

Sino.

Vra. COS.

CoBec'nt

Tang.

Co tang.

Secant.

Vrs. sin.

Cosine.

M.

.08715

.91284

11.474

.08749

11.430

1.0038

.00380

.99019

60

1

. 8744

. 1255

1.436

. 8778

1.392

.0038

. 0383

. 9017

59

2

. 8773

. 1226

1.398

. 8807

1.354

.0039

. 0386

. 9614

68

3

. 8802

. 1197

1.360

. 8837

1.316

.0039

. 0388

. 9612

57

4

. 8831

. 1168

1.323

. 8866

1.279

.0039

. 0391

. 9609

56

5

.08800

.91139

11.286

.08895

11.242

1.0039

.00393

.99607

55

6

. 8889

. 1110

1.249

. 8925

1.205

.0010

. 0396

. 9604

54

7

. 8918

. 1082

1.213

. 8954

1.1G8

.0040

. 0398

. 9601

53

8

. 8947

. 1053

1.176

. 8983

1.132 .

.0040

. 0401

. 9599

52

9

. 8976

. 1024

1.140

. 9013

1.095

.0040

. 0404

. 9596

51

10

.09005

.90995

11.104

.09042

11.059

1.0041

.00106

.99594

50

11

. 9031

. 09G6

1.069

. 9071

1.024

.0011

. 0109

. 9591

19

12

. 9063

. 0937

1.033

. 9101

0.988

.0011

. 0111

. 9588

18

13

. 9092

. 0908

0.998

. 9130

0.953

.0041

. 0111

. 9586

17

14

. 9121

. 0879

0.963

. 9159

0.918

.0042

. 0117

. 9583

16

15

.09150

.90850

10.929

.09189

10.883

1.0042

.00119

.99580

15

16

. 9179

. 0821

0.894

. 9218

0.848

.0012

. 0122

. 9578

14

17

. 9208

. 0792

0.860

. 9247

0.814

.0013

. 0125

. 9575

13

18

. 9237

. 0763

0.826

. 9277

0.780

.0013

. 0127

. 9572

42

19

. 9266

. 0734

0.792

. 9306

0.746

.0013

. 0130

. 9570

11

20

.09295

.90705

10.758

.09335

10.712

1.0013

.00133

.99567

40

21

. 9324

. 0676

0.725

. 9365

0.678

.0014

. 0436

. 9564

39

22

. 9353

. 0647

0.692

. 9394

0.645

.0011

. 0138

. 9562

38

23

. 9382

. 0618

0.659

. 9423

0.612

.0011

. 0111

. 9559

37

24

. 9411

. 0589

0.626

. 94.53

0.579

.0011

. 0144

. 9556

36

25

.09440

.90560

10.593

.09482

10.546

1.0045

.00416

.99553

35

26

. 9469

. 0531

0.561

. 9511

0.514

.0045

. 0149

. 9551

34

27

. 9498

. 0502

0.529

. 9541

0.481

.0045

. 0152

. 9548

33

28

. 9527

. 0473

0.497

. 9570

0.449

.0046

. 0155

. 9545

32

29

. 9556

. 0444

0.465

. 9599

0.417

.0046

. 0158

. 9542

31

80

.09584

.90415

10.433

.09629

10.385

1.0046

.00160

.99M0

30

81

. 9613

. 0386

0.402

. 9658

0.354

.0046

. 0163

. 9537

29

32

. 9642

. 0357

0.371

. 9088

0.322

.0047

. 0466

. 9534

28

33

. 9671

. 0328

0.340

. 9717

0.291

.0047

. 0169

. 9531

27

34

. 9700

. 0300

0.309

. 9746

0.260

.0017

. 0472

. 9528

26

35

.09729

.90271

10.278

.09776

10.229

1.0048

.00171

.99525

25

86

. 9758

. 0242

0.248

. 9805

0.199

.0048

. 0177

. 9523

24

37

. 9787

. 0213

0.217

. 9834

0.168

.0048

. 0180

. 9520

23

38

. 9816

. 0184

0.187

. 9864

0,138

.0048

. 0183

. 9517

22

39

. 9845

. 0155

0.157

. 9893

0.108

.0049

. 0486

. 9514

21

40

.09874

.90126

10.127

.09922

10.078

1.0049

.00489

.99511

20

41

. 9903

. 0097

0.098

. 9952

0.048

.0049

. 0191

. 9508

19

42

. 9932

. 0068

0.068

. 9981

0.019

.0050

. 0494

. 9505

18

43

. 9961

. 0039

0.039

.10011

9.9893

.0050

. 0197

. 9503

17

44

. 9990

. 0010

0.010

. 0010

.9601

.0050

. 0500

. 9500

16

45

.10019

.89981

9.9812

.10069

9.9310

1.0050

.00503

.99497

15

46

. 0048

. 9952

.9525

. 0099

.9021

.0051

. 0506

. 9494

14

47

. 0077

. 9923

.9239

. 0128

.8734

.0051

. 0509

. 9191

13

48

. 0106

. 9894

.8955

. 0158

.8448

.0051

. 0512

. 9188

12

49

. 0134

. 9865

.8672

. 0187

.8164

.0052

. 0515

. 9185

11

50

.10163

.89836

9.8391

.10216

9.7882

1.0052

.00518

.99182

10

51

. 0192

. 9807

-..8112

. 0246

.7601

-.0052

. 0521

. 9179

9

52

. 0221

■. 9779

.7834

. 0275

.7322

.0053

. 0524

. 9176

8

53

. 0250

. 9750

.7558

. 0305

.7044

.0053

. 0527

. 9473

7

54

. 0279

. 9721

.7283

. 0334

.6768

.0053

. 0530

. 9470

6

55

.10308

.89692

9.7010

.10363

9.6493

1.0053

.00533

.99467

5

56

. 0337

. 9663

.6739

. 0393

.6220

.0051

. 0536

. 9461

4

57

. 0366

. 9634

.6469

. 0422

.5949

.0054

. 0539

. 9161

3

58

. 0395

. 9605

.6200

. 0452

.5679

.0054

. 0542

. 9458

2

59

. 0424

. 9576

.5933

. 0481

.5411

.0055

. 0545

. 9455

1

60

. 0453

. 9547

.5668

. 0510

.5144

.0055

. 0548

. 9152

M.

Coaine.

Vre. sin.

Secant.

Cotang.

Tang.

Cosec'nt

Vrs. COS.

Sine.

M.

95°

84°

328

NATURAL FUNCTIONS.

Table a.

6°

Natural Trigonometrical Functions.

173°

M.

Sine.

Vrs. COS.

Cosec'nt

Tang.

Co tang.

Secant.

Vro. sin.

Cosine.

31.

.10453

.89547

9.5668

.10510

9.5144

1.0056

.00548

.99452

60

1

. 0482

. 9518

.5404

. 0540

.4878

.0055

. 0561

. 9449

69

2

. 0511

. 9489

.5141

. 0569

.4614

.0056

. 0554

. 9446

58

3

. 0540

. 9460

.4880

. 0599

.4351

.0056

. 0557

. 9443

57

4

. 0568

. 9431

.4020

. 0628

.4090

.0056

. 0560

. 9440

56

5

.10597

.89402

9.4362

.10657

9.3831

1.0057

.00563

.99437

55

6

. 0626

. 9373

.4105

. 0687

.3572

.0057

. 0666

. 9434

64

7

. 0655

. 9345

.3850

. 0716

.3315

.0057

. 0569

. 9431

53

8

. 0684

. 9316

.3596

. 0746

.3060

.0057

. 0.i72

. 9428

52

9

. 0713

. 9287

.3343

. 0775

.2806

.0058

. 0575

. 9424

51

10

.10742

.89258

9.3092

.10805

9.2553

1.0058

.00579

.99421

50

11

. 0771

. 9229

.2842

. 0834

.2302

.0068

. 0582

. 9418

49

12

. 0800

. 9200

.2593

. 0863

.2051

.0059

. 0585

. 9415

48

13

. 0829

. 9171

.2346

. 0893

.1803

.0059

. 0588

. 9412

47

14

. 0858

. 9142

.2100

. 0922

.1655

.0069

. 0591

. 9409

46

15

.10887

.89113

9.1855

.10952

9.1309

1.0060

.00594

.99406

45

16

. 0916

. 9084

.1612

. 0981

.1064

.0060

. 0597

. 9402

44

17

. 0944

. 9055

.1370

. 1011

.0821

.0060

. 0601

. 9399

43

18

. 0973

. 9026

.1129

. 1040

.0579

.0061

. 0604

. 9396

42

19

. 1002

. 8998

.0890

. 1069

.0338

.0061

. 0607

. 9393

41

20

.11031

.88969

9.0651

.11099

9.0098

1.0061

.00610

.99390

40

21

. 1060

. 8940

.0414

. 1128

8.9860

.0062

. 0613

. 9386

39

22

. 1089

. 8911

.0179

. 1158

.9623

.0062

. 0617

. 9383

38

23

. 1118

. 8882

8.9944

. 1187

.9387

.0062

. 0620

. 9380

37

2i

. 1147

. 8853

.9711

. 1217

.9152

.0063

. 0623

. 9377

36

25

.11176

.88824

8.9479

.11246

8.8918

1.0063

.00626

.99373

35

26

. 1205

. 8795

.9248

. 1276

.8686

.0063

. 0630

. 9370

34

27

. 1234

. 8766

.9018

. 1305

.8455

.0064

. 0633

. 9367

33

28

. 1262

. 8737

.8790

. 1335

.8225

.0064

. 0636

. 9364

32

29

. 1291

. 8708

.8663

. 1364

.7996

.0064

. 0639

. 9360

31

30

.11320

.88680

8.8337

.11393

8.7769

1.0065

.00643

.99357

30

31

. 1349

. 8651

.8112

. 1423

.7542

.0005

. 0646

. 9354

29

32

. 1378

. 8622

.7888

. 14.52

.7317

.0065

. 0649

. 9350

28

33

. 1407

. 8593

.7665

. 1482

.7093

.0066

. 0653

. 9347

27

34

. 1436

. 8564

.7414

. 1511

.6870

.0066

. 0656

. 9344

26

35

.11465

.88535

8.7223

.11641

8.6648

1.0066

.00659

.99341

25

36

. 1494

. 8506

.7004

. 1570

.6427

.0067

. 0663

. 9337

24

37

. 1523

. 8477

.6786

. 1600

.6208

.0067

. 0666

. 9334

23

38

. 1551

. 8448

.6569

. 1629

.5989

.0067

. 0669

. 9330

22

39

. 1580

. 8420

.6353

. 1659

.5772

.0068

. 0673

. 9327

21

40

.11609

.88391

8.6138

.11688

8.6555

1.0068

.00676

.99324

20

41

. 1638

. 8362

.5924

. 1718

.5340

.0068

. 0679

. 9320

19

42

. 1667

. 8333

.5711

. 1747

.5126

.0069

. 0683

. 9317

18

43

. 1696

. 8304

.5499

. 1777

.4913

.0069

. 0686

. 9314

17

44

. 1725

. 8272

.5289

. 1806

.4701

.0069

. 0690

. 9310

16

45

.11754

.88246

8.5079

.11836

8.4489

1.0070

.00693

.99307

15

46

. 1783

. 8217

.4871

. 1865

.4279

.0070

. 0696

. 9303

14

47

. 1811

. 8188

.4603

. 1895

.4070

.0070

. 0700

. 9300

IS

48

. 1840

. 8160

.4457

. 1924

.3862

.0071

. 0703

. 9296

12

49

. 1869

. 8131

.4251

. 1954

.3655

.0071

. 0707

. 9293

11

50

.11898

.88102

8.4046

.11983

8.3449

1.0071

.00710

.99290

10

51

. 1927

. 8073

.3843

. 2013

.3244

.0U72

. 0714

. 9286

9

52

. 1956

. 8044

.3640

. 2042

.3040

.0072

. 0717

. 9'283

8

53

. 1985

. 8015

.3139

. 2072

.2837

.0073

. 0721

. 9279

7

54

. 2014

. 7986

.3238

. 2101

.2635

.0073

. 0724

. 9276

6

65

.12042

.87957

8.3039

.12131

8.2434

1.0073

.00728

.99272

5

56

. 2071

. 7928

.2840

. 2160

.2234

.0074

. 0731

. 9269

4

67

. 2100

. 7900

.2642

. 2190

.2035

.0074

. 0735

. 9265

3

58

. 2129

. 7871

.2446

. 2219

.1837

.0074

. 0738

. 9262

2

69

. 2158

. 7842

.2250

. 2249

.1640

.0075

. 0742

. 9258

1

.60

2187

. 7813

.20.55

. 2278

.1443

.0075

. 0745

. 9265

M.

Cosine.

Vrs. sin.

Secant.

Co tang.

Tang.

Cosec'nt

Vrs. COB.

Sine.

M.

96°

83°

Table 3.

NATUKAL FUNCTIONS.

329

7°

Natural Trigonometrical Functions.

172°

M.

Sine.

Vre. COS.

Cosec'nt

Tang.

Co tang.

Secant.

Vrs. siu.

Cosine.

M.

.12187

.87813

8.2055

.12278

8.1443

1.0075

.00745

.99255

60

1

. 2216

. 7787

.1861

. 2308

.1248

.0075

. 0749

.9251

59

2

. 2245

. 7755

.1668

. 2337

.1053

.0076

. 0752

. 9247

58

3

. 2273

.7726

.1476

. 2367

.0860

.0076

. 0756

. 9244

.57

4

. 2302

. 7697

.1285

. 2396

.0667

.0076

. 0760

. 9240

56

5

.12331

.87669

8.1094

.12426

8.0476

1.0077

.00763

.99237

55

6

. 2360

. 7640

.0905

. 2456

.0285

.0077

. 0767

. 9233

.54

7

. 2389

. 7611

.0717

. 2485

.0095

.0078

. 0770

. 9229

53

8

. 2418

. 7582

.0529

. 2515

7.9906

.0078

. 0774

. 9226

52

9

. 2447

7553 :

.0342

. 2544

.9717

.0078

. 0778

. 9222

51

10

.12476

.87524

8.0156

.12574

7.9.130

1.0079

.00781

.99219

50

11

. 2504

. 7495

7.9971

. 2603

.9.344

.0079

. 0785

. 9215

49

12

. 2533

. 7467

.9787

. 2633

.9158

.0079

. 0788

. 9211

48

13

. 2662

. 74.38

.9604

. 2662

.8973

.0080

. 0792

. 9208

47

14

. 2591

. 7409

.9421

. 2692

.8789

.0080

. 07%

. 9204

46

15

.12620

.87:380

7.9240

.12722

7.8606

1.0080

.00799

.99200

45

16

. 2G49

. 7351

.9059

. 2751

.8424

.0081

. 0803

. 9197

44

17

. 2678

. 7322

.8879

. 2781

.8243

.0081

. 0807

. 9193

43

18

. 2706

. 7293

.8700

. 2810

.8062

.0082

. 0810

. 9189

42

19

. 2735

. 7265

.8522

. 2840

.7882

.0082

. 0814

. 9186

41

20

.12764

.87236

7.8344

.12869

7.7703

1.0082

.00818

.99182

40

21

. 2793

. 7207

.8168

. 2899

.7525

.0083

. 0822

. 9178

39

22

. 2822

. 7178

.7992

. 2928

.7348

.0083

. 0825

. 9174

38

23

. 2851

. 7149

.7817

. 2958

.7171

.0084

. 0829

. 9171

37

24

. 2879

7120

.7642

. 2988

.6996

.0084

. oass

. 9467

36

25

.12908

.87091

7.7469

.13017

7.6821

1.0084

.00837

.99163

35

26

. 2937

. 7063

.7296

. 3047

.6646

.0085

. 0840

. 9160

34

27

. 2966

. 7034

.7124

. 3076

.6473

.0085

. 0844

. 9156

33

28

. 2995

. 7005

.6953

. 3100

.6300

.0085

. 0848

. 9152

32

29

. 3024

. 6976

.6783

. 3136

.6129

.0086

. 0852

. 9148

31

30

.13053

.86947

7.6613

.13165

7..5957

1.0086

.00855

.99144

30

31

. 3081

. 6918

.6414

. 3195

.5787

.0087

. 0859

. 9141

29

32

. 3110

. 6890

.6276

. 3224

.5617

.0087

. 0863

. 9137

28

33

. 3139

. 6861

.6108

. 3254

.5449

.0087

. 0867

. 9133

27

34

. 3168

. 6832

.5942

. 3284

.5280

.0088

. 0871

9129

26

35

.13197

.86803

7.5776

.13313

7.5113

1.0088

.00875

.99125

25

36

. 3226

. 6774

..5611

. 3343

.4946

.0089

. 0878

. 9121

24

37

. 3254

. 6745

.5446

. 3372

.4780

.0089

. 0882

. 9118

23

38

. 3283

. 6717

.5282

. 3402

.4615

.0089

. 0886

. 9114

22

39

. 3312

. C688

.5119

. 3432

.4451

.0090

. 0890

. 9110

21

40

.13341

.86659

7.4957

.13461

7.4287

1.0090

.00894

.99106

20

41

. 3370

. G630

.4795

. 3491

.4124

.0090

. 0898

. 9102

19

42

. 3399

. 6601

.4634

. 3520

.3961

.0091

. 0902

. 9098

18

43

. 3427

. 6572

.4474

. 3550

;3800

.0091

. 0905

. 9094

17

44

. 3456

. 6544

.4315

. 3580

.3639

.0092

. 0909

. 9090

16

45

.13485

.86515

7.4156

.13609

7.3479

1.0092

.00913

.99086

15

46

. 3514

. 6486

.3998

. 3639

.3319

.0092

. 0917

. 9083

14

47

. 3543

. 6457

.3840

. 3669

.3160

.0093

. 0921

. 9079

13

48

. 3571

. 6428

.3683

. 3698

.3002

.0093

. 0925

. 9075

12

49

. 3600

. 6400

.3527

. 3728

.2844

.0094

. 0929

. 9070

11

50

.13629

.86371

7.3372

.13757

7.2687

1.0094

.00933

.99067

10

51

. 3658

. 6342

.3217

. 3787

.2531

.0094

. 0937

. 9063

9

52

. 3687

. 6313

.3063

. 3817

.2375

.0095

. 0941

■ . 9059

8

53

. 3716

. 6284

.2909

. 3846

.2220

.0095

. 0945

. 9055

7

54

. 3744

. 6255

.2757

. 3876

.2066

.0096

. 0949

. 9051

6

55

.13773

.86227

7.2604

.13906

7.1912

1.0096

.00953

.99047

5

56

. 3802

. 6198

.2453

. 3935

.1759

.0097

. 0957

. 9043

4

57

. 3831

. 6169

.2302

. 3965

.1607

.0097

. 0961

. 9039

3

58

. 3860

. 6140

.2152

. 3995

.1455

.0097

. 0965

. 9035

2

59

. 3888

. 6111

.2002

. 4024

.1304

.0098

. 0969

. 9031

1

60

. 3917

. 6083

.1853

. 4054

.1154

.0098

. 0973

. 9027

M.

Cosine.

Vrs. Bin.

Secant.

Cotang.

Tang.

CoBec'nt

Vrs. COS.

Sine.

M.

P7°

82°

330

NATURAL FUNCTIONS.

Table 3.

8°

Natural Trigonometrical Functions.

171°

M.

Sine.

Vrs. COS.

Cosec'nt

Tiing.

Cotang.

Secant.

Vrs. Bin.

Cosine.

M.

.13917

.86083

7.1853

.14054

7.1154

1.0098

.00973

.99027

60

1

. 3946

. 6054

.1704

. 4084

.1004

.0099

. 0977

. 9023

59

2

. 3975

. 6025

.1557

. 4113

.0854

.0099

. 0981

. 9019

58

3

. 4004

. 5996

.1409

. 4143

.0706

.0099

. 0985

. 9015

.57

4

. 4032

. 5967

.1263

. 4173

.0558

.0100

. 0989

. 9010

56

5

.14001

.85939

7.1117

.14202

7.0410

1.0100

.00993

.99006

55

6

. 4090

. 5910

.0972

. 4232

.0264

.0101

. 0998

. 9002

54

7

. 4119

. 5881

.0827

. 4262

.0117

.0101

. 1002

. 8998

63

8

. 4148

. 5852

.0683

. 4291

6.9972

.0102

. 1006

. 8994

62

9

. 4176

. 5823

.0539

. 4321

.9827

.0102

. 1010

. 8990

51

10

.14205

.85795

7.0396

.14351

6.9682

1.0102

.01014

.98986

50

11

. 4234

. 5766

.0254

. 4;wo

.9538

.0103

. 1018

. 8982

49

12

. 4263

. 5737

.0112

. 4410

.9395

.0103

. 1022

. 8978

48

13

. 4292

. 5708

6.9971

. 4440

.9252

.0104

. 1026

. 8973

47

14

. 4320

. 5679

.9830

. 4470

.9110

.0104

. 1031

. 8969

46

15

.14349

.85651

6.9690

.14499

6.8969

1.0104

.01035

.98965

45

16

. 4378

. 5622

.9550

. 4529

.8828

.0105

. 1039

. 8961

44

17

. 4407

. 5593

.9411

. 4.559

.8687

.0105

. 1043

. 8957

43

18

. 4436

. 5564

.9273

. 4588

.8547

.0106

. 1047

. 8952

42

19

. 4464

. 5536

.9135

. 4618

.8408

.0106

. 1052

. 8948

41

20

.14493

.85507

6.8998

.14048

6.8269

1.0107

.01056

.98944

40

21

. 4522

. 5478

.8861

. 4677

.8131

.0107

. 1060

. 8940

39

22

. 4551

. 5449

.8725

. 4707

.7993

.0107

. 1064

. 8936

38

23

. 4579

. 5420

.8589

. 4737

.7856

.0108

. 1068

. 8931

37

24

. 4608

. 5392

.8454

. 4767

.7720

.0108

. 1073

. 8927

36

25

.14637

.85363

6.8320

.14796

6.7584

1.0109

.01077

.98923

35

26

. 4666

. 5334

.8185

. 4826

.7448

.0109

. 1081

. 8919

34

27

. 4695

. 5305

.8052

. 4856

.7313

.0130

. 1085

. 8914

33

28

. 4723

. 5277

.7919

. 4886

.7179

.0110

. 1090

. 8910

32

29

. 4752

. 5248

.7787

. 4915

.7045

.0111

. 1094

. 8906

31

SO

.14781

.85219

6.7655

.14945

6.6911

1.0111

.01098

.98901

30

31

. 4810

. 5190

.7523

. 4975

.6779

.0111

. 1103

. 8897

29

32

. 4838

.5161

.7392

. 5004

.6646

.0112

. 1107

. 8893

28

33

. 4867

. 5133

.7262

. 5034

.6514

.0112

. nil

. 8889

27

34

. 4896

. 5104

.7132

. 5064

.6383

.0113

. 1116

. 8884

28

35

.14925

.85075

6.7003

.15094

6.0262

1.0113

.01120

.98880

25

36

. 4953

. 5046

.6874

. 5123

.6122

.0114

. 1124

. 8876

24

37

. 4982

. 6018

.6745

. 5153

.5992

.0114

. 1129

. 8871

23

38

. 5011

. 4989

.6617

. 5183

.6863

.0115

. 1133

. 8867

22

39

. 5040

. 4960

.6490

. 5213

.5734

.0116

. 1137

. 8862

21

40

.15068

.84931

6.6363

.15243

6.5605

1.0U6

.01142

.98858

20

41

. 5097

. 4903

.6237

. 5272

.5478

.0116

. 1146

. 8854

19

42

. 5126

. 4874

.6111

. 5302

.5350

.0116

. 1151

. 8849

18

43

. 5155

. 4845

.5985

. 5332

.5223

.0117

. 1155

. 8845

17

44

. 5183

. 4816

.5860

. 5362

.5097

.0117

. 1159

. 8840

16

45

.15212

.84788

6,5736

.15391

6.4971

1.0118

.01164

.98836

15

46

. 5241

. 4759

.6612

. 5421

.4845

.0118

. 1168

. 8832

14

47

. 5270

. 4730

.5488

. 5451

.4720

.0119

. 1173

. 8827

13

48

. 5298

. 4701

.5365

.5481

.4696

.0119

. 1177

. 8823

12

49

. 5328

. 4672

.5243

. 5511

.4472

.0119

. 1182

. 8818

11

50

.15356

.84644

6.6121

.15540

6.4348

1.012U

.01186

.98814

10

51

. 5385

. 4615

.4999

. 5570

.4225

.0120

. 1190

. 8809

9

52

. 5413

. 4586

.4878

. 5600

.4103

.0121

. 1195

. 8805

8

53

. 5442

. 4558

.4757

. 5630

.3980

.0121

. 1199

. 8800

7

54

. 5471

. 4529

.4637

. 5659

.3859

.0122

. 1204

. 8796

6

55

.15500

.84500

6.4517

.15689

6.3737

1.0122

.01208

.98791

5

66

. 5528

. 4471

.4398

. 5719

.3616

.0123

. 1213

. 8787

4

57

. 6557

. 4443

.4279

. 5749

.3496

.0123

. 1217

. 8782

3

58

. 5586

. 4414

.4160

. 5779

.3376

.0124

. 1222

. 8778

2

59

. 5615

. 4385

.4042

. 5809

.3257

.0124

. 1227

. 8773

1

60

. 5643

. 4366

.3924

. 6838

.3137

.0125

. 1231

. 8769

M.

Cosine.

Vrs. sin.

Secant.

Cotang.

Tang.

Cosec'nt

Sine.

Vrs. COS.

M.

Table 3.

NATURAL FUNCTIONS.

331

90

Natural Trigonometrical Functions.

170°

M^

Sine.

Vrs. COS.

Cosec'nt

Tang.

Cotang.

Secant.

Vrs. Bin.

Cosine.

M.

