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Full text of "Surveying manual; a manual of field and office methods for the use of students in surveying"

BOUGHT WITH THE INCOME 
FROM THE 

SAGE ENDOWMENT FUND 

THE GIFT OF 

1891 



1.3.p.5L..i(>.i i).nj.i>. 




Cornell University 
Library 



The original of this book is in 
the Cornell University Library. 

There are no known copyright restrictions in 
the United States on the use of the text. 



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



TABLE OF CONTENTS. 



-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 



X TABLE OF CONTENTS. 

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 



TABLE OF CONTENTS. xi 

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 

Grade Lines 69 

Cross-sectioning 69 

Running Lines 70 

Practical Hints 70 

Adjustment of Wye Level 72 

Adjustment of Dumpy Level 75 

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 



xii TABLE OF CONTENTS. 

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 

Grade Lines 101 

Adjustment of Transit 102 

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 Stadia 139 

The Plane Table 143 

The Sextant 146 



TABLE OF CONTENTS. xiii 

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 

CHAPTER VIII.— RAILROAD SURVEYING. 

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 



xiv TABLE OF CONTENTS. 

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. 

Sunshade. — ^Always attach the sunshade regardless of 
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- 
proximately level. Give the tripod legs additional spread 
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 
expressly for adjustment, do not disturb the adjusting 
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 



/• 










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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; 
in headings, " upright," about 1 : 25 to left.) 

(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. 

(e) To be made freehand. 

( 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 
graduation. 




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 
made of spring steel wire. 

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 
steps ahead, and reads the fraction at the pin, noting the 
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 
reading fractions. It is best to read the fraction first 'by 
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 
refined work, read by scale. 

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 
radius is commonly used, although good results may be had 
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 




Railroad Juncf-fon • 



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 
swung slowly about the head. 

(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 
following heads : 

(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- 


Sec- 


FhptrSec 


Z 


137-4 






Ag. 


inst " 


Veryshvi 


714-6 




(s) 




(B) 


3 


159-0 






ft 


't 


tt >r 


717-8 


Zll-70 


t-fS 


If7 


Z-^O 


4 


137-1 








n » 


Slow 


16S-0 




(k) 




(h) 


S 


131-0 






n 


f* 


n 


167-S 


le7-7S 


7-3i 


111 


3-60 


( 


139-0 








It Pf 


mara/ 


139-4 




ra 




re) 


7 


137-3 






tt 


ti 


" 


137-S 


13S-4S 


Z-S9 


71 


S-63 


! 


J39-0 








M n 


3'Focf 


133-3 




I'd) 




0J 


9 


J3S-0 






f 


ff 


T, 


1S3-6 


133-4! 


3-00 


77 


S-ZO 


10 

S-FictI 
2 


139-3 


I3S-T( 


Z-S9 


With tl 
A,, 


'nsf- " 


Fast 


174-7 
lZS-3 


IZS-OO 


fe) 
3-ZO 


SS 


(e) 

e-90 


(jl-e) 

J3J 




1 


I3Z-6 


o:* 




























} 


J 33-0 






}* 






















d 




^ 


■-t 
e 


4 

S 
6 
7 
t 
9 
K 


I3i-^ 
134-0 
133-3 
13Z-0 
133-0 
133-3 
132->(S 






W 


ft tr 
















b 




- 




\ 
































m J. 


S^ 


«j 


^ 


a 












































/ 


























t 




























M) 


1 Z $ 4 5 e 7 1 


II 


133-0 


133-02 


3-01 


" 


n 


Speed oF Wal 


ing, Ft- per Sec>;S* | 




(lO-l) 




























y 



r 


DiSTA 


ICES 


BY ?A 


CINS 








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 












i--# 


/ 






134-0 






1 \ 


/ 


"^ 


11 


z 






1340 


134-0 


37S 


1 


\ 




e-9 


1 






Z17-0 






1 


\ / 




n 


2 






Zlg-S 


Z17S 


eio 


1 


\/ 






/ 
z 












M/scot/nf- 1 










mo 










tt 


3 






llS-5 


Ug-Z 


331 


1 


/ \ 




HI 


1 






Z7a-o 






1 / 
/ / 




S 


» 


Z 






ZtO-E 


Z7S-g 


7S5 


^■^y 


1-9 


t 






Z09-0 






/ / 


""'^ / 


If 


Z 






ZIO-0 


Z09-S 


saz 


1 / ^' 


/ 


s-s 


1 






zse-0 








/--^' 


/ 


If 


z 






ZMS 


Z87-Z 


g04- 


9 




/ 


s-i 


1 






140-0 










/ " 


tt 


z 






141-0 


140-S 


393 






/ / 


k 


t-l 


1 






ies-0 








/ 






tt 


z 






me 


117- g 


sze 




/ 


W- 


-E 
















/ 


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 
head chainman; head chainman, with spring balance at- 
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 



^ 














NeadCAain ~J?-J?oe 






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 


^NS3iZ, LacicerSS 






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 


ic/r was graduated 






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 
only, usin0 SO- ft- radius' 

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 




Head Chainma 


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 










tape by cAof 1 metJiodflO^ft-j-adlc/s, 














and cheeked 


lymeasurj'n^ snffle be- 


Tape 


99-SgO 










fweenASaadi 


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 
checked results. Follow the form. 



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. 
Follow the form. 



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 
the arc, chords ^rad-xsin-P 

'chord . so ,a-F<, - 



rad-' __ ^ 

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 

Had- 
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 
gradually decreased In length, causing 
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 
obtain duplicate needle readings on each sighting. Kead 
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. 

ADJUSTMENTS AND TESTS. 

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- 
toread zero • 
on n7er/i//an sf" 
•f and read 
fo Sm/niftes 
'oj?e-/i3/F decree) f 
and 
Observed 
m/nt/te • 

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



^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, 
reading glass. 

(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 
minutes. Follow the form. 

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 

hettveen read/. 

(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 


Adjusted 


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-£ 


Made needJeread .. 


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 


Read bean'nffs tvi 


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 


read H- End oF 


Heedls- 


£ 


E-P 


iZZ^S'i 


S7'S0' 












£-A 


IfSO'jf^ 




mWe 


241.6 


N A 




S4S'PS' 














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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 
additions as a check. (7) Transcribe field notes and ad- 
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 



Adjusted 
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- 



Adjusted 

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 



Adjusted 
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) 

e© 

(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 is read from the micrometer head for a given 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) 
aclf -reading or speaking rods which are read directly by 
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 


s.in 


100.00 mi 
t4U B.i. 


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


Ad) 


^■442 


lOUl 






HO 


10447 /f/ 
-1.16 F.5. 




ei 






-1.16 


103.26 


340 


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


Peq, KEcanJ. 6reen'3 lot. 


m) 


t4V 


ior.B 






m 














\ 


-3.551:5. 




01 






-}.U 


104.01 


1300 


104.08 ez 

fi.9l B.5. 


5i'c/em/k,F. qatepost.J.Doe. 


m) 


■fISI 


lOSJ} 






ml 


105.93 H3 
-2.47) F.5. 




mi 






-Z.40 


I03JS 


300 


103.59 mi 


n.W.bolt, (nickeJ)H3terplaa 




+10.70 




■ -7.11 


100.00 


940 




S.E. car. Hiqh and East Sts . 
(Bal3ifcedB.5.andF.5 0/st. 




-7.11 


<^'J-- ► 


t3.!9 


\\ 






+ -i.i3 


-«-'";' 




m 




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. 

Grade Lines. — The establishment of grade lines is usu- 
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. 

Adjustments. — Keep the instrument in good adjustment 
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- 
ting before reading the rod. Read the rod at least twice, 
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.) 

ADJUSTMENT OE THE WYE LEVEL. 

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 
the purpose of adjustment there should be added to these : 
(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- 



ADJUSTMENT OF WYE LEVEL. 73 

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- 
hair ring. Gradually perfect the adjustment until the in- 
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- 
trated in (a). Pig. 13. The wye adjustment should follow 
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. 



ADJUSTMENT OF DUMPY LEVEL. 75 

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



ADJUSTMENT OP THE DUMPY LEVEL. 

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 
establishing an adjusting base. The adjustments of a good 
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. 



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» opposite eatliollc Cliureli,Reyni7ds- 


»S 


SIS 


t99-39 


i-40 


694^ZI 


m. 


m 


fi 


99 


4S4 


197-ie 


(-07 


(93-3Z 


300, 


300 


" e side tr'k, iff men M- poles ZZ83-9e- 


ajff 


411 


e9S-iz 


4-SS 


(93-31 


300. 


300 


it 


»// 


48Z 


69S-86 


4-08 


694-04 


300, 


TOO 


neiidstr'k,l,etivMn fefipole3ZZ56-57 


BIfZ. 






0-7& 


((95-10. 


\ 


po. 


3-tf-onoBk,MlpoleZZ9S, SO' e- track- 
Peg 3t tel.'pole Z30ff. 


BIZ 


410 


(91-00 


4-96 


693-90 


300. 


300 


ai3 


3-6S 


69113 


B-SZ 


69Z48 


3(0. 


300 


It ri 17 ft Z305- 


el4- 


Z-71 


69Z-9Z 


S-9Z 


690-ZI 


ilO, 


xo 


11 


eJS 


3ZI 


690-03 


6-10 


686-az 


360 


3(0 


11 11 II /> Z313,e-side track- 


BM-3 






Q-IJ) 


(6l(-Si. 


\ 


'l5i 


H£-car.parapet ivall, Bridge //s ^7-4- 


»I6 


4(S 


(90-13 


3-lS 


6S6-IS 


w 


W 


'^'nonJ^-K-B-7l-{'f^,-lfi-^^P''\ ^^^^ 
Peg \iO'e-trk;a—figg-IZ 


8/r-4- 
t/7 


4SI 


691-21 


&-Z9) 
4-13 


(eis-5i. 
(se-zo 


V 
360. 


