MEYER
ROUTE SURVEYING
ENGIN. LIB.
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445
MGZ
1962
A 1,051,851
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ROUTE
SURVEYING
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SCIENTIA VERITAS
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9.2
INTERNATIONAL TEXTBOOKS IN CIVIL ENGINEERING
Consulting Editor
RUSSELL C. BRINKER
Professor of Civil Engineering
New Mexico State University
ROUTE SURVEYING

ROUTE
SURVEYING

CARL F. MEYER
Professor of Civil Engineering
Worcester Polytechnic Institute
THIRD EDITION
INTERNATIONAL TEXTBOOK COMPANY
Scranton, Pennsylvania
Engin. Library
TA
7
625
Ben ist hat d
Copyright, 1962, 1956, 1946, by International Textbook Company.
All rights reserved. Printed in the United States of America by The Haddon
Craftsmen, Inc., at Scranton, Pennsylvania. Library of Congress Catalog
Card Number: 62-12773.
PREFACE TO THE THIRD EDITION
The primary objective in the third edition of this book was
to modernize every chapter while retaining the basic approach
that made earlier editions so popular. Another goal was to
improve the book's teachability by a clear rewriting of many
of the more difficult subjects and especially by the inclusion
of almost three times as many problems-all of them new.
The most important addition is the new chapter “Automa-
tion in Location and Design." This chapter brings into focus
the electronic applications that have revolutionized many
aspects of field and office work. The treatment clarifies terms
such as data procurement, data reduction, and data processing
for practicing engineers and students alike. The applications
of wave mechanics and electronics to distance measurement;
the fundamentals of modern computers; and the new auto-
mated methods of data recording, data transmission, and
data plotting are explained. Special emphasis is given to the
automatic integration of photogrammetry and electronic
computer in mapping and route location.
The chapter "Aerial Photography in Route Surveying" has
been up-dated by the inclusion of new material on analytic
aerotriangulation and on orthophotography and its product-
the photo contour map. Recent cost data on photogrammetric
mapping for highway location have been added and cost
trends analyzed.
Significant additions have been made in the chapters on
curves. These include the theory and practice of staking
parallel offset curves and spirals, and the solution of special
curve problems by analytic geometry.
The new problems have been carefully chosen to serve in
fixing the principles of curves and earthwork. Answers are
given to many of the problems; but the development of
students' self-reliance is promoted by omitting some answers
so as to encourage checking the work by an independent
method.
vii
viii
PREFACE
As a prelude to the problems, new material on proper
methods of computation and use of significant figures are
given in Articles 2-16 and 2-17. This material fills a need
rarely satisfied in a surveying textbook. Students are urged
to weigh the accompanying suggestions before doing problems
and to refer to these articles frequently until the procedures
become habitual.
Worcester, Massachusetts
April, 1962
C.F.M.
PART I: BASIC PRINCIPLES
1. Route Location..
2. Simple Curves.
3. Compound and Reverse Curves.
4. Parabolic Curves..
5. Spirals..
6. Earthwork……….
CONTENTS
List of Tables...
•
•
PART II: PRACTICAL APPLICATIONS
7. Special Curve Problems.
8. Curve Problems in Highway Design..
9. Railroad Surveys.
10. Highway Surveys
11. Surveys for Other Routes.
12. Aerial Photography in Route Surveying.
13. Automation in Location and Design.
•
·
·
•
APPENDIX: Partial Theory of the Spiral.
INDEX.
ix
PART III: TABLES
·
•
•
•
•
•
•
1
12
50
65
83
115
159
185
262
297
313
321
343
367
659
663
ROUTE SURVEYING
PART I
BASIC PRINCIPLES
CHAPTER 1
ROUTE LOCATION
1-1. Introduction. This chapter comprises an outline of
the basic considerations affecting the general problem of route-
location. The material is non-mathematical, but it is neces-
sary for a clear understanding of the purposes served by the
technical matters in the remaining chapters of Part I. Specific.
practical applications of these basic considerations to the loca-
tion of highways, railroads, and other routes of transportation
and communication are given in Part II.
1-2. Definition of Route Surveying.-Route surveying
includes all field work and requisite calculations (together with
maps, profiles, and other drawings) involved in the planning
and construction of any route of transportation. If the word
transportation be taken to refer not only to the transportation
of persons but also to the movement of liquids and gases and
to the transmission of power and messages, then route survey-
ing is a very broad subject. Among the important engineer-
ing structures thus included are: highways and railroads;
aqueducts, canals, and flumes; pipe lines for water, sewage,
oil, and gas; cableways and belt conveyors; and power, tele-
phone, and telegraph transmission lines.
Though this definition of route surveying serves to distin-
guish the subject from other branches of surveying, it is
customary to assume that projects designated as route surveys
have considerable magnitude. The setting of a few telephone
poles along a highway or the staking out of several blocks of
city street scarcely fits the definition. There must be definite
termini a considerable distance apart. In such a situation,
route surveys are for two purposes: (1) determining the best
general route between the termini; (2) fixing the alignment,
grades, and other details of the selected route. Sound engi-
neering principles require that the route be chosen in such a
way that the project may be constructed and operated with
the greatest economy and utility.
1
2
ROUTE LOCATION
1-3. Relation of Project to Economics.-Every route-sur-
veying project involves economic problems both large and
small. By far the most important question is whether or not to
construct the project. Essentially, this decision is based on a
comparison of the cost of the enterprise with the probable
financial returns or social advantages to be expected. In some
cases the question can be answered after a careful preliminary
study without field work; in others, extensive surveys and cost
estimates must first be made.
However simple or complex the project may be, it is rarely
possible for the engineer alone to answer this basic economic
question. To his studies must be added those of the persons
responsible for the financial and managerial policies of the
organization. In the case of a public project the broad social
and political objectives also carry weight.
The engineer responsible for conducting route surveys is
not solely a technician. In addition to his indispensable aid
in solving the larger economic problems, he is continually con-
fronted with smaller ones in the field and office. For example,
the relatively simple matter of deciding which of several
methods is to be used in developing a topographic map of a
strip of territory is, basically, an economic problem that in-
volves the purpose of the survey, the terrain, and the equip-
ment and personnel available.
1-4. Relation of Project to Design.-Design problems in
route location are closely related to route surveying. Some
matters of design must precede the field work; others are de-
pendent on it. For example, in order that the field work for
a proposed new highway may be done efficiently, the designers
must have chosen-at least tentatively-not only the termini
and possible intermediate connections but also such design de-
tails as the number of traffic lanes, width of right-of-way, maxi-
mum grade, minimum radius of curve, and minimum sight
distance. On the other hand, considerable field work must be
done before the designers can fix the exact alignment, grade
elevations, shoulder widths, and culvert locations to fit the
selected standards safely and with the greatest over-all
economy. The interrelationship between modern highway
surveying and design is outlined in Art. 10-2.
ROUTE LOCATION
3
1-5. Basic Factors of Alignment and Grades.-In route
location it is usually found that the termini and possible inter-
mediate controlling points are at different elevations. More-
over, the topography and existing physical features rarely
permit a straight location between the points. These circum-
stances invariably require the introduction of vertical and
horizontal changes in direction; therefore, grades, vertical
curves, and horizontal curves are important features of route
surveying and design.
Curvature is not inherently objectionable. Though a
straight line is the shortest distance between two points, it is
also the most monotonous-a consideration of some aesthetic
importance in the location of scenic highways. The device
of curvature gives the designer limitless opportunities to fit a
location to the natural swing of the topography in such a way
as to be both pleasing and economical. Excessive or poorly
designed curvature, however, may introduce serious operating
hazards, or may add greatly to the costs of constructing, main-
taining, or operating over the route.
Steep grades are likely to have the same effects on safety
and costs as excessive curvature. It should be emphasized,
nevertheless, that problems of curves and grades are ordinarily
interrelated. Thus, on highway and railroad location it is
often the practice to increase the distance between two fixed
points in order to reduce the grade. This process, known as
"development," necessarily adds to the total curvature. It is
not always a feasible solution, for the added curvature may be
more objectionable than the original steep grade.
The aim of good location should be the attainment of con-
sistent conditions with a proper balance between curvature
and grade. This is especially true in highway location, owing
to the fact that each vehicle is individually operated and the
driver often is unfamiliar with the particular highway. Many
highway accidents occur at a point where there is a sudden and
misleading variation from the condition of curvature, grade,
or sight distance found on an adjacent section of the same high-
way. To produce a harmonious balance between curvature
and grade, and to do it economically, requires that the engi-
neer possess broad experience, mature judgment, and a
thorough knowledge of the objectives of the project.
4
ROUTE LOCATION
1-6. Influence of Type of Project.-The type of route to be
built between given termini has a decided influence on its
location. As an example, the best location for a railroad
would not necessarily be the most suitable one for a power-
transmission line. A railroad requires a location having
fairly flat grades and curves. Moreover, there are usually
intermediate controlling points such as major stream and
highway crossings, mountain passes, and revenue-producing
markets. In contrast, power is transmitted as readily up a
vertical cable as along a horizontal one. Grades, therefore,
have no significance, and river and highway crossings present
no unusual problems. Where changes in direction are needed,
they are made at angle towers. Consequently, the alignment
is as straight as possible from generating station to sub-station.
1-7. Influence of Terrain. The character of the terrain
between termini or major controlling points is apt to impress
a characteristic pattern upon a route location, particularly in
the case of a highway or a railroad. Terrain may be generally
classified as level, rolling, or mountainous.
1
In comparatively level regions the line may be straight for
long distances, minor deviations being introduced merely to
skirt watercourses, to avoid poor foundations, or possibly to
reduce land damages. On an important project, however, the
artificial control imposed by following section lines or other
political subdivisions should not be permitted to govern.
In rolling country the location pattern depends on the
orientation of the ridges and valleys with respect to the
general direction of the route. Parallel orientation may result
in a valley line having flat grades, much curvature, frequent
culverts and bridges, and fill in excess of cut; or it may permit
a ridge line (from which the word highway originated) on which
the alignment and drainage problems are simpler. To connect
two such situations, and also in case the ridges are oblique to
the general direction of the route, there may be a side-hill line.
This has the characteristics of uniformly rising grades, curva-
ture fitted to the hillsides, and relatively light, balanced grad-
ing.
Where the ridges and valleys are approximately at right
angles to the general direction of the route, the typical pattern
*
ROUTE LOCATION
5
which results may be called a cross-drainage line. Here the
location of passes through the ridges and the location of cross-
ings over the major streams constitute important controlling
points between which the line may be of the side-hill type.
Generally, a cross-drainage line involves steep grades, heavy
grading with alternate cuts and fills, expensive bridges, and
curvature considerably less than that on a valley line.
Mountainous terrain imposes the severest burden upon the
ingenuity of the locating engineer. No simple pattern or set
of rules fits all situations. Short sections of each of the types
of lines previously described must be inserted as conditions
require. "Development," even to the extent of switchbacks
and loops, may be the only alternative to expensive tunnel
construction.
1-8. Sequence of Field and Office Work. The definition
of route surveying stated in Art. 1-2 referred to field work
and calculations concerning both planning and construction.
Though these operations vary with different organizations, and
particularly with the nature and scope of the project, the
following is a typical outline of the sequence of field and office
work:
For Planning
(a) The conception of the project, and preliminary office
studies regarding its desirability and feasibility.
(b) Field reconnaissance of the terrain between the termini,
followed by further office studies and recommendation
report.
(c) Preliminary surveys over one or more locations along
the general route recommended in the reconnaissance
report.
(d) Office studies consisting of preparation of a map from
preliminary survey data; projection of a tentative align-
ment and profile; and preliminary estimate of quantities
and cost.
(e) Location survey involving staking of projected location,
complete with curves; minor adjustment of alignment
and grades; cross-sectioning for more accurate earth-
work estimate; ties to property lines and existing im-
6
ROUTE LOCATION
provements; and field measurements for design of mis-
cellaneous structures.
For Construction
(f) Office work including preparation of specifications and
drawings covering all details of the project; negotiations
for acquiring right-of-way or easement; and advertising
for bids.
(g) Construction surveys including reestablishment of final
location; setting reference stakes, grade stakes, and
slope stakes for controlling the construction; making
periodic measurements and estimates of work done, to
serve as a basis for partial payments to the contractor;
taking final cross-sections and other measurements of
the completed project, to serve as a basis both for final
payment and for preparation of "record" plans; and
setting right-of-way monuments in accordance with pre-
pared legal descriptions.
1-9. Importance of the Reconnaissance.-Second in im-
portance to the primary question whether or not to build
the project-is the selection of the general route between the
termini. This is usually determined by the reconnaissance.
The statement by Wellington,* "The reconnaissance must
not be of a line, but of an area," is a most apt one. The extent
of the area depends, of course, on the type of project and the
nature of the terrain, but the area must be broad enough to
cover all practicable routes joining the termini. Of particular
importance is the need for guarding against the natural
tendency to favor an obviously feasible location. It is
possible that country which is covered with tangled under-
growth, or is otherwise rough for foot travel on reconnaissance,
may hide a much better location than is available in more
settled or open territory.
With regard to the importance of the "art of reconnaissance"
and the attitude of the engineer toward it, nowhere will more
effective comments be found than in Wellington's classic
treatise.* Though written by that author in 1887 for the
*Reprinted by permission from Economic Theory of the Location of Rail-
ways by A. M. Wellington, published by John Wiley & Sons, Inc., 1915
ROUTE LOCATION
7
instruction of engineers on railroad location, the following
statements are timeless in their application to all types of
route location:
""
there is nothing against which a locating engineer will find it
necessary to be more constantly on his guard than the drawing of hasty
and unfounded conclusions, especially of an unfavorable character,
from apparent evidence wrongly interpreted. If his conclusions on
reconnaissance are unduly favorable, there is no great harm done-
nothing more at the worst will ensue than an unnecessary amount of
surveying; but a hasty conclusion that some line is not feasible, or that
further improvements in it cannot be made, or even sometimes—often
very absurdly that no other line of any kind exists than the one which
has chanced to be discovered these are errors which may have dis-
astrous consequences.
• •
•
"On this account, if for no other, the locating engineer should culti-
vate what may be called an optimistic habit of mind. He should
not allow himself to enter upon his work with the feeling that any
country is seriously difficult, but rather that the problem before him is
simply to find the line, which undoubtedly exists, and that he can only
fail to do so from some blindness or oversight of his own, which it will
be his business to guard against.
•
"For the reason that there is so much danger of radical error in the
selection of the lines to be surveyed (or, rather, of the lines not to be
examined), it results that THE WORST ERRORS OF LOCATION
GENERALLY ORIGINATE IN THE RECONNAISSANCE. This
truth once grasped, the greatest of all dangers, over-confidence in one's
own infallibility, is removed."
If, as often happens, the reconnaissance is entrusted to
one engineer, he should have mature experience in the pro-
motional, financial, and engineering aspects of similar projects.
It is not enough that he be an experienced locating engineer,
for such a man is likely to concentrate upon the purely physical
possibilities of a route and to overlook the related commercial
or social values. Furthermore, he should be able to sense the
significance of present trends and their probable effect upon
the future utility of the project, or to realize when to seek a
specialist's advice in such matters.
1-10. Purposes of Preliminary Surveys.-A preliminary
survey follows the general route recommended in the recon-
naissance report. The most important purpose of such a
survey is to obtain the data for plotting an accurate map of a
strip of territory along one or more promising routes. This
8
ROUTE LOCATION
map serves as the basis for projecting the final alignment and
profile, at least tentatively. Enough data are also obtained
from which to make an estimate of earthwork quantities, of the
sizes of drainage structures, and needed right-of-way. Taken
together, these data permit the compilation of a fairly close
cost estimate.
Preliminary surveys differ greatly in method and precision.
Invariably, however, there is at least one traverse (compass,
stadia, or transit-and-tape) which serves as a framework for
the topographical details. Elevations along the traverse line
and tie measurements to existing physical features are essen-
tial. Accurate contours may or may not be needed, the re-
quirement depending on the type of the project.
Detailed methods of running preliminary surveys adapted
to particular types of routes will be found in Chapters 9, 10,
and 11.
1-11. Proper Use of Topography.--On new locations of
routes over which grades are particularly important, an
accurate contour map is indispensable. A relocation of an
existing route, such as a highway, may sometimes be made by
revising the preliminary survey directly on the ground.
This method, termed “field location" or "direct location,” is
not recommended for a new line. It is true that some engi-
neers seem to have uncanny ability for locating a satisfactory
line-though not necessarily the best one-by direct field
methods. This natural gift is not to be belittled, but it should
be subordinated in difficult terrain to careful office studies
aided by a contour map.
The primary purpose of the contour map is to serve as a
basis for making a "paper location" of the final center line.
On such a map the locating engineer is able to scan a large
area at once. By graphical methods he can study various
locations in a small fraction of the time required for a field
party to survey the lines on the ground. Furthermore, he is
not subject to the natural optical illusions which often mislead
even the most experienced engineer in the field. An added
advantage of the contour map, provided it is extensive enough,
is to supply visible evidence that no better line has been over-
looked.
ROUTE LOCATION
9
It is possible, however, to put too much reliance upon map
topography. Particularly to be avoided is the temptation to
control the work from the office by making such a meticulous
paper location, even to the extent of complete notes for stak-
ing all curves, that the field work of final location becomes a
mere routine of carrying out "instructions from headquarters."
No contour map, no matter how accurate it may be, can im-
press upon the mind more forcibly than field examination such
details as the true significance of length and depth of cuts and
fills; the nature of the materials and foundations; susceptibility
to slides, snow drifting, and other maintenance difficulties; or
the aesthetic values of the projected location. At best, the
map facilitates making what might be termed a "semi-final
location," which is to be further revised in minor details dur-
ing the location survey.
1-12. Function of Location Survey. The purpose of the
location survey is to transfer the paper location, complete with
curves, to the ground. It is too much to expect that this line
on the ground will conform to the paper location in every
detail. It is almost certain that there will be minor deviations,
resulting usually from errors in the preliminary traverse or in
the taking or plotting of the topography. An exact agree-
ment does not assure the excellence of the location; it merely
proves the geometric accuracy of the field and office work.
Consequently, regardless of the "fit" with the paper location,
the engineer should be constantly on watch for opportunities
to make those minor adjustments in alignment or grades which
only close observation of the field conditions will reveal.
When staked, the final location is usually cross-sectioned
for closer determination of earthwork quantities. In addition,
tie measurements to property lines are made to serve for pre-
paring right-of-way descriptions, and all necessary field data
are obtained to permit the detailed design of miscellaneous
structures.
1-13. Relation of Surveying to Engineering. Before we
leave these basic considerations to study the technical aspects
of route surveying, it should be pointed out that surveying and
mapping, as ordinarily practiced, are not engineering; they are
merely methods of obtaining and portraying data needed as
10
ROUTE LOCATION
a prelude to the design and construction of engineering
works.
During the study of the chapters which follow, it will be
natural for the student to concentrate on the geometrical and
instrumental techniques. However, the course in Route Sur-
veying will not reach its potential value unless it is more than
drill in field and office practice. The student, stimulated by
the instructor's examples and illustrations, should attempt to
look beyond the technical details and gain some insight into
the factors which lead to the conception of a particular project.
Knowledge of those factors will give him a better appreciation
of the engineering surveys-their planning, the controlling
specifications, and the usefulness of the data to the designers.
To be of the greatest usefulness, without being unduly
costly, the surveys, maps, and computations should be only
as complete and accurate as needed for the ultimate purpose.
For some purposes the utmost in accuracy is required. For
others, extreme niceties are too costly and time-consuming;
they may be replaced by approximate methods and short cuts.
As an example, much time is often wasted in "exact" calcula-
tion of yardage estimates prior to construction, only to find
that shrinkage, compaction, overbreak, or stripping allowances
change the estimated values by large amounts. This is not to
imply that grading calculations may always be done by
approximate methods. Accuracy is always required, for
example, in determination of yardage for payment to con-
tractors. One trait of a good engineer is his judgment of the
degree of precision required in obtaining data and computing
values for use.
Surveying and mapping are essential prerequisites to engi-
neering design for mass transportation. Despite the growing
file of good maps, the rapid modernization of our transporta-
tion systems is creating expanding demands for surveying and
mapping services. In the field of highway engineering alone,
the total distance covered by surveys and resurveys for a re-
cent year was estimated to reach 26,000 miles. In designing
a large transportation project, extensive surveying operations
are involved in the early reconnaissance, in the detailed pre-
liminary and location surveys, and in all the work leading to
the preparation of topographic maps, profiles, cross sections,
ROUTE LOCATION
11
and other working drawings. If to these there are added the
construction layout and "as-built" record surveys, it is
apparent that a large portion of the total engineering costs is
absorbed by surveying and mapping.
In contrast to the leisurely pace of highway construction
in the early part of this century, wherein the ordinary piece-
meal survey served the purpose, we now have vast and costly
projects. Noteworthy among these are the heavily-traveled
California Freeways and the toll highways in Massachusetts,
Connecticut, New York, New Jersey, Pennsylvania, Ohio,
Indiana, Illinois, Kansas, and Oklahoma. Even these are
dwarfed by the Interstate Highway System now under con-
struction. Already conceived is the proposed Mississippi
River Parkway, for which map reconnaissance of 50,000 square
miles has yielded over 8,000 miles of alternate routes. Survey
and mapping methods must be designed to keep pace with
the advanced design and construction techniques used on such
vast projects.
Applications of photogrammetry (Chapter 12) and auto-
mation (Chapter 13) represent important advances in route
surveying and design. In addition to reducing costs, photo-
grammetry and automation save time-a factor that can
substantially reduce interest charges during the construction
of a major project.
In fitting surveying and mapping into the plans for a trans-
portation project, saving in time by use of short-cut methods
should not be achieved at the expense of reduction in the
ultimate required accuracy. Time saved by short cuts in
control surveys, for example, may be lost many times over in
transferring the paper location to the ground or in monument-
ing the right of way. During construction, one field change
caused by poor original surveys may delay the work longer
than the time saved earlier by the short-cut survey methods.
By keeping in mind the ultimate accuracies required for the
various phases of a project and adopting well-planned, skill-
fully executed methods of surveying and mapping, even at the
expense of some extra time, the whole project will have a firmer
base around which the design and construction operations can
be planned. This kind of surveying and mapping is true
engineering.
SIMPLE CURVES
2-1. Foreword.-Every route has a calculable geometric
alignment. The form may be a simple series of straight lines
and angles, as in the case of a power transmission line, or it
may be an intricate combination of straight lines and curves.
The primary purpose of any curve is to provide the required
change in direction in the form best suited to the operating
characteristics. Secondary considerations are reasonable
economy in construction cost and ease of staking the curve in
the field.
Direction of
(T.C.) A
Back
Tangent
survey
(P.I.) V
CHAPTER 2

T
R
K
90° CM
Kźl
I
E
L.C.
T
12
90°
Ź 1 →
/R
O
Fig. 2-1. Simple-curve layout
B(C.T.)
Forward
Tangent
Horizontal curves are usually arcs of circles or of spirals.
Generally, the circular arc makes up the greater portion of a
curve. The arcs of varying radii, or spirals, provide a gradual
transition between the circular arc and the tangents.
SIMPLE CURVES
13
Vertical curves joining straight sections of grade line are
invariably parabolic arcs. When vertical and horizontal
curves overlap, their study is simplified by considering them
separately.
A
2-2. Definitions and Notation.-A simple curve consists
of a circular arc tangent to two straight sections of a route.
Though spiral transitions are commonly used at the ends of
circular arcs on modern highways and railroads, a thorough
knowledge of the simple curve-its basic geometry, calcula-
tion, and method of staking-is necessary for an understand-
ing of more complex curve problems.

T
R
~I~
Fig. 2-1(a)
E
OK
10
There is no universally accepted notation. That shown in
Fig. 2-1, however, is commonly used in recent practice. (In
this diagram, parts of the curve layout frequently surveyed
in the field are drawn as solid lines, whereas geometric con-
struction lines and curve parts infrequently surveyed are
shown dotted.) The intersection of the tangents at V is
called the vertex, or point of intersection, abbreviated P.I.
14
SIMPLE CURVES
The deflection angle between the tangents is denoted by I; it
is equal to the central angle of the curve.
For a survey
progressing in the direction indicated, the tangent up to the
P.I. is called the initial tangent, or back tangent; that beyond
the P.I. is the forward tangent. The beginning of the circular
arc at A is known as the T.C. (tangent to curve); the end at
B, as the C.T. (curve to tangent). In a simple curve the
T.C. and the C.T. are equidistant from the P.I. The T.C. is
sometimes designated as the P.C. (point of curve) or B.C.
(beginning of curve). Corresponding terms for the C.T. are
P.T. (point of tangent) and E.C. (end of curve).
Certain lines on the curve layout are very useful in calcula-
tions or for field work. Those shown in Fig. 2-1 are: the dis-
tance from the P.I. to the T.C. (or C.T.), known as the tangent
distance, T; the distance from the P.I. to the mid-point K of
the curve, called the external distance, E; and the radius of the
circular arc, designated by R. Also shown, though of lesser
importance, are the long chord, L.C., which is the distance be-
tween the T.C. and the C.T.; and the middle ordinate, M,
or the distance from the mid-point C of the long chord to the
mid-point K of the curve.
2-3. General Formulas. Two trigonometric functions,
used rarely in plane surveying, are especially convenient in
route surveying formulas. These are the versed sine (vers)
and the external secant (exsec).
By definition, for any angle A, vers A = 1−cosA and exsec A
=sec A-1. (The student should study the graphical signifi-
cance of these relations shown on page 637).
From Fig. 2-1 the following basic formulas may be written
practically by inspection:
T-R tan I
E=R exsec I
L.C. 2 R sin I
M=R vers I
(2−1)
(2-2)
(2–3)
(2-4)
Some other useful expressions may be derived by combining
the basic formulas and using trigonometric conversions.
SIMPLE CURVES
15
However, their derivation directly from sketches will illustrate
further interesting properties of the simple curve.
A portion of Fig. 2-1 is reproduced in Fig. 2-1(a). If arc
VA' is drawn with radius OV(A' being on OA produced), the
triangle A'OV will be isosceles and angle A' must equal
90° - I. Therefore the angle at V in the right triangle A'AV
must equal I, from which
E = T tan I
(2-5)
In Fig. 2-1 the circle inscribed within triangle ABV must
have its radius equal to M and its center located at point K.
From this fact it follows that
and
Other useful relations are -
Full sta.
100 ft
100
Do
R\
Dc
90
M = E cos I
E=R tan I tan I
M=R(1-cos
I)
*
/
Full sta.
Full sta.
100 ft
C
KDax
1/
४
(a)
(b)
Fig. 2-2. Definitions of degree of curve
/R
(2-6)
(2-7)
(2-8)

Full sta.
2-4. Degree of Curve.-The curvature of a circular arc is
perfectly defined by its radius. However, where the radius
is long, as on modern highway and railroad alignment, the
16
SIMPLE CURVES
center of the curve is inaccessible or remote. In this case
the radius is valueless for surveying operations, though it is
still needed in certain computations; it must be replaced by a
different characteristic of the curve which is directly useful
in the field. The characteristic commonly used is known as
the degree of curve, D. Though several definitions of degree
of curve may be found, all are based upon the fact that a circle
is a curve having a constant angular change in direction per
unit of distance. The two most widely used are the chord
definition and the arc definition of D.
According to the chord definition the degree of curve is the
central angle subtended by a 100-foot chord. It is denoted by
De, as indicated in Fig. 2–2(a).
According to the arc definition the degree of curve is the
central angle subtended by a 100-foot arc. It is denoted by
Da, as indicated in Fig. 2-2(b).
By inspection of Fig. 2–2(a), it is seen that
In Fig. 2-2(b),
from which
sin Dc
Da:100=360°:2 π R
Da=
-
=
50
R
Ꭰ
(2-10)
It is rarely necessary to use equation 2-9 or equation 2-10.
When D is given, the value of R or log R should be taken from
Table I. For approximate calculations using either defini-
tion of D,
(2-11) Approx.
Neither definition of D is perfectly adapted to all phases of
calculation and field work. Both involve slight approxima-
tions when doing field work the usual way, though high
precision is obtainable by introducing certain small corrections.
The chord definition has been used almost invariably by the
railroads. It is sometimes called the "railroad definition” of
D. On the other hand, highway practice has tended toward
greater use of the arc definition. However, modern standards
of alignment for high-speed operation over both railroads and
5,729.58
R
(2-9)
5,730
Ꭱ
SIMPLE CURVES
17
highways have practically reduced to an academic matter all
controversy over the merits of a particular definition.
A 2° curve computed and staked according to either defini-
tion is substantially the same curve on the ground. Never-
theless, it will usually be necessary for the engineer to conform
to the definition of D used by the organization with which he is
associated. Moreover, he may frequently be required to use
the other definition of D, as in performing calculations for
fitting a relocation to existing alignment. For these reasons
the theoretical treatment in this book covers both definitions.
The tables in Part III are exceptionally complete, as they are
designed to simplify curve calculations based upon either
definition of D.

I
Sta. 16+00
d
D
iy
//
لا
Sta. 16+50
Fig. 2-3. Subchords
Sta. 16+80
Sta. 17+00
2-5. Measurements on Curves.-Regardless of whether the
chord definition or the arc definition of D is used, measure-
ments along a curve must be made by taping a series of chords.
An isolated curve may be staked conveniently by dividing the
central angle into an equal number of parts. The resulting
chords with their directions determined by any appropriate
method then form an inscribed polygon of equal sides. How-
ever, on most route surveys the curves are parts of the con-
18
SIMPLE CURVES
tinuous alignment over which it is convenient to carry the
regular survey stationing without a break. In consequence,
a curve rarely begins or ends at a whole station; accordingly,
chords less than 100 feet long, called subchords, will adjoin
the T.C. and the C.T. The methods of treating these sub-
chords are responsible for the slight approximations or correc-
tions mentioned in Art. 2-4.
Fig. 2–3 represents a portion of a circular arc having certain
stakes located thereon. For the chord definition of D, sta-
tions 16 and 17 will be exactly 100 feet apart as measured
along the chord joining those stations. For the arc definition,
however, the stations will be slightly closer together since, by
definition, they are separated by exactly 100 feet as measured
around the arc.
It is frequently necessary to set stakes on a curve at closer
intervals than at whole stations, as at sta. 16+50, for example.
The only logical position for this stake is exactly midway be-
tween stations 16 and 17, regardless of the definition of D.
Thus, sta. 16+50 would be separated from the adjacent whole
stations by chords which are of equal length but are not 50
feet long. They would be slightly greater or slightly less than
50 feet, the actual length depending on the definition of D.
Such chords are loosely referred to as “50-foot subchords.”
On precise work, however, it is advisable to use a more
definite terminology. All uncertainty is eliminated by calling
the actual length of a chord its true length, and designating as
the nominal length the value found by taking the difference
between the stationing at the ends of the chord. The nominal
length may be further illustrated by imagining the arc in
Fig. 2-3 between stations 16 and 17 to be divided into exactly
100 equal parts. Theoretically, sta. 16+80 is at the 80th
division, and it is joined to sta. 16 by an 80-foot nominal sub-
chord; but it is located in the field by taping a true subchord of
80 feet from sta. 16.
A summary on lengths of subchords follows:
For chord definition of D-True subchords are longer
than nominal subchords.
For arc definition of D-True subchords are shorter
than nominal subchords; nominal subchords are equal to
true lengths of arc.
SIMPLE CURVES
19
If d, Fig. 2-3, is the central angle subtended by any nominal
subchord cn, then d: D=cn: 100, or
The true subchord is
or
d=
I).
CnD
100
ct=2 R sin d
(2-13)
Note the analogy of formula 2-13 to formula 2-3. True
subchords are given in Table II.
2-6. Length of Curve. The length of curve, L, is the
nominal distance around the curve. It equals the difference
between the stationing of the C.T. and the T.C.
For either definition of D,
L:100=I:D
L
(2-12)
1001
D
(2-14)
For the arc definition of D, formula 2-14 obviously gives
the true length of the total circular arc; whereas, for the chord
definition, it gives the total length of the 100-foot chords and
nominal subchords between the T.C. and C.T. In the latter
case the true length of arc is slightly greater than L. If
needed, it may be obtained theoretically from the relation
L=
but a more practical method is to use Table II
57.2958'
or Table VI.
Example.—Given: I=34°14′; D.=9°; R=637.275 (Table
34°=0.593412 rad. (Table VI)
14'=0.004072 rad. (Table VI)
Sum=0.597484 rad.
=380.76
For check of above by Table II see example on page 20.
2–7. Use of Tables.-The formulas developed thus far
may be solved with the aid of tables of logarithmic or natural
trigonometric functions. However, certain calculations are
expedited, or even eliminated, by using special curve tables.
The example on page 20, given in outline form, illustrates the
use of some of the tables given in Part III.
True length of arc
=0.597484X637.275
20
SIMPLE CURVES
Example.-Given: I=34°14′; D=9°.
For chord definition of D
T=1764.5÷÷9
= 196.06
+ 0.19
T=196.25
E=265.5÷9
= 29.50
+ 0.03
E=29.53
L.C.=2R sin I
Result, using Tables I and
XX (or XXII): L.C.=375.12
M = R vers 1
Result, using Tables I and
XX (or XXIII): M=28.23
(Table VIII)
(Table IX)
1007 100X34.2333
L=
D
9
or, to avoid decimals of de-
grees, convert I and D to
minutes, giving
L:
205,400
540
100
50
82.48
(Table VIII)
(Table X)
True
For true length of arc (if
needed) use arc for 1 sta.
=100.103 (Table II). Then,
excess arc in L stations
0.103×3.80=0.39.
100
50.04
82.51
Therefore, true arc length-
380.37+0.39 = 380.76
Examples of true subchords
Nom-
inal
380.37
=
(Table II)
(Table III)
For arc definition of D
T=1764.5÷9 (Table VIII)
= 196.06
E=265.5÷9
= 29.50
L.C.=2R sin I
Result, using Tables I and
XX (or XXII): L.C.=374.74
M=R vers I
Result, using Tables I and
XX (or XXIII): M=28.20
(Table VIII)
L=
**********
1001
Ꭰ
Compute L as shown in ad-
jacent column, giving
L=380.37
100
50
82.48
From definition of Da, the
true length of arc equals L as
computed above.
Examples of true subchords
Nom-
inal
True
99.90
49.99
82.42
(Table II)
(Table II)
(Table IV)
SIMPLE CURVES
21
The example on page 20 not only shows the utility of certain
tables but also gives an idea of the effect of using different
definitions of D. Whether the differences between the com-
puted values of corresponding parts of the curve are in-
significant or important depends on the circumstances. If
the curve is one staked on semi-final location prior to grading,
they are probably unimportant; if the curve is staked on final
location, for setting track centers or forms for concrete pav-
ing, the differences are large enough to warrant close adherence
to a particular definition of D in all calculations. On modern
high-speed alignment where D is 3° or less, the two definitions
of D give practically the same results.
2-8. Locating the T.C. and C.T.-In locating a curve on
a projected alignment, the tangents are run to an intersection
at the P.I., the angle I is measured, and the stationing is
carried forward along the back tangent as far as the P.I.
(see Art. 7-3 for a description of the process when the P.I. is
inaccessible). The degree of curve will usually have been
selected from the paper-location study. If it has not been, a
suitable value may be determined by measuring the approxi-
mate E (or T′) needed to give a good fit with the topography.
The tabulated E (or T) for a 1° curve (Table VIII) divided
by the approximate E (or T) gives a value of D which will fit
the field conditions. Usually this D is rounded off to a figure
convenient for calculation, and of course it must be within the
limiting specifications for the project.
The values of T and L are next computed, and from them
the stationing of the T.C. and C.T. are determined as follows:
Sta. P.I.-T=Sta. T.C.
Sta. T.C.+L (nominal) = Sta. C.T.
For staking and checking the curve it is necessary to set
hubs at the T.C. and the C.T. This is done by taping the
calculated T backward and forward from the P.I. Sometimes
it is more convenient to set the T.C. by taping from a tacked
hub (P.O.T.) on the back tangent. In case the curve is long
or the terrain is difficult for taping, it is also advisable to set a
check hub at the mid-point of the curve by taping the exact
value of E from the P.I. along the bisector of the angle.
22
SIMPLE CURVES
2-9. Deflection-Angle Method. The convenient deflec-
tion-angle method of locating points on a simple curve is based
on a proposition in geometry: an angle formed by a tangent
and a chord is measured by one-half the intercepted arc.
In Fig. 2-4, a, b, and c represent 100-ft stations on a portion
of a simple curve, a being less than one station beyond the T.C.
By definition, each central angle between full stations is equal
to D; the angle subtended by the first subchord is d₁. The
angles VAa, VAb, and VAC are known as the deflection angles,
or total deflections, to the stations on the curve. From the
foregoing proposition they are equal to one-half the corre-
sponding central angles. Thus, the deflection angle to locate
c from a set-up at the T.C. is }(d₁+2D).
The subdeflection for the fractional station is d₁. Once this
first subdeflection has been computed, the total deflections to
succeeding points are found by adding the inscribed angles
aAb, bAc, ... . Since an inscribed angle equals one-half its
central angle, the total deflections to full stations are found
by adding successive increments of D. A second subchord
c2 falls between the C.T. and the preceding full station. Its
subdeflection d½ should be added to the total deflection for
the last full station on the curve. If the sum equals 1, the
arithmetic is checked.
In practice, subdeflections are usually small and are best
computed in minutes from either of the following formulas:
d (in minutes)=0.3 cn D°
(2-15)
d (in minutes) = cn X defl. per ft*
(2-16)
The distances taped in connection with the deflection angles
are not the rays from the T.C. to the various points on the
curve. Theoretically they are the successive true chords from
point to point, starting at the T.C. Therefore, once the deflec-
tion angles have been figured, there is practically no added
computation. However, the non-coincidence of tape and line
of sight may be slightly confusing in the field work.
*In formula 2-16, defl. per ft means "deflection in minutes per foot of
nominal chord" (or per foot of station). Thus, if D=9°=540', }D=270′
and defl. per ft=2.7'. Values of defl. per ft are listed in Table I.
SIMPLE CURVES
23
Example. Given: Sta. P.I.=24+63.8; I=34°14'. Stakes
to be set for semi-final highway location; curve to pass approxi-
mately 30 ft inside vertex; stakes needed at full stations and
half-stations; distances to tenths and deflections to nearest
minute adequate. Use arc definition of D.
265.5
Approx. Da= =8.85° (Table VIII).
Round off to 9°
30
curve with defl. per ft=2.7'. The problem now becomes that
in Art. 2-7, from which T=196.1, L=380.4, and E=29.5.
A (T.C.)
To V (PI)

|
kd₁ D
|
т
Sta. P.I.=24+63.8
-T= 1+96.1
Sta.T.C.=22+67.7
Sta.C.T.=26+48.1
---|-·
*
D
上
​Fig. 2–4. Deflection-angle method
b
C₁ = 32.3
+L= 3+80.4 23 to 26=300.0
ź (d₁+2D)
d₁=2.7×32.3= 1°27′
DX 3 =13°30′
C₂ = 48.1 d₂=2.7×48.1= 2°10'
Sum=380.4
= L
(check)
Sum=17°07'
= 10
(check)
There follows a typical form for setting up the notes on the
left-hand page of the field book. Notes run upward in order
to simplify orienting sketches on the right-hand page, where
the center line is used to represent the survey line.
24
SIMPLE CURVES
27
26
25
24
23
22
Sta.
+50
+50
Point
+50
Form of notes for simple curve
Total
Defl.
Calc.
Bearing
+50
Da=9° rt.
+48.1 OC.T. 17°07′ N50°41′E N50°E P.I.=24+63.8
14°57'
T=196.1
12°42'
L=380.4
E = 29.5
10°27'
8°12'
Defl. per ft
5°57'
= 2.7'
3°42'
1°27'
+67.7 OT.C. 0°00′ N16°27′E | N161°E
+50
Mag.
Bearing
Set-up at
sta. 23+50
Curve
Data
2-10. Transit Set-ups on Curve.-In the preceding descrip-
tion the implication is that the entire curve is staked from a
set-up at the T.C. This is often true. However, there are
T.C.
23
23+50
I=34°14'

24
To P.I.
Tangent 342
0°00"
-5°57′
Fig. 2-5. Moving up on curve
circumstances (see Art. 2-11) which make this procedure im-
practicable or even impossible. Should such be the case,
deflections are computed as though for a set-up at the T.C. By
SIMPLE CURVES
25
proper manipulation of the transit it can be set up at any
staked point on the curve, and the staking can then continue
from that set-up without altering the previously computed
deflections.
Fig. 2-5 shows the first portion of the curve calculated in
Art. 2-9. It is assumed that the curve has been staked as
far as sta. 23+50 from a set-up at the T.C., and that condi-
tions make it difficult to sight beyond from that set-up. If,
with the transit moved to sta. 23+50, the vernier is set to 0º
and the telescope is inverted and then backsighted to the T.C.
and plunged, the condition will be as represented by the dotted
line from the T.C. through sta. 23+50. If now the upper-
motion is loosened and the telescope is turned clockwise, the
line of sight will be on the local tangent when the vernier reads
3°42′ and will be in the correct direction for locating sta. 24
when the vernier reads 5°57'. These facts follow directly from
the geometric principle noted at the beginning of Art. 2-9.
An extension of Fig. 2-5 would show that, for a set-up
at sta. 25 with the transit to be oriented by a backsight to
the previous set-up at sta. 23+50, the vernier should first be
set at 3°42′ on the correct side of 0°. Backsighting with the
telescope reversed and then plunging would permit sighting
the remaining points on the curve by using the total deflections
opposite those stations in the transit notes. No added calcu-
lations are required if this method is followed; the only record
of the set-ups is the symbol O in the Point column of the notes
on page 24.
The preceding description may be summarized in a rule-of-
thumb:
To move up on curve and retain original notes, occupy
any station and backsight to any other station with the
vernier set to the total deflection of the station-sighted.
Occasionally it may be desired to have the vernier read 0°
when the line of sight is on the local tangent to the curve at a
new set-up, as at the C.C. of a compound curve (see Art.
3-12). To do this the vernier must be set at the difference
between the tabulated deflections for station sighted and station
26
SIMPLE CURVES
occupied.* As in applying the rule-of-thumb, the surveyor
must be careful to set the vernier reading for the backṣight on
the correct side of 0°.
2-11. Comments on Field Work.-Transit set-ups for
staking the curve must be started at the T.C. or C.T. For
short curves which are entirely visible from one set-up, it is
preferable to occupy the C.T. and to tape the chords toward it
from the T.C. By so doing, the longer sights are taken before
possible settlement of the transit occurs. Moreover, one set-
up is eliminated, for the transit is then in position for lining
in the stakes along the forward tangent. For a set-up at the
C.T., the transit is oriented by backsighting to the T.C. with
the vernier reading 0° or to the P.I. with the vernier set at
1. In either case it is wise to check the angle to the other
point, in order to verify the equality of the tangent distances.
In case the field conditions require one or more set-ups
on the curve, it is good practice to occupy the T.C. first and
to tape in a forward direction, moving up the transit according
to the rule-of-thumb stated in Art. 2-10. The final portion
of the curve is best located by setting up at the C.T. and tap-
ing backward to the previous set-up. This practice insures
good tangencies at the T.C. and C.T., and throws any slight
error into the curve where it is more easily adjusted.
•