.15643

.84356

6.3924

.15838

6.3137

1.0125

.01231

.98769

60

1

. 5672

. 4328

.3807

. 5868

.3019

.0125

. 1236

. 8764

59

2

. 5701

. 4299

.3690

. 5898 -

.2901

.0125

. 1240

. 8760

58

3

. 5730

. 4270

.3574

. 5928

.2783

.0126

. 1245

. 8755

57

4

. 5758

. 4242

.3458

. 5958

.2665

.0126

. 1249

. 8750

56

5

.15787

.84213

6.3343

.15987

6.2548

1.0127

.01254

.98716

56

6

. 5816

. 4184

.3228

. 6017

.2432

.0127

. 1259

. 8741

54

7

. 5844

. 4155

.3113

. 6047

.2316

.0128

. 1263

. 8737

53

8

. 5873

. 4127

.2999

. 6077

.2200

.0128

. 1268

. 8732

52

9

. 5902

. 4098

.2885

. 6107

.2085

.0129

. 1272

. 8727

51

10

.15931

.84069

6.2772

.16137

6.1970

1.0129

.01277

.98723

50

11

. 6959

. 4041

.2659

. 6167

.1856

.0130

. 12S2

. 8718

49

12

. 5988

. 4012

.2546

. 6196

.1742

.0130

. 1286 ■

. 8714

48

IS

. 6017

. 3983

.2434

. 6226

.1628

.0131

. 1291

. 8709

47

14

. 6045

. 3954

.2322

. 6256

.1515

.0131

. 1296

. 8704

46

15

.16074

.83926

6.2211

.16286

6.1402

1.0132

.01300

.98700

46

16

. 6103

. 3897

.2100

. 6316

.1290

.0132

. 1305

. 8695

44

17

. 6132

. 3868

.1990

. 6346

.1178

.0133

. 1310

. 8690

43

18

. 6160

. 3840

.1880

. 6376

.1066

.0133

. 1314

. 8685

42

19

. 6189

. 3811

.1770

. 6405

.0955

.0134

. 1319

. 8681

41

20

.16218

.83782

6.1661

.16435

6.0844

1.0134

.01324

.98676

40

21

. 6246

. 3753

.15.52

. 6465

.0734

.0135

. 1328

. 8671

39

22

. 6275

. 3725

.1443

. 6495

.0624

.0135

. 1333

. 8lili7

38

23

. 6304

. 3696

.1335

. 6525

.0514

.0136

. 1338

. 8i;(i2

37

24

. 6333

. 3667

.1227

. 6555

.0405

.0136

. 1343

. 8657

36

26

.16361

.83639

6.1120

.16585

6.0296

1.0136

.01347

.9Si;52

35

26

. 6390

. 3610

.1013

. 6615

.0188

.0137

. 1352

. 8648

34

27

. 6419

. 3581

■ .0906

. 6644

.0080

.0137

. 1357

. 8643

33

28

. 6447

. 3553

.0800

. 6674

5.9972

.0138

. 1362

. 8638

32

29

. 6476

. 3524

.0694

. 6704

.9865

.0138

. 1367

. 8633

31

30

.16505

.83495

6.0588

.16734

5.9758

1.0139

.01371

.98628

30

31

. 6533

. 3466

.0483

. 6764

.9651

.0139

. 1376

. 8624

29

32

. 6562

. 3438

.0379

. 6794

.9545

.0140

. 1381

. 8619

28

33

. 6591

. 3409

.0274

. 6824

.9439

.0140

. 1386

. 8614

27

34

. 6619

. 3380

.0170

. 6854

.9333

.0141

. 1391

. 8609

26

35

.16648

.83852

6.0066

.16884

5.9228

1.0141

.01395

.98604

25

36

. 6677

. 3323

5.9963

. 6911

.9123

.0142

. 1400

. 8600

24

37

. 6706

. 3294

.9860

. 6944

.9019

.0142

. 1405

. 8595

23

38

. 6734

. 3266

.9758

. 6973

.8915

.0143

. 1410

. 8590

22

39

. 6763

. 3237

.9655

. 7003

.8811

.0143

. 1415

. 8585

21

40

.16791

.83208

5.9554

.17033

5.8708

1.0144

.01420

.98580

20

41

. 6820

. 3180

.9452

. 7063

.8605

.0144

. 1425

. 8575

19

42

. 6849

. 3151

.9351

. 7093

.8602

.0145

. 1430

. 8570

18

43

. 6878

. 3122

.9250

. 7123

.8400

.0145

. 1434

. 8566

17

44

. 6906

. 3094

.9150

. 7153

.8298

.0146

. 1439

. 8560

16

45

.16935

.83065

5.9049

.17183

5.8196

1.0146

.01444

.98556

15

46

. 6964

. 3036

.8950

. 7213

.8095

.0147

. 1449

. 8551

14

47

. 6992

. 3008

.8850

. 7243

.7994

.0147

. 1454

. 8546

13

48

. 7021

. 2979

.8751

. 7273

.7894

.0148

. 1459

. 8541

12

49

. 7050

. 2950

.8652

. 7803

.7793

.0148

. 1464

. 8536

11

50

.17078

.82922

5.8554

.17333

5.7694

1.0149

.01469

.98531

10

61

. 7107

. 2893

.8456

. 7363

.7594

.0150

. 1474

. 8526

9

52

. 7136

. 2864

.8358

. 7393

.7495

.0150

. 1479

. 8521

8

53

. 7164

. 2836

.8201

. 7423

.7396

.0151

. 1484

. 8516

7

54

. 7193

. 2807

.8163

. 7463

.7297

.0151

. 1489

. 8511

6

55

.17221

.82778

5.8067

.17483

5.7199

1.0152

.01494

.98506

6

56

. 7250

. 2750

.7970

. 7513

.7101

.0152

. 1499

. 8501

4

67

. 7279

. 2721

• .7874

. 7543

.7004

.0153

. 1604

. 8496

3

58

. 7307

. 2692

.7778

. 7573

.6906

.0153

. 1509

. 8491

2

59

. 7336

. 2664

.7683

. 7603

.6809

.0154

. 1514

. 8486

1

60

. 7365

. 2635

.7588

. 7633

.6713

.0154

. 1519

. 8481

M.

Cosine.

Vra. ein.

Secant.

Cotang.

Tang.

Cosec'nt

Vrs. COS.

Sine.

M.

332

NATURAL FUNCTIONS.

Table 3.

10°

Natural Trigonometrical Functions.

«69<^

M.

Sine.

Vra. C03.

Cosec'nt

Tang.

Cotang.

Secant.

Vrs. sin.

Cosine.

M.

.17365

.82635

5.7588

.17633

5.6713

1.0164

.01519

.98481

60

1

. 7393

. 2606

.7193

. 7663

.6616

.0155

. 1524

. 8476

59

2

. 7422

. 2578

.7398

. 7693

.6520

.0155

. 1529

. 8471

58

3

. 7451

. 2549

.7304

. 7723

.6126

.0156

. 1534

. 8465

57

4

. 7479

. 2521

.7210

. 7753

.6329

.0156

. 1539

. 8460

56

5

.17508

.82192

6.7117

.17783

5.6234

1.0157

.01544

.98455

55

6

. 7537

. 2463

.7023

. 7813

.6140.

.0167

. 1550

. 8450

54

7

. 7565

. 2435

.6930

. 7813

.6045

.0158

. 1656

. 8445

53

8

. 7594

. 2106

.6838

. 7873

.5951

.0158

. 1.560

. 8140

52

9

. 7622

. 2377

.6745

. 7903

.6867

.0169

. 1566

. 8435

51

10

.17651

.82349

5.6653

.17933

5.6764

1.0169

.01570

.98130

50

11

. 7680

. 2320

.6561

. 7963

.5670

.0160

. 1575

. 8425

49

12

. 7708

. 2291

.6470

. 7993

.5578

.0160

. 1680

. 8419

48

13

. 7737

. 2263

.6379

. 8023

.5186

.0161

. 1686

. 8414

47

14

. 7766

. 2234

.6288

. 8063

.6393

.0162

. 1591

. 8109

46

15

.17794

.82206

5.6197

.18083

5.5301

1.0162

.01596

.98404

46

16

. 7823

. 2177

.6107

. 8113

.6209

.0163

. 1601

. 8399

44

17

. 7852

. 2148

.6017

. 8113

.5117

.0163

. 1C06

. 8394

43

18

. 7880

. 2120

.6928

. 8173

.5026

.0161

. 1611

. 8388

42

19

. 7909

. 2091

.5838

. 8203

.4936

.0164

. 1617

. 8383

41

20

.17937

.82062

5.5719

.18233

5.1845

1.0166

.01622

.98378

40

21

. 7966

. 2031

.6660

. 8^63

.4755

.0165

. 1627

. 8373

39

22

. 7995

. 2005

.5672

. 8293

.1665

.0166

. 1632

. 8368

38

23

. 8023

. 1977

.5184

. 8323

.4575

.0166

. 1638

. 8362

37

24

. 8052

. 1948

.5396

. 8363

.4186

.0107

. 1613

. 8367

36

25

.18080

.81919

6.5308

.18383

5.1396

1.0167

.01618

.98352

36

26

. 8109

. 1891

■ .6221

. 8413

.4308

.0168

. 1653

. 8347

34

27

. 8138

. 1862

.5134

. 8144

.1219

.0169 ■

1659

. 8341

33

28

. 8166

. 1834

.5017

. 8474

.1131

.0169

. 1661

. 8336

32

29

. 8195

. 1805

.1960

. 8501

.4043

.0170

. 1669

. 8331

31

30

.18223

.81776

6.1874

.18531

6.3955

1.0170

.01674

.98326

30

31

. 8252

. 1748

.1788

. 8561

.3868

.0171

. 1680

. 8320

29

32

. 8281

. 1719

.1702

. 8591

.3780

.0171

. 1685

. 8315

28

33

. 8309

. 1691

.1617

. 8624

.3691

.0172

. 1690

. 8309

27

34

. 8338

. 1662

.1532

. 8654

.3607

.0172

. 1696

. 8304

26

35

.18366

.81633

6.4117

.18684

5.3521

1.0173

.01701

.98299

26

36

. 8395

. 1605

.4362

. 8714

.3134

.0174

. 1706

. 8293

24

37

. 8424

. 1576

.4278

. 8745

.3349

.0174

. 1712

. 8288

23

38

. 8452

. 1518

.1194

. 8775

.3263

.0176

. 1717

. 8283

22

39

. 8481

. 1519

.4110

. 8805

.3178

.0175

. 1722

. 8277

21

40

.18509

.81190

5.4026

.18836

5.3093

1.0176

.01728

.98272

20

41

. 8538

. 1162

.3943

. 8866

.3008

.0176

. 1733

. 8267

19

42

. 8567

. 1133

.3860

. 8896

.2923

.0177

. 1739

. 8261

18

43

. 8595

. 1405

.3777

. 8926

.2839

.0177

. 1714

. 8266

17

44

. 8624

. 1376

.3695

. 8955

.2755

.0178

. 1719

. 8250

16

45

.18652

.81318

6.3612

.18985

5.2671

1.0179

.01756

.98245

15

46

. 8681

. 1319

.3530

. 9016

.2588

.0179

. 1700

. 8240

14

47

. 8709

. 1290

.3449

. 9046

.2606

.0180

. 1766

. 8234

13

48

. 8738

. 1262

.3367

. 9076

.2422

.0180

. 1771

. 8229

12

49

. 8767

. 1233

.3286

. 9106

.2339

.0181

. 1777

. 8223

11

50

.18795

.81205

5.3205

.19136

6.2'257

1.0181

.01782

.98218

10

51

. 8824

. 1176

.3124

. 9166

.2174

.0182

. 1788

. 8212

9

52

. 8852

. 1117

.3044

. 9197

.2092

.0182

. 1793

. 8207

8

53

. 8881

. 1119

.2963

. 9227

.2011

.0183

. 1799

. 8201

7

54

. 8909

. 1090

.2883

. 9257

.1929

.0181

. 1804

. 8196

6

55

.18938

.81062

5.2803

.19287

5.1818

1.0184

.01810

.98190

5

56

. 8967

. 1033

.2721

. 9317

.1767

.0185

. 1815

. 8185

4

57

. 8995

. 1005

.2615

. 9347

.1686

.018S"

. 1821

. 8179

3

58

. 9024

. 0976

.2566

. 9378

.1606

.0186

. 1826

. 8174

2

59

. 9052

. 0918

.2487

. 9108

.1525

.0186

. 1832

. 8168

1

00

. 9081

. 0919

.2108

. 9138

.1145

.0187

. 1837

. 8163

M.

CoBiue.

Vrs. sin.

Secant.

Cotaug.

Tang.

Cosec'nt

Vrs. COS.

Sine.

M.

100°

79°

Table 3.

NATTJRAL FUNCTIONS.

333

11°

Natural Trigonometrical Functions,

168°

mT

Sine.

Vrs. COS.

Cosec'nt

Tang.

Cotang.

Secant.

Vrs. Bill.

Cosine.

M.

.19081

.80919

6.2408

.19438

5.1445

1.0187

.01837

.98163

60

1

. 9109

. 0890

.2330

. 9468

.1366

.0188

. 1843

. 8157

59

2

. 9138

. 0862

.2262

. 9498

.1286

.0188

. 1848

. 8152

58

3

. 9166

.0833

.2174

. 9629

.1207

.0189

. 1854

. 8146

57

4

. 9195

. 0805

.2097

. 9559

.1128

.0189

. 1859

. 8140

56

5

.19224

.80776

5.2019

.19589

5.1049

1.0190

.01866

.98135

55

6

■. 9252

. 0748

.1942

. 9619

.0970

.0191

. 1871

. 8129

54

7

. 9281

.0719

.1866

•. 9649

.0892

.0191

. 1876

. 8124

53

8

. 9309

. 0691

.1788

. 9680

.0814

.0192

. 1882

. 8118

52

9

. 9338

. 0662

.1712

. 9710

.0736

.0192

. 1887

. 8112

61

10

.19366

.80634

5.1636

.19740

5.0658

1.0193

.01893

.98107

50

11

. 9395

. 0605

.1560

. 9770

.0581

.0193

. 1899

. 8101

49

12

. 9423

. 0576

.1484

. 9800

.0504

.0194

. 1904

. 8096

48

13

. 9452

. 0548

.1409

. 9831

.0427

.0196

. 1910

. 8090

47

14

. 9480

. 0519

.1333

. 9861

.0350

.0196

. 1916

. 8084

46

15

.19509

.80491

5.1268

.19891

5.0273

1.0196

.01921

.98078

45

16

. 9637

. 0462

.1183

. 9921

.0197

.0196

. 1927

. 8073

44

17

. 9566

. 0434

.1109

. 9952

.0121

.0197

. 1933

. 8067

43

18

. 9595

. 0406

.1034

. 9982

.0045

.0198

. 1938

. 8061

42

19

. 9623

. 0377

.0960

.20012

4.9969

.0198

. 1944

. 8066

41

20

.19652

.80348

5.0886

.20042

4.9894

1.0199

.01950

.98060

40

21

. 9680

. 0320

.0812

. 0073

.9819

.0199

. 1956

. 8044

39

22

. 97C9

. 0291

.0739

. 0103

.9744

.0200

. 1961

. 8039

38

23

. 9737

. 0263

.0666

. 0133

.9669

.0201

. 1967

. 8033

37

24

. 9766

. 0234

.0593

. 0163

.9694

.0201

. 1973

. 8027

36

25

.19794

.80206

5.0520

.20194

4.9520

1.0202

.01979

.98021

35

26

. 9823

. 0177

.0447

. 0224

.9446

.0202

. 1984

. 8016

34

27

. 9861

. 0149

.0375

. 0254

.9372

.0203

. 1990

. 8010

33

28

. 9880

. 0120

.0302

. 0285

.9298

.0204

. 1996

. 8004

32

29

. 9908

. 0092

.0230

. 0315

.9225

.0204

. 2002

. 7998

31

30

.19937

.80063

5.0158

.20345

4.9151

1.0206

.02007

.97992

30

31

. 9965

. 0035

.0087

. 0375

.9078

.0205

. 2013

. 7987

29

32

. 9994

. 0006

.0015

. 0406

.9006

.0206

. 2019

. 7981

28

83

.20022

.79978

4.9944

. 0436

.8933

.0207

. 2025

. 7975

27

34

. 0051

. 9949

.9873

. 0466

.8860

.0207

. 2031

. 7969

26

35

.20079

.79921

4.9802

.20497

4.8788

1.0208

.02037

.97963

25

36

. 0108

. 9892

.9732

. 0527

.8716

.0208

. 2042

. 7957

24

37

. 013B

. 9863

.9661

. 05C7

.8644

.0209

. 2048

. 7952

23

38

. 0165

. 9835

.9591

. 0688

.8573

.0210

. 2054

. 7946

22

39

. 0193

. 9807

.9621

. 0618

.8501

.0210

. 2060

. 7940

21

40

.20222

.79778

4.9452

.20648

4.8430

1.0211

.02066

.97934

20

41

. 0250

. 9760

.9382

. 0679

.8359

.0211

. 2072

. 7928

19

42

. 0279

. 9721

.9313

. 0709

.8288

.0212

. 2078

. 7922

18

43

. 0307

. 9693

.9243

. 0739

.8217

.0213

. 2084

. 7916

17

44

. 0336

. 9664

.9175

. 0770

.8147

.0213

. 2089

. 7910

16

45

.20364

.79636

4.9106

.20800

4.8077

1.0214

.02095

.97904

15

46

. 0393

. 9607

.9037

. 0830

.8007

.0215

. 2101

. 7899

14

47

. 0421

. 9679

.8969

. 0861

.7937

.0216

. 2107

. 7893

13

48

. 0450

. 9660

.8901

. 0891

.7867

.0216

. 2113

. 7887

12

49

. 0478

. 9622

.8833

. 0921

.7798

.0216

. 2119

. 7881

11

50

.20506

.79493

4.8765

.20952

4.7728

1.0217

.02125

.97875

10

61

. 0535

. 9466

.8697

. 0982

.7659

.0218

. 2131

. 7869

9

52

. 0563

. 9436

.8630

. 1012

.7691

.0218

. 2137

. 7863

8

53

. 0592

. 9408

.8563

. 1043

,7522

.0219

. 2143

. 7857

7

64

.0620

. 9379

.8496

. 1073

.7453

.0220

. 2149

. 7851

6

55

.20649

.79361

4.8429

.21104

4.7385

1.0220

.02165

.97845

6

66

. 0677

. 9323

.8362

. 1134

.7317

.0221

. 2161

. 7839

4

57

. 0706

. 9294

.8296

. 1164

.7249

.0221

. 2167

. 7833

3

58

. 0734

. 9266

.8229

. 1196

.7181

.0222

. 2173

. 7827

2

69

. 0763

. 9237

.8163

. 1226

.7114

.0223

. 2179

. 7821

1

60

. 0791

. 9209

.8097

. 1256

.7046

.0223

. 2185

. 7815

m7

Cosine.

Vra. an.

Secant.

Cotang.

Tang.

Cosec'nt

Vrs. COS.

Sine.

M.

101°

78°

334

NATURAL FUNCTIONS.

Table b.

12

D

Natural Trigonometrical Functions.

167°

M.

Sine.

Vrs. COS.

Cosec'nt

Tang.

Co tang.

Secant.

Vrs. sin.

Cosine.

M.

.20791

.79209

4.8097

.21256

4.7046

1.0223

.02185

.97815

60

1

. 0820

. 9180

.8032

. 1286

.6979

.0224

. 2191

. 7809

69

2

. 0848

. 9152

.7966

. 1316

.6912

.0225

. 2197

. 7803

68

3

. 0876

. 9123

.7901

. 1347

.6845

.0225

. 2203

. 7806

57

4

. 0905

. 9105

.7835

. 1377

.6778

.0226

. 2209

. 7790

56

5

.20933

.79006

4.7770

.21408

4.0712

1.0226

.02215

.97784

55

6

. 0962

. 9038

.7706

. 1438

.6646

.0227

. 2222

. 7778

54

7

. 0990

. 9010

.7641

. 1468

.6580

.0228

. 2228

. 7772

53

8

. 1019

. 8981

.7576

. 1499

.6514

.0228

. 2234

. 7766

62

9

. 1047

. 8953

.7512

. 1529

.6448

.0229

. 2240

. 7760

51

10

.21076

.78924

4.7448

.21560

4.6382

1.0230

.02246

.97754

50

U

. 1104

. 8896

.7384

. 1590

.6317

.0230

. 2252

. 7748

49

12

. 1132

. 8867

.7320

. 1621

.6252

.0231

. 2258

. 7741

48

13

. 1161

. 8839

.7257

. 1651

.6187

.0232

. 2264

. 7735

47

14

. 1189

. 8811

.7193

. 1682

.6122

.0232

. 2271

. 7729

46

15

.21218

.78782

4.7130

.21712

4.6057

1.0233

.02277

.97723

45

16

. 1246

. 8754

.7067

. 1742

.5993

.0234

. 2283

. 7717

44

17

. 1275

. 8726

.7004

. 1773

.5928

.0234

. 2289

. 7711

43

18

. 1303

. 8697

.6942

. 1803

.5864

.0235

. 2295

. 7704

42

19

. 1331

. 8668

.6879

. 1834

.5800

.0235

. 2302

. 7698

41

20

.21360

.78640

4.6817

.21864

4.5736

1.0236

.02308

.97692

40

21

. 1388

. 8612

.6754

. 1895

.6673

.0237

. 2314

. 7686

39

22

. 1117

. 8583

.6692

. 1925

.6609

.0237

. 2320

. 7680

38

23

. 1445

. 8555

.6631

. 1956

.6546

.0238

. 2326

. 7673

37

24

. 1473

. 8526

.6569

. 1986

.5483

.0239

. 2333

. 7667

36

25

.21502

.78508

4.6507

.22017

4.5420

1.0239

.02339

.97661

35

26

. 1530

. 8470

.6446

. 2047

.5357

.0240

. 2345

. 7655

34

27

. 1.559

. 8441

.6385

. 2078

.5294

.0241

. 2351

. 7648

33

28

. 1587

. 8413

.6324

. 2108

.5232

,0241

. 2358

. 7642

32

29

. 1615

. 8384

.6263

. 2139

.5169

.0242

. 2364

. 7636

31

30

.21644

.78356

4.6202

.22169

4.5107

1.0243

.02370

.97630

30

31

. 1672

. 8328

.6142

. 2200

.5045

.0243

. 2377

. 7623

29

32

. 1701

. 8299

.6081

. 2230

.4983

.0244

. 2383

. 7617

28

33

. 1729

. 8271

.6021

. 2261

.4921

.0245

. 2389

. 7611

27

34

. 1757

. 8242

.5961

. 2291

.4860

.0245

. 2396

. 7604

26

35

.21786

.78214

4.5901

.22322

4.4799

1.0246

.02402

.97598

25

36

. 1814

. 8186

.5841

. 2353

.4737

.0247

. 2408

. 7592

24

37

. 1843

. 8154

.5782

. 2383

.4676

.0247

. 2415

. 7585

23

38

. 1871

. 8129

.5722

. 2414

.4615

.0248

. 2421

. 7.579

22

39

. 1899

. 8100

.5663

. 2444

.4555

.0249

. 2427

. 7573

21

40

.21928

.78072

4.5C04

.22475

4.4494

1.0249

.02434

.97566

20

41

. 1956

. 8043

.55J5

. 2505

.4434

.0250

. 2440

. 7560

19

42

. 1985

. 8015

.5486

. 2536

.4373

.0251

. 2446

. 7553

18

43

. 2013

. 7987

.5428

. 2566

.4313

.0251

. 24.53

. 7547

17

44

. 2041

. 7959

.5369

. 2597

.4263

.0252

. 2459

. 7541

16

45

.22070

.77930

4.5311

.22628

4.4194

1.0253

.02466

.97634

15

46

. 2098

. 7902

.5253

. 2658

.4134

.0253

. 2472

. 7528

14

47

. 2126

. 7873

.5195

. 2689

.4074

.0254

. 2479

. 7521

13

48

. 2155

. 7845

.5137

. 2719

.4015

.0265

. 2485

. 7515

12

49

. 2183

. 7817

.5079

. 2750

.3956

.0255

. 2491

7608

11

50

.22211

.77788

4.5021

.22781

4.3897

1.0256

.02498

.97502

10

51

. 2240

. 7760

.4964

. 2811

.3838

.0257

. 2504

. 7495

9

52

. 2268

. 7732

.4907

. 2842

.3779

.0257

. 2511

. 7489

8

63

. 2297

. 7703

.4850

. 2872

.3721

.0268

. 2517

. 7483

7

64

. 2325

. 7675

.4793

. 2903

.3662

.0259

. 2524

. 7476

6

55

.22353

.77647

4.4736

.22934

4.3604

1.0260

.02530

.97470

5

56

. 2382

. 7618

.4679

. 2964

.3646

.0260

. 2537

. 7463

4

57

. 2-110

7590

.4623

. 2995

.3488

.0261

. 2543

. 7457

3

58

2438

. 7561

.4566

. 3026

.3430

.0262

. 2550

. 7450

2

59

. 2467

. 7533

.4510

. 3056

•.3372

.0262

. 2556

. 7443

1

60

. 2495

. 7505

.4454

. 3087

.3315

.0263

. 2563

. 7437

C

M.

Cosine.

Vrs. sin.

Secant.

Co tang.

Tang.

Cosec'ntl

Vrs. COB.

Sine.

M.

102°

77"

Table 3.

NATUEAL FUNCTIONS.

335

13°

Natural Trigonometrical Functions.

166°

M.

Sine.

Vrs. COS.

Cosec'nt

Tang.

Cotang.

Secant.

Vrs. sin.

Cosine.

M.

.22495

.77505

4.4454

.23087

4.3315

1.0263

.02563

.97437

60

1

. 2523

. 7476

.4398

. 3137.

.3257

.0264

. 2569

. 7430

59

2

. 2552

. 7448

.4312

. 3118

.3200

.0264

. 2576

. 7424

58

3

. 2580

. 7420

.4287

.3179

.3143

.0265

. 2.583

. 7417

57

4

. 2608

. 7391

.4231

. 3209

.3086

.0266

. 2589

. 7411

66

5

.22637

.77363

4.4176

.23240

4.3029

1.0266

.02596

.97404

65

6

. 2665

. 7335

.4121

. 3270

.2972

.0267

. 2602

. 7398

54

7

. 2693

. 7306

.4065

. 3301

.2916

.0268

. 2609

. 7391

63

8

. 2722

. 7278

.4011

. 3332-

.2859

.0268

. 2616

. 7384

52

9

. 2750

. 7250

.3956

. .3363

.2803

.0269

. 2622

. 7378

51

10

.22778

.77221

4.3901

.23393

4.2747

1.0270

.02629

.97371

50

11

. 2807

. 7193

.3847

. 3424

.2691

.0271

. 2635

. 7364

49

12

. 2835

. 7165

.3792

. 3455

.2635

.0271

. 2642

. 7368

48

13

. 2863

. 7136

.3738

. 3485

.2579

.0272

. 2649

. 7351

47

14

. 2892

. 7108

.3684

. 3516

.2524

.0273

. 2655

. 7344

46

15

.22920

.77080

4.3630

.23547

4.2468

1.0273

.02662

.97338

45

16

. 2948

. 7052

.3676

. 3577

.2413

.0274

. 2669

. 7331

44

17

. 2977

. 7023

.3522

. 3608

.2358

.0275

. 2675

. 7324

43

18

. 3005

. 6995

.3469

. 3639

.2303

.0276

. 2682

. 7318

42

19

. 3033

. 6967

.3415

. 3670

.2218

.0276

. 2689

. 7311

41

20

.23061

.76938

4.3362

.23700

4.2193

1.0277

.02695

.97304

40

21

. 3090

. 6910

.3309

. 3731

.2139

.0278

. 2702

. 7298

39

22

. 3118

. 6882

.3256

. 3762

.2084

.0278

. 2709

. 7291

38

23

. 3146 ■

. 6853

.3203

. 3793

.2030

.0279

. 2716

. 7284

37

24

. 3175

. 6825

.3150

. 3823

.1976

.0280

. 2722

. 7277

36

25

.23203

.76797

4.3098

.23854

4.1921

1.0280

.02729

.97271

35

26

. 3231

. 6769

.3045

. 3885

.1867

.0281

. 2736

. 7264

34

27

. 3260

. 6740

.2993

. 3916

.1814

.0282

. 2743

. 7257

33

28

. 3288

. 6712

.2941

. 3946

.1760

.0283

. 2749

. 7250

32

29

. 3316

. 6684

.2888

. 3977

.1706

.0283

. 2756

. 7244

31

30

.23344

.76655

4.2836

.24008

4.1663

1.0284

.02763

.97237

30

31

. 3373

. 6627

.2785

. 4039

.1600

.0285

. 2770

. 7230

29

32

. 3401

. 6599

.2733

. 4069

.1516

.0285

. 2777

. 7223

28

33

. 3429

. 6571

.2681

. 4100

.1493

.0286

. 2783

. 7216

27

34

. 3458

. 6542

.2630

. 4131

.1440

.0287

. 2790

7210

26

35

.23486

.76514

4.2579

.24162

4.1388

1.0288

.02797

.97203

25

36

. 3514

. 6486

.2527

. 4192

.1335

.0288

. 2804

. 7196

■24

37

. 3542

. 6457

.2476

. 4223

.1282.