3(0 


til 


file 


(91-37 


S-00 


eie-zi 


300, 


3(0 


II 


BI9 






S-13 


6lt-Z4 




300 


n at tell pole Z33a- 

J 




ss-7e 


89-34 
85-76 


689-8Z 


=B-M-I 




.-3-58 
'Check 


^ 


-3-S8- 




_ 



80 



THE LEVEL. 



5t3 

Ml 

m 

■tfP-S 
SMZi 

■flM 
■H41 

m-B 

Mi 
tTO 
fi4 
■tS4( 




Survey for a Drain from 

Description 
Bed eF Btreafl? • 
SuitsUs eufUf-furdrsln, /f'fV- af 

tV-face of sUne 3rch lirfdge- 
Break of N-bankj Boneyard Branch' 
5- edge oF 7' drive- 
Crosses drain from Con^erv^i-oty- 
5 team pipe iine fo Conseri^^hiry- 
Cenfar J^drive fo ConsarvsfOry- 

}kaih of main track, U-^C-Sf-fiy- 
(Cafoff f/irouff/i University frOt/ntfe^ 
1 Rslis ofsidefrackf Urt^ans and 

Cframpaign Fiectric Ry- 
firainis 2 L-Cf) of sfake- 
liims 5- in iV-parJeinfff Bcrrr/iiAve- 
'iZ'Asfi tree 6' to JK- 

Cement watk ^-s/de Svrriil Avff- 

Crosses tf/Iiiary tiaii Steam Pipe Line- 
/point 3'W-SZ'3-of//WCor- of^ 
frt^lad- Line ri/ns firence W~ 
parallel to S 9'S-of S-Jine oP 
Spring field Ave- StaJcas are set 
Z'fo rig/it c/^ pirofosed frvttch 
for drain J wifii ieveiirfff pe^s 
flirsij with ground Beside sfakes- 



Ensineerins Laboratory- 

fiead Cfiainman, J-Doe^ 

Rear C//ainman, ^-Moe - 

Apr 2e, im. (Z Hours) CJoi/dy S Cod/- 

idCft- Stee/Tape, HiZ7S, ioclcerJf^3S- 




<4 




Steam Pipe 



StRyCvtoff 



Springfieid A ve 



r 












leveier, R-Roe- Rodman, J-Poe- \ 


Ll 


VEL 1 


lOTES 


FOR 


A Dl 


;Am 


FROM ENSINEERINS LABORATORY- 


5t8. 


B-5- 


H-I- 


F-5- 


Elev- 

loo-ao 


Srade 


Cut 


Oct 23, '14^ CZItrsj Clear and Cool- 
S-Endslvnesiif, IV-door, Fn^- Lab- 


;r 


f-i-Z3 


ICi-Zi 










Station stakes-are Z'R- proposed trench- 









iOS 


9S-IS 


9400 


418 


Peg drii/en flush mth gremd Besidesfake. 


t^O 






iZZ 


0S-CI 


93-20 


481 


n 


tS4i 






3-Z 


SSC 


93-13 


4-9 


Oroiiitd, 6'Sfeami'ipe,10'cBsmg,Jop 2!6 deep- 


t$i 






3-Zi 


SSCH 


9Z-94 


S-16 


Cement ivalkf P-side Burr/IiAve- 


1 






3-38 


97-SS 


92-40 


S-4S 


Peg- 


tZi-B 






}-7Z 


nsi 


9206 


S4S 


Oralntums S-in IV- parking Surrii/Ave- 


tso 






41S 


n-ss 


91-eo 


498 


Peg 


z 






5-71 


9SSZ 


90SO 


4-72 




tSO 






6-3i 


94-92 


9000 


492 


}* 


etss 






-s-ss 


9S-3S 


S9Se 


552 


{Turnrngfei/rj ff-Rai/, Main Track, l/-d£C-/?y. 


w 


tO-13 


95-51 












i 






ZHg 


9343 


8920 


423 


Peg 


tss 






3-fB 


nci0 


ISM 


see 


)7 Fauth Wye lei/e/- 


4 






4-3-I- 


9117 


S7-60 


3S7 


?' Phils- Rod, Ikr-ZO- 


fSt 






s-so 


9001 


lllO 


3-21 


f» 


5 






SSZ 


3999 


» seoo 


3-99 


37 


t4Zi 






»■! 


19-4- 


SS91 


3-5 


Sr<!u/7d,4''SAram.^pff,S''^Sff7a,fop2'/ff cfeep- 


HO 






ez6 


19-ZS 


SS-90 


3-3S 


Peg- 


fMS 






$-f 


g9-l 


SSH 


32 


6rot/nd,4''v/MfJed dram, Tap3-4deep* 


i 






711 


sua 


S580 


z-es 


Peg. 


fZ3 






.7-0 


lis 


15-7S 


2-7 


ffrvak 0p/f Aank, Boney^nd Branch' 


tu 






lO-S 


14 6 






BedoPSfreamj tveivr /^"deep* 

J 



PEOBLEMS. 81 

PROBLEM C4. EAILEOAD PEOFILE LEVELING. 

(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. 



^ 


(Profi 


.E LE 


fEL N 


)TES, 


Srovjni 


Elevatioms to 0-1 Foot-) 


s- 


+ 


T^i 


— 


R- 


E- 




20$ 




118-33 




S-O 


713-3 


In Brown Sf- f Unimproved) 


no 








4-7 


713-6 


fi It 


?//„ 


6-73 


9 
723-ri 


1-23 


3-9 


714-4 


11 ff 

Water Plu0, If- bolt, ll-W- Cor-, Smwn-Ci/rfis ■ 


8-M-Z6 


(717:01 


ZIZ 








7-« 


716-3 


Ground, Brown Sf. 


?I3 








i-4 


717-S 


n " 


214- 








S-9 


7IS-0 


.. 


HO 








4-0 


719-7 


n « 


ZIB 








6-1 


717-S 


fi " 


Z/6 








BO 


7IS-9 


ft t1 


Z/7 








t-S 


7IS-4 


„ •> 


718 








10-3 


713-6 


ft » 


?I9. 


9-22 


6 
721-64 


JI-4S 


12-2 


711-7 


In Corn Field 
Stake, St3-219- 
Corn Fj'e/d- 


eftakt 


(712-41 


2?0 


713-0 


221 








■4-4 


717-2 


'• 


222 








2-7 


718-9 


M 


223 








2-9 


711-7 


t« 


224 








2-3 


719-3 


t1 


22S 








3-4 


718-2 


Timber Pasture- 


t-m 








12-4 


709-2 


Sully 


226 








ll-Z 


710-4 




r3S 








6-0 


71S-6 


Break eF bank, Plum Rji/er- 


g-M-27 


2-04 


7/3-S2 
7IS-33 


10-/6 

-tl-SS 
1-18-05 


6-0 
Check 


{711-41 


B-M-,nol-,U'elm,72'e., Sta-?2e-t6S- 

,„, Cs,- station- - = fDre S/s/tt- 

K, J" . <■ ■ 6ack Sielit R- gtd (Umptilalw) 


i-SO 


707-S 




+ia.os 


-4-S, 


^ 


.-4-B, 



Pron/e of Locat/on Line. 

A.B.&.C.RR. 




-Blue. Water level; notes relative to same. 



^lack. 5ui^race line, station nurr?erals, etc.- 






210 ' ' 



SI-.. 






Z£0 



'^roocx'F^z^d- 



19c. 



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- 
ting grade stakes.) 



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 
dividing the grade angle by the assigned or adopted change 
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? 


rCorrfclitms 


By Chord Grad 


ients. 


Sbabion. 


oF Grade 


Tanqenb 


Curve 


Chord 6ra 


dienfcs. 


Curve 




Tanqent . 


Correction. 


Elevation. 


DifF. 


Oradient. 


Elevation. 




Fb. 


Fb. 


Fb. 


Percent. 


Percent. 


Fb. 


84 


108.00 








f-I.OO) 




d5(P.Q 


107.00 


tO.OO 


107.00 


+0.10 


-0.90 


107.00 


86 


106.00 


i-O.IO 


106.10 


+0.20 


-0.70 


106.10 


87 


10^.00 


f0.40 


105.40 


fO.ZO 


-0.50 


105.40 


88 


m.OB 


+0.90 


104.90 


+0.20 


-O.iO 


104.90 


89 


JOi.OO 


+ 1.60 


104.60 


+0.20 


-0.10 


m.60 


90(Apex) 


102.00 


+i.50 


104.50 


+0.20 


+010 


104.50 


91 


103.00 


+ 1.60 


104.60 


+0.20 


+0.30 


104.60 


9Z 


I04.OO 


+ 0.90 


104.90 


+0.20 


+0.50 


104.90 


di 


105.00 


+ 0.40 


105.40 


+0.20 


+0.70 


105.40 


94 


MOO 


+ 0.10 


106.10 


+ 0.20 


■ +0.90 


106.10 


95(P.T.) 
96 


107.00 
708.00 


tP.oo 


107,00 


+0.10 


l+I.OO) 


107.00 


+ 2:00=A 



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 
railroad, canal, etc., as assigned. 

(c) Methods. — (Follow the methods described in Chap- 
ter VTII, "Eailroad Surverying," under the head of " Cross- 
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. 



r 

Leve 

Shtlen 



4-K>t 

*76 

3 
7! 

t 

Z 
/ 

f 

7! 

7! 



7! 



.5 FDt 
B-S- 



JforfA / raperfy 

Center 

South 



Line, 
iYoper^ 



Hcrfh 
3-Ii 



Ml 
3-Qi 



PROF|iLE AMp QUAHTlTIEf 
T-P- 



H-l- 



' fy. 



PfldofB 
7li-0S 



line J / saly Si 
/{eely 
Line, 
'dge 



South Ciil sF Br, 



t/trfh a operty Wne, S/ ren 
OSS 



7/e-u 

7Z0-/1 
7?}-<l3 



F-5- 



/. 



Hesty 
BfttBy. rdCnek 



7-IS 



4-Zf 
3-S* 



St. 