A long curve, or one having a large central angle, may
justify intermediate set-ups even though the entire curve is
visible from either end. The consideration here is the required
degree of accuracy in setting the points on the curve. When
the tape and the line of sight intersect at a large angle, it is
possible for the chainman to swing one end of the tape through
a certain distance without detection by the transitman, thus
throwing off the position of the point and introducing an
accumulative taping error. The amount of this swing-and
its relative importance on the particular work-is best deter-
*When this book is used as a textbook for a course in Route Surveying, it
is recommended that students be given the opportunity to compute and
stake at least one simple curve at this point in their study. Doing so will
help to fix the principles more firmly in mind and contribute toward a better
appreciation of the practical suggestions which follow.
SIMPLE CURVES
27
mined by trial in the field. As a general guide, the deflection
angle from the transit to the taping point should be kept under
25° or 30°.
Field checks should be made at every convenient oppor-
tunity. In locating the T.C. and the C.T., it is good insur-
ance to double-chain the tangent distances, at least until the
chainmen have demonstrated the reliability of their first
measurements. Less time will be lost if the check chaining is
done while curve deflections are being computed. Setting a
check point at the middle of the curve may also be advisable
(see Art. 2–8).
sight
b
རིམ་དུ་གྱུར་པས་ཚོགས་པ་ཡོད།
sight
C

sight
m
Fig. 2-6. Check of stake positions
d
Before definitely setting each stake on the curve, the head
chainman should sight to the second stake back and verify the
middle ordinate m to the intervening stake. (See Fig. 2–6.)
A sudden change in middle ordinate (for equidistant points on
the curve) will reveal a mistake in setting off the deflection
angle. Otherwise, such an error might go undetected at the
time or even fail to show up in the final closing check if the
curve is flat.
The final check occurs at the extremity of the last chord
taped. This may be at the T.C., at the C.T., or at any inter-
mediate stake previously set. The best method is to mark an
independent point by measuring the required distance and
deflection from the final set-up. The distance between this
point and its theoretical equivalent at the stake previously
set is a direct measure of the error of closure in the traverse
around the tangent distances and curve chords. This error
should be consistent with the survey methods, the difficulty of
the terrain, and the requirements of the project.
Chord lengths used in running curves depend principally
on the degree of curve and on the purpose of the survey. On
28
SIMPLE CURVES
flat curves in easy terrain, full stations suffice for the field work
and for most construction purposes. Thus, for De true chords
between full stations are exactly 100 feet long, and the end
subchords may be considered equal to their nominal lengths
(see Tables II and III). For Da the end subchords may also
be taken equal to their nominal (arc) lengths, but the chord
for each of a long series of full stations may require a slight
correction in order to reduce the systematic error in taping
(see Tables II and IV).
Stake...
---Flag

-sight-
Pin
Fig. 2-7. Setting station stakes
On sharp curves, or in difficult terrain requiring frequent
cross-sections, additional points between full stations are
needed. Whether it is necessary to use true subchords instead
of nominal subchords depends on the accuracy required.
Table II shows at a glance the effect of the sharpness of the
curve on the true lengths of nominal full chords and sub-
chords. When Table II is used in conjunction with Tables III
and IV, any degree of accuracy is readily obtainable. Study
of Table II shows in general that chord corrections are unneces-
sary on curves flatter than about 5°. On sharper chord-defini-
tion curves, corrections may be eliminated by using 100-ft
chords; whereas, on sharper arc-definition curves, shorter
nominal subchords serve the same purpose.
Stakes are set on curves where needed for subsequent field
work or for controlling construction. A tacked hub, with a
guard stake designating the point and stationing, is set at each
transit station. Important points, such as the P.I., T.C., and
C.T., are carefully referenced. Points not occupied by the
SIMPLE CURVES
29
transit are identified by station stakes, which are driven and
marked as in profile leveling.
Station stakes used for cross-sectioning prior to construc-
tion need not be individually centered by the transit. Doing
this is of no practical value, and it wastes time if the stakes
are hidden by grass or brush. A rapid method is shown in
Fig. 2-7. The head chainman first obtains line from the
transitman and then sets a range pole (flag) off line about the
width of a stake and a few tenths of a foot beyond the station.
He next verifies the middle ordinate (see Fig. 2-6). Then he
stretches the tape tangent to the range pole and sets a chain-
ing pin at the exact distance. Finally, he uses the point of
the range pole to make a hole beside and slightly behind the
point where the pin enters the ground, and proceeds to the
next station with the pole and tape. The stakeman imme-
diately drives a marked station stake into the hole at the pin,
at the same time keeping the moving tape to one side with his
foot. The pin should not be disturbed in the process; it
remains in place until pulled by the rear chainman after the
next forward pin is set.
By following the foregoing method on tangents and flat
curves, station stakes will always be close enough in both line
and distance. On sharp curves where the angle between tape
and line of sight is large, it may be slightly better to set the
range pole practically on the line of sight.
In setting points on curves the head chainman can expedite
the work by placing himself quickly on line at the point to
be set. One method of doing this, especially suited to open
country, is for the head chainman to range forward the proper
chord distance along a line through the last two stakes, plac-
ing himself in position as at d' in Fig. 2-6. By then pacing
inward the known chord-offset distance (Art. 2-14), he will
be in approximately the correct position for setting the re-
quired point at d. With experience the head chainman will
invariably be in the field of view by the time the transitman
sets off the deflection angle and sights.
Many variations are possible in the technique described.
The range pole may be dispensed with for short sights in
open country. Another device used where frequent plus
30
SIMPLE CURVES
points are required is to set those station stakes directly at the
proper division on the tape and to move the tape forward at
each full station. The important factors to consider are speed,
accuracy consistent with the needs, and avoidance of taping
errors. In all cases, carrying the taping forward by chaining
from pin to pin is advisable.
After grading, all points on curves which are set to control
construction details must be more carefully centered than in
the methods just described.
On pavements or in other locations where stakes cannot be
driven, the points are chiseled or marked with keel, or a nail
may be driven flush through a small square of red cloth. Guard
stakes marked with the stationing are set at a convenient off-
set distance to one side.
and
Minor obstacles to setting calculated points on curves are
often met in running location prior to clearing or grading.
These should not be permitted to delay the field work.

T.C.
To P.I.
23
+50
+90
24 (Omit)
+20
-Tree
Fig. 2-8. Passing minor obstacle
Fig. 2-8 shows another sketch of the first portion of the
curve calculated in Art. 2-9. If the line of sight to sta. 24
were to hit a large tree, for example, it could be backed off
to the left of the tree for setting a stake at any convenient plus
point. Thus, if sta. 23+90 were in the clear, its deflection
would be 40×2.7' greater than that just used to set sta. 23+50.
The calculations-40X2.7' 108'1°48' and 3°42′+1°48′=
5°30′-may be made mentally in a few seconds. After the
station stake at 23+90 is set, another new plus point to the
right of the tree (such as 24+20) could be staked by following
a similar process; or else the taping could proceed from sta.
23+90 directly to the next calculated point at sta. 24+50.
SIMPLE CURVES
31
A minor obstacle on the curve itself (such as a large boulder
at sta. 24) could be by-passed by the same method.
Major obstacles are treated in Chapter 7.
This reduces to
2-12. Even-Radius Curves.-Some organizations, includ-
ing a few highway departments, do not use the degree of curve
designation either in calculations or in field work. Instead,
a curve is chosen on the basis of its radius, which usually is a
round number. Deflections are calculated theoretically from
the following proportion:
deflection: arc length=360°:2 π R
defl. (in minutes)
1,718.873
R
=
Xarc length
(2-17)
This method has the advantage that radii may be selected
which are convenient for calculating curve parts. The radius
would also mean more to the average person than the degree,
should it become the practice to post warning values of R
at the approaches to highway curves. The calculation of
deflections, however, is much less simple than with a rounded
value of D, and their fractional values are inconvenient in
the field.
The process of calculating and staking these “even-radius
curves" is expedited by use of Tables V and VI. The latter
table is used to compute L when I and R are given. Other
curve parts are found from the usual formulas (see Art. 2-3).
Table V gives directly most of the data concerning deflections
and chord lengths needed in the field. Subchord deflections
are readily computed from the tabulated deflections per foot.
2-13. Metric Curves.-In countries having the metric
system, distances are usually measured with a 20-meter tape,
and a "station" is nominally a distance of 10 meters. On route
surveys either the radius in meters or the degree of curve
may be used in selecting and staking curves. However, the
customary metric "degree of curve" Dm differs from those
already defined not only in unit of measurement but in angle.
and length of chord (or arc) as well. Specifically, Dm is the
32
SIMPLE CURVES
deflection angle for a chord (or arc) of 20 meters. Thus, by
analogy from equations 2-9 and 2-10,
sin Dm (chord def.)
and
Dm (arc def.)
572.958
Rm
Also, by analogy from equation 2-14,
Im
(2-20)
It is not necessary to use special metric-system tables to
compute and stake metric curves. The foregoing simple rela-
tions used in connection with the tables in Part III of this
book will enable an engineer to handle any metric-curve prob-
lem. If Table VIII is used, for example, the tangents and
externals for a 1° metric curve are one-tenth the tabulated
values (for the same central angle). This fact follows from
equations 2-10 and 2–19.
For a quick comparison between metric and foot curves, it
is well to remember that for curves of the same radius the ratio
between D and Dm is about 3 to 1 (actually 3.048 to 1). In
other words, a curve of given Dm is about 3 times as sharp as
one having the same numerical value of D.
=
107
Dm
**********
10
Rm
(a) Chord offsets
(b) Middle ordinates
(c) Tangent offsets
(d) Ordinates from a long chord
(2-18)
(2-19)
2-14. Staking by Offsets.-There is no good substitute for
the deflection-angle method when points must be set quickly
and accurately on long-radius curves. It is mathematically
possible, however, to devise several other methods based either
upon angles from the P.I. or upon offsets. The former basis is
rarely practicable, but there are occasions where an offset
method is very useful.
The offset methods* are known as:
*Following the derivation of some needed mathematical relations, only
the chord-offset method is described in detail. Abbreviated descriptions of
the remaining methods are enough to serve as a basis for applying them.
SIMPLE CURVES
33
The chord offset, tangent offset, and middle ordinate for a
full station are designated by C.O., T.O., and M.O.; cor-
responding terms for any other distance are c.o., t, and m.
The true chord for a full station is 100 feet or C, the length de-
pending on the definition of D.

Tangent
and
C or 100'-
KD
|ŹD
1
C or 100'
D
Also, by similar triangles,
Hence,
M.O.
LR
3-
Fig. 2-9. Offsets for full station
From Fig. 2-9 the following exact relations may be written
practically by inspection:
C=2 R sin Da
C.O. (chord def.) = 200 sin De
C.O. (arc def.) = 2 C sin Da
T.O. = C.O.
M.O.= R vers D
C.O.:100 (or C) = 100 (or C): R.
1002
R
C.O. (chord def.):
______
C.O. (arc def.) =
=
C2
R
C.O.
Four useful approximate relations are:
M.O. = C.O.
C.O.=13 D
T.O.
(2-21)
(2-22)
(2-23)
(2–24)
(2-25)
(2–26)
(2–27)
(2-28) Approx.
(2–29) Approx.
34
SIMPLE CURVES
7
t = s² D (s is chord in stations)
m (in inches for a 62-ft chord) =D
(2-30) Approx.
(2-31) Approx.
The approximate 8:4:1 ratio of the chord offset, tangent
offset, and middle ordinate is worth memorizing (mnemonic:
"eight for one").
Offsets for any subchord (or subarc) may be found by first
determining the values for a full station and then applying
the principle that c.o., t, and m are proportional to the squares
of the subchords (or subarcs). They may also be found from
the exact relations:
$1
A
C.O.
=
C2
t
R
C2
2R
m = R vers d
(2-33)
(2-34)
Table I contains values of certain of the full-station offsets.
The explanatory notes preceding that table show its useful-
ness in obtaining any offsets accurately.
Chord offsets, sometimes called "deflection distances,"
provide a means of setting station stakes rapidly when a
transit is not available. If the tapeman sights past his plumb-
bob cord, surprising accuracy is possible. It is invariably
adequate for surveys made prior to clearing and grading.
C1
T.C. C₂ 2
た
​Bi
im
---
(2-32)

1
Fig. 2-10. Staking by offsets
To P.I
C.O.
SIMPLE CURVES
35
In Fig. 2-10 (curvature greatly exaggerated) the P.I. and
the tangent points are assumed to have been set by the usual
methods. If the transitman and some of the field party now
proceed along the forward tangent for purposes of setting
stakes or measuring the angle I for the next curve, three men
(or even two) can set the station stakes on the curve left
behind.
The procedure follows: Knowing the stationing of the T.C.,
calculate the subchords c₁ and c₂ to the adjacent full stations
at A and B (sta. A is on the curve produced backward). Then
calculate t₁ and t2, which are the tangent offsets for the sub-
chords. The field work may now be started after range poles
have been set at convenient points on the tangent. Establish
a temporary point at A by the "swing offset" method. Locate
the station stake at B similarly. Range the line AB ahead a
full-station chord to C', and locate the next station stake at C
by the C.O. Continue around the curve by full-chord offsets
(obviously, half-stations or quarter-stations could be used
instead) until the full station preceding the C.T. has been set.
Make the final check by measuring the last subchord and the
swing offset to the forward tangent; check them against their
theoretical values.
In an exceptional situation, a curve may be selected to fit
the topography and may be completely staked without use of
the transit or tables of any kind; it is assumed that the prin-
ciples are understood and that the approximate relations 2–29
and 2-30 have been memorized. For example, suppose that
the back tangent has been partially cleared, but the P.I. has
not been established and D has not been chosen. To run the
curve, set the T.C. at a point fitting the topography, tape
toward the P.I. a certain distance (possibly 200 feet), take a
right angle by eye, and measure t to a suitable random point on
the tentative curve. For all practical purposes the chord dis-
tance c from this point to the T.C. may be taken equal to the
distance along the tangent unless t is greater than 20 per cent
of the tangent measurement. Now compute D from the
relation 2-30, and round off the result to a convenient value.
Obtain the C.O. from 2-29, compute the values of t₁ and të for
the subchords at the T.C., and stake the complete curve by
chord offsets as previously described. No check can be made
36
SIMPLE CURVES
at the C.T. unless I (and consequently L and the stationing of
the C.T.) can be determined by some means.
Middle ordinates are also useful in staking a curve by tape
alone. Their most valuable use, however, is in checking and
realigning existing curves on railroads. (See Art. 9-17.)
In staking problems, first locate full stations either side of
the T.C. by swing tangent offsets, as in the chord-offset method
just described. Then set the next full station by sighting from
the rear station along a line m feet inside the forward station,
and measure the full chord distance from the forward station
(see Fig. 2-10). Continue this process to the stake before the
C.T., and make the final check as in the chord-offset method.
(Fractional stations could be used instead of full stations.)
It should be noted that, when stakes are set at full stations,
the distance m is the middle ordinate for two stations; con-
sequently, m=T.O. = 1 C.O.
Topographical conditions sometimes make this method
preferable to chord offsets. It is probably better on sharp
curves, regardless of topography, because the offsets are only
half as long.
Tangent offsets provide a handy means of setting station
stakes by tape alone. The method is essentially one of rec-
tangular coordinates, the T.C. being used as the origin and the
tangent being used as an axis.
The deflection d to the first regular station is first com-
puted. The tangent distance to this station is c cos d, and
its offset is c sin d. Coordinates of remaining full stations
are obtained by adding increments of C cos (d+1D) and C' sin
(d+1D), C cos (d+3D) and C sin (d+3D), etc., where C is
the chord for a full station. Since in most cases C=100 feet,
the computation is very simply performed by using natural
sines and cosines. As a check on the computations, the co-
ordinates of the C.T. should equal R sin I and R vers I. If
tangent distances fall beyond the P.I., it is probably better to
calculate the second half of the curve with reference to the C.T.
and the forward tangent.
In the field, set temporary stakes at the calculated tangent
distances. Then locate the station stakes at the intersections
of the tangent offsets and the chords from the preceding
stations.
SIMPLE CURVES
37
Ordinates from a long chord are also based upon rectangular
coordinates. As in the tangent-offset method, the origin is
at the T.C., but the axis is along a convenient long chord.
If the chord is the L.C. of the curve, simple relations for the
coordinates of the station stakes may be expressed (these will
not be derived). However, it is difficult to imagine many
practical situations in which one of the foregoing methods
would not be better than this.

C
B
01 02
03
A_%Dq=g
Fig. 2-11. Ordinates from a long chord
0%
• ·
الا
Perhaps the most useful application of this method is in
setting stakes at odd plus points between a series of minor
obstacles on a curve. Fig. 2-11 illustrates such a situation.
Station E is set by turning a deflection angle with the transit
at A and taping the chord AE, which is 2 R sin AOE.“
Stakes set sufficiently close for cross-sectioning are then located
by taping the offsets 01, 02, at convenient points along AE.
If the segments of AE at any ordinate are designated by si
and 82 (units are stations),then the length of the ordinate in
feet is found from the relation
E
0 3 S1 S2 D
8
Note the resemblance between formulas 2-30 and 2–35.
(2-35) Approx.
2-15. Parallel Curves. It is often necessary to determine
the length of a curve parallel to a curved center line. This
problem occurs in connection with finding lengths of curb,
guardrail, or right-of-way along curved alignment. More-
over, stakes are often set along a parallel "offset" line in order
to avoid traffic or construction operations. Fig. 2-12 shows
offset curves outside and inside a center-line curve of length L.
I°R
57.2958'
Since the true length of a circular arc equals
38
SIMPLE CURVES
Lo
1° (R+p.)
57.2958
and
and Li
P。
I° (R— pi)
57.2958
INC
Pi
Lo=L+;
(2-37)
Though an offset curve may be staked without reference to
the stationing along the center line, it is more practical to set
stakes at points radially opposite the center-line stakes. This
is done either by the deflection-angle method or by one of the
offset methods described in Art. 2-14. In either case the
'true chords to be taped must be found.
L。
I°po
57.2958
Li=L-
from which
I°pi
57.2958
L
L;
1º
R
I
Fig. 2-12. Parallel curves
(2-36)

C.IT.
True chord lengths can always be found by means of formula
2-13; they can also be taken from Table II with sufficient
precision for most purposes.
As an illustration, refer to the example on page 20 and
assume that it is required to stake one offset curve 60 feet
SIMPLE CURVES
39
Stakes on
outside and another 25 feet inside the given curve.
the offset curves are to be set radially opposite the full and
half stations along the center line. The length of curve L
equals 380.370 ft along the center line. From formulas 2-36
and 2-37 (Table VI is useful), L, equals 416.219 ft and Li
equals 365.434 ft. The lengths of the true chords depend on
whether the 9° curve conforms to the chord definition or arc
definition of D. Using formula 2-13 with d equal to 2°15′,
the true half-station chords are as follows:
True Chords for Half-Station Along Center Line
Chord Def. of D
Arc Def. of D
54.750
54.699
50.039
49.987
48.076
48.024
Outside curve
Center-line curve
Inside curve
2-16. Methods of Computation.-Proper methods of com-
putation and use of significant figures are so important that
this and the following article are inserted here so that the
suggestions can be weighed and then used, wherever appli-
cable, in problems throughout the book.
Route surveying computations require relatively simple
mathematics—arithmetic, geometry, and trigonometry. The
solution of a problem is often tedious in that it involves many
repetitions of the same mathematical operations. As a result,
students (particularly) find the manipulation of surveying
instruments much more interesting. They are likely to con-
clude that surveying field work is of primary importance,
whereas computations are a chore to be done with as quickly
as possible. This is a serious mistake. Field work and
computations are equally important.
Surveying field work alone is of little value; it is always put
to use in some practical form that involves computations.
Unless the computations are made correctly, they negate the
field work and may cause serious errors when used as a basis
for the design or construction of engineering works. The
student can take as much pride in obtaining the answer to a
tedious surveying problem—the correct answer, with the
proper number of significant figures and with the decimal
40
SIMPLE CURVES
point in the right place—as he does in his ability to "run a
transit" rapidly and accurately. There is no better place to
develop the habits of computation which insure this result
than in a course in surveying.
Surveying computations are made algebraically, graphically,
and mechanically, or by a combination of these methods.
Useful tables reduce much of the drudgery (see Part III for
examples of tables).
A surveying problem can usually be solved by more than
one exact method. This should always be done on important
work. Moreover, all computations should be checked, pre-
ferably by a different computer. If an exact algebraic solu-
tion cannot be found, the unknowns may be scaled from an
accurate drawing on which the measured data have been
carefully plotted. This method is reliable to three significant
figures if the scale of the drawing is fairly large.
Among the mechanical aids are the slide rule, planimeter,
calculating machine, and electronic computer. The slide rule
is an excellent device for checking the first three significant
figures. The planimeter will measure irregular areas (see
Art. 6-6) to an accuracy within 1 per cent.
Important surveying computations require accuracy to at
least five significant figures. This degree of refinement may
be obtained with the aid of logarithms (Tables XXI and
XXII) or by calculating machine and natural trigonometric
functions (Table XX). Logarithms are especially useful when
surveying computations must be made "on the spot" in the
field. But desk calculating machines are preferred in sur-
veying offices because of their greater speed.
The electronic computer (Art. 13-4) is used by large organ-
izations where the number of repetitive operations is great
enough to justify the high cost of computer purchase or rental.
Such a machine is not a "brain." It does no thinking, but
only follows coded commands fed into it. Instead of making
the individual obsolete as a computer, the electronic com-
puter requires the surveyor to be even more expert in order
that he may "program" the computer to operate with the
highest efficiency.
Some practical suggestions for making surveying computa-
tions are:
SIMPLE CURVES
41
1. Do all computations on "squared paper" to simplify the
making of sketches and the arranging of work in columns.
2. After transferring data from field book to computation
sheet, check the copying. Mistakes made at this point are
common and are not usually disclosed by the computing
operations.
3. Draw a small sketch, approximately to scale, showing
the known data and the unknowns sought. Aided by the
sketch, make an estimate of the answers to the problem and
record them.
4. Show the formula at the head of the block of computa-
tions to which it applies.
5. Adopt a clear and logical arrangement of computations;
a tabular form fits many problems (see examples in Art. 4-4).
Avoid an arrangement that requires numbers to be added or
subtracted horizontally. Line up digits and decimal points
in a column of numbers.
6. Wherever possible, perform computations in a systematic
order, e.g., addition, subtraction, multiplication, and division.
Do no work on "scratch paper." Incidental computations
may be written in a margin on the computation sheet.
7. Use a straight edge in order to avoid taking a value from
the wrong line of a table. Check additions, subtractions,
extractions of tabular values, etc. as they are made and
indicate by a check mark.
8. Follow good practice in the use of significant figures and
rounding off. In general, carry one extra significant figure in
intermediate calculations. (See Art. 2-17 for detailed sug-
gestions.)
9. Take advantage of all automatic checks on the accuracy
of calculations (see example on page 23 for illustrations).
Make a slide-rule check at convenient points; this will verify
the arithmetic to three significant figures though it will not
disclose errors resulting from the use of the wrong formula.
Any such blunder, or a great error in arithmetic, will usually
be caught by applying a graphical check.
10. Label the final answers and all intermediate results
clearly. Compare the answers with the estimated values
obtained from the sketch and do not accept the work unless
the agreement is reasonable.
42
SIMPLE CURVES
2-17. Significant Figures.-Both exact numbers and round
numbers are used in computations. Exact numbers come
from tallies and theoretical considerations; round numbers
come from measurements. For example, the number 2 in the
formula for L.C. (equation 2-3) is an exact number that
comes from geometric theory. On the contrary, the measured
length of a line is a round number; it can never be exact, for
it would have to be correct to an infinite number of digits.
In computations it is often desirable to change a round
number to an equivalent one having fewer digits. This
process is called "rounding off." Common practice in
rounding off is as follows:
When dropping one or more digits, round off to the
nearer of the two round numbers between which the
given round number lies. If the digit to be dropped
is 5, use the nearest even number for the preceding
digit. (Consider zero to be an even number.)
According to this practice, the round number 18.1827 is
rounded off to 18.183, 18.18, or 18.2, the result depending on
the number of digits desired. Also, the round numbers 85.155
and 85.165 are rounded off to 85.16; the numbers 85.095 and
85.105 are rounded off to 85.10. When properly rounded off,
a measured quantity contains the number of significant figures
consistent with the data.
The significant figures in any round number are those digits
(including zeros) which have real meaning. The maximum
number of such figures includes those digits which are certain,
plus one that is estimated. In making measurements the
number of significant figures sought is governed by economic
considerations; whereas the number obtained is limited by the
precision with which the measurements are made.
Confusion as to which digits are significant is avoided by
observing the following principles (in each example there are
five significant figures and all significant digits and zeros are
printed in italics):
1. The digits 1 through 9 are always significant.
SIMPLE CURVES
43
2. Zeros which lie between any two digits are significant,
regardless of the position of the decimal point. Examples
are 21006 and 21.006.
3. Zeros which lie to the left of the leftmost digit are not
significant, regardless of the position of the decimal point.
Examples are 0.71700 and 0.00071700.
4. Zeros which lie to the right of any digit and also to the
right of the decimal point are significant. An example is
717.00.
5. Zeros which lie to the right of any digit and also to the
left of the decimal point may or may not be significant. For
example, in 71700. the zeros are significant if they were
actually measured. If they were not measured, doubt as to
the number of figures which are significant is resolved by
writing the number as 717 ≈ 102.
6. The number of significant figures in any round number is
independent of the units in which it is expressed. An example
is 586.08 ft 0.11100 miles.
=
It is a waste of time, as well as misleading, to carry com-
puted results beyond the precision inherent in the measure-
ments on which the computations are based. A few examples
will clarify this statement.
Example of Addition or Subtraction.-Add the round
numbers: 9.47, 241.3, 7.6625, and 0.891.
Solution and comments:
Given
Number
No. of
Sig. Figs.
9.47
241.3
7.6625
0.891
3
4
5
3
Wrong
Method
9.47
Rounded answer =
241
7.66
0.891
Sum=259.021
=259
Common Preferred
Method
Method
9.47
9.5
241.3
7.7
0.9
259.4
259.4
241.3
7.66
0.89
259.32
259.3
The “wrong method" comes from a mistaken interpretation
of the adage that a chain is no stronger than its weakest link.
The fault lies in assuming that each number, before adding,
must be rounded off to the least number of significant figures
44
SIMPLE CURVES
possessed by any of the numbers (3 significant figures in this
example), and also that the sum must be similarly rounded off.
In the "common method" the numbers are first rounded
off to the limit of accuracy of the least accurate number
(tenths in this example). Their sum is accepted without
further rounding.
In the "preferred method" the numbers are first rounded off
to one decimal place beyond that of the least accurate number.
Their sum is then rounded off to the nearest digit in the
preceding decimal place (nearest tenth in this example). This
method recognizes the fact that the principal sources of error
lie in the numbers that are not rounded off. In this example,
the final 6 and the final 9 in the last two numbers are certain.
Consequently, the final 2 in the sum may not be far in error,
and the rounded answer 259.3 has a somewhat higher proba-
bility than the answer 259.4.
Example of Multiplication. Multiply the round numbers
362.56 and 2.13.
Solution and comments: When two or more round numbers
are to be multiplied, the common method is to round off each
number to the same number of significant figures as appears
in the least accurate number. Then multiply, and round off
each intermediate product and the final answer to this same
number of significant figures. As applied to this example, the
common method is 363 X2.13=773.19, after which 773 is used
as the intermediate product or the final answer.
A preferred method is to round off each number to one more
significant figure than appears in the least accurate number.
Then multiply, and round off each intermediate product to
this number of significant figures. Finally round off the
answer to the number of significant figures in the least accurate
number. As applied to this example, the preferred method is
362.6×2.13=772.3 as the intermediate product or 772 as the
final answer.
Example of Division.-Divide the round number 675.41
by the round number 87.5.
Solution and comments: As in multiplication, the common
method is 675÷87.5=7.71 as the final answer. A preferred
SIMPLE CURVES
45
method is to round off the divisor to the same number of
significant figures as appears in the dividend, or round off the
dividend to one more significant figure than appears in the
divisor (whichever case applies). Then divide, and round off
the final answer to the number of significant figures in the
rounded divisor. One more significant figure is temporarily
carried in the answer if the quotient is to be used in subsequent
computation involving other round numbers. As applied to
this example, the preferred method is 675.4÷87.5=7.719 as
the intermediate quotient or 7.72 as the final answer.
The difference between the results of the common and
preferred methods will never be large. Both methods recog-
nize that final computed results can be no more accurate than
the least accurate factor used to obtain them. The preferred
method also incorporates the practice, favored by experienced
computers, of carrying one extra significant figure in inter-
mediate calculations so as to avoid the build-up of errors due
solely to rounding off.
PROBLEMS
NOTE. Certain of the practical problems in Chapter 7 may also be
assigned, particularly numbers 7-1 and 7-9.
2-1. Given the following values of I and Dc. After taking
Rc (or log Rc) from Table I, compute T, E, L.C., M, and L
from basic formulas. Find true arc length by Table VI.
Verify answers for T, E, and true arc length by Tables VIII,
IX, X, and II. Pa=
(a) I=16°22′; De=1°40′. Answers: T=494.38; E=35.365;
L.C.=978.70; M =35.005; true length=982.04.
(b) I=32°44'; D=3°20′.
(c) I=29°18′42″; D.=4°00′00″.
(d) I=91°24'; Dc=16°.
2-2. Given the following values of I and Da. After taking
Ra (or log Ra) from Table I, compute T, E, L.C., M, and L
from basic formulas. Verify answers for T, E, and L by
Tables VIII and VI.
(a) I=20°34′; Da=2°05′. Answers: T'=498.97; E=44.897;
L.C.=981.91; M =44.176; L=987.20.
46
SIMPLE CURVES
(b) I=41°08′; Da=4°10′.
(c) I=25°44′18″; Da=3°00′00″.
(d) I=89°38'; Da=15°.
2-3. Prepare a set of field notes for staking each of the
following curves of Prob. 2-1 by the deflection-angle method:
(a) Curve 2-1(a). Sta. P.I.=47+62.6. Set full stations.
Carry distances to tenths and deflections to nearest minute.
Partial answers: sta. T.C. =42+68.2; first deflection =0°16′;
sta. C.T.=52+50.2; true chord for full station = 100.0 ft.
(b) Curve 2-1(b). Sta. P.I.=83+12.7. Set full stations.
Carry distances to tenths and deflections to nearest minute.
(c) Curve 2-1 (c). Sta. P.I. =121+05.81. Set full and half
stations. Carry distances to hundredths and deflections to
nearest 10 seconds.
(d) Curve 2-1(d). Sta. P.I.=89+21.79. Set full, half,
and quarter stations. Carry distances to hundredths and
deflections to nearest minute.
2-4. Prepare a set of field notes for staking each of the
following curves of Prob. 2-2 by the deflection-angle method:
(a) Curve 2-2(a). Sta. P.I.=23+72.2. Set full stations.
Carry distances to tenths and deflections to nearest minute.
Partial answers: Sta. T.C.=18+73.2; first deflection =0°17′;
sta. C.T.=28+59.4; true chord for full station = 100.0 ft.
(b) Curve 2-2(b). Sta. P.I.=19+87.1. Set full stations.
Carry distances to tenths and deflections to nearest minute.
(c) Curve 2-2(c). Sta. P.I.=79+58.71. Set full and half
stations. Carry distances to hundredths and deflections to
nearest 10 seconds.
(d) Curve 2-2(d). Sta. P.I.=52+62.96. Set full, half,
and quarter stations. Carry distances to hundredths and
deflections to nearest minute.
2-5. Refer to the field notes for Prob. 2-3(a):
(a) By the simplest method (page 30), compute the total
deflection to sta. 47 +70. Answer: 3°36′+70×0.5′ =4°11′.
(b) With transit at sta. 48 on the curve, what vernier reading
must be set before backsighting to sta. 44 in order for the
SIMPLE CURVES
47
vernier to read 6°56' when the telescope is plunged and sighted
at sta. 51? Answer: 1°06'.
(c) With transit at sta. 48, what vernier reading must be
set before backsighting to sta. 44 in order for the vernier to
read 0°00' when the line of sight is turned along the local
tangent? Answer: 3°20'.
(d) With transit at the C.T., what vernier reading must be
set before backsighting to sta. 48 in order for the vernier to
read 0°00' when the line of sight is turned to the P.I.?
Answer: 3°45'.
2-6. Refer to the field notes for Prob. 2-4(a).
(a) Compute the total deflection to sta. 23 +40.
(b) With transit at sta. 25 on the curve, what vernier
reading must be set before backsighting to sta. 21 in order for
the vernier to read I when the telescope is plunged and
sighted at the C.T.?
(c) With transit at sta. 25, what vernier reading must be
set before backsighting to sta. 21 in order for the vernier to
read 0°00' when the line of sight is turned along the local
tangent?
(d) With transit at the C.T., what vernier reading must be
set before backsighting to sta. 23 in order for the vernier to
read 0°00' when the line of sight is turned to the P.I.?
2-7. Find D for each of the following curves, given I and
the approximate E or T.
(a) I=36°25′; E = ±65 ft.
(b) I=17°52′; T=±320 ft.
=2°45'.
(c) I=55°48′; E = ±120 ft.
(d) I=13°10'; 7±185 ft.
Answer: Da to nearest 1°=44°.
Answer: De to nearest 15′
Required: De to nearest 10'.
Required: Da to nearest 5'.
2-8. Compute T and L, given the following values of I
and R:
(a) I=33°42′; R=2,000 ft. Answers: T=605.74; L (Table
VI) = 1,176.35.
(b) I=41°27'; R=3,000 ft.
48
SIMPLE CURVES
(c) I=68°18'; R=800 ft.
(d) I=52°57′20″; R=500 ft.
2-9. Prepare a set of field notes for staking each of the
following curves of Prob. 2-8 by the deflection-angle method:
(a) Curve 2--8(a). Sta. P.I.=30+57.9. Set full stations.
Carry distances to tenths and deflections to nearest minute.
Partial answers: Sta. T.C.=24+52.2; first deflection =0°41′;
sta. C.T.=36+28.5; true chord for full station = 100.0 ft.
Sta. P.I.=142+63.6. Set full stations.
Carry distances to tenths and deflections to nearest minute.
(c) Curve 2-8(c). Sta. P.I.=46+66.66. Set full and half
stations. Carry distances to hundredths and deflections to
nearest 10 seconds.
(b) Curve 2-8(b).
(d) Curve 2-8(d). Sta. P.I.=69+83.52. Set full, half,
and quarter stations. Carry distances to hundredths and
deflections to nearest minute.
2-10. Compute the tangent offset t to sta. 23 for the curve
on page 24. Use 4 different methods: (1) formula 2-29 for
C.O., then t=}(C.O.)(0.323)²; (2) formula 2–30; (3) formula
2-33; (4) t=c sin 1°27′.
2-11. Find the indicated offsets to hundredths for the fol-
lowing curves by the most efficient method (refer to page 2
of Tables):
(a) D.=4°. Find T.O., M.O., and t for a 50-ft subchord.
(b) Da=4°. Find T.O., M.O., and m for a 50-ft subchord.
(c) D.=20°. Find T.O., M.O., and t for a 20-ft subchord.
(d) Da=20°. Find T.O., M.O., and m for a 20-ft subchord.
2-12. Compute a set of offsets for staking full stations on
each of the following curves by the tangent-offset method.
Carry offsets to hundredths. Set up the calculations in
systematic tabular form and make an independent check of
the coordinates of the C.T. (a) Curve of Prob. 2-3(c).
(b) Curve of Prob. 2-4(c).
2-13. Find D to the nearest 10' for a curve which will pass
closest to a point 20 feet inside the initial tangent and 250 feet
beyond the T.C. Check the result by an independent method.
SIMPLE CURVES
49
2-14. Find D to the nearest 10' for a curve which will pass
closest to a point 40 feet inside the initial tangent and 500 feet
beyond the T.C. Check the result by an independent method.
2-15. Compute L, L., Li, and the true full-station chords
needed for staking:
(a) Curves 100 feet outside and 50 feet inside the curve of
Prob. 2-1(b).
(b) Curves 60 feet outside and 120 feet inside the curve of
Prob. 2-2(b).
2-16. Derive: (a) formula 2-30, (b) formula 2-31, and
(c) formula 2-35.
CHAPTER 3
COMPOUND AND REVERSE CURVES
3-1. Definitions.-Stated generally, a compound curve
consists of two or more consecutive curves that are tangential.
In the terminology of route surveying, however, a compound
curve is a two-arc simple curve having its centers on the same
side of the common tangent at the junction; whereas a reverse
curve is one having its centers on opposite sides. A multi-
compound curve has three or more centers on the same side of
the curve.
COMPOUND CURVES
3-2. Use.-Owing to the inequality of their tangent dis-
tances, compound curves permit the fitting of a location to
the topography with much greater refinement than do simple
curves. Conditions often occur in railroad and highway loca-
tion where the changes in direction between established tan-
gents can only be accomplished economically by compound
curves. This is true in mountainous terrain or along a large
river winding close to a rock bluff.
The flexibility of compound curves may tempt the locating
engineer to use them merely to reduce grading quantities or
to expedite the field work (as in Art. 3–14). This is not good
practice, as it complicates design details related to super-
elevation and introduces certain permanent operating dis-
advantages. A compound curve should not be used where a
simple curve is practicable.
A
3-3. Requirements for a Rigid Solution.-Fig. 3-1 shows
a compound curve, the notation for which is self-explanatory.
The layout has seven important parts: Ts, TL, RS, RL, IS, IL,
and I. However, since I = Is+IL, there are only six inde-
pendent variables, namely, the four lengths and any two angles.
Trial with compass and ruler will show that:
;
For a rigid solution four parts must be known, including
at least one angle and at least two lengths.
50
COMPOUND CURVES
51
Of the many possible combinations of four known parts
giving a rigid solution, some are more readily solved than
others. In practice the more difficult problems may often be
converted to simpler cases by measuring one more angle or
distance. Certain combinations rarely occur in the field.
(T.C.) A
The treatment which follows is not exhaustive; yet it is
complete enough to serve as a basis for solving any com-
pound-curve problem.
R57
کاره
Os
·Ts= AV
Vs

Is *
(C.C.) C
"
V (P.I.)
R₂
·T₁ = VB
Fig. 3-1. Compound-curve layout
VL
B(C.T.)
20
3-4. Solution Through Vertex Triangle.-The most obvious
method of solving a compound-curve problem is by means of
the triangle VVSV L, Fig. 3-1, formed by lines joining the P.I.
and the vertices of the two simple curves. (The base of the
triangle is the common tangent at the C.C.) If this vertex
triangle can be solved, all unknown parts of the layout are
easily determined. For example, if Is, IL, Rs, and RL are
known, solve first for the tangent distances of the individual
simple curves. Their sum equals the base of the vertex tri-
angle. Solve that triangle for the other sides by two applica-

52
COMPOUND CURVES
tions of the sine law. Then find the missing compound-curve
tangents, Ts and TL, by adding those sides to the proper
individual tangent distances. Of course, I=Is+Il.
It should be emphasized that the vertex-triangle method
of solution is possible only if at least two of the known com-
pound-curve parts are angles (fixing its shape), and then
only if one side of the triangle can be found (fixing its size).
The method is not applicable if, along with any two angles, the
parts T's and TL, RS and TL, or RL and Ts are known.
A
F
T
V

R₂
|
Ok=
Et
14ㅅ
​YOU
Ğţ
.L
iOS.
$
B`
Tangent
at C
Fig. 3-2. Solution by construction
3-5. Solution by Special Construction.-One of the most
useful cases occurs when I has been measured, and topo-
graphical conditions fix the T.C. and the C.T. within narrow
limits at unequal distances from the P.I. If one more variable
is assumed, the layout is fixed, but the problem may not be
solved easily. Should the assumed variable be the degree of
the sharper curve (a practical situation, since D is often limited
by specifications), then Rs is the fourth known part, and solu-
tion through the vertex triangle is impossible.
The problem is solved by means of Fig. 3-2. From the
construction, chords AC and DOS are parallel, and point G
must lie on chord AC. Therefore triangles DEOS and AFG
are congruent.
COMPOUND CURVES
53
tan DOSE=
Since angle DOSE=I L,
tan IL
or
or
=
from which (RL-Rs)
Or else
JB-GH
DE FG
EOS AF AV+VJ−BH
Also,
Is=I-IL
The missing unknown R may be obtained either by the
vertex-triangle method of Art. 3-4 or directly from Fig. 3-2.
Thus,
T's sin I-Rs vers I
TL+TS cos I - Rs sin I
EOS=AF=AV+VJ¬BH
(RL-RS) sin IL=TL+Ts cos I – Rs sin I
TL+TS COS I - Rs sin I
sin IL
from which
=
DE=FG=JB-GH
(R1-Rs) vers IL=Ts sin I-Rs vers I
Ts sin I-Rs vers I
vers IL
(3-1)
=
(RL-RS)
(3-3)
L
When I is small, relation 3-3 is preferable to 3-2. For
moderate angles either formula is satisfactory. Though the
formulas look complicated, actual computation is simple
because the numerators of formulas 3-2 and 3-3 appear in
formula 3-1.
(3-2)
Fig. 3-2 applies to any rigid case in which both T's and Rs
are known. At least two of the three general relations, 3–1,
3–2, and 3-3, are needed; the particular unknowns determine
the choice of relations, as well as their form and order of use.
If both TL and RL are known, the diagram for direct solu-
tion results from drawing the construction lines in a reverse
way. However, the correct formulas may also be obtained
by merely changing the subscripts in formulas 3-1, 3-2, and
3-3.
3-6. Solution by Traverse.-The general relations de-
veloped in Art. 3-5 are used too infrequently to justify
memorizing them. Moreover, if the textbook is not available,
it may be difficult to derive them when needed, owing to the
complicated geometric construction. Fortunately, the same
54
COMPOUND AND REVERSE CURVES
formulas may be derived by a method which requires no con-
struction lines and involves only the elementary truth that
the algebraic sum of the latitudes and the algebraic sum of the
departures of a closed traverse must each equal zero.
А
T₂