.0289

. 2811

. 7189

23

38

. 3571

. 6129

.2425

. 4254

.1230

.0290

. 2818

. 7182

22

39

. 3599

. 6401

.2375

. 4285

.1178

.0291

. 2824

. 7175

21

40

.23627

.76373

4.2324

.24316

4.1126

1.0291

.02831

.97169

20

41

. 3655

. 6344

.2273

. 4346

.1073

.0292

. 2838

. 7162

19

42

. 3684

. 6316

.2223

. 4377

.1022

.0293

. 2846

. 7165

18

43

. 3712

. 6288

.2173

. 4408

.0970

.0293

. 2852

. 7148

17

44

. 3740

. 6260

.2122

. 4439

.0918

.0294

. 2869

. 7141

16

45

.23768

.76231

4.2072

.24470

4.0867

1.0295

.02866

.97134

15

46

. 3797

. 6203

.2022

. 4501

.0815

.0296

. 2873

. 7127

14

47

. 3825

. 6175

.1972

. 4531

.0764

.0296

. 2880

. 7120

13

48

. 3853

. 6147

.1923

. 4562

.0713

.0297

. 2886

. 7113

12

49

. 3881

. 6118

.1873

. 4693

.0662

.0298

. 2893

. 7106

11

50

.23910

.76090

4.1824

.24624

4.0611

1.0299

.02900

.97099

10

51

. 3938

. 6062

.1774

.4655

.0560

.0299

. 2907

. 7092

9

52

. 3966

. 6034

.1725

.4686

.0509

.0300

. 2914

. 7086

8

5S

. 3994

. 6005

.1676

. 4717

.0458

.0301

. 2921

. 7079

7

54

. 4023

. 5977

.1627

. 4747

.0408

.0302

. 2928

. 7072

6

55

.24051

.75949

4.1578

.24778

4.0368

1.0302

.02935

.97065

5

66

. 4079

. 5921

.1529

. 4809

.0307

.0303

. 2942

. 7058

4

57

. 4107

. 5892

.1481

. 4840

.0257

.0304

. 2949

. 7051

3

58

. 4136

. 5864

.1432

. 4871

.0207

.0305

. 2956

. 7044

2

59

. 4164

. 5836

.1384

. 4902

.0157

.0305

. 2963

. 7037

1

60

. 4192

. 5808

.1336

. 4933

.0108

.0306

. 2970

. 7029

M.

CosiDe.

Vrs. sin.

Secant.

Cotang.

Tang.

Cosec'nt

Vrs. COS.

Sine.

M.

103°

76°

336

NATURAL FUNCTIONS.

Table 3.

14°

Natural T

rigonometrical

Functions.

1

55°

5L

Sine.

Vrs. COS.

Cosec'nt

Tang.

Cotang,

Secant.

"Vrs. sin.

Cosine.

m!

.24192

.75808

4.1336

.24933

4.0108

1.0306

.02970

.97029

60

1

. 4220

. 5779

.1287

. 4964

.0058

.0307

. 2977

. 7022

59

2

. 4249

. 5751

.1239

. 4995

.0009

.0308

. 2984

. 7015

58

3

. 4277

. 5723

.1191

. 5025

3.9959

.0308

. 2991

. 7008

57

4

. 4305

. 5695

.1144

. 5056

.9910

.0309

. 2999

. 7001

56

5

.24333

.75667

4.1096

.25087

3.9861

1.0310

.03006

.96994

55

6

. 4361

. 5638

.1048

. 5118

.9812

.0311

. 3013

. 6987

54

7

. 4390

. 5610

.1001

. 5149

.9763

.0311

. 3020

. 6980

53

8

. 4418

. 5582

.0953

. 5180

.9714

.0.312

. 3027

. 6973

.52

9

. 4146

. 5564

.0906

. 5211

.9665

.0313

. 3034

. 6966

51

10

.24474

.75526

4.0859

.25242

3.9616

1.0314

.03041

.96959

50

11

. 4602

. 5497

.0812

. 6273

.9.668

.0314

. 3048

. 6952

49

12

. 4531

. 5469

.0765

. 6304

.9520

.0315

. 3055

. 6944

48

13

. 4559

. 5441

.0718

. 5336

.9471

.0316

. 3063

. 6937

47

14

. 4587

. 5413

.0672

. 5366

.9423

.0317

. 3070

. 6930

46

15

.24615

.75385

4.0625

.25397

8.9375

1.0317

.0.3077

.96923

45

16

. 4643

. 6366

.0579

. 5128

.9327

.0318

. 3084

. 6916

44

17

. 4672

. 6328

.0532

. 6459

.9279

.0319

. 3091

. 6909

43

18

. 4700

. 6300

.0486

. 5490

.9231

.0320

. 3098

. 6901

42

19

. 4728

. 5272

.0440

. 5521

.9184

.0320

. 3106

. 6894

41

20

.24756

.75244

4.0394

.25562

3.9136

1.0321

.03113

.96887

40

21

. 4784

. 5215

.0348

. 5583

.9089

.0322

. 3120

. 6880

39

22

. 4813

. 6187

.0302

. 5614

.9042

.0323

. 3127

. 6873

38

23

. 4841

. 5159

.0266

. 5645

.8994

.0323

. 3134

. 6865

37

24

. 4869

. 5131

.0211

. 5676

.8947

.0324

. 3142

. 6858

36

26

.24897

.75103

4.0165

.25707

3.8900

1.0.326

.03149

.96851

35

26

. 4925

. 5075

.0120

. 5738

.8853

.0326

. 3166

. 6844

34

27

. 4963

. 5046

.0074

. 5769

.8807

.0327

. 3163

. 6836

33

28

4982

. 6018

.0029

. 5800

.8760

.0327

. 3171

. 6829

32

29

. 5010

. 4990

3.9984

. 5831

.8713

.0328

. 3178

. 6822

31

30

.25038

.74962

3.9939

.25862

3.8667

1.0329

.03185

.96815

30

31

5066

. 4934

.9894

. 5893

.8621

.0330

. 3192

. 6807

29

32

. 6094

. 4906

.9850

. 5924

.8574

.0330

. 3200

. 6800

28

33

. 5122

. 4877

.9805

. 5965

.8528

.0331

. 3207

. 6793

27

34

. 5151

. 4849

.9760

. 5986

.8482

.0332

. 3214

. 6785

26

35

.25179

.74821

3.9716

.26017

3.8436

1.0333

.03222

.96778

26

36

. 5207

. 4793

.9672

. 6048

.8390

.0334

. 3229

. 6771

■24

37

. 5235

. 4765

.9627

. 6079

.8345

.0334

. 3236

. 6763

23

38

. 5263

. 4737

.9583

. 6110

.8299

.0335

. 3244

. 6766

22

39

. 5291

. 4709

.9539

. 6141

.8254

.0336

. 3251

. 6749

21

40

.25319

.74680

3.9495

.26172

3.8208

1.0337

.03258

.96741

20

41

. 5348

. 4652

.9451

. 6203

.8163

.0338

. 3266

. 6734

19

42

. 5376

. 4624

.9408

. 6234

.8118

.0338

. 3273

. 6727

18

43

. 6404

. 4596

.9364

. 6266

.8073

.0339

. 3281

. 6719

17

44

. 6432

. 4568

.9320

. 6297

.8027

.0340

. 3'288

. 6712

16

45

.26460

.74540

3.9277

.26328

3.7983

1.0341

.03295

.96704

15

46

. 5488

. 4612

.9234

. 6369

.7938

.0341

. 3303

. 6697

14

47

. 5516

. 4483

.9190

. 6390

.7893

.0342

. 3310

. 6690

13

48

. 5544

. 4465

.9147

. 6421

.7848

.0343

. 8318

. 6682

12

49

. 5573

. 4427

.9104

. 6462

.7804

.0344

. 3325

. 6675

11

50

.25601

.74399

3.9061

.26483

3.7759

1.0345

.03332

.96667

10

51

. 5629

. 4371

.9018

. 6514

.7715

.0345

. 3340

. 6660

9

52

. 5657

. 4344

.8976

. 6546

.7671

.0346

. 3347

. 6652

8

53

. 6685

. 4315

.8933

. 6577

.7027

.0347

. 3355

. 6645

7

54

. 5713

. 4287

.8890

. 6608

.7583

.0348

. 3362

. 6638

6

55

.25741

.74269

3.8848

.26639

3.7539

1.0349

.03370

.96630

6

56

. 5769

. 4230

.8805

. 0670

.7495

.0349

. 3377

. 6623

4

57

. 6798

. 4202

.8763

. 6701

.7461

.0360

. 3385

. 6615

3

58

. 6826

. 4174

.8721

. 6732

.7407

.0361

. 3392

. 6608

2

59

. 5864

. 4146

.8679

. 6764

.7364

.0352

. 3400

. 6600

1

60

. 5882

. 4118

.8637

. 6796

.7320

.0353

. 3407

. 6592

U.

Cosine.

Vrs. siu.

Secant.

Co tang.

Taug.

Cosec'ntI

Vre. COS.

Sine.

M.

104°

75°

Table 3.

NATURAL FUNCTIONS.

337

15

3

Natural Trigonometrical Functions.

164°

m7

Sine.

Vts. cos.

Ooeec'nt

Tang.

Ootting.

Secant.

■Vrs. Bin.

Cosine.

M.

.25882

.74118

3.8637

.26795

3.7820

1.0353

.03407

.96592

60

1

. 5910

. 4090

.8595

. 6826

.7277

.0353

. 3115

. 6585

59

2

. 5938

. 4062

.8553

6857

.7234

.0354

. 3422

. 6577

58

3

. 5966

. 4034

.8512

. to88

.7191

.0355

. 3430

. 6570

57

4

. 5994

. 4006

.8-170

. 6920

.7117

.0356

. 3438

. 6562

56

5

.26022

.73978

3.8428

.26951

3.7104

1.0367

.03445

.96.555

55

6

. 6050

. 3949

.WS7

. 6982

.7062

.0368

. 3453

. 6.547

54

7

. 6078

. 3921

.8346

. 7013

.7019

.0358

. 3460

. 6540

53

8

. 6107

. 3893

.8304

. 7044

.6976

.0359

. 3468

.6532

52

9

. 6135

. 3865

.8263

. 7076

.6933

.0360

. 3475

. 6524

51

10

.26163

.73837

3.8222

.27107

3.6891

1.0361

.03483

.96517

50

U

. 6191

. 3809

.8181

. 7138

.6848

.0362

. 3491

. 6609

49

12

. 6219

. 3781

.8140

. 7169

.6806

.0362

. 3498

. 6502

48

13

. 6247

. 3753

.8100

. 7201

.6764

.0363

. 3506

. 6494

47

M

. 6275

. 3725

.8059

. 7232

.6722

.0364

. 3514

. 6486

46

15

.26303

.73697

3.8018

.27263

3.6679

1.0365

.03521

.96479

45

16

. 6331

. 3669

.7978

. 7294

.6637

.0366

. 3529

. 6471

44

17

. 6359

. 3641

.7937

. 7326

.6596

.0367

. 3536

. 6463

43

18

. 6387

. 3613

.7897

. 7357

.6554

.0367

. 3544

. 6456

42

19

. 6415

. 3585

.7857

. 7388

.6512

.0368

. 3552

. 6448

41

20

.26443

.73556

3.7816

.27419

3.6470

1.0369

.03560

.96440

40

21

. 6471

. 3528

.7776

. 7451

.6429

.0370

. 3567

. 6433

39

22

. 6499

. 3500

.7736

. 7482

.6387

.0371

. 3575

. 6425

38

23

. 6527

. 3472

.7697

. 7513

.6346

.0371

. 3583

. 6117

37

24

. 6556

. 3444

.7657

. 7544

.6305

.0372

. 3590

. 6409

36

25

.26584

.73416

3.7617

.27576

3.6263

1.0373

.03598

.96402

35

26

. 6612

. 3388

.7577

. 7607

.6222

.0374

. 3606

. 6394

34

27

. 6640

. 3360

.7538

.7638

.6181

.0375

. 3614

. 6386

33

28

. 6668

. 3332

.7498

. 7670

.6140

.0376

. 3621

. 6378

32

29

. 6696

. &304

.7459

. 7701

.6100

.0376

. 3629

. 6371

31

30

.26724

.73276

3.7420

.27732

3.6059

1.0377

.03637

.96363

30

31

. 6752

. 3248

.7380

. 7764

.6018

.0378

. 3645

. 6355

29

32

. 6780

. 3220

.7341

. 7795

.5977

.0379

. 3652

. 6347

28

83

. 6808

. 3192

.7302

. 7826

.5937

.0380

. 3660

. 6340

27

34

. 6835

. 3164

.7263

. 7858

.6896

.0381

. 3668

. 6332

26

35

.26864

.73136

3.7224

.27889

3.5856

1.0382

.03676

.96324

25

36

. 6892

. 3108

.7186

. 7920

.5816

.0382

. 3684

. 6316

24

37

. 6920

. 3080

.7147

. 7952

.5776

.0383

. 3691

. 6308

23

38

. 6948

. 3052

.7108

. 7983

.5736

.0384

. 3699

. 6301

22

39

. 6976

. 3024

.7070

. 8014

.5696

.0385

. 3707

. 6293

21

40

.27004

.72996

3.7031

.28046

3.5656

1.0386

.03715

.96285

20

41

. 7032

. 2968

.6993

. 8077

.5616

.0387

. 3723

. 6277

19

42

. 7060

. 2940

.6955

. 8109

.5576

.0387

. 3731

. 6269

18

43

. 7088

. 2912

.6917

. 8140

.5536

.0388

. 3739

. 6261

17

44

. 7116

. 2884

.0878

. 8171

.5497

.0389

. 3746

. 6253

16

45

.27114

.72856

3.6810

.28203

3.5457

1.0390

.03754

.96245

15

46

. 7172

. 2828

.6802

. 8234

.5418

.0391

. 3762

. 6238

14

47

. 7200

. 2800

.6765

. 8266

.5378

.0392

. 3770

. 6230

13

48

. 7228

. 2772

.6727

. 8297

.5339

.0393

. 3778

. 6222

12

49

. 7256

. 2744

.6689

. 8328

.5300

.0393

. 3786

. 6214

11

50

.27284

.72716

3.6651

.28360

3.5261

1.0394

.03794

.96206

10

51

. 7312

. 2688

.6614

. 8391

.5222

.0395

. 3802

. 6198

9

52

. 7340

. 2660

.6576

. 8423

.5183

.0396

. 3810

. 6190

8

53

. 7368

. 2632

.6539

. 84S4

.5144

.0397

. 3818

. 6182

7

54

. 7396

. 2604

.6502

. 8486

.5105

.0398

. 3826

. 6174

6

55

.27424

.72576

3.6464

.28517

3.5066

1.0399

.03834

.96166

5

56

. 7452

. 2548

.6427

. 8519

.5028

.0399

. 3842

. 6158

4

57

. 7480

. 2520

.6390

. 8580

.4989

.0400

. 3850

. 6150

3

58

. 7508

. 2492

.6353

. 8611

.4951

.0401

. 3858

. 6142

2

59

. 7536

. 2464

.6316

. 8643

.4912

.0402

. 3866

. 6134

1

60

. 7564

. 2436

.6279

. 8674

.4874

.0403

. 3874

. 6126

M.

Cosine.

Vrs. Bin.

Secant.

Cotang,

Tang.

Cosec'nt

Vrs. COS.

Sine.

M.

105°

23

74°

338

NATURAL J<'U^'CT1UJN«.

16°

Natural Trigonometrical

Functions.

163°

M.

Sine.

Vrs. COB.

Cosec'nt

Tang.

Cotang,

Secant.

Vrs. sin.

Cosine.

M.

.27564

.72436

3.6279

.28674

3,4874

1,0403

.o;k-4

.96126

60

1

. 7592

. 2408

.6243

. 8706

.4836

.0404

. 3,882

. 6118

59

2

. 7620

. 2380

.6206

. 8737

.4798

.0405

. 3890

. 6110

58

3

. 7648

. 2352

.6109

. 8769

.4760

.0406

. 3898

. 6102

57

4

. 7675

. 2324

.6133

. 8800-

.4722

.0406

. 3906

. 6094

56

5

.27703

.72296

3.6096

.28832

3,4684

1.0407

.03914

.90086

55

6

. 7731

. 2268

.6060

. 8863

.46-16

.0108

. 3922

. 6078

54

7

. 7759

. 2240

.6024

. 8895

.4608

.0409

. 3930

. 6070

53

8

. 7787

. 2213

.5987

. 8926

.4.570

.0410

. 39.38

. 6062

52

9

. 7815

. 2185

.5951

. 8958

.4533

.0411

. 3946

. 6054

51

10

.27843

.72157

3.5915

.28990

3.4495

1.0412

.03954

.96045

60

11

. 7871

. 2129

.5879

. 9021

.4458

.0413

. 3962

. 6037

49

12

. 7899

. 2101

.5843

. 9053

.4420

.0413

. 3971

. 6029

48

13

. 7927

. 2073

.5807

. 9084

.4383

.0414

. 3979

. 6021

47

14

. 7955

. 2045

.5772

. 9116

.4346

Mlb

. 3987

. 6013

46

15

.27983

.72017

3.5736

.29147

3,4308

1.0416

.03995

.96005

45

16

. 8011

. 1989

.5700

. 9179

.4271

.0417

. 4003

. 5997

44

17

. 8039

. 1961

.5665

. 9210

.4234

.0418

. 4011

. 5989

43

18

. 8067

. 1933

.5629

. 9242

.4197

.0419

. 4019

. 5980

42

19

. 8094

. 1905

.5594

. 9274

.4160

.0420

. 4028

. 5972

41

20

.28122

.71877

3.5559

.29305

3,4124

1.0420

.04036

.95964

40

21

. 8150

. 1849

.5523

. 9337

.4087

.0421

. 4014

. 5956

39

22

. 8178

. 1822

.5488

. 9368

.4050

.0422

. 4052

. 5948

38

23

. 8206

. 1794

.5453

. 9400

.4014

.0123

. 4060

. 5940

37

24

. 8234

. 1766

.5418

. 9432

.3977

.0124

. 4069

. 5931

36

25

.28262

.71738

3.5383

.29463

3.3941

1,0125

.04077

.95923

36

26

. 8290

. 1710

.5348

. 9495

.3904

.0426

. 4085

. 5915

34

27

. 8318

. 1682

.5313

. 9526

.3868

.0427

. 4093

. 5907

33

28

. 8346

. 1654

.5279

. 9558

.3832

.0428

. 4101

. 5898

32

29

. 8374

. 1626

.5244

. 9.590

.3795

.0428

. 4110

. 5890

31

30

.28401

.71608

8,5209

.29621

3.3759

1,0429

.04118

.9.5882

30

31

. 8429

. 1570

.5175

. 9653

.3723

,0430

. 4126

. 6874

29

32

. 8457

. 1543

.5140

. 9685

.3687

,0431

. 4131

. 5865

28

33

. 8485

. 1515

.5106

. 9716

.3651

,0432

. 4143

. 5857

27

34

. 8513

. 1487

.5072

. 9748

.3616

,0433

. 4151

. 5849

26

35

.28541

.71459

3,5037

.29780

3,3580

1,04.34

.04159

.9.5840

25

36

. 8569

. 1131

.5003

. 9811

,3514

,0135

. 4168

. 5832

24

37

. 8597

. 1403

.4969

. 9843

.3509

.0436

. 4176

. 5824

23

38

. 8624

. 1375

.4935

. 9875

.3473

.0437

. 4184

. 5816

22

39

. 8652

. 1347

.4fl01

. 9906

.3438

,0438

. 4193

. 5807

21

40

.28680

.71320

8,4867

.29938

3.3402

1,0438

.04201

.95799

20

41

. 8708

. 1292

.4833

. 9970

.3367

,0439

. 4209

. 5791

19

42

. 8736

. 1204

.4799

.30001

.3332

.0440

. 4218

. 5782

18

43

. 8764

. 1236

.4766

. 0033

.3296

.0441

. 4226

. 5774

17

44

. 8792

. 1208

.4732

. 0065

.3261

.0442

. 4234

. 5765

16

45

.28820

.71180

3.4698

.30096

3.3226

1,0443

.04243

.96757

15

46

. 8847

. 1152

.4665

. 0128

.3191

,0144

. 4251

. 5749

14

47

. 8875

. 1125

.4632

. 0160

.3156

,0445

. 4260

. 5740

13

48

. 8903

. 1097

.4598

. 0192

.8121

,0446

. 4268

. 5732

12

49

. 8931

. 1069

.4565

. 0223

.3087

.0447

. 4276

. 5723

U

50

.28959

.71041

3.4532

.30255

3.3052

1.0448

.04285

.95715

10

51

. 8987

. 1013

.4498

. 0287

.3017

.0448

. 4293

. 5707

9

52

. 9014

. 0985

.4465

. 0319

.2983

.0449

. 4302

. 5698

8

53

. 9042

. 0958

.4432

. 0350

.2948

.0150

. 4310

. 5690

7

64

. 9070

. 0930

.4399

, 0382

.2914

.0451

. 4319

. 5681

6

55

.29098

.70902

3.4366

,30414

3,2879

1,04.52

.04327

.95673

6

56

. 9126

. 0874

.4334

0446

,2845

,0453

. 4335

. 5664

4

57

. 9154

. 0846

.4301

. 0178

.2811

,0454

. 4344

. 5656

3

58

. 9181

. 0818

.426S

. 0509

,2777

,0455

. 43.52

. 5647

2

59

. 9209

. 0791

.42:-;o

. 0541

.2712

.0456

. 4361

. 5639

I

60

. 9237

. 0763

.4203

. 0573

.2708

.0457

. 4369

. 5630

M.

Cosine.

Vrs. sin.

Secant.

Co tang.

Tung,

Cosec'nt

Vrs. COS.

Sine.

M.

106°

73°

Table 3.

NATUEAL FUNCTIONS.

339

,70

Natural Trigonometrical Functions.

162°

M.

Sine.

Vra. COB.

Cosec'nt

Tang.

Cotang.

Secant.

Vrs. sin.

Cosine.

M.

.29237

.70763

3.4203

.30573

3.2708

1.0457

.04369

.95630

60

1

. 9265

. 0735

.4170

. 0605

.2674

.0468

. 4378

. 6622

59

2

. 9293

. 0707

.4138

. 0637

.2640

.0459

. 4386

. 5613

58

3

. 9321

. 0679

.4106

. 0668

.2607

.0460

. 4395

. 5606

67

4

. 9348

. 0651

.4073

. 0700

.2573

.0461

. 4404

. 5596

55

5

.29376

.70624

3.4041

.30732

3.2639

1.0461

.04412

.96688

55

6

. 9404

. 0596

.4009

. 0764

.2605

.0462

. 4421

. 6579

54

7

. 9432

. 0568

.3977

. 0796

.2472

.0463

. 4426

. 6671

63

8

. 9460

. 0540

.3945

. 0828

.2438

.0404

. 4438

. 5562

.52

9

. 9487

. 0512

.3913

. 0859

.2405

.0465

. 4446

. 5564

51

10

.29515

.70485

3.3881

.30891

3.2371

1.0466

.04455

.95545

50

11

. 9543

. 0457

.3849

. 0923

.2338

.0467

. 4463

. 5536

49

12

. 9571

. 0429

.3817

. 0955

.2305

.0468

. 4472

. 5628

48

13

. 9598

. 0401

.3785

. 0987

.2271

.0469

. 4481

. 5519

47

14

. 9626

. 0374

.3754

. 1019

.2238

.0470

. 4489

. 5511

46

15

.29654

.70346

3.3722

.31051

3.2205

1.0171

.04498

.95502

45

16

. 9682

. 0318

.3690

. 1083

.2172

.0472

. 4507

. 5493

44

17

. 9710

. 0290

.3669

. 1115

.2139

.0473

. 4615

. 5485

43

18

. 9737

. 0262

.3627

. 1146

.2106

.0474

. 4624

. 5476

42

19

. 9765

. 0235

.3596

. 1178

.2073

.0475

. 4532

. 5467

41

20

.29793

.70207

3.3565

.31210

3.2041

1.0476

.04541

.96469

40

21

. 9821

. 0179

.3534

. 1242

.2008

.0477

. 4550

. 5450

39

22

. 9848

. 0151

.3502

. 1274

.1975

.0478

. 4558

. 6441

38

23

. 9876

. 0124

.3471

. 1306

.1942

.0478

. 4567

. 5433

37

24

. 9904

. 0096

.3440

. 1338

.1910

.0479

. 4576

. 5424

36

25

.29932

.70068

3..S409

.31370

3.1877

1.0480

.04585

.96416

35

26

. 9959

. 0040

.3378

. 1402

.1845

.0481

. 4593

. 5407

34

27

. 9987

. 0013

.3347

. 1434

.1813

.0482

. 4602

. 5398

33

28

.30015

.69982

.3316

. 1466

.1780

.0483

. 4611

. 5389

32

29

. 0043

. 9967

.3286

. 1498

.1748

.0484

. 4619

. 5380

31

30

.30070

.69929

3.3265

.31530

3.1716

1.0486

.04628

.95372

30

31

. 0098

. 9902

.3224

. 1662

.1684

.0486

. 4637

. 6363

29

32

. 0126

. 9874

.3194

. 1594

.1652

.0487

. 4646

. 5354

28

33

. 0154

. 9846

.3163

. 1626

.1620

.0488

. 4654

. 6345

27

34

. 0181

. 9818

.3133

. 1658

.1588

.0489

. 4663

. 5337

26

35

.30209

.69791

3.3102

.31690

3.1556

1,0490

.04672

.95328

25

36

. 0237

. 9763

.3072

. 1722

.1524

.0491

. 4681

. 5319

24

37

. 0265

. 9735

.3042

. 1754

.1492

.0192

. 4690

. 5310

23

38

. 0292

. 9707

.3011

. 1786

.1460

.0493

. 4698

. 5301

22

39

. 0320

. 9680

.2981

. 1818

.1429

.0494

. 4707

. 5293

21

40

.30348

.69652

3.2951

.31850

3.1397

1.0496

.04716

.96284

20

41

. 0375

. 9624

.2921

. 1882

.1366

.0496

. 4725

. 5276

19

42

. 0403

. 9597

.2891

. 1914

.1334

.0497

. 4734

. 6266

18

43

. 0431

. 9569

.2861

. 1946

.1303

.0498

. 4743

.5257

17

44

. 0459

. 9541

.2831

. 1978

.1271

.0499

. 4751

. 6248

16

45

.30486

.69513

3.2801

.32010

3.1240

1.0500

.04760

.95239

15

46

. 0514

. 9486

.2772

. 2042

.1209

.0501

. 4769

. 5231

14

47

. 0542

. 9458

.2742

. 2074

.1177

.0502

. 4778

. 5222

13

48

. 0569

. 9430

.2712

. 2106

.1146

.0503

. 4787

. 5213

12

49

. 0597

. 9403

.2683

. 2138

.1115

.0604

. 4796

. 5204

11

50

.30625

.69375

3.2653

.32171

3.1084

1.0505

.04805

.95195

10

51

. 0653

. 9347

.2624

. 2203

.1053

.0506

. 4814

. 5186

9

52

. 0680

. 9320

.2594

. 2235

.1022

.0507

. 4823

. 5177

8

53

. 0708

. 9292

.2565

. 2267

.0991

.0508

. 4832

. 5168

7

54

. 0736

. 9264

.2535

. 2299

.0960

.0509

. 4840

. 6169

6

55

.30763

.69237

3.2506

.32331

3.0930

1.0510

.04849

.95150

5

56

. 0791

. 9209

.2477

. 2363

.0899

.0511

. 4868

. 5141

4

57

. 0819

. 9181

.2448

. 2395

.0868

.0612

. 4867

. 5132

3

58

. 0846

. 9154

.2419

. 2428

.0838

.0513

. 4876

. 5124

2

59

. 0874

. 9126

.2390

. 2460

.0807

.0514

. 4885

. 6115

1

60

. 0902

. 9098

.2361

. 2492

.0777

.0515

. 4894

. 5106

M.