7IS-36 



B-M- 



FOR Pavement on Wrisht St- 

Leveigr,J'Poe. ^odman.R-Jloe. Ctaiiimen,S'l'-Keen 

LEVELS ON THE anif SHi-Smfl. 
L-Prop L-S-Wlll|L-6utt«r C«nterR-6uHer R-5WlWR-Prop. 
L- 40 Ft- L-37 Ft- L-IO Ft- R-IOFt- R-37Ft- MJFt. 



70S-9S 
Saffeys. r^ Creek 



ISM. 



4-7 

701-0 



1QS:4 



707-1 
S-0 



7II-I 
71}-} 



701-0 



77 



'i 



TSIO 



37 
701-4 



74 

711-0 



7/4-0 



IT 



2013 
4T 

707-1 



m 



70S-6 
3S 



ss 

70/-1 
1-6 
7/0-1 



IT 



2!li. 



im. 



3-S 
70S-6 



3-S 

70!-g 



3-3 



70S.S 



7//-3 



'7T 



w 






s-o 

707S 



706-i 
'7PS 

7or-e 
TT 
71/-/ 

713-3 
IT 



mi 



If? 

TSSi 



^ 



70S-I 



7-0 



706-1 



7//-9 



7/4-9 



'W 






206:2- 



7T 



7oe-3 



70t-2 

S-6 

7/1-3 



w 



Hay 7,1914. (3 hours) Wsrm and Wwdy- 
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. 




Leve 


A FC 


e. Coi 


iTOUR 


h OH 


Proposed Park Site- 


s- 


+ 


71 


- 


%■ 


E- 


Oct ie;i4- Clear, mrm.B-SB-Wye Level, puis. llo<7- 


B-M- 


6-61 


l06-e7 






100-00 


BwUer,Sfs-AH-45, 46'Ji- o/> kmll. 


AO 








II* 


}S-i 


Srovnd' (All leve/s From si'/y/e- settih^-) 


AJ 








14 


S$i 




AH-4S 
Alt4S 


C46'») 






e-9 


SS-4 


A 


1 2 3 4 fSO 




A 


Ai 








9-7 


}!-0 


%, 


, /I'velK 




Ai 

Ai'SO 


K!<t}e3 






S-6 


lOI-l 


B 


/ \ i / 


B 






3-7 




^h/ "^l V 3/ A k -- 


A* 

A4U0 
BO 








2-4 

OS 
lO-S 


104-3 
m-2 
it-2 


C 


C 




gOtJB 

SI 

BlUI 


/Oi^tl 
fullyj 






1-3 
1-9 
l-l 


i7-i 
97-S 
31-6 


ff 


'Ar/^.-^-w^-E 


1 


/'A/ 2 3 4 *S0 


B2 


Sully Z 






9-7 


97-0 




B2*U 


Wdiiei 






l-Z 


91-S 




B2m 


Suliyi 






t-l 


97-9 




Bi 








l-O 


91-7 


}tt stakes only sf Sfs-O end -ftSO 


B* 








S-3 


101-4 


on each li'ne For future rvFsr- 


f4m 








3-1 


W2-9 


ance- Ujfd c/)3i'n/n0 pins For 


CO 








y/-7 


9S-0 


inttr/nodiate foinfs- 


COtU 


Kidjtl 






11-2 


9B-! 




COfIS 


SuUyl 






ll-B 


94-S 




CI 


(Cant. 


'nvrd on 


Mlcw, 


11-3 


OS'* 





J5J Line A J8.3 




M.2 



1019 



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. 
Follow the form. 




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. 



r 












JPoeiK-Koe, Obs, 


rt/ers- \ 




Deli 


:acy 


OF Bu 


BBLE ■ 


IIAL, 


B-SB- Wye Lev 


:L- 1 


IsJ.- 
No. 


Mathod 


, With 
Lsvel 


Stale* 


ester. 


■enccs 


2nd Method, 


With 


Telescope. 




Microm- 


Ciffe 


Length 




1.. 






Rearfm^ 


A End 


BEnd 


A End 


BEnd- 


Bubble 





' 


U_fir},t 
J. mot-ement 


/ 


7 


9-! 


si-e 








ipr^- 




i 


17 


J4-2 


47-2 


4-4 


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4-3 


4-S 


61 -Z 


11 / 






4 


i7 


ti-0 


31-S 


4-3 


4-Z 


61-B 




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S 


47 


27-0 


34-4 


4-0 


4-1 


$1-4 






e 


S7 


31-5 


30-0 


43 


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6I-B 


1 \ .' 




7 


67 


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4-3 


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at 100 ft- (D) 


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21-1 


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47 


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For Bubble Movement 


5 


4-7 


76-7 


341 


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4-4 


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■f 


37 


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(*Lsv 
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= in Ft. 


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J 



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- 
lowing adjustments (but do not disturb the adjusting 
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 
adjustments. 



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) 
the adjustment. 



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 



Raadinq 

" 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. 



Keadinq 



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. 
Found radius oFcur 

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 
for suggested graduations.) 

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 
tripod head. The alidade includes the telescope, magnetic 
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 
read fractional parts of the main graduated scale or limb. 
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. 
Add the reading of the vernier to the reading of the limb. 

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 
the eye. Notice adjacent and end graduations. 

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 
with the A vernier reading 180°, reversing the alidade in 
azimuth and locating stations ahead with the A vernier 
reading zero. A third method employs two points ahead 
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 
around the point are read to the left. The readings are 
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 
to stadia and railroad work. Azimuths can be easily 
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. 

Grade Lines. — Grade lines may be established with the 
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. 

ADJUSTJIENTS OF THE TRANSIT. 

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. 



ADJUSTMENTS OF THE TRANSIT. 



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- 
justment is made approximately and the hair is made ver- 
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 
collimation adjustment is then made precisely. 

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 



ADJUSTMENTS OF THE TRANSIT. 



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 
vernier of the vertical circle should be made to read zero 
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. 
Follow the prescribed form. 



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- 
was made by siglitfng 
'ower motion^ the 
xero} then sf^ht/n^ 
i^per matio/j, and 



fvas mads by 
^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 
AB:AD::BC:DF 



(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 
A 


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 


\ 


C 


im'si'r 


A 


W'JJ'R 


(Check, 


) 


<i 


\ 


C-D 


tiesls'i^ 










^^-^ 


\ 


D 


7f'l>}'ji 










^\ 


. \ 


D-e- 


M'4}'e 












>l. 


V 




















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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 


Adjostci) 


Eefced 


COMPl 


TATlOt 


OF L/ 


fit 


UDE3 


COWh 


TATIOt 


OF DE 


PAR 


rUEES 




Bearmg 


Distance 


Multipli- 


Lo^ar- 


Computed 


Lat. 


Adjusted 


Multipli- 


Logar- 
ithms 


Computed 


Dep. 


Adjusted 








cation 


itlims 


Latitude 


Cor. 


Latitude 


cation 


)eparfure 


Cor. 


)eparture 






Ft. 






Ft. 


Ft- 


Ft- 






Ft. 


Ft. 


Ft. 


AB 


S-OCV 


33S-05 






S-33S-Ci 


■10 


5-334^93 






OMO 


-00^ 


00-00 


SC 


5-J0'l3'E 


464-SB 


"'Vv 


9-9330i 


5-457-1} 


■To 


5-437-31 


t«|?. 


Z-i(744 
9-Z4m 


B-iZ-fl 


-f3 


l-SZ-SO 


HM 


i-eeoso 


I-9I63Z 








je 


(4S7-6J) 




■i- 




:ikk 


(SZ-47) 








*B7-iiie 


CD 


/t-6sisi/ 


413.7Z 


^S 


US459 
9-eiill 


H-20I-Z4 


■OS 


H-Z0I-Z9 


4eB-7Z 


Z-6S4SS 
S-9SS73 


W43}-n 


■18 


m39-l9 








ZSOZ^ 


1-30370 
(70I-Z4 




■f- 




'II 


Z-64330 
(439-87) 




^ 




zoisW 


f-3S-S7S 


DE 


H-I0'43B 


ew-si 


■",m 


Z-7199S 
9-99Z}l 


17-61378 


■IS 


H-60S-91 


lil'/f. 


Z-7S99S 
S-Z6940 


f-m-64 


■OS 


E-II46S 


7-78731 


Z-OSSiS 








■'Sf 


(6BS-71) 




+ 




TT?:i-fir 


(114-64) 




■I- 




61IS-777 


EA 


5-!6'3li. 


Z4Z-S4 


'/5 


7-3SS3Z 
8-78361 


5- 1476 


■00 


5- 14-76 


%'M 


Z-38S3Z 
9-9S9Z0 


BZ4Z-40 


-10 


£Z4Z-S0 


H6S93 
(J4-76) 


Z-384SZ 




F. 




/f-7se 








^^Iff 


(Z4Z-40) 








2143-n 


H-S07-OZ 


H-807-ZZ 


5439-31 


1-430,6} 


Error 
Actu 
Perm' 


op Clos 
.1 


ure 
55 ft. 
6 Ft. 


Line 
AB 

ec 

CO 


2 
2 
1 


Cor 

-/O 

-/a 

■OS 


5-107-K 


■40 


S-807-2Z 


Line 
AS 
BC 
CO 



I 
10 


Cor. 
-00 
•03 
-11 


W-43M1 


■36 


W-43M3 


5- 0-40 


0-00 


W- 0-36 


0-00 








(St 


Pisgrs 


«; 


PS 
£A 


} 



■IS 


■f- 


<S 


PS 
tA 


4 


'OS 


■^I'-on J 


\. 




T 


-*0 


,^__ 




71 


3r 


__ 




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 
tower and read the vertical angle, as before. (7) Eead the 
level rod on the base of the tower as before. Each angle 
and rod reading is to be based on duplicate readings. Fol- 
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) . 


'ooI^^t/U 


/•• 








AS 


m'47h> 


'47'47'2II 


'47'ZI!" 