Side
OLA
AV
VB
BOS
OSOL
RL
or
Fig. 3-3.
OS
Length Azimuth
RL
0°
TL
90°
T's
90°+I
Rs
180°+I
RL-RS
180°+IL
kas,
Then, from Σ latitudes,
Fig. 3-3 is the same as Fig. 3-2 except that construction
lines have been omitted. Proceed clockwise around the
selected closed traverse, assuming 0° azimuth in the direction
OLA. Consider that I<90°; recall that latitude-length
cosine of azimuth and departure=length sine of azimuth.
AX
Departure
0
TL
T's cos I
Solution by traverse
ду
Latitude
-Rs
RL
0
Stat
Ts sin I
-Rs cos I
-(RL-RS) cos IL
-Ts
B
0°
-
Rs sin I
—(Rī— Rs) sin IL
RL-Ts sin I-Rs cos I-(RL-RS) cos IL=0
cos IL=
RL-Ts sin I-Rs cos I
RL-RS
(3-4)
17
COMPOUND CURVES
55
The term RL in the numerator may be eliminated and the
relation may be converted to formula 3-3 by subtracting both
sides from 1, canceling RL, and converting cosines to ver-
sines. The result is
vers IL
or
From Z departures,
TL+T'S COS I-Rs sin I-(RL-Rs) sin IL=0
sin IL
0°
vers A
But
tan A (for proof see page 637).
sin A
Therefore, if formula 3-3 is divided by 3-2, the result is
tan IL
OL
A
Rs
S
Ts sin I-Rs vers I
RL-RS
OS
TL+TS COS I - Rs sin I
RL-RS
=
In applying the traverse method for solving any problem, it
should be obvious that the traverse must be drawn so as to
include all six independent variables (four lengths and any two
angles). Moreover, any direction may be taken as 0° azimuth;
Ts
T's sin I-Rs vers I
TL+TS COS I - Rs sin I
R₁
Fig. 3-3R
(3-3)
V
(3-2)
(3-1)

56
COMPOUND AND REVERSE CURVES
but, in order to obtain workable relations directly, the axes
must be assumed parallel and perpendicular to an unknown
length. Summarizing,. the general rules of procedure are:
1. Draw traverse to include all independent variables.
2. Take 0° azimuth parallel or perpendicular to an unknown
length and proceed clockwise around traverse.
3. Set Z latitudes and Σ departures equal to zero, obtaining two
equations.
Side
Os A
AV
VB
or
To illustrate the directness and simplicity of the traverse
method, Fig. 3–3R shows another compound curve in which
the layout starts (for variety) with the shorter-radius arc.
Another set of general equations will be developed by taking 0°
azimuth in the direction OsA.
BOL
OLOS
or
+
4. Solve the equation containing one unknown. If both equa-
tions have two unknowns, divide one by the other in such a
way as to eliminate an unknown.
Length
Rs
Ts
TL
RL
RL-RS
But
Then, from Σ latitudes,
Azimuth
0°
90°
90°+I
180°+I
Is
cos Is=
vers A
sin A
(3-4R)
The term Rs in the numerator may be eliminated by sub-
tracting both sides from 1, canceling Rs, and converting
cosines to versines. The result is
From departures,
RS-TL sin I-RL cos I+(RL-Rs) cos Is=0
TL sin I+RL COS I - Rs
RL-RS
vers Is=
Latitude
Rs
0
- TL sin I
- RL COS I
(RL-RS) cos Is
sin Is
=
Departure
0
Ts
TL COS I
RL sin I
(RL-RS) sin Is
RL vers I-T sin I
RL-RS
TS+TL cos I - RL sin I+(RL—Rs) sin Is=0
RL sin I-TL cos I-Ts
RL-RS
tan A (for proof see page 637).
(3–3R)
(3-2R)
COMPOUND CURVES
57
Thus, if formula 3-3R is divided by 3-2R, the result is
tan Is
RL vers I-TL sin I
RL sin I-TL cos I-Ts
(3-1R)
Inspection of the four formulas just derived shows that
they are the reversed forms of 3-1 through 3-4; that is, they
may be derived by changing the subscripts in the first set of
formulas. Thus the statement made at the end of Art. 3-5
is verified.
and the ways
The traverse method is valuable not only in the "standard"
compound-curve problems just described but also in any
special-curve problem which must otherwise be solved by
means of complicated geometric construction. (Chapter 7
contains some practical examples.)
3–7. Summary of Compound Curves.—In a practical prob-
lem in which only one of the four known independent variables
is an angle, it will be the angle I. If any two angles are known,
the first step will always be to compute the dependent angle
from I=Is+IL. After this the method of solution will
depend on the particular combination of unknowns.
Case
No.
The following tabulation shows how to determine the fifth
independent variable for the ten practical cases in which I is
known, either directly or from given values of Is and IL.
123 TH LOCO 7 ∞
4
5
6
8
9
10
Given
2 angles, Ts, Rs
2 angles, TL, RL
2 angles, RL, Rs
2 angles, Ts, RL
2 angles, TL, Rs
2 angles, TL, Ts
I, TL, Ts, Rs
I, TL, TS, RL
I, Ts, RL, Rs
I, TL, RL, RS
Recommended Solution
Vertex-triangle, or 3-3 for RL
Vertex-triangle, or 3–3R for Rs
Vertex-triangle, or 3–3R for TL
3-3 or 3-4 for Rs
3-3R or 3-4R for RL
3-1 for Rs, or 3-1R for RL
3-1 for IL
3-1R for Is
L
3-3 or 3-4 for Il
3-3R or 3-4R for Is
58
COMPOUND AND REVERSE CURVES
After only one independent variable remains unknown, the
solution can be performed either by the vertex-triangle method
or by use of one of the eight general equations developed in
Arts. 3-5 and 3-6.
3-8. Example of Calculation.-Given: I-75°21'; Ts=
1,175.42; Tz=2,000 (±50 ft); Ds=6°. Determine a con-
venient value for DL which fits the conditions, and determine
the exact values of IL, Is, and TL for the selected DL.
NOTE. This problem illustrates the practical case noted in Art.. 3-5.
It also shows the utility of Table XX, which lists all needed natural func-
tions in one table.
Find I by formula 3-1, which is
tan IL
T's sin I=1,175.42×0.967489 = 1,137.21
Rs vers I= 955.366×0.747086 = 713.74
tan L=
TL
T's cos I = 1,175.42×0.252914=
-Rs sin I= 955.366×0.967489-924.31
RL-RS:
From 3-3,
N
D
T's sin I-Rs vers I
TL+TS COS I - Rs sin I
=
Diff.
= 0.308434
RL-RS:
Now apply formula 3-3, which is
Alg. sum= 1,372.97 = Denominator, D
IL-17°08'30"
=
N
0.173738
Tentative IL=34°17'
vers IL:
=
423.47 = Numerator, N
€2,000.00
297.28
=2,437.41
Rs 955.37
Sum=RL=3,392.78 and Dь=1° 41′+
Round off DL to 1°40' for convenience in staking.
Exact RL=3,437.87
―
Ts sin I-Rs vers I
vers IL
=
Rs 955.37
RL-RS-2,482.50
N
2,482.50
=0.170582
COMPOUND CURVES
59
and
By formula 3-2,
TL=(RL-Rs) sin IL+Rs sin I-T's cos I
(RL-Rs) sin Iь=2,482.50×0.558590= 1,386.70
924.31
-T's cos I
-
-297.28
TL=alg. sum = 2,013.73
A
IL=33°57′30″
I =75°21′00″
Is=41°23'30"
form.
3-9. Multi-Compound Curves.-In especially difficult ter-
rain, a compound curve may be made to fit the situation better
by using more than two circular arcs. Such a multi-compound
curve is located most simply by trial (see Art. 3-14). The
layout is rarely symmetrical; practical examples are given in
Art. 7-14.
ů
Rs sin I=
www.c
P.I. A'

Fig. 3-4. Reverse curve
B
REVERSE CURVES
B'
1
Symmetrical multi-compound curves have many uses out-
side of route surveying, especially in architecture. The
"three-centered" oval, or "basket-handle" arch, is a common
3–10. Limitations and Uses.-When conditions do not per-
mit a simple curve AB inside the P.I. of established tangents
(see Fig. 3—4), the change in direction may be accomplished by
locating a reverse curve A'CB' in the area beyond the P.I.
Points A' and B' may lie on either side of the P.I. and point B,
the positions depending on the radii.
60
COMPOUND AND REVERSE CURVES
On routes where speeds are high, reverse curves are in-
admissible. This is particularly true on highways and rail-
roads because opposite superelevation at the point of reversal
cannot be provided. If the area beyond the P.I. must be
used for the location, the two arcs of the reverse curve should
be separated by a tangent long enough to permit proper operat-
ing conditions.
Reverse curves may be used to advantage on closed conduits
such as aqueducts and pipe lines; on flumes and canals where
erosion is no problem; and on local roads, in railroad yards, or
in any similar location where speeds are low.
P
Į
R
and
A
1
E --
1
+
/
VOL
Fig. 3-5. Parallel tangents
Osi
ka
10

F
B
Rs
3-11. Case of Parallel Tangents.-The simplest case of a
reverse curve occurs when the tangents are parallel (see Fig.
3-5).
With the aid of the perpendicular dropped from C to the
radii, it is clear that p=AE÷FB and AD=EC+CF, or
p=(RL+Rs) vers I
(3-5)
AD=(RL+Rs) sin I
(3-6)
Usually, p is known and two more variables must be
assumed. More commonly, RL = Rs, which reduces the
number of variables to four, of which two must be known or
assumed.
REVERSE CURVES
61
3-12. Case of Non-Parallel Tangents.-In essential theory,
the general case of a reverse curve between non-parallel
tangents is no different from a compound curve.
The same
seven parts are present, and the identical requirement for a
rigid solution must be met (see Art. 3-3). The problem may
be treated as two separate simple curves, provided sufficient
data are available. More complicated cases may be solved by
the construction method of Art. 3-5 or by the simpler traverse
method described in Art. 3-6; the final relations resemble
formulas 3-1, 3–2, and 3–3 closely. (See Art. 7–15.)
COMMENTS ON NOTES AND FIELD WORK
3-13. Notes.-Notes for staking compound and reverse
curves by deflection angles are computed by treating the
branches as separate simple curves, though they are set up in
continuous stationing according to the form given in Art. 2-9.
Owing to the change in curvature at the C.C. (or at the point
of reversal in curvature, or C.R.C., on a reverse curve), that
point must be occupied in moving up the transit from the T.C.
The deflection angle would not be correct for a sight spanning
the C.C. (or C.R.C.). Consequently, the safest form of notes
is that in which the deflection at the C.C. (or C.R.C.) is 0°
for starting the second arc. Some other method may be used,
but it is likely to confuse the transitman.
3-14. Field Work.-Set-ups and field checks follow the
general scheme outlined in Art. 2-11. If a set-up is made at
the C.C. (or C.R.C.) and the deflections are recorded as just
described, the transit is oriented so that the vernier will read 0°
when the line of sight is along the common tangent. Gen-
erally, set-ups are best made at the T.C. and C.T., with the
final check occurring at the C.C. (or C.R.C.).
Location by trial in the field often expedites fitting com-
pound curves to particular situations. An example of an
obstacle on the tangent is given in Art. 7-6.
Another use of the trial method is in running semi-final
location in mountainous or otherwise difficult terrain not
accurately mapped. Fig. 3-6 represents a situation in which
62
COMPOUND AND REVERSE CURVES
the direction of the back tangent and the location of the T.C.
at A have been fixed by the topography. The direction of
the forward tangent is indefinite, though it should pass near
a distant prominent point. The back tangent plunges into
inaccessible territory toward the P.I
inaccessible
To P.I.
шишишни
A
с
B
To distant point
Ed
B' Tangent at B'

\s+d+
|
Jo
Fig. 3-6. Location by trial
The procedure is as follows: From the relation t=} s²D,
select the degree of an initial curve fitting the conditions, as
described on page 35. Continue this curve until it requires
compounding (as at C) to fit the topography. Select the
degree of the second arc by the same method, and repeat. Do
this as many times as necessary. When the distant point is
visible from a set-up on the last arc (as at B'), measure the
angle d. To find the stationing of the C.T. at B, consider that
the directions to the distant point from B and B' are parallel.
100 d
where Ds is the degree of the
Ds
Then, sta. B-sta. B'.
final arc.
Check the field work, if desired, by locating B and
closing back to A by a convenient random traverse.
COMMENTS ON NOTES AND FIELD WORK,
63
PROBLEMS
——
NOTE. Certain of the practical problems in Chapter 7 may also be
assigned, particularly numbers 7-2 and 7-6.
3-1. Given four parts of each of the following compound
curves, compute the remaining three parts.
(a) I=74°55'; Is=50°12'; Ts=576.24; Ds (chord def.) =
8°10′. Answers: I1=24°43′; DL (chord def.)=5°11′*; T½=
696.61.
(b) Is=51°06′; IL=26°17′; TL=938.64; Dı (chord def.)
3°30'.
(d) I=81°08'; T'z=907.18; Ts=579.21; Ds (arc def.)
10°34'.
=
(c) I=78°55′; Ts=802.37; DL (chord def.)=3°05′; Ds
(chord def.)=6°40′.
=
(e) I=80°15′40″; T'L=1,037.46; 7's=731.89; Dī (arc def.) =
3°.
(ƒ) I=81°22′15″; T=1,442.07; R1=2,000; Rs=1,000.
==
3-2. Prepare a set of deflection angles for staking each of
the following compound curves of Prob. 3-1 by the deflection-
angle method. Carry distances to hundredths and deflections
to nearest minute.
(a) Curve 3-1(a). Sta. P.I.=33+38.52. Curve begins
with Ds. Set full stations.
(b) Curve 3-1(d). Sta. P.I.=54+11.73. Curve begins
with Ds. Set full and half stations.
(c) Curve 3-1(f). Sta. P.I.=91+13.51. Curve begins
with DL. Set full and half stations.
3-3. Compute angle I and distance AD (Fig. 3-5) for a
common-radius 20° (chord def.) reverse curve between parallel
tangents exactly 50 feet apart. Answers: I=24°03′07″; AD=
234.71 ft.
3-4. Find the flattest common-radius reverse curve (to the
nearest 10, arc def.) that can be inserted between parallel
tangents exactly 100 feet apart without the distance AD
(Fig. 3-5) exceeding 800 feet.
3-5. Reverse curve (refer to Fig. 7-13). Given: IL=16°42′;
64
COMPOUND AND REVERSE CURVES
Is=75°26'; D=1°35'; Ds=4°25′ (both arc def.). Find:
T₁(AV) and Ts(BV). See Prob. 7-6(a) for an extension of
this problem. Answers: TL=2,268.52; Ts=487.40.
3-6. Reverse curve beginning with DL. Given: I1=93°32′;
Is=31°54'; D=4°30′; Ds=7°30′ (both arc def.).
(a) Find distances T.C. to P.I. (A'V) and P.I. to C.T. (VB').
(b) Assume the reverse curve is replaced by a 3º (arc def.)
simple curve joining the same tangents. Find the distances
AA' and BB' between the T.C.'s and C.T.'s of the two layouts.
CHAPTER 4
PARABOLIC CURVES
4-1. Uses.-Parabolic, instead of circular, arcs may be
used for horizontal curves. Where the curves are flat, there
is no discernible difference between the two types. However,
a parabola cannot be staked out readily by the deflection-
angle method. Moreover, the determination of the radius of
curvature at any point requires higher mathematics, thus
complicating superelevation and related calculations. Also,
parabolic alignment does not permit making simple right-of-
way descriptions. For these reasons parabolic arcs on hori-
zontal curves are restricted to such locations as park drives
and walks, where they may be easily located by tape alone.
For curves in a vertical plane the situation is the reverse.
Here, parabolic arcs are almost always used because elevations
can be computed much more easily than on circular arcs.
The vertical curve used is a portion of a vertical-axis parabola,
the particular parabola and portion being chosen with regard
to certain practical considerations.
4-2. Equal-Tangent Vertical Curve. From analytic geom-
etry the equation of a vertical parabola with respect to
rectangular coordinate axes is y = ax²+bx+c (Fig. 4-1). The
magnitude of a controls the sharpness of the parabola; its
sign controls the orientation. When a is positive, the parabola
is turned upward; when negative, downward. In a practical
situation the gradients (tangents) at the ends of the portion of
a parabola used are known. Selection of a then fixes the
sharpness of the parabola and the distance between tangent
points.
A useful property of a vertical parabola is the fact that
tangents drawn from any two points on the curve always
intersect midway between the points of tangency. Moreover,
there is a constant change in gradient (direction) between all
pairs of tangents to the same parabola, provided that the
portions of the parabola between tangent points are the same
length, measured horizontally. (See the two shaded angles in
65
66
PARABOLIC CURVES
Fig. 4-1.) These properties lend great flexibility to the
selection of an “equal-tangent" vertical curve for route
alignment.
For purposes of computation, the length L of a vertical
curve is not the distance along the curve; it is the difference
in stationing (foot units) between the ends of the curve,
regardless of the signs or numerical values of the tangent
gradients. Thus, in Fig. 4-2 the length L is 600 feet.
It is convenient to use the beginning of a vertical curve
(point A) as the origin of coordinates. Term c then dis-
appears and the equation of the curve becomes y=ax²+bx.
dy
The slope of any tangent is 2 ax+b. But at the origin,
dx
x=0; consequently, the slope equals b. However, the slope at
the origin is also expressed by the tangent gradient G₁. The
practical formula for an equal-tangent vertical curve is
therefore
y=aX²+G₁X
(4-1).
where y is in feet, X is in stations, G₁ is in per cent, and a is
in units of per cent per station. The usual algebraic signs
apply to the coordinates. Grades rising in the direction of
stationing are positive; falling grades are negative.
12/2/L
X-
L-
1/2/2 L
Axis of
Parabola

Y
1/1/1 1 + 1 = 1/1 12
L -
الله
-L -
Origin of
Coordinates
Fig. 4-1. Vertical parabola
-X
PARABOLIC CURVES
67
The terms in formula 4-1 have the graphical significance
shown in Fig. 4-2. The rule of offsets (coming from the term
aX2) is particularly important:
Vertical offsets from a tangent to a parabola are pro-
portional to the squares of the distances from the point
of tangency.
For the 600-foot curve in Fig. 4-2, the offsets from AB'
are obviously a, 4a, 9a, 16a, 25a, and 36a; those from BV
(extended) are the same. Moreover, the offsets from a tangent
at M (not shown) are a, 4a, and 9a.
In Fig. 4-2, VC=1B'B (from similar triangles). From the
rule of offsets, VM=B'B. Consequently, VM = MC. Thus,
a parabola has the important characteristic that the external
distance and the middle ordinate are equal. (This is not
quite true for a circular arc. See Fig. 2-1.)
The term G2-G₁ is the rate, in per cent, at which the tangent
grades diverge. It is analogous to the angle I of a simple
Some formulas are simplified by replacing G₂-G₁ by
the term A, defined as the algebraic difference in grades. In
Fig. 4-2, B'B equals the amount the grades diverge in the
curve.
X
distance L, or B'B= - (†) (156¹), L being in foot units.
100

차
​4a
ax 2.
aX
2
ΤΙ
HG, X
3
ic
Axis
M
9a
L-
stations
4
169
G1
5
1
Fig. 4-2. Equal-tangent vertical curve
G₂
25a
Ex
I
B
36a
B'
68
PARABOLIC CURVES
Since VMB'B, the external distance, designated as E, is
L
(850) (G₂-G₁) =
(4-2)
E =
The term a is useful in computations. In general, the
second derivative of y = ax2+bx+c equals the constant 2a.
This means that a tangent to a vertical-axis parabola changes
direction a constant amount for equal increments of horizontal
distance. (By analogy, a tangent to a circular arc changes
direction a constant amount for equal increments of distance
along the arc.) Therefore, the rate of change of grade on an
equal-tangent vertical curve is constant, and equals 2a per cent
per station.
On a vertical curve the total change in direction between
the profile grades is G2-G1 (or A) per cent. If this change. is
accomplished on a curve L feet long, the constant rate of
change must be
100(G₂-G₁)
L
2a:
AL
800
K =
(4-3)
It is convenient to think of 2a as a measure of the sharpness
of a vertical curve. In this respect it is analogous to D for
a circular curve. Another measure of curvature that is used
in sight-distance calculations (see Chapter 8) is the term K,
which equals L÷A. This is the horizontal distance in feet
required to effect a 1% change in gradient on a vertical curve.
Therefore
L 50
a
100A
L
(4-4)
4-3. Methods of Calculation of Vertical Curves. The
object of vertical-curve calculations is to determine the eleva-
tions at specified stations on the designed grade line. These
elevations are needed for cross-sectioning prior to grading and
for setting construction grade stakes.
Before starting the calculations, a simple sketch should be
drawn which shows G₁ and G₂ in their correct relations with
regard to sign and magnitude. This will show to which of the
six possible types the vertical curve conforms, and will help
in making a common-sense check of the results.
PARABOLIC CURVES
69
The calculation is simpler than for horizontal circular
curves; no trigonometric formulas or special curve tables are
needed. Either chord gradients or tangent offsets may be used.
The method of chord gradients (or "chord offsets") is based
upon the rate-of-change principle; it is a "self-checking"
method. The tangent-offset method utilizes the rule of off-
sets.
4-4. Chord-Gradient Method.-Chords between full sta-
tions on a curve such as that in Fig. 4-2 must have the succes-
sive gradients G₁+a, G₁+3a, G₁+5a..., differing (algebra-
ically) by 2a. Since the offsets to the curve from BV are the
same as those from AV, the calculated chord gradients are
correct if the last chord gradient plus a equals G2.. For the
600-foot curve in Fig. 4-2, for example, G₁+12a should
equal G2; it does when L=600 in formula 4-3. If the calcula-
tions are started with the known elevation of A, the successive
addition of chord gradients gives the elevations of the full
stations on the curve. All elevations are checked if the
elevation at B equals its value as computed around the tan-
gent grades AVB. This method has two pronounced advan-
tages: simple theory and automatic checks.
On railroads, grades are so slight that it is common practice
to place the vertex at a full station and use an even number of
stations in the curve. (On high-speed main tracks, the
A.R.E.A.* recommends that 2a should not exceed 0.05 in
sags nor 0.10 on summits.) Consequently, the calculation is
simple and requires no elaborate form; it may even be done
mentally. See example on page 70.
Elevations at plus points on vertical curves are frequently
needed for controlling cross-sections or for locating culverts
and other construction details. Moreover, an even-length
curve with the vertex at a full station does not always conform
well with the topography, particularly in hilly country where
grades are steep. In consequence, vertical curves must often
have odd lengths with vertices at plus points, especially in
highway work. (See Chapter 8 for considerations affecting
lengths of vertical curves on highways.) The tangent-offset
method has usually been used in such situations. However,
*American Railway Engineering Association.
70
PARABOLIC CURVES
Example. Special case-railroad type
Given: Sta. V=18+00; Elev. V=624.25; G₁= -0.15%; G₂ = +0.09%; L=600 feet.
Station
On tang.
(A) 15
16
17
18
19
20
(B) 21
On tang.
Chord Gradient
G₁
a
G₁+a
2a
G₁+3a
2a
G₁+5a
2a
G₁+7a
2a
G₁+9a
G₁+1la
a
= -0.15
+0.02
-0.13
G₁+12a
=
==
+0.04
+0.03
2a = +0.04
+0.07
+0.02
+0.09
www
+0.04
-0.09
=
+0.04
-0.05
+0.04
-0.01
=
Curve
Elev.
624.70
-0.13
624.57
-0.09
624.48
-0.05
624.43
-0.01
624.42
+0.03
624.45
+0.07
624.52
G2 (check)
=
B (check)
2a
Preliminary
Calculations
=
0.09-(-0.15)
6
= +0.04% per sta.
2aL
100
= G₁+12a
G₂=G1+
Elev. A=624.25
+3×0.15=624.70
Elev. B 624.25
+3×0.09624.52
PARABOLIC CURVES
71
the chord-gradient method can also be made to apply by
calculating the gradients of the end subchords according to
the following principles:
1. The change in gradient between a tangent and a chord
equals "a" times the station length of the chord.
2. The change in gradient between two adjacent chords
equals “a” times the sum of their station lengths.
An application of these principles is given on page.72.
The foregoing relations may be easily remembered by noting
the analogy to a horizontal circular curve. The change in
direction per station (D) corresponds to the change in grade
per station (2a). On a circular arc the first subdeflection
equals D times the nominal subchord length, and the deflec-
tion from any backsight chord to a foresight chord equals D
times the sum of their nominal lengths (in stations).
The curve elevation at any plus point may also be found by
the chord-gradient method. The procedure is to find the
gradient of the subchord to the plus point (by principle 2),
and then to calculate the elevation to the required plus point.
As an illustration, the calculations to find the curve elevation
at sta. 74+30 in the foregoing example are as follows:
= +2.92%
Chord gradient sta. 73 to 74
Change in gradient at 74-1.30a
Subchord gradient 74 to 74+30
= -0.52
= +2.40%
252.79
Elevation at sta. 74
Change in elevation = +2.40X0.30+ 0.72
Required elevation at sta. 74+30 253.51
It is not essential to compute the complete vertical curve in
order to find the elevation at any plus point. Principle 1
can be used in this same example as follows:
Gradient at 70+65=G₁
=+5.20%
Change in gradient 70+65 to 74+30=3.65a= −1.46
Chord gradient 70+65 to 74+30
= +3.74%
Elevation at 70+65
Change in elevation = +3.74×3.65
Elevation at 74+30
*******
= 239.86
=+13.65
= 253.51
1
72
PARABOLIC CURVES
Example.-General case-highway type
Given: Sta. V=73+40; Elev. V=254.16; G₁=+5.2%; G2=+0.8%; L=550 feet.
Station
On tang.
(A) 70+65
71
72
73
74
75
76
(B) 76+15
On tang.
Chord Gradient
G1 +5.20
0.35a = -0.14
G₁+0.35α
1.35a
G₁+1.7a
2a
G₁+3.7a
2a
G₁+5.7a
2a
G₁+7.7a
2a
G₁+9.7a
1.15a
G₁+10.85a
=
=
=
+4.52
-0.80
+3.72
= -0.80
=
=
=
=
=
Me+
+5.06
-0.54
G
===
p
+2.92
-0.80
+2.12
0.80
+1.32
-0.46
+0.86
0.15a -0.06
G₁+lla
prada
Chord in
Stations
=
0.35
1
1
1
1
1
0.15
+0.80 G2 (check)
Curve
Elev.
239.86
+1.77
241.63
+4.52
246.15
+3.72
249.87
+2.92
252.79
+2.12
254.91
+1.32
256.23
2α
trav
P
Preliminary
Calculations
G
6
0.8-5.2
5.5
-0.8% per sta.
2aL
100
= G₁+11a
G₂ = G₁+:
Elev. A=254.16
-2.75×5.2=239.86
+0.13
256.36 B (check)
Elev. B=254.16
+2.75X0.8=256.36
PARABOLIC CURVES
73
А
4-5. Tangent-Offset Method. The tangent-offset method
is based upon the rule of offsets. For a special case, such
as that in Fig. 4-2 VM is first computed from the external
formula 4-2. Then the elevations at full stations along the
tangent grades AV and VB are computed, and the symmetrical
offsets-simple multiples of a-are applied to those elevations
to give the curve elevations.
To show the application of this method to a general case,
the problem worked on page 72 is set up as on page 74.
Though the foregoing method is perfectly general, it is
not inherently self-checking, as is the chord-gradient method.
This defect may be remedied partially by testing the second
differences between elevations at equidistant points along the
curve. Theoretically the second differences must be con-
stant; they equal 2a for points 100 feet apart.
L
L₂

V₁ G1
G₂
£2
V2
Fig. 4-3, Unequal-tangent vertical curve
4-6. Unequal-Tangent Vertical Curve.-A vertical parab-
ola having tangent grades of unequal station lengths is
analogous to a compound curve (Chapter 3). It consists of
two (or more) equal-tangent vertical curves having a common
tangent where they join; and it is used where a single equal-
tangent vertical curve cannot be made to fit imposed condi-
tions so well.
Fig. 4-3 shows an unequal-tangent vertical curve. It
approximates a parabola having an inclined axis. However,
for ease in calculating elevations, it is best treated as two con-
secutive equal-tangent vertical curves AM and MB. Since
18
74
PARABOLIC CURVES
Station
(A) 70+65
(5.2x0.35)
71
72
73
74
75
76
(5.2×0.15)
(B) 76+15
·
11
II
Tangent
Elev. =G₁X
239.86
+1.82
241.68
+5.20
246.88
+5.20
252.08
+5.20
257.28
+5.20
262.48
+5.20
267.68
+0.78
268.46
Offsets from Av=aX²
Curve
Elev.
-0.4X4.352=-7.57
239.86
-0.4×0.352=-0.05 241.63
-0.4X1.35²=-0.73
-0.4X2.352=-2.21
-0.4X3.352=-4.49
246.15
249.87
252.79
254.91
-0.4×5.352=-11.45 256.23
-0.4×5.50²=-12.10 256.36
Check
1st Diff.
-4.52
-3.72
- 2.92
-2.12
- 1.32
B (check)
2nd Diff.
-0.8
-0.8
-0.8
−0.8
a
Preliminary
Calculations
100 (G2-G1)
2L
α
a = -0.4
From page 59
Elev. A=239.86
Elev. B 256.36
PARABOLIC CURVES
75
•
the vertices V₁ and V₂ of the separate parabolas are at the mid-
1
2
points of AV and VB, it follows that VM = MC.
The gradients of AB and of the parallel tangent V₁V₂ are
G₁L₁+G₂L2
both equal to
Consequently, V₁M and V₁V
L₁+ L2
diverge at the rate of
G₁L₁+G2L2
L₁+ L2
in the distance L₁, equals
duces to
equals (G
L₁ (G₁L₁+G2L2
G₁; and VM, the divergence
G₁), which
200 L₁+L2
E=
Xt=
LILA
200L
(4-5)
In the preceding formula the lengths are in feet, gradients
always being in per cent.
re-
Either chord gradients or tangent offsets may be used to
calculate elevations. The curve must be treated as two
separate parabolas having different values of 2a. Tangent
offsets, if used, are calculated from AV for the first parabola
and from BV for the second; they are not symmetrical as in
an equal-tangent vertical curve.
4-7. Lowest or Highest Point on Vertical Curve. The
lowest or highest point on a vertical curve is sometimes needed.
(This point is sometimes called the turning point on the curve.)
For installing a culvert at the low point, the approximate
stationing may be determined quickly by interpolating be-
tween calculated elevations. The stationing of a high point
or low point may be computed more accurately either by
applying the chord-gradient principle or from specific formulas.
t
From the definition of K, the distance X, in feet from the
beginning of a vertical curve to the turning point (where the
gradient is zero) must be - KG₁. This may be derived mathe-
dy
matically by setting
equal to zero in the expression
dX
y=aX²+G₁X. Thus, 2aX+G₁=0 at the turning point.
The result of solving this equation and expressing X in foot
-100G₁
units is X+=
which reduces to
2a
LG1
G₁-G₂
-KG₁
(4-6)
76
PARABOLIC CURVES
After Xt has been computed, the elevation of the turning
point may be found from the fact that the gradients from the
ends of the curve to the turning point are exactly one-half the
gradients on the tangents. This comes from Principle 1, which
requires the gradient from A to the turning point to be G₁+
aXt
(algebraically). This is equivalent to
100
100 (G₂-G1)
2L
G₂+ [10
and
which reduces to G₁. From the turning point to B the
gradient is G2.
LG
100(G1-G2)
On an unequal-tangent vertical curve the turning point
may occur on either of the two parabolas, its position depend-
ing on the tangent grades. The final relations are:
X₁
21) ITC
X2
- (LG₁₂) (1)
2
LG 2
=
=(
·(10%) (1)
G2
(4-7)
(4-8)
where X1 and X2 are the distances (in feet) to the right and
left from A and B; they cannot exceed L₁ and L2, respectively.
4-8. Vertical Curve to Pass Through Fixed Point.-In the
office work of grade-line design it is often necessary to find
the length of a vertical curve that will join given tangents and
will pass through a fixed point. The fixed point P (Fig. 4–4)
may be a road intersection or the minimum clearance over a
culvert or rock outcrop. Given values will usually be the
gradients G₁ and G2, the elevation and stationing of the fixed
point P, and the elevation and stationing either of the begin-
ning of the vertical curve or of the vertex.
For any given set of conditions, the unknown length of
curve may always be found by substituting known values in
formula 4-1. Careful attention must be paid to algebraic
signs.
Example.-Given: G₁=-4.2%; G₂ = +1.6%; Sta. P=
17+00; Elev. P=614.00.
Case A-Beginning of curve known: Sta. = 13+00; Elev.
624.53.
PARABOLIC CURVES
77
y=aX²+G₁X, where y=614.00–624.53 and a=
(4) ² — (4.2) (4)
Substituting, - 10.53 =
=
from which
and
-4.2L
200
6.27L=4,640
L=740 feet
Case B-Vertex of curve known: Sta. = 16+70; Elev.=
-G₁L
608.99. In this case y:
=
+(614.00–608:99).
Or
200
2
M
+5.01 = (530) (256+0.3) - (4.2) (2016 +0.3)
2L
G
1/1/14
L
2
IN
2L
which reduces to
L²-744.83L+3,600=0
A rapid slide-rule solution for L is performed by first con-
verting the quadratic to the form L(L-744.83) = −3,600.
Then, by trial, L=740. (This is the same curve as in Case A.)
// L
X
h
P
те.
у
7
100 (G₂-G₁)
2L

G2
V
Fig. 4-4. Parabola through point P
B
The need for solving a quadratic equation may be eliminated
by using the following alternate solution.*
From the rule of offsets and the construction shown in
*From suggestion by Max Kurtz, P.E., Brooklyn, N.Y.
78
PARABOLIC CURVES
DP
(L+h)²
Fig. 4-4, EPL — h)²'
But DP V-
*****
V
G₁h
100
1
V.
for L gives
G₂h
and EP-V-
Each term in these
100
expressions is known. To simplify, therefore, replace
In this example,
C @=
which reduces to
G₁h
100
by the constant C. Substituting and solving
G₂h
100
L
5.01
2h(C+1)
C-1
5.01
(−4.2)(30)
100
(1.6) (30)
100
√ D
DP L+2h
EP L-2h

crest vertical curves, C is found from C=
=
= 1.1765
1
2X30X2.1765
Therefore L:
=740 feet as before.
0.1765
In formula 4-9, h is positive when the station of P exceeds
the station of V and negative when the reverse is true.
For
G₁h
100
G₂h®
√ +
100
(4-9)


√+:
Case C-Fixed point is highest or lowest point: In this
practical case the elevation of the turning point (such as the
clearance over the invert of a culvert) and the vertex of the
curve are known. Although the stationing of the turning
point can be scaled approximately from the profile, it is
assumed to be unknown and is not used in finding the required
curve.
If the position of the vertex is the same as in Case B and
the turning point is at elevation 613.28, then the value of
LG₁
X in stations is
(from 4-6). By substitution,
100 (G1-G2)
PARABOLIC CURVES
79
-4.2L
200
which reduces to
and
+4.29=
A
G₁
580
−4.2L\ 2
') -
·) — — (4.2) (
2L
- 580
0.0058 L=4.29
L=740 feet
4-9. Reversed Vertical Curve.-In highway location where
the terrain is rolling, and also on interchange ramps, it is often
necessary to insert reversed vertical curves. Fig. 4-5 shows a
simple example of this problem.
V20 G3 → C
G₂
B
4.2L
V₁
Fig. 4-5. Reversed vertical curve
- 580

A large number of cases are possible, the conditions de-
pending on the particular combination of given values. Some
cases have little practical value. However, all of them may be
solved by proper use of formula 4-3. In one practical situa-
tion the given values will be the gradient G₁ and the elevation
and stationing of points A, B, and C. A numerical example of
this case is given in Prob. 4–8.
4-10. Laying Out Parabola by Taping.-Though it is incon-
venient to stake a horizontal parabola by the deflection-angle
method, laying out such a curve by tape alone is very simple.
Fig. 4-6 shows two taping methods applied to the general
case of an unequal-tangent parabola. The "middle-ordinate”
method (shown by solid lines) is the simpler in field work and
arithmetic; it requires only division by 2. The "tangent-
offset” method (shown dotted) is based upon the rule of offsets.
One variation of the middle-ordinate method follows:
Set stakes at the desired controlling points A, V, and B.
Measure AV and VB; divide by 2 and set the mid-points V₁
and V2. Measure V₁V2; divide by 2 and set the point M on
the axis of the parabola (VM may be measured, but it is not
essential). Then measure AM and MB; divide by 2 and set
80
PARABOLIC CURVES

A
My
سلم
&
Axi
587.32
18+60 426.18
M
Cz
Fig. 4-6. Horizontal parabola by taping
d
G1
-2.7%
+1.5%
1
2
1
the mid-points C1 and C2. Set two more points, M1 and M2,
on the parabola at the mid-points of C₁V₁ and C₂V₂ (C₁V₁ and
C2V2 both equal MV). If more points are required to define
the curve, measure the chords AM₁, M₁M, MM2, and M₂B;
divide by 2 and set four more points by middle ordinates as
before. When conditions make the long chord AB inacces-
sible, the middle-ordinate method is particularly suitable.
V/2
In the tangent-offset method the long chord AB is divided
into an even number of equal parts. The tangents AV and
VB are also divided into equal parts, their number being the
same as on the long chord. Then CV is measured and bi-
sected, and the point M on the axis of the parabola is thus
located. Offsets to the parabola from the points on the tan-
gents are measured in the direction of the corresponding
points on the long chord; the distances are simple fractions of
VM, computed according to the rule of offsets (see Art. 4-2).
In approximate work, points on the long chord may be dis-
pensed with, except for C; the offsets from the tangents are
computed as before, but are aligned by eye.
PROBLEMS
4-1. Compute grade elevations to the nearest hundredth
at all full stations on the following equal-tangent vertical
curves. Include all checks. (Note: In some cases it may be
necessary to carry chord gradients to 3 or 4 decimal places.)
Case Sta. V
Elev. V
L
(a) 24+25
(b)
M₂
Gr
+4.3%
-2.5%
700 ft
500 ft
B
PARABOLIC CURVES
81
Case
(c)
(d)
(e)
(ƒ)
Sta. Y
32+00
67 +82
129 +50
212+65
(a)
(b)
24+50
18+50
(c) 32+65
(d). 67+40
130+75
213+18.6
I
(e)
(f)
Elev. V
352.57
853.64
664.91
247.86
G₁
-0.27%
+4.54%
+0.16%
-0.17%
4-2. Compute the values of K for each of the vertical
curves in Prob. 4-1. Assuming that all cases except (c) occur
on highway alignment, determine the approximate highway-
design speeds for which the five highway curves give proper
sight distances (see Figs. 8-27 and 8-30).
G2
-0.15%
+1.00%
+4.83%
-4.50%
4-3. Compute and check independently the curve elevation
at a specified plus point on the following curves in Prob. 4-1:
Case
Plus Point
Answer
593.68
423.73
·
L
400 ft
850 ft
450 ft
650 ft
4-4. Compute the station and elevation of (a) the lowest
point on curve of Prob. 4-1(a), and (b) the highest point on
curve of Prob. 4–1(b).
4-5. Unequal-tangent vertical curve. Given: sta. V=
63+50; elev. V=572.46; G₁ = −3%; G₂ = +5%; L₁ =300 ft;
L2=500 ft. Find: the external VM, and the station and
elevation of the lowest point on the curve. (Also compute
elevations at all full stations if specified by instructor.)
Answers: VM=7.5 ft; sta. of lowest point=62+30, elev. =
578.76.
4-6. Unequal-tangent vertical curve. Given: sta. V=
46+75; elev. V=872.13; G₁ = +4%; G₂ = -3%; L₁ =800 ft;
L2=500 ft. Find: the external VM, and the station and
elevation of the highest point on the curve. (Also compute
elevations at all full stations if specified by the instructor.)
82
PARABOLIC CURVES
4-7. Given the following data for six vertical curves:
Sta. A Elev. A Sta. V Elev. V
G1
G₂
(a) -2.0% +3.1% 28+00
(b)
-3.8% -0.54% 56+50
(c) -2.7% +4.3%
(d) +1.5% -2.5%
(e) -2.2% +4.6%
(f)
+2.5% -1.2%
Find L (to the nearest foot) for curves which will also pass
through these fixed points:
(a) Sta. P=33+00; elev. P=447.82. Answer: 822 ft.
(b) Sta. P=60+00; elev. P=708.50.
(c) Sta. P=25+00; elev. P=594.76. Answer: 752 ft.
(d) Sta. P=17+50; elev. P=423.64.
(e) Turning point Pt, elev. 332.68. Answer: 695 ft.
(f) Turning point Pt, elev. 207.66.
I I I
Sta, and
Elev. of A
53+00
265.40
4-8. Given the following data for two reversed vertical
curves:
0+00
621.41
Sta, and
Elev. of B
450.06
719.86
61 +00
269.40
►
24+25 587.32
18+60 426.18
75+00
327.51
45+25 212.47
Sta. and
Elev. of C
65+00
275.40
(a)
(b)
+1.0%
=
In curve (a) find: G2, G3, and the separate values of 2a.
Answers: G2 +2.0%; G3 = +1.0%; 2a = +0.375 and -0.25%
per station. In curve (b) the reversed curves must be of
equal length and the crest curve must be twice as sharp as the
sag curve, as measured by 2a. Find: G2, G3, 2a, and the
sta. and elev. of B.
Gi
17+00
612.06
-1.0%
CHAPTER 5
SPIRALS
5-1. Foreword.-In high-speed operation over alignment
on which the curves are circular arcs, an abrupt change from a
straight path to a circular path is required at the T.C. of the
curve. It is obviously impossible to make this change in-
stantaneously. Smooth, safe operation around railroad and
highway curves requires a gradual transition between the uni-
form operating conditions on tangents and the different (but
also uniform) operating conditions on circular curves.
Any
curve inserted to provide such a transition is called an "ease-
ment" curve.
Fig. 5-1 shows a simple curve A'E'B'. The only way in
which easement curves can be inserted at the tangents, while
still preserving the radius R, is to shift the curve inward to a
position represented by the parallel curve KEK'. It is im-
possible, however, to use a circular arc having the same length
as the original circular arc. The portions KC and C'K' must
be deducted in order to provide room for the easement curves
AC and C'B.
The easement curve AC is tangent to the initial tangent at
A, at which point its radius of curvature is infinite. (The
tangent may also be thought of as a curve of infinite radius.)
At successive points along AC, the radius of curvature de-
creases until it becomes equal to R at point C, where the ease-
ment AC and the circular arc CC' have a common center at 0.
Thus, instead of abrupt changes in direction at A' and B' on
the original simple curve, there are now gradual transitions
between the tangents and the simple curve CC' by means of
the easement curves AC and C'B.
In Fig. 5-1 the curved layout starts at the T.S. (tangent
to spiral) and ends at the S.T. (spiral to tangent). The
approach spiral AC joins the circular arc at the S.C. (spiral
to curve), and the circular arc joins the leaving spiral C'B at
the C.S. (curve to spiral). It should be observed that the
total central angle I is unchanged. However, the central
83
84
SPIRALS
angle of the circular arc is less than that of the original simple
curve A'B' by the amount used up by the two spirals. Thus,
the central angle Ic of the arc CC' equals I-2A. From
theory to be developed, the spiraled curve AB will be shown
to be exactly one spiral length longer than curve A'B'.
V(P.I.)
A(T.S.)
A'(T.C.)
D