Cosine.

Vrs. sin.

Secant.

Cotang.

Tang.

Cosec'nt

Vrs. cos.

Sine.

iT

107°

72°

340

NATURAL FUNCTIONS.

Table 3.

18

o

Natural Trigonometrical

Functions.

161°

M.

Sine.

Vrs. COB.

Cosec'nt

Tanpr.

Cotang.

Secant.

Vrs. sin

Cosine.

M.

.30902

.69098

3.2361

.32492

3.0777

1.0515

.04894

.96106

60

1

. 0929

. 9071

.2332

. 2524

.0746

.0516

. 4903

. 5097

59

2

. 0957

. 9043

.2303

. 2656

.0716

.0517

. 4912"

. 5088

58

3

. 0985

. 9015

.2274

. 2588

.0686

.0518

. 4921

. 5079

bl

4

. 1012

. 8988

.2245

. 2621

.0655

.0519

. 4930

. 5070

56

5

.31040

.68960

3.2216

.32653

3.0625

1.0520

.04939

.95061

55

6

. 1068

. 8932

.2188

. 2685

.0595

.0521

. 4948

. 5051

54

7

. 1095

. 8905

.2159

. 2717

.0565

.0622

. 4957

. 5042

53

8

. 1123

. 8877

.2131

. 2749

.0535

.0523

. 4966

. 5033

52

9

. 1160

. 8849

.2102

. 2782

.0505

.0524

. 4975

. 5024

51

10

.31178

.68822

3.2074

.32814

3.0475

1.0525

.04985

.95015

60

11

. 1206

. 8794

.2045

. 2846

.0445

.0626

. 4994

. 5006

49

12

. 1233

. 8766

.2017

. 2878

.0415

.0.527

. 5003

. 4997

48

13

. 1261

. 8739

.1989

. 2910

.0385

.0528

. 5012

. 4988

47

14

. 1289

. 8711

.1960

. 2943

.0356

.0529

. 5021

. 4979

46

15

.31316

.68684

3.1932

.32975

3.0326

1.0530

.05030

.94970

45

16

. 1344

. 8656

.1904

. 3007

.0296

.0531

. 5039

. 4961

44

17

. 1372

. 8628

.1876

. 3039

.0267

.0532

. 5048

. 4952

43

18

. 1399

. 8601

.1848

. 3072

.0237

.0533

. 5057

. 4942

42

19

. 1427

. 8573

.1820

. 3104

.0208

.0534

. 6066

. 4933

41

20

.31151

.68645

3.1792

.33136

3.0178

1.0535

.06076

.94924

40

21

. 1482

. 8518

.1764

. 3169

.0149

.0536

. 5085

. 4915

39

22

. 1510

. 8190

.1736

. 3201

.0120

.0537

. 5094

. 4906

38

23

. 1537

. 8163

.1708

. 3233

.0090

.0538

. 5103

. 4897

.37

24

. 1565

. 8435

.1681

. 3265

.0061

.0539

. 5112

. 4888

36

25

.31592

.68407

3.1653

.33298

3.0032

1.0540

.06121

.94878

35

26

. 1620

. 8380

.1625

. 3330

.0003

.0641

. 5131

. 4869

34

27

. 1648

. 8352

.1598

. 8362

2.9974

.0642

. 5140

. 4860

33

28

. 1675

. 8325

.1570

. 3395

.9945

.0543

. 6149

. 4851

32

29

. 1703

. 8297

.1543

. 3427

.9916

.0544

. 6158

. 4841

31

30

.31730

.68269

3.1615

.33459

2.9887

1.0645

.05168

.94832

30

31

. 17S8

. 8242

.1488

. 3492

.9858

.0546

. 5177

. 4823

29

32

. 1786

. 8214

.1461

. 3524

.9829

.0547

. 5186

. 4814

28

S3

. 1813

. 8187

.1433

. 3557

.9800

.0648

. 5195

. 4805

27

34

. 1841

. 8159

.1106

. 3589

.9772

.0549

. 5205

. 4795

26

35

.31868

-68132

3.1379

.33621

2.9743

1.0560

.05214

.94786

25

36

. 1896

. 8104

.1352

. 3654

.9714

.0561

. 5223

. 4777

24

37

. 1923

. 8076

.1325

. 3686

.9686

.0552

. 5232

. 4767

23

38

. 1951

. 8049

.1298

. 3718

.9657

.0553

. 5242

. 4758

22

39

. 1978

. 8U21

.1271

. 37.61

.9629

.0654

. 5251

. 4749

21

40

.32006

.67994

3.1244

.33783

2.9600

1.0555

.05260

.94740

20

41

. 2034

. 7966

.1217

. 3816

.9672

.0556

. 5270

. 4730

19

42

. 2061

. 7939

.1190

. 3848

.9544

.0557

. 5279

. 4721

18

43

. 2089

. 7911

.1163

. 3880

.9515

.0658

. 5288

. 4712

17

44

. 2116

. 7884

.1137

. 3913

.9487

.0559

. 5297

. 4702

16

45

.32144

.67866

3.1110

.33945

2.9459

1.0660

.05307

.94693

15

46

. 2171

. 7828

.1083

. 3978

.9431

.0561

. 5316

. 4684

14

47

. 2199

. 7801

.1057

. 4010

.9403

.0562

. 5326

. 4674

13

48

. 2226

. 7773

.1030

. 4043

.9375

.0563

. 5335

. 4665

12

49

. 2254

. 7746

.1004

. 4075

.9347

.0565

. 5344

. 4655

11

60

.32282

.07718

3.0977

.34108

2.9319

1.0566

.05354

.94646

10

51

. 2309

. 7691

.0951

. 4140

.9291

.0.567

. 6363

. 4637

9

52

. 2337

. 7663

.0925

. 4173

.9263

.0568

. 5373

. 4627

8

53

. 2364

. 7636

.0898

. 4205

.9235

.0569

. 5382

. 4618

7

64

. 2392

. 7008

.0872

. 4238

.9208

.0570

. 5391

. 4608

Q

55

.32419

.67581

3.0846

.34270

2.9180

1.0571

.05401

.94599

5

56

. 2447

. 7653

.0820

. 4303

.9162

.0672

. 5410

. 4590

4

57

. 2474

. 7526

.0793

. 4335

.9326

.0573

. 5420

. 4580

3

58

. 2502

. 7498

.0767

. 4368

.9097

.0574

. 5429

. 4571

69

2529

7471

.0741

. 4400

.9069

.0575

. 5439

4561 i

60

. 2557

. 7443

.0715

. 4433

.9042

.0576

,6448

.' 4552

M.

Cosine.

Vrs. sin.

Secant.

Colang.

Tang.

;Josec'nt

Vrs. cos.

Sine. Im.

108

o

7

1°

Table 3.

NATURAL FUNCTIONS.

341

19°

Natural Trigonometrical Functions.

160°

M.

Sine.

Vra. COS.

Cosec'nt

Tang.

Cotang.

Secant.

Vrs. sin.

Cosine.

M.

.32557

.67443

3.0715

.34433

2.9042

1.0576

.06448

.94552

60

1

. 2584

. 7416

.0690

. 4465

.9015

.0577

. 6458

. 4542

59

2

. 2612

. 7388

.0664

. 4498

.8987

.0578

. 6467

. 4533

68

3

. 2639

.7361

.0638

. 4530

.8960

.0579

. 5476

. 4523

57

4

. 2667

. 7383

.0612

. 4563

.8933

.0580

. 5486

. 4614

56

5

.32694

.67306

3.0586

.34595

2.8905

1.0581

.05495

.94604

55

6

. 2722

. 7278

.0561

. 4628

.8878

.0582

. 5505

. 4495

54

7

. 2749

. 7251

.0535

. 4661

.8851

.0584

. 5515

. 4485

63

8

. 2777

.7223

.0509

. 4693

.8824

.0585

. 5524

. 4476

52

9

. 2804

. 7196

.0484

. 4726

.8797

.0586

. 5534

. 4466

51

10

.32832

.67168

3.0458

.34758

2.8770

1.0587

.05643

.94467

60

11

. 2859

. 7141

.0433

. 4791

.8743

.0588

. 5553

. 4447

49

12

. 2887

. 7113

.0407

. 4824

.8716

.0589

. 5562

. 4438

48

13

. 2914

. 7086

.0382

. 4856

.8689

.0590

. 5572

. 4-128

47

14

. 2942

. 7058

.0357

. 4889

.8662

.0591

. 6581

. 4418

46

15

.32969

.67031

3.0331

.34921

2.8636

1.0392

.05591

.94409

45

16

. 2996

. 7003

.0306

. 4954

.8609

.0693

. 5601

. 4399

44

17

. 3024

. 6976

.0281

. 4987

.8582

.0594

. 6610

. 4390

43

18

. 3051

. 6948

.0256

. 5019

.8555

.0695

. 5620

. 4380

42

19

. 3079

. 6921

.0281

. 5052

.8529

.0596

. 5629

. 4370

41

20

.33106

.66894

3.0206

.35085

2.8502

1.0598

.06639

.94361

40

21

. 3134

. 6866

.0181

. 5117

.8476

.0599

. 6649

. 4351

39

22

. 3161

. 6839

.0156

. 5150

.8449

.0600

. 6658

. 4341

38

23

. 3189

. 6811

.0131

. 5183

.8423

.0601

. 5668

. 4332

37

24

. 3216

. 6784

.0106

. 5215

.8396

.0602

. 6678

. 4322

36

25

.33243

.66756

3.0081

.35248

2.8370

1.0603

.05687

.94313

35

26

. 3271

. 6729

.0066

. 5281

.8344

.0604

. 6697

. 4303

34

27

. 3298

. 6701

.0081

. 6314

.8318

.0605

. 6707

. 4293

33

28

. 3326

. 6674

.0007

. 5346

.8291

.0606

. .5716

. 4283

32

29

. 3353

. 6647

2.9982

. 5379

.8265

.0607

. 5726

. 4274

31

30

.33381

.66619

2.9957

.35412

2.8239

1.0608

.05736

.94264

30

31

. 3408

. 6592

.9933

. 5445

.8213

.0609

. 6745

. 4254

29

32

. 3435

. 6564

.9908

. 5477

.8187

.0611

. 5766

. 4245

28

33

. 3463

. 6537

.9884

. 5510

.8161

.0612

. 5765

. 4285

27

34

. 3490

. 6510

.9859

. 5543

.8136

.0613

. 5776

. 4225

26

35

.33518

.66482

2.9835

.35576

2.8109

1.0614

.05784

.94215

25

36

. 3545

. 6455

.9810

. 6608

.8083

.0615

. 6794

. 4206

24

37

. 3572

. 6427

.9786

. 5641

.8057

.0616

. 6804

. 4196

23

38

. 3600

. 6400

.9762

. 5674

.8032

.0617

. 5814

. 4188

22

39

. 3627

. 6373

.9738

. 5707

.8006

.0618

. 5823

. 4178

21

40

.33655

.66345

2.9713

.35739

2.7980

1.0619

.05833

.94167

20

41

. 3682

. 6318

.9689

. 5772

.7964

.0620

. 5843

. 4157

19

42

. 3709

. 6290

.9665

. 5805

.7929

.0622

. 6853

. 4147

18

43

. 3737

. 6263

.9641

. 6838

.7903

.0623

. 5863

. 4137

17

44

. 3764

. 6236

.9617

. 5871

.7878

.0624

. 6872

. 4127

16

45

.33792

.66208

2.9593

.35904

2.7852

1.0626

.06882

.94118

15

46

. 3819

. 6181

.9569

. 5936

.7827

.0626

. 5892

. 4108

14

47

. 3846

. 6153

.9545

. 5969

.7801

.0627

. 5902

. 4098

13

48

. 3874

. 6126

.9521

. 6002

.7776

.0628

. 5912

. 4088

12

49

. 3901

. 6099

.9497

. 6035

.7751

.0629

. 6922

. 4078

11

50

.33923

.66071

2.9474

.36068

2.7726

1.0630

.06932

.94068

10

51

. 3956

. 6044

.9450

. 6101

.7700

.0632

. 6941

. 4058

9

52

. 3983

. 6017

.9426

. 6134

.7675

.0633

. 5951

. 4049

8

53

. 4011

. 5989

.9402

. 6167

.7650

.0634

. 5961

. 4039

7

54

. 4038

. 5962

.9379

. 6199

.7625

.0636

. 6971

. 4029

6

55

.34065

.65935

2.9355

.36232

2.7600

1.0636

.05981

.94019

5

56

. 4093

. 5907

.9332

. 6265

.7574

.0637

. 5991

. 4009

4

57

. 4120

. 5880

.9308

. 6298

.7549

.0638

. 6001

. 3999

3

58

. 4147

. 5853

.9285

. 6331

.7524

. .0639

. 6011

. 3989

2

59

. 4175

. 5825

.9261

. 6364

.7500

.0641

. 6021

. 3979

1

60

. 4202

. 5798

.9238

. 6397

.7475

.0642

. 6031

. 3969

mT

Cosine.

Vrs. sin.

Secant.

Ootang.

Tang.

Cosec'nt

Vrs. cos.

Sine.

M^

109°

70°

342

NATURAL FUNCTIONS.

Table 3.

20

3

Natural Trigonom

etrical Functions.

159°

M.

Sine.

Vrs. COS.

Cosec'nt

Tang.

Cofcing.

Secant.

Vrs. gin

Cosine.

M.

.34202

.65798

2.9238

.36397

2.7475

1.0642

.06031

.93969

60

1

. 4229

. 5771

.9215

. 6430

.7450

.0643

. 6041

. 3959

59

2

. 4257

. 5743

.9191

. 6463

.7426

.0644

. 6051

. 3949

58

3

. 4284

. 5716

.9168

. 6196

.7400

.0645

. 6061

. 3939

57

4

. 4311

. 5689

.9115

. 6529

.7376

.0646

. 6071

. 3929

56

5

.34339

.65661

2.9122

.36562

2.7351

1.0647

.06080

.93919

55

6

. 4366

. 5634

.9098

. 6595

.7326

.0643

. 6090

. 3909

54

7

. 4393

. 5607

.9075

. 6628

.7302

.0650

. 6100

. 3899

53

8

. 4421

. 5579

.9052

. 6661

.7277

.0651

. 6110

. 3889

52

9

. 4448

. 5552

.9029

. 6694

.7262

.0652

. 6121

. 3879

61

10

.34475

.65525

2.9006

.36727

2.7228

1.0653

.06131

.93869

50

11

. 4502

. 5497

.8983

. 6700

.7204

.0654

. 6141

. 3859

49

12

. 4530

. 5470

.8960

. 6793

.7179

.0655

. 6151

. 3849

48

13

. 4557

. 5443

.8937

. 6826

.7155

.0656

. 6161

. 3839

47

14

. 4584

. 5415

.8915

. 6859

.7130

.0658

. 6171

. 3829

46

15

.34612

.65388

2,8892

.36892

2.7106

1.0659

.06181

.93819

45

16

. 4639

. 5361

.8869

. 6925

.7082

.0660

. 6191

. 3809

44

17

. 4666

. 5334

.8846

. 6958

.7058

.0661

. 6201

. 3799

43

18

. 4693

. 5306

.8824

. 6991

.7033

.0662

. 6211

. 3789

42

19

. 4721

. 5279

.8801

. 7024

.7009

.0663

. 6221

. 3779

41

20

.34748

.05252

2.8778

.37057

2.6985

1.0664

.06231

.93769

40

21

. 4775

. 5226

.8756

. 7090

.6961

.0666

. 6241

. 3758

39

22

. 4803

. 5197

.8733

. 7123

.6937

.0667

. 6251

. 3748

38

23

. 4830

. 5170

.8711

. 7156

.6913

.0668

. 6262

. 3738

37

24

. 48.57

. 5143

.8688

7190

.6889

.0669

. 6272

. 3728

36

25

.34884

.65115

2.8666

.37223

2.6865

1.0670

.06282

.93718

35

26

. 4912

. 5088

.8644

. 7256

.6841

.0671

. 6292

. 3708

34

27

. 4939

. 5061

.8621

. 72X9

.6817

.0673

. 6302

. 3698

33

28

. 4966

. 5034

.8599

. 7322

.6794

.0674

. 6312

. 3687

32

29

. 4993

. 5006

.8577

. 7356

.6770

.0675

. 6323

. 3677

31

30

.35021

.64979

2.8554

.37388

2.6746

1.0676

.06333

.93667

30

31

. 5048

. 4952

.8532

. 7422

.6722

.0077

. 6343

. 3657

20

82

. 5075

. 4926

.8510

. 7155

.6699

.0678

. 6353

. 3647

28

33

. 5102

. 4897

.8488

. 7488

.6675

.0679

. 6363

. 3637

27

34

. 5130

. 4870

.8466

. 7521

.6652

.0681

. 6373

. 3626

26

35

.35157

.6-1843

2.8444

.37554

2.6628

1.0682

.06384

.93616

25

36

. 5184

. 4816

.8422

7587

.6604

.0683

. 6394

. 3606

24

37

. 5211

. 4789

.8400

7G21

.6581

.0681

. 6404

. 3596

23

38

. 5239

. 4761

.8378

. 7654

.6558

.0686

. 6414

. 3585

22

39

. 5266

. 4734

.8356

7687

.6534

.0686

. 6425

. 3575

21

40

.35293

.64707

2.8334

.37720

2.6511

1.0688

.06435

.93565

20

41

. 5320

. 4680

.8312

77M

.6487

.0689

. 6445

. 3555

19

42

. 5347

. 4652

.8290

. 7787

.6464

.0690

. 6456

. 3544

18

43

. 5375

. 4625

.8269

. 7820

.6441

.0691

. 6466

. 3534

17

44

. 5402

. 4598

.8247

. 7853

.6418

.0692

. 6476

. 3524

16

45

.35429

.64571

2.8225

.37887

2.6394

1.0694

.06486

.93513

15

46

. 5456

. 4544

.8204

. 7920

.6371

.0695

. 6497

. 3503

14

47

. 5483

. 4516

.8182

. 7953

.6348

.0696

. 6507

. 3493

13

48

. 5511

. 4489

.8160

. 7986

.6325

.0697

. 6517

. 3482

12

49

. 5538

. 4462

.8139

. 8020

.6302

.0698

. 6528

. 3472

11

50

.35665

.64435

2.8117

.38053

2.6279

1.0699

.06638

.93462

10

51

. 5592

. 4408

.8096

. 8086

.6266

.0701

. 6548

. 3451

9

52

. 5619

. 4380

.8074

. 8120

.6233

.0702

. 6559

. 3441

8

53

. 5647

. 4353

.8053

. 8153

.6210

.0703

. 6569

. 3431

7

54

. 5674

. 4326

.8032

. 8186

.6187

.0704

. 6579

. 3420

6

55

.35701

.64299

2.8010

.38220

2.6164

1.0705

.06590 ■

.93410

5

56

. 5728

. 4272

.7989

. 8263

.6142

.0707

. 6600

. 3400

4

57

. 5755

. 4245

.7968

. 8286

.6119

.0708

. 6611

. 3389

3

58

. 5782

4217

.7947

. 8320

.6096

.0709

. 6621

. 3379

2

59

. 5810

. 4190

.7925

. 8353

.6073

.0710

. 6631

. 3368

1

60

. 5837

. 4163

.7904

. 8386

.6051

.0711

. 6642

. 3358

M.

Cosine.

Vrs. sin.

Secant.

Cotang.

Tang.

('osec'nt

Vrs. COS.

Sine.

M.

110

o

X

no

Table 3.

NATURAL FUNCTIONS.

343

21°

Natural Trigonometrical Functions.

158°

mT

Sine.

Yre. C06.

Cosec'nt

Tang.

Cotang.

Secant.

Vrs. bin.

Cosine.

M.

.35837

.64163

2.7904

.38386

2.6051

1.0711

.06642

.93368

60

1

. 5864

. 4136

.7883

. 8420

.6028

.0713

. 6652

. 3348

59

2

. 5891

. 4109

.7862

. 8453

.6006

.0714

. 6663

. 3337

58

3

. 5918

. 4082

.7841

. 8486

.5983

.0715

. 6673

. 3327

57

4

. 5945

. 4055

.7820

. 8520

.5960

.0716

. 6684

. 3316

66

5

.35972

.64027

2.7799

.38553

2.5938

1.0717

.06694

.93306

55

6

. 6000

. 4000

.7778

. 8587

.5916

.0719

. 6705

. 3295

54

7

. 6027

. 3973

.7757

. 8620

.6893

.0720

. 6715

. 3285

53

8

. 6054

. 3946

.7736

. 8654

.5871

.0721

. 6726

. 3274

52

9

.6081

. 3919

.7715

. 8687

.5848

.0722

. 6736

. 3264

51

10

.36108

.63892

2,7694

.38720

2.6826

1.0723

.06747

.93253

50

11

. 6135

. 3865

.7674

. 8754

.5804

.0725

. 6757

. 3243

49

12

. 6162

. 3837

.7653

. 8787

.5781

.0726

. 6768

. 3232

48

13

. 6189

. 3810

.7632

. 8821

.5759

.0727

. 6778

. 3222

47

14

. 6217

. 3783

.7611

. 8854

.5737

.0728

. 6789

. 3211

46

15

.36244

.63756

2.7591

.38888

2.5715

1.0729

.06799

.93201

45

16

. 6271

. 3729

.7570

. 8921

.5693

.0731

. 6810

. 3190

44

17

. 6298

. 3702

.7550

. 8955

.6671

.0732

. 6820

. 3180

43

18

. 6325

. 3675

.7629

. 8988

.5640

.0733

. 6831

. 3169

42

19

. 6352

. 3648

.7609

. 9022

.5627

.0734

. 6841

. 3158

41

20

.36379

.63621

2.7488

.39055

2.5605

1.0736

.06862

.93148

40

21

. S406

. 3593

.7468

. 9089

.5583

.0737

. 6863

. 3137

30

22

. 6433

. 3566

.7447

. 9122

.5661

.0738

. 6873

. 3127

30

23

. 6460

. 3539

.7427

. 9156

.5639

.0739

. 6884

. 3116

37

24

. 6488

. 3512

.7406

. 9189

.5517

.0740

. 6894

. 3105

36

26

.36515

.63485

2.7386

.39223

2.6496

1.0742

.06905

.93095

36

26

. 6542

. 3458

.7366

. 9267

.6473

.0743

. 6916

. 3084

34

27

. 6569

. 3431

.7346

. 9290

.6451

.0744

. 6926

. 3074

33

28

. 6596

. 3404

.7325

. 9324

.5430

.0746

. 6937

. 3063

32

29

. 6623

. 3377

.7305

. 9357

.5408

.0747

. 6947

. 3052

31

30

.36660

.63360

2.7285

.39391

2.5386

1.0748

.06958

.93042

30

31

. 6677

. 3323

.7265

. 9425

.6366

.0749

. 6969

. 3031

29

32

. 6704

. 3296

.7216

. 9468

.6343

.0750

. 6979

. 3020

28

33

. 6731

. 3269

.7226

. 9492

.5322

.0751

. 6990

. 3010

27

34

. 6758

. 3242

.7205

. 9525

.5300

.0753

. 7001

. 2999

26

35

.36785

.63214

2.7185

.39559

2.5278

1.0754

.07012

.92988

25

36

. 6812

. 3187

.7165

. 9593

.5257

.0756

. 7022

. 2978

24

37

. 6839

. 3160

.7145

. 9626

.5236

.0756

. 7033

. 2967

23

38

. 6866

. 3133

.7126

. 9660

.5214

.0758

. 7044

. 2956

22

39

. 6893

. 3106

.7106

. 9694

.5193

.0769

. 7054

. 2946

21

40

.36921

.63079

2.7085

.39727

2.6171

1.0760

.07065

.92935

20

41

. 6948

. 3052

.7065

. 9761

.6150

.0761

. 7076

. 2924

19

42

. 6975

. 3025

.7046

. 9796

.5129

.0763

. 7087

. 2913

18

43

. 7002

. 2998

.7026

. 9828

.5108

.0764

. 7097

. 2902

17

44

. 7029

. 2971

.7006

. 9862

.5086

.0765

. 7108

. 2892

16

45

.37056

.62944

2.6986

.39896

2.5066

1.0766

.07119

.92881

15

46

. 7083

. 2917

.6967

. 9930

.5044

.0768

. 7130

. 2870

14

47

. 7110

. 2890

.6947

. 9963

.5023

.0769

. 7141

. 2859

13

48

. 7137

. 2863

.6927

. 9997

.5002

.0770

. 7151

. 2848

12

49

. 7164

. 2836

.6908

.40031

.4981

.0771

. 7162

. 2838

11

50

.37191

.62809

2.6888

.40065

2.4960

1.0773

.07173

.92827

10

51

. 7218

. 2782

.6869

. 0098

.4939

.0774

. 7184

. 2816

9

52

. 7215

. 2755

.6849

. 0132

.4918

.0775

. 7195

. 2805

8

53

.7272

. 2728

.6830

. 0166

.4897

.0776

. 7205

. 2794

7

54

. 7299

. 2701

.6810

. 0200

.4876

.0778

. 7216

. 2784

6

55

.37326

.62674

2.6791

.40233

2.4855

1.0779

.07227

.92773

5

56

. 7353

. 2647

.6772

. 0267

.4834

.0780

. 7238

. 2762

4

57

. 738C

. 2620

.6762

. 0301

.4813

.0781

. 7249

. 2751

3

68

. 7407

. 2593

.6733

. 0336

.4792

.0783

. 7260

. 2740

2

59

. 7434

. 2566

.6714

. 0369

.4772

.0784

. 7271

. 2729

1

CO

. 7461

. 2539

.6695

. 0403

.4761

.0785

. 7282

. 2718

M.