47'47'ZD" 






Si'ng/e 










ss'si'zt 


za'si'n 


' S6'20' 


ISI'SI'Z^' 


47'47'I6" 




SXaps- 




lip 


UF) 


AS 


fWdn' 


mVis 


' OO'M" 
















AS 


*7'4T6l 


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' 47'0(l' 


47'47'm" 






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l3Si(ifi 


SS'JlW 


■ S^'4<)" 


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' 


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43°ZZ'Z4" 




S£<ips- 




u 


L 


AS 


m'M'Ki 


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'4}1f2l' 


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it'siki 


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' Sl'40' 


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43'ZZ'Z^'' 


43'ZZ'ZZ" 


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D 


R 


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i^'tsii!'' 


rn'mit 


me^ 
















Ae 


n'mii' 


m'siit 


' so'zs' 


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Single 










l4'lZ'0 


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444'JZ'4(>'' 


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u 


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t'ltio' 


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AS 


m'si'a 


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' Sl'Zl' 


s/'so'zo" 






Single 










Z64'/l'Z<l 


'M'li'ze 


' Il'ZH' 


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7 exceecfIS 


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ISP'W'IO" 


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) 



(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 
exceed 15". Follow the form. 



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 
RslJroad Time i> a z''jS-o" 

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 
Standard, or railroad time, by adding or subtracting 4 min- 
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 




(add 


1.6 up to Mar 


chl 


1922 


add 3.1 




1916-^ subtract 


2.3 on and aft 


er 


1923 


add 4.5 




1 


March 1 




1924 


/add 5.9 up to 
\ add 2.0 on and 


March 1 


1917 subtract 


0.7 




after March 1 


191S add 


0.9 




1925 


add 3.3 




1919 add 


2.5 




1926 


add 4.6 




(add 


4.0 up to Mar 


chl 


1927 


add 5.9 




1920 -(subtract 


0.1 on and aft 


er 


1928 


f add 7.2 up to 
1 add 3.3 on am 


Warch 1 


1 


March 1 




after March 1 


1921 add 


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 


1915, add 0.34 m 


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" 
road or Standard T/mCj axe- 

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 
1915," published by the Department of Interior, General 
Land Office. Tables I, II and IV were compiled from " Prin- 
cipal Pacts of the Earth's Magnetism," published by the U. 
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 
may be obtained from standard or railroad time by adding 
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^ 
watch settolceep JSaJiroad Time, Se/ar 
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-?) 
Pead azimuth -Pepeated until lOvaiues 
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 


Horizontal Circle Readings 


Vertical Circle 
i?cadings 


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 




which had been } 


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 
chain, tape, or stadia, is used on railroad and similar sur- 
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 
table and stadia. The transit and stadia are used by the 
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 



THE STADIA. 139 

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. 

THE STADIA. 

Description. — The stadia is a device for measuring dis- 
tances by reading an intercept on a, graduated rod. The 
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 
on a separate reticule and made adjustable. It is, how- 
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. 

Stadia Rods. — Stadia rods are always of the self reading 
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 



THE STADIA. 141 

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. 

Adjustments. — The adjustments of the plane table are : 
(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 
read the vernier. This reading is the index error which 
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 
telescope nearly level, and read the upper stadia-hair. (4) 
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. 

PROBLEM E2. STADIA REDUCTION TABLE. 

(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 



3TADl>fi 
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 
and reading i/p^ ^er hafr- 

Measured d/sfanci from cenfer af 
frans/f fo eacJ ' p/n tvifb sfeeS 
tape fa nearas/- d-07 ft- 

Wif/i objecfff/assj'ocusedo/jad/sfani' 
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- 

H-IZVC- 
H3'/ijV- 

usVe. 

H-4°0W- 

MWe- 
ti-z'zsi- 

M'ZlfW- 

i-is'is'n 

S4'0SB- 



Distance 
Ft. 



43Z 
6?t 



iZ4 
499 



7SS 



7S6 



eis 

473 



47S 
434 



Vertical 
Angle 

to'zo' 

-0'40' 

*0°3S' 
f-O'SO' 

-O'SO' 
-I'W 

tl'iz' 

-C'S6' 
i-0°S4' 

-0'S4' 

-o'zo' 



Elevation 

Ft. 
7/g-OO 

(-7-Z) 



Error 
Alhnsifle Erroh 



7I0-! 
12:31 



7IS-I 



70Z-7 

9-/0-I) 

712-8 



tlA 



7za.z 

(-Z-B) 



7/7-7 
711-0 



0-i 
S-Sft- 



' Soar t 



Observers, J^Doe il? R- 

Transit and 

Pec-/5,/m,f3fiou 
Used BofF S Bergi 

and Stadia 
Stadia Constants 
5i^/7tedat target i 

Angle. 
Oriented tiie transij 

1 



l^oe- > 

Stadia- 

<) Clear and Warm- 
Transit^ Locker /fo-lZ, 
'/lo-6- 
\tF=H7ft., k= 100-00 
itatM-I-,for Vert- 

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. 



Stadia Reduction Table I 


(c+F)= 1-20 Feet- 


k=ll5-,75 


D=kS + (c+F) = D'+Cc+F) 1 


Stadia 
Reading 


Distance 
D'=kS 


Stadia 
Reading 


Distance 
D'=k.5 


Stadia 
Reading 

»3 


[)ist^nce 
D'=k.5 


0-0/ 


l-Z 


0-1 


//■6 


hO 


IIS-S 


■OZ 


Z-i 


■Z 


25-Z 


z 


2ih5 


■03 


3-5 


•3 


34-7 


3 


347-Z 


■04 


4^6 


■4 


46^3 


4 


463-0 


■05 


5-8 


■B 


S7^3 


S 


S78-8 


•06 


£■3 


■6 


6S^4 


6 


634-5 


■07 


8-1 


■7 


Sl-0 


7 


SW-Z 


■08 


3-2 


■8 


3Z^6 


S 


9Z6-0 


■09 


I0^4 


■3 


104-Z 


3 


1041 -S 


■10 


11-6 


1-0 


1I5-8 


10 


JI57-S 



(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 
AND STADIA. 



(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 
BY RADIATION. 

(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 
radiation with the plane table. 

(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 



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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 
been made of Invar Steel.) 

(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 
AND STADIA. 

(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 
right"; (c) read vertical angle; (d) read azimuth.) (5) 
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 
station, move the instrument ahead to a new stadia station, 
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 
TABLE AND STADIA. 

(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 
either the transit and stadia or the plane table and stadia, 
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 
meridians is made whenever necessary. 



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- 
ship are made to close on the section corners already es- 
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 



n ade i 



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 
Originai surxvey was made about JSZZ' 
' 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 

of the road . ^or ma. 7/ years- About IS94- 
' 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- 
Inirr ask J2Ins. cl/sm.,/f>f3'm,J?3 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$ 



S-F-Kingsley, Head Cliainman' F-Hotigmanj Trausif/m 
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 
of road graders, or the failure to lower the corner sufB- 
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) 
and made theFoHow- 
ihg witnesses to tfie 
corner, VIZ-: Adoubfe 
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; 


I Unitad States 5urv< 


>" 


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 


H-betwimSSS 


ZS-00 


Broak leading If- thence 


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_ 



5H-5mith, Head^ainmen' L-B-Brow/7, Axman- 
/■£■ 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 
proceedings. Follow the form. 

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 
(OE ADDITION). 

(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' 

Public Road for J-D-Clark- 

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- 

"MissioH Ridse" Road. 

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 
grading both the street and adjacent property. Residences 
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. 
RAILROAD StTRVEYING. 



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 



184 KAILROAD SURVEYING. 

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. 

Transitman. — (Transit, reading glass, adjusting pin, 
transit note book, railroad curve text book, figuring pad.) 
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 
movement and note the adjacent graduations. If the tran- 
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 



186 EAILEOAD SURVEYING. 

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- 
nier must always be set to read the index-reading for that 
station; and (2) when pointing on tangent at any station, 
the vernier must be set to read the index-reading for that 
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 



r~ 

5ta. 
86 

85 

84 

83 

82 

81 o 

80 

79 ^ 

78 

77 

76 

75 

7i 

note. 

V 



( Transit Notes For 



Defl. 



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Magnetic Back Siglit h needle resct- 
inq on Mcit'anqejit pro/onqed. 



ROAD PRELIMINARY SURVEYj 
(Organization ^ „!^ of PartyJ 

(Var.^S'iZ'J S^'^^^'' 



(rar.=S°2f) %Zl 

(Van^rifj^^^'^^y 
W/reTence s=; 




(Yar=3°3/l') 

\y'(C3lc.Brq.at last ^ 
deflection point.) § 

(Calculated bearings are based on a 
true meridian.) ' 



^ 


Transit 


1 
Motes Tor 


Railroad Locatiom Survey 


Sta. 


Corves 1 Index 


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Needle. 






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70 
















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62 




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188 EAILROAD SURVEYING. 

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 
preliminary line, the head chainman hastens ahead the in- 
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 



190 RAILROAD SURVEYING. 

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- 
rection of the head chainman. 

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 
about flush and usually receive a tack ; they are properly 
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 



EAILROAD SUEVEYING. 







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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 



194 RAILEOAD SURVEYING. 

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 
grade crossings ; other railroad lines, with the same point 
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. 



r 

Sta. 
I3e 

138 

131 

136 

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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. 



200 KAILEOAD SURVEYING. 

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 
purpose.) The profile should show the grade line, grade 
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 
line on maximum gradients at each deflection point. 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- 



202 RAILROAD SURVEYING. 

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- 
tion of the station about to be cross-sectioned, reading the 
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 



Sta. 

130 
*40 

129 

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Renrar-kft 

(i-levei section //i cuf') 
(Level secf-f9n In cvf) 

(Orade point, L, Csmtl) 
{2 leveJ sect/on in Fill) 
{Levd sect/on in fill) 
li-and stringer, 8r.0J8- 
(Head of Dump) 
(Toe of Dump) 
Bridge its IS ]jtS*34 
6,J4sp3/?3 \{B7+£0 

(Head oF Dump) 
S-end sMngeri Sr-IKIS 
Ditch 2-'4'-4'T- 33'- 
(3 level section In Fill) 
(Srsde point- right) 
(Srade point center) 
(Srsde point leFt) 
(3 level section In ct/f) 
(level section in cut) 
(4 level section in cut) 

(S level section in cut) 
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Typical Cases 

L^vel Sections. 