1
1
1
1
1
1
Κ KIT-
11
Radius R
Infinite Radius
-A
(S.C.)
1-24
Ei
MARINE
"E"
(C.S.)
K'
B'(C.T.)
{B(S.T.)
Fig. 5-1. Simple curve with spirals
The effect of an easement curve is to introduce centrifugal
force gradually, thus reducing shock to track and equipment
on railroads and making high-speed "streamliner" operation
attractive to passengers. Moreover, the easement curve tends
to "build safety into the highways" by following the natural
path of the vehicle between tangent and circular arc, in that
way reducing the tendency to veer from the traffic lane.
An easement curve serves several incidental purposes,
the most important of which is to provide a logical place
for accomplishing the gradual change from zero to full super-
elevation. It also simplifies, on highway curves, the addition
of the extra pavement width found to be needed on curves for
mechanical and psychological reasons..
Finding a suitable easement curve is not difficult. On
SPIRALS
85
the contrary, the problem confronting engineers has been to
decide which of several available forms should be selected.
Many forms have been used. Some, including the cubic
parabola, the lemniscate, and the clothoid, have definite
mathematical equations; others, such as the Searles spiral (a
multi-compound curve) and the A.R.E.A. ten-ehord spiral
(see Art. 5-14), are empirical. Within the limits used in
practice, all these easements give substantially the same curve
on the ground. However, consideration of their relative
merits from three important viewpoints-mathematical
simplicity, adaptability to a variety of conditions, and ease of
staking out in the field-has led most American engineers to
favor the clothoid, or the spiral first investigated by the Swiss
mathematician, Leonard Euler.
The clothoid (called the Euler spiral, the American spiral,
or the transition spiral) is adopted in this book. For simplicity
it will be referred to hereafter as "the spiral." All spiral
tables (Tables XI through XVI-C) are based upon this spiral;
some are set up with small corrections which enable rapid con-
version from the arc definition to the chord definition of D on
the circular arc, thus changing the spiral practically to the
A.R.E.A. ten-chord form used in the past on many railroads.
Therefore, the tables are adaptable to wide usage.
5–2. Basic Geometry of the Spiral.—The spiral obeys a
simple, exact law:
The radius of the spiral at any point is inversely
· proportional to its length.
Although the spiral is a mathematically rich curve with
applications in other fields of pure and applied science, its
use in route surveying involves spiral parts which have clear
graphical significance, simple formulas, and easily remembered
analogies to the parts of a simple curve. Only a few relations
require use of calculus in their derivations. Tabulations of
spiral functions are given so completely in Part III that the
actual calculation and staking of a spiral-curve layout requires
only slightly more trigonometry and very little more field work
than are needed for a simple curve. On the other hand, it is
possible to compute a satisfactory transition spiral without
referring to special tables of any kind (see Art. 5–13).·
86
SPIRALS
All the relations needed for computing and staking a
spiraled curve stem from Fig. 5-1 and the law of the spiral.
In contrast to a circular arc, the spiral is a curve of variable
radius, or variable degree of curve. At any point on the
spiral, however, the inverse relationship between R and D is
still correctly represented by equation 2-10. Since the radius
of the spiral is infinite at the T.S., its degree of curve at that
point must be zero. But the law of the spiral shows an inverse
relationship between the radius at any point and the distance
to that point from the T.S. The following statement is
therefore true:
The degree of curve of the spiral increases at a uni-
form rate from zero at the T.S. to the degree D of the
circular arc at the S.C.
The constant rate of increase in degree of curve per station
along a spiral is represented by k. The basic formula for k is
derived from its definition by dividing the total change in
degree of curve, D, by the length of the spiral in stations.
Since this length is
Ls
100'
100D
Ls
k =
(5-1)
The constant k is useful in a variety of problems. As an
illustration, suppose a spiral is required in making the transi-
tion between arcs of a compound curve on which D=3° and
Ds=9°. In such a case a limitation would be placed on the
sharpness of the transition by specifying, for example, that
the value of k should not exceed 2° per station. This condi-
tion would be met by a 300-ft spiral (k=2°) or by a 400-ft
spiral (k=11°). These lengths are the latter portions of
complete spirals that are 450 ft and 600 ft long.
An extremely important element of a spiral is its central
angle, or spiral angle, A. For a curve of constant radius (a
circular arc), equation 2-14 shows that the central angle
equals the length of curve in stations times the degree of curve,
LD
or I =
Since the spiral is a curve of uniformly changing
100
degree of curve, it follows that its central angle equals the
length of spiral in stations times the average degree of curve, or
SPIRALS
87
K
(5-2)
The foregoing central angle is exactly half that of a simple
curve of the same length and degree (see formula 2–14).
The S.C. and the C.S. are comparable to the C.C. of a com-
pound curve; at least one of them must be occupied by the
transit in staking the layout by deflection angles. In con-
sequence, the coordinates of the S.C. are two of the most
important spiral parts. Their values, which are X = AD and
Y = DC, have been computed from exact relations derived in
Appendix A and tabulated in the spiral tables of Part III; it is
never necessary for the surveyor to compute them from their
theoretical formulas.
J B
~|~
Δ
Fig. 5-2 is a sketch of the first part of Fig. 5-1. The
original simple curve has been omitted, however, and certain
construction lines have been added to aid in deriving necessary
formulas.
A
L&D
200

R
T
Fig. 5-2
F V
G
Tangent at C
The point K (the theoretical point where a tangent to the
circular curve produced backward becomes parallel to the
tangent AV) is known as the offset T.C. It is sometimes
needed in the field; but is used more often in computations.
If a perpendicular is dropped from C to OK, the coordinates
of the offset T.C., which are X。= AJ and o=JK, may be
written by inspection. Thus,
88
SPIRALS
(5-3)
(5-4)
The distance o is often called the "throw" (in Great Britain,
the "shift"). It is the distance through which the circular
curve must be moved inward in order to provide clearance for
inserting the spiral. The shift E'E at the middle of the curve
(Fig. 5–1) is also called the throw. Obviously, E'E = o sec ≥1.
In any problem the particular distance referred to as the throw
will be clear from the context.
A (T.S.)
snipbu aluyut
k
Consequently,
and
By integration,
Xo=X-R sin ▲
No-Y-R vers ▲
0 =

L
δ
dot
of
Fig. 5-3
ds
|
However, from the law of the spiral,
r: R=L,:l
1 dl
R Ls
dd=
Fig. 5-3 represents a spiral on which P is any point located
by the spiral angle 8, the radius r, and the length 1. Dif-
ferentials being used,
፩ =
dl
r
72
2 R L
A=TH
Δ
Pdx
dl
2 R
in which the angles are in radians.
R
To P.I.
dy
S.C.
(5-5)
(5-6)
SPIRALS
89
When 5-5 is divided by 5-6 and the resulting equation is
solved for 8, it is found that
Therefore,
-(₁₂)'s
Δ
sin d=
d=
which expresses the following important property:
Spiral angles are directly proportional to the squares
of the lengths from the T.S.
(On a simple curve, central angles are directly proportional to
the first powers of the lengths from the T.C.)
Also, from Fig. 5–3,
dy=8 (approximately)
dl
dy = 3 dl - (RL)dl
2
Y
a
By integration,
This relation shows that:
Tangent offsets are closely proportional to the cubes
of the lengths from the T.S.
If the deflection angle to any point P be designated by a,
sin a==a (approximately)
When the value in 5-8 is substituted for y,
12
6R L
73
Ꮾ Ꭱ Ꮮ.
Hence, A, the deflection angle to the S.C., is
Ls
6 R
(5-7)
A
α
(5-10) Approx.
If relation 5-9 is divided by 5-10 and the resulting equation
is solved for a,
(5-11) Approx.
=
2
a-(+)'4
(5-8) Approx.
(5-9) Approx.
This equation expresses another important property:
Deflection angles are closely proportional to the squares
of the lengths from the T.S.
90
SPIRALS
(On a simple curve, deflection angles are exactly proportional
to the first powers of the lengths from the T.C.)
It follows from the foregoing that
a = 3/31 8
(5-12) Approx.
A = A
(5-13) Approx.
Relations 5-12 and 5-13 are correct for most practical pur-
poses. Theoretically, the relations produce values which are
slightly too large. Should exact deflection angles be needed
(as on a very long, sharp spiral), they are given in Table XV
for any ten-chord spiral, and may be obtained quickly from
Tables XVI and XVI-C for a spiral staked with any number of
chords up to twenty. The derivation of the small correction
which, if subtracted, would make equations 5-12 and 5–13
exact, is given in Appendix A.
and
The triangle ABC in Fig. 5-2 is analogous to that formed
by the vertex and tangent points of a simple curve. The
simple curve has equal tangents and equal angles at the long
chord; the spiral, on the contrary, cannot have equal local
tangents or equal angles. From formula 5-13, the angle BCA
must be almost exactly equal to 3 A.
The three lengths AC, AB, and BC are occasionally useful
in field work; they are called the long chord (L.C.), long
tangent (L T.), and short tangent (S.T.) of the spiral. When
needed, their values are taken from tables, though they may
readily be computed when X, Y, and ▲ are known. For the
flat spirals used on modern alignment, the L.T. and the S.T.
are approximately in a 2:1 ratio.
5-3. Simple Curve with Spirals.-Theoretical relations
for laying out any spiral, once the T.S. has been located, were
given in Art. 5-2. The T.S. is usually staked, as in a simple
curve, by measuring the calculated tangent distance T, from
the P.I. The tangent distances will be equal in the usual case
of equal spirals at the tangents. There is rarely any justifica-
tion for using unequal spirals, except in realigning existing
railroad track.
If equal spirals are assumed in Fig. 5-2 and a line is drawn
parallel to AV from K to G, T, is made up of the three seg-
SPIRALS
91
ments AJ, KG, and FV, which are Xo, R tan I, and o tan
I. Therefore,
or
(5-14)
Ts=(R+0) tan 11+X。
Ts=T+X。+o tan 11
(5-15)
By the same construction, the external distance E, (VE)
may be divided into the two segments EG and GV, which are
R exsec I and o sec I. Therefore,
Es=E+o sec I
(5-16)
By means of a trigonometric conversion, this equation may be
written in the form
(5-17)
E₁ = (R+0) exsec 11+0
For the rare case of unequal spirals,
781=T+X01+
and
02-01 COS I
sin I
(5-15a)
T32T+X02+
(5-15b)
In these formulas the subscripts 1 and 2 refer to the initial and
the final spirals and the resulting unequal values of T.. The
last term in each formula may be positive or negative,
depending on the magnitudes of 01, 02, and I. (See the end
of Art. 5-9 for general versions of these formulas as applied to
a completely-spiraled compound curve.)
01
02 cos I
sin I
Calculations and field work for a spiral follow the general
pattern described in Chapter 2 for a simple curve; the varia-
tions are in details only. In the usual case, sta. P.I., I, and D
are known. Briefly, the sequence of the remaining work is as
follows:
(a) Select Ls to fit the imposed conditions (Art. 5–17 con-
tains reference to the choice of L.).
(b) Calculate A from formula 5–2.
(c) Take X and Y from tables, and calculate X。 and o from
formulas 5-3 and 5-4. (This is theoretical; in practice, Xo and
o also may be taken from tables.)
(d) Calculate T, from formula 5-14 or formula 5–15.
8
(e) Calculate the stationing of the T.S., which is sta.
92
SPIRALS
P.I.-T.; the stationing of the S.C., which is sta. T.S.+Ls;
the stationing of the C.S., which is sta. S.C.+100 (1-2 A),
and the stationing of the S.T., which is sta. C.S.+Ls.
D
(f) Calculate deflection angles at selected points on the
approach spiral, using formulas 5-13 and 5-11. (This is
theoretical; for regularly spaced points on the spiral, tables or
abbreviated relations may be used.)
(g) Set hubs at the T.S. and the S.T. by measuring Ts from
the P.I.
(h) Occupy the T.S. and stake the approach spiral to the
S.C. by deflection angles. (L, ordinarily equals the sum of the
chords used to lay it out; corrections to chords are necessary
only for fairly long chords near the end of long, sharp spirals.)
(i) Occupy the S.C. and backsight to the T.S. with (A—A)
set off on the proper side of 0°; A—A is almost exactly equal to
2 A. Then, plunge the telescope and stake the simple curve
to the chosen check point by the usual methods. (The check
point may be the C.S. or any point on the simple curve.)
(j) Occupy the S.T. and run in the leaving spiral to the C.S.
(For regularly spaced points, the deflections are the same as
those on the approach spiral.)
(k) Make the final check at the selected check point, which
should preferably be near the middle of the simple curve. A
set-up at the C.S. is required if this is done, but on final loca-
tion the resulting smooth junction at the C.S. justifies this
method.
Conditions frequently warrant varying the foregoing pro-
cedure. For example, instead of following steps (h) and (j),
the S.C. and the C.S. may be located (or checked) by any one
of three other methods: (1) by measuring the X and Y co-
ordinates; (2) by measuring the long tangent (L.T.), the angle
▲, and the short tangent (S.T.) of the spiral; (3) by measuring
A and the long chord (L.C.).
Tables may be used to expedite some of the operations just
described.
Table XI: See explanation on page 410.
SPIRALS
93
Table XII: See explanation on page 410.
Table XIII: This table is explained in Art. 5–11.
Table XIV: This table is explained in Art. 5–12.
Table XV: This table gives exact deflection angles for any
spiral up to = 45° staked with ten equal chords.
A
If a table such as Table XV is not available, it is not neces-
sary to use formulas 5-13 and 5-11 when points are spaced
regularly. The following method is quicker:
Let a₁=the deflection angle (in minutes) to the first regular
point. Then
(5-18)
a₁ (in min.)
=
20 A(in degrees)
n²
where n = the number of equal chords on the spiral.
Calculate a₁ for given values of ▲ and n. Then, to find the
remaining deflection angles, simply multiply a₁ by the squares
of the chord-point numbers. Thus, if A were 6° and n were 10,
a₁=1.2 min. The remaining deflections are 4 α1,
9 α1, 16 α1,
25 ai, etc. (The student should check these against Table
XV.)
Table XVI: This table is an extension of the principle just
described (see the examples on pages 438 and 439). The
"corrections" mentioned are the result of the slight approxima-
tion in the relation a= d; their source is explained in Appen-
dix A. The theory underlying deflections for a set-up on the
spiral is outlined in Art. 5-7.
5-4. Locating Any Intermediate Point on Spiral.-Though
the spiral is usually laid out with equal chords, the number
commonly being ten, such a process does not serve all purposes.
For example, on location prior to grading, earthwork esti-
mates are made more rapidly if cross-sections are taken at
regular full stations and possibly half-stations. Furthermore,
important "breaks" requiring cross-sectioning may fall be-
tween regularly spaced points. During construction it may
be necessary to set points on a spiral at trestle bents or on
bridge piers. For these reasons it is convenient to have a
simple formula for determining the deflection angle to any
point at a distance l feet beyond the beginning of a spiral. The
94
SPIRALS
following relation serves this purpose:
k l2
1,000
(5-19)
The constant k may be computed from formula 5-1. It is
recommended that an integral value of k be chosen in order to
simplify the computation.
Sta.
a (in min.):
5-5. Field Notes.-Notes for staking a spiraled curve are
set up in continuous stationing according to a modification
of the form in Art. 2-9. Deflection angles for the spirals
should start with 0° at the T.S. and S.T. The comments in
Art. 3-13 are pertinent.
Form of Notes for Spiraled Curve
Total
Point Defl.
44+00
43+50
43+00
42+-50
=

46+55.8
47+15.8 OS.T. 0°00′ | At S.T., orient by
0°04' sighting to P.I.
0°14' with 0°00′ on
45+95.8
45+35.8
0°32'
vernier. Make
44+75.8
0°58'
44+15.8 C.S. 1°30'
42+43.6
S.C. 1°30'
41+83.6
0°58'
41+23.6
0°32'
40+63.6
0°14'
40+03.6
0°04'
39+43.6 OT.S. 0°00'
Field
Procedure
44+15.8 C.S. 2°35' At S.C., orient by
2°21'
sighting to T.S.
1°36'
with 3°00' on
0°51'
proper side of
vernier.
0°06'
42+43.6 OS.C. 0°00'
final check at
C.S.
At T.S., orient by
sighting to P.I.
with 0°00' on
vernier.
Curve
Data
I=14° 10'
D=3°
P.I.=43+31.1
T₁=387.5
Ls=300
SPIRALS
95
In the accompanying form, the columns for calculated and
magnetic bearings have been omitted in order to insert
explanatory notes. Otherwise the form is a typical example
of the left-hand page of a field book. Distances are here com-
puted to tenths of a foot, and deflection angles are taken to the
nearest minute. In final location a higher degree of precision
would be required.
In the foregoing example both spirals were run as five-
chord spirals. The deflection angles could be computed by
finding a₁ from formula 5-18 and multiplying by the squares of
the chord-point numbers; or they could be taken directly from
Table XV for every other chord point of the ten-chord spiral.
To illustrate how easily spiral deflections may be computed
to fit any desired conditions, the following alternate notes are
given for staking the approach spiral.
Alternate Notes for Approach Spiral

Case A: 4-chord spiral. L'se
formula 5-18 and Table XVI.
Sta.
Point
S.C.
42+43.6
41+68.6
40+93.6
40+18.6
39+43.6 T.S.
Total
Defl.
1°30'
0°51'
0°22'
0°06'
0°00'
Case B: stakes at regular full
stations. Use formula 5-19.
Sta.
Point
S.C.
42+43.6
42+00
41+00
40+00
39+43.6 T.S.
Total
Defl.
1°30'
1°06'
0°24'
0°03'
0°00'
5-6. Principle of the Osculating Circle.-In Fig. 5–4, P. is
any point on a spiral just as in Fig. 5-3. The circular arc
drawn tangent to the spiral at point P has a radius r equal to
the radius of curvature of the spiral at the point of tangency.
Any such tangential arc is called an osculating circle. From
the law of the spiral, the osculating circle at any point must
lie inside the spiral toward the T.S. and outside the spiral
toward the S.C.

At the T.S. a spiral has an infinitely long radius of curvature
and a degree of curve equal to zero. Therefore, the osculating
96
SPIRALS
circle at the T.S. is a straight line coinciding with the main
tangent. But any circle is a curve of constant degree, whereas
a spiral is a curve of uniformly changing degree. It follows,
as an important principle, that:
A spiral departs in both directions from any osculating
circle at the same rate as from the tangent at the T.S.
In order to conform to the osculating-circle principle, any
complete spiral is bisected by its throw o, and vice versa.
That is, line JK and spiral AC (Fig. 5-2) bisect each other.
Since tangent offsets are closely proportional to the cubes of
the spiral lengths, this means that
Y
4
Another useful relation is found by substituting
Y (from formula 5-8), giving
1
1
1
1
T
0
L
(5-21) Approx.
24 R
The mutual bisection of spiral and throw is also true for any
portion of a spiral. In Fig. 5-4, for example, assume that P
A(T.S.)
1
0 =
2
Osculating Circle at A
To P.I.
P
(5–20) Approx.
2
L'
6R
3-
for

Fig. 5-4. Osculating-circle principle
Osculating
Circle at P
SPIRALS
97
is the second of several regularly-spaced chord points on a
spiral. The offsets to chord point 1 from the osculating
circles at A and P must be equal, and each is equal to the
offset at chord point 3.
An interesting consequence of the foregoing principles is
that the three shaded portions of Fig. 5-4 are equal in area.
This property has practical value in computing areas of high-
way pavement on spiraled curves. See Art. 8-21.
5-7. Transit Set-ups on Spiral.-Field conditions sometimes
require a transit set-up on a spiral. The osculating-circle
principle supplies the basis for locating any other point on a
spiral from such a set-up.
In Fig. 5-5 assume that the spiral has been staked as far
as point t from a set-up at the T.S., and that obstructions to
the line of sight prevent further staking from that set-up.
A(T.S.)
To P.I.

b
"Osc.
Local Tangent
/t
Osc.o
Fig. 5-5. Transit set-up on spiral
Qs
The problem is how to orient the transit at point t and stake
the rest of the spiral from the new set-up. Point b is any
point on the spiral (including the T.S.) to which a backsight
is taken in order to orient the transit. Point f is any point
on the spiral beyond point t (including the S.C.) which is to
be located by means of a foresight from t. The angle o is the
orientation angle turned between the backsight to b and the
98
SPIRALS
local tangent, and of is the deflection angle to the foresight
point f. Spiral lengths from the T.S. to these points are
denoted by l, lt, and lƒ.
Angle & equals the deflection angle for a circular arc having
a length of (ll) and a degree of curve equal to that of the
osculating circle minus the angular departure between the
osculating circle and the spiral in the distance tb. Since
D is the degree of curve of the spiral at the S.C., the degree
of the osculating circle at t equals (1)
b=
1/1t-lv
1/lt
( 5b)(4)-(4-6)²s. (Refer to formulas 2-12,
2\ 100
Ls
Ls
200A
5-7, and 5-12.) When D is replaced by its equivalent, Lis
this expression may be simplified to
$b=(2lt+lb) (lt-lb)
Δ.
D. In general,
(5–22) Approx.
3L2
3
Angle of equals the deflection angle to the same osculating
circle for a length of (-l) plus the angular departure
between the osculating circle and the spiral in the distance tf.
1/ls - li
Ls
2
That is, 4, (1) (12) + (1 - 4) s,
)A, which may
2 100
simplified to
Δ
$ƒ=(ls+2 le) (ls—le)
3L2
(in min.) = (2n₁+nь) (nı− Nь)
may be
(5-23) Approx.
The total deflection angle from backsight to foresight
equals ø¿+$ƒ. When formulas 5–22 and 5–23 are added, the
simplified result is
(5-24) Approx.
$=(lo+le+ls) (ls—lo)
312
Where stakes on a spiral are spaced regularly and a set-up
is required on a regular chord point, the foregoing formulas
may be expressed in terms of chord point numbers. Since
L, equals n equal chords and the length l of any portion of
Ls
times the chord point number,
the spiral sighted over equals
n
it follows that
20A(in deg.) (5–25) Approx.
n?
SPIRALS
99
Also
øƒ(in min.) = (nƒ+2nı) (ns−ni)
and
(in min.) = (no+ni+ns) (ns−no)
Preliminary computations:
LSD 900X10
Δ
200
200
Sight at Coefficient
Point
1
2
3
20▲ (in deg.)
n2
The second terms in the six foregoing formulas represent the
spiral lengths sighted over in performing the indicated oper-
ations. In the last three formulas the final term is the deflec-
tion angle in minutes to the first regular chord point (see
formula 5-18); the products of the first two terms yield the
coefficients given in Table XVI. These formulas give accu-
rate results except for long sights over long sharp spirals,
where the effect of the slight approximation in formula 5–12
may become noticeable. Exact results may be obtained by
applying the corrections found in Table XVI-C.
45
Example.-Given: Da=10°; Ls=900 ft. Spiral to be staked
using fifteen 60-ft chords. Assume that obstructions require
set-ups at chord points 5 and 12 in addition to the T.S.
204(in deg.) (5–27) Approx.
n2
45° a1
Computation of Deflections
Deflection
(Table XVI) X a₁ = to Spiral
Δ -
1
4'
4
4'
9
4'
16 4' (64')1°04'
25
4'
(100')1°40′
2
· (5/5)² >
0°04'
0°16'
0°36'
(5–26) Approx.
204 20X45
n²
152
}
Transit at T.S.
(Orient by sighting along tangent with 0°00′ on vernier.)
4'
X45°=5°
Comments
These same deflections
obtainable from
Table XV at every
other chord point by
entering Table with
100
SPIRALS
Sight at Coefficient
Point (Table XVI)X a₁ = to Spiral
Deflection
0 (T.S.) 50
6
7
8
9
10
11
12
Transit at Point 5
(Orient by backsight to T.S. with 3°20′ on vernier.)
4'
16 4'
34
4'
54
4'
76
4'
100
4'
126
4'
154
4′
5
203
13
37
14
76
15 (S.C.) 117
4'
4'
4'
4'
Comments
(200') 3°20′
(64') 1°04′
(136') 2°16′
(216') 3°36'
(304') 5°04'
(400') 6°40′
(504) 8°24′
(616′)10°16′
Orientation angle
Transit at Point 12
(Orient by backsight to point 5 with 13°32′ on vernier.)
(812')13°32'
(148') 2°28′
(304') 5°04'
(468′) 7°48′
equals twice the
deflection from the
T.S. Plunge tele-
scope after backsight
and set deflections
for points 6 to 12 on
the other side of 0°00'.
Plunge telescope after
backsight and set
deflections for points
13 to 15 on the other
side of 0°00'.
If this spiral were staked entirely from one set-up at the
T.S., the deflections to the distant chord points would have to
be decreased by slight corrections. Table XVI-C shows that
these corrections exceed minute when the ratio of spiral
length sighted over to length Ls is 0.7 or greater. In this
example the correction for chord point 12 would be -1.25
minutes; for the 900-ft-long sight to the S.C., it would be
-4.77 minutes. Obviously, one beneficial effect of setting
up at points 5 and 12 is the elimination of such corrections,
for the longest sight (between points 5 and 12) has an l¸: Ls
ratio of less than 0.5 and a negligible correction.
Regardless of the location of transit set-ups used in staking
a spiral, the actual chords taped should (theoretically) be
slightly shorter than their nominal arc lengths. For most
practical purposes, L, equals the sum of the nominal chords
used to stake the spiral. Chord corrections can ordinarily be
SPIRALS
101
neglected; they are necessary only for fairly long chords near
the end of a long sharp spiral. In the preceding example the
position of the S.C. would overrun its theoretical position by
only 0.04 ft if the spiral were staked entirely from the T.S.
with fifteen 60-ft chords. If necessary to allow for this small
discrepancy, each of the last four chords could be taped as·
59.99 feet long. Chord corrections may be found with the
aid of Tables II and IV.
5-8. Spiral Between Arcs of Compound Curve.-If either
arc´of a compound curve is sharp enough to require insertion
of a spiral at the main tangent, a spiral between the two
arcs may also be justified.
In Fig. 5-6 arc KC is moved inward to provide room for
inserting spiral AC (the main tangents are omitted to simplify
the sketch). The radius of the spiral at A equals RL, instead
of infinity as in Fig. 5-1; that at C equals Rs. In other words,
AC is a portion of a spiral cut to fit between curves of degree
D1 and Ds. An additional value-usually the length AC but
sometimes the offset JK-must be assumed. The problem is
then determinate; only one spiral will satisfy the given con-
ditions.

J
А
=
Local A Tangent
I
R2
|
1
K
Ost
Local & Tangent

XRS
|
Fig. 5-6. Spiral between arcs of compound curve
From the principle of the osculating circle, the offset or
"throw" JK is the value of o for a spiral of length AC and
102
SPIRALS
terminal degree Ds-DL. It also follows that, for all practical
purposes, JK and AC bisect each other. Accordingly, the
problem of calculating deflections for running in the spiral AC
is identical with that of calculating the deflections for staking
the remainder of a spiral from a set-up at any point, as
explained in Art. 5–7.
If the length of the spiral AC be designated by ls, its true
central angle equals the sum of the central angles for the
simple curves produced, that is, for the arcs AJ and KC.
From formula 2-14,
true ▲=
which reduces to true ▲=
ᎠᏞ
(b) (D6) + (1) (1Ds)
100
100
(200) (DL+Ds)
which reduces to
For computing the spiral deflections from the local tangent
at A or C, the nominal value of A is used. This is defined
as the difference between the true central angle of the con-
necting spiral and that of the osculating circle at either end.
ls DL
Since the osculating circle at A has a central angle of
100
in the distance ls,
ls D L
100
(Ds-DL)
nominal A = true A
Is
200
nominal A
•
(5-28)
(5–29)
(The nominal A may be taken from Table XI if the table is
entered with l, as L, and with Ds-DL as D.)
Example.—Given: Dı=2°; Ds=7°; AC=l,=300 ft.
Here Ds-Dɩ=5°; consequently, o=3.27 (Table XI).
From formula 5-1 the value of k for this spiral is
100 (7° -2°)
300
= 13° per station, which means that l, is the last
7°
300 ft of a complete spiral that is X100=420 ft long.
k
From formula 5-28, true A=13°30′.
From formula 5-29 (or Table XI), nominal ▲=7°30′.
To compute spiral deflections, use the nominal A in for-
mula 5-18. Thus, if the spiral AC were to be divided into
four equal parts,
SPIRALS
103
20×7.5°
-9.375'
42
and the deflections to the spiral from the osculating circle at
A are a1, 4 a1, 9 a₁, and 16 a₁. These are added to the deflec-
tions from the tangent to the corresponding points on the
osculating circle, namely, 0°45', 1°30′, 2°15′, and 3°00′. The
final deflections are: 0°54', 2°08′, 3°39', and 5°30'. For
running in the spiral from the sharper arc (set-up at C), the
required deflections from the tangent would be 2°28′, 4°38′,
6°28', and 8°00'. Note: This spiral is one of three used in
Prob. 5-5(a).
T.S.
a1=
5-9. Completely Spiraled Compound Curve.-Fig. 5-7
shows the geometric relations between the nonspiraled com-
pound curve ACB and the same curve provided with three
XOL AA
=tL

0°
1
1
5.1
R₁
OS
I
C.15.2
KH.
S.20.2
S
R
b
J
C.25.3
Os
FIL
Jo
OL
B
B
S.T
B
OL
Fig. 5-7. Completely-spiraled compound curve
S
Xos
104
SPIRALS
transition spirals. The spirals themselves are omitted.from
the sketch, and the offsets needed to accomodate them are
exaggerated for the purpose of clarity. For this case, in
which the layout begins with the flatter arc, point a is the
offset T.C. for the first spiral. Its throw A'a is denoted
by oi. The corresponding throw for the third spiral is B'b,
denoted by os. The throw oc for the combining spiral between
the two arcs is JK (see also Fig. 5-6).
Assume the given values to be sta. V, I, IL, RL, Rs, and the
lengths (or the k-values) of the spirals. The problem is to
find the stationing of all curve points (T.S., S.1C.1, C.1S.2, etc.)
so that the layout may be computed and staked. The best
solution directly from the given data is to find VB' and A'V
by the traverse method. Use the traverse OLA'VB'OsOĩ, and
follow the procedure developed in Art. 3-6. (Note the simi-
larity between the following tabulation and that on page 54.)
Length Azimuth
RL+OL
0°
A'V
90°
VB'
90°+I
-VB' sin I
— (Rs+os) cos I
Rs+os
180° +1.
RL-RS-oc 180°+IL -(RL-Rs-oc) cos IL
From Σ latitudes,
VB':
Latitude
RL+OL
0
C
Departure
0
A'V
VB' cos I
G
− (Rs+os) sin I
(RL-RS—oc) sin IL
(RL+0L)-(Rs+os) cos I−(R-Rs-oc) cos Iĩ
sin I
(5-30)
Then from Σ departures,
A'V= (Rs+os) sin I+(RL-Rs-oc) sin IL-VB' cos I (5-31)
The tangent distances T. and Tss are X。z+A'V and
VB'+X。s. Stations of the curve points are then found in
the usual way by subtracting TÎ from sta. V to obtain sta.
T.S. and adding successive spiral and arc lengths to sta. T.S.
It is necessary to use the principles brought out in Art. 5-8
in computing the true central angles of the circular arcs.
If the layout begins with the sharper arc, formulas 5-30 and
5-31 remain the same. The only difference is that sta. T.S.
SPIRALS
105
equals sta. V minuș Tâs, and, of course, the stationing increases
in the direction opposite to that in Fig. 5–7.
This problem may be solved by another method that is
particularly convenient in case the given values pertain to a
nonspiraled compound curve already computed. In such a
case the tangent distances (TLAV and Ts=VB) are known
from appropriate formulas derived in Art. 3-6. If it is now
required to modify the curve so as to insert three spirals,
XoL and Xos are found readily from Tables XI or XII. The
additional corrections needed to convert TL and Ts to TsL
and Ts are the distances AA' and BB', denoted by tɩ and ts.
The arcs AC and CB of the nonspiraled curve are shifted to
their locations aJ and Kb on the spiraled curves as the result
of movements of translation involving the throws oL, oc, and
The combined effects of these translations yield the
distances tɩ and ts, which may be written by means of the
enlarged sketch in Fig. 5-7 as
Os.
tL
os-oL cos I—oc cos Is
sin I
OL-OS COS I+oc cos IL
sin I
and
(5-32)
ts:
(5-33)
Both tz and ts will normally be positive quantities. If
either is negative, points A and A'(or B and B') are inter-
changed in Fig. 5-7, and the t-correction must be subtracted.
One advantage of this method is that the terms in formulas
5-32 and 5-33 are so small that the t-corrections may be
found by slide-rule calculation.
If the combining spiral is omitted, oc equals zero and the
formulas yield the t-corrections either for a compound curve
with spirals at the main tangents only or for a simple curve
with unequal spirals. (Compare the resulting t-correction
formulas with the final terms in formulas 5-15a and 5-15b.)
Furthermore, for a simple curve with equal spirals, o1=
os=0, and tr=ts. In this case the t-correction becomes
-cos I
(0) (1 - 00s 1)
which equals o tan I (see formula 5–15).
sin I
5-10. Fitting Spiraled Curve to Specified Es or Ts.—As in
the case of a simple curve, it may be necessary to fit a spiraled
curve to definite field conditions such as a specified E, or T..
106
SPIRALS
8
There are voluminous tables* giving convenient combinations
of D and L, which approximate a specified value for E, or Ts;
the exact values of E, and T, for a selected combination are
obtained by interpolation.
The tables in this book may also be used to select a spiral-
curve layout fitting given conditions.
Example.—Given: I=27°00′; E,=approximately 34 ft.
Determine suitable values of D and L. (slide-rule calculation).
First use Table VIII to obtain a fairly close value of D.
Thus,
D
163
34
=
4.8° (roughly)
Ls D
shows two unknowns.
200
Consequently, there are any number of combinations of A and
Ls which will fit the conditions. A glance at Table XI might
suggest a suitable combination.
Suppose that Ls is assumed to be 250 ft. Then
250
Δ ×4.8°=6.0° (tentatively)
200
Consideration of the relation A
-
From Table XII, o=0.00872×250=2.2 (approx.).
A close value of R+o is then found from formula 5–17.
34-2.2
Thus,
R+o
= 1,119
0.02842
==
Therefore, R=1,117 and D=5°08′ (by Table I).
If D were chosen to be 5°, the 250-ft spiral is one listed in
"selected spirals" of Table XI. For this combination, Es
would differ slightly from 34 ft, but a rigid requirement per-
mitting no deviation from a fixed E, is rarely met (it would
require an odd value of either D or L、).
If the approximate value of Ts were fixed, the first trial
calculation may also be made by using Table VIII. This will
give a rough value of D, but will allow selection of a suitable
value for Ls. Since o is very small in comparison with R,
formula 5-14 can be expressed approximately as T's=T+1Ls,
which can be solved for T.
Suppose that T', is fixed at approximately 400 ft in this
example. From Table VIII,
SPIRALS
107
1,376
400
D=
=3.4° (roughly)
Trying L, as 250 ft, T=T¸—½L¸-400-125-275 ft (approx.);
and (by Table VIII) D=1,376
275
-
-5° as before.
5-11. Spiraled-Curve Formulas with Radius as Parameter.
In connection with the use of even-radius curves (Art. 2-12)
it is convenient to express formulas for needed spiraled-curve
parts in terms of the radius of the circular arc.
In Fig. 5-2, certain of the distances may be related to R
by means of coefficients N, P, M, and S, which have the
following significance:
NR=the distance AD=X
PR=the distance DC=Y
M_R=the distance OJ=R+0
SR the distance AJ = X。
Values of these coefficients for R=1 have been computed
for various values of ▲ (see Table XIII). When these values
are substituted in the following relations, parts needed for
computing and staking the layout result:
X=RN
Y=RP
T= R(M tan I+S)
E.= R(M sec I−1) = R[M exsec I+(M-1)] (5–37)
8
(5-34)
(5-35)
(5–36)
A=[2(2 5]
p
2(2—p).
L₁ =2 R ▲ (radians)
(5-6)
In applying this procedure to practical problems in the
highway field, the "proportion rule" has been suggested* for
choosing Ls. The "proportion of transition" p is the ratio
of the length of the two spirals to the total length of the
spiraled curve. It may be proved that
(5-38)
*Leeming, J.J., "The General Principles of Highway Transition Curve
Design," Transactions, ASCE, Vol. 113, 1948, pp. 877 ff.
108
SPIRALS
It is not possible (except by coincidence) to have integral
values of R, Ls, and ▲ (in degrees); one, at least, must be
an odd value. When it is desired to use Table XIII without
interpolation, the odd variable is L、.
As an illustration the example of Art. 5-10 is solved by this
method. For a trial, assume that p=about 0.6 (evidence
shows that the natural value of p ranges between 0.4 and 0.7
for high-speed operation on highways). Then
^=[26
0.6
¸2(2−0.6).
6)] X27°=5.8°
Round off to 6° for convenience in using Table XIII.
From formula 5-37, R=1,130 (approx.). Use R=1,100,
which is listed in Table V.
Finally, 2 A=12°=0.20944 radians (Table VI) and L¸=
0.20944×1,100=230.4 ft.
If Ls were rounded to a convenient value (say 250 ft),
then A would be the odd value, but the circular arc could still
be staked readily by means of the deflections listed in Table V.
When the chord definition of D is used, ▲ is obtained from
5-38; Rc, from 5-37; De, from Table I; and L,, from 5-2.
The coefficients in formulas 5-34 through 5-37 apply
whether or not a central circular arc is present; that is, they
also can be used for the double-spiral curve described in the
next article.
5-12. Double-Spiral Curve. Some engineers advocate
making curves transitional throughout by using a double spiral,
which consists of two equal spirals placed end to end with the
curvature changing from increasing to decreasing at the middle
of the layout. This simplifies the selection of a curve that
will have a specified value of E, or Ts; since ▲=11, it follows
that Ts=X+Y tan I and E. = Y sec 11.
Es
8
Table XIV gives values of T's and E, for L,=1. Multiply-
ing the proper tabulated values by the actual length L, gives
Ts and E¸ for a double-spiral curve having the given value of I.
Obviously, a double-spiral curve of given I is longer than a
spiraled simple curve having the same D; hence there is a
much greater length over which operating conditions-cen-
trifugal force and superelevation, in particular-are variable.
For this reason, the all-transitional curve should be used only
in exceptional cases, at least on highway work.
SPIRALS
109
5-13. Calculating Spiral Without Special Tables. It is
possible to compute a spiral without reference to special tables
(if the engineer will do this occasionally, he will understand
the spiral better). The method suggested is based upon cer-
tain approximations the significance of which may be seen by
examining Fig. 5-2 in connection with Table XI. All calcu-
lations may be made by slide rule.
Procedure Recommended for Flat Spirals (up to A=5°)
(a) Compute Y from Y=L, sin 34
Y2
2 Ls
(b) Compute X from X=L,-
Y
(c) Compute o from o=
(d) Compute Xo from Xo
*
X
(e) Compute deflections by applying the principle that
deflections are proportional to the squares of the distances.
If the foregoing method is applied to a 200-ft spiral joining
a 3° curve, the computed values are: Y=3.49, X=199.97,
0=0.87, and Xo=99.98. For all practical purposes these are
the same as the values listed in Table XI.
As A increases beyond 5°, more significant differences arise
between the exact and approximate values of these four spiral
parts. The greatest difference will be found in the value of X.
However, X is not often needed in the field. The values of o
and X。 are of more practical use, for they are needed in obtain-
ing T. by formula 5-15. The following table shows that a
relatively small error in T, would result from using the approxi-
mate method even when A is as large as 15°.
0
Given
Ls=300
D=10°
(A=15°)
Y=
X=
0 =
X。=
Values by
Approx.
Method
26.1
298.86
6.52
149.43
Exact Values from Table XI
Da=10°
Dc=10°
26.05
297.95
6.53
149.66
26.05
297.95
6.51
149.47
5-14. The A.R.E.A. Ten-Chord Spiral.-The ten-chord
spiral of the American Railway Engineering Association has
been used by many American railroads since about 1912.
This spiral is an approximation of equation 5-8 in which L、
110
SPIRALS
is measured by ten equal chords instead of around the spiral
itself. It is commonly used in connection with the chord
definition of D.
The spiral in this book may be converted practically to
the A.R.E.A. spiral by applying the corrections marked with
an asterisk, as explained in Part III on page 44. (See Appen-
dix A for the source of those corrections.) Consequently,
there is no need to master the details of the A.R.E.A. spiral or
to use tables prepared for it alone, as demonstrated by the
following comparison in the case of an exceptionally sharp
railroad curve:
X
Y
Xo
0
A
Dc Ls
10° 300 297.95 26.05 149.47 6.51 4°59.8′
10° 300 297.96 26.05 149.48 6.51 4°59.8'
5-15. Laying Out Spiral by Taping.-A spiral may be staked
by using tangent offsets, chord offsets, or middle ordinates.
The operations in the field are similar to those described in
Art. 2-14. However, the calculations differ slightly from
those for a simple curve owing to the variable curvature on
a spiral.
Spiral in this book
A.R.E.A. spiral
Tangent offsets for selected points on the spiral are com-
puted from the cube law (Art. 5-2). It is sufficiently accurate
for the flat spirals used on modern highway and railroad align-
ment to assume that the spiral and the throw o bisect each
Y
other, and that o= (equation 5-20). The throw is com-
4
2
puted from the relation
(equation 5-21) or is taken from
24 R
tables, after which the offsets from the tangent to the mid-
point of the spiral (at equidistant points) are found from the
principle that offsets are proportional to the cubes of the
distances. The same offsets are then used to locate the second
half of the spiral by measuring them radially from the circular
arc (osculating circle) produced backward from the S.C. For
example, the five offsets required to locate a ten-chord spiral
are found by multiplying the throw by 0.004, 0.032, 0.108,
0.256, and 0.500. The advantage of this method, in compari-
son with measuring all offsets from the tangent, is that the
measurements usually come well within the limits of the graded
roadbed.
SPIRALS
111
Chord offsets and middle ordinates are approximately
proportional to the lengths from the T.S. Needed values
D₁+ D₂
for D in the various
may be obtained by substituting 2
simple-curve offset formulas, D1 and D2 being the degrees of
curve of a spiral at the ends of a particular chord. The
values of C.O. and M.O. in Table I facilitate the computation.
5-16. Parallel Spirals. For the same reasons given in
Art. 2-15 it is often necessary to stake an offset curve parallel
to a spiral along the center line. This matter is more complex
than on circular curve alignment. It is also closely related
to the subjects of edge lengths and widening of highway
pavements. For these reasons parallel spirals are discussed
in detail in Art. 8-20 as a special curve problem in highway
design.
5-17. Length of Spirals.-A spiral need not have a par-
ticular length, but it should be at least long enough for the
transition to be made safely and comfortably. Design speed
and the rate of attaining superelevation are controlling factors
in this respect.
On railroads, practice has been fairly well standardized
for a number of years, though the operation of high-speed
streamliners is adding new problems. Chapter 9 contains
examples and recommendations.
Practice on highways is not so definite as is that on railroads.
Lack of standardization is due to the later adoption of spirals,
the more diverse operating conditions, and the greater number
of administrative units involved. Recent practice is outlined
in Chapter 8.
PROBLEMS
5-1. For each of the following spirals, compute values of
X, Y, X。, 0, L.T., S.T., and L.C. by use of Table XII. Use
Table XI to verify answers to (a) and (d) exactly and, by
interpolation, to verify the remaining answers approximately.
(a) Da=3°00′; L¸=200
(b) Da=1°45'; L,=160
(c) Da=7°12'; L,=250
(d) Dc=8°00′; L,=300
(e) D.=3°20'; L=240
(f) Dc=5°15'; L₁ = 160
112
SPIRALS
5-2. Assume simple curves with equal spirals corresponding
to those in Prob. 5-1. For each layout, compute values of
Ts and Es to hundredths using formulas 5-14 and 5-17.
Check the results to tenths using formulas 5-15 and 5-16.
Given values of I are:
(a) I=18°42'
(b) I=27°13'
minute.
(á) 18+57.12
(b) 26+34.37
(c) I=54°10′20″
(d) I=62°44'
5-3. Prepare sets of field notes for staking the spiraled
curves in Prob. 5-2 with set-ups at the T.S., S.C., and S.T.
Compute approach spirals as 5-chord spirals. Between the
S.C. and C.S. compute deflections for full stations on flat
curves (less than D=4°) and for full and half stations on
sharp curves. On all leaving spirals compute deflections for
full and half stations. All deflections should be carried to the
nearest
Stations of T.S. are:
(c) 35+61.41
(d) 57+72.28
(e) I=31°53'
(f) I=40°05'30"
5-4. Prepare alternate notes for staking certain of the
approach spirals of Prob. 5–3 from set-ups other than at the
T.S. Carry deflections to the nearest minute. Supply
explanatory comments as in the example on page 95.
(e) 71+17.59
(ƒ) 84+81.28
Spiral in (a): 5-chord spiral run backward from set-up
at C.S.
Spiral in (c): 10-chord spiral; set-ups at S.C. and point 6.
Spiral in (d): 20-chord spiral; set-ups at S.C. and points
8 and 14.
Spiral in (e): 16-chord spiral; set-ups at S.C. and points
5 and 12.
First Arc
5-5. Completely-spiraled compound curves. Find station-
ing of all curve points.
I
Second Arc
I
Sta. V
Each Spiral
D
D
k=13°
(a) 42+24.08 2°(Da) 23°48′ 7°(Da) 48°02′
(b)* 56+34.87 13°(D) 33°57′ 6°(Dc) 41°23′ 150-ftlong
*This is the same compound curve used in Art. 3-8 (with sta. V added).
Solve with the aid of formulas 5-32 and 5-33 using the exact values of
TL and Ts given in Art. 3-8.
SPIRALS
113
(c) 63+18.51 9°(Da) 63°14′ 3°(Da) 21°33′ k=2°
(d) 75+91.24 12° (De) 52°30' 6°(Dc) 34°18' 200-ftlong
Answers to (a): T.S.=27+95.49; S.1C.1=29+15.49; C.1S.2=
38+95.49; S.2C.2=41+95.49; C.2S.3=45+21.68; S.T.=49+
41.68.
5-6. Compute the deflections to the nearest minute for
staking the combining (middle) spirals of Prob. 5–5.
Spiral in (a): use a 10-chord spiral.
Spiral in (b): use a 5-chord spiral.
Spiral in (c): use a 6-chord spiral.
Spiral in (d): use a 4-chord spiral.
5-7. For each of the following simple curves with equal
spirals list all values of Da and L, found in Table XI that
conform to the stated conditions, without using spirals shorter
than 200 ft or longer than 400 ft:
(a) I=40°; Ts=700±25 ft.
Answers: Da=3°30' with
L.=200 or 250 ft; Da=4° with L、=350 or 400 ft.
(b) I=40°; E¸=180±10 ft.
(c) I=32°15′; T's=800±15 ft.
(d) I=54°27′; E,=85±5 ft.
5-8. Use the method of Art. 5-11 to find a simple curve
with equal spirals for each of the following conditions:
(a) I=18°42′; E,=exactly 27 ft. Find: the exact value of
Ls which, with an integral value of R listed in Table V, will
make p equal approximately 1. Answer: L=198.97 ft with
R=1,900 ft.
(b) I=62°44'; T, = exactly 600 ft. Find: the exact value of
L, which, with an integral value of R in Table V, will make p
equal approximately 0.55.
(c) For the curve in Prob. 2-8(a), choose an integral value
of L, so that p will equal approximately 1.
(d) For the curve in Prob. 2-8(c), choose an integral value
of L, so that p will equal approximately 0.4.
5-9. For each of the following simple curves with equal
spirals determine the added length resulting from substituting
114
SPIRALS
a double-spiral curve having the same radius at the mid-point:
(a) I=18°42′; Da=3°; Ls=200.
Answer: 423.34.
Answer: 485.44.
(b) I=62°44′; Dc=8°; Ls=300.
(c) I=54°10'20"; Da=7°12'; L=250.
(d) I=40°05'30"; D.=5°15′; Ls=160.
5-10. Using a slide rule and the approximate formulas in
Art. 5-13, compute Y, X, o, and X, for any of the spirals in
Prob. 5-1. Compare the results with those found in Prob. 5–1.
5-11. Derive:
(a) Formula 5-18 from the basic relations in Art. 5–2.
(b) Formula 5-19 from the basic relations in Art. 5–2.
(c) Formulas 5-36 and 5-37.
(d) Formula 5–38.
(e) The approximate formulas in Art. 5-13, using a sketch
to show the nature of the approximations made.
CHAPTER 6
EARTHWORK
6-1. Foreword.-Payment for grading is usually based on
a bid price per cubic yard for excavation measured in place as
computed from survey notes. The unit price ordinarily includes:
hauling excavated material (cut) from within the limits of the
roadway or moving in other material (borrow) from outside
areas; building the embankments (fill) to specified form; dis-
posing of surplus material (waste); and performing such opera-
tions as forming earth shoulders, trimming slopes, and prepar-
ing the subgrade for ballast or pavement.
Separate unit prices for different types of material excavated
may be used. There are advantages, however, in reducing
the number of classifications to two-"rock excavation" and
"common excavation" or even to a single type called
"unclassified excavation" (see Art. 6–5).
Fill quantities are important in grade-line design, though
they are not paid for directly in the usual contract. However,
on projects consisting wholly of embankment-such as levees—
the payment is based on a unit price for fill as computed from
survey notes.
Operations included under the general heading of earthwork
are (see relation to Art. 1-8):
(a) Office work of making preliminary estimate of grading
quantities by scaling depths of cut and fill at regular intervals
along one or more paper locations.
(b) Field work of taking cross-sections along located line
prior to construction.
(c) Office work of calculating volumes more accurately than
in (a) from data obtained in (b) and, possibly, of making a
distribution analysis (economical grading schedule) based upon
a mass diagram.
(d) Field work of setting stakes for controlling the construc-
tion and of making measurements needed for computing
partial and final payments.
(e) Office work of calculating all final quantities.
The same principles apply to computing volumes of mate-
rials other than earthwork, such as riprap and concrete built
into structures, material stock piles, and reservoir volumes.
115
116
EARTHWORK
6-2. Types of Cross-Sections. The exact determination of
earthwork quantities is usually based upon field cross-sections
taken in a specified manner before and after grading.
On highway and railroad work, cross-sections are vertical
and at right angles to the survey center-line. Every section
is an area formed by the subgrade (or base), the side slopes,
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EARTHWORK
117
and the original ground surface. Except as noted in Art. 6–5,
the base is flat and level, and the side slopes at a section are
uniform from the edge of the base to the ground surface. The
base is usually wider in cuts than on fills, to provide room for
side ditches.
Fig. 6-1 shows a portion of a graded roadbed passing from
fill to cut (side ditches are omitted for simplicity). The
sketch illustrates several cross-sections, the types depending
on their shape and the number of rod readings used to deter-
mine them.
At any cross-section a rod reading is always taken at the
center-line stake. Two additional readings are usually taken
at the intersections of the side slopes and the ground surface;
if stakes are driven at these points they are called slope stakes.
Additional rod readings are taken where conditions require
them.
The section at H is a regular three-level section in cut, so
named because three rod readings are used to fix it-one at
the center stake and the others at the slope-stake locations.
This type (in cut or fill) occurs more often than any other.
The sections at D and G are special cases of a three-level
section, each having a grade point (point D or G) at one corner.
The section at B is a five-level section in fill. This is a
modification of a three-level section in which two additional
readings are taken directly below (or above) the edges of the
base.
The section at A is an irregular section in fill. A large
number of rod readings are required to fix it—in the case
shown, there are four readings at points r in addition to the
three at the slope stakes and center stake.
The section at E is a side-hill section, having cut on one
side and fill on the other side of a grade point at E. In the
case illustrated the grade point is on the center line, but in
general it may fall anywhere between the edges of the base.
The section at I is a level section, so designated because the
ground is level transversely and only one rod reading at the
center line is sufficient.
6-3. Location of Cross-Sections.-For convenience in
calculations and field work, cross-sections are usually taken
at each full-station (or half-station) stake on the survey center-
118
EARTHWORK
line. They are also taken at curve points and at additional
plus-points where important "breaks" in the topography occur.
Where grading is very heavy or where unit costs are high, as
in rock excavation, cross-sections are taken at closer intervals.
D