Cosine.

Vrs. Bin.

Secant.

Cotang.

Tang.

Cosec'nt

Vrs. COS.

Sine.

M.

11°

68°

344

NATUEAL FUNCTIONS.

Table 3.

22°

Natural Trigonometrical Punctions.

1

57°

M.

Sine.

Vra. coa.

Cosec'nt

Tang.

Co tang.

Secant.

Vrs. sin

Cosine.

il.

.G7461

.62639

2.6695

.40103

2.4761

1.0785

.07282

.92718

60

1

. 7488

. 2612

.6675

. 0136

.4730

.0787

. 7292

. 2707

69

2

. 7614

, 2485

.6656

. 0470

.4709

.0788

. 7303

. 2696

58

3

. 7641

. 2458

.6637

. 0504

.4689

.0789

. 7314

. 2686

57

4

. 7568

. 2431

.6618

. 0538

.4668

.0790

. 7325

. 2675

56

5

.37595

.62404

2.6599

.40572

2.4647

1.0792

.07336

.92664

55

6

. 7622

. 2377

.6680

. 0606

.4627

.0793

. 7347

. 2653

54

7

. 7649

. 2351

.6561

. 0640

.4606

.0794

. 7358

. 2642

63

8

. 7676

. 2324

.6542

. 0673

.4586

.0795

. 7369

. 2631

62

9

. 7703

. 2297

.6523

. 0707

.4566

.0797

. 7380

. 2620

51

10

.37730

.62270

2.6504

.40741

2.4515

1.0798

.07391

.92609

50

11

. 7757

. 2243

.6485

. 0775

.4525

.0799

. 7402

. 2598

49

12

. 7784

. 2216

.6466

. 0809

.4504

.0801

. 7413

. 2587

48

13

. 7811

. 2189

.6447

. 0843

.4484

.0802

. 7424

. 2676

47

14

. 7838

. 2162

.6428

. 0877

.4463

.0803

. 7436

. 2565

46

15

.37865

.62135

2.6410

.40911

2.4443

1.0804

.07446

.92554

45

16

. 7892

. 2108

.6391

. 0945

.4423

.0806

. 7457

. 2543

44

17

. 7919

. 2081

.6372

. 0979

.4403

.0807

. 7468

. 2532

43

18

. 7946

. 2054

.6353

. 1013

.4382

.0808

. 7479

. 2521

42

19

. 7972

. 2027

.6335

. 1047

.4362

.0810

. 7490

. 2610

41

20

.37999

.62000

2.6316

.41081

2.4342

1.0811

.07501

.92499

40

21

. 8026

. 1974

.6297

. 1116

.4322

.0812

. 7512

. 2488

39

22

. 8063

. 1947

.6279

. 1119

.4302

.0813

. 7523

. 2477

38

23

. 8080

. 1920

.6260

. 1183

.4282

.0815

. 7534

. 2466

37

24

. 8107

. 1893

.6242

. 1217

.4262

.0816

. 7546

. 2455

36

25

.38134

.61866

2.6223

.41251

2.4242

1.0817

.07556

.92443

35

26

. 8161

. 1839

.6206

. 1285

.4222

.0819

. 7567

. 2432

34

27

. 8188

. 1812

.6186

. 1319

.4202

.0820

. 7679

. 2421

33

28

. 8214

. 1786

.6168

. 1353

.4182

.0821

. 7690

. 2410

32

29

. 8241

. 1758

.6150

. 1387

.4162

.0823

. 7601

. 2399

31

30

.38268

.61732

2.6131

.41421

2.4142

1.0824

.07612

.92388

30

31

. 8295

. 1705

.6113

. 1465

.4122

.0825

. 7623

. 2377

29

32

. 8322

. 1678

.6095

. 1489

.4102

.0826

. 7634

. 2366

28

33

. 8349

. 1651

.6076

. 1524

.4083

.08'28

. 7645

. 2354

27

34

. 8376

. 1624

.6058

. 1558

.4063

.0829

7667

. 2343

26

35

.38403

.61597

2.6040

.41592

2.4043

1.0830

.07668

.92332

25

36

. 8429

. 1570

.6022

. 1626

.4023

.0832

. 7679

. 2321

24

37

. 8456

. 1514

.6003

. 1660

.4004

.0833

. 7690

. 2310

23

38

. 8483

. 1617

.6985

. 1694

.3984

.0834

. 7701

. 2299

22

39

. 8510

. 1490

.5967

. 1728

.3964

.0836

. 7712

. 2287

21

40

.38537

.61463

2.6919

.41762

2.3945

1.0837

.07724

.92276

20

41

. 8564

. 1436

.6931

. 1797

-.3925

.0838

. 7735

. 2265

19

42

. 8591

. 1409

.6913

. 1831

.3906

.0840

. 7746

. 2254

18

43

. 8617

. 1382

.6895

. 1865

.3886

.0841

. 7757

. 2242

17

44

. 8644

. 1366

.6877

. 1899

.3867

.0842

. 7769

. 2231

16

45

.38671

.61329

2.6859

.41933

2.3847

1.0814

.07780

.92220

15

46

. 8698

. 1302

.5841

. 1968

.3828

.0846

7791

. 2209

14

47

. 8725

. 1275

.5823

. 2002

.3808

.0816

. 7802

. 2197

13

48

. 8751

. 1248

.5805

. 2036

.3789

.0817

. 7814

. 2186

12

49

. 8778

. 1222

.5787

. 2070

.3770

.0849

. 7826

. 2175

11

50

.38805

.61195

2.5770

.42105

2.3760

1.0850

.07836

.92164

10

61

. 8832

. 1168

.5762

. 2139

.3731

.0861

. 7847

. 2152

9

52

. 8869

. 1141

.6734

. 2173

.3712

.0863

. 7869

. 2141

8

53

. 8886

. 1114

.6716

. 2207

.3692

.0854

. 7870

. 2130

7

54

. 8912

. 1088

.5699

. 2242

.3673

.0855

. 7881

. 2118

6

55

.38939

.61061

2.5681

.42276

2.3654

1.0857

.07893

.92107

5

66

. 8966

. 1034

.5663

. 2310

.3636

.0858

. 7904

. 2096

4

57

. 8993

. 1007

.5646

. 2344

.3616

.0859

. 7915

. 2084

3

68

. 9019

. 0980

.6628

. 2.379

.3597

.0861

. 7927

. 2073

2

59

. 9046

. 0954

.6610

. 2413

.8677

.0862

. 7938

. 2062

1

60

. 9073

. 0927

.6593

. 2447

.3558

.0864

. 7949

. 2050

M.

Cosine.

Vrs. sin.

Secant.

Co tang.

Tang.

Cosec'nt

Vrs. COS.

Sine.

M.

Table 3.

NATURAL FUNCTIONS.

345

23°

Natural Trigonometrical Functions.

156°

M.

Sine.

Vr8. COS.

Oosec'nt

Tang.

Cotang.

Secant.

Vrs. sin.

Cosine.

M.

.39073

.60927

2.5593

.42447

2.3658

1.0864

.07949

.92050

60

1

. 9100

. 0900

.5575

. 2482

.3539

.0865

. 7961

. 2039

59

2

. 9126

. 0873

.5558

. 2616

.3520

.0866

. 7972

. 2028

58

3

. 9153

. 0846

.5540

. 2550

.3501

.0868

. 7984

. 2016

67

4

. 9180

. 0820

.5523

. 2581

.3482

.0869

. 7995

. 2005

56

5

.39207

.60793

2.5506

.42619

2.3463

1.0870

.08006

.91993

65

6

. 9234

. 0766

.5488

. 2654

.3445

.0872

. 8018

.1982

54

7

. 9260

. 0739

.5471

. 2688

.3426

.0873

. 8029

. 1971

53

8

. 9287

. 0713

.5453

. 2722

.3407

.0874

. 8041

. 1959

52

9

. 9314

. 0686

.5436

. 2757

.3388

.0876

. 8052

. 1948

51

10

.39341

.60659

2.5419

.42791

2.3369

1.0877

.08063

.91936

50

11

. 9367

. 0632

.5402

. 2826

.3350

.0878

. 8075

. 1925

49

12

. 9394

. 0606

.5384

. 2860

.3332

.0880

. 8086

. 1913

48

13

. 9421

. 0579

.5367

. 2894

.3313

.0881

. 8098

. 1902

47

14

. 9448

. 0552

.5350

. 2929

.3294

.0882

. 8109

. 1891

48

15

.39474

.60526

2.5333

.42963

2.3276

1.0884

.08121

.91879

45

ii;

. 9501

. 0499

.5316

. 2998

■• .3257

.0885

. 8132

. 1868

44

17

. 9528

. 0472

.5299

. 3032

.3238

.0886

. 8144

. 1856

43

18

. 9554

. 0445

.5281

. 3067

.3220

.0888

. 8155

. 1845

42

19

. 9581

. 0419

.5264

. 3101

.3201

.0889

. 8167

. 1833

41

20

.39608

.60392

2.5247

.43136

2.3183

1.0891

.08178

.91822

40

21

. 9635

. 0365

.5230

. 3170

.3164

.0892

. 8190

. 1810

39

22

. 9661

. 0339

.5213

. 3205

.3145

.0893

. 8201

. 1798

38

23

. 9688

. 0312

.5196

. 3239

.3127

.0895

. 8213

. 1787

37

24

. 9715

. 0285

.5179

. 3274

.3109

.0896

. 8224

. 1775

36

25

.39741

.60258

2.5163

.43308

2.3090

1.0897

.08236

.91764

35

26

. 9768

. 0232

.5146

. 3343

.3072

.0899

. 8248

. 1752

34

27

. 9795

. 0205

.5129

. 3377

.3053

.0900

. 8259

. 1741

33

28

. 9821

. 0178

.5112

. 3412

.3035

.0902

.8271

. 1729

32

29

. 9848

. 0152

.5095

. 3447

.3017

.0903

. 8282

. 1718

31

30

.39875

.60125

2.5078

.43481

2.2998

1.0904

.08294

.91706

80

31

. 9901

. 0098

.5062

. 3516

.2980

.0906

. 8306

. 1694

29

32

. 9928

. 0072

.5045

. 3550

.2962

.0907

. 8317

. 1683

28

33

. 99f)5

. 0045

.5028

. 3585

.2944

.0908

. 8329

. 1671

27

34

. 9981

. 0018

.6011

. 3620

.29-26

.0910

. 8340

. 1659

26

sr-,

.40008

.59992

2.4995

.43654

2.2907

1.0911

.08352

.91648

25

36

. 0035

. 9965

.4978

. 3689

.2889

.0913

. 8364

. 1636

24

37

. 0061

. 9938

.4961

. 3723

.2871

.0914

. 8375

. 1625

23

38

. 0088

. 9912

.4945

. 3758

.2853

.0915

. 8387

. 1613

22

39

. 0115

. 9885

.4928

. 3793

.2835

.0917

. 8399

. 1601

21

40

.40141

.59858

2.4912

.43827

2.2817

1.0918

.08410

.91590

20

41

. 0168

. 9832

.4895

. 3862

.2799

.0920

. 8422

. 1578

19

42

. 0195

. 9805

.4879

. 3897

.2781

.0921

. 8434

. 1566

18

43

. 0221

. 9778

.4862

. 3932

.2763

.0922

. 8445

. 1554

17

44

. 0248

. 9752

.4846

. 3966

.2745

.0924

.8457

. 1643

16

45

.40275

.59725

2.4829

.44001

2.2727

1.0925

.08469

.91631

15

4G

. 0301

. 9699

.4813

. 4036

•2709

.0927

. 8480

. 1519

14

47

. 0328

. 9672

.4797

. 4070

.2691

.0928

. 8492

. 1508

13

48

. 0354

. 9645

.4780

. 4105

.2673

.0929

. 8504

. 1496

12

49

. 0381

. 9619

.4764

. 4140

.2655

.0931

. 8516

. 1484

11

50

.40408

.59592

2.4748

.44176

2.2637

1.0932

.08527

.91472

10

51

. 0434

.9566

.4731

. 4209

.2619

.0934

. 8639

. 1461

9

52

. 0461

. 9539

.4715

. 4244

.2602

.0935

.8551

. 1449

8

53

. 0487

. 9512

.4699

. 4279

.2584

.0936

. 8563

. 1437

7

54

. 0514

. 9486

.4683

. 4314

.2566

.0938

. 8575

. 1425

6

55

.40541

.59459

2.4666

.44349

2.2548

1.0939

.08586

.91414

6

66

.0567

. 9433

.4660

. 4383

.2531

.0941

. 8598

. 1402

4

57

. 0594

. 9406

.4634

. 4418

.2513

.0942

. 8610

. 1390

3

58

. 0620

. 9379

.4618

4453

.2495

.0943

. 8622

.1378

2

59

. 0647

. 9353

.4602

. 4488

.2478

.0945

. 8634

. 1366

1

60

. 0674

. 9326

.4586

. 4523

.2460

.0946

. 8646

. 1354

M.

Cosine,

Vrs. sin.

Secant.

Cotang.

Tang.

Cosec'nt

Vrs. COS.

Sine.

M.

346

NATURAL FUNCTIONS.

Table 3.

24°

Natural Trigonometrical Functions.

iSS°

M.

Sine.

Vrs. COS.

Cosec'nt

Tang.

Cotang.

Secant.

Vra. sin.

Cosine.

M.

.40674

.59326

2.4586

.44523

2.2460

1.0946

.08645

.91354

60

1

. 0700

. 9300

.4570

. 4558

.2443

.0948

. 8657

. 1343

59

2

. 0727

. 9273

.4554

. 4593

.2425

.0949

. 8669

. 1331

58

3

. 0753

. 9247

.4538

. 4627

.2408

.0951

. 8681

. 1319

57

4

. 0780

. 9220

.4622

. 4662

.2390

.0952

. 8693

. 1307

56

5

.40806

.59193

2.4506

.44697

2.2373

1.0953

.08705

.91295

55

6

. 0833

. 9167

.4490

. 4732

.2355

.0955

. 8716

. 1283

54

7

. 0860

. 9140

.4474

. 4767

.2338

.0956

. 8728

. 1271

53

8

. 0886

. 9114

.4458

. 4802

.2320

.0958

. 8740

. 1260

62

9

. 0913

. 9087

.4442

. 4837

.2303

.0959

. 8752

. 1248

51

10

.40939

.59061

2.4426

.44872

2.2286

1.0961

.08764

.91236

50

11

. 0966

. 9034

.4411

. 4907

.2268

.0962

. 8776

. 1224

49

12

. 0992

. 9008

.4395

. 4942

.2251

.0963

. 8788

. 1212

48

13

. 1019

. 8981

.4379

. 4977

.2234

.0965

. 8800

. 1200

47

14

. 1045

. 8955

.4363

. 5012

.2216

.0966

. 8812

. 1188

46

15

.41072

.58928

2.4347

.45047

2.2199

1.0968

.08824

.91176

45

16

. 1098

. 8901

.4332

. 5082

.2182

.0969

. 8836

. 1164

44

17

. 1125

. 8875

.4316

. 5117

.2165

.0971

. 8848

. 1152

43

18

. 1151

. 8848

.4300

. 5152

.2147

.0972

. 8860

. 1140

42

19

. 1178

. 8822

.4285

. 5187

.2130

.0973

. 8872

. 1128

41

20

.41204

.58795

2.4269

.45222

2.2113

1.0975

.08884

.91116

40

21

. 1231

. 8769

.4254

. 5257

.2096

.0976

. 8896

. 1104

39

22

. 1257

. 8742

.4238

. 5292

.2079

.0978

. 8908

. 1092

38

23

. 1284

. 8716

.4222

. 5327

.2062

.0979

. 8920

. 1080

37

24

. 1310

. 8689

.4207

. 5362

.2045

.0981

. 8932

. 1068

36

25

.41337

.58663

2.4191

.45397

2.2028

1.0982

.08944

.91056

36

26

. 1363

. 8636

.4176

. 5432

.2011

.0984

. 8956

. 1044

34

27

. 1390

. 8610

.4160

. 5467

.1994

.0985

. 8968

. 1032

33

28

. 1416

. 8584

.4145

. 5502

.1977

.0986

. 8980

. 1020

32

29

. 1443

. 8557

.4130

. 5537

.1960

.0988

. 8992

. 1008

31

30

.41469

.58531

2.4114

.45573

2.1943

1.0989

.09004

.90996

30

31

. 1496

. 8504

.4099

. 5608

.1926

.0991

. 9016

. 0984

29

32

. 1522

. 8478

.4083

. 6643

.1909

.0992

. 9028

. 0972

28

33

. 1549

. 8451

.4068

. 5678

.1892

.0994

. 9040

. 0960

27

34

. 1575

. 8425

.4053

. 5713

.1875

.0995

. 9052

. 0948

26

35

41602

.58398

2.4037

.45748

2.1S59

1.0997

.09064

.90936

25

36

. 1628

. 8372

.4022

. 5783

.1842

.0998

. 9076

. 0924

24

37

. 1654

. 8345

.4007

. 5819

.1825

.1000

. 9088

. 0911

23

38

. 1681

. 8319

.3992

. 5854

.1808

.1001

. 9101

. 0899

22

39

. 1707

. 8292

.3976

. 5889

.1792

.1003

. 9113

. 0887

21

40

.41734

.58266

2.3961

.45924

2.1775

1.1004

.09125

.90875

20

41

. 1760

. 8240

.3946

. 5960

.1758

.1005

. 9137

. 0863

19

42

. 1787

. 8213

.3931

. 5995

.1741

.1007

. 9149

. 0851

18

43

. 1813

. 8187

.3916

. 6030

.1725

.1008

. 9161

. 0839

17

44

. 1839

. 8160

.3901

. 6065

.1708

.1010

. 9173

. 0826

16

45

.41866

.58134

2.3886

.46101

2.1692

1.1011

.09186

.90814

15

46

. 1892

. 8108

.3871

. 6136

.1675

.1013

. 9198

. 0802

14

47

. 1919

. 8081

.3856

. 6171

.1658

.1014

. 9210

. 0790

13

48

. 1945

. 8055

.38'11

. 6205

.1642

.1016

. 9222

. 0778

12

49

. 1972

. 8028

.3826

. 6242

.1625

.1017

. 92.34

. 0765

11

50

.41998

,58002

2.3811

.46277

2.1609

1.1019

.09247

.90753

10

51

. 2024

. 7975

.3796

. 6312

.1592

.1020

. 9259

. 0741

9

52

. 2051

. 7949

.3781

. 6348

.1576

.1022

. 9271

. 0729

8

53

. 2077

. 7923

.3766

. 6383

.1559

.1023

. 9283

. 0717

7

54

. 2103

. 7896

.3751

. 6418

.1543

.1025

. 9296

. 0704

6

55

.42130

.57870

2.3''36

.46454

2.1.527

1.1026

.09308

.90692

5

56

. 2156

. 7844

.3'21

. 6-189

.1510

.1028

. 9320

. 0680

4

57

. 2183

. 7817

.3706

. 6524

.1494

.1029

. 93.32

. 0668

3

58

. 2209

. 7791

.3691

. 6560

.1478

.1031

. 9.S45

. 0655

2

59

. 2235

. 7764

.3677

. 6595

.1461

.1032

. 9357

. 0643

1

60

. 2262

. 7738

.3662

. 6631

.1445

.1034

. 9369

. 0631

M.

CoKine.

Vrs. Bin.

Secant.

Cotang.

Tang.

Coeec'nt

Vrs. COB.

Sine.

M.

114°

65°

Table 3.

NATUKAL FUNCTIONS.

347

25°

Natural Trigonometrical Functions.

J 54°

M.

Sine.

Vrs. COS.

Cosec'nt

Tang.

Cotang.

Secant.

Vrs. sin.

Cosine.

M.

.42262

.57738

2.3662

.46631

2.1445

1.1034

.09369

.90631

60

1

. 2288

. 7712

.3647

. 6666

.1429

.1035

. 9381

.n618

59

2

. 2314

. 7685

.3632

. 6702

.1412

.1037

. 9394

. 0606

58

3

. 2341

. 7659

.3618

. 6737

.1396

.1038

. 9106

. 0594

67

4

.2367

. 7633

.3603

. 6772

.1380

.1040

. 9118

. 0681

56

5

.42394

.57606

2.3588

.46808

2.1364

1.1041

.09131

.90569

55

C

. 2420

. 7580

.3574

. 6843

.1348

.1013

. 9413

. 0557

54

7

. 2446

. 7554

.3559

. 6879

.1331

.1044

. 9165

. 0611

53

8

. 2473

. 7527

.3544

. 6914

.1315

.1046

. 9468

. 0532

52

9

. 2499

. 7501

.3530

. 6950

.1299

.1047

. 9180

. 0520

51

10

.42525

.57475

2.3515

.46985

2.1283

1.1049

.09192

.90507

50

U

. 2552

. 7418

.3601

. 7021

.1267

.1050

. 9605

. 0195

49

12

. 2578

. 7422

.3486

. 7056

.1251

.1062

. 9617,

. 0183

48

13

. 2604

. 7396

.3472

. 7092

.1235

.1053

. 9530

. 0170

47

14

. 2630

. 7369

.3457

. 7127

.1219

.1056

. 9542

. 0458

46

15

.426.57

.57343

2.3443

.47163

2.1203

1.1056

.09551

.90115

45

16

. 2683

. 7317

.3428

. 7199

.1187

.1058

. 9567

. 0433

44

17

. 2709

. 7290

.3414

. 72.34

.1171

.1059

. 9579

. 0421

43

18

. 2736

. 7264

.3399

. 7270

.1155

.1061

. 9592

. 0408

42

19

. 2762

. 7238

.3385

. 7306

.1139

.1062

. 9604

. 0396

41

20

.42788

.57212

2.3371

.47341

2.1123

1.1061

.09617

.90383

40

21

. 2815

. 7185

.3356

. 7376

.1107

.1065

. 9629

. 0371

39

22

. 2841

. 7159

.3342

. 7412

.1092

.1067

. 9641

. 0358

3S

23

. 2867

. 7133

.3328

. 7448

.1076

.1068

. 9661

. 0316

37

24

. 2893

. 7106

.3313

. 7483

.1060

.1070

. 9666

. 0333

36

25

.42920

.57080

2.3299

.47519

2.1014

1.1072

.09679

.90321

35

26

. 2946

. 7054

.3285

. 7555

.1028

.1073

. 9691

. 0308

34

27

. 2972

. 7028

.8271

. 7590

.1013

.1075

. 9704

. 0296

33

28

. 2998

. 7001

.3256

. 7626

.0997

.1076

. 9716

. 0283

32

29

. 3025

. 6975

.3242

. 7662

.0981

.1078

. 9729

. 0271

31

30

.43051

.66949

2.3228

.47697

2.0966

1.1079

.09741

.90258

30

31

. 3077

. 6923

.3214

. 77.S3

.0950

.1081

. 9764

. 0216

29

32

. 3104

. 6896

.3200

. 7769

.0934

.1082

. 9766

. 0233

28

33

. 3130

. 6870

.3186

. 7805

.0918

.1081

. 9779

. 0221

27

34

. 3156

. 6844

.3172

. 7810

.0903

.1085

. 9792

. 0208

26

35

.43182

.56818

2.3158

.47876

2.0887

1.1087

.09804

.90196

25

36

. 3208

. 6791

.3143

. 7912

.0872

.1088

. 9817

. 0183

24

37

. 3235

. 6765

.3129

. 7948

.0856

.1090

. 9829

. 0171

23

38

. 3261

. 6739

.3115

. 7983

.0840

.1092

. 9842

. 0158

22

39

. 3287

. 6713

.3101

. 8019

.0825

.1093

. 9854

. 0115

21

40

.43313

.56685

2.3087

.18055

2.0809

1.1095

.09867

.90133

20

41

. 3340

. 6660

.3073

. 8091

.0794

.1096

. 9880

. 0120

19

42

. 3366

. 6634

.3069

. 8127

.0778

.1098

. 9892

. 0108

18

43

. 3392

. 6608

..3046

. 8162

.0763

.1099

. 9905

. 0095

17

44

. 3418

. 6582

.3032

. 8198

.0747

.1101

. 9917

. 0082

16

45

.43444

.56555

2.3018

.48234

2.0732

1.1102

.09930

.90070

15

46

. 3471

. 6529

.3004

. 8270

.0717

.1101

. 9913

. 0057

14

47

. 3497

. 6503

.2990

. 8306

.0701

.1106

. 9956

. 0014

13

48

. 3523

. 6477

.2976

. 8342

.0686

.1107

. 9968

. 0032

12

49

. 8549

. 6451

.2962

. 8378

.0671

.1109

. 9981

. 0019

11

SO

.43575

.56424

2.2949

.48414

2.0655

1.1110

.09993

.90006

10

61

. 3602

. 6398

.2936

. 8449

.0640

.1112

.10006

.89991

9

52

. 3628

. 0372

. .2921

. 8485

.0625

.1113

. 0019

. 9981

8

53

. 3654

. 6346

.2907

. 8521

.0609

.1115

. 0031

. 9968

7

54

. 3680

. 6320

.2894

. 8557

.0891

.1116

. 0044

. 9956

6

55

.43706

.56294

2.2880

.48593

2.0579

1.1118

.10057

.89943

5

56

. 3732

. 6267

.2866

. 8629

.0564

.1120

. 0070

. 9930

4

57

. 3759

. 6241

.2853

. 8665

.0518

.1121

. 0082

. 9918

3

58

. 3^85

. 6215

.2839

. 8701

.0633

.1123

. 0095

. 9905

2

59

. 3811

. 6189

.2825

. 8737

.0518

.1124

. 0108

. 9892

1

60

. 3837

. 6163

.2812

. 8773

.0503

.1126

. 0121

. 9879

mT

Cosine.

Vrs. sin.

Secant.

Cotang.

Tang.

Cosec'nt

Vrs. coo.

Sine.

M.

348

NATURAL FUNCTIONS.

Table 3.

26<:

Natural Trigonometrical Functions.

153°

M.

Sine.

Vre. CO.S.

Cosec'nt

Tang.

Co tang.

Secant.

Vrs. Bin.

Cosine.

M.