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Grade PoinbCw/th Diaijonal Contour) 
Riqht. Center. Lefi^ 



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204 



EAILEOAD SUEVEYING. 




Cross-SectionatSlatm /E7-f-53 
headofDump 

TT — \ ? 

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M Z9I- -->^-—/9.6- 



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? 



206 RAILEOAD SUEVEYING. 

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 
the center grade point, although the side grade point stakes 
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 
about a vertical axis through a corner of the top roadbed 
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 



208 KAILEOAD SUKVEYING. 

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. 

PROBLEMS IN RAILROAD SURVEYING. 
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 
work. The adjustments will be made under the persona] 
direction of the instructor. The student should attempt to 
be speedy as well as accurate in testing and making the 
adjustments. 

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 



210 EAILKOAD SUEVEYING^ 

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 
reading to 30 seconds. 



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 
closed polygon or about a point, varies about as the square 
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 


Baden Instructions. 


/: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 
added at one time. 

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 

adapted to the addition of long columns where one is liable 
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 
etc. All products are added mentally, only the final result 
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 
made. 



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 




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/ 














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TELfTEMlL 


IK 


// 


Ml 


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^t^ 


H 


•i 


// 


WY 








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cLinU'E.^'niii 


j-kt 


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77 


7 


I' 


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r 


77 


V 


7 




^ 1 






1 




/ 




]_ 






/ 


zAaH^^TfE^T- 


T't 


t 


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777 


77 


?7 


-J-' 


T 


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J 












whTLU-EJEirELTi 


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JZZlfEET^R 


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SBLDIEEHI 


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Mr 


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P 


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K 


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1 


11 


V 






























SEEDIEEII 


IK 


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P 


Q 


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F 


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7 






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h: icdEtqin 


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T 


THT 


1 


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. 



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 
Denver,Colorado. 

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 
Denuer, Colorado. 

•(oPOGRAPH/c Map 

OFTHE 

City OF Boulder,Colorado 

Surveyed by the 

Class in Topographic Surveying 

University of Colorado 

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 
Denver, Coiorado. 

RmihitofWay Map 
New YORK& Denver KK 

St a tion 33U55 to Station 511 f 10 
Scatelin-rdOOft. January 1,1915 

Office of Chief Engineer 
Denver, Colorado. 

lojjographic Map 
ClTYOFBOULDER,COLORADO 

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. 

Pub/ic and Mam Roada 7-/ 

Private and Secondary Roads 

Trails 

Street and Public Road Crossings - // //' 

Pnvate Road Crossing // 

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 


.2813 


. 1967 


. 8043 


18 


43 


. 2547 


. 7453 


.5988 


. 0163 


.2475 


.2816 


. 1975 


. 8025 


17 


44 


. 2570 


. 7430 


.5982 


. 0211 


.2467 


.2819 


. 1993 


. 8007 


16 


45 


.62592 


.37408 


1.5976 


.80268 


1.2460 


1.2822 


.22011 


.77988 


15 


46 


. 2615 


. 7385 


.5971 


. 0306 


.2462 


.2825 


. 2030 


. 7970 


14 


47 


. 2638 


. 7362 


.5965 


. 0354 


.2445 


.2828 


. 2048 


. 7952 


13 


48 


. 2660 


. 7340 


.5969 


. 0402 


.2437 


.2831 


. 2066 


. 7934 


12 


49 


. 2683 


. 7317 


.5953 


. 0460 


.2430 


.2834 


. 2084 


. 7915 


11 


50 


.62708 


.37294 


1.5947 


.80498 


1.2423 


1.2837 


.22103 


.77897 


10 


61 


. 2728 


. 7272 


.5942 


. 0546 


.2415 


.2840 


. 2121 


. 7879 


9 


52 


. 2751 


. 7249 


.6936 


. 0594 


.2408 


.2843 


. 2139 


. 7861 


8 


53 


. 2774 


. 7226 


.6930 


. 0642 


.2400 


.2846 


. 2157 


. 7842 


7 


54 


. 2796 


.7204 


.6924 


.0690 


.2393 


.2849 


. 2176 


. 7824 


6 


55 


.62819 


.37181 


1.5919 


.80738 


1.2386 


1.2862 


.22194 


.77806 


5 


56 


. 2841 


. 7158 


.6913 


. 0786 


.2378 


.2865 


. 2212 


. 7788 


4 


57 


. 2864 


. 7136 


.6907 


. 0834 


.2371 


.2858 


. 2230 


. 7769 


3 


58 


. 2887 


. 7113 


.6901 


. 0882 


.2364 


.2861 


. 2249 


. 7751 


2 


59 


. 2909 


. 7090 


.5896 


. 0930 


.2356 


.2864 


. 2267 


. 7733 


1 


60 


. 2932 


. 7068 


.6890 


. 0978 


.2349 


.2867 


. 2285 


. 7715 





M. 


Coeine. 


Vrs. sin. 


Secant. 


Cotang. 


Tang. 


Cosec*nt 


Vrs. COS. 


Sine. 


M. 



128° 



Si" 



Table 3. 



NATURAL FUNCTIONS. 



361 



39° 


Natural Trigonometrical Functions. 


140° 


M. 


Sine. 


Vrs. COB. 


Cosec'nt 


Tang. 


Co tang. 


Secant. 


Vrs. Bin. 


Cosine. 


M. 





.62932 


.37068 


1.5890 


.80978 


1.2349 


1.2867 


.22285 


.77715 


60 


1 


. 2955 


. 7045 


.5884 


. 1026 


.2342 


.2871 


. 2304 


. 7696 


59 


2 


. 2977 


. 7023 


.5879 


. 1076 


.2334 


.2874 


. 2322 


. 7678 


68 


3 


. 3000 


. 700O 


.5873 


. 1123 


.2327 


.2877 


. 2340 


. 7660 


57 


4 


. 3022 


. 6977 


.5867 


. 1171 


.2320 


.2880 


. 2359 


. 7641 


56 


5 


.63045 


.36955 


1.5862 


.81219 


1.2312 


1.2883 


.22377 


.77623 


55 


C 


. 3067 


. 6932 


.5856 


. 1268 


.2305 


.2886 


. 2395 


. 7605 


64 


7 


. 3090 


. 6910 


.5850 


. 1316 


.2297 


.2889 


. 2414 


. 7586 


53 


8 


.3113 


. 6887 


.5845 


. 1364 


.2290 


.2892 


. 2432 


. 7568 


52 


9 


. 3135 


.6865 


.5839 


; 1413 


.2283 


.2895 


. 2450 


. 7549 


51 


10 


.63158 


.36512 


1.5833 


.81461 


1 9716 


1.2898 


.22469 


.77531 


50 


11 


. 3180 


. 6820 


.5828 


. 1509 


.2268 


.2901 


. 2487 


. 7513 


49 


12 


. 3203 


. 6797 


.5822 


. 1558 


.2261 


.2904 


. 2505 


. 7494 


48 


13 


. 3225 


. 6774 


.5816 


. 1606 


.2254 


.2907 


. 2524 


. 7476 


47 


14 


. 3248 


. 6752 


.6811 


. 1655 


.2247 


.2910 


. 2542 


. 7458 


46 


15 


.63270 


.36729 


1.5805 


.81703 


1.2239 


1.2913 


.22561 


.77439 


45 


16 


. 3293 


. 6707 


.6799 


. 1752 


.2232 


.2916 


. 2579 


. 7421 


44 


17 


. 3315 


. 6684 


.5794 


. 1800 


.2225 


.2919 


. 2597 


. 7402 


43 


18 


. 3338 


. 6662 


.5788 


. 1849 


2218 


.2922 


. 2616 


. 7384 


42 


19 


. 3360 


. 6639 


.5783 


. 1898 


.2210 


.2926 


. 2634 


. 7366 


41 


20 


.63383 


.36617 


1.5777 


.81946 


1.2203 


1.2929 


.22653 


.77347 


40 


21 


. 3405 


. 6594 


.5771 


. 1995 


.2196 


.2932 


. 2671 


. 7329 


39 


22 


. 3428 


. 6572 


.5766 


. 2043 


.2189 


.2935 


. 2690 


. 7310 


38 


23 


. 3450 


. 6549 


.5760 


. 2092 


.2181 


.2938 


. 2708 


. 7292 


37 


24 


. 3473 


. 6527 


.5755 


. 2141 


.2174 


.2941 


. 2727 


. 7273 


36 


25 


.63495 


.36504 


1.5749 


.82190 


1.2167 


1.2944 


.22745 


.77265 


36 


26 


. 3518' 