hi
T
dr-
OTI
TI
f1
bib-
2
+dr
I
IOT
Fig. 6-2
S
fr
1
Stations
in Fig. 6-1
-I
·I·
↓
hr -H-
क

-G-

-E

-B
EARTHWORK
119
If the transition between cut and fill occurs on a side hill,
as many as five cross-sections may be needed. In Fig. 6-1
these sections are located at C, D, E, F, and G. Theoretically,
complete cross-sections are not necessary at C and F, but their
stationing is needed to locate the apexes of the pyramids hav-
ing triangular end bases at E. Thus, the cross-sections at the
transition are reduced to three: (1) at the fill-base grade point
D, (2) at the center-line grade point E, and (3) at the cut-base
grade point G. The points C and F are often so close to D and
G that they are omitted from the notes and the apexes of the
transition pyramids are assumed to fall at D and G. Where
the three sections at the transition are very close together, the
grade contour DEG is assumed to be at right angles to the
center line; there are then wedged-shaped solids on either side
of the grade contour. See page 130.
6-4. Formulas for End Areas.-Fig. 6-2 shows the areas at
certain cross-sections in Fig. 6-1. The common notation is
spread among the several sketches. The distance c is always
the vertical distance (cut or fill) between ground and grade at
the center line, and hɩ (or h,) is the vertical distance between
ground and grade at the slope stake. Distances between
ground and grade at other points are denoted by ci and c, in
cut, and by fi and fr in fill (as at section B). The inclination s
of the side slopes is expressed by the ratio of horizontal dis-
tance to vertical distance (as unity).
The horizontal distance from the survey center-line to any
slope stake is
(6-1)
or
dı (or d,) = b+s hɩ (or s h,)
The area of a level section (as at I) is
A₂ = c(b + D)
AL
(6-2)
2
AL=c (b+cs)
(6-3)
The area of a regular three-level section (as at H) is found by
adding the areas of the two cross-hatched triangles to the
areas of the two triangles having the common base c.
Thus,
Asc (di+dr) + 1 b (h₂+h,)
120
EARTHWORK
Substitution of H for hi+h, and D for di+d, reduces the
relation to
c D. b H
-+
(6-4)
2
4
Another convenient formula for the area of a regular three-
level section is found by extending the side slopes to an inter-
section at the center line so as to form a triangle, called the
grade triangle, below (or above) the base. The dimensions
of the grade triangle are constant until the base or slope
b2
Consequently, the "grade-tri-
A3:
changes; its fixed area is
angle formula" is
=
4 s'
4₁ = 1 (0+ 2015) - 10
A3=
s 4 s
(6–5)
Formula 6-5 is slightly more convenient than 6-4 for com-
puting a long series of regular three-level sections in cut or fill,
b
owing to the constant terms and
2 s
b2
4 s
The special three-level section having a grade point at the
ground surface (as at D or G, Fig. 6-1) is also determined by
formula 6-4. One of the four triangles disappears; there-
fore, D=dı (or d,)+1b, and H=hɩ or hr. The grade-triangle
formula also applies if properly modified, but its use is not
recommended at the transition between cut and fill.
At a side-hill section (as at E) the end areas for cut and fill
are kept separate. Obviously, both are triangles. In the
general case, with the grade point not at the center line, each
area is
Ar=wh
(6-6)
where w is the actual base width of the triangle. At section E
b
in Fig. 6–2, w=2-
The area of a five-level section (as at B) is found by combin-
ing the indicated triangles having common bases. The final
relation is
A5= (cb+fidi+fr dr)
(6-7)
If the section is in cut, c, and c, are substituted for fɩ and fr.
The area of an irregular section (as at A, Fig. 6-1) is best
found by a coordinate method. Thus, the coordinates of the
corners of the area-the origin being taken at the center of
the base and the order of the coordinates being determined by
EARTHWORK
121
progressing clockwise from A and repeating those at A-are
arranged as in the first form on page 122.
It will be recalled from plane surveying that the area of a
closed figure whose coordinates are set up in the preceding
form is equal to one-half the difference between the algebraic
sums of the products indicated by the full diagonal lines and
those indicated by the dotted diagonal lines.
A method of making the foregoing calculations in such a
way as to reduce chances of errors in signs is to set up the
coordinates as in the second form on page 122.
The coordinates start at the center of the base as origin,
and then proceed clockwise around the left portion of the section
and counter-clockwise around the right portion in the form of a
figure 8 on its side. Algebraic signs are omitted. All products
indicated by the solid diagonal lines have the same sign; all
those indicated by the dotted diagonal lines have the opposite
sign. As before, the end area is one-half the difference be-
tween the sums of the two sets of products.
Machine calculation of areas is done best with coordinates
set up in the second form. In using the machine, multiply
the figures connected by the solid diagonal lines, accumulating
the products on the proper dial. Then reverse-multiply the
figures connected by the dotted diagonal lines, thus subtract-
ing those products. The final result remaining on the dial is
the double area. For cut areas, the sum of the products
indicated by dotted diagonal lines is larger than the sum of
the products indicated by full diagonal lines; for fill areas,
the reverse is true. When using the calculating machine the
products making up the larger sum are set up first.
Some computers prefer to use machine calculation for all
areas, even the standard types represented by formulas 6–2
through 6-7. This is done by a generalization of the method
just described. In all cases omit algebraic signs. For areas
lying entirely to the left of the survey center-line, use the
coordinates in clockwise order; for areas to the right, use
counter-clockwise order of the coordinates. It makes no dif-
ference which coordinate is used first, so long as it is repeated
at the end, i.e., so long as the traverse is closed. For areas
cut by the survey center-line, use the figure-8 construction
described for section A, starting at a point on the center
line.
122
EARTHWORK
Ordinate -hi
Abscissa
-dr
$%-t
—h, --f4
+記 ​+d,
+ds
X
※
XXXXXX
-c
-f3
+ds!
Double Area by Coordinates-First Form
-f2 -fi
-d2 -d
fa
da ds
fi hi
xxxxxxxxxxxx
dr dı b
2
Double Area by Coordinates-Second Form
2
-hi
-d
EARTHWORK
123
After a little practice it will be found unnecessary actually
to write down the coordinates. Their values may be trans-
ferred directly from field notes to machine in proper sequence.
Original earth surface

JEME MEME ME
EXE
ROCK
Rock surface
ハミ
​Fig. 6-3. Compound section
VEIFIEIF/F
TER
Earth
FA
EE
6-5. Compound and Other Irregular Sections.-In locations
where rock lies between ground and grade, a compound section
occurs, as in Fig. 6-3. If there are different unit prices for
rock and earth ("common") excavation, it is necessary to
determine the quantities of each material. This is done by
first cross-sectioning the original surface, and then doing the
same to the rock surface after stripping.
In modern highway construction the trend is toward
denoting all excavation as "unclassified." There is only one
unit price, regardless of whether the material excavated is
ledge rock, loose rock, earth of any type, or a combination of
such materials with boulders and (as on much reconstruction)
old pavement, building foundations, railroad ties, or mis-
cellaneous scrap deposited in dumps. Even in a clear-cut
situation such as in Fig. 6-3, the contractor may prefer to
expedite the work by drilling through the earth cover and
blasting the underlying rock without having to clean the rock
surface for separate cross-sectioning. Obviously, the unclas-
sified specification speeds the work and eliminates arguments
between, contractor and engineer as to the pay classification of
material excavated.
Railroad roadbeds on curves are not usually crowned or
banked; the superelevation is adjusted in the rock ballast.
124
EARTHWORK
Earthwork quantities in the drainage ditches are usually com-
puted as a separate item.
On highway roadbeds the subgrade may be crowned on
tangents and is usually banked parallel to the surface on
curves. Moreover, the drainage ditches and earth shoulders
are usually considered part of the cross-sectional area. (See
Fig. 6-4.) The resulting irregular areas may be found by the
coordinate method (possibly employing a calculating machine
or an electronic computer) or by graphical methods.
6-6. End Areas by Graphical Methods.-End areas, no
matter how irregular, are easily found by plotting them to
scale and running a planimeter around the boundaries. This
method is widely used in highway work, especially if ditches
and shoulders are part of the cross-section proper. It is
particularly adapted to projects on which a permanent
graphical record of the cross-sections is desired, as in tunnel
construction through rock. Another distinct advantage of
plotted sections is their value in studying the effects of minor
changes in alignment or grade elevation. (See Chapter 10.)
In order to obtain precision consistent with the field work,
areas are plotted to a fairly large scale, usually 1 inch = 10 feet
or larger; consequently, the file of cross-section sheets is
voluminous.
Another graphical method, which is especially useful for
shallow areas plotted accurately to scale, is to mark off on the
edge of a paper strip the continuous summation of verticals at
1-foot intervals across the area. Each vertical is the area of
a trapezoidal strip 1 foot wide; the total length, applied
against the scale of the cross-section, is the desired end area.
A special scale (Avol rule) can replace the paper strip.
6-7. Methods of Cross-Sectioning.-The term cross-sec-
tioning is loosely used to include any vertical and horizontal
measurements made on a transverse section. On routes such
as railroads, highways, and canals, two methods of cross-
sectioning are in use: (1) cross-section leveling, and (2) slope
staking.
Cross-section leveling is used when end areas are to be
determined graphically, as by planimeter. It is also the
method which must be used to obtain whatever cross-profiles
EARTHWORK
125
are needed in the office work of grade-line design. The field
process is essentially that of profile leveling, the difference
being that intermediate foresights are taken at breaks in the
transverse profile in addition to breaks along the survey
center-line. The left-hand page of the notes is similar to
a page in differential leveling, but the notes run up the page.
On the right-hand page the cross-section notes are entered in
the form of fractions, the numerator of each fraction indicating
the rod reading (intermediate foresight) and the denominator
indicating the transverse distance to the rod from the survey
center-line.
Portion of
left-hand page
The hand level is a useful supplement to the engineer's
level, especially on steep transverse slopes. Rod readings
on ground higher than the H.I. are recorded as negative;
Form of Notes for Cross-Section Leveling
Station
(T.C.)
71+76.2
H.I.
71+00 472.58
473.4
468.3 469.6
Right-hand page
2470.5
466.9468.4
466.4 467.9A
| 467.3
465.8
0 12
465.6467.2
Rel
H.I.=472.6
-0.8 3.0 4.2 4.7 5.3 5.4 7.5 8.1
464.1465.1
462.8 2464.5
2.1 4.3 5.7 6.2 6.8 7.0 8.5 9.8
40 20
12 0 12
24
they must be added to the H.I. elevation (see the extreme left
reading at sta. 71+76.2 in the notes.
The form of notes represents a portion of the left-hand
page and complete notes for two cross-sections on the right-
hand page. The sections are plotted and used in office studies
as explained further in Art. 10–6.
-
126
EARTHWORK
KY///
•
Original surface

•
Fig. 6-4. Side-hill section on curve
Slope staking* is a special form of leveling used only after
the grade line and the form of cross-section have been decided
upon. The cross-sections are not usually plotted; either areas
or volumes are computed directly from the field notes.
This method may be used solely for the purpose of obtaining
data for calculating volumes without actually setting slope-
stakes. It is also used preceding construction where grading
is heavy and slope stakes are needed to control the work.
The process of slope staking at any cross-section consists
of finding and recording the positions where the designed
side slopes will intersect the ground surface and of recording
breaks in the transverse ground profile between slope stakes.
The work may be done in connection with profile leveling over
the located line or as a separate process after profile leveling.
The latter is the simpler and speedier method.
When profile levels have already been run, the slope-
staking party is provided with the ground and subgrade eleva-
tions at each station stake. Leveling is done with an engi-
neer's level, a hand level, a level board, or a combination of
these instruments. The form of notes resembles that used
in cross-section leveling. But the numerator of the fraction,
instead of being the rod reading, is the difference between the
ground-surface elevation and the center-line grade elevation
at the particular station. Cuts are designated C or +; fills,
For
No record is kept of the actual rod readings used in
the process. Also, the extreme left-hand and right-hand
entries in the notes are for the slope-stake locations.
The form on page 127 is a record of slope-staking notes
for sections along the 2-lane road shown in Fig. 6-1.
* See traverse method of slope staking in Art. 10-7.
EARTHWORK
127
In the example illustrated, elevations will have been sup-
plied for the full stations only, since the exact locations of
the grade points are not known in advance. Grade points are
Slope-Staking Notes-First Example
Sta.
Bases: 40 ft in cut; 30 ft in fill.
Side slopes: 14:1; Gradient: +1.2%.
¢

Surface Grade
Elev. Elev.
(I)-63+00 570.3 562.84
(H) 62+00 568.4 561.64
(G) 61+48 565.1 561.02
(E) 61+36 560.9 560.87
(D) 61+25 557.1 560.74
(B) 61+00 554.6 560.44
L
C7.6
31.4
C7.6
C7.0
30.5
C4.8
27.2
C2.6
23.9
0
15
F5.4
F6.9
23.1 15.0
F7.2 F8.8 F7.0
25.8 20.0 14.7
C6.8
C4.1
F3.6
F5.8
F8.0
R
C7.6
31.4
C6.4
29.6
0
20
F2.0
18.0
F5.8
23.7
F8.2
27.3
F8.1 F9.3 F9.4
18.0 24.0 29.1
F6.7
15.0
(A) 60+00 551.2 559.24
found by trial in the field, their grade elevations are computed,
and their surface elevations are obtained by leveling from any
convenient known point. It will be noted that the stations
at C and F were omitted. These points are assumed to fall
at D and G, as explained in Art. 6–3.
In the field the process of finding the slope-stake locations
is a matter of trial and error. It is one of those procedures
in which a detailed numerical illustration of the method
appears more complicated than it actually is. Briefly, the
process follows:
(1) Record the cut or fill at the center, found by taking the
difference between the given elevations.
(2) Observe whether the ground slopes up or down trans-
128
EARTHWORK
versely, estimate the cut or fill h at the probable slope-stake
location, and calculate the corresponding distance d from the
center line by the relation d=b+s h.
(3) Take a rod reading at the computed distance and find
the actual h
(4) If the actual and estimated values of h differ by more
than 0.1 foot, make a new estimate, being guided by the first
result; then repeat the process until d and h satisfy formula 6–1.
Distances and depths are determined to the nearest tenth.
Slope staking is done rapidly by the foregoing method.
With a little experience many of the points are located close
enough for the purpose on the first trial. More than two
trials are seldom needed.
Slope stakes, if set, are driven aslant. The cut or fill is
marked on the side facing the center line, and the stationing is
marked on the back.
If profile levels have not been run or if it is desired to verify
them in slope staking, the engineer's level is used to carry
continuous elevations along the line by usual leveling methods.
In this case it is convenient to use a device known as the grade
rod (see Fig. 6-5), which is the imaginary reading on a rod
held on the finished subgrade. If the line of sight is below
Ground
Rod
1
Cut hi
H.I. #1
AV
Grade
Rod (+)

Grade Elev.
또
​Fig. 6-5
Ground
Rod
Cut C
Fill hr
/
Grade
Rod (-)
Ground
Rod
H. I. #2
Д
12%
EARTHWORK
129
subgrade elevation, the rod is assumed to be read downward
and the grade rod is given a negative sign. Various com-
binations are possible, the sign of the result depending on the
type of section (cut or fill) and the relative elevations of
instrument and subgrade. The following rules always give
correct results if algebraic signs are strictly observed:
(1) H.I. minus grade elevation equals grade roa.
(2) Grade rod minus ground rod equals cut or fill at the
ground rod (the sign + signifies cut; the sign indi-
cates fill).
If desired, grade-rod readings may be entered in a separate
column in the leveling notes on the left-hand page of the field
book, with slope-staking notes on the right-hand page. Once
the center cut or fill has been determined, the slope-staking
process is the same as that previously described.
In brush or on steep slopes the field work may be expedited
by using a hand level or a level board to supplement the
engineer's level.
The second example of slope-staking notes along a single-
track railroad shows irregular ground with more-complex
transitions between excavation and embankment. Included
also is an example of the grade contour at right angles to the
center line.
6-8. Volume by Average End Areas.*-Except where the
solid between cross-sections is a pyramid (as between E and F
in Fig. 6-1), it is usually considered a prism whose right cross-
sectional area is the average of the end areas. For sections
having areas of A1 and A2 square feet and L feet apart, the
average-end-area formula for volume in cubic yards is
L / A₁+ A ₂
(
+4₂2)
Ve=
27
(6-8)
This formula is exact only when the end areas are equal.
For other cases it usually gives volumes slightly larger than
their true values. If it were to be applied to a pyramid, for
example, the error would be the maximum and would be
equal to 50 per cent of the correct volume. In practice, how-
ever, the total error in a long line is rarely more than 2 per
* See "contour grading” in Art. 10–7.
130
EARTHWORK
cent. Also, calculation of the errors or corrections (see Art.
6–11) is much more complicated than determining the average-
end-area volumes themselves. In consequence, the average-
end-area method is almost always used; it is invariably ruled
to apply in the absence of specifications to the contrary.
Slope-Staking Notes-Second Example
Bases: 20 ft in cut; 16 ft in fill.
Side slopes: 11:1; Gradient: -0.75%.

Sta.
20+15
Surface Grade
Elev.
Elev.
662.4
19+81 675.9 675.9
19+65 681.9 676.0
675.6
19+00 688.3 676,5
18+47 677.8 676.9
17+83
18+37 677.0 677.0
18+28 676.2 677.0
16+48
17+98 677.3 677.3
678.0 677.4
17+05 678.0 678.0
677.6
678.4
L
F11.4
25.1
00
108
C12.4
28.6
C13.0
29.5
C2.8
14.2
C1.4
12.1
C1.4 0
12.1 6
C1.8
12.7
C2.4
13.6
C1.8
12.7
0100
8
CO
&
F13.2
0
C10.0
C11.8
C0.9
F0.8
C0.6
F0.8
R
F12.4
26.6
00
8 10
C9.8
24.7
C8.6
22.9
0
10
F0.6
8.9
F1.4
10.1
F1.0
9.5
0
10
F1.2
9.8
F3.0
12.5
EARTHWORK
131
In applying the average-end-area formula the simplest
method is to add the end areas (determined by calculation or
by planimeter) and multiply their sum by Table XIX
54*
facilitates the process.
L
6-9. Example of Earthwork Calculation. The tabulation
on page 132 gives the results of earthwork calculations for
the notes on page 127. Areas were computed from formulas
given in Art. 6-4; and volumes from the average-end-area
formula (except for the two pyramids).
The number of significant figures used in computing the
areas and volumes is inconsistent with the principles set forth
in Arts. 2-16 and 2-17. Both the rod readings and measured
distances are round numbers; consequently, the computed
volumes are reliable to only two significant figures. Yet it is
conventional practice to consider the recorded measurements
as exact numbers and to compute volumes to the nearest
cubic yard (or to the nearest cu yd as in this example).
6-10. Earthwork Tables.-Table XIX is especially adapted
to highway work or other projects on which cross-sectional
areas are obtained graphically. Two other tables in Part III
are useful in earthwork computations:
Table XVII gives cubic yards per 100 feet for level sections
having various base widths and side-slope ratios. It is very
useful in making preliminary estimates of grading quantities
by scaling center cuts and fills from a projected paper location.
(See Art. 9-11.)
Table XVIII gives cubic yards per 50 feet for triangular
prisms having various widths and heights. It is the most
useful earthwork table for general purposes, since practically
all solids met in route surveying may be broken up into con-
stituent triangular prisms.
If w is the width (base) of any triangle and h is the height
(altitude), the volume in cubic yards for a triangular prism
50 feet long is
(6-9)
√50'
50
54
wh
132
EARTHWORK
Comparison with the average-end-area formula shows that
the volume between two end sections with areas A1 and A2 and
spaced 100 feet apart is equal to the sum of the volumes of two
50-ft prisms having the given right cross-sectional areas.
Even more generally, if the area of any type of section is
converted into the form A=1×(the product of two quanti-
Sta.
63+00
62+00
61+48
61+36
61+25
61+00
End Areas
Formula
I I I I I
6-4
6-7
60+00 coordinates
Sq Ft
390.64
*
338.34
144.76
C 26.00
F 15.00
113.16
258.15
338.16
Volumes, Cu Yd
Cut
1,350.0
465.2
37.9
3.5
Totals=1,856.6
Fill
2.2
26.11
171.9
1,104.3
1,304,5
ties), the volume in cubic yards between sections 100 feet
apart may be found simply by adding two values taken from
Table XVIII, each one found by entering the table with the
proper given quantities. If the distance between the stations
is less than 100 feet, the sum of the tabulated quantities is
multiplied by the ratio of the actual spacing to 100. By means
of this principle, volumes may be computed directly from
slope-staking notes without separate computation of areas.
The following computations show the use of Table XVIII
in determining three of the volumes previously found
(Art. 6-9) by the average-end-area method.
EARTHWORK
133
Sta.
Formula
Converted to
Form wh
63+00 (2 c) (b+c s)
62+00 c D+1(2)(H)
61+48 1 c D+1(2)(hi)
61+36w hɩ
w=2c=15.2
h=b+cs=51.4
Table XVIII
Entries
w=c=6.8
h=D=60.1
b
w=2
=20
h=H=13.4
w=c=4.1
h=D=47.2
พ
b
2
h=hr=4.8
W
20
b
=2/2=200
h=hr=2.6
Cu Yd
per
50 Ft
723.4
626.6
268.1
48.1
Take out under 5 (X10) 703.7
14.1
Take out under 1
Take out under 4 (×0.1)
5.6
Sum=cu yd per 50 ft=723.4
Cu Yd
Cut
Between
Stations
1,350.0
465.2
In entering Table XVIII either of the given quantities may
be taken as the height. By proper shifting of the decimal
point all the separate values making up a given product may
be taken from the same line in the table. For example, at
sta. 63 the given quantities are 15.2 and 51.4.
Enter left-hand column with 15.2
37.9
At sta. 62, instead of entering the table with 20.0 and taking
out the three values under 1(X10), 3, and 4(×0.1), it is
quicker to enter with 13.4 and take out the result in one
operation under 2(X10). Any slight error caused by multi-
plying a tabular value by 10 can be eliminated, if desired, by
adding proper values. For example, a result found under
134
EARTHWORK
2(X10) is obtained more accurately by adding the tabular
values under 6, 7, and 7.
6-11. Prismoidal Volumes and Corrections.-As noted in
Art. 6-8, the average-end-area formula usually gives volumes
slightly larger than their true values. When a precise value
is necessary-and the field measurements are refined enough
to warrant it-the solid between cross-sections is considered
to be a prismord rather than an average-end-area prism.
The prismoidal formula for volume in cubic yards is
L/A₁+4 Am+A 2`
(1₁+
Am+₁₂)
Vp
27
(6-10)
where Am is the area of a section midway between A1 and A2
and the other terms have the same meanings as in formula 6-8.
In route surveying the prismoidal formula applies to any
solid generated by a straight line passing around the sides of
plane parallel end-bases. Accordingly, it fits warped-surface
solids as well as plane-surface solids, provided that the warp
is continuous between the ends. The formula also applies to
a wide variety of solids seldom found in earthwork calcula-
tions, such as the frustums of prisms, cylinders, and cones.
Owing to the need for computing the area of the mid-section
A m, direct determination of volumes from the basic prismoidal
formula is inconvenient. It is easier to apply a prismoidal
correction Cp to the average-end-area volume. By definition,
(6–11)
Cp=Ve-Vp
When the values given by formulas 6-8 and 6-10 are sub-
stituted in formula 6-11 and the resulting formula is reduced,
the general prismoidal-correction formula is
Cp=
(A1-2 Am+A₂)
CT=
L
3X27
(6-12)
More convenient working formulas for solids commonly met
in practice are found by calculating Am in terms of the given
dimensions of A1 and A2 and substituting in formula 6-12.
(Note: Am is not the mean of A1 and A2, but its dimensions
are the means of corresponding dimensions at the end sections.)
For a solid having triangular end areas the result is
(W1—W2) (h1—h2)
L
12×27
(6–13)
EARTHWORK
135
Formula 6-13 can be made to fit any type of end area by
dividing the area into triangles. However, the prevalence of
three-level sections makes the following formula valuable:
(D1-D2) (C1-C2)
(6–14)
Although formula 6-14 is derived from the dimensions of
three-level sections, it also fits a solid having level-section end
areas and a solid having a triangular section at one end.
The prismoidal correction is applied with the sign indicated
in formula 6–11; that is, it is normally subtracted from the
average-end-area volume. In case the sign of Cr or C3 should
happen to be negative (rare, but possible), the prismoidal cor-
rection is added.
T
The corrections to the three volumes computed in Art. 6–10,
in cubic yards, are:
Sta. 62+00 to 63+00: C3=
C3
L
12×27
-
Sta. 61+48 to 62+00:С'3=
100
12×27
52
12×27
12
12X27
(62.8-60.1) (7.6-6.8)=0.7
(60.1-47.2) (6.8-4.1)=5.6
Sta. 61+36 to 61+48:C3=
(47.2-23.9) (4.1—0.0)=3.5
Prismoidal corrections may also be determined by means of
Table XVIII. The tabulated values come from formula 6–9,
which may be written in the following general form:
50
V50' = X(product of two quantities)
54
When L=100 the prismoidal-correction formulas 6–13 and
6-14 may also be written as follows:
Cp= (3) (4) (product of two quantities)
Consequently, the prismoidal correction for sections 100
feet apart is one-third the value found by entering Table
XVIII with (w₁—w₂) and (h₁-h2), or (D₁— D2) and (cı—c2), as
the given height and width. The three corrections previously
computed are verified by Table XVIII to be:
X2.00
=0.7
X0.52 X32.25=5.6
X0.12X88.5 =3.5
136
EARTHWORK
The foregoing corrections are 0.05 per cent, 1.2 per cent,
and 9.2 per cent of the respective average-end-area volumes.
It is evident that prismoidal corrections are insignificant,
except at transitions between cut and fill. Since, normally,
these locations account for only a small percentage of the total
yardage, it is obvious that volumes determined by the average-
end-area method are adequate for all but rare situations.
6-12. Correction for Curvature. Where conditions warrant
calculation of prismoidal corrections, they may also justify
correcting the prismoidal volumes on curves for the slight
error involved in assuming the center line to be straight.
On curves, cross-sections are taken radially. The true volume
between two such sections is a curved solid with plane, non-
parallel ends, as portrayed by Fig. 6-6(a). But when curva-
ture is ignored, the computed volume is that represented by
Fig. 6-6(b). The curvature correction is the difference between
these volumes, i.e.,
Cc=VComputed - VTrue
Add.com
(6-15)
Fig. 6-6(c) represents a typical cross-section at a station
on a curve. The center of gravity of the total end area is at
point G, located at a distance e from the survey center line.
The volume generated by revolving the end area is the product
of the area and the length of path described by its center of
gravity (theorem of Pappus). Obviously, if the end area were

Ź
(a)
20
R
S
I
B'
¢
(b)
Fig. 6-6
- dr dr→
dr-
(c)
2
335
&
dit dr
2
B
EARTHWORK
137
π
180
shaped so that G fell on the survey center line, the curvature
correction would be zero.
Let A denote the total end area of the section in Fig. 6-6(c).
The curvature correction per station is AX1 sta. -A (R − e)
π
D°. Since RD° equals 1 station, the curvature cor-
π
180
180
rection is AeD° per station. When A is in square feet and
e is in feet, this reduces to
AeD°
Cc=
cu yd per sta.
(6-16)
1,550
Three-level sections occur so often that it is convenient to
have a special version of formula 6-16 in which A and e are
replaced by the notation used in three-level sections. Fig.
6-6(c) is a three-level section in cut with the slope stake S on
the inside of the curve. If B' is drawn on the inner side slope
at the same elevation as slope stake B, the non-shaded portion
of the end area is symmetrical about the survey center line
and there is no curvature correction for that portion. The
remaining shaded area SB'C has its center of gravity at g,
which is two-thirds the distance from C to the mid-point
of SB'.
The curvature correction per station is
Area SB'CX1 sta. —Area SB'C [R—}(dı+d,)]
T
180
Since RD° equals 1 station, the curvature correction is
D°
π
180
π
Area SB'C (dı+d,) D° per station. But Area SB'C equals
1 (bsc) (hi-h,) sq ft. Therefore,
180
D°
C. = (3b+xo) (hi−ha) (di+dr) (1,000) cu yd per sta. (6-17)
For irregular sections, C. may be found by plotting the
sections to scale, drawing for each an equivalent three-level
section by estimation, and scaling the values needed in formula
6-17. If this method is not considered accurate enough (as it
may not be for highly-eccentric sections in rock), the following
procedure may be applied: An irregular section may be divided
into triangles; each triangular area may be multiplied by the
distance from its center of gravity to a vertical axis at the
138
EARTHWORK
survey center-line; and the algebraic sum of the products may
be divided by the total area of the irregular section. The
result is the eccentricity e of the total cross-section, or the dis-
tance from the survey center-line to the center of gravity G
(see Fig. 6-6). This is the familiar method of moments.
The curvature correction is applied as indicated in formula
6-15. The sign of Ce, however, may be positive or negative.
Where the end area is unsymmetrical about the survey center
line and has excess area on the inside of the curve, Ce has a
positive sign and is subtracted from the volume as computed
by ignoring the curvature. Where the excess area is on the
outside of the curve, the curvature correction is added.
In applying either of the formulas to find the curvature
correction for a solid between two different cross-sections L
feet apart, the results are averaged and multiplied by the
ratio of L to 100. For example, if a 10° curve to the left is
assumed, calculations for the curvature correction between
stations 61 +48 and 62+00 in the notes in Art. 6-7 give:
Sta. 62+00....C. 1.17 cu yd per sta.
Sta. 61+48....Cc-6.37
Sum=7.54 cu yd per sta.
Avg=3.77 cu yd per sta.
Cc=0.52×3.77=2.0 cu yd
The final corrected volume equals 457.6 cu yd, found as.
follows:
Ve=465.2 cu yd (Art. 6–10)
(Art. 6-11)
-C₂ = −5.6
-Сp=
-Cc=-2.0
V=457.6 cu yd
6-13. Borrow Pits.-When the quantity of material within
the theoretical limits of excavation is not enough to make the
fills, it is necessary to provide additional material, termed
borrow. It is most convenient to obtain borrow by widening
the cuts adjacent to the fills where the material is needed.
When this can be done within the right-of-way limits (and
without interfering with existing or planned structures), it has
the added advantages of permitting wider shoulders (on
EARTHWORK
139
highways), of “daylighting” curves, of reducing slope erosion,
and of minimizing snow drifting on the traveled way.
Material taken from borrow pits adjacent to the mair
construction may be measured by extending the regular cross-
sections and adding intermediate ones where necessary. The
work is conveniently done by the cross-section-leveling
method (see Art. 6–7).
Borrow pits located away from the route are cross-sectioned
independently of the survey stationing. A convenient method
is to stake out over the area a system of rectangles referenced
to points outside the limits of the work. By leveling at the
stakes before and after excavation, data are obtained from
which to compute the volume of borrow taken from the pit.