.4S837

.56163

2.2812

.48773

2.0503

1.1126

.10121

.89879

60

1

. 3863

. 6137

.2798

. 8809

.0488

.1127

. 0133

. 9867

59

2

. 3aS9

. 6111

.2784

. 8845

.0473

.1129

. 0146

. 9854

58

8

. 3915

. 0084

.2771

. 8881

.0458

.1131

. 0159

. 9841

57

4

. 3942

. 0058

.2757

. 8917

.0443

.1132

. 0172

. 9828

56

5

.43968

.56032

2.2744

.48953

2.0427

1.1134

.10184

.89815

55

6

. 3994

. 6006

.2730

. 8989

.0412

.1135

. 0197

. 9803

54

7

. 4020

. 5980

.2717

. 9025

.0397

.1137

. 0210

. 9790

53

8

. 4046

. 5954

.2703

. 9062

.0382

.1139

. 0223

. 9777

52

9

. 4072

. 5928

.2690

. 9098

.0367

.1140

. 0236

. 9764

51

10

.44098

.55902

2.2676

.49134

2.0352

1.1142

.10248

.89751

50

11

. . 4124

. 5875

.2663

. 9170

.0338

.1143

. 0261

. 9739

49

12

. 4150

. 5849

.2650

. 9206

.0323

.1145

. 0274

. 9726

48

IS

. 4177

. 5823

.2636

. 9242

.0308

.1147

. 0287

. 9713

47

14

. 4203

. 5797

.2623

. 9278

.0293

.1148

. 0300

. 9700

46

15

.44229

.55771

2.2610

.49314

2.0278

1.1150

.10313

.89687

45

16

. 4255

. 5745

.2.596

. 9351

.0263

.1151

. 0326

. 9674

44

17

. 4281

. 5719

.2583

. 9387

.0248

.1153

. 0338

. 9661

43

18

. 4307

. 5693

.2570

. 9423

.0233

.1155

. 0351

. 9619

42

19

. 4333

. 5667

.2556

. 9459

.0219

.1156

. 0364

. 9636

41

20

.14359

.55641

2.2543

.49495

2.0204

1.1158

.10377

.89623

40

21

. 4385

. 5615

.2530

. 9.532

.0189

.1159

. 0390

. 9610

39

22

. 4411

. 5S89

.2517

. 9668

.0174

.1161

. 0403

. 9697

38

23

. 4437

. 5562

.2503

. 9604

.0159

.1163

. 0416

. 9684

37

24

. 4463

. 5536

.2490

. 9640

.0145

.1164

. 0429

. 9571

36

25

.44489

.55510

2.2477

.49077

2.0130

1.1166

.10442

.89658

35

26

. 4516

. 5484

.2464

. 9713

.0115

.1167

. 0455

. 9515

34

27

. 4542

. 5458

.2451

. 9749

.0101

.1169

. 0468

. 9532

33

28

. 4568

. 5432

.2438

. 9785

.0086

.1171

. 0481

. 9519

32

29

. 4594

. 5406

.2425

. 9822

.0071

.1172

. 0493

. 9.506

31

30

.44620

.55380

2.2411

.49858

2.0058

1.1174

.10606

.89493

30

31

. 4646

. 5354

.2398

. 9894

.0042

.1176

. 0619

. 9480

29

32

. 4672

. 5328

.2385

. 9931

.0028

.1177

. 0532

. 9467

28

33

. 4698

. 6302

.2372

. 9967

.0013

.1179

. 0545

. 9454

27

34

. 4724

. 5276

2359

.50003

1.9998

.1180

. 0558

. 9441.

26

35

.44750

.65250

2^2346

.50040

1.9984

1.1182

.10571

.89428

25

36

. 4776

. 5224

.2333

. 0076

.9969

.1184

. 0584

. 9415

24

37

. 4802

. 5198

.2320

. 0113

.9955

.1185

. 0598

. 9402

23

38

. 4828

. 6172

.2307

. 0149

.9940

.1187

. 0611

. 9389

22

39

. 4854

5146

.2294

. 0185

.9926

.1189

. 0624

. 9376

21

40

.44880

.55120

2.2282

.50222

1.9912

1.1190

.10637

.89363

20

41

. 4906

. 6094

.2269

. 0258

.9897

.1192

. 0650

. 9350

19

42

. 4932

. 5068

.2256

. 0295

.9883

.1193

. 0663

. 9337

18

43.

. 4958

. 5042

.2243

. 0331

.9868

.1195

. 0676

. 9324

17

44

. 4984

. 5016

.2230

. 0368

.9854

.1197

. 0689

. 9311

16

45

.45010

.54990

2.2217

.50404

1.9840

1.1198

.10702

.89298

15

46

5036

. 4964

.2204

. 0441

.9825

.1200

. 0715

. 9285

14

47

. 5062

. 4938

.2192

. 0477

.9811

.1202

. 0728

. 9272

13

48

5088

4912

.2179

. 0514

.9797

.1203

. 0741

. 9258

12

49

. 5114

. 4886

.2166

. 0550

.9782

.1206

. 0754

. 9215

11

50

.45140

.54860

2.2153

.50587

1.9768

1.1207

.10768

.89232

10

51

. 5166

. 4834

.2141

. 0623

.9754

.1208

. 0781

. 9219

9

52

. 5191

. 4808

.2128

. 0660

.9739

.1210

. 0794

. 9206

8

53

. 5217

. 4782

.2115

. 0696

.9725

.1212'

. 0807

. 9193

7

54

. ,5243

. 4756

.2103

. 0733

.9711

.1213

. 0820

. 9180

6

65

.45269

.54730

2.2090

.50769

1.9697

1.1215

.10833

.89166

5

56

. 6295

. 4705

.2077

. 0806

.9683

.1217

. 0846

. 9153

4

57

. 5321

. 4679

.2065

. 0843

.9668

.1218

. 0860

. 9140

3

58

5347

. 4653

.2052

. 0879

.9654

.12-20

. 0873

. 9127

2

59

. 5373

. 4627

.2039

. 0916

.9640

12,22

. 0886

. 9114

1

60

. 5399

. 4601

.2027

. 0952

.9626

.1223

. 0899

. 9101

M.

Cosine.

Vrs. sin.

Secant.

Co tang.

Tang.

Cosec'nt

VrB. cos.

Sine.

M.

116°

63°

Table 3.

NATURAL FUNCTIONS.

349

27°

Natural Trigonometrical Functions.

152"

M.

Sine.

Vre. COB.

Cosec'nt

Taug.

Co tang.

Secant.

Vi-a. sin.

Cosine.

M.

.45399

.54601

2.2027

.50952

1.9626

1 1223

.10899

.89101

60

1

. 5425

. 4575

.2014

. 0989

.9612

.1225

. 0912

. 9087

59

2

. 5451

. 4549

.2002

. 1026

.9598

.1226

. 0926

. 9074

58

3

. 5477

. 4523

.1989

. 1062

.9684

.1228

. 0939

. 9061

57

4

. 5503

. 4497

.1977

. 1099

.9570

.1230

. 0952

. 9048

56

5

.45528

.54-171

2.1964

.51136

1.9656

1.1231

.10965

.89034

55

6

. 6554

. 4145

.1962

. 1172

.9542

.1233

. 0979

. 9021

54

7

. 5580

. 4420

.1939

. 1209

.9528

.1235

. 0992

. 9008

53

8

. 5606

. 4394

.1927

. 1246

.9514

.1237

. 1005

. 8995

52

9

. 5632

. 4368

.1914

. 1283

.9500

.1238

. 1018

. 8981

51

10

.45658

.51342

2.1902

.51319

1.9486

1.1240

.11032

.88968

50

11

. 6684

. 4316

.1889

. 1356

.9472

.1242

. 1045

. 8955

49

12

. 5710

. 4290

.1877

. 1393

.9458

.1243

. 1058

. 8942

48

13

. 5736

. 4264

.1865

. 1430

.9444

.1245

. 1072

. 8928

47

14

. 5761

. 4238

.1852

. 1466

.9430

.1247

. 1085

. 8915

46

15

.45787

.54213

2.1840

.51603

1.9416

1.1248

.11098

.88902

45

16

. 6813

. 4187

.1828

. 1540

.9402

.1250

. 1112

. 8888

44

17

. 5839

. 4161

.1815

. 1677

.9388

.1252

. 1125

. 8875

43

18

. 6865

. 4135

.1803

. 1614

.9375

.1253

. 1138

. 8862

42

19

. 6891

. 4109

.1791

. 1651

.9361

.1255

. 1152

. 8848

41

20

.45917

.54083

2.1778

.51687

1.9347

1.1267

.11165

.88835

40

21

. 5942

. 4057

.1766

. 1724

.9333

.1258

. 1178

, 8822

39

22

. 5968

. 4032

.1754

. 1761

.9319

.1260

. 1192

. 8808

38

23

. 5994

. 4006

.1742

. 1798

.9306

.1202

. 1205

. 8795

37

24

. 6020

. 3980

.1730

. 1835

.9292

.1264

. 1218

. 8781

36

25

.46046

.53954

2.1717

.51872

1.9278

1.1265

.11232

.88768

35

26

. 6072

. 3928

.1705

. 1909

.9264

.1267

. 1245

. 8765

34

27

. 6097

. 3902

.1693

. 1946

.9251

.1269

. 1259

. 8741

33

28

. 6123

. 3877

.1681

. 1983

.9237

.1270

. 1272

. 8728

32

29

. 6149

. 3851

.1669

. 2020

.9223

.1272

. 1285

. 8714

31

30

.46175

.53825

2.1657

.52057

1.9210

1.1274

.11299

.88701

30

31

. 6201

. 3799

.1645

. 2094

.9196

.1275

. 1312

. 8688

29

32

. 6226

. 3773

.1633

. 2131

.9182

.1277

. 1326

. 8674

28

33

. 6252

. 3748

.1620

. 2168

.9169

.1279

. 1339

. 8661

27

34

. 6278

. 3722

.1608

. 2205

.9155

.1281

. 1353

. 8647

26

35

.46304

.53696

2.1596

.52242

1.9142

1.1282

.11366

.88634

25

36

. 6330

. 3670

.1584

. 2279

.9128

.1284

. 1380

. 8620

24

37

. 6355

. 3645

.1572

. 2316

.9115

.1286

. 1393

. 8607

23

38

. 6381

. 3619

.1560

. 2353

.9101

.1287

. 1407

8593

22

39

. 6407

. 3593

.1548

. 2390

.9088

.1289

. 1420

. 8580

21

40

.46433

.53567

2.1536

.52427

1.9074

1.1291

.11434

.88666

20

41

. 6458

. 3541

.15'25

. 2464

.9061

.1293

. 1417

. 8563

19

42

. 6484

. 3516

.1513

. 2501

.9047

.1294

. 1461

. 8539

18

43

. 6510

. 3490

.1501

. 2638

.9034

.1296

. 1474

. 8526

17

44

. 6536

. 3464

.1489

. 2675

.9020

.1298

. 1488

. 8512

16

45

.46561

.53438

2.1477

.52612

1.9007

1.1299

.11501

.88499

15

46

. 6587

. 3413

.1465

. 2660

.8993

.1301

. 1515

. 8485

14

47

. 6613

. 3387

.1453

. 2687

.8980

.1303

. 1528

. 8472

13

48

. 6639

. 3361

.1441

. 2724

.8967

.1305

1642

. 8458

12

49

. 6664

. 3336

.1430

. 2761

.8953

.1306

. 1555

. 8444

11

60

.46690

.53310

2.1418

.52798

1.8940

1.1308

.11569

.88431

10

51

. 6716

. 3284

.1406

. 2836

.8927

.1310

. 1583

. 8417

9

52

. 6741

. 3258

.1394

. 2873

.8913

.1312

. 1596

. 8404

8

63

. 6767

. 3233

.1382

. 2910

.8900

.1313

. 1610

. 8390

7

64

. 6793

. 3207

.1371

. 2947

.8887

.1315

. 1623

. 8376

6

55

.46819

.53181

2.1359

.52984

1.8873

1.1317

.11637

.88363

5

66

. 6844

. 3156

.1347

. 3022

.8860

.1319

. 1651

. 8349

4

67

. 6870

. 3130

.1335

. 3059

.8847

(1320

. 1664

. 8336

3

68

. 6896

. 3104

.1324

. 3096

.8834

.1322

. 1678

. 8322

2

59

. 6921

. 3078

.1312

. 3134

.8820

.1324

. 1691

. 8308

1

60

. 69J7

. 3053

.1300

. 3171

.8807

.1326

. 1705

. 8295

M^

Cosine.

Vrs. sin.

Secant.

Cotang.

Tang.

Cosec'nt

Yrs. cos.

Sine.

M.

Ii7°

62°

S50

NATDEAL FUNCTIONS.

Table 3.

28<^

Natural Trigonometrical Functions.

JS1°

M.

Sine.

Vrs. COS.

Cosec'ut

Tang.

Cotang.

Secant.

Vrs. sin.

Cosine,

M.

.46947

.53053

2.1300

.53171

1.8807

1.1326

.11705

,88295

60

1

. 6973

. 3027

.1289

. 3208

.8794

.1327

. 1719

. 8281

59

2

. 6998

. 3001

.1277

. 3245

.8781

.1329

. 1732

. 8267

58

3

. 7024

. 2976

.1266

. 3283

.8768

.1331

. 1746

. 8254

57

4

. 7050

. 2950

.1264

. 3320

.8754

.1333

. 1760

. 8240

56

5

.47075

.52924

2.1242

.63358

1.8741

1.1334

.11774

.88226

55

6

. 7101

. 2899

.1231

. 3395

.8728

.1336

. 1787

. 8213

54

7

. 7127

. 2873

.1219

. 3432

.8715

.1338

. 1801

. 8199

5S

8

. 7152

. 2847

.1208

. 3470

.8702

.1340

. 1815

. 8185

52

9

. 7178

. 2822

.1196

. 3507

.8089

.1341

. 1828

. 8171

51

10

.47204

.52796

2.1185

.53545

1.8676

1.1343

.11842

.88158

50

11

. 7229

. 2770

.1173

. 3582

.8603

.1345

. 1856

. 8144

49

12

. 7255

. 2745

.1102

. 3619

.8650

.1347

. 1870

. 8130

48

13

. 7281

. 2719

.1150

. 3657

.8637

.1349

. 1883

. 8117

47

14

. 7306

. 2694

.1139

. 3694

.8624

.1350

. 1897

. 8103

46

15

.47332

.62668

2.1127

.53732

1.8611

1.1362

.11911

■88089

45

16

. 7367

. 2642

.1116

. 3769

.8598

.1364

. 1925

. 8075

44

17

. 7383

. 2617

.1104

. 3807

.8585

.1356

. 1938

. 8061

43

18

. 7409

. 2591

.1093

. 3844

.8572

.1357

. 1952

. 8048

42

19

. 7434

. 2565

.1082

. 3882

.8569

.1359

. 1966

. 8034

41

20

.47460

.52540

2.1070

.53919

1.8546

1.1361

.11980

.88020

40

21

. 7486

. 2514

.1069

. 3957

.8533

.1363

. 1994

. 8006

39

22

. 7511

. 2489

.1048

. 3996

.8520

.1365

. 2007

. 7992

38

23

. 7537

. 2463

.1036

. 4032

.8507

.1366

. 2021

. 7979

37

24

. 7562

. 2437

.1025

. 4070

.8495

.1368

. 2035

. 7965

36

25

.47588

.52412

2.1014

.51107

1.8482

1.1370

.12049

.87951

35

26

. 7613

. 2386

.1002

. 4145

.8469

.1372

. 2063

. 7937

34

27

. 7639

. 2361

.0991

4183

.8456

.1373

. 2077

. 7923

33

28

. 7665

. 2335

.0980

. 4220

.8443

.1375

. 2090

. 7909

32

29

. 7690

. 2310

.0969

. 4268

.8430

.1377

. 2104

. 7895

31

30

.47716

.52284

2.0957

.54296

1.8418

1.1379

.12118

.87882

30

31

. 7741

. 2258

.0946

. 4333

.8405

.1381

. 2132

. 7868

29

32

. 7767

. 2233

.0935

. 4371

.8392

.1382

. 2146

. 7854

28

33

. 7792

. 2207

.0924

. 4409

.8379

.1384

. 2160

. 7840

27

34

. 7818

. 2182

.0912

. .4446

.8367

.1386

■ . 2174

. 7826

26

36

.47844

.52156

2.0901

.54484

1.8354

1.1388

.12188

.87812

25

36

. 7869

. 2131

.0890

. 4522

.8341

.1390

. 2202

. 7798

24

37

. 7895

. 2105

.0879

. 4659

.8329

.1391

. 2216

. 7784

23

38

. 7920

. 2080

.0868

. 4597

.8316

.1393

. 2229

. 7770

22

39

. 7946

. 2054

.0867

. 4635

.8303

.1395

. 2243

. 7756

21

40

.47971

.52029

2.0846

.54673

1.8291

1.1397

.12257

.87742

20

41

. 7997

. 2003

.0835

. 4711

.8278

,1399

. 2271

. 7728

19

42

. 8022

. 1978

.0824

. 4748

.8265

.1401

. 2285

. 7715

18

43

. 8048

. 1952

.0812

. 4786

.8253

.1402

. 2299

. 7701

17

44

. 8073

. 1927

.0801

. 4824

.8240

.1404

. 2313

. 7687

16

45

.48099

.51901

2.0790

.54862

1.8227

1.1406

.12327

.87673

16

46

. 8124

. 1876

.0779

. 4900

.8215

.1408

. 2341

. 7659

14

47

. 8160

. 1850

.0768

. 4937

.8202

.1410

. 2355

. 7645

13

48

. 8175

. 1825

.0757

. 4975

.8190

.1411

. 2369

. 7631

12

49

. 8201

. 1799

.0746

. 5013

.8177

.1413

. 2383

. 7617

11

50

.48226

.61774

2.0736

.55051

1.8165

1.1415

.12397

.87603

10

61

. 8252

. 1748

.0726

6089

.8152

.1417

. 2411

. 7688

9

52

. 8277

. 1723

.0714

. 6127

.8140

.1419

. 2425

. 7574

8'

63

. 8303

. 1697

.0703

. 6165

.8127

.1421

. 2439

. 7560

7

64

. 8328

. 1672

.0692

. 5203

.8115

.1422

. 2453

. 7546

6

85

.48354

.51646

2.0681

.55241

1.8102

1,1424

.12468

.87532

5

66

. 8379

. 1621

.0670

. 5279

.8090

.1426

. 2482

. 7518

4

67

. 8405

. 1695

.0659

. 5317

.8078

.1428

. 2496

. 7504

3

58

. 8430

. 1670

.0648

. 5355

.8065

.1430

. 2510

. 7490

2

59

. 8455

. 1644

.0637

. 6393

.8063

,1432

. 2524

. 7476

1

60

. 8481

. 1519

.0627

. 6431

.8040

.1433

. 2538

. 7462

M.

Cosine.

Vrs, sin.

Secant.

Ootang.

Tang.

Gosec'nt

Vrs, cos.

Sine.

jE

118°

6J°

Table 3.

NATURAL FUNCTIONS.

351

29°

Natural Trigonom

etrical Functions. .

J 50°

M.

Sine.

Vrs. COS.

Cosfc'nt

Tang.

Cotang.

Secant.

Vrs. sin.

Cosine.

M.

.48481

.51519

2.0627

.55431

1.8040

1.1433

.12538

.87462

60

1

. 8506

. 1493

.0616

. 5469

.8028

.1435

. 2552

. 7448

59

2

. 8532

. 1468

.0605

. 5507

.8016

.1437

. 2566

. 7434

58

3

. 8557

. 1443'

.0594

. 5545

.8003

.1439

. 2580

. 7420

57

4

. 8583

. 1417

.0683

. 5583

.7991

.1441

. 2594

. 7406

66

5

.48608

.51392

2.0573

.55621

1.7979

1.1443

.12609

.87391

55

6

. 8533

. 13G6

.0562

. 6659

.7966

.1445

. 2623

. 7377

64

7

. 8659

. 1341

.0551

. 5697

.7964

.1446

. 2637

. 7363

53

8

. 8684

. 1316

.0540

. 5735

.7942

.1448

. 2661

. 7349

52

9

. 8710

. 1290

.0530

. 5774

.7930

.14,50

. 2665

. 7335

51

10

.48735

.51265

2.0519

.55812

1.7917

1.1452

.12679

.87320

50

11

. 8760

. 1239

.0508

. 6850

.7905

.1454

. 2694

. 7,306

49

12

. 8786

. 1214

.0498

. 5888

.7893

.1456

. 2708

. 7292

48

13

. 8811

. 1189

.0487

. 5926

.7881

.1458

. 2722

. 7278

47

14

. 8837

. 1163

.0476

. 6964

.7868

- .1459

. 27,36

. 7264

46

15

.48862

.51138

2.0466

.56003

1.7866

1.1461

.127.50

.87250

45

16

. 8887

. 1112

.0465

. 6041

.7844

.1463

. 2765

. 7235

44

17

. 8913

. 1087

.0444

. 6079

.7832

.1465

. 2779

. 7221

43

18

. 8938

. 1062

.0434

. 6117

.7820

.1467

. 2793

. 7207

42

19

. 8964

. 1036

.0423

. 6156

.7808

.1469

. 2807

. 7193

41

20

.48989

.51011

2.0413

.66194

1.7795

1.1471

.12821

.87178

40

21

. 9014

. 0986

.0402

. 6232

.7783

.1473

. 2836

. 7164

39

22

. 9040

. 0960

.0.392

. 6270

.7771

.1474

. 2850

. 7150

38

23

. 9065

. 0935

.0381

. 6309

.7759

.1476

. 2864

. 7136

37

24

. 9090

. 0910

.0370

. 6347

.7747

.1478

. 2879

. 7121

36

25

.49116

.60884

2.0360

.66385

1.7735

1.1480

.12893

.87107

35

26

. 9141

. 0859

.0349

. 6424

.7723

.1482

. 2907

. 7093

34

27

. 9166

. 0834

.0339

. 6462

.7711

.1484

. 2921

. 7078

33

28

. 9192

. 0808

.0329

. 6500

.7699

.1486

. 2936

. 7064

32

29

. 9217

. 0783

.0318

. 6539

.7687

.1488

. 2950

. 7050

31

30

.49242

.60758

2.0308

.56577"

1.7675

1.1489

.12964

.87035

30

31

. 9268

. 0732

.0297

. 6616

.7663

.1491

. 2979

. 7021

29

32

. 9293

. 0707

.0287

. 6654

.7651

.1493

. 2993

. 7007

28

33

. 9318

. 0682

.0276

. 6692

.7639

.1495

. 3007

. 6992

27

34

. 9343

. 0656

.0266

. 6731

.7627

.1497

. 3022

. 6978

26

35

.49369

.50631

2.0256

.66769

1.7615

1.1499

.13036

.86964

25

36

. 9394

. 0606

.0245

. 6808

.7603

.1501

. 3050

. 6949

24

37

. 9419

. 0580

.0235

. 6846

.7591

.1503

. 3065

. 6935

23

38

. 9445

. 0565

.0224

. 6886

.7579

.1505

. 3079

. 6921

22

39

. 9470

. 0530

.0214

. 6923

.7567

.1607

. 3094

. 6906

21

40

.49495

.50505

2.0204

.66962

1.7565

1.1608

.13108

.86892

20

41

. 9521

. 0479

.0194

. 7000

.7544

.1610

. 3122

. 6877

19

42

. 9M6

. 0454

.0183

. 7039

.7532

.1512

. 3137

. 6863

18

43

. 9571

. 0429

.0173

. 7077

.7520

.1614

. 3151

. 6849

17

44

. 9596

. 0404

.0163

. 7116

.7608

.1516

. 3166

. 6834

16

45

.49622

.50378

2.0152

.67165

1.7496

1.1518

.13180

.86820

15

46

. 9647

. 0363

.0142

. 7193

.7484

.1520

. 3194

. 6805

14

47

. 9672

. 0328

.0132

. 7232

.7473

.1522

. 3209

. 6791

13

48

. 9697

. 0303

.0122

. 7270

.7461

.1524

. 3223

. 6776

12

49

. 9723

. 0277

.0111

. 7309

.7449

■ .1526

. 3238

. 6762

11

50

.49748

.60252

2.0101

.67.348

1.7437

1.1528

.13252

.86748

10

51

. 9773

. 0227

.0091

. 7386

.7426

.1530

. 3267

. 6733

9

52

. 9798

. 0202

.0081

. 7425

.7414

.1531

. 3281

. 6719

8

53

. 9823

. 0176

.0071

. 7464

.7402

.1633

. 3296

. 6704

7

54

. 9849

. 0151

.0061

. 7602

.7390

.1535

. 3310

. 6690

6

55

.49874

.50126

2.00.50

.57541

1.7379

1.1637

.13325

.86675

5

56

. 9899

. 0101

.0040

. 7580

.7367

.1539

. 3339

. 6661

4

57

. 9924

. 0076

.0030

. 7619

.7365

.1541

. 3354

. 6646

3

58

. 9950

. 0050

.0020

. 7657

.7344

.1543

. 3368

. 6632

2

69

. 9975

. 0025

.0010

. 7696

.7332

.1645

. 3383

. 6617

1

60

.50000

. 0000

.0000

. 7735

.7320

.1547

. 3397

. 6602

mT

Cosine.

Vrs. sin.

Secant.

Cotang.

Tang.

Cosec'nt

Vrs. COS.

Sine.

M.

U9°

60°

352

NATURAL FUNCTIONS.

Table 3.

30

3

Natural Trigonometrical Functions.

149°

M.

Sine.

Vrs. COS.

Cosec'nt

Tang.

Co tang.

Secant.

Vrs. sin.

Cosine.

M.