. 6482 


.5743 


. 2238 


.2160 


.2947 


. 2763 


. 7236 


34 


27 


. 3540 


. 6469 


.6738 


. 2287 


.2152 


.2950 


. 2782 


. 7218 


33 


28 


. 3563 


. 6487 


.5732 


. 2336 


.2145 


.2953 


. 2800 


. 7199 


32 


29 


. 3585 


. 6415 


.5727 


. 2385 


.2138 


.2956 


. 2819 


. 7181 


31 


30 


.63608 


.36392 


1.5721 


.82434 


1.2131 


1.2960 


.22837 


.77162 


30 


31 


. 3630 


. 6370 


.6716 


. 2482 


.2124 


.2963 


. 2856 


. 7144 


29 


32 


. 3653 


. 6347 


.5710 


. 2531 


.2117 


.2966 


. 2874 


. 7125 


28 


33 


. 3675 


. 6325 


.5705 


. 2580 


.2109 


.2969 


. 2893 


. 7107 


27 


34 


. 3697 


. 6302 


.5699 


. 2629 


.2102 


.2972 


. 2912 


. 7088 


26 


35 


.63720 


.36280 


1.5694 


.82678 


1.2096 


1.2975 


.22930 


.77070 


25 


36 


. 3742 


. 6258 


.5688 


.2727 


.2088 


.2978 


. 2949 


. 7051 


24 


37 


. 3765 


. 6235 


.5683 


. 2776 


.2081 


.2981 


. 2967 


. 7033 


23 


38 


. 3787 


. 6213 


.5677 


. 2825 


.2074 


.2985 


. 2986 


. 7014 


22 


39 


. 3810 


. 6190 


.6672 


. 2874 


.2066 


.2988 


. 3004 


. 6996 


21 


40 


.63832 


.36168 


1.5666 


.82923 


1.2059 


1.2991 


.23023 


.76977 


20 


41 


. 3854 


. 6146 


.6661 


. 2972 


.2052 


.2994 


. 3041 


. 6958 


19 


42 


. 3877 


. 6123 


.5655 


. 3022 


.2045 


.2997 


. 3060 


. 6940 


18 


43 


. 3899 


. 6101 


.5650 


. 3071 


.2038 


.3000 


. 3079 


. 6921 


17 


44 


. 3921 


. 6078 


.5644 


. 3120 


.2031 


.3003 


. 3097 


. 6903 


16 


45 


.63944 


.36056 


1.6639 


.83169 


1.2024 


1.3006 


.23116 


.76884 


15 


46 


. 3966 


. 6034 


.5633 


. 3218 


.2016 


.3010 


. 3134 


. 6865 


14 


47 


. 3989 


. 6011 


.5628 


. 3267 


.2009 


.3013 


. 3153 


. 6847 


13 


48 


. 4011 


. 5989 


.5622 


. 3317 


.2002 


.3016 


. 3172 


. 6828 


12 


49 


. 4033 


. 5967 


.6617 


. 3366 


.1995 


.3019 


. 3190 


. 6810 


11 


50 


.64056 


.35944 


1.6611 


.83415 


1.1988 


1.3022 


.23209 


.76791 


10 


51 


. 4078 


. 5922 


.5606 


. 3465 


.1981 


.3026 


. 3227 


. 6772 


9 


52 


. 4100 


. 5900 


.5600 


. 3514 


.1974 


.3029 


. 3246 


. 6754 


8 


53 


. 4123 


. 5877 


.5595 


. 3663 


.1967 


.3032 


. 3265 


. 6735 


7 


54 


. 4145 


. 5855 


.5590 


. 3613 


.1960 


.3035 


. 3283 


. 6716 


6 


55 


.64167 


.35833 


1.5584 


.83662 


1.1953 


1.3038 


.23302 


.76698 


5 


56 


. 4189 


. 6810 


.6579 


. 3712 


.1946 


.3041 


. 3321 


. 6679 


4 


57 


. 4212 


. 5788 


.6573 


. 3761 


.1939 


.3044 


. 3339 


. 6660 


3 


58 


. 4234 


. 6766 


.5568 


. 3811 


.1932 


.3048 


. 3358 


. 6642 


2 


69 


. 4256 


. 6743 


.5663 


. 3860 


.1924 


.3051 


. 3377 


. 6623 


1 


60 


. 4279 


. 5721 


.5557 


. 3910 


.1917 


.3054 


. 3395 


. 6604 





M. 


Cosine. 


Vrs. Bin. 


Secant. 


Cotang. 


Tang. 


CoBec'nt 


Vrs. cos. 


Sine. 


M. 



129° 



50° 



362 



NATUEAL FUNCTIONS. 



Table 3. 



40° 


Natural Trigonometrical 


Functions. 


139° 


M. 


Sine. 


Vrs. COS. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Yrs. sin. 


Cosine. 


M. 





.64279 


.35721 


1.5557 


.83910 


1.1917 


1.3054 


.23395 


.76604 


60 


1 


. 4301 


. 5699 


.5552 


. 3959 


.1910 


.3057 


. 34] <1 


. 6686 


59 


2 


. 4323 


. 5677 


.5546 


. 4009 


.1903 


.3060 


. 3433 


. 6567 


68 


3 


. 4345 


. 5654 


.5541 


. 4059 


.1896 


.3064 


. 3462 


. 6548 


57 


4 


. 4368 


. 5632 


.5536 


. 4108 


.1889 


.3067 


. 3470 


. 6530 


56 


5 


.64390 


.35610 


1.5530 


.84158 


1.1882 


1.3070 


.23489 


.76611 


55 





. 4412 


. 5588 


.5525 


. 4208 


.1875 


.3073 


. 3508 


. 6492 


54 


7 


. 4435 


. 5565 


.5520 


. 4267 


.1868 


.3076 


. 3627 


. 6473 


53 


8 


. 4457 


. 5543 


.5514 


. 4307 


.1861 


.3080 


. 3545 


. 6455 


52 


9 


. 4479 


. 5521 


.5509 


. 4357 


.1854 


.3083 


. 3564 


. 6436 


,51 


10 


.64601 


.35499 


1.6503 


.81407 


1.1847 


1.3086 


.23583 


.76417 


50 


11 


. 4523 


. 5476 


.5498 


. 4457 


.1840 


.3089 


. 3602 


. 6398 


49 


12 


. 4516 


. 5-164 


.5493 


. 4506 


.1833 


.3092 


. 3620 


. 6380 


48 


13 


. 4568 


. 5432 


.5187 


. 4556 


.1826 


.3096 


. 8639 


. 6361 


47 


U 


. 4590 


. 5410 


.5482 


. 4606 


.1819 


.3099 


. 3658 


. 6342 


46 


15 


.61612 


.36388 


1.5477 


.84656 


1.1812 


1.3102 


.23677 


.76323 


45 


16 


. 4635 


. 5365 


.5471 


. 4706 


.1805 


.3105 


. 3695 


. 6304 


44 


17 


. 4657 


. 5343 


.5466 


4756 


.1798 


.3109 


. 3714 


. 6286 


43 


18 


. 4679 


. 6321 


.5161 


. 4806 


.1791 


.3112 


. 3733 


. 6267 


42 


19 


. 4701 


. 5299 


.5456 


. 4856 


.1785 


.3115 


. 3752 


. 6248 


41 


20 


.64723 


.35277 


1.5450 


.84906 


1.1778 


1.3118 


.23771 


.76229 


40 


21 


. 4745 


. 6254 


.6445 


. 4956 


.1771 


.3121 


. 3790 


. 6210 


39 


22 


. 4768 


. 5232 


.5440 


. 5006 


.1764 


.3125 


. 3808 


. 6191 


38 


23 


. 4790 


. 5210 


.5134 


. 5056 


.1757 


.3128 


. 3827 


. 6173 


37 


24 


. 4812 


. 5188 


.5429 


. 5107 


.1750 


.3131 


. 3846 


. 6154 


36 


26 


.64834 


.35166 


1.5424 


.85157 


1.1743 


1.3134 


.23865 


.76135 


35 


26 


. 4856 


. 5144 


.5419 


. 5207 


.1736 


.3138 


. 3884 


. 6116 


34 


27 


. 4878 


. 5121 


.5413 


. 5257 


.1729 


.3141 


. 3903 


. 6097 


33 


28 


. 4900 


. 6099 


.5408 


. 5307 


.1722 


.3144 


. 3922 


. 6078 


32 


29 


. 4923 


. 6077 


.5403 


. 6358 


.1715 


.3148 


. 3940 


. 6059 


31 


30 


.64945 


.35055 


1.5398 


.85408 


1.1708 


1.3151 


.23959 


.76041 


30 


31 


. 4967 


. 5033 


.6392 


. 5458 


.1702 


.3154 


. 3978 


. 6022 


29 


32 


. 4989 


. 5011 


.6387 


. 6509 


.1695 


.3157 


. 3997 


. 6003 


28 


33 


. 5011 


. 4989 


.6382 


. 6559 


.1688 


.3161 


. 4016 


. 5984 


27 


34 


. 6033 


. 4967 


,5377 


. 6609 


.1681 


.3164 


. 4035 


. 5965 


26 


35 


.65055 


.34945 


1.5371 


.85660 


1.1674 


1.3167 


.24054 


.76946 


25 


36 


. 6077 


. 4922 


.5366 


. 6710 


.1667 


.3170 


. 4073 


. 5927 


24 


37 


. 5099 


. 4900 


.5361 


. 6761 


.1660 


.3174 


. 4092 


. 5908 


23 


38 


. 5121 


. 4878 


.5356 


. 6811 


.1653 


.3177 


. 4111 


. 5889 


22 


39 


. 5144 


. 4856 


.5351 


. 5862 


.1647 


.3180 


. 4130 


. 5870 


21 


40 


.65166 


.34834 


1.6345 


.85912 


1.1640 


1.3184 


.24149 


.75851 


20 


41 


. 5188 


. 4812 


.6340 


. 5963 


.1633 


.3187 


. 4168 


. 5832 


19 


42 


. 5210 


. 4790 


.6335 


. 6013 


.1626 


.3190 


. 4186 


. 5813 


18 


43 


. 5232 


. 4768 


.5330 


. 6064 


.1619 


.3193 


. 4205 


. 5794 


17 


44 


. 5254 


. 4746 


.5325 


. 6115 


.1612 


.3197 


. 4224 


. 5775 


16 


45 


.66276 


.34724 


1.5319 


.86165 


.1.1605 


1.3200 


.24243 


.75766 


15 


46 


. 5298 


. 4702 


.5314 


. 6216 


.1599 


.3203 


. 4262 


. 5737 


14 


47 


. 5320 


. 4680 


.5309 


. 6267 


.1592 


.3207 


. 4281 


. 5718 


13 


48 


. 6342 


,. 4658 


.5304 


. 6318 


.1685 


.3210 


. 4300 


. 5699 


12 


49 


. 5364 


. 4636 


.5299 


. 6368 


.1578 


.3213 


. 4319 


. 5680 


11 


50 


.65386 


.34614 


1.5294 


.86419 


1.1571 


1.3217 


.24338 


.75661 


10 


51 


. 6408 


. 4592 


.5289 


. 6470 


.1565 


.3220 


. 4357 


. 5642 


9 


52 


. 5430 


. 4570 


.5'283 


. 6521 


.1558 


.3223 


. 4376 


. 5623 8 


63 


. 5452 


. 4548 


.5278 


. 6672 


.1551 


.3227 


. 4396 


. 5604 7 


54 


. 5474 


. 4526 


.5273 


. 6623 


.1544 


.3230 


. 4415 


. 5585 6 


55 


.65496 


.34504 


1.5268 


.86674 


1.1637 


1.3233 


.24434 


.75566 5 


'56 


. 6518 


. 4482 


.5263 


. 6725 


.1531 


.3237 


. 4453 


. 5547 


4 


57 


. 5640 


. 4460 


.5258 


. 6775 


.1524 


.3240 


. 4472 


. 5528 


3 


58 


. 5662 


. 4438 


.5253 


. 6826 


.1517 


.3243 


. 4491 


. 6509 


2 


59 


. 5584 


. 4416 


.6248 


. 6878 


.1510 


.3247 


. 4510 


. 5490 


1 


60 


. 5606 


. 4394 


.6242 


. 6929 


.1504 


.3260 


. 4529 


. 5471 





M. 