a
2
3
49
5
e
b.
6
|
7
8
OC
9
10
Id
13
14
// 15
17
18 8 22
19
2/
20
23
24
25
26
27
28
12
16
Fig. 6-7. Borrow pit
Fig. 6-7 shows a borrow-pit area over which 28 squares were
originally staked out. The cross-hatched line represents the
limits of the excavation. Squares are of such size that no
important breaks, either in the original ground surface or in
the pit floor, are assumed to lie between the corners of squares
or between the edge of the excavation and the nearest interior
corner. Those squares falling completely within the excava-
tion are outlined by a heavy line. Within that line each
square excavated to the pit floor is the volume of a truncated
square prism. Square 7, for example, has the surface corner
points b, c, d, and e; after excavation, corresponding points on
the pit floor are b', c', d', and e' (see Fig. 6-8). The volume
of the resulting prism is the product of the right cross-sectional
140
EARTHWORK
dd'
area A and the average of the four corner heights bb', cc',
and ee'. In cubic yards,
A
V₁=-
4X27
(bb'+cc'+dd'+ee′)
exactly
Each similar complete prism might be computed by the
preceding method. However, when a number of such prisms
adjoin one another, it is quicker to use the following relation
which gives the total volume of any number of complete prisms:
A
4X27
V=
(Σh₁+2 Zh₂+3 Σh3+4 Σhs)
(6-17)
1
In formula 6-17, A is the right cross-sectional area of the
unit rectangular prism, not the total area of all the complete
prisms; h₁ is a corner height found in only one prism; h₂ is one
common to two prisms; h, is one common to three prisms; and
h is one common to four prisms. For example, ee' is an hı,
dd' is an h₂, and cc' is an h.
f
a
b

B
C
2'
9
Fig. 6-8
The total borrow-pit quantity also includes the wedge-
shaped volumes lying between the complete prisms and the
limits of the excavation.
The portion of square 3, Fig. 6-7 excavated to the near
face of prism 7 is shown in Fig. 6-8 to be a wedge-shaped mass
with the cutting edge fa and the trapezoidal base ebb'e'. For
all practical purposes its volume is one-half the product of
the area of the base and the average of the horizontal dis-
tances ab and fe.
At a corner the portion of the square excavated is composed
approximately of two quarter-cones, base to base. For
example, as shown in Fig. 6-8, the mass in square 4 has one
quarter-cone with base fxg and altitude xe, and another quarter-
EARTHWORK
141
cone with the same base but with altitude xe'. The radius. r
of the circular base may be taken as the average
of fx and gx
Consequently, the volume at the corner equals (1) Xee
4
cu ft. If the height ee' is designated by h, the volume reduces
r² h
to approximately cu yd.
103
6-14. Shrinkage, Swell, and Settlement.-On many routes,
one object of the paper-location study is to design the grade
line so that total cut within the limits of the work will equal.
total fill. If it is assumed economical to haul all excavated
material to the embankments, the result is that borrow and.
waste are eliminated. Attainment of this ideal is prevented.
by many factors, one of which is the uncertainty regarding
shrinkage or swell of the material.
Shrinkage denotes the fact-commonly noticed—that 1
cubic yard of earth as measured by cross-sectioning before
excavation will occupy less than a cubic yard of space when
excavated, hauled to an embankment, and compacted in place.
This difference is due principally to the combined effects of loss
of material during hauling and compaction to a greater than.
original density by the heavy equipment used in making the
embankment. Shrinkage is small in the case of granular
materials such as sand and gravel; larger in ordinary earth
containing appreciable percentages of silt, loam, or clay;
and very high (possibly as much as 30 per cent) for shallow
cuts containing humus which is discarded as being unsuitable
for building embankments.
Since payment for grading is usually based upon excavation
quantities (see Art. 6-1), the shrinkage allowance in grade-
line design is made by adding a percentage to the calculated
fill quantities.
Swell is the term used in referring to a condition which is
the reverse of shrinkage. It occurs rarely, and then usually
in the case of broken rock blasted from solid beds and mixed.
with little, if any, earth in making embankments.
Swell is apt to be fairly uniform for a given material.
Shrinkage, however, varies not only with changes in the soil
constituents but also with fluctuations in moisture content
when compacted and with the type of construction equipment.
142
EARTHWORK
used. Consequently, a percentage allowance assumed in
design may eventually prove to be 5 per cent or more in error.
A common shrinkage allowance is 10 to 15 per cent for ordinary
earth having little material unsuitable for fills.
The term settlement refers to subsidence of the completed
embankment. It is due to slow additional compaction under
traffic and to gradual plastic flow of the foundation material
beneath the embankment. On railroad fills, small settlement
can be corrected by tamping more ballast beneath the ties as
routine maintenance work. In highway construction, new
fills are sometimes built higher than the designed subgrade
elevation and the placing of permanent pavement is deferred
until most of the settlement has taken place. With modern
construction methods, however, involving good foundations
and compaction at optimum moisture content, settlement of
fills is rarely serious.
DISTRIBUTION ANALYSIS
On projects in which embankments are built from material
excavated and hauled from cuts within the limits of the right-
of-way, mere calculation of separate cut and fill quantities
does not provide enough information. The distribution of
the earthwork, which involves the quantity, direction, and
distance hauled, is also important both in planning the work
and in computing extra payment in case the contract contains
an overhaul clause.
6-15. Haul, Free Haul, and Overhaul.-The word haul
has several definitions. In earthwork analyses, however,
it means either the distance over which material is moved or
(in a more technical sense) the product of volume and distance
moved, the units being station-yards.
The contract sometimes contains a clause providing extra
payment for hauling material a distance greater than a
specified amount, known as the limit of free haul. In this
case there is one unit price, per cubic yard, for earth excavation
and another unit price, per station-yard, for overhaul. The
former price includes hauling within the free-haul limit and
forming the embankments either inside or outside that limit
(see Art. 6-1). Short hauls are never averaged with those
DISTRIBUTION ANALYSIS
143
longer than the free-haul limit; therefore, there is no need to
calculate the station-yards of free haul.
TEXEIR
6-16. Limit of Economic Haul.-With an overhaul clause
in effect, there is obviously a certain distance beyond which
the cost of overhaul exceeds the cost of excavation without
overhaul. This limit of economic haul equals the limit of free
haul plus the quotient found by dividing the unit price for
borrow (or for excavation, if there is no separate price for
borrow) by the unit price for overhaul. Thus, if the free-haul
limit were 1,000 feet and unit prices for excavation and over-
haul were $0.60 per cu yd and $0.05 per sta.-yd, respectively,
the limit of economic haul would be 10+2=22 stations.
Balance points,

BA
с
L
ANRIFIED
G
Grade Line
AV
Ľ
Fig. 6-9. Station-to-station method
6-17. Balance Points. The principal problem in making a
distribution analysis is locating the stationing of balance points
between which excavation equals fill plus shrinkage allowance.
On a small job the primary balance points may be found by
making separate sub-totals of the cuts and corrected fills,
balance points being located where the two sub-totals are
equal. On important work this method is inadequate. It
does not fix intermediate balance points; neither does it give
data for computing overhaul, nor show how to schedule the
work. More detailed analyses may be made by the station-to-
station method or by the mass-diagram method.
Regardless of how complete an analysis is made, it is
fairly common practice to show balance points on the plans,
together with estimated quantities of cut, fill, borrow, and
waste. It is advisable to label such balance points as "approxi-
Mad
F
Α
144
EARTHWORK
mate," in order to avoid controversy with the contractor in
case the balance points should prove to be in error because of
variable shrinkage.
6-18. Station-to-Station Method.-Making a distribution
analysis by the station-to-station method is a numerical
process. The steps are illustrated by Fig. 6–9 which shows a
portion of the profile along a route center-line. A grade
point G is first located in the notes. Then balance points A
and A', a distance apart equal to the limit of economic haul,
are found by adding computed cuts and fills (plus shrinkage
allowance) in opposite directions from G. Balance points L
and L', spaced at the limit of free haul, are located similarly.
Excavation between A and L, which just equals that portion
of the fill between L' and A', is subject to payment for over-
haul. The average distance over which that excavation is
hauled is assumed to be the distance between the center of
gravity of the cut mass and the center of gravity of the fil
mass. Deducting the free-haul limit LL' from that distance
and multiplying by the yardage hauled gives the overhaul in
station-yards
In this method the center of gravity of each cut solid and
each fill solid between adjacent cross-sections (usually one
station apart) is assumed to lie midway between the sections.
Overhaul on each solid is the product of its volume and the
distance between its center of gravity and the beginning of
the free-haul limit. Thus, in Fig. 6-9, C is at the center of
gravity of the individual cross-hatched cut volume and F
is at the center of gravity of the indicated fill volume. The
overhaul on the cut is its volume times CL; the overhaul on
the fill is its volume times L'F. The total overhaul is the
sum of the products found by multiplying each volume of cut
between A and L by the distance between its center of gravity
and station L, plus those products found by multiplying each
volume of fill (plus shrinkage) between L' and A' by the dis-
tance between its center of gravity and station L'.
If B is an economic balance point following an earlier
grade point, the quantity of excavation between B and A is
not used in making embankment; it represents waste.
DISTRIBUTION ANALYSIS
145
WIM
6-19. Mass-Diagram Method. Though the numerical
method just described is quite simple and rapid, it is not
adapted to making a broad study of grading operations by
analyzing the effects upon the over-all economy produced by
various shifts in balance points. This is best done by a semi-
graphic method in which the mass diagram is used.
The earthwork mass-diagram is a continuous graph of
net cumulative yardage. It is plotted with stations as
abscissas and algebraic sums of cut and fill as ordinates.
Customarily, a cut volume is given as a plus sign: a fill volume
(plus shrinkage allowance) is given a minus sign.
Profile
А

1cuyd
-X stations-
G Grade Line
1 cu yd
وا
1 cu yd
Mass Diagram
Fig. 6-10
Á'
Haul in station-yards is measured by areas on the mass
diagram. In Fig. 6-10, suppose 1 cu yd of excavation at A on
the profile is moved X stations to A' in the embankment.
The haul is obviously X station-yards, which is shown graphi-
cally on the mass diagram by the cross-hatched trapezoidal
If the remaining excavation between A and G were to
be moved to the embankment between G and A', the haul for
each cubic yard would be shown on the mass diagram as a
area.
146
EARTHWORK
stack of trapezoidal areas above the one indicated. The total
haul in station-yards between A and A' would be the area aga'.
The profile illustrated in Fig. 6-9 is repeated (to reduced
scale) in the upper sketch of Fig. 6-11. Directly below is a
representation of the corresponding mass diagram.
By reference to Fig. 6-11 the following characteristics of a
mass diagram are apparent:
BA
T
Va
Pb
Profile
L
HELENINTE

GL
وا
е
K
Mass Diagram
Fig. 6-11
h
A'
La
(a) Any horizontal line (as aa') intersecting the mass dia-
gram at two points is a balance line; total cut and total fill are
equal between the stations at the intersections (as A and A').
(b) Any ordinate between two balance lines (as kl) is a
measure of the yardage between the stations at the extremities
of the balance lines (as between A and L or between L' and A').
Stated more generally, the vertical distance between two
points on the diagram (as a and b) is a measure of the yard-
age between the corresponding stations.
(c) The highest point of a loop (as at g) indicates a change
from cut to fill (in the direction of the stationing); conversely,
the lowest point represents a change from fill to cut. Such
points may not fall exactly at the stationing of center-line
DISTRIBUTION ANALYSIS
147
grade points if there is a side-hill transition (as at sta. E in
Fig. 6-1).
(d) The area between the diagram and any balance line is a
measure of the haul in station-yards between the stations at
the extremities of the balance line. If this area were divided
by the maximum ordinate between the balance line and the
mass diagram, the result would be the average haul in stations.
In Fig. 6-11 the area bounded by aga' measures the total haul
between A and A'; that bounded by lgl' measures the total
haul between L and L'. Since the latter is free haul, as is also
the station-yards represented by the rectangle kl', the differ-
ence between total haul and free haul is the overhaul between
A and A'. This overhaul is represented by the two cross-
hatched areas.
When the portions of a mass diagram on the sides of two
related overhaul areas are fairly smooth (as al and l'a'), even
though not straight, the sum of the two areas may be found by
drawing a horizontal line midway between the two balance
lines, deducting the free-haul distance from its length, and
multiplying the difference by the ordinate between the balance
lines. For example, hh' bisects kl. The points h and h' are
approximately at the centers of gravity of the volumes between
A and L and between L' and A'. Consequently, the overhaul
between A and A'is (hh'—ll')×kl.
In case the mass diagram is very irregular between balance
lines, the overhaul may be determined either by planimeter or
by the method of moments. In the first of these methods the
overhaul is found directly by planimetering the areas repre-
senting overhaul and applying the necessary scale factors to
convert areas to station-yards. If needed, the distance to the
center of mass of the yardage overhauled could be found by
dividing the overhaul by the volume. Thus, in Fig. 6-11, the
station of the center of mass of the yardage between A and L
area alk
is sta. A+
kl
In the method of moments each separate volume is multi-
plied by its distance from a selected station, and the sum of the
products is divided by the sum of the volumes. The result is
the distance from the selected station to the center of mass.


148
Fig. 6-12
NET CUMULATIVE YARDAGE
€
こ
​Stations
MASS DIAGRAM
391
I
ELEVATIONS
| | PROFILE |
| |
Borrow
Waste
Waste
Borrow
Waste
DISTRIBUTION ANALYSIS
149
As in any other method, overhaul=yardage (distance between
centers of mass — free-haul distance).
Other useful principles in making a distribution analysis by
mass diagram are illustrated by Fig. 6-12, which represents
the profile and mass diagram of a continuous section of line.
Balance lines equal in length to the limit of economic haul
(as aa' and cc') are first drawn in the larger loops.
1=
Between a' and c the most economical position of the balance
line is at bb', which is drawn so that bb₁ = b₁b' with neither
segment longer than the limit of economic haul. That this
is the best position may be shown by imagining bb' lowered to
coincide with the horizontal plotting axis. There would be
no change in the total waste; the waste at b would be decreased
by the increase at b'. However, the total haul would be
increased by the area shown cross-hatched diagonally and
would be decreased by the area cross-hatched vertically. Since
these areas have equal bases bb₁ and b₁b', there is a net increase
in area, or haul. Shifting the balance line higher than bb'
would obviously have the same effect.
The balance line dd' is adjusted so that (ddı+d½d' — d₁d2}
is equal to the limit of economic haul and no segment is greater
than that limit. An analysis similar to that made for bb'
would prove that raising or lowering dd' from the position
shown also increases the cost.
In general, the most economical position for a balance line
cutting any even number of loops is that in which the sum of the
segments cutting convex loops equals the sum of the segments
cutting concave loops, no segment being longer than the
limit of economic haul. The most economical position for a
balance line cutting any odd number of loops is that in which
the sum of the segments cutting convex (or concave) loops
less the sum of the segments in loops of opposite form equals
the limit of economic haul, no segment being longer than that
limit.
Theoretically, the foregoing principles are unaffected by the
length of free haul. For example, if the alternate positions of
balance line bb' produced segments longer than the free-haul
limit, overhaul would be increased with consequent increase in
the payment to the contractor. If the balance lines were
shorter than the free-haul limit, there would be no actual pay-
150
EARTHWORK
ment for overhaul in either case. Nevertheless, the total
haul in station-yards would be increased, thus adding to the
contractor's cost of doing the work and, possibly, influencing
him to submit slightly higher bid prices.
In drawing balance lines, one note of caution should be
mentioned: adjacent balance lines must not overlap. The effect
is to use part of the mass diagram twice-an obvious impossi-
bility except by borrowing an extra mass of earthwork meas-
ured by the distance between the overlapping balance lines.
Figs. 6-13 and 6-14 show two solutions for a case not found
in Fig. 6-12. This is the case in which there is an intermediate
loop that is not cut by a balance line equal in length to the limit
of economic haul.
Borrow.
(+)
O
(-)
X
C
Profile
Yardage K

Yardage KI
||
C
D
Mass Diagram
Fig. 6-13. Two-way hauling from intermediate cut
Waste

B
In both solutions, AB is the limit of economic haul and CD
is the free-haul distance. The total overhaul in Fig. 6-13 is
the sum of the two numbered cross-hatched triangles on the
mass diagram (found in the usual way) plus the overhaul
(shaded) on yardage K, which equals (XY-CD) XK station-
yards.
DISTRIBUTION ANALYSIS
151
In the solution shown by Fig. 6-14, the total overhaul is the
sum of the four numbered cross-hatched triangles plus the over-
haul (shaded) on yardage K', which equals (X'Y' −CD)×K'
station-yards.
Borrow
Theoretically, the solution in Fig. 6-14 is the more eco-
nomical one because it has less overhaul, more free haul, and
the same amounts of borrow and waste. Yet, in practice, the
solution in Fig. 6-13 might be preferred because of the two-
way hauling and the shorter haul distances.
(+
(-)
1'0
Profile
-Yardage K

-Yardage K
3°0
Mass Diagram
2
Fig. 6-14. Uni-directional hauling
Ό
<
Waste

Making a distribution analysis by mass diagram is not the
purely mechanical process implied in the preceding discussion.
Factors other than obtaining theoretical maximum economy
enter into the planning of grading operations. For example,
on steep grades the contractor prefers loaded hauls to be down
hill. Moreover, he may prefer to haul more of a particular
cut in a certain direction than is indicated on the plans.
Again, there may be one fill which acts as a bottle-neck.
Building it ahead of schedule, possibly by using extra borrow
152
EARTHWORK
or longer hauls than those theoretically needed, may save time
and money. These preferences may be realized by exercising
good judgment in altering the theoretical balance lines. The
result may be submission of lower bid prices. Even if the
bid is not lower, the contractor is better satisfied-a condition
which should produce a better job and friendlier relations with
the owners.
Even if a grading contract contains no overhaul clause (this
practice is becoming more common), the mass diagram is still
very useful in the work of grade-line design. Approximate
balance points are shown on the final plans to indicate the
grading schedule to the contractor. It is then his responsibility
to calculate or estimate the hauls and to adjust his bid prices
for excavation to cover their cost.
PROBLEMS
(NOTE.-Problems 6-1 through 6-8 refer to the notes on page 127.)
6–1. Compute grade-rod and ground-rod readings at:
(a) Sections at A and B. Assume H.I.=557.65.
G. Assume H.I.=566.14.
Assume H.I. =575.32.
(b) Sections at D, E, and
(c) Sections at H and I.
6-2. Verify any of the end areas specified by the instructor,
using the formulas shown in the table on page 132. Check the
results at stations 61+25, 61+48, 62+00, and 63+00 by use
of formula 6-5.
6-3. Verify any of the volumes specified by the instructor,
assuming the areas tabulated on page 132 to be correct. Use
formula 6-8 (except for the two pyramids) and check the
results by use of Table XIX.
6-4. Verify the calculations tabulated on page 133.
6-5. Verify the prismoidal corrections for the three cut
solids between stations 61+36 and 63+00 as found by the
two methods used in Art. 6-11.
-11. Then find values of V, by
applying these corrections (with proper sign) to the values
of Ve tabulated on page 132.
6-6. Use formula 6-10 to compute values of V, for any
p
DISTRIBUTION ANALYSIS
153
(or all) of the cut solids between stations 61 +36 and 63+00.
Compare the results with Prob. 6-5 and decide which of the
two methods of finding V, is preferable.
6-7. Assume that all sections lie on a curve to the left. By
inspection of the notes, determine the sign of the curvature
correction for each solid.
6-8. Using formula 6-17 and assuming a 12° curve to the
left, determine the curvature corrections for any (or all) of
the cut volumes between stations 61+36 and 63+00. Indi-
cate whether the corrections should be added or subtracted.
(NOTE.-Problems 6-9 through 6-15 refer to the notes on page 130.)
6-9. Compute grade-rod and ground-rod readings at:
(a) Stations 16+48 through 17+83.
(b) Stations 17 +98 through 18+37.
(c) Stations 18+47 through 19+65.
(d) Station 20+15. Assume H.I.=667.6.
(a) 16+48 to 17+83.
(b) 17+83 to 18+28.
(c) 18+28 to 18+47.
(d) 18+47 to 19+65.
(e) 19+65 to 20+15.
Assume H.I. =683.4.
Assume H.I.=681.7.
Assume H.I.=689.7.
6-10. Compute yardages of cut and fill between the indi-
cated stations. Use formula 6-8 supplemented by Table XIX
for all solids except pyramids. Verify the results by use of
Table XVIII.
Partial answer: Fill-31.0 cu yd.
Partial answer: Cut=14.4 cu yd.
Partial answer: Fill=2.7 cu yd.
Partial answer: Fill=0.
Partial answer: Cut=111.9 cu yd.
6-11. What condition must exist in formulas 6-13 or 6-14
in order for the prismoidal volume to (a) exceed the average-
end-area volume and (b) equal the average-end-area volume?
Which solids in the notes conform to either of these conditions?
6-12. Compute the prismoidal corrections for the solids
between the following stations, and state whether the cor-
rections should be added to, or subtracted from, the average-
end-area volumes:
(a) 16+48 to 17+05 (fill). Answer: Subtract 1.5 cu yd.
154
EARTHWORK
(b) 17+83 to 17+98 (cut). Answer: Subtract 0.3 cu yd.
(c) 18+47 to 19+00 (cut).
(d) 19+00 to 19+65 (cut).
(e) 19+81 to 20+15 (fill).
6-13. Use formula 6-10 to compute values of V, for any
(or all) of the solids in Prob. 6-12. Compare the results with
those obtained by combining Probs. 6-10 and 6–12.
6-14. Assume that all sections lie on a curve to the right.
By inspection of the notes, determine the sign of the curvature
correction for each solid.
6-15. Using formula 6-17 and assuming a 20° curve to the
right, determine the curvature corrections for any (or all) of
the solids in Prob. 6-12. Indicate sign. Partial answers:
(a) subtract 0.5 cu yd; (c) add 2.8 cu yd.
6-16. Plot a mass diagram from the data in Table A or
Table A: Free-haul limit 800 ft. Cost of excavation $0.70
per cu yd; borrow $0.80 per cu yd; overhaul $0.08 per sta.-yd.

Sta.
O12 M † 10 CO I ∞ ∞
0
3
4
5
6
7
8
9
10
11
12
13
Net Cumu-
lative
Yardage
0
-386
- 1352
-2881
-4668
-6175
-7290
-8108
-8652
-8994
-9303
-9382
-9278
-9090
Sta.
14
15
16
17
18
19
20
21
2223
24
25
26
27
Net Cumu-
lative
Yardage
-8745
- 8264
-7837
-7290
-6524
-5605
-4112
- 2606
0
2992
5115
6784
7992
8263
Sta.
28
29
30
31
32
33
34
34+70
35
36
37
38
39
40
Net Cumu-
lative
Yardage
8091
7100
5408
2996
1707
1220
1046
1018
1092
1387
2175
3290
4464
5985
DISTRIBUTION ANALYSIS
155
Table B using a horizontal scale of 1"=5 sta. and a vertical
scale of 1″=5,000 cu yd. (An 8X11 inch sheet of cross-
section paper will suffice.) Show a hypothetical profile above
the diagram, as in Fig. 6-12. Establish balance points and
compute the cost of grading. On the profile, show the sepa-
rate amounts of waste, borrow, and yardage excavated; indi-
cate disposition of excavation by arrows.
Sta.
Table B: Free-haul limit 800 ft. Cost of excavation $1.00
per cu yd; borrow $0.80 per cu yd; overhaul $0.10 per sta.-yd.

012 30 + 10 67
4
5
8
9
10
11
12
12+50
13
14
Net Cumu-
lative
Yardage
0
284
691
1090
1672
2393
3205
4184
5351
6904
7922
8465
8732
8860
8614
8035
Sta.
15
16
17
18
19
20
21
22
23
24
25
25+60
26
27
28.
29
Net Cumu-
lative
Yardage
7100
5993
4481
1573
-315
-2064
-3182
-4057
-4876
-5221
-5402
-5483
-5470
-5252
-4773
-3588
G
Sta.
282383
30
31
34
35
36
37
38
39
40
41
42
43
44
45
Net Cumu-
lative
Yardage
-1206
+620
+1021
+810
-208
-2182
-3889
-4750
-5015
-4823
-4462
-3924
-3348
-2180
-605
+1400
PART II
PRACTICAL APPLICATIONS
:
CHAPTER 7
SPECIAL CURVE PROBLEMS
7-1. Foreword.—In a subject as utilitarian as route survey-
ing, there is hardly a strict division between basic principles
and practical applications. Though these headings are used
in this book, many practical features have already been re-
ferred to in Part I; moreover, some problems involving addi-
tional theory will be found in Part II. Nevertheless, Part I
is complete in itself; it does not require this additional material
in order to understand and apply the theory to any practical
problem met in route surveying.
Part II contains specific applications of basic theory to
some of the common problems and survey procedures found
in practice. It is not a detailed compilation of instructions
covering field and office work; such instructions are well taken
care of in the manuals published by most state highway de-
partments and various other organizations for the guidance
of their chiefs of party. Part II is in the nature of a supple-
ment to such manuals. Technical knowledge of route survey-
ing being assumed, the explanations are briefer than those
given in Part I and the simpler proofs are omitted.
In practice, special curve problems occur in such great
variety that it is not possible to include a large number of
them in the space allotted to this chapter. Doing so, if
possible, would have questionable merit, since the "textbook
type" of problem is less apt to occur than some perplexing
variation of it. To serve the purpose, a few of the more
common problems will be described and general methods of
approach will be outlined. These, combined with a thorough
grounding in the basic principles of Part I, should enable the
engineer to develop the skill needed for solving any special
curve problem.
7-2. Methods of Solution.-In solving special curve prob-.
lems there are four general methods of approach: (1) an exact.
geometric or trigonometric solution, (2) a cut-and-try calcula-
159
160
SPECIAL CURVE PROBLEMS
tion method, (3) a graphical method, and (4) a cut-and-try
field method.
Generally the first method is preferred. However, if the
solution cannot be found or is very complicated, cut-and-try
calculation often provides a fairly quick solution. In case a
problem is not readily solvable by either of these methods, the
unknowns may sometimes be scaled from a careful drawing;
the scale must be fairly large to give adequate precision. Some
problems are adaptable to cut-and-try solution in the field
(see Art. 3–14 for an example previously described). Addi-
tional examples of these methods of solution follow.
OBSTACLE PROBLEMS
7-3. P.I. Inaccessible.-Simple Curve: This problem is of
frequent occurrence. Conditions are shown in Fig. 7-1. The
problem is to locate the curve points A and B.

L
X
J
===
1
B
Fig. 7-1
Find any convenient line XY cutting the established
tangents.
Measure distance XY and deflection angles x and y. Angle
[=x+y.
Calculate XV and VY by using the sine law. Subtract
their values from T (or T、), giving the required distances to
the beginning and end of the curve.
Compound Curve, Fig. 7-2: In this case the cut-off line is
chosen so as to establish the T.C. and C.T. at X and Y. One
OBSTACLE PROBLEMS
161
From Σ latitudes:
more variable must be known. Assume Rs, since the maxi-
mum D is often fixed by specifications. Thus, the four known
values are XY, Rs, and the angles x and y.
Use the closed traverse shown by heavy lines and solve by
the traverse method described in Art. 3-6.
Angle I=x+y
RL-XY sin x-Rs cos I—(RL-RS) cos IL=0

X
R₁
OL
Y
Os
R
tan IL:
Is
Fig. 7-2
If the hint preceding equation 3-3 in Art. 3-6 is used, this
relation reduces to
vers IL:
=
XY sin x-Rs vers I
RL-RS
(7-1)
Similarly, the relation based on Σ departures reduces to
XY cos x-Rs sin I
RL-RS
sin IL
(7-2)
When equation 7-1 is divided by 7-2, the result is
XY sin x-Rs vers I
XY cos x-Rs sin I
(7-3)
Solve equation 7-3 for IL. Then obtain RL-Rs from
equation 7-1 (or 7-2). Finally, Is=I—IL.
Use the same method if RL were assumed; the final relations
differ only in signs and subscripts.
162
SPECIAL CURVE PROBLEMS
7-4. T.C. or C.T. Inaccessible.-In Fig. 7-3, assume В
to be the C.T. The problem is to stake the computed curve
and to check the work for alignment and stationing.
Set a check point F on the forward tangent by measuring
VF, using right-angle offsets to get around the obstacle. Sta.
F=Sta. C.T.+VF−T.
a
Stake the curve from A to a station (as P) from which a
sight parallel to the forward tangent would clear the obstacle.
Occupy P and deflect angle I-2 (i.e., angle I minus the
tabulated deflection for the station occupied), thereby placing
the line of sight parallel to the tangent. Set point E on this
line by measuring PE=Sta. F-Sta. B+R sin (I—a).
A
R1
1
kay
V

Fig. 7-3
E
012
Occupy E, turn 90° from EP, and check the distance and
direction to F. The offset EF should equal R vers (I-a).
(If EF is small, it may be computed by slide rule from formula
2-30, in which arc PB is used for s.)
While the transit is at A (or P), set stakes on the curve
for cross-sectioning between P and the obstacle, leaving one
stake at the plus point marking the beginning of the obstacle.
If B were the T.C., run in the part AP of the curve back-
ward from the C.T. The rest of the process is similar in
principle to that just described.
OBSTACLE PROBLEMS
163
If the curve is spiraled and the S.T. or T.S. is made inacces-
sible, such as by the obstacle at position 1 in Fig. 7-4, the field
procedure is the same but the computations differ somewhat.
In the general case, assume that the line of sight from the
C.S., pointed parallel to the forward tangent, would be cut off
by the obstacle at the S.T. As before, run in the layout to a
station P on the circular arc, a being the central angle (twice
the deflection difference) between P and the C.S. at C.

a→
or
P_CTC.S.) Δ
X
त
Į
Δ
R
2010
KILWA
E
-Position 2
sta. F-sta. P+
(5.7.) QB
w's
F
-pa
-Position I
Fig. 7-4
Occupy P, turn the line of sight parallel to the forward
tangent, and measure the distance PE to any convenient point
beyond the obstacle.
Point K is the offset C.T. (KJ=0; BJ=X。).
The offset PQ (=EF)=0+} (PK)² D, the value for PK
being taken equal to the difference in stationing between P
and C plusLs. If the offset is large, compute it from the
relation PQ=0+R vers (a+A).
For checking out on the tangent, the relation is:
sta. F-sta. P+PC+CB+BF
100 a
D
+L+[PE-Xo-R sin (a+A)]
If both the P.I. and the T.C. (or C.T.) are inaccessible, a
combination of the foregoing procedures will provide the solu-
tion.
164
SPECIAL CURVE PROBLEMS
7-5. Obstacle on Curve. For methods of by-passing
obstacles preventing sights to curve points or obstacles on the
curve itself, see Arts. 2-11 and 2-14 along with Figs. 2-8 and
2-11.
you
++++B' Main
B' Main B X Track V
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est

1
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od
05 کار
Proposed
Warehouse
Fig. 7-5
F 1
t +
2
22살
​Rest
If the obstacle cuts the spiral and the tangent but does
not obstruct the S.T. (or T.S.), as at position 2 in Fig. 7-4,
first run in the layout to the C.S. (or S.C.). Then set up at
the C.S. (or S.C.) and locate stakes on the spiral which are
needed for cross-sectioning between the set-up and the
obstacle. Finally, check the position of the S.T. (or T.S.)
and the direction of the main tangent, by measuring the spiral
coordinates X and Y and turning the necessary right angles.
An alternate procedure, which may be used if the obstacle
interferes with the sight from C pointed parallel to the main
tangent, is to place the line of sight on the local tangent at C
CHANGE-OF-LOCATION PROBLEMS
165
and to measure the spiral short tangent to a point X on the
main tangent. The field work may then be checked by
occupying point X, deflecting angle A from a backsight to
point C, and measuring (possibly using right-angle offsets
around the obstacle) the spiral long tangent to a check at the
S.T. (or T.S.) of the layout. See also Art. 5-2 and Fig. 5-2.
7-6. Obstacle on Tangent.-For the general method of by-
passing an obstacle on a tangent at a point unaffected by a
curve, see Art. 9-13.
Examples of obstacles spanning both a tangent and a curve
are given in Arts. 7-4 and 7-5.
Special problems affecting tangent distances of curves have
infinite variety. Fig. 7-5 represents a case in which it was
required to run a spur track to a proposed warehouse. Con-
ditions (not shown) fixed the warehouse location as indicated.
It was impossible to run in a simple curve AB without inter-
fering with existing buildings and a turnout track. The solu-
tion was a compound curve APB'. See Prob. 7-9 for the
numerical values involved.
CHANGE-OF-LOCATION PROBLEMS
After part of the paper-location alignment has been staked,
desirable minor adjustments often become apparent (see Art.
1-12). Frequently the data obtained and the stakes already
set for the original curve can be used to advantage in making
the revision. A few common cases coming under this heading
are outlined in the following articles. More extensive adjust-
ments are best made by locating the new P.I. and staking the
revised curve irrespective of the original layout.
If the line has been staked and cross-sectioned beyond a
location change, it is necessary to determine and mark clearly
the station equation at the point common to both layouts.
7-7. Practical Suggestions.-Skill in solving change-
of-location problems does not come from memorizing certain
"textbook solutions" but is developed by identifying the key
steps in those solutions and applying them to the unusual prob-
lems that arise in practice. The following practical hints,
numbered for ease of reference, are helpful.
166
SPECIAL CURVE PROBLEMS
Hint 1: Draw a careful sketch, which is not necessarily to
scale. Exaggerate small distances to make their effects clear.
Preserve right angles. Do not make other angles close to 90°;
otherwise, a special case might result. If only one graphical
solution is possible when the known data are used, the problem
is definite and determinate.
Hint 2: If the problem involves a revision of some kind,
such as the shift of a tangent, the solution must contain that
known revision. Try to connect known revisions to known
points on the original layout by simple geometric construction,
especially by triangles. Considering a triangle containing
curve centers or vertices often leads to the solution.
A A V V

оо
Fig. 7-6
SB
Jok
Hint 3: A known linear revision may often be expressed
as the difference between a known part of the original layout
and a similar unknown part of the revised scheme. Also, an
unknown linear revision may equal the difference between
similar known parts of the two layouts.
Hint 4: If a point is common to both layouts, perpendicu-
lars dropped from that point to tangents or radii frequently
disclose the key to the solution.
Hint 5: Although there is only one correct set of numerical
answers to a definite problem, there may be several correct
geometric solutions. If a certain solution cannot produce
adequate precision, determine the reason for the lack of preci-
sion and search for a better solution.
CHANGE-OF-LOCATION PROBLEMS
167
Hint 6: If a solution by means of simple construction can-
not be found, recall that a solution by traverse is usually
applicable. (See Art. 3-6.)
7-8. Simple Curve; New Parallel Tangent; Same D.
Assume that the forward tangent is to be shifted outward
parallel to itself a small distance p in order to reduce grading
or to improve the approach to the next curve.
A
오
​0°

Rs
S
0% 11 R₁
B
skew shift
b
!B'
P
sin I
P
Fig. 7-7
In Fig. 7-6, the skew shift obviously equals AA'=00'=
VV'= BB'. From a triangle at any one of these positions
(see Hint 2, Art. 7–7),
(7-4)
Tape the skew shift from A and B to locate hubs at the
new curve points A' and B'.
For a tangent shifted inward, use A'B' as the original curve.
7-9. Compound Curve; New Parallel Tangent; Same D's.
This is the same problem as the one in Art. 7-8, except that the
curves are compounded at C and C' (Fig. 7-7).
Although a solution by construction is possible (Hints 3
and 4 being used), it may not be readily apparent. Therefore,
use the traverse method (see Hint 6).
In the closed traverse bBО10s0′ıb,
Σ departures=-RL+(RLRs) cos IL
— (RL-Rs) cos I'+(Rь−р)=0
-
L
Ꮮ
168
SPECIAL CURVE PROBLEMS
from which
(7-5)
The distance bB needed for locating the new C.T. at B' is
found by setting Σ latitudes equal to zero and reducing. The
result is
cos I'L=cos IL-
bB=(RL-Rs) (sin I'L-sin IL)
(7-6)
(Observe that equation 7-6 also comes directly from Hints 3
and 4 by dropping perpendiculars from Os to the radii.)
There are four variations of this problem, the solution
depending on whether the layout starts with the sharper or
flatter arc and on the direction (outward or inward) of the
tangent shift. The final relations contain different signs and
subscripts.
A
v v
o's
R
1
p
RL-RS

B
Hot R⋅
т
q
p
R' = R+
vers I
B'
Ok
Fig. 7-8
7-10. Simple Curve; Parallel Tangent; Same T.C.-In
contrast to the situation in Art. 7-8, assume that the original
T.C. must be preserved, thereby requiring a curve of new D,
as indicated in Fig. 7–8.
From the triangle at the vertex (Hint 2), VV'= the skew
p
shift = and the new tangent distance T' equals T+VV'.
sin I'
=
Then D' is found from Table VIII, or, if preferred (Hints 4
and 3), from
(7-7)
CHANGE-OF-LOCATION PROBLEMS
169
For setting the new C.T. at B', notice that B' must lie on
AB produced. Since angle bBB' equals 1,
P
sin I
(7-8)
When the tangent is shifted inward, use AB' as the original
curve.
R
Q
BB':
7-11. Simple Curve; New C.T. Opposite Original C.T.-If
conditions do not permit moving the C.T. forward (as in
Fig. 7-8), it may be kept on the same radial line opposite the
original position, as in Fig. 7–9.
A Á V V
X
R'A K
=


Api
A A'
RARG
0%____
P
exsec I
Fig. 7-9
Fig. 7-10
For the triangle at the vertex (Hint 2), VK=p cot I, and
the new tangent distance T' equals T-VK.
**********
Then D' is found from Table VIII, or (if preferred) BB' =
BX-B'X (Hint 3). That is, p= (R-R') exsec I, from which
R' = R-
(7-9)
For setting the new T.C. at A', notice that AA' does not
equal VV'. Find AA' from the triangle at the centers (Hint
2). This triangle gives
AA'=(R—R') tan I
(7-10)
When the tangent is shifted inward, use A'B' as the original
curve.
7-12. Simple Curve; New Direction From C.T.-Changing
the direction of the forward tangent after a curve has been
staked may place the alignment on more favorable ground
170
SPECIAL CURVE PROBLEMS
than by shifting the tangent parallel to its original position.
Fig. 7-10 represents a case in which the tangent is swung in-
ward through a measured angle a, the C.T. at B being pre-
served. The new central angle I' equals I+a.
Using Hint 4, drop perpendiculars (not shown) from B to
the tangent AV produced and to the radii OA and O'A'.
Then, by inspection, R' vers l'= R vers I, and
R vers I
R'
vers I'
Also, from Hint 3,
AA'R sin I - R' sin I'
(7-12)
If preferred, solve triangle VBV'' (Hint 2) for T'. Thus,
T sin I
(7-13)
sin I'
Then, obtain D' from Table VIII.
T'
-
(7-11)
7-13. Modification for Spiraled Curve.-In the preceding
examples illustrating change-of-location problems, the curves
were not spiraled. Spiraling complicates field adjustments to
a certain extent. The best general method is to locate on the
ground the positions of the offset T.C.'s and the offset P.I.,
thereby converting the problem to one involving unspiraled
curves.
In a simple problem it may not be necessary actually to
stake the offset T.C.; but the calculations must then take the
spiral into account. (See Fig. 7-4 for an example.)
RELOCATION PROBLEMS
Major relocations of existing highways are continually
being made in order to bring them into conformity with
modern standards. To a lesser extent, some large sections
of railroad line are being relocated, especially in mountainous
terrain where the amount of traffic originally expected did
not warrant low grades and expensive alignment. In such
work little, if any, use is made of the existing alignment
records.
Minor relocations, both on highways and railroads, are
even more common; they will probably continue to outrank
RELOCATION PROBLEMS
171
major relocation projects in total mileage and construction
cost. The shorter the relocation the more convenient it
becomes to tie the survey work closely to the original align-
ment. Survey and design problems are closely related to
those having to do with obtaining new right-of-way and
abandoning old right-of-way.
of
A
Two typical minor relocation problems are outlined in
the following articles. References to some major projects
are given in Chapters 9 and 10.

pc.s.
S.C
B
C
-pl
Is Rs FR FRL
yó
H
Fig. 7-11
|
C.S.
S.C.
7-14. Replacement of Broken-Back Curve.-A "broken-
back" curve consists of two curves in the same direction
separated by a tangent shorter than the sum of the distances
needed to run out the superelevation. Formerly, such align-
ment was often used for reasons of economy, but present
standards rarely justify the practice. When the entire lay-
out is visible, it is very unsightly; even though obscured, it is
apt to be dangerous (on highways).
Elimination of the tangent between the curves is a common
relocation problem. Occasionally it can be done by inserting
a single curve between the outer tangents of the existing curves.
02
172
SPECIAL CURVE PROBLEMS
A more general method is illustrated in Fig. 7-11, which
shows the original tangent BC separating curves with centers
at Os and OL. A new curve with its center at O is sprung
between points A and D on the existing curves, thus forming a
three-centered compound curve (Art. 3-9). The problem is
to locate points A and D.
If a value is assumed for the radius R of the new curve, the
positions of A and D can be found by solving the right tri-
angle OsHOL and the triangle Os00L, and then calculating
the angles Is and IL, which locate A and D.