.50000

.50000

2.0000

.57735

1.7320

1.1547

.13397

.86602

60

1

. 0025

.49975

1.9990

. 7774

.7309

.1549

. 3412

. 6588

69

2

. 0050

. 9950

.9980

. 7813

.7297

.1551

. 3426

. 6573

58

S

. 0075

. 9924

.9970

. 7851

.7286

.1553

. 3441

. 6559

67

4

. 0101

. 9899

.9960

. 7890

.7274

.1565

. 3456

. 6544

56

5

.50126

.49874

1.9950

.57929

1.7262

1.1567

.13470

.86630

55

6

. 0151

. 9849

.9940

. 7968

.7261

.1559

. 3485

. 6516

54

7

. 0176

. 9824

.9930

. 8007

.7239

.1561

. 3499

. 6500

53

8

. 0201

. 9799

.9920

. 8046

.7228

.1562

. 3514

. 6486

62

9

. 0226

. 9773

.9910

. 8085

.7216

.1564

. 3529

. 6171

61

10

.60252

.49748

1.9900

.58123

1.7205

1.1666

.13543

.86457

.50

11

. 0277

. 9723

.9890

. 8162

.7193

.1568

. 3568

. 6442

49

12

. 0302

. 9698

.9880

. 8201

.7182

.1570

. 3572

. 6427

48

13

. 0327

. 9673

.9870

. 8240

.7170

.1672

. 3587

. 6413

47

14

. 0352

. 9648

.9860

. 8279

.7169

.1574

. 3602

. 6398

46

15

.50377

.49623

1.9850

.58318

1.7147

1.1576

.13616

.86383

45

16

. 0402

. 9597

.9840

. 8357

.7136

.1578

. 3631

. 6369

44

17

. 0428

. 9572

.9830

. 8396

.7124

.1680

. 3646

. 6354

43

18

. 0453

. 9547

.9820

8435

.7113

.1582

. 3660

. 6339

42

19

. 0478

. 9522

.9811

. 8474

.7101

.1584

. 3675

. 6325

41

20

.50503

.49497

1.9801

.58513

1.7090

1.1.586

.13690

.86310

40

21

. 0528

. 9472

.9791

. 8552

.7079

.1588

. 3704

. 6295

39

22

. 0553

. 9447

.9781

. 8591

.7067

.1590

. 3719

. 6281

38

23

. 0578

. 9422

.9771

. 8630

.7056

.1592

. 3734

. 6266

37

24

. 0603

. 9397

.9761

. 8670

.7044

.1594

. 3749

. 6251

36

25

.50628

.49371

1.9752

.58709

1.7033

1.1596

.13763

.86237

35

26

. 0653

. 9346

.9742

. 8748

.7022

.1698

. 3778

. 6222

34

27

. 0679

. 9321

.9732

. 8787

.7010

.1600

. 3793

. 6207

33

28

. 0704

. 9296

.9722

. 8826

.6999

.1602

. 3807

. 6192

32

29

. 0729

. 9271

.9713

. 8865

.6988

.1604

. 3822

. 6178

31

30

.50754

.49246

1.9703

.58904

1.6977

1.1606

.13837

.86163

30

31

. 0779

. 9221

.9693

. 8944

.6965

.1608

. 3852

. 6148

29

32

. 0804

. 9196

.9683

. 8983

.6954

.1610

. 8867

. 6133

28

33

. 0829

. 9171

.9674

. 9022

.6943

.1612

. 3881

. 6118

27

34

. OS.'M

. 9146

.9664

. 9061

.6931

.1614

. 3896

. 6104

26

35

.50879

.49121

1.9654

.59100

1.6920

1.1616

.13911

.86089

25

36

. 0904

. 9096

.9645

. 9140

.6909

.1618

. 3926

. 6074

24

37

. 0929

. 9071

.9635

. 9179

.6898

.1620

. 3941

. 6059

23

38

. 0954

. 9040

.9625

. 9218

.6887

.1622

. 3955

. 6044

22

39

. 0979

. 9021

.9616

. 9258

.6875

.1624

. 3970

. 6030

21

40

.6a0O4

.48996

1.9006

.59297

1.6864

1.1626

.13985

.86015

20

41

. 1029

. 8971

.9596

. 93.36

.6853

.1628

. 4000

. 6000

19

42

. 1054

. 8946

.9587

. 9376

.6842

.1630

. 4015

. 5985

18

43

. 1079

. 8921

.9577

. 9415

.6831

.1632

. 4030

. 5970

17

44

. 1104

. 8896

.9568

. 9454

.6820

.1634

. 4044

. 5965

16

45

.51129

.48871

1.9558

.59494

1.6808

1.1636

.14059

.85941

16

46

. 1154

. 8846

.9549

. 9533

.6797

.1638

. 4074

. 5926

14

47

. 1179

. 8821

.9539

. 9572

.6786

.1640

. 4089

. 5911

13

48

. 1204

. 8796

.9530

. 9612

.6775

.1642

. 4104

. 5896

12

49

. 1229

. 8771

.9520

. 9651

.6764

.1644

. 4119

. 6881

11

60

.51254

.48746

1.9510

.59691

1.6753

1.1646

.14134

.85866

10

51

. 1279

. 8721

.9501

. 9730

.6742

.1648

. 4149

. 5851

9

52

. 1304

. 8696

.9491

. 9770

.6731

.1650

. 4164

. 5836

8

53

1329

. 8671

.9482

. 9809

.6720

.1652

. 4178

. 5821

7

54

. 1354

. 8646

.9473

. 9849

.6709

.1654

. 4193

. 5806

6

55

.51379

.48621

1.9463

.59888

1.6698

1.1656

.14208

.85791

5

56

. 1404

. 8596

.9454

. 9928

.6687

.1658

. 4223

. 5777

4

57

. 1429

. 8571

.9444

. 9967

.6676

.1660

4238

. 5762

3

58

. 1454

. 8546

.9435

.60007

.6665

.1662

. 4253

. 5747

2

69

. 1479

. 8521

.9425

. 0046

.6654

.1664

. 4268

. 5732

1

60

. 1504

. 8496

.9416

. 0086

.6643

.1666

. 4283

. 5717

M,

Cosine.

Vrs. sin.

Secant.

Co tang.

Tang.

Cosec'nt

Vrs. COS.

Sine.

M.

120°

59°

Table 3.

NATURAL FUNCTIONS.

353

31°

Natural Trigonometrical Functions.

148°

M.

Sine.

Vrs. COS.

CoBoc'nt

Tang.

Co tang.

Secant.

Vrs. sin.

Cosine.

M.

.51504

.48496

1.9416

.60086

1.6643

1.1666

.14283

.85717

60

1

. 1529

. 8471

.9407

. 0126

.6632

.1668

. 4298

. 5702

59

2

.1554

. 8446

.9397

. 0165

.6621

.1670

. 4313

. 5687

58

3

. 1578

. 8421

.9388

.0205

.6610

.1672

. 4328

. 5672

57

4

. 1603

. 8396

.9378

. 0244

.6599

.1674

. 4343

. 5657

56

5

.51628

.48371

1.9369

.60284

1.6588

1.1676

.14358

.85642

55

6

. 1653

. 8347

.9360

. 0324

.6577

.1678

. 4373

. 5627

54

7

. 1678

. 8322

.9350

. 0363

.6566

.1681

. 4388

. 5612

53

8

. 1703

. 8297

.9311

. 0403

.6555

.1683

. 4403

. 6597

52

9

. 1728

. 8272

.9332

. 0443

.6544

.1685

. 4418

. 5582

51

10

.51753

.48247

1.9322

.60483

1.6534

1.1687

.14433

.85566

50

11

. 1778

. 8222

.9313

. 0522

.6523

.1689

. 4418

. 5551

49

12

. 1803

. 8197

.9304

. 0562

.6512

.1691

. 4163

. 5536

48

13

. 1827

. 8172

.9295

. 0602

.6501

.1693

. 4479

. 5521

47

14

. 1852

. 8147

.9285

. 0642

.6490

.1695

. 4494

. 5606

46

15

.51877

.48123

1.9276

.60681

1.6479

1.1697

.14509

.85491

45

16

. 1902

. 8098

.9267

. 0721

.6469

.1699

. 4524

. s-ne

44

17

. 1927

. 8073

.9258

. 0761

.6468

.1701

. 4539

. 5461

43

18

. 1962

. 8048

.9248

. 0801

.6447

.1703

. 4554

. 5446

42

19

. 1977

. 8023

.9239

. 0841

.6436

.1705

. 4569

. 5431

41

20

.52002

.47998

1.9230

.00881

1.6425

1.1707

.14581

.86416

40

21

. 2026

. 7973

.9221

. 0920

.6415

.1709

. 4599

. 6400

39

22

. 2051

•. 7949

.9212

. 0960

.6404

.1712

. 4615

. 5385

38

23

. 2076

. 7924

.9203

. 1000

.6393

.1714

. 4630

'. 5370

37

24

. 2101

. 7899

.9193

. 1040

.6383

.1716

. 4645

. 5355

36

25

.52126

.47874

1.9184

.61080 -

1.6372

1.1718

.14660

.85340

35

26

. 2151

. 7849

.9175

. 1120

.6361

.1720

. 4675

. 5325

34

27

. 2175

. 7824

.9166

. 1160

.6350

.1722

. 4690

. 5309

33

28

. 2200

. 7800

.9157

. 1200

.6340

.1724

. 4706

. 5294

32

29

. 2225

. 7775

.9148

. 1240

.6329

.1726

. 4721

. 5279

31

30

.52250

.47760

1.9139

.61280

1.6318

1.1728

.14736

.85264

30

31

. 2275

. 7725

.9130

. 1320

.6308

.1730

. 4751

. 5249

29

32

. 2299

. 7700

.9121

. 1360

.6297

.1732

. 4766

. 5234

28

33

. 2324

. 7676

.9112

. 1400

.6286

.1734

. 4782

. 5218

27

34

. 2349

. 7651

.9102

. 1440

.6276

.1737

. 4797

. 6203

26

35

.52374

.47626

1.9093

.61480

1.6265

1.17.39

.14812

.86188

25

36

. 2398

. 7601

.9084

. 1520

.6255

.1741

. 4827

. 6173

'24

37

. 2423

. 7577

.9075

1560

.6244

.1743

. 4842

. 5157

23

38

. 2448

. 7552

.9066

. 1601

.6233

.1745

. 4858

. 5142

22

39

. 2473

. 7527

.9057

. 1611

.6223

.1747

. 4873

. 5127

21

40

.52498

.47502

1.9048

.61681

1.6212

1.1749

.14888

.85112

20

41

. 2522

. 7477

.9039

. 1721

.6202

.1751

. 4904

. 6096

19

42

. 2547

. 7453

.9030

. 1761

.6191

.1753

. 4919

. 6081

18

43

. 2572

. 7428

.9021

. 1801

.6181

.1756

. 4934

. 6066

17

44

. 2597

. 7403

.9013

. 1842

.6170

.1758

. 4949

. 6050

16

45

.52621

.47379

1.9004

■ .61882

1.6160

1.1760

.14965

.85035

15

46

. 2616

. 7354

.8995

. 1922

.6149

.1762

. 4980

. 6020

14

47

. 2671

. 7329

.8986

. 1962

.6139

.1764

. 4995

. 5004

13

48

. 2695

. 7304

.8977

. 2004

.6128

.1766

. 5011

. 4989

12

49

. 2720

. 7280

.8968

. 2043

.6118

.1768

. 5026

. 4974

11

50

.52745

.47255

1.8959

.62083

1.6107

1.1770

.15041

.84959

10

61

. 2770

. 7230

.8950

. 2123

.6097

.1772

. 5067

. 4943

9

52

. 2794

. 7205

.8941

. 2164

.6086

.1775

. 6072

. 4928

8

53

. 2819

. 7181

.8932

. 2204

.6076

.1777

. 5087

. 4912

7

54

. 2844

. 7156

.8924

. 2244

.6066

.1779

. 5103

. 4897

6

55

.52868

.47131

1.8915

■ .62285

1.6055

1.1781

.15118

.84882

5

56

. 2893

. 7107

.8906

. 2325

.6045

.1783

. 6133

. 4806

4

57

. 2918

. 7082

.8897

. 2366

.6034

.1785

. 5149

. 4851

3

58

. 2942

. 7057

.8888

. 2406

.6024

.1787

. 5164

. 4836

2

59

. 2967

. 7033

.8879

. 2416

.6014

.1790

. 5180

. 4820

1

60

. 2992

. 701'S

.8871

. 2487

.6003

.1792

5195

. 4805

mT

Cosine.

Vrs, Bin,

Secant.

Co tang.

Tang.

Coeec'nt

Vrs. cos.

Sine.

M.

121°

58°

354

NATURAL FUNCTIONS.

Table 3.

32<:

Natural Trigonometrical Functions.

147°

M.

Sine.

Vrs. COS.

Cosec'nt

Tang.

Co tang.

Secant.

Vrs. sin.

Cosine.

M.

.52992

.47008

1.8871

.62487

1.6003

1.1792

.15195

.84805

60

1

. 3016

. 6983

.8862

. 2627

.5993

.1794

. 5211

4789

59

2

. 3041

. 6959

.8853

. 2568

.5983

.1796

. 5226

. 4774

58

3

. 3066

. 6934

.8844

. 2608

.5972

.1798

. 5241

. 4758

57

4

. 3090

. 6909

.8836

. 2649

.5962

.1800

. 5267

. 4743

56

5

.53115

.46885

1.8827

.62689

1.5952

1.1802

.15272

.84728

'55

6

. 3140

. 6860

.8818

. 2730

.5941

.1805

. 5288

. 4712

54

7

. 3164

. 6835

.8809

. 2770

.5931

.1807

. 5303

. 4697

53

8

. 3189

. 6811

.8801

. 2811

.5921

.1809

. 5319

. 4681

62

9

. 3214

. 6786

.8792

. 2851

.5910

.1811

. 5334

. 4666

51

10

.53238

.46762

1.8783

.62892

1.5900

1.1813

.15350

.84650

50

11

. 3263

. 6737

.8775

. 2933

.5890

.1815

. 5365

. 4635

49

12

. 3288

. 6712

.8766

. 2973

.5880

.1818

. 5381

. 4619

48

13

. 3312

. 6688

.8757

. 3014

.5869

.1820

. 5396

. 4604

47

14

. 3337

. 6663

.8749

. 3056

.5869

.1822

. 5412

. 4588

46

15

.53361

.46638

1.8740

.63095

1.5849

1.1824

.15427

.84673

45

16

. 3386

. 6614

.8731

. 3136

.5839

.1826

. 5443

. 4557

44

17

. 3111

. 6589

.8723

. 3177

.5829

.1828

. 5458

. 4542

43

18

. 3435

. 6565

.8714

. 3217

.5818

.1831

. 5474

. 4526

42

19

. 3460

. 6540

.8706

. 3258

.5808

.1833

. 5489

. 4511

41

20

.53484

.46516

1.8697

.63299

1.5798

1.1835

.16505

.84495

40

21

. 3509

. 6191

.8688

. 3339

.5788

.1837

. 5520

. 4479

39

22

. 3533

. 6466

.8680

. 3380

.5778

.1839

. 5536

. 4464

38

23

. 3558

. 6442

.8671

. 3121

.5768

.1841

. 5582

. 4448

37

24

. 3583

. 6417

.8663

. 3462

.5757

.1844

. 5567

. 4433

36

25

.53607

.40393

1.8654

.63603

1.5747

1.1846

.15583

.84417

35

26

. 3632

. 6368

.8646

. 8643

.5737

.1848

. 6698

4402

,34

27

. 3656

. 6344

.8637

. 3684

.5727

.1850

. 5614

4386

33

28

. 3681

. 6319

.8629

. 3625

.5717

.1862

. 5630

. 4370

32

29

. 3705

. 6294

.8620

. 8666

.6707

.1865

. 5645

. 4355

31

30

.53730

.46270

1.8611

.63707

1.5697

1.1857

.15661

.84339

30

31

. 3754

. 6245

.8603

. 3748

.5687

.1859

. 5676

. 4323

29

32

. 3779

. 6221

.8595

. 3789

.5677

.1861

. 5692

. 4308

28

33

. 3803

. 6196

.8586

. 3830

.5667

.1863

. 5708

. 4292

27

34

. 3828

. 6172

.8578

. 3871

.6657

.1866

. 5723

. 4276

26

35

.53852

.46147

1.8569

.63912

1.5646

1.1868

.15739

.84261

25

36

. 3877

. 6123

.8561

. 3953

.5636

.1870

. 5755

. 4245

24

37

. 3901

. 6098

.8552

. 3994

.5626

.1872

. 5770

. 4229

23

38

. 3926

. 6074

.8544

. 4035

.6616

.1874

. 5786

. 4214

22

39

. 3950

. 6049

.8535

. 4076

.6606

.1877

. 5802

. 4198

21

40

.53975

.46025

1.8527

.64117

1.5596

1.1879

.15817

.84182

20

41

. 3999

. 6000

.8519

. 4168

.5586

.1881

. 5833

. 4167

19

42

. 4024

. 5976

.8510

. 4199

.6577

.1883

. 5849

. 4151

18

43

. 4048

. 5951

.8502

. 4240

.5567

.1886

. 5865

. 4135

17

44

. 4073

. 5927

.8493

. 4281

.5557

.1888

. 5880

. 4120

16

45

.54097

.45902

1.8485

.64322

1.5547

1.1890

.16896

.84104

15

46

. 4122

. 5878

.8477

. 4363

.5537

.1892

. 5912

. 4088

14

47

. 4146

. 5854

.8468

. 4404

.5527

.1894

. 5927

. 4072

13

48

. 4171

. 5829

.8460

. 4446

.5517

.1897

. 5943

. 4057

12

49

. 4195

. 5805

.8452

. 4487

.5607

.1899

. 5959

. 4041

11

50

.54220

.45780

1.8443

.64528

1.5497

1.1901

.15975

.84025

10

51

. 4244

. 5756

.8435

. 4569

.5487

.1903

. 5991

. 4009

y

52

. 4268

. 5731

.8427

4610

.5477

.1906

. 6006

. 3993

8

53

. 4293

. 5707

.8418

. 4052

.6467

.1908

. 6022

. 3978

7

54

. 4317

. 5682

.8410

. 4693

.5458

.1910

. 6038

. 3962

6

55

.54342

.45658

1.8402

.64734

1.5448

1.1912

.16064

.83946

5

66

. 4366

. 5634

.8394

. 4775

.5438

.1915

. 6070

. 3930

4

57

. 4391

. 5609

.8385

. 4817

.5428

.1917

. 6085

. 3914

3

58

. 4415

. 5585

.8377

. 4868

.5418

.1919

. 6101

. 3899

2

59

. 4439

. 5560

.8369

. 4899

.5408

.1921

. 6117

. 3883

1

60

. 4464

. 5536

.8361

. 4941

.6399

.1922

. 6133

. 3867

H.

Cosine.

Vre. sin.

Secant.

Cotang.

Tang.

Cosec'nt

Vrs. COB.

Sine.

M.

122°

57°

Table 3.

NATURAL FUNCTIONS.

355

33°

Natural Trigonometrical Functions.

146°

M.

Sine.

Vrs. COS.

Oosec'nt

Tang.

Cotang.

Secant.

Vrs. Bin.

Cosine.

M.

.54464

.45536

1.8361

.64941

1.5399

1.1924

.16133

.83867

60

1

. 4488

. 5512

.8352

. 4982

.5389

.1926

. 6149

. 3851

59

2

. 4513

. 5487

.8344

. 5023

.5379

.1928

. 6165

. 3835

58

3

. 4537

. 5463

.8336

. 6065

.5369

.1930

. 6180

. 3819

57

4

. 4561

. 5438

.8328

. 5106

.5359

.1933

. 6196

. 3804

56

5

.54586

.45414

1.8320

.65148

1.5350

1.1935

.16212

.83788

55

6

. 4610

. 5390

.8311

. 5189

.5340

.1937

. 6228

. 3772

54

7

. 4634

. 5365

.8303

. 5231

.5330

.1939

. 6244

. 3756

53

8

. 4659

. 5341

.8295

. 5272

.5320

.1942

. 6260

. 3740

52

9

. 4683

. 5317

.8287

. 5314

.5311

.1944

. 6276

. 3724

51

10

.54708

.45292

1.8279

.65355

1.5301

1.1946

.16292

.83708

50

11

. 4732

. 5268

.8271

. 5397

.5291

.1948

. 6308

. 3692

49

12

. 4756

. 5244

.8263

. 5438

.5282

.1951

. 6323

. 8676

48

13

. 4781

. 5219

.8255

. 5480

.5272

.1953

. 6339

. 3660

47

14

. 4805

. 5195

.8246

. 5521

.5262

.1955

. 6355

. 3644

48

15

.54829

.45171

1.8238

.65563

1.5262

1.1958

.16371

.83629

45

10

. 4854

. 5146

.8230

. 5604

.5243

.1960

. 6387

. 3613

44

17

. 4878

. 5122

.8222

. 5646

.5233

.1962

. 6403

. 3597

43

18

. 4902

. 5098

.8214

. 5688

.5223

.1964

. 6419

. 3581

42

19

. 4926

. 5073

.8206

. 5729

.5214

.1967

. 6435

. 3565

41

20

.54951

.45049

1.8198

.65771

1.6204

1.1969

.16451

.83549

40

21

. 4975

. 5025

.8190

. 5813

.5195

.1971

. 6467

. 3533

39

22

. 4999

. 5000

.8182

. 5864

.6185

.1974

. 6483

. 3617

38

23

. 5024

. 4976

.8174

. 5896

.5175

.1976

. 6499

. 3601

37

24

. 5048

. 4952

.8166

. 5938

.5166

.1978

. 6515

. 3485

36

25

.55072

.44928

1.8158

.65980

1.6156

1.1980

.16531

.83469

35

26

. 5097

. 4903

.8150

. 6021

.5147

.1983

. 6547

. 3453

34

27

. 5121

. 4879

.8142

. 6063

.6137

.1985

. 6563

. 3437

33

28

. 5145

. 4855

.8134

. 6105

.6127

.1987

. 6679

. 3421

32

29

. 5169

. 4830

.8126

. 6147

.6118

.1990

. 6595

. 3405

31

30

.55194

.44806

1.8118

.66188

1.5108

1.1992

.16611

.83388

30

31

. 5218

. 4782

.8110

. 6230

.6099

.1994

. 6627

. 3372

29

32

. 5242

. 4758

.8102

. 6272

.6089

.1997

. 6643

. 3356

28

33

. 5266

. 4733

.8094

. 6314

.6080

.1999

. 6660

. 3340

27

34

. 5291

. 4709

.8086

. 6356

.5070

.2001

. 6676

. 3324

26

35

.55315

.44685

1.8078

.66398

1.5061

1.2004

.16692

.83308

25

3G

. 5339

. 4661

.8070

. 6440

.5051

.2006

. 6708

. 3292

24

37

. 5363

. 4637

.8062

. 6482

.5042

.2008

. 6724

. 3276

23

38

. 5388

. 4612

.8054

. 6524

.5032

.2010

. 6740

. 3260

22

39

. 5112

. 4588

.8047

. 6666

.6023

.2013

. 6756

. 3244

21

40

..55436

.44564

1.8039

.66608 ■

1.5013

1.2015

.16772

.83228

20

41

. 5460

. 4540

.8031

. 6650

.5004

.2017

. 6788

. 3211

19

42

. 5484

. 4515

.8023

. 6692

.4994

.2020

. 6804

. 3195

18

43

. 5509

. 4491

.8015

. 6734

.4985

.2022

. 6821

. 3179

17

44

. 5533

. 4467

.8007

. 6776

.4975

.2024

. 6837

. 3163

16

45

.55557

.44443

1.7999

.66818

1.4966

1.2027

.16853

.83147

15

46

. 5581

. 4419

.7992

. 6860

.4957

.2029

. 6869

. 3131

14

47

. 5605

. 4395

.7984

. 6902

.4947

.2031

. 6885

. 3115

13

48

. 5629

. 4370

.7976

. 6944

.4938

.2034

. 6901

. 3098

12

49

. 5654

. 4346

.7968

. 6986

.4928

. 6918

. 3082

11

50

.55678

.44322

1.7960

.67028

1.4919

1.2039

.16934

.83066

10

51

. 5702

. 4298

.7953

. 7071

.4910

!2041

. 6950

. 3050

9

52

. 5726

. 4274

.7945

. 7113

.4900

.2043

. 6966

. 3034

8

53

. 5750

. 4250

.7937

. 7155

.4891

.2046

. 6982

. 3017

7

54

. 5774

. 4225

.7929

. 7197

.4881

.2048

. 6999

. 3001

6

55

.55799

.44201

1.7921

.67239

1.4872

1.2050

.17015

.82985

5

56

. 5823

. 4177

.7914

. 7282

.4863

.2053

. 7031

. 2969

4

57

. 5847

. 4153

.7906

. 7324

.4853

.2055

. 7047

. 2962

3

58

. 5871

. 4129

.7898

. 7366

.4844

.2057

. 7064

. 2936

2

59

. 5895

. 4105

.7891

. 7408

.4835

.2060

- 7080

. 2920

1

60

. 5919

. 4081

.7883

. 7451

.4826

.2062

. 7096

. 2904

M.

Cosine.

Vrs. sin.

Secant.

Cotang.

Tang.

Cosec'nt

Vrs. cos.

Sine.

M.

356

NATUnAL FUNCTIONS.

Table 3.

34°

Natural Trigonometrical Functions.

145°

M.

Sine.

Vrs. C08.

Cosec'nt

Tang.

Cotang.

Secant.

Vre. sin.

Cosine.

M.