Cosine. 


Vrs. sin. 


Secant. 


Co tang. 


Tang. 


Cosec'nt 


Vrs. COB. 


Sine. 


M. 



130° 



49° 



Table 3. 



NATURAL FUNCTIONS. 



363 



4«° 




Natural Trigonometrical Functions. 


138° 


mT 


Sine. 


Vra. COB. 


C!osec'nt 


Tang. 


Cotang. 


Secant, 


Yrs. sin. 


Cosine, 


M. 





.65606 


.34394 


1.5242 


.86929 


1.1504 


1,3250 


.21529 


.75471 


60 


1 


. 5628 


. 4372 


.6237 


. 6980 


.1497 


.3253 


.4648 


. 5462 


59 


2 


. 5650 


. 4350 


.5232 


.7031 


.1490 


.3257 


. 4567 


. 5133 


58 


3 


.6672 


. 4328 


.5227 


. 7082 


.1483 


.3260 


. 4586 


. 5414 


57 


A 


. 5694 


. 4306 


.6222 


. 7133 


.1477 


.3263 


. 4605 


. 5394 


56 


C 


.65716 


.34284 


1.5217 


.87184 


1.1470 


1.3267 


.21624 


.75375 


65 


6 


. 5737 


. 4262 


.5212 


. 7235 


.1463 


.3270 


.4644 


. 6356 


54 


7 


. 5759 


. 4210 


.5207 


. 7287 


.1456 


.3271 


. 4663 


. 5337 


53 


8 


. 5781 


.4219 


.5202 


. 7338 


.1450 


.3277 


. 4682 


. 5318 


52 


9 


. 5803 


. 4197 


.5197 


. 7389 


.1443 


,3280 


. 4701 


. 5299 


51 


10 


.65825 


.31175 


1.6192 


.87441 


1,1436 


1.3284 


.21720 


,75280 


50 


11 


. 5847 


. 4153 


.5187 


. 7192 


.1430 


.3287 


. 1739 


. 5261 


49 


12 


. 5869 


. 4131 


.5182 


. 7513 


.1423 


.3290 


. 4758 


. 5241 


48 


13 


. 5891 


. 4109 


.5177 


. 7595 


.1416 


.3294 


. 4778 


. 5222 


47 


14 


. 5913 


. 4087 


.5171 


. 7616 


.1409 


.3297 


. 4797 


. 5203 


46 


15 


.65934 


.31065 


1.5166 


.87698 


1.1103 


1.3301 


.24816 


.76184 


45 


16 


. 5956 


. 4043 


.5161 


. 7719 


.1396 


.3304 


.4835 


. 5165 


44 


17 


. 5978 


. 4022 


.5156 


. 7801 


.1389 


.3307 


. 4861 


. 5146 


43 


18 


.6000 


. 4000 


.5151 


. 7852 


.1383 


.3311 


. 4873 


. 5125 


42 


19 


. 6022 


. 3978 


.5146 


. 7904 


.1376 


.3314 


. 1893 


. 5107 


41 


20 


.66044 


.33956 


1.5141 


.87955 


1.1369 


1.3318 


.21912 


,7.5088 


40 


21 


. 6066 


. 3931 


.6136 


. 8007 


.1363 


.3321 


. 4931 


. 5069 


39 


22 


. 6087 


. 3912 


.5131 


. 8058 


.1356 


.3324 


. 1950 


. 5049 


38 


23 


. 6109 


. 3891 


.5126 


. 8110 


.1319 


.3328 


. 4970 


. 5030 


37 


24 


. 6131 


. 3869 


.5121 


. 8162 


.1313 


.3331 


. 1989 


. 6011 


36 


25 


.66153 


.33847 


1.5116 


.88213 


1.1336 


1.3335 


.25008 


.74992 


35 


26 


. 6175 


. 3825 


.5111 


. 8265 


.1329 


.3338 


, 5027 


. 4973 


34 


27 


. 6197 


. 3803 


.5106 


. 8317 


.1323 


.3342 


. 5017 


. 4953 


33 


28 


. 6218 


. 3781 


.5101 


. 8369 


.1316 


.3345 


. 5066 


. 4934 


32 


29 


. 6240 


. 3760 


.5096 


. 8121 


.1309 


.3318 


. 5085 


. 4915 


31 


30 


.66262 


.33738 


1.8092 


.88172 


1.1303 


1.3352 


.25104 


.74896 


30 


31 


. 6284 


. 3716 


.6087 


. 8521 


.1296 


.3355 


. 5124 


. 4876 


29 


32 


. 6305 


. 3694 


.5082 


. 8576 


.1290 


.3359 


. 5143 


. 4857 


28 


33 


. 6327 


. 3673 


.5077 


. 8628 


.1283 


.3362 


, 5162 


. 4838 


27 


34 


. 6349 


. 3651 


.5072 


. 8680 


,1276 


.3366 


. 5181 


. 1818 


26 


35 


.66371 


.33629 


1.5067 


.88732 


1.1270 


1,3369 


,25201 


.74799 


25 


36 


. 6393 


. 3607 


.6062 


. 8781 


.1263 


,3372 


. 5220 


. 4780 


24 


37 


. 6414 


. 3586 


.5057 


. 8836 


.1257 


.3376 


. 5239 


. 4760 


23 


38 


. 6436 


. 3564 


.5052 


. 8888 


.1250 


.3379 


. 5259 


. 4741 


22 


39 


. 6158 


. 3542 


..5017 


. 8940 


,1243 


.3383 


. 5278 


. 4722 


21 


40 


.66479 


.33520 


1.5012 


.88992 


1.1237 


1.3386 


.25297 


.74702 


20 


41 


. 6501 


. 3499 


.5037 


. 9041 


.1230 


.3390 


. 5317 


. 4683 


19 


42 


. 6523 


. 3477 


.5032 


. 9097 


.1224 


.3393 


. 5336 


. 4664 


18 


43 


. 6545 


. 3455 


.5027 


. 9119 


.1217 


.3397 


. 5355 


. 4644 


17 


44 


. 6566 


. 3433 


.5022 


. 9201 


.1211 


.3400 


. 5375 


. 4626 


16 


45 


.66588 


.33412 


1.5018 


.89253 


1.1204 


1.3404 


.26394 


.74606 


15 


46 


. 6610 


. 3390 


.5013 


. 9306 


.1197 


.3407 


. 6414 


. 4586 


14 


47 


. 6631 


. 3368 


.5008 


. 9358 


.1191 


.3411 


. 5433 


. 1567 


13 


48 


. 6653 


.3347 


.6003 


. 9110 


.1184 


.3414 


. 5462 


. 4518 


12 


49 


. 6675 


. 3325 


.1998 


. 9163 


.1178 


.3418 


. 6472 


. 1528 


11 


50 


.66697 


.33303 


1.1993 


.89515 


1.1171 


1.3421 


.25491 


.71509 


10 


51 


. 6718 


. 3282 


.4988 


. 9567 


.1165 


,3126 


. 5510 


. 4489 


9 


52 


. 6740 


. 3260 


.4983 


. 9620 


.1158 


.3428 


. 5530 


. 4170 


8 


63 


. 6762 


. 3238 


.4979 


. 9672 


.1152 


.3432 


. 5619 


. 4450 


7 


54 


. 6783 


. 3217 


.4971 


. 9725 


.U45 


.3435 


. 5569 


. 4131 


6 


55 


.66805 


.33195 


1.4969 


.89777 


1.1139 


1.3439 


.25588 


.74412 


5 


56 


. 6826 


. 3173 


.4964 


. 9830 


.1132 


.3442 


. 5608 


. 4392 


4 


57 


. 6848 


. 3152 


.4959 


. 9882 


.1126 


.3446 


. 5627 


. 4373 


3 


58 


. 6870 


. 3130 


.4954 


. 9935 


,1119 


.3449 


. 5617 


. 4353 


2 


69 


. 6891 


. 3108 


.4949 


. 9988 


.1113 


.3153 


. 5666 


. 4334 


1 


60 


. 6913 


. 3087 


.4945 


.90040 


.1106 


.3156 


. 5685 


. 4314 





M. 


Cosine. 


Vrs. sin. 


Secant, 


Co tang. 


Tang. 


Cosec'nt 


Vrs. COB, 


Sine, 


M. 



131° 



48° 



364 



NATURAL FUNCTIONS. 



Table 3. 



42° 


Natural Trigonometrical Functions. 


137° 


M^ 


Sine. 


Vrs. COS. 


Cosec'nt 


Tang. 


Cotang. 


Secant. 


Yrs. Bin. 


Cosine. 


M. 