A
Is
Os
B
Rs
1-
R-
OL
X
R
Fig. 7-12
The maximum offset PK between the original and revised
layouts should be computed to see if it falls within the limit
permitted by topography.
PK=(R − Rs) vers Is
(7-14)
If spirals are required between the curves, the problem
then becomes a special case of inserting spirals between arcs
of a compound curve. The center O is preserved and the
radius of the new curve is increased to R+KJ, where KJ ·
The two spirals usually have different lengths, since
=
01-02.
RELOCATION PROBLEMS
173
the differences between the degrees of the new and original
The theory of the process is
curves will rarely be the same.
outlined in Arts. 5-8 and 5-9.
Side
DX
ΧΟ
00s
Os B
BC
COL
OLD
Fig. 7-12 illustrates how a change in specifications some-
times complicates a problem. The existing layout is a broken-
back curve similar to that in Fig. 7-11, but field conditions
require that the new curve start at A (the T.C. of the original
shorter-radius curve) and end on the existing forward tangent
at an unknown point X. The problem is to find the new radius
R and the distance DX. No solution is possible by merely
solving two triangles, as was the case in Fig. 7-11. To avoid
more-intricate construction, the traverse method is used.
(Hint 6, Art. 7–7).
The closed traverse DXO0sBCOLD is imagined to be
oriented so that direction DX is 0° azimuth (Rule 2, Art. 3–6).
The data for the traverse are then as follows:
Length
DX
R
R-Rs
R$
BC
ᎡᏞ
RL
Azimuth
0°
270°
90°-I
90°- IL
180°- IL
270°— IL
90°
R =
Setting Σ departures equal to 0 and reducing gives
Rs (cos IL-cos I)+BC sin IL+RL vers IL
vers I
Similarly, setting Σ latitudes equal to 0 gives
DX= (RL-Rs) sin IL-(R-Rs) sin I+BC cos IL (7-16)
As in Fig. 7-11, the maximum offset is expressed by formula
7-14.
(7-15)
The foregoing formulas are solved quickly by calculating
machine and Table XX, in which all needed natural functions
appear in the same table. (See Prob. 7-5.)
7-15. Replacement of Reverse Curve.-Fig. 7-13 shows an
existing reverse curve ACB which is to be replaced by a new
simple curve starting at the same T.C. at A. The problem is
174
SPECIAL CURVE PROBLEMS
to find the new radius R and distance XB. The following
solution is by the traverse method employed in Fig. 7-12, 0°
azimuth being taken parallel to OX in the closed traverse
OXBO SOLO. Angle I=Is-IL.
Setting Σ latitudes equal to 0 and reducing gives
RL (cos I-cos Is)+Rs vers Is
vers I
R=
Then,
(See Prob. 7-6.)
X
Setting departures equal to 0 and reducing gives
XB=(RL+R) sin I−(RL+Rs) sin Is
(7-18)
To find the maximum offset PK, first calculate angle a
from the relation
-R
tan a=
Offset PK=(R-Rs) exsec a
B V
PR
XB
R-Rs
Pos
C
Fig. 7-13
A
(7-17)
O
(7-19)
(7-20)

O₁
7-16. General Method for Major Relocation.-Even in the
case of a long relocation supplanting several curves and their
intervening tangents, it is useful to tie the survey to the exist-
ing alignment and to compute the resulting closed traverse by
means of the old alignment data on file. This not only gives
a check on the field work without extra surveying but also
MISCELLANEOUS PROBLEMS
175
provides the coordinates needed for drawing a map of the two
layouts.
Relocations often result in surprisingly simple alignment.
Fig. 7-14* shows a case in which a single 1° curve and two
tangents replaced seventeen curves having total central angles
of more than 900°.
7-17. Curve Through Fixed Point.-A curve may be made
to pass through (or close to) a fixed point either by trial
methods or by an exact geometric solution.
-BRISTOW
RELOCATION OF U.S.66
NEAR BRISTOW, OKLAHOMA
To
MISCELLANEOUS PROBLEMS
Begin
relocation
NEW LOCATION
(2.9 miles; 1 curve)
SAPULPA
Fig. 7-14
OLD LOCATION
(3.8 miles; 17 curves)
U.S.66.

LEnd
relocation
A method of accomplishing the result by a trial field method
from a selected T.C. was described in Art. 2-14 under the
heading of chord offsets.
This problem is more intricate where the P.I. and the
directions of the tangents have been established. In this
situation a trial field method is apt to be unduly time-con-
suming. Either a trial calculation or an exact geometric
method is better. Fig. 7-15 shows this situation, P being the
point fixed by ties to the P.I. The tie measurements may be
the distances VP' and P'P or the distance VP and the angle a.
Either pair of values may be obtained from the other by
computation. The problem is to find the unknown radius R,
after which all remaining data are easily computed.
* Concrete Highways, Vol. XX, No. 1, Jan.-Feb., 1939.
176
SPECIAL CURVE PROBLEMS
The key to the trigonometric solution is the triangle OPV
(the interior angles of which are denoted by o, p, and v).
Although only one side (PV) and one angle (v) are known,
the relation between the two unknown sides is known; there-
fore, the triangle can be solved.
A

XXX
1
R
21+0
The ratio of the unknown sides OP and OV is R:R sec 11,
which equals cos 1. Therefore, the law of sines can be
used in the key triangle, giving sin p:sin v=OV:OP, from
which
(7-21)
Fig. 7-15
sin P
R=
=
Since all angles of the triangle are now known, R may be
found from the sine law. However, if angles v and o are very
small, R may not be calculable to sufficient precision owing to
the rapid variation in the sines of the angles (see Hint 5,
Art. 7-7). In this case, solve for R from the relation
*****
sin v
cos I
(7-22)
The problem just described may be used to illustrate how
a quick slide-rule solution by a trial calculation method often
gives results close enough for the purpose. For example, sup-
pose that I=28°16′, VP'=150 ft, and P'P=20 ft. It is
required to find D to the nearest 10 minutes.
T₁o (Table VIII) = 1,442.7
PV sin (I+a)
vers (1+0)
MISCELLANEOUS PROBLEMS
177
Try D=3°
00:
AV
VP'
AP'diff.
B
1,442.7
3
=
T.O.3° (Table I)=2.62
and
PP'3° =3.31²×2.62=28.7
3
(PP'₂° may be checked by formula 2-30.) Since PP';°
exceeds 20 feet, the correct D is greater than 3°.
Try D=4°
X
From a similar process, PP'4° = 15.4
Hence, by interpolation, the required value of D is approxi-
mately equal to 3°+13.3×60′=3°39′ (say 3°40′).
8.7
о
a
481
= 150
=331

R
C
ए
Fig, 7-16
The 3°40' curve will not pass exactly through point P.
If a closer value of D is required, interpolation between
another pair of values for D=3°30′ and 3°40′ should give D
to the nearest minute.
7-18. Intersection of Straight Line and Curve.-Fig. 7-16
represents a more general case of Fig. 7-15 in which a straight
line XP' cuts the tangent AV of a given curve. The problem
178
SPECIAL CURVE PROBLEMS
is to locate the intersection of the line and the curve at P.
Among other places this problem occurs in right-of-way work,
as in defining the corner of the piece of property shown shaded.
All curve data are known and the survey notes also provide
the angle a and the distance AP' (or P'V).
This information being given, the problem could be con-
verted to the preceding one by drawing another tangent (not
shown) from P' to the curve, thereby making P' correspond to
point V in Fig. 7-15. The problem could also be converted
to one in analytic geometry (see Art. 7-19) by finding the
equations of the line and of the circle and solving them
simultaneously. However, the traverse solution is more direct.
If 0° azimuth is assumed in the direction PP', setting
Σ departures equal to 0 in the traverse PP'AOP gives
cos (a+c)=
R cos a-P'A sin a
Ꭱ
(7-23)
from which the central angle c and the stationing of P may be
computed.
From Σ latitudes =0,
PP' = R sin a+P'A cos a−R sin (a+c)
(7-24)
This is the type of problem which is solved readily by
graphical methods. For example, if AP' is known, the exact
tangent offset at P' is computed and laid off at right angles
to a line representing the tangent AV. A large scale should
be used (say 1 inch = 1 foot). Then, at a point 10 feet closer
to A, another tangent offset is computed and laid off parallel
to the previous one and 10 inches away. A spline fitted to
three such points is used for drawing in the curve, and the
line XP' is then drawn to the intersection at P. Finally,
the required distance P'P is scaled to the nearest 0.01 inch,
the ground distance thus being determined with an error hardly
greater than 0.01 foot.
In the case of a field problem, rather than one met in the
office, the point P may be found by setting two points close
together on the curve either side of where the straight line
comes through. Point P is then located by "string intersec-
tion."
MISCELLANEOUS PROBLEMS
179
Y
7-19. Solutions by Analytic Geometry.-When a curve
problem cannot be solved readily by any other method,
recourse may be had to analytic geometry. This procedure
relies on a basic mathematical tool and has the virtue that
computations can be arranged so as to be self-checking.
Analytic geometry is also the basis for the COGO system of
programming problems for solution by electronic digital com-
puter (Art. 13–5).
To review some of the principles of analytic geometry, the
equation of a line cutting the Y-axis (as line 1 in Fig. 7-17) is
X
Y=Iy±tan @
(7-25)
For a line cutting the X-axis (line 2),
X=IY tan 0
Y Linel
8
X+
Y
X-
Ix
Fig. 7-17
Line 2
X
(7-26)

To find the equation of a line from the coordinates of
two points on the line, first find the bearing from 0=
X difference
arc tan
Then find I, or I by substituting the
Y difference*
coordinates of one of the points in equation 7-25 or 7-26.
When 0 falls between zero and 10°, equation 7-26 is preferred;
between 80° and 90°, equation 7-25 is preferred. In either
180
SPECIAL CURVE PROBLEMS
case, it is advisable to check the result by using the coordinates
of the second point in the same equation.
The equation of a circle, as shown in Fig. 7-18, is
(X-h)²+(Y-k)² = R²
(7-27)
Any of the problems previously solved by the traverse
method might have been solved by analytic geometry. In
fact, analytic geometry can also be applied to computing the
locations of critical points along a simple-curve or spiraled-
curve layout. This method is particularly useful where all
details of a route alignment are tied to a state plane coordinate
system.
Y
Y

R
h→
-X·
Fig. 7-18
P
X
Analytic geometry is especially adapted to solving a problem
that requires finding the locations of points of intersection of
straight lines or curves. The problem in Art. 7-18 is such a
case. Fig. 7-19 illustrates the solution of this problem by
analytic geometry for the general case where the coordinate
axes are not rotated or translated so as to simplify the solution.
The same basic data are known as in the traverse solution of
Art. 7-18. These data are the coordinates of V (or A), the
bearings of the tangents, all geometric-curve data, the distance
VP'(or AP'), and the bearing 0 of the intersecting line. To
find the desired coordinates of the intersection P, proceed
as follows:
MISCELLANEOUS PROBLEMS
181
Solve for the unknown constants h and k from h=X₁+
R sin 0₁ and k=YA-R cos 01. The equation of the simple
curve (formula 7-27) is then known. Commence with the
given coordinates of V (or A) and find the coordinates of P'.
This fixes the coordinates of one point on the intersecting line.
Y

A
k
ト
​نا
B&
h
02
p'
Р
||
||
Fig. 7-19
11
XA
01
LR
1
=0
о
X
XP'
tan 02
Find the value of 1, from Iy=Yp'+; The equation of
the intersecting line (formula 7-25) is then known. Finally,
obtain the required coordinates of P by solving the equations
of the curve and the intersecting line simultaneously.
For additional applications of analytic geometry to practical
problems, see "Analytic Geometry in Highway Design and
Layout," Proceedings ASCE, Vol. 86, Paper 2548, July, 1960
and “Drafting-Room Problems Solved by Analytic Geometry"
by S. L. Goldberg, Civil Engineering, Vol. 30, No. 12,
December, 1960, pp. 68–70.
182
SPECIAL CURVE PROBLEMS
PROBLEMS
(NOTE:-Deflection angles for staking the following curves are not called
for. These may be specified by the instructor if desired.)
7-1. Simple curve with P.I. inaccessible (Fig. 7-1). Find:
sta. T.C. (A), sta. C.T. (B), and distance YB.
(a) Sta. X=172+50; XY=366.28; x=32°30'; y=23°22′;
Da=6°. Answers: T.C. = 169+19.19; C.T. = 178+50.30; YB =
268.56.
(b) Sta. X=67+00; XY=411.57; x=25°12′10″; !=
39°36′30″; Dc=5°. Answers: T.C.=62+62.35; C.T.=75+
58.57; YB = 533.94.
(c) Sta. X=82+40.6; XY=550.2; x=57°18'; y=16°35';
R=500 ft.
(d) Sta. X=275+00; XY=1,034.7; x=32°00′; y-92°26';'
Dc=10°.
7-2. Compound curve with P.I. inaccessible (Fig. 7-2).
Find: sta. C.T.(Y), IL, Is, and the missing radius.
(a) Sta. T.C.(X)=48+00; XY=1,985.2; x=29°33′; y=
47°25'; Rs 1,000. Answers: C.T. = 69+32.6; IL=30°24′38″;
Is=46°33'22"; RL=2,487.1.
(b) Sta. T.C.(X) = 432+00; XY=1,562.73; x=33°52′40″;
y=52°04′10″; Ds (chord def.) =8°.
7-3. Change-of-location problems. Assume that original
curves are staked and that field revisions conform to the indi-
cated figures. Find: the station of the new C.T. and the
distances needed to set hubs at the new T.C. and C.T. by
taping from their original positions.
(a) Fig. 7-6: Sta. A=63+84.2; I=41°24′; Da=5°; p=20.
(b) Fig. 7-8: Sta. A =82+10.6; I = 54°15′; R = 1,000; p=45.
(c) Fig. 7-9: Sta. A=164+25.91; I=36°24′30″; D.=3°;
p=10.
(d) Fig. 7-10: Sta. A=287+68.34; I = 65°42′50″; D.=41°;
a=5°.
7-4. Broken-back curve (Fig. 7-11). Find: PK and the
station equation at point D.
(a) Sta. B=92+41.26; Rs=500; R1=800; R=1,000; BC=
MISCELLANEOUS PROBLEMS
183
385.47. Answers: PK=61.84; sta. D 103+00.76 revised
sta. 102+73.28 original.
(b) Sta. B=36+82.53; Rs=800; RL=1,200; R=1,500;
BC=270.19.
7-5. Broken-back curve (Fig. 7-12). Find: R, the maxi-
mum offset, and the station equation at point D.
(a) Sta. A=73+48.2; Rs=310; RL=500; Is=29°35′; IL=
22°04'; BC=269.6. Answers: R=613.54; max. offset =39.6;
sta. D 79+84.5 revised 79+70.4 original.
=
(b) Sta. A =31+82.56; Ds=10° and D=8° (both are def.);
Is=42°48′20″; IL=32°30′10″; BC=264.37.
7-6. Replacement of reverse curve (Fig. 7-13). Find: R,
PK, and the station equation at point X.
(a) Sta. T.C. (4) = 18+52.34; D=1°35′ and Ds=4°25′
(both are def.); Iz=16°42′; I=75°26′. (Note: This is the
reverse curve used in Prob. 3-5. Answers: R=4,031.47;
PK=528.97; sta. X 59+84.96 revised = 63+96.13 original.
(b) Sta. T.C.(A)=71+83.1; RL=1,000; Rs=500; IL=
24°03′; Is = 55°35'.
7-7. Curve through fixed point (Fig. 7–15). Find: Da (to
nearest minute) and sta. T.C.(4). Verify answers by the
trial-calculation method.
(a) Sta. P.I.(V) = 53+47.6; I =58°10'; VP' =490.7; P'P =
26.4. Answers: Da=4°12'; sta. T.C. =45+88.8.
(b) Sta. P.I.(V) = 18+35.7; I=42°00′; VP'=179.2; P'P =
50.4.
(c) Sta. P.I.(V)=76+02.46; I=53°40′20″; VP=298.59;
a =5°32′10″".
7-8. Intersection of line and curve (Fig. 7-16). Find:
distance P'P, sta. P, and the coordinates of P by the method
of Art. 7-18. Verify answers by the method of Art. 7-19.
(a) Sta. P.I.(V)=81+92.47; coordinates of V=N 6,055.22,
E 5,409.63; bearings AVS 52°42′20″W, VBS 8°08′40″E,
XP'=S_80°37'10" E; lengths AV(T) = 587.28, AP'=480.27,
R=1,000. Answers: PP' =115.41; sta. P=86+05.17; coordi-
nates of PN 6,101.25, E 5,608.63.
184
SPECIAL CURVE PROBLEMS
(b) Sta. T.C. (A) = 226+82.54; coordinates of A = N 4,368.27,
E 11,836.52; bearings AVS 84°28′45″ E, VB = S 19°26′30″ E,
XP' =S3°02′45″ E; lengths AV (T) =1,912.59, AP' =1,191.73,
R=3,000.
(NOTE: The remaining problems are "original" in the sense that no
special formulas for solving them appear in this book. The suggestions
found in Arts. 7-2 and 7-7 are especially pertinent. Problems appear in
the order of increasing difficulty. Answers are purposely omitted as a chal-
lenge to the computer to verify his work by a different method.)
7-9. Fig. 7-5. Given: angles XVA=100°, VXP=36°,
VAP=22°; distances VA = 1,200, AOs=1,000; sta. A=12+
00. Find: sta. B'(along curve APB') and distance VB'.
7-10. Two independent simple curves have the following
data:.
Curve 1. R=2,500; sta. T.C.=75+16.21; coordinates of
T.C.=N 20,782.65, E 36,030.62; bearing initial_tangent =
N 73°04′10″ W, final tangent = N 23°56′40″ W.
Curve 2. R=1,000; sta. T.C.=27+84.35; coordinates of
T.C.=N_22,290.14, E 32,780.44; bearing initial tangent
N 6°10′20″ W, final tangent = N 80°56′00″ W. A connection
in the form of a common tangent is to be run from point X on
curve 1 to point Y on curve 2. Find: length and bearing of
XY, sta. X on curve 1, and sta. Y on curve 2.
=
7-11. A simple curve AB has R = 1,000, I=55°27', and sta.
T.C.(A)=47+21.63. A location change requires the curve
to be compounded at sta. 51 +50 so as to join a new forward
tangent located exactly 100 feet outward parallel to its origi-
nal position. Find: R of the second arc, sta. of the new
C.T. (B'), distance BB', and the angle between BB' and the
original tangent at B.
7-12. Reverse curve between non-parallel tangents. Given:
angle between tangents at P.I.=30°19′10″; Ts(P.I. to T.C.) =
438.62; Tı(P.I. to C.T.) =2,037.54; sta. T.C.=429+81.67.
Conditions require that both arcs have exactly the same
radius. Find: R, sta. of point of reversal, and sta. of C.T.
CHAPTER 8
CURVE PROBLEMS IN HIGHWAY DESIGN
8-1. Foreword.-Because of the public nature of highway
traffic, highway curves have a greater effect upon safety of
operation than do curves on railroad lines. Railways,
operating over fixed track on private right-of-way, are able
to control the volume and spacing of traffic and to enforce
slow orders on dangerous curves. Such restrictions are not
practicable on the public highway. Consequently, it is neces-
sary to "build safety into the highways" by proper location
and design. It would be difficult enough to meet this require-
ment if conditions were static, but the continuous improve-
ments in vehicle design and in highway construction, both of
which encourage ever-increasing speeds, make safety the high-
way designer's paramount engineering problem-as yet
unsolved in several important respects.
The importance and variety of curve problems in highway
design warrant devoting a separate chapter to aspects of these
problems not fully covered in the preceding chapters. Though
certain physical and geometric principles are reasonably well
established, the numerical recommendations controlling design
are in many cases only tentative. Illustrative examples are
taken from recent research or from current design "policies"
or "standards." Revision of policies and standards can be
expected as conditions change and as research discloses facts
not yet known.
Until recently the various elements entering into highway-
alignment design had been fixed largely by rule-of-thumb
methods, and there was little agreement among State highway
departments. This difference in practice was partly due to
the mushroom growth of traffic, which forced the highway
engineer to concentrate upon meeting the resulting demands
quickly by any methods that seemed adequate at the time.
However, the principal reason was lack of basic research
concerning the human and mechanical factors which con-
tribute to safe operation at high speeds. Research during
185
186
CURVE PROBLEMS IN HIGHWAY DESIGN
the decade preceding World War II supplied much of the
missing information; from time to time additional research
continues to reveal new facts needed to bring all design ele-
ments into harmony with a chosen design speed.
C. M. Noble has listed¹* the design elements affected by
speed as follows:
(1) Over-all Width of Highway
(a) Median strip, width and treatment
(b) Shoulders, width and treatment
(c) Paved lane width
Sight Distance
(a) Vertically
(b) Horizontally
(c) Determination of proper friction factor and reac-
tion time
(3) Minimum Radius for Horizontal Curvature
(4) Rates of Superelevation for Horizontal Curves
(a) Determination of proper friction factor (un-
balanced centrifugal ratio)
(b) Correlation of superelevation rate with require-
ments of slow and fast vehicles
(5) Length of Spiral Curves
(6) Length of Superelevation Runout Beyond Spiral
(7) Length of Profile Tangents
(8) Distance Between Horizontal Curves
(a) Same direction
(b) Reverse direction
(9) Spacing of Points of Access and Exit
(10) Length of Acceleration and Deceleration Lanes at
Points of Access and Exit
(11) Design of Turnouts at Points of Exit
(12) Signs
(a) Size of letters
(b) Maximum number of words in sign message
*Superscript numbers refer to the bibliography at the end of this chapter.
SIGHT DISTANCE
187
(c) Reflectorization
(d) Position
(13) Type of Pavement
Of the items in the foregoing list, those from (2) to (8) inclu-
sive come logically within the scope of this chapter; the others
belong more properly to the field of highway construction and
design.
An important step toward the incorporation in practice
of design features which will result in the maximum degree of
safety and utility was taken in 1937 by the American Associa-
tion of State Highway Officials with the organization of a
"Special Committee on Administrative Design Policies."
Since that year several brochures have been approved by the
States after thoughtful research, discussion, criticism, and
final revision. One of the most useful was "A Policy on Sight
Distance for Highways," approved in 1940. This policy,
along with six others on various aspects of geometric design,
played an important role in the gradual replacement of rule-
of-thumb methods by scientific design based on research.
Later, the material in these seven policies was revised, ex-
panded, and brought up to date in the form of a single volume²
entitled "A Policy on Geometric Design of Rural Highways."
This latest policy, adopted May 3, 1954, is a monumental
work which is bound to have a good influence on geometric
highway design for many years. In the following articles,
references to this publication are denoted, for brevity, by
AASHO Policy.
SIGHT DISTANCE
8-2. Speeds.-Speed and sight distance are closely related.
Several definitions of speed are used in the AASHO Policy.
Over-all travel speed is the speed over a specified section of
highway, being the distance divided by the over-all travel
time. The term running speed refers to the distance divided
by the time the vehicle is in motion. In either case, the
average speed for all traffic, or component thereof, is the
summation of distances divided by the summation of running
(or over-all travel) times. The most useful concept of speed
is design speed, which is the maximum safe speed that can be
188
CURVE PROBLEMS IN HIGHWAY DESIGN
maintained over a specified section of highway when conditions
are so favorable that the design features of the highway govern.
8-3. Definitions.-Two definitions of sight distance are in
use, known as "stopping" and "passing."
Stopping sight distance should be long enough to permit a
vehicle traveling at the assumed design speed to stop safely
before reaching a stationary object in its path. At horizontal
curves and at crest vertical curves, the height of the driver's
eye is assumed to be 4.5 feet, and the height of the object is
taken as 4 inches (AASHO Policy). At no point on a high-
way should the sight distance be less than the stopping value.
Passing sight distance on a tangent is the shortest distance
required for a vehicle safely to pull out of a traffic lane, pass a
vehicle traveling in the same direction, and return to the
correct lane without interfering either with the overtaken
vehicle or with opposing traffic. At horizontal curves and at
crest vertical curves, passing sight distance is the length of
road that must visibly be free of obstructions in order to permit
a vehicle moving at the design speed to pass a slower-moving
vehicle. For these cases the height of both eye and object is
taken to be 4.5 feet (AASHO Policy). Highways on which
passing must be accomplished on lanes that may be occupied
by opposing traffic should be provided with frequent safe pass-
ing sections on which the sight distance is not less than the
passing value for the assumed design speed.
Sight distances on overlapping horizontal and vertical
curves are determined independently for each type of curva-
ture. The critical sight distance at any point is then taken
as the smaller of the two.
8-4. Stopping Sight Distance.-Stopping sight distance is
the sum of two distances: (1) that traversed during perception
plus brake reaction time; (2) that required for stopping after
brakes are applied.
Numerous scientifically controlled tests have been made to
determine perception time and brake reaction time. As might
be expected, the results vary according to vehicle speed, age
and natural aptitude of the driver, and the conditions accom-
panying the test. Brake reaction time is assumed to be 1
second, this having been found to be the value sufficient for

SIGHT DISTANCE
189
most drivers; perception time is selected as slightly greater
than that required by most drivers, and is assumed to be 1.5
seconds (AASHO Policy).
For a speed of V miles per hour and perception plus brake
reaction time of t seconds, the total reaction distance in feet is
D, 1.47 V t
(8–1)
(The conversion factor 1.47 may be recalled more readily by
means of the exact relation: 60 mph=88 ft per sec.)
Braking distance may be determined from fundamental
principles of mechanics. The force causing a vehicle to stop
after application of the brakes equals mass times decelera-
tion, or
W
F=Ma= a
g
v2
If the coefficient of friction ƒ is assumed to be uniform during
deceleration, F =W ƒ; hence, a =ƒ g. Since the distance trav-
ersed in decelerating from a velocity v to rest is the brak-
2 a'
v2
ing distance is When g is taken as 32.2 ft per sec², and
2 fg
the speed is converted to V in miles per hour, the braking dis-
tance in feet reduces to
Do=
=
V2
30f
(8-2)
Actually the coefficient of friction is not constant during
deceleration, but assuming it to be constant introduces no
error so long as the proper equivalent uniform value is assumed.
to fit the speed in effect at the beginning of the operation. The
coefficients of friction used in the AASHO Policy apply to
normal clean wet pavements that are free of mud, snow, or ice.
In Table 8-1, values from 30 to 70 mph, inclusive, are from
the AASHO Policy; others are obtained by extrapolation.
The speed for wet conditions is taken to be slightly less than
the design speed so that the greater proportion of traffic,
traveling at yet lower speeds, will enjoy an additional safety
factor. If the full design speed were used along with the
coefficients of friction for dry pavements (almost double the
tabulated values), the required stopping sight distances would
:
190
CURVE PROBLEMS IN HIGHWAY DESIGN
be somewhat less than those for the assumed wet conditions.
Therefore, the critical design values are those in Table 8–1.
TABLE 8-1
MINIMUM STOPPING SIGHT DISTANCE-WET PAVEMENTS

Design
Speed
mph
888888
30
40
50
60
70
80
90
100
Assumed
Speed
for
Condi-
tion
mph
V
28
36
44
52
59
67
75
888
82
Perception
plus
Reaction
Time
sec
t
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
Dist.
ft
Dr
103
132
161
191
216
245
275
300
Coef-
ficient
of
Friction
f
Braking
Dis-
tance
on
Level
V2
Do=
30 (f±G)
ft
Do
Stopping
Sight Distance
Com-
puted
ft
Dr+ Do
176
263
369
492
616
780
945
Rounded
for
Design
ft
0.36
73
0.33
131
0.31
208
0.30
301
0.29
400
0.28
535
0.28
670
0.27 830 1,130 1,150
200
275
350
475
600
775
950
Theoretically, stopping distances are affected slightly
by grades. If G is the per cent grade divided by 100, the
formula for braking distance becomes
(8–3)
In practice the sight distance is usually longer on down-
grades than on upgrades, a fact that automatically compen-
sates for the greater braking distances on downgrades required
by formula 8-3. Exceptions would be one-way lanes on
divided highways having independent profiles for the two
roadways.
8-5. Passing Sight Distance.—In the AASHO Policy the
minimum passing sight distance for two-lane highways is the
SIGHT DISTANCE
191
sum of the following four distances which are shown in Fig.
8-1:
d₁=distance traversed during the preliminary delay period
(the distance traveled during perception and reaction
time and during the initial acceleration to the point
of encroachment on the left lane).
d₂ = distance traveled while the passing vehicle occupies any
part of the left lane.
d3=distance between the passing vehicle at the end of its
maneuver and the opposing vehicle.
d4= distance traversed by an opposing vehicle for two-thirds
of the time the passing vehicle occupies the left lane,
or 3d2.
The preliminary delay period d is a complex one. For
purposes of analysis it may be broken into two components:
(1) time for perception and reaction, and (2) an interval dur-
ing which the driver accelerates his vehicle from the trailing
speed V-m to the passing speed V at the point of encroach-
ment on the left lane. The distance traversed is expressed as
(8-4)
The distance traveled while the passing vehicle occupies the
left lane is
d2 = 1.47 Vt2
(8-5)
Reference to Fig. 8-1 shows that during the first part of
the passing maneuver the driver can still return to the right
lane if he sees an opposing vehicle. From experience, this
"uncommitted" distance is about d₂. Since the opposing
and passing vehicles are assumed to be traveling at the same
speed, d4=3d2.
d₁ = 1.47 t₂ (V−m+
at
m+ali)
Basic data used in establishing the design curves in Fig.
8-1 are summarized in Table 8-2. The values of a, t₁, ta, da, and
average passing speed come from a report on extensive field
observations of driver behavior during passing maneuvers.
The average value of m is taken as 10 mph.
192
CURVE PROBLEMS IN HIGHWAY DESIGN
Elements of Passing Sight Distance-Feet
d
d₁
2,500
2,000
1,500
1,000
500
/d2
30
Passing Vehicle
A
30
3d2
d2
40
FIRST PHASE
SECOND PHASE
d3
Opposing vehicle appears
when passing vehicle
reaches point A
Design Speed-mph
50
d2
Total di+d₂+da+ds
d4
d4
di
d3
60.
40
50
Average Speed of Passing Vehicle-mph
Fig. 8-1. Passing sight distance (2-lane highways)
B

70

60
HORIZONTAL ALIGNMENT
193
TABLE 8-2
ELEMENTS OF SAFE PASSING SIGHT DISTANCE-
2-LANE HIGHWAYS

Speed group, mph
Average passing speed, V, mph
Preliminary delay:
a = avg. acceleration, mph per
second
t₁ = time, seconds
d₁=distance traveled, feet
Occupation of left lane:
t₂ = time, seconds
d₂-distance traveled, feet
Clearance length:
d3=distance traveled, feet
Opposing vehicle:
d4-distance traveled, feet
Total distance, d₁+d₂+dз+ds,
feet
30-40
34.9
1.40
3.6
145
9.3
475
100
315
40-50
43.8
1.43
4.0
215
HORIZONTAL ALIGNMENT
10.0
640
180
425
1,035 1,460
50-60
52.6
1.47
4.3
290
10.7
825
250
550
1,915
Table 8-3 contains a summary of passing sight distances
determined from the foregoing analysis. Also included are
values tentatively suggested in the AASHO Policy for the
case of 3-lane highways. However, these values are not based
on actual field observations and cannot be considered so
reliable as those for 2-lane highways. The 3-lane values come
from computation by omitting the distance d4.
8-6. Superelevation Theory.-Fig. 8-2 shows the forces
W (weight of vehicle) and F (centrifugal force) acting through
the center of gravity c of a vehicle traveling at a speed v
around a curve of radius R, when the pavement is super-
elevated at an angle with the horizontal (0=tan¯¹ e).
194
CURVE PROBLEMS IN HIGHWAY DESIGN
TABLE 8-3
MINIMUM PASSING SIGHT DISTANCE
Design speed, mph
Passing speed, V, mph
Minimum passing sight
distance, feet:
Fig. 8-1
Rounded
e
Calculated
Rounded
unity
(3)
مرحوم
(1)
33333
30
30
810
800
40
40
50
48
2-Lane Highways
1,265
1,265 1,675 2,040 2,310
1,300 1,700 2,000 2,300
3-Lane Highways
W Wp
Wn
(2)
Possible directions
of resultant
1,190 1,450 1,650
1,200 1,400 1,600
FP
Ө
↓
60
55
F= Wv2
GR
Fn
Fig. 8-2. Superelevation theory
70
60

R
In order to simplify the analysis, the two forces are resolved
into their components normal and parallel to the pavement.
The resultant of the forces must take one of three possible
general directions:
HORIZONTAL ALIGNMENT
195
(1) When W₂=Fp, the resultant is perpendicular to the
pavement and no centrifugal sensation is felt by the occupants
of the vehicle. The speed which produces this effect is called
"equilibrium speed."
(2) When Wp>Fp, the resultant is inclined to the pavement
down the slope. Consequently, there is a tendency for the
vehicle to slide inward, and this tendency is resisted by a
lateral force acting up the slope at the contact between the
wheels and the road surface. Obviously, there is also a
clockwise overturning moment causing the vehicle to tilt
inward.
(3) When W₂<Fp, the resultant is inclined to the pavement
up the slope. The effects are then opposite to those in (2); the
resisting lateral force acts down the slope and the tilt is cut-
ward.
At equilibrium speed, W₂=Fp, or
W sin 0
or
е
=
tan 0
Solving for 0 gives the following basic equilibrium formula:
22
9 R
(8-6)
W v2
Ꭱ
ទ
The equilibrium formula is expressed in more usable form
by replacing v in feet per second by V in miles per hour and
by substituting 32.16 ft per sec² for g. The result is
e
е
0.067 V2
Ꭱ
M
COS O
V2
15 R
V2 Da
85,700
(8-8)
When WpFp, the magnitude of the tendency for the vehicle
to move laterally may be denoted by the term f, which is the
ratio of the lateral component of the resultant to its normal
component. If all forces are assumed to have positive signs,
the value of ƒ is
f=
Fp-Wp F cos 0-W sin 0
Fn+Wn F sin 0+W cos ℗
(8-7)
Since the superelevation angles used in practice are small
(0 is rarely permitted to exceed 7°), it is sufficiently accurate to
196
CURVE PROBLEMS IN HIGHWAY DESIGN
assume that F sin 0=0, this term being very small in compari-
son with W cos 0. Therefore, ƒ is approximately equal to
F cos 0-W sin 0
W cos 0
F
tan 0, which reduces to
W
v2
9 R
f=
The resulting working relations are:
When Wp>Fp (or V is less than equilibrium speed),
e-f=
e
e+f=
When W₂<Fp (or V exceeds equilibrium speed),
V2 Da
85,700
Ve V² Da
15 R 85,700
1
(a) Favorable crown
V2
15 R
(8-9)
Fig. 8-3
(8-10)
It should be observed that ƒ is necessarily proportional to,
and opposite in direction to, the resisting side friction which
automatically comes into action at other than equilibrium
speed.
Curve to right
Curve to left
(8-11)


1
(b) Adverse crown
In solving problems for various operating conditions, con-
fusion in signs may be avoided by considering the terms in the
foregoing equations to be positive quantities that are propor-
tional to the actual forces acting on the vehicle. Since the
forces themselves act in definite directions, these same direc-
V2
tions may be given to the terms e, f, and
Fig. 8-3
15 R*
and the accompanying analysis illustrate this procedure.
HORIZONTAL ALIGNMENT
197
Fig. 8-3(a)-Favorable crown
(1) V Equilibrium speed.
=
Superelevation counterbalances the
centrifugal force. There is no
side friction.
(2) V<Equilibrium speed
Side friction and centrifugal force
assist each other in counter-
balancing excessive supereleva-
tion.
(3) V>Equilibrium speęd
Side friction and superelevation
assist each other in counter-
balancing centrifugal force.
Fig. 8-3 (b)—Adverse crown
Only one possible case regardless
of speed. Side friction alone
holds car on road.
V2
15 R
<
f +
→
(Formula 8-7)
V2
15 R
e
V2
15 R
Compressing
(Formula 8-10)
f
e
(Formula 8-11)
e + f
→
= e +
V2
15 R
The term ƒ has been called the "lateral ratio" in research
done by Leeming in Great Britain, and this appears to be the
most logical expression. In the United States, f has been
termed "unbalanced centrifugal ratio," "cornering ratio” (at
General Motors Proving Ground), "unbalanced side friction,"
and "side friction factor." Because of the widespread use of the
last expression, it will be adopted in this book—with the warn-
ing that it should not be confused with the "coefficient of
friction" as understood in dynamics.
8-7. Dynamics of Vehicle Operation on Curves.-On a
curve having accurately built constant superelevation, there
is a particular speed, known as the equilibrium speed or “hands-
off" speed, at which a car steers itself around the curve.
this speed WFp, and the value of ƒ is zero.
At
198
CURVE PROBLEMS IN HIGHWAY DESIGN
At other than equilibrium speed, equations 8-10 and 8-11
seem to indicate that safe operation around a given curve is
entirely within the control of the driver. He has only to
adopt any desired speed and rely upon the automatic develop-
ment of whatever value of ƒ is needed to make up for the lack of
V2
balance between e and
However, the matter is not so
15 R
simple as this. On the contrary, it is very complex; and
further research is required before curve design for high-speed
operation can be placed upon a sound scientific basis.
When WpFp, the car tends to creep out of the traffic lane.
To offset this tendency the driver must exert force at the
steering wheel and must steer slightly toward or away from the
center of curvature, the direction depending on whether V is
greater or less than equilibrium speed. As a result, each pair
of tires must "slip" across the surface at a definite angle
between the path of travel and the longitudinal axis of the
wheels. Fig. 8-4 illustrates the normal condition when V
exceeds the equilibrium speed.
Front Slip Angle

·Steering
Angle
===|
Path of front
of car
3.
Path of rear
of car
Rear Slip Angle
Fig. 8-4. Understeering action
Front and rear slip angles are rarely equal. When the
front slip angle is greater than the rear slip angle, the car is
said to be "understeering," as in Fig. 8-4; when the reverse
is true, the car is "oversteering." Whether a car understeers
or oversteers depends principally on its design and partly
on factors within the control of the operator. For example,
HORIZONTAL ALIGN MENT
199
research at the General Motors Proving Ground shows that
it is possible to make a car either highly oversteering or highly
understeering by merely varying front and rear tire pressures
within certain limits.
car.
The understeering car is somewhat more stable and sus-
ceptible to control than the oversteering type. Perhaps
this is true because an increase in speed on a curve requires an
increase in the steering angle of the understeering car (an
operation that is instinctively natural), whereas a decrease in
the steering angle is necessary in the case of the oversteering
Above a certain critical speed the front wheels of the
understeering car start to slide off the road, but by careful
braking the driver can generally regain control and return to
a fairly fixed course at a speed below the critical value. It is
more difficult to hold an oversteering car on a fixed path even
at moderate speeds. Above a certain critical speed the rear
end of the car starts to slide off the road, and any slight
application of brakes is apt to put the car into a spin.
At the same time at which a car develops certain slip angles
in rounding a curve, there is some tilt or "body roll." The
roll angle is a linear function of f, at least up to the limits of ƒ
considered safe. Roll angles are not large. Tests show that,
when f=0.20, body-roll angles vary between 1.8° and 3.5°,
the value depending on the make and model year of the car.
Though body roll has less effect upon a car's general "road-
ability" than do the slip angles, it is a factor that must be
allowed for in the accurate use of the ball bank indicator (see
Art. 8-8).
Valuable research on the relation of slip angles, steering
angles, and body roll to safe speed on curves has been pub-
lished by Fox (low-speed tests) and by Stonex and Noble
(high-speed tests on the Pennsylvania Turnpike). The facts
brought out show why all cars do not handle alike on the same
curves. Information is also available on the special problems
involved in test-track design.24
8-8. Side Friction Factors.-The value of ƒ at which side
skidding is imminent depends principally upon the speed of
the vehicle, the condition of the tires, and the characteristics
of the roadway surface. Moyer's work indicated maximum
200
CURVE PROBLEMS IN HIGHWAY DESIGN
values of ƒ of about 0.50 at relatively low speeds, with a
reduction to approximately 0.35 at high speeds. The tests
at very high speed on the Pennsylvania Turnpike (speeds up
to 105 mph) showed maximum values as low as 0.30, even
though the cars were driven by skilled test drivers.
An important problem in curve design-especially on
curves to be marked with safe-speed signs—is to determine
the percentage of the maximum side friction that can be
utilized safely by the average driver. The resulting values
of ƒ used in design should give posted speeds that have an
ample margin of safety even when the pavement is wet.
Furthermore, when the posted speed is exceeded, the added
unbalanced centrifugal force should be enough to produce
an uncomfortable sensation and an instinctive reduction in
speed.
The simplest device yet developed for determining maxi-
mum safe speeds and their relation to side friction factors
is the ball bank indicator, apparently first used by the Missouri
State Highway Department in 1937. It consists of a sealed
curved glass tube containing liquid and a steel ball slightly
smaller than the bore of the tube. When the indicator is
mounted on the dash by means of rubber suction cups, the
ball is free to roll transversely under the influence of the forces
acting upon it. The liquid produces enough damping effect
to hold the ball fairly steady, even when the car is driven
around a curve at high speed on a slightly rough surface.
Readings are taken on a scale graduated in degrees with the
0° mark at the center of the tube.
Where the ball bank indicator is to be used, it is first
mounted on the dash with the ball at the 0° reading when the
car is in a stationary level position. Obviously, all observers
who are to be in the car during the test run must be in their
assigned positions when the indicator is set in place.
The indicator is used for two purposes: (1) to determine
the ball bank angle and side friction factor at the maximum
comfortable speed on a particular curve, and (2) to determine
the speed on a curve required to produce a specified ball bank
angle.
In the first use, the body roll p of the test car is determined
by stopping the car on the curve and reading the ball bank
HORIZONTAL ALIGNMENT
201
angle B. When the car is stationary, p=ß-0, as shown in
Fig. 8-5. (Obviously the superelevation @ must be measured.)
Then trial runs are made around the curve at various constant
speeds until that speed is reached which first produces an
uncomfortable centrifugal sensation. Simultaneous readings
of speed and ball bank angle are then made, and the averages
are taken. Thus, one of the desired values, B, is measured by
direct observation.
Axis
kp
0=tan'e
of car
-Ł
Fig. 8-5. Ball bank indicator-car stopped
Ball bank
-indicator


To find the desired side friction factor f, the following
analysis, based on Fig. 8-6, is made:
22
9 R
When a car rounds a superelevated curve at equilibrium
speed, the resultant of the forces is perpendicular to the pave-
ment surface; there is no body roll and the ball bank indicator
reads 0°. At a speed greater than equilibrium speed, side
friction comes into play, acting inward. Were it not for the
body roll, the ball bank angle would be practically a direct
function of this side friction. However, the outward body
roll tilts the car toward the horizontal, thereby increasing the
reading of the ball bank indicator. The condition is shown
in Fig. 8-6, from which
B-p=tan-1
Ꮎ
(8-12)
202
CURVE PROBLEMS IN HIGHWAY DESIGN
But equation 8-9 shows that
v2
g R
tan-¹f=tan-1
or
Ball bank
indicator
Centrifugal
force
f=
e=tan 'e
H
HP
р
|
кон
кв-
ト
​-1
tan-¹ƒ=tan-¹
CAR
اماد
v2
9 R
g
ik
AXIS
e
v2
g R
-e)