.55919

.44081

1.7883

.67451

1.4826

1.2062

.17096

.82904

60

1

. 5943

. 4057

.7875

. 7493

.4816

.2064

. 7112

. 2887

59

2

. 5967

. 4032

.7867

. 7535

.4807

.2067

. 7129

. 2871

58

3

. 5992

. 4008

.7860

. 7578

.4798

.2069

. 7145

. 2855

57

4

. 6016

. 3984

.7852

. 7620

.4788

.2072

. 7161

. 2839

56

5

.56040

.43960

1.7844

.67663

1.4779

1.2074

.17178

,82822

55

6

. 6064

. 3936

.7837

. 7705

.4770

.2076

. 7194

. 2806

54

7

. 6088

. 3912

.7829

. 7747

.4761

.2079

. 7210

. 2790

53

8

. 6112

. 8888

.7821

. 7790

.4751

.2081

. 7227

. 2773

52

9

. 6136

. 3864

.7814

. 7832

.4742

.2083

. 7243

. 2757

51

10

.56160

.43840

1.7806

.67875

1.4733

1.2086

.17259

.82741

50

11

. 6184

. 3816

.7798

. 7917

.4724

.2088

. 7276

. 2724

49

12

. 6208

. 3792

.7791

. 7960

.4714

.2091

. 7292

. 2708

48

13

. 6232

. 3768

.7783

. 8002

.4705

.2093

. 7308

. 2692

47

14

. 6256

. 3743

.7776

. 8045

.4696

.2095

. 7325

. 2675

46

15

.56280

.43719

1.7768

.6S087

1.4687

1.2098

.17341

.82659

45

16

. 6304

. 3695

.7760

. 8130

.4678

.2100

. 7357

. 2643

44

17

. 6323

. 3671

.7753

. 8173

.4669

.2103

. 7374

. 2626

43

18

. 6353

. 3647

.7745

. 8215

.4659

.2105

. 7390

. 2610

42

19

. 6377

. 3623

.7738

. 8258

.4650

.2107

. 7406

. 2593

41

20

.56101

.43599

1.7730

.68301

1.4641

1.2110

.17423

.82,577

40

21

. 6425

. 3575

.7723

. 8343

.4632

.2112

. 7439

. 2561

39

22

. 6449

. 3551

.7715

. 8386

.4623

.2115

. 7456

. 2544

38

23

. 6473

. 3527

.7708

. 8429

.4614

.2117

. 7472

. 2528

37

24

. 6497

. 3503

.7700

. 8471

.4605

.2119

. 7489

. 2611

36

25

.56521

.43479

1.7093

.68514

1.4,595

1.2122

.17505

.82495

35

26

. 6545

. 3455

.7685

. 8.557

.4586

.2124

. 7521

. 2478

34

27

. 6569

. S4S1

.7678

. 8600

.4577

.2127

. 7538

. 2462

33

28

. 6593

. 3407

.7670

. 8642

.4568

.2129

. 7554

. 2445

32

29

. 6617

. 3383

.7663

. 8685

.4559

.2132

. 7571

. 2429

31

30

.56641

.43359

1.7655

.68728

1.4550

1.2134

.17587

.82413

30

31

. 6664

. 3335

.7648

. 8771

.4541

.2136

. 7604

. 2396

29

32

. 6688

. 3311

.7640

. 8814

.4532

.2139

. 7620

. 2380

28

33

. 6712

. 3287

.7633

. 8857

.4523

.2141

. 7637

. 2363

27

34

. 6736

. 3263

.7625

. 8899

.4514

.2144

. 7653

. 2.347

26

35

.56760

.43239

1.7618

.68942

1.4505

1.2146

.17670

.82330

25

36

. 6784

. 3216

.7610

. 8985

.4496

.2149

. 7686

. 2314

24

37

. 6808

. 3192

.7603

. 9028

.4487

.2151

. 7703

. 2297

23

38

. 6832

. 3168

.7596

. 9071

.4478

.2153

. 7719

. 2280

22

39

. 6856

. 3144

.7588

. 9114

.4469

.2156

. 7736

. 2264

21

40

.56880

.43120

1.7581

.69157

1.4460

1.2158

.17752

.82247

20

41

. 6904

. 3096

.7573

. 9200

.4451

.2161

. 7769

. 22,31

19

42

. 6928

. 3072

.7566

. 9243

.4442

.2163

. 7786

. 2214

18

43

. 6952

. 3048

.7559

. 9286

.4433

.2166

. 7802

. 2198

17

44

. 6976

. 3024

.7551

. 9329

.4424

.2168

. 7819

. 2181

36

45

.57000

.43000

1.7514

.69372

1.4415

1.2171

.17835

.82165

15

46

. 7023

. 2976

.7537

. 9415

.4406

.2173

. 7852

. 2148

14

47

. 7047

. 2952

.7529

. 9459

.4397

.2175

7868

. 2131

13

48

. 7071

. 2929

.7522

. 9502

.4388

.2178

. 7885

. 2115

12

49

. 7095

. 2905

.7514

. 9.545

.4379

.2180

. 7902

. 2098

11

50

.57119

.4'2881

1.7507

.69588

1.4370

1.2183

.17918

.82082

10

61

. 7113

. 2857

.7500

. 9631

.4361

.2185

. 7935

. 2066

9

52

. 7167

. 2833

.7493

. 9674

.4352

.2188

. 7951

. 2048

8

63

. 7191

. 2809

.7485

. 9718

.4343

.2190

. 7968

. 2032

7

54

. 7214

. 2785

.7478

. 9761

.4335

.2193

. 7985

. 2015

6

55

.57238

.42761

1.7471

.69804

1.4326

1.2195

.18001

.81998

5

56

. 7262

. 2738

.7463

. 9847

.4317

.2198

. 8018

. 1982

4

57

. 7286

. 2714

.7456

. 9891

.4308

.2200

. 8035

. 1965

3

68

. 7310

. 2690

.7449

. 99.S4

.4299

.2203

. 8051

. 1948

2

59

. 7334

. 2666

.7412

. 9977

.4290

.2205

. 8068

. 1932

1

60

. 7358

. 2642

.7434

.70021

.4281

.2208

. 8085

. 1915

M.

Cosine.

ViB. sin.

Secant.

Co tang.

Tang.

CoBec'nt

Vrs. COB.

Sine.

M.

J 24°

55°

Table 3.

NATURAL FUNCTIONS.

357

35'

Natural Trigonometrical Functions.

144°

M.

Sine.

Vre. COS.

CoBec'nt

Tang.

Cotang.

Secant.

Yrs. Bin.

Cosine.

M.

.57358

.42642

1.7434

.70021

1.4281

1.2208

.18085

.81915

60

1

. 7:«1

. 2618

.7427

. 0064

.4273

.2210

. 8101

. 1898

59

2

. 7405

. 2595

.7420

. 0107

.4264

.2213

. 8118

. 1882

58

3

. 7429

. 2571

.7413

. 0151

.4255

.2215

. 8135

. 1865

57

4

. 7453

. 2547

.7405

. 0194

.4246

.2218

. 8151

. 1848

56

5

.57477

.42523

1.7398

.70238

1.4237

1.2220

.18168

.81832

55

6

. 7500

. 2499

.7391

. 0281

.4228

.2223

. 8185

. 1815

54

7

. 7524

. 2476

.7384

. 0325

.4220

.2225

. 8202

. 1798

53

8

. 7548

. 2452

.7377

. 0368

.4211

22?«

. 8218

. 1781

52

9

. 7572

. 2428

.7369

. 0412

.4202

.2230

. 8235

. 1765

51

10

.57596

.42404

1.7362

.70455

1.4193

1.2233

.18252

.81748

50

11

. 7619

. 2380

.7355

. 0499

.4185

.2235

. 8269

. 1731

49

12

. 7643

. 2357

.7348

. 0542

.4176

.2238

. 8285

. 1714

43

13

. 7667

. 2333

.7341

. 0586

.4167

.2240

. 8302

. 1698

47

14

. 7691

. 2309

.7334

. 0629

.4158

.2243

. 8319

. 1681

46

15

.57714

.42285

1.7327

.70673

1.4150

1.2245

.18336

.81664

45

16

. 7738

. 2262

.7319

. 0717

.4141

.2248

. 8353

. 1647

44

17

. 7762

. 2238

.7312

. 0760

.4132

.2250

. 8369

. 1630

43

18

. 7786

. 2214

.7305

. 0804

.4123

.2253

. 8386

. 1614

42

19

. 7809

. 2190

.7298

. 0848

.4115

.2255

. 8403

. 1597

41

20

.57833

.42167

1.7291

.70891

1.4106

1.2258

.18420

.81580

40

21

. 7857

. 2143

.7284

. 0935

.4097

.2260

. 8437

. 1563

39

22

. 7881

. 2119

.7277

. 0979

.4089

.2263

. 8453

. 1546

38

23

.7904

. 2096

.7270

. 1022

.4080

.2265

. 8470

. 1530

37

24

. 7928

. 2072

.7263

. 1066

.4071

.2268

. 8487

. 1513

36

25

.57952

.42048

1.7256

.71110

1.4063

1.2270

.18504

.81496

35

26

. 7975

. 2024

.7249

. 1154

.4054

.2273

. 8521

. 1479

34

27

. 7999

. 2001

.7242

. 1198

.4045

.2276

. 8538

. 1462

33

28

. 8023

. 1977

.7234

. 1241

.4037

.2278

. 8555

. 1445

32

29

. 8047

. 1953

.7227

. 1285

.4028

.2281

. 8571

. 1428

31

30

.58070

.41930

1.7220

.71329

1.4019

1.2283

.18588

.81411

30

31

. 8094

. 1906

.7213

. 1373

.4011

.2286

. 8605

. 1395

29

32

. 8118

. 1882

.7206

. 1417

.4002

2?88

. 8622

. 1378

28

33

. 8141

. 1859

.7199

. 1461

.3994

.2291

. 8639

. 1361

27

84

. 8165

. 1835

.7192

. 1505

.3985

.2293

. 8656

. 1344

20

35

.58189

.41811

1.7185

.71549

1.3976

1.2296

.18673

.81327

25

36

, 8212

. 1788

.7178

. 1593

.3968

.2298

. 8690

. 1310

24

37

. 8236

. 1764

.7171

. 1637

.3959

.2301

. 8707

. 1293

23

38

. 8259

. 1740

.7164

. 1681

.3951

.2304

.8724

. 1276

22

39

. 8283

. 1717

.7157

. 1725

.3942

.2306

. 8741

. 1259

21

40

.58307

.41693

1.7151

.71769

1.3933

1.2309

.18758

.81242

20

41

. 8330

. 1669

.7144

. 1813

.3925

.2311

. 8775

. 1225

19

42

. 8354

. 1646

.7137

. 1857

.3916

.2314

. 8792

. 1208

18

43

. 8378

. 1622

.7130

. 1901

.3908

.2316

. 8809

. 1191

17

44

. 8401

. 1599

.7123

. 1945

.3899

.2319

. 8826

. 1174

16

45

.58425

.41575

1.7116

.71990

1.3891

1.2322

.18843

.81157

15

46

. 8448

. 1551

.7109

. 2034

.3882

.2324

. 8860

. 1140

14

47

. 8472

. 1528

.7102

. 2078

.3874

.2327

. 8877

. 1123

13

48

. 8496

. 1504

.7095

. 2122

.3865

.2329

. 8894

. 1106

12

49

. 8519

. 1481

.7088

. 2166

.3857

.2332

. 8911

. 1089

11

60

.58543

.41457

1.7081

.72211

1.3848

1.2335

.18928

.81072

10

51

. 8566

. 1433

.7075

. 2'255

.3840

.2337

. 8945

. 1055

9

62

. 8990

. 1410

.7068

. 2299

.3831

.2340

. 8962

. 1038

8

53

. 8614

. 1386

.7061

. 2344

.3823

.2342

. 8979

. 1021

7

54'

. 8637

. 1363

.7054

. 2388

.3814

.2345

. 8996

. 1004

6

55

.58661

.41339

1.7047

.72432

1.3806

1.2348

.19013

.80987

5

56

. 8684

. 1316

.7040

. 2477

.3797

.2350

. 9030

. 0970

4

57

. 8708

. 1292

.7033

. 2521

.3789

.2353

. 9047

. 0953

3

58

. 8731

. 1268

.7027

. 2565

.3781

.2355

. 9064

. 0936

2

59

. 8755

. 1245

.7020

. 2610

.3772

.2358

. 9081

. 0919

1

60

. 8778

. 1221

.7013

. 2654

.3764

.2361

. 9098

. 0902

jr.

CosiDe.

Vrs. sin.

Secant.

Cotang.

Tang.

Cosec'nt

Vrs. COS.

Sine. M.

125°

358

NATURAL FUNCTIONS.

Table 3.

36°

Natural Trigonometrical Functions.

143°

M.

Sine.

Vrs. coe.

Coscc'nt

Tang.

Cotang.

Secant.

Vrs. sin.

Cosine.

M.

.58778

.41221

1.7013

.72654

1.3764

1.2361

.19098

.80902

60

1

. 8802

. 1198

.7006

. 2699

.3755

.2363

. 9115

. 0885

59

2

. 8825

. 1174

.6999

. 2743

.3747

.2366

. 9132

. 0867

58

3

. 8849

. 1151

.6993

. 2788

.3738

.2368

. 9150

. 0850

57

4

. 8873

. 1127

.6986

. 2832

.3730

.2371

. 9167

. 0833

56

5

.58896

.41104

1.6979

.72877

1.3722

1.2374

.19184

.80816

55

6

. 8920

. 1080

.6972

. 2921

.3713

.2376

. 9201

. 0799

54

7

. 8943

. 1057

.6965

. 2966

.3705

.2379

. 9218

. 0782

53

8

.8967

. 1033

.6959

. 3010

.3697

.2382

. 9235

. 0765

52

9

. 8990

. 1010

.6952

. 3055

.3688

.2384

. 9252

. 0747

51

10

.59014

.40986

1.6945

.73100

1.3680

1.2387

.19270

.80730

50

H

. 9037

. 0963

.6938

. 3144

.3672

.2389

. 9287

. 0713

49

12

. 9060

. 09.39

.6932

. 3189

.3663

.2392

. 9304

. 0696

48

13

. 9084

. 0916

.6925

. 8234

.3655

.2395

. 9321

. 0679

47

14

. 9107

. 0892

.6918

. 8278

.3647

.2397

. 9338

. 0662

46

15

.59131

.40869

1.6912

.73323

1.3638

1.2400

.19365

.80644

45

16

. 9164

. 0845

.6905

. 3368

.3630

.2403

. 9373

. 0627

44

17

. 9178

. 0822

.6898

. 3412

.3622

.2405

. 9390

. 0610

43

18

. 9201

. 0799

.6891

. 3457

.3613

.2408

. 9407

. 0593

42

19

. 9225

. 0775

.6885

. 3502

.3005

.2411

. 9424

. 0576

41

20

.69248

.40752

1.6878

.73547

1.3597

1.2413

.19442

.80558

40

21

. 9272

. 0728

.6871

. 3592

.3588

.2416

. 9459

. 0641

39

22

. 9295

. 0705

.6865

. 3637

.3580

.2419

. 9476

. 0524

38

23

. 9318

. 0681

.6858

. 3681

.3672

.2421

. 9493

. 0507

37

24

. 9342

. 0658

.6851

. 3726

.3564

.2424

. 9511

. 0489

36

25

.59365

.40635

1.6845

.73771

1.3655

1.2127

.19528

.80472

35

26

. 9389

. 0611

.6838

. 3816

.3547

.2429

. 9545

. 0455

34

27

. 9412

. 0588

.6831

. 3861

.3539

.2432

. 9562

. 0437

33

28

. 9435

. 0564

.6825

. 3906

.3531

.2435

. 9580

. 0420

32

29

. 9459

. 0541

.6818

. 3951

.3522

.2437

. 9597

. 0403

31

30

.59482

.40518

1.6812

.73996

1.3514

1.2440

.19614

.80386

30

31

. 9506

. 0494

.6805

. 4041

.3506

.2443

. 9632

. 0368

29

32

. 9529

. 0471

.6798

. 4086

.3498

.2445

. 9649

. 0351

28

33

. 9562

. 0447

.6792

. 4131

.3489

.2448

. 9666

. 0334

27

34

. 9576

. 0424

.6785

. 4176

.3481

.2451

. 9683

. 0316

26

35

.59599

.40401

1.6779

.74221

1.3473

1.2453

.19701

.80299

25

36

. 9622

. 0377

.6772

. 4266

.3465

.2456

. 9718

. 0282

24

37

. 9646

. 0354

.6766

. 4312

.3457

.2459

. 9736

. 0264

23

38

. 9669

. 0331

.6759

. 4357

.3449

.2461

. 9753

. 0247

22

39

. 9692

. 0307

.6752

. 4402

.3440

.2464

. 9770

. 0230

21

40

.59716

.40284

1.6746

.74447

1.3432

1.2467

.19788

.80212

20

41

. 9739

. 0261

.6739

. 4492

.3424

.2470

. 9805

. 0195

19

42

. 9762

.0237

.6733

. 4538

.3416

.2472

. 9822

. 0177

18

43

. 9786

. 0214

.6726

. 4583

.3408

.2475

. 9840

. 0160

17

44

. 9R09

. 0191

.6720

. 4628

.3400

.2478

. 9867

. 0143

16

45

.59832

.40167

1.6713

.74673

1.3392

1.2480

.19875

.80125

15

46

. 9856

. 0144

.6707

. 4719

.3383

.2483

. 9892

. 0108

14

47

. 9879

. 0121

.6700

. 4764

.3375

.2486

. 9909

. 0090

13

48

. 9902

. 0098

.6694

. 4809

.3367

.2488

. 9927

. 0073

12

49

. 9926

. 0074

.6687

. 4855

.3359

.2491

. 9944

. 0056

11

60

.59949

.40051

1.6681

.74900

1.3351

1.2494

.19962

.80038

10

51

. 9972

. 0028

.6674

. 4946

.3343

.2497

. 9979

. 0021

9

52

. 9995

. 0004

.6668

. 4991

.3335

.2499

. 9997

. 0003

8

53

.60019

.39981

.6661

. 5037

.3327

.2502

.20014

.79986

7

54

. 0042

. 9958

.6655

. 5082

.3319

.2505

. 0031

. 9968

6

55

.60065

.39935

1.6648

.75128

1.3311

1.2508

.20049

.79951

5

56

. 0088

. 9911

.6642

. 5173

.3303

.2510

. 0066

. 9933

4

57

. 0112

. 9888

.6636

. 5219

.3294

.2513

. 0084

. 9916

3

68

. 0135

. 9866

.6629

. 5264

.3286

.2516

. 0101

. 9898

2

59

. 0158

. 9842

.6623

. 6310

.3278

.2519

. 0119

. 9881

1

60

. 0181

. 9818

.6616

. 6355

.3270

.2521

. 0136

. 9863

M.

Cosine.

Vrs. sin.

Secant.

Cotang.

Tang.

Cosec'nt

Vrs. COS.

Sine.

M,

126°

53°

Table 3.

NATURAL FUNCTIONS.

359

37=

Natural Trigonometrical Functions.

142°

mT

Sine.

Vra. COS.

Cosec'nt

Tang.

Cotang.

Secant.

Vrs. sin.

Cosine.

M.

.60181

.39818

1.6616

.75355

1.3270

1.2521

.20136

.79863

60

1

. 0205

. 9795

.6610

. 5401

.3262

.2524

. 0154

. 9846

59

2

. 0228

. 9772

.6603

. 5447

.3254

.2527

. 0171

. 9828

58

3

.0251

. 9749

.6597

. 5492

.3246

.2530

. 0189

. 9811

57

4

. 0274

. 9726

.6591

. 5538

.3238

.2532

. 0206

. 9793

56

5

.60298

.39702

1.6584

.75584

1.3230

1.2535

.20224

.79776

55

6

.0320

. 9679

.6578

. 5629

.3222

.2538

. 0242

. 9758

54

7

. 0344

. 9656

.6572

. 5675

.3214

.2541

. 0259

. 9741

53

8

. 0367

. 9633

.6565

. 5721

.3206

.2543

. 0277

. 9723

52

9

.0390

. 9610

.6559

. 5767

.3198

.2546

. 0294

. 9706

51

10

.60413

.39586

1.6552

.75812

1.3190

1.2549

.20312

.79688

50

11

. 0437

. 9563

.6546

. 5858

.3182

.2552

. 0329

. 9670

49

12

. 0460

. 9540

.6540

. 5904

.3174

.2554

. 0347

. 9653

48

13

. 0483

. 9517

.6533

. 5950

.3166

.2557

. 0365

. 9635

47

14

. 0506

. 9494

.6527

. 5996

.3159

.2560

. 0382

. 9618

46

15

.60529

.39471

1.6521

.76042

1.3151

1.2563

.20400

.79600

45

16

. 0552

. 9447

.6514

. 6088

.3143

.2565

. 0417

. 9582

44

17

. 0576

. 9424

.6508

. 6134

.3135

.2568

. 0435

. 9565

43

18

. 0599

. 9401

.6502

. 6179

.3127

.2571

. 0453

. 9547

42

19

. 0622

. 9378

.6496

. 6225

.3119

.2574

. 0470

. 9530

41

20

.60645

.39355

1.6489

.70271

1.3111

1.2577

.20488

.79512

40

21

. 0668

. 9332

.6183

. 6317

.3103

.2579

. 0505

. 9494

39

22

. 0691

. 9309

.6477

. 6364

.3095

.2582

. 0523

. 9477

38

23

. 0714

. 9285

.6470

. 6410

.3087

.2585

. 0541

. 9459

37

24

. 0737

. 9262

.6464

. 6156

.3079

.2588

. 0558

. 9441

36

25

.60761

.39239

1.6458

.76502

1.3071

1.2591

.20576

.79424

35

26

. 0784

. 9216

.6152

. 6548

.3064

.2593

. 0594

. 9406

34

27

. 0807

. 9193

.6445

. 6594

.3056

.2596

. 0611

. 9388

33

28

. 0830

. 9170

.6439

. 6640

.3048

.2599

. 0629

. 9371

32

29

. 0853

. 9147

.6433

. 6686

.3040

.2602

. 0647

. 9353

31

30

.60876

.39124

1.6427

.76733

1.3032

1.2605

.20665

.79335

30

31

. 0899

. 9101

.6420

. 6779

.3024

.2607

. 0682

. 9318

29

32

. 0922

. 9078

.6414

. 6825

.3016

.2610

. 0700

. 9300

28

33

. 0945

. 9055

.6408

. 6871

.3009

.2613

. 0718

. 9282

27

34

. 0963

. 9031

.6402

. 6918

.3001

.2616

. 0735

. 9264

26

35

.60991

.39008

1.6396

.76964

1.2993

1.2619

.20753

.79247

25

36

. 1014

. 8985

.6389

. 7010

.2985

.2622

. 0771

. 9229

24

37

. 1037

. 8962

.6383

. 7057

.2977

.2624

. 0789

. 9211

23

38

. 1061

. 8939

.6377

. 7103

.2970

.2627

. 0806

. 9193

22

39

. 1084

. 8916

.6371

. 7149

.2962

.2630

. 0824

. 9176

21

40

.61107

.38893

1.6365

.77196

1.2954

1.2633

.20842

.79158

20

41

. 1130

. 8870

.6359

. 7242

.2946

.2636

. 0860

. 9140

19

42

. 1153

. 8847

.6352

. 7289

.2938

.2639

. 0878

. 9122

18

43

. 1176

.8824

.6346

. 7335

.2931

.2641

. 0895

. 9104

17

44

. 1199

. 8801

.6340

. 7382

.2923

.2644

. 0913

. 9087

16

45

.61222

.38778

1.6334

.77428

1.2915

1.2647

.20931

.79069

15

46

. 1245

. 8755

.6328

. 7475

.2907

.2650

. 0949

. 9051

14

47

. 1268

. 8732

.6322

.7521

.2900

.2653

. 0967

. 9033

13

48

. 1290

. 8709

.6316

. 7568

.2892

.2656

. 0984

. 9015

12

49

. 1314

. 8686

.6309

. 7614

.2884

.2659

. 1002

. 8998

11

50

.61337

.38663

1.6303

.77661

1.2876

1.2661

.21020

.78980

10

51

. 1360

. 8640

.6297

. 7708

.2869

.2664

. 1038

. 8962

9

52

. 1383

. 8617

.6291

. 7754

.2861

.2667

. 1056

. 8944

8

53

. 1405

. 8594

.6285

. 7801

.2853

.2670

. 1074

. 8926

7

54

. 1428

. 8571

.6279

. 7848

.2845

.2673

. 1091

. 8908

6

55

.61451

.38548

1.6273

.77895

1.2838

1.2676

.21109

.78890

5

56

. 1474

.8525

.6267

. 7941

.2830

.2679

. 1127

. 8873

4

57

. 1497

. 8503

.6261

. 7988

.2822

.2681

. 1145

. 8855

3

58

. 1520

. 8480

.6255

. 8035

.2815

.2684

. 1163

. 8837

2

59

. 1543

. 8457

.6249

. 8082

.2807

.2687

. 1181

. 8819

1

60

. 1566

. 8434

.6243

. 8128

.2799

.2690

. 1199

. 8801

M.

Cosine.

Vrs. flin.

Secant.

Cotang.

TanK.

Cosec'nt

Vrs. COS.

Sine.

M.

360

NATURAL FUXCTTONS.

Table 3.

38'

Natural Trigonometrical Functions.

141°

M.

Sine.

Vrs. COS.

Cosec'nt

Tang.

Cotang.

Secant.

Vrs. ein.

Cosine.

M.

.61566

.38434

1.6243

.78128

1.2799

1.2690

.21199

.78801

60

1

. 1589

. 8411

.6237

. 8175

.2792

.2693

. 1217

. 8783

59

2

. 1612

. 8388

.6231

. 8222

.2784

.2696

. 1235

. 8765

58

3

. 1635

. 8365

.6224

. 8269

.2776

.2699

. 1253

. 8747

67

4

. 1658

. 8342

.6218

.8316

.2769

.2702

. 1271

. 8729

56

5

.61681

.38319

1.6212

.78363

1.2761

1.2705

.21288

.78711

55

6

. 1703

. 8296

.6206

. 8410

.2753

.2707

. 1306

. 8693

54

7

. 1726

. 8273

.6200

. 8457

.2746

.2710

. 1324

. 8675

63

8

. 1749

. 8251

.6194

. 8504

.2738

.2713

. 1342

. 8657

52

9

. 1772

. 8228

.6188

. 8561

.2730

.2716

. 1360

. 8640

61

10

.61795

.38205

1.6182

.78598

1.2723

1.2719

.21378

.78622

50

11

. 1818

. 8182

.6176

. 8645

.2715

.2722

. 1396

. 8604

49

12

. 1841

. 8159

.6170

. 8692

.2708

.2726

. 1414

. 8586

48

13

. 1864

. 8136

.6164

. 8739

.2700

.2728

. 1432

. 8568

47

H

. 1886

. 8113

.6159

. 8786

.2692

.2731

. 1450

. 8550

46

15

.61909

.38091

1.6153

.78834

1.2685

1.2734

.21468

.78532

45

16

. 1932

. 8068

.6147

. 8881

.2677

.2737

. 1486

. 8514

44

17

. 1955

. 8045

.6141

. 8928

.2670

.2739

. 1504

. 8496

43

18

. 1978

. 8022

.6135

. 8975

.2662

.2742

. 1622

. 8478

42

19

. 2001

. 7999

.6129

. 9022

.2655

.2745

. 1540

. 8460

41

20

.62023

.37976

1.6123

.79070

1.2647

1.2748

.21558

.78441

40

21

. 2046

. 7954

.6117

. 9117

.2639

.2751

. 1576

. 8423

39

22

. 2069

. 7931

.6111

. 9164

.2632

.2754

. 1694

. 8405

38

23

. 2092

. 7908

.6105

. 9212

.2624

.2757

. 1612

. 8387

37

24

. 2115

. 7885

.6099

. 9259

.2617

.2760

. 1631

. 8369

86

25

.62137

.37862

1.6093

.79306

1.2609

1.2763

.21649

.78351

35

26

. 2160

. 7840

.6087

. 9354

.2602

.2766

. 1667

. 8333

34

27

. 2183

. 7817

.6081

. 9401

.2594

.2769

. 1685

. 8315

33

28

. 2206

. 7794

.6077

. 9449

.2587

.2772

. 1703

. 8297

32

29

. 2229

. 7771

.6070

. 9496

.2579

.2776

. 1721

. 8279

31

30

.62251

.37748

1.6064

.79543

1.2572

1.2778

.21739

.78261

30

31

. 2274

. 7726

.6058

. 9591

.2564

.2781

. 1767

. 8243

29

32

. 2297

. 7703

.6052

. 9639

.2557

.2784

. 1775

. 8224

28

33

. 2320

. 7680

.6046

. 9686

.2549

.2787

. 1793

. 8206

27

34

. 2312

. 7657

.6040

. 9734

.2542

.2790

. 1812

. 8188

26

35

.62365

.37635

1.6034

.79781

1.2534

1.2793

.21830

.78170

25

36

. 2388

. 7612

.0029

. 9829

.2527

.2795

. 1848

. 8152

24

37

. 2411

. 7589

.6023

. 9876

.2519

.2798

. 1866

. 8134

23

38

. 2433

. 7566

.6017

. 9924

.2612

.2801

. 1884

. 8116

22

39

. 2456

. 7544

.6011

. 9972

.2604

.2804

. 1902

. 8097

21

40

.62479

.37521

1.6005

.80020

1.2497

1.2807

.21921

.78079

20

41

. 2501

. 7498

.6000

. 0067

.2489

.2810

. 1939

. 8061

19

42

. 2524

. 7476

.5994

. 0115

.2482

.```