.66913 


.33087 


1.4945 


.90040 


1.1106 


1.3456 


.26685 


.74314 


00 


1 


. 6936 


. 3065 


.4940 


. 0093 


.1100 


.3460 


. 5705 


. 4295 


59 


2 


. 6956 


. 3044 


.4935 


. 0146 


.1093 


.3463 


. 5724 


. 4275 


58 


3 


. 6978- 


. 3022 


.4930 


. 0198 


.1086 


.3467 


. 5744 


. 4256 


.67 


4 


. 6999 


. 3000 


.4925 


. 0251 


.1080 


.3470 


. 5763 


. 4236 


56 


5 


.67021 


.32979 


1.4921 


.90304 


1.1074 


1.3474 


.25783 


.74217 


55 


6 


7043 


. 2957 


.4916 


. 0357 


.1067 


.3477 


. 5802 


. 4197 


54 


7 


. 7064 


. 2936 


.4911 


. 0410 


.1061 


.3481 


. 5822 


. 4178 


.53 


8 


. 7086 


. 2914 


.4906 


. 0463 


.1054 


.3485 


. 5841 


. 4168 


52 


9 


. 7107 


. 2893 


.4901 


. 0515 


.1048 


.3488 


. 5861 


. 4139 


51 


10 


.07129 


.32871 


1.4897 


.90568 


1.1041 


1.3492 


.25880 


.74119 


60 


11 


. 7150 


. 2849 


.4892 


. 0621 


.1035 


.3495 


. 5900 


. 4100 


49 


12 


. 7172 


. 2828 


.4887 


. 0674 


.1028 


.3499 


. 5919 


. 4080 


48 


13 


. 7194 


. 2806 


.4882 


. 0727 


.1022 


.3502 


. 5939 


. 4061 


47 


14 


. 7215 


. 2785 


.4877 


. 0780 


.1015 


.3506 


. 5959 


. 4041 


46 


15 


.07237 


.32763 


1.4873 


.90834 


1.1009 


1.3509 


.25978 


.74022 


45 


10 


. 7258 


. 2742 


.4868 


. 0887- 


.1003 


.3513 


. 5998 


. 4002 


44 


17 


. 7280 


. 2720 


.4863 


. 0940 


.0996 


.3517 


. 6017 


. 3983 


43 


la 


. 7301 


. 2699 


.4868 


. 0993 


.0990 


.3520 


. 6037 


. 3963 


42 


19 


. 7323 


. 2677 


.4864 


. 1046 


.0983 


.3524 


. 6056 


. 3943 


41 


20 


.67344 


.32656 


1.4849 


.91099 


1.0977 


1.3527 


.26076 


.73924 


40 


21 


. 7366 


. 2634 


.4844 


. 1153 


.0971 


.3531 


. 6096 


. 3904 


39 


22 


. 7387 


. 2613 


.4839 


. 1206 


.0964 


.3534 


. 6115 


. 3885 


38 


23 


. 7409 


. 2591 


.4835 


. 1259 


.0953 


.3538 


. 6135 


. 3865 


37 


24 


. 7430 


. 2570 


.4830 


. 1312 


.0951 


.3542 


. 6154 


. 3845 


36 


25 


.67452 


.32548 


1.4825 


.91366 


1.0945 


1.3545 


.26174 


.73826 


35 


26 


. 7473 


. 2527 


.4821 


. 1419 


.0939 


.3549 


. 0194 


. 3806 


34 


27 


. 7495 


. 2505 


.4816 


. 1473 


.0932 


.3552 


. 6213 


. 3787 


33 


28 


. 7516 


. 2484 


.4811 


. 1526 


.0926 


.3556 


. 6233 


. 3767 


32 


29 


.7537 


. 2462 


.4806 


. 1580 


.0919 


.3560 


. 6253 


. 3747 


31 


30 


.67559 


.32441 


1.4802 


.91633 


1.0913 


1.3563 


.26272 


.73728 


30 


31 


. 7580 


. 2419 


.4797 


. 1687 


.0907 


.3567 


. 6292 


. 3708 


29 


32 


. 7602 


. 2398 


.4792 


. 1740 


.0900 


.3571 


. 6311 


. 3688 


28 


33 


. 7623 


. 2377 


.4788 


. 1794 


.0894 


.3574 


. 6331 


. 3669 


27 


34 


. 7645 


. 2355 


.4783 


. 1847 


.0888 


.3578 


. 6351 


. 3649 


26 


35 


.67666 


.32334 


1.4778 


.91901 


1.0881 


1.3581 


.26371 


.73629 


25 


36 


. 7688 


. 2312 


.4774 


. 1955 


.0875 


.3585 


. 6390 


. 3610 


24 


37 


. 7709 


. 2291 


.4769 


. 2008 


.0868 


.3589 


. 6410 


. 3590 


23 


38 


. 7730 


. 2269 


.4764 


. 2062 


.0862 


.3592 


. 6430 


. 3570 


22 


39 


. 7752 


. 22J8 


.4760 


. 2116 


.0856 


.3596 


. 6449 


. 3551 


21 


40 


.07773 


.32227 


1.4755 


.92170 


1.0849 


1.3600 


.26169 


.73631 


20 


41 


. 7794 


. 2205 


.4750 


. 2223 


.0843 


.3603 


. 6489 


. 3511 


19 


J2 


. 7816 


. 2184 


.4746 


. 2277 


.0837 


.3607 


. 6508 


. 3491 


18 


43 


. 7837 


. 2163 


.4741 


. 2331 


.0830 


.3611 


. 6528 


. 3472 


17 


44 


. 7859 


. 2141 


.4736 


. 2385 


.0824 


.3614 


. 6548 


. 3452 


16 


45 


.67880 


.32120 


1.4732 


.92439 


1.0818 


1.3618 


.26568 


.73432 


15 


40 


. 7901 


. 2098 


.4727 


. 2493 


.0812 


.3622 


. 6587 


. 3412 


14 


47 


. 7923 


. 2077 


.4723 


. 2547 


.0805 


.3625 


. 6607 


. 3393 


13 


48 


. 7944 


. 2056 


.4718 


. 2601 


.0799 


.3629 


. 6627 


. 3373 


12 


49 


. 7965 


. 2034 


.4713 


. 2655 


.0793 


.3633 


. 6647 


. 3353 


11 


60 


.67987 


.32013 


1.4709 


.92709 


1.0786 


1.3636 


.26066 


.73333 


10 


51 


. 8008 


. 1992 


.4704 


. 2703 


.0780 


.3640 


. 6686 


. 3314 


9 


52 


. 8029 


. 1970 


.4699 


. 2817 


.0774 


.3644 


. 6700 


. 3294 


8 


53 


. 8051 


. 1949 


.4695 


. 2871 


.0767 


.3647 


. 6726 


. 3274 


7 


54 


. 8072 


. 1928 


.4690 


. 2926 


.0761 


.3651 


. 6746 


. 3254 


6 


65 


.68093 


.31907 


1.4686 


.92980 


1.0755 


1.3655 


.26765 


.73234 


5 


66 


. 8115 


. 1885 


.4681 


. 3034 


.0749 


.3658 


. 6785 


. 3215 


4 


57 


. 8136 


. 1864 


.4676 


. 3088 


.0742 


.3662 


. 6805 


. 3195 


3 


68 


. 8157 


. 1843 


.4672 


. 3143 


.0736 


.3666 


. 6825 


. 3175 


2 


r.9 


. 8178 


. 1821 


.4667 


3197 


.0730 


.3669 


. 6845 


. 3155 


1 


60 


. 8200 


. 1800 


.4663 


. 3251 


.0724 


.3673 


. 6865 


. 3135 





M. 


Conine. 


VrB. ein. 


Secant. 


Cotang. 


Tang. 


Coeec'nt 


IVrB. COB. 


Sine. 


M. 



132° 



470 



Table 3. 



NATURAL FUNCTIONS. 



365 



43° 




Natural Trigonometrical Functions. 


136° 


mT 


Sinn. 


Vrs. COS. 


Cosec'nt 


Tang. 


Co tang. 


Secant. 


Yrs. sin. 


Cosine. 


M. 





.68200 


.31800 


1,4663 


.93251 


1.0724 


1.3673 


.26865 


.73135 


60 


1 


. 8221 


. 1779 


.4658 


. 3306 


.0717 


.3677 


. 6884 


. 3115 


59 


2 


. 8242 


. 1758 


.4654 


. 3360 


.0711 


.3681 


.6904 


. 3096 


58 


3 


. 8264 


. 1736 


.4649 


.3415 


.0705 


.3684 


. 6924 


. 3076 


67 


4 


. 8285 


. 1715 


.4614 


. 3469 


.0699 


.3688 


. 6944 


. 3056 


56 


5 


.68306 


.31694 


1.4610 


.93524 


1.0692 


1.3692 


.26964 


.73036 


55 


6 


.8327 


. 1673 


.4635 


. 3578 


.0686 


.3695 


. 6984 


. 3016 


54 


7 


. 8349 


. 1651 


.4631 


. 3633 


.0680 


.3699 


. 7004 


. 2996 


53 


8 


. 8370 


. 1630 


.4626 


. 3687 


.0674 


.3703 


. 7023 


. 2976 


62 


9 


. 8391 


. 1609 


.4622 


. 3742 


.0667 


.3707 


. 7043 


. 2966 


51 


10 


.68412 


.31588 


1.4617 


.93797 


1.0661 


1.3710 


.27063 


.72937 


50 


11 


. 8433 


. 1566 


.4613 


. 3851 


.0665 


.3714 


. 7083 


. 2917 


49 


12 


. 8455 


. 1545 


.4608 


. 3906 


.0649 


.3718 


. 7103 


. 2897 


48 


13 


. 8476 


. 1524 


.4604 


. 3961 


.0643 


.3722 


. 7123 


. 2877 


47 


14 


. 8497 


. 1503 


.4599 


. 4016 


.0636 


.3725 


. 7143 


. 2857 


46 


15 


.68518 


.31482 


1.4595 


.94071 


1.0630 


1.3729 


.27163 


.72837' 


45 


16 


. 8539 


. 1460 


.4590 


. 4125 


.0624 


.3733