Fig. 8-6. Ball bank indicator when V > equilibrium speed
tan' y²
GR
Yook
Because of the small angles involved, it is sufficiently
accurate to assume that
22
g R
-tan-¹ e=tan-1
Substituting from equation 8-12 gives
tan-¹ƒ=ẞ-p
(8-13) Approx.
The work may be checked by assuming that p is zero and
solving equation 8-12 for ß, thus obtaining the theoretical ball
HORIZONTAL ALIGNMENT
203
bank angle on the assumption of zero body roll. The difference
between the calculated and observed ball bank angles is the
body roll, which should agree with the value measured while
For any
the car is stationary on the superelevated curve.
one test car there should be a linear relation between the values
p and f.
This first use of the ball bank indicator represents its
application in research. After the ball bank angle correspond-
ing to the maximum safe speed has been determined, the
second use of the indicator is in connection with establishing
posted speeds on curves. In this application it is not neces-
sary to measure the radius or the superelevation. The speed
of the test car is gradually increased until the specified ball
bank angle is reached, and this maximum speed is recorded.
The practical value of the ball bank indicator rests in
its simplicity and in the fact that there is a surprisingly
close agreement among the various States using it in regard
to the numerical value, 10°, most widely adopted to indicate
maximum safe speed. A canvass made by Moyer and Berry⁹
showed that, after curves were first marked with safe-speed
indications in Missouri in 1937, the plan was quickly adopted
in 25 States and the ball bank indicator was the device most
commonly used for determining posted speeds. As a result
of this study, which also included an analysis of driver reaction
to posted speeds, the authors arrived at the following conclu-
sions:
To obtain the driver's respect for the speed on the sign over a wide
range of speed, the following ball bank angles are recommended: 14°
for speeds below 20 mph, 12° for speeds of 25 and 30 mph, and 10° for
speeds of 35 mph and higher.
For speeds up to 50 or even 60 mph, a ball bank angle of 10° has
been found to be quite satisfactory but for speeds above 60 mph a
lower value should be used.
If the foregoing recommendations are converted into side
friction factors by means of equation 8-13 (in which average
values of body roll reported by the General Motors Proving
Ground are used), the resulting values of ƒ range from about
0.21 at 20 mph to 0.14 at 50 mph.
In earlier tests reported by Barnett¹º a safe side friction
factor of 0.16 had been recommended for speeds up to 60 mph,
204
CURVE PROBLEMS IN HIGHWAY DESIGN
with a reduction of 0.01 in ƒ for each 5 mph increase in speed
above 60 mph. However, of the 900 tests involved in this
research only a few were made at speeds of 60 or more mph.

f=Side Friction Factor
0.22
0.20
0.18
0.16
0.14
0.12
0.10
0.08L
10
H.R.B. 1940
-Moyer & Berry
20
Arizona
Meyer 1949
30
Assumed for Curve Design'
H.R.B. 1936 Barnett
H.R.B. 1940
Moyer & Berry
40
50
Speed-mph
Fig. 8-7
Meyer 1949
60
Arizona
H.R.B. 1940
Stonex & Noble
70
80
As a result of the high-speed tests on the Pennsylvania
Turnpike, Stonex and Noble' concluded: "f=0.10 is the
maximum that should be utilized in curve design on public
highways at speeds of 70 mph and higher."
The slight inconsistencies in the recommended values of
f noted in the preceding paragraphs merely reflect the dif-
HORIZONTAL ALIGNMENT
205
ferences in judgment as to what constitutes incipient insta-
bility or uncomfortable centrifugal sensation. Further
research and a more scientific means of measuring discomfort
should narrow the range of disagreement.
For reasons to be explained in Art. 8-9, it is not practicable
to use sufficient superelevation to counteract centrifugal force
completely, except possibly on sharp curves designed for low-
speed operation. On flat curves on which vehicles are operated
at high speeds, drivers place more reliance upon side friction
than upon superelevation. It is therefore logical to design
with a greater margin of safety at high speeds, owing to the
greater difficulty in steering a true course and to the longer
stopping distance required in case of an emergency. The
practice may be stated differently as follows: A smaller
percentage of the available side friction should be used at
high speeds than at low values.
Design
Speed
V
mph
22243
20
25
30
35
40
45
50
55
60
65
70
80
90
100
RECOMMENDED VALUES FOR USE IN DESIGN
Side
Friction
Factor
f
TABLE 8-4
0.17
0.165
0.16
0.155
0.15
0.145
0.14
0.135
0.13
0.125
0.12
0.11
0.10
0.09
tan-¹ƒ
or
(B-p)°
9.6
9.4
9.1
8.8
8.5
8.3
8.0
7.7
7.4
7.1
6.8
6.3
5.7
5.2
Body
Roll
Angle
3.0
2.7
2.5
2.4
2.2
2.1
2.0
1.9
1.8
1.7
1.6
1.5
1.4
1.3
Ball
Bank
Angle
B°
12.6
12.1
11.6
11.2
10.7
10.4
10.0
9.6
9.2
8.8
8.4
7.8
7.1
6.5
206
CURVE PROBLEMS IN HIGHWAY DESIGN
In view of the maximum values of ƒ noted in the first
paragraph of this article, it will be observed that the values
of ƒ summarized in Fig. 8-7 (from A ASHO Policy) embody this
principle of a greater safety factor at high speeds. The
straight-line relation assumed for curve design represents a
reasonable compromise based on the safe values recommended
by various investigators; it was purposely kept lower at the
low design speeds in order to compensate for the tendency of
drivers to overdrive on highways with low design speeds.
Table 8-4 gives recommended side friction factors cor-
responding to the straight-line relation in Fig. 8-7. The
ball bank angles were found by calculation, average body roll
angles reported by General Motors Proving Ground being
used. The final ball bank angles are accurate enough for
determining posted speeds on curves, even though test-car
roll angles differ somewhat from those listed. In the range
of speeds-45 to 55 mph-within which the safe-speed signs
are most frequently used, the recommended side friction factors
give ball bank angles very close to the value 10° used by most
States.
8-9. Maximum Superelevation Rates.-Because of the
presence of both slow-moving and fast traffic, and the varia-
tions in weather conditions over the seasons, it is impossible
to design a highway so that the superelevation is ideal for all
traffic at all times. Safety is the paramount consideration.
This requires a fairly low superelevation rate, with the result
that road speeds are usually greater than the equilibrium
speed.
When the speed is less than the equilibrium value, the
resultant (Fig. 8-2) acts inward and the driver must steer
slightly away from the center of curvature in order to maintain
a true course in the traffic lane. The effect is somewhat like
the action of an oversteering car. Since steering outward on
a curve is not a natural operation, there is a tendency for
slow-moving vehicles to "edge in" toward the shoulder or
toward the inner traffic lane. Moreover, if the side friction
factor is greater than about 0.05 when the road is icy, vehicles
may slide inward despite the driver's efforts to steer a true
course. On the other hand, if the superelevation rate is too
HORIZONTAL ALIGNMENT
207
e = Superelevation in Feet per Foot
low, design speeds on curves are limited by the safe side fric-
tion factors to values less than considered practicable in
modern design. Therefore, a compromise between these con-
flicting requirements is necessary.
0.13

0.12
0.11
0.10
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
Michigan
1
2
Washington-max
3
Washington
and Idaho
California
Arkansas
Colorado
4
5
6
D= Degree of Curve
Fig. 8-8
7
Washington-normal
J
Idaho
1-4
Michigan
8
9
10
There is fairly general agreement that a superelevation rate
of 0.10 (approximately 14 inches per foot) is about the maxi-
mum that should be used in regions where snow and ice are
encountered. Where exceptionally adverse winter conditions
are likely to prevail for several months, a maximum super-
208
CURVE PROBLEMS IN HIGHWAY DESIGN.
elevation rate of 0.07 or 0.08 is recommended. A maximum
as high as 0.13 is used in localities free from snow or ice.
Fig. 8-8 shows superelevation practice in several States
when it was customary to ignore design speed. This was the
traditional method. Since publication of the AASHO Policy
an increasing number of States vary superelevation with
design speed. In general the same maximum superelevation
TABLE 8-5
MAXIMUM CURVATURE
Design
Speed
V, mph
30
40
50
60
70
30
40
50
60
70
30
40
50
60
70
30
40
50
60
70
Maximum Maximum Total
е
ƒ
e+f
88888 888 88 89 9
.06
.06
.06
.06
.06
.08
.08
.08
.08
.10
.10
.10
.10
.10
.12
.12
.12
.12
.16
.15
.14
.13
.12
.16
.15
.14
.13
.12
.16
.15
.14
.13
.12
.16
.15
.14
.13
.12
2229* ***** ***** *222 2
.21
.20
.19
.18
.24
.23
.25
.22
.28
.25
.24
Minimum Maximum
D
Ꭱ
Ft
Deg
273
508
833
1,263
1,815
250
464
758
1,143
1,633
231
427
694
1,043
1,485
214
395
641
960
1,361
21.0
11.3
6.9
4.5
3.2
22.9
12.4
7.6
5.0
3.5
24.8
13.4
8.3
5.5
3.9
26.7
14.5
8.9
6.0
4.2
HORIZONTAL ALIGNMENT
209
is used at all speeds, but at the higher design speeds the
maximum is reached on flatter curves.
e+f=Rate of Superelevation Plus Side Friction Factor
In the A ASHO Policy it is “concluded that (a) several rates
rather than a single rate of maximum superelevation should be
0.32
0.28
0.24
0.20
0.16
0.12
0.08
0.04
of
8886 8ង្គ៩ ៖
BO
000
V = 70
4
V = 60
V = 50
8
R = Radius in Feet
V = 40
-400
Safe Maximum D When e=0.12.
e = 0.10
e = 0.08
e = 0.06
V = Design Speed = 30 mph
300
12
16
D = Degree of Curve
Fig. 8-9
Formula 8-11
v2
15R
e+f=
-250

or D₂=85,700 (e+f)
20
24
28
recognized in establishing design controls for highway curves,
(b) a rate of 0.12 should not be exceeded, and (c) at the other
extreme a rate of 0.06 is applicable for urban design. Con-
sistent with current practice, values for the 0.10 rate are
referred to as generally desirable or nationally representative."
210
CURVE PROBLEMS IN HIGHWAY DESIGN

10
Fig. 8-10. Design superelevation rates
D = Degree of Curve
15
20
25
0.01
0.02
e = Superelevation Rate
00888
8880
= 30
0.08
0.09
V = 70
V = 60
V = 50
V = 40
3.86°
5.49°
8.25°
13.43°
24.83°
0.10
-5,000
3,000
-2,000
1,500
-1,000
-900
800
-700
-600
500
-400
-300
212
R = Radius of Curve-Feet
HORIZONTAL ALIGNMENT
211
8-10. Maximum Degree of Curve.-The use of any recom-
mended maximum rate of superelevation in combination with
a particular design speed results in a maximum degree (or
minimum radius) of curve. If sharper curvature were used
with the stated design speed, either the superelevation rate or
the side friction factor would have to be increased beyond
recommended safe limits. Thus, this maximum degree of
curve is a significant value in alignment design.
Table 8-5 (from AASHO Policy) lists limiting values of
curvature for four maximum rates of superelevation. The
final values are consistent with formula 8-11 in which are sub-
stituted the safe side friction factors from Table 8-4.
The relationships given by formula 8-11 are shown graphi-
cally in Fig. 8–9 (from AASHO Policy) by the straight lines
for each design speed. Superimposed on the graph are the
values of maximum curvature, as taken from Table 8-5.
8-11. Superelevation Rate Over Range of Curvature.-It
is not necessary to superelevate very flat curves; the normal
crown is carried around the curve unchanged. (This matter is
treated in Art. 8-12.) However, it is not logical to change
abruptly from zero superelevation on a very flat curve to
maximum superelevation at some arbitrary value of D or R.
There should be a transition range of curvature within which
maximum superelevation increases in a rational manner from
zero to the full value permitted by climatic conditions. Fig.
8-8 indicates how some States have handled this matter.
As a result of careful analysis of the dynamic factors
involved in four different methods of approach to this problem,
"it was concluded (AASHO Policy) that a parabolic form,
with the horizontal distance governing, represents a practical
distribution [of superelevation] over the range of curvature."
This is the type of relationship that was proposed in the first
edition of this textbook as "a rational suggestion for general
design."
8-12. Superelevation Rates for Design. Fig. 8-10
(AÁSHO Policy) shows recommended design superelevation
rates for the case where the maximum rate of superelevation
equals 0.10. The maximum values of curvature at e=0.10
correspond to those in Table 8-5. On curves flatter than the
212
CURVE PROBLEMS IN HIGHWAY DESIGN
maximum, the superelevation rates lie on parabolic curves of
the form recommended in Art. 8-11. In practice similar
design curves could be constructed for other maximum super-
elevation rates, such as those given in Table 8-5.
On two-lane highways, if the pavement cross-section
normally used on tangents is carried around a horizontal curve
unchanged, traffic entering a curve to the right has the benefit
of some favorable superelevation from the crowned pavement.
On the other hand, traffic entering a curve to the left meets an
adverse crown, as in Fig. 8-3 (b). This leads to a considera-
tion of the maximum curvature for which the normal crowned
cross-section is suitable.
As a general rule, it is recommended that the minimum rate
of superelevation on any curve (except at a reverse transition)
should be about 0.012 and that the particular value should
correspond to the average rate of cross slope used on tangents.
With an average adverse cross slope of 0.012, the correspond-
ing degree of curve (rounded) for each design speed in Fig.
8-10 is shown in the third column of Table 8-6 (from AASHO
Policy). It should be observed that the resulting side friction
required to counteract both adverse superelevation and cen-
trifugal force is very small. Obviously, if the curves were
made sharper, the required side friction factors would increase,
Design
Speed
V, mph
MAXIMUM CURVATURE FOR NORMAL CROWN SECTION
30
40
50
60
70
Average
Running Maximum
Speed
D
mph
AAA ON
27
34
TABLE 8-6

40
45
49
1°30'
0°45'
0°30'
0°20'
0°15'
Minimum
R
ft
3,800
7,600
11,450
17,200
22,900
Resulting Side Friction
Factor f When Adverse
Crown e=0.012
At Design At Running
Speed Speed
.028
.026
.027
.026
.027
.025
.023
.022
.020
.019
HORIZONTAL ALIGNMENT
213
and a point would be reached where a favorable slope across
the entire pavement would be desirable.
In the AASHO Policy it is recommended that a plane slope
across the pavement should be used wherever a curve is sharp
enough to require a superelevation rate in excess of about 0.02.
This practical limit corresponds to degrees of curve ranging
from 2°30′ at 30 mph to 0°45′ at 70 mph. For curves between
these values and those in Table 8-6 a compromise could be
made, in the interest of construction economy, by rotating the
normal crown slightly toward the inside of the curve. How-
ever, a change to a plane slope across both lanes would be pre-
ferable, at least at the higher design speeds.
Table 8-7 shows the resulting design superelevation rates
for the case in which e=0.10; it is the tabular form of Fig. 8-10.
Similar tables for three other values of maximum e are found
in the AASHO Policy. The basis for selecting the tabulated
runoff or spiral lengths is discussed in Arts. 8-13 and 8-15.
8-13. Length of Spiral.-The purposes served by an ease-
ment curve and the reasons for choosing the spiral as the ease-
ment were stated succinctly in Art. 5–1.
Safe operation at high speeds requires that curves be
designed to fit natural driver-vehicle behavior. It is obviously
impossible, when traveling at any appreciable speed, to change
instantaneously from a straight to a circular path at the T.C.
of an unspiraled curve. On such alignment the driver makes
his own transition as a matter of necessity, usually by starting
to steer toward the curve in advance of the T.C. In so doing,
there is bound to be some deviation from the traffic lane. If
the curve is sharp or if the speed is high, the deviation may
result in dangerous encroachment on the shoulder or on the
adjacent traffic lane (see Fig. 8–11).
Though the operational and aesthetic advantages of spirals
are generally recognized, their adoption in the United States
(since about 1925) has been very gradual; as of 1962 more than
one-half of the State highway departments used them. Among
the reasons for this are the cumbersome, highly-mathematical
treatments often presented, which result in the belief that
tedious computations and awkward field work are inherent in
the use of the spiral; the mistaken belief that a spiral of
214
CURVE PROBLEMS IN HIGHWAY DESIGN

D
0° 15' 22918'
0° 30'
0°.45'
1° 00'
11459'
7639'
5730'
1° 30'
2° 00'
2° 30′
3° 00'
3° 30'
4° 00'
5° 00'
6° 00'
7° 00'
8° 00'
9° 00'
10° 00'
11° 00'
12° 00'
13° 00′
14° 00'
16° 00'
18° 00'
20° 00'
22° 00'
R
24° 00'
24.8°
3820' RC
2865' RC
Q
2292' .020
1910'
.024
819'
716′
NC
NC
NC
NC
1637' .027
1432'❘ .030
1146' .038
955' | .044
637'
573'
239'
231'
.050
.055
.061
.065
521'
.070
477' .074
441' .078
409'
358'
.087
318'
.093
286'
.096
260' .099
V = 30
L
.100
.100
2-lane
OOOO
0
0
100
100
100
100
100
100
100
100
100
100
130
130
140
.082 150
110
120
160
170
170
180
4-lane
Oooo
Ο
100
100
100
100
140
150
100
.045
100
.050
100
.060
120 .068
190
200
210
220
240
250
260
270
e
180
180
D max = 24.8°
160
.089
180 .093
270
270
.020
.027
.033
.038
NC
0
NC
0
RC 125
RC
125
V = 40
.096
.098
.099
2-lane 4-lane
125
125
125
125
.076 160
.084
180
125
125
130
140
190
200
L
200
210
210
.100 210
D max
e max = 0.10
V
0
0
125
125
125
125
125
125
140
160
190
210
240
260
280
290
300
310
310
320
13.4°
NC
RC
RC
.018
0
150
150
150
2-lane 4-lane
.027 150
.036
150
.045
150
.054
150
.063
150
.070 170
.083 200
.093
220
50
.097
230
.100 240
.100 240
D max
L
0
150
150
150
150
150
160
190
230
250
300
330
350
360
360
8.3°
е
NC
RC
:018
.022
.034
.046
.059
.070
V = 60
L
.081
.090
.099
.100
2-lane 4-lane
0
175
175
175
175
175
175
190
0
175
175
175
175
190
240
280
220
240
270
270
D max = 5.5°
330
360
400
400
е
RC
RC
.020
.028
.042
.055
.069
.083
V = 70
.096
.100
2-lane 4-lane
200
200
200
200
200
200
L
210
250
200
200
200
200
200
250
290
300
D max =3.9°
-
310
370
.430
450
NOTES. NC=normal crown section. RC remove
adverse crown, superelevate at normal crown slope.
Spirals desirable but not as essential above heavy
line. Lengths rounded in multiples of 25 or 50 feet
permit simpler calculations.
TABLE 8-7.-DESIGN VALUES FOR RATE OF ELEVATION (e)
AND MINIMUM LENGTH OF RUNOFF OR SPIRAL CURVE
(I IN FEET)
HORIZONTAL ALIGNMENT
215
particular length is required for each different value of curve
radius or of design speed; and inertia-a reluctance to change
existing rule-of-thumb practices.
Ineffective pavement

−1 . Konkrete
T.C.
BAR
Fig. 8-11. Vehicle paths on unspiraled curve
It is hoped that the simple presentation of the basic geo-
metry of the spiral in Chapter 5, in which easily remembered
analogies to the corresponding parts of a circular curve are
emphasized, will assist in dispelling the bugaboo of mathe-
matical complexity. Moreover, the various spiral tables in
Part III are so complete and so readily adapted to basing
the circular arc either upon the radius or upon the chord or
the arc definition of degree of curve that the required trigo-
nometry and field work are no more complicated for the
calculation and staking of a spiral-curve layout than for an
unspiraled curve. (As a matter of interest, it is the author's
practice to require that students be able to compute and stake
an acceptable spiral without referring to tables of any kind.
The method-given in Art. 5-13-is based upon a sketch
showing the basic geometry of the spiral, from which all
needed approximate relations may be derived by inspection;
slide-rule calculation is adequate.)
The belief that the proper length of spiral is closely cal-
culable adds to the impression of mathematical complexity
This belief is probably based upon early research done by
Shortt" in the field of railroad practice. Later, the Shortt
formula was applied to highways by Royal-Dawson¹ and
others, although the physical conditions differed in some
important respects. Until recently, no proof of the correct-
216
CURVE PROBLEMS IN HIGHWAY DESIGN
ness of this transfer existed, although observation of highway
driving practices led many engineers to doubt its validity.
Fortunately, certain fallacies have been disclosed by recent
research, the outcome of which promises to be a simpler
method of selecting spirals.
It is significant that Shortt's research was on unsuper-
elevated transition curves on railway track. Briefly, the result
of his work seemed to indicate that the length of transition
for comfortable operation was a definitely calculable variable.
The reasoning was as follows: At constant speed, the length
of time required to traverse an easement curve of length L, is
Ls
seconds. On the curve, the acceleration toward the center
ย
v2
is ft per sec². Consequently, the average (assumed con-
R
ft per
stant) rate of change in centripetal acceleration is
23
R LS
sec.³ Shortt denoted this constant rate by C. Converting v
to V in miles per hour and solving for L. gives the basic Shortt
formula, which is
8
Ls
=
Ls
3.15 V3
Ꭱ Ꮯ
Shortt concluded from these experiments on unsuperelevated
railroad track that a value for C of 1 ft per sec³ was the maxi-
mum that would go unnoticed.
In transplanting the Shortt formula to highway practice,
it has been customary to overlook the possible effect of super-
elevation and also to use a larger value of C. Tentative sug-
gestions that 2 might be a suitable value of C for highways
led to the following version of the Shortt formula:
***
(8-14)
1.58 V³
Ꭱ
-
(8-15)
Unfortunately, equation 8-15 has sometimes been used
quite literally, as though to deviate from it would represent
a departure from correct spiral theory as applied to highways.
Arbitrary use of this equation involves two fallacies. One is
neglect of the superelevation. Actually, the presence of
superelevation cancels out part or all of the centrifugal force
(as far as its effect upon comfort is concerned), thereby
invalidating any mathematical relation that is based upon
HORIZONTAL ALIGNMENT
217
unsuperelevated curves. Review of the geometry of the spiral
shows that it is a curve of uniformly increasing degree. If
superelevation on a curve were increased uniformly, it follows
from basic superelevation theory (Art. 8-6) that the side fric-
tion factor ƒ would be zero when traversing the spiral at the
so-called equilibrium speed for the circular arc. In other
words, it would be theoretically possible for a car to steer itself
on a true course around the complete spiral-curve layout with-
out the driver touching the wheel.
It may be shown that the correct form of the Shortt formula
for superelevated curves is
1/v3
L. ===√(
CR
cos 0-v g sin
sin e
L₂ = 3.15 V (V²-15 e)
Ls
с R
Since the values of used in practice are small, it is close
enough to assume that cos 0=1 and that sin 0=tan 0=e.
When these substitutions are made and v is converted to V
in miles per hour, equation 8-16 becomes¹³
Ls
(8-16)
(8-17)
When e=0, formula 8-17 reduces to the Shortt formula
for unsuperelevated curves; for this reason it has been called²
the "modified Shortt formula.”
47. V f
с
In practice the speed on the circular arc is usually greater
than the equilibrium value, resulting in a uniform value of ƒ
that must fit equation 8-11. Both comfort and safety at
high speed make it desirable to approach ƒ at a uniform rate
along some form of easement curve. The spiral, when banked
at a uniformly increasing rate, produces this condition.
A simpler form of equation 8-17 for the case in which V is
greater than the equilibrium value is found by substituting
15(e+f) for
V2
R
The result is
(8-18)
A relation of this basic form was given by Haile,14 though
he recommended that it should not be used because of the
short spiral lengths produced when ƒ is small. There is no
218
CURVE PROBLEMS IN HIGHWAY DESIGN
need to discard the theoretically correct relation represented
by equation 8-18 solely for this reason. It is important,
however, to consider the rotational effect. Though an
extremely short spiral to a superelevated curve will result in
no centrifugal sensation (if traversed at equilibrium speed),
yet the rotational change about the axis of the car may be too
rapid for comfort or safety.
Long years of operating experience on spiraled superelevated
railroad track have led the American Railway Engineering
Association to recommend that minimum spiral lengths be
based upon attaining superelevation across standard-gage
track at a desirable maximum rate of 1½ inches per second of
time. Highway and railroad operation are by no means
analogous; but, in the absence of research on the motor-
vehicle rotational rate at which discomfort begins, there is
logic in tentative adoption of the same rate used successfully
on railroads. (Haile¹ and other engineers advocated consid-
eration of the rotational effect as early as 1936.)
A rise of 1.25 inches per second across the track gage of
4 ft 8 in. is equivalent to about 0.022 ft per ft per sec. Con-
sequently, L. = 1.47 V
(0.622)
"
or
Ls=67 e V
(8-19)
The two formulas for minimum desirable spiral length are
separately inconsistent for certain conditions. Equation
8-18 gives values which are too short when ƒ is small (4,=0
at equilibrium speed), whereas equation 8-19 gives values
which are too short when e is small (L, =0 for unsuperelevated
curves). However, by using each formula within its proper
range, the resulting minimum spiral lengths will produce
neither unsafe values of ƒ nor an uncomfortable rate of angular
rotation.
47 V f
C
The spiral length at which the change from equation 8-18
to 8-19 occurs is found by equating the two values of L.
Thus,
governs choice of minimum L, when e<0.7 £
HORIZONTAL ALIGNMENT
219
67 e V governs choice of minimum L, when e>0.7
The value of e is found from the relation
V2
15 R.-).
V2 Da
85,700
Rotational change, rather than side friction and centripetal
acceleration, is likely to govern minimum spiral lengths on the
f
open highway. This is because e will exceed 0.7 wherever
C
the actual superelevation rate is greater than about 60 per
cent of the recommended maximum for the given degree of
curve and design speed.
-f (or
Table 8-8 (from AASHO Policy) shows values of spiral
length on the sharpest curves recommended when e=0.10,
as determined from the several formulas just described. The
variations serve to emphasize that there is no basis for insist-
ing on great precision in calculating lengths of spirals for
design.
TABLE 8-8
CALCULATED LENGTHS OF SPIRAL FOR
MAXIMUM CURVATURE
Basis
e=0.10
The practical value of the rotational-change method of
choosing minimum spiral lengths is well-illustrated in the
case of the Pennsylvania Turnpike. The spirals on this high-
way (Table 8-10) were designed on the basis of a rotational
change of about 0.02 ft per ft per second-practically, the value
Shortt formula, C=1
Shortt formula, C=2
Modified Shortt formula, C=1
Modified Shortt formula, C=1.35
Meyer, L=67 e V
Ls Minimum length of
spiral in feet for design
speed, mph, of:
30 40 50 60 70
370 470 570 650 730
185 235 285 325 365
230 285 330 370 400
170 210 250 275 295
200 270 335 400 470
220
CURVE PROBLEMS IN HIGHWAY DESIGN
7
used in deriving equation 8-19. The generally favorable
operating characteristics of these spirals, as observed on
high-speed tests, are undoubtedly due in no small measure to
the constant (and comfortable) rate of rotational change
experienced on successive curves.
Until recently, valid objection to this method of selecting
spiral lengths might have been made on the grounds that it
does not necessarily produce a constant value of C. The
impression that C must be constant on a highway spiral—
as it must of necessity be in the case of a spiral on railroad
track-is another fallacy which has been disclosed by research.
This assumption is equivalent to stating that the driver of a
motor vehicle who is free to choose his own path in making
the transition from a tangent to a circular curve always turns
his wheel (1) at a constant rate on any given transition and
(2) at the same rate on all transitions.
The careful field tests and statistical analyses made by
Leeming and Black, 15 summarized and amplified in Leeming's
later report,16 verify the first part of this assumption. (This,
in effect, validates the spiral as a curve of suitable shape but
does not involve any condition as to its length.) However,
the research, made with a recording accelerometer on more
than 2,000 curves, shows that C varies between such wide
limits that there is no "natural rate of turning" the wheel,
such as is represented by C=2 or by any other constant value.
Though these investigators found no correlation between C
and V nor between C and R, there was a definite tendency for
C to increase as the value of ƒ reached on any curve becomes
greater. Since the higher values of ƒ were recorded on the
sharper curves, this trend is equivalent to stating that a driver
turns his wheel faster upon entering a sharp curve than upon
entering a flat one. Though contrary to the Shortt formula,
this conclusion is in accordance with observation of general
driver behavior, and has been verified in tests by Warren¹ and
by Welty.18
From Leeming's experiments it appears that safety and
comfort depend mainly on the value of f; and that a driver
slows down to reduce ƒ, not to reduce C. Relatively large
values of C (far in excess of 2 ft per sec³) were frequently
recorded as producing an "imperceptible" degree of discom-
HORIZONTAL ALIGNMENT
221
fort, as long as the accompanying values of ƒ were moderate
(in the neighborhood of the design values suggested in Fig.
8-7). ·
The length of spiral may also be made equal to the length
required for superelevation runoff. Runoff length is deter-
mined by the rate at which the pavement cross slope is
changed, or rotated, subject to some modification on the basis
of appearance as viewed by the driver. Thus, the resulting
formula for runoff (and spiral) length would have the same
general form as 8-19. This subject is treated in greater detail
in Art. 8-15.
8-14. Minimum Curvature for Use of Spirals.-Neither
superelevation nor spirals are required on extremely flat curves.
It would appear logical to use spirals approaching all curves on
which a plane cross slope is used across the entire pavement.
In Art. 8-12 the superelevation requiring such design was fixed
(AASHO Policy) at the rate e=0.02 or greater. However, at
this limiting rate a selected spiral would have such a small o-
distance (offset, or throw, at the offset T.C.) that the spiral
would have little significance. It could be omitted without
hampering the ability of the driver to keep well within the
traffic lane; if used, it would serve principally as a graceful
method of changing from the normal crown to the super-
elevated cross-section.
In view of the fact that minimum curvature for use of spirals
TABLE 8-9
MINIMUM DEGREE OF CURVE FOR USE OF SPIRALS

Design
Speed
mph
30
40
50
60
70
Minimum
Curvature
3°30'
2°15'
1°45'
1°15'
1°15'
Assumed Mini-
mum Length of
Spiral
ft
100
125
150
175
200
Calculated
o-Distance
ft
.25
.26
.29
.26
.36
222
CURVE PROBLEMS IN HIGHWAY DESIGN
must be set arbitrarily, the control values selected in the
AASHO Policy were taken as rounded values obtained from
Fig. 8–10 at the points where e=0.03, approximately. The
resulting recommendations are given in Table 8-9, the throws
in the last column being obtainable from Table XI. Mini-
mum lengths of spirals approximate the distances traveled in
2 seconds at the design speeds.
8-15. Length of Superelevation Runoff.-Superelevation
runoff is the general term denoting the transition from the
normal crown section on a tangent to the fully superelevated
section on a curve. The runoff should be pleasing in appear-
ance to the driver and effected smoothly over a length that is
safe and comfortable when the vehicle is operated at design
speed.
There is no completely rational method of determining
length of runoff. It has been made equal to the length of
spiral, as calculated from one of the formulas in Art. 8-13. If
this method is used, the most logical formula would be one
having the same general form as 8-19, since this formula is
based on restricting the rate of angular change in supereleva-
tion.
To possess a pleasing appearance as viewed by the driver,
the edge profiles should not appear to be distorted. Control
of runoff length from this standpoint has also been used; the
numerical controls must, of course, be empirical. In the
A ASHO Policy the values suggested are as follows:
Design speed, mph...... 30 40 50 60 70
Max. relative slope in %
gradient between edges of
2-lane pavement....
1.33 1.14 1.00 0.89 0.80
On spiraled curves there is no sound basis for using dif-
ferent lengths of spiral and runoff. Simplicity in construction
is gained by using identical lengths. Since length of runoff is
applicable to all superelevated curves, whether spiraled or not,
it is concluded (AASHO Policy) that runoff lengths, as deter-
mined from the foregoing appearance controls, should also be
used for minimum lengths of spirals. On 4-lane highways the
runoff lengths are taken to be 1.5 times the lengths for 2-lane
highways, purely on an empirical basis. The resulting spiral
• •
HORIZONTAL ALIGNMENT
223
and runoff lengths are listed in Table 8-7. It should be
emphasized that these are minimum values; high-type align-
ment or the attainment of proper pavement drainage may
justify the use of greater lengths.
8-16. Methods of Attaining Superelevation.-The transi-
tion from the normal crowned section on a tangent to the fully
superelevated section on a curve should be pleasing in appear-
ance and inherently safe and comfortable for the operation of
vehicles at the highway design speed. In addition, it should
be relatively simple to calculate and stake out the transition.
Spiraled curves may be superelevated by the method shown
in Fig. 8-12. In this method the normal profile grade of the
center line is unchanged. The outer lane between sections
a-a and b-b is gradually warped from the normal crowned
section to a straight level section at the T.S.; beyond the T.S.
the section is rotated at a uniform rate about the survey center-
line until it reaches full superelevation at the S.C. (section d-d).
The normal profile grade of the inner edge of the pavement is
continued as far as section c-c. Between sections b-b and c-c
the normal convex crown (if any) on the inner lane is gradually
converted to a straight inclined section at c-c, where the rate
of superelevation equals that on the outside lane. (The pave-
ment areas over which the crown is taken out are shown cross-
hatched in the plan view.) Between sections c-c and d-d
there is a uniformly increasing one-way bank across both lanes.
The same method is used on a curve to the left and on the
leaving spiral.
In Fig. 8-12 the edge profiles are shown as straight lines
merely to illustrate the basic design; the edge breaks would
actually be rounded in construction. Some agencies obtain
the effect of short vertical curves by eye adjustments of the
stakes or forms; others insert true vertical curves at the breaks;
and a few States use reversed vertical curves. Graphical
determination of edge profiles by means of splines is an excel-
lent and economical method in office design.
The preceding method of rotating a section about the center
line should not be adhered to rigidly. Practical considerations,
as well as aesthetics, are poorly served by such stereotyped
design. Drainage conditions, for example, may not permit
224
CURVE PROBLEMS IN HIGHWAY DESIGN
depressing the inside edge of the pavement by the amount
required by this method. In such a case the pavement can be
rotated about the normal profile grade of the inside edge or
about a line a short distance from that edge and parallel to it.
Where summit vertical curves and horizontal curves overlap,


لوں
b
소
​a
لوين
T.S
a
Ł
R
b
a
S.C
t
CROSS-
SECTIONS
-
utside_edge
}
I
Grade
Tangent
Runout
Profile
sto
I's
\s.c.
PLAN
Fig. 8-12. Attaining superelevation at spiraled curve
Inside edge
Crown
PROFILE
rotation about the normal profile grade of the outside edge
may be the obvious method of reducing the unsightly hump
produced along that edge by either of the preceding methods.
No one method is best for all situations; each case should be
studied individually.
Multi-lane highways having a median strip present an
HORIZONTAL ALIGNMENT
225
especially difficult problem in runoff design. Here, considera-
tion of all important factors-aesthetics, drainage conditions,
economics of grading, riding comfort, and safety—is neces-
sary in arriving at a harmonious solution.
Tangent
Runout
The design finally adopted on the original section of the
Pennsylvania Turnpike is especially instructive in these
respects. On this highway the rather narrow median strip
(10 ft wide) and frequent curves-aggregating 50 miles in
the 160-mile distance-were principally responsible for the
decision to keep the edges of the paved roadway nearest the
median strip in the same horizontal plane at all times. Each
24-ft roadway slopes away from these edges at a rate of
in. per ft. Consequently, on tangents there is surface
drainage over the roadways from the median strip. This
disadvantage, however, is offset by the simpler method of
runoff design made possible. The method is essentially that
of Fig. 8-12, adapted to rotation about the edges closest to
the median strip. Fig. 8-13 shows the runoff details for a
curve to the left; specific curve data are given in Table 8-10.
120°
V.C.
T.S.
+ Length of Spiral
1
Profile along outer edge of
24' pavement-Right roadway
S.C.

V.C.
150'
V.C.
Profile along outer edge of
24 pavement-Left roadway
Fig. 8-13. Runoff design on Pennsylvania Turnpike
1-3" Profile grade along inner
edge adjacent to median strip
·
| |
Full
Super.
נוך
It should be observed that the level-inclined section, cor-
responding to b-b in Fig 8-12, occurs in advance of the T.S.
in Fig. 8-13, and that the inclined section corresponding to
c-c in Fig. 8-12 occurs at the T.S. in Fig. 8-13. These modifica-
Super.
226
CURVE PROBLEMS IN HIGHWAY DESIGN
Degree
of
Curve
1°45'
2°00'
2°15'
2°30'
2°45'
3°00'
3°15'
3°30'
4°00'
4°30'
5°00'
5°15'
5°30'
TABLE 8-10
PENNSYLVANIA TURNPIKE SPIRALS
Rate of
Superelevation
per Foot
of Width
Inches
1
155/65
0100
7
16
9
16
11 34 HH 700
11
16
13
16
8
116
3
11%
3
116
1136
Length
of
Spiral
Feet
150
150
200
200
250
250
280
300
350
370
410
400
400
Tangent
Runout
Feet
260
210
210
164
162
160
158
157
154
148
146
145
141
tions are required by the one-way bank over the roadways on
tangents.
(C
Observations of high-speed driving over spiraled curves
built with and without a tangent runout on the outer lane
indicate that the runout is a desirable feature of superelevation
runoff design. Used in combination with a suitable length of
spiral, it is a pronounced aid in entering horizontal curva-
ture. This is quite noticeable at night during adverse visi-
bility conditions. The car appears to steer itself into the
curve before the operator is aware of its presence. This
quality of self-steering eliminates the element of surprise
when entering curves travelling at speeds in excess of visibility
requirements during adverse weather."
Unspiraled circular curves are superelevated by various
rule-of-thumb methods. Obviously, no method can be com-
pletely rational, since it is impossible to have full supereleva-
HORIZONTAL ALIGNMENT
227
tion between the T.C. and the C.T. (where it belongs) without
placing the runoff entirely on the tangent (where it does not
belong). On the other hand, the runoff cannot be accom-
plished completely on the curve without having inadequate
and variable superelevation over a substantial portion of the
distance. No agencies are known to follow the latter proce-
dure, although several States (Massachusetts among them)
place all the runoff on the tangent. The method adopted is
usually a compromise, in which the runoff starts on the tangent
and ends at a point some distance beyond the T.C.
In the foregoing method there is invariably a section of
tangent at each end of the curve over which the cross-section
varies from the normal crown on tangents to a one-way bank
at the T.C. and C.T., where the rate is between seven-tenths
and nine-tenths of the full superelevation value. There is no
standard practice as to details. For example, some State
highway departments provide full superelevation a fixed
distance beyond the T.C., regardless of the length of runoff
(Washington, 50 feet). The distance may or may not coincide
with the point at which full widening (Art. 8-18) is attained.
Other States provide full superelevation beyond the T.C. at a
variable distance equal to a certain fraction of the length of
runoff. In Michigan and California the fraction is one-third.
Rotation may be about the center-line or about any other line
parallel to the center-line; the modifying circumstances that
need to be considered are the same as those in the case of
spiraled curves.
The objection inherent in all methods of designing super-
elevation runoff for unspiraled curves is the inevitable viola-
tion of sound dynamic principles. In traversing that portion
of the runoff on the entering tangent, the driver-if he is to
maintain a straight course-must steer against the gradually
increasing superelevation. However, as soon as he reaches
the T.C. of a curve involving the usual combination of an
understeering car and a speed greater than the equilibrium
value, he must steer toward the center of curvature (see
Art. 8-7). This reversal in steering direction is neither
natural nor obtainable instantaneously. Consequently, dur-
ing the approach to the T.C. the vehicle usually creeps toward
the shoulder or toward the inner lane. Near the T.C. it
228
CURVE PROBLEMS IN HIGHWAY DESIGN
traverses a reverse curve in getting back into the traffic lane.
Upon leaving the curve similar effects are produced.
The effects just described are not particularly important
where vehicles are operated at low speeds. But at high design
speeds they may have consequences serious enough to justify
universal adoption of center-line spiraling. In fact, one of
the conclusions of Stonex and Noble as a result of the care-
fully-instrumented high-speed tests on the Pennsylvania Turn-
pike was: "the use of spirals in modern highway design is
imperative if inherent safety is to be provided.”
8-17. Pavement Widening on Curves.-The practice of
widening pavements on sharp curves is well established,
although there is little uniformity among State highway
departments as to the amount of widening required.

L
Fig. 8-14
R
When a vehicle travels at equilibrium speed around a curve,
the rear wheels track inside the front wheels by an amount
equal to R- √R²- L² (see Fig. 8-14). At other than equilib-
rium speed, the rear wheels track further in or out, the posi-
tions depending on whether the speed is less or greater than the
equilibrium value. When the speed is high, though still
within safe limits, the rear wheels may even track outside the
front wheels (see Fig. 8-4). There is no way of determining
the exact amount of extra lane widening required to compen-
sate for the non-tracking effect, except at equilibrium speed,
since it depends on the particular slip angles developed by
HORIZONTAL ALIGNMENT
229
each vehicle. However, measurements of actual wheel paths
on two-lane highways show that drivers sense the need for
greater clearance between opposing traffic, and that they
instinctively increase the clearance when the speed is higher
or the curvature is sharper.
One of the earliest formulas giving the recommended total
extra width for a two-lane highway was that suggested by
Voshell.19 This formula is
พ
35
=2[R- √ R²—L³] +
(8-20)
VR
Formula 8-20, with L taken as 20 ft, has been used by
several agencies. The expression is not entirely rational,
however, since the first term applies only at equilibrium speed
and the second term is purely empirical.
Other simpler
empirical formulas give results fully as satisfactory. Among
these are: w= VD, w=
D
3
+1, and w=
10
It is customary to limit the widening for a 2-lane pavement
to between 1.5 and 6.0 feet, approximately, there being no
widening on curves flatter than 5° or 6°. There are many
exceptions to these values, however. Some States determine
the widening only to the nearest foot; others work as close as
the nearest 0.1 foot.
Ꭰ
5°
In an attempt to rationalize the subject, the A ASHO Policy
contains an analysis in which four factors enter into the
formula for widening. These are (1) the track width of the
design vehicle (a single-unit truck or bus); (2) the lateral
clearance per vehicle; (3) the width of front overhang of the
vehicle on the inner lane; and (4) an extra width allowance
that depends on the sharpness of the curve and the design
speed. Accompanying the analysis is the recommendation
that design values for widening should be multiples of foot
and the minimum value should be 2 feet. On this basis no
widening is required on 2-lane pavements that are normally
24 feet wide or greater.
Table 8-11 contains a summary of values for widening
obtained by three representative methods. The term W is
the normal width of the 2-lane pavement.
Widening is no longer the important subject that it was
when sharp curves and 8-foot or 9-foot lane widths were preva-
230
CURVE PROBLEMS IN HIGHWAY DESIGN
+1 67∞
Degree
of
Curve Formula
D
8-20
L=20
4
5
8
10
13
16
20
24
TABLE 8-11
WIDENING ON CURVES-TWO-LANE PAVEMENTS
1.2
1.4
1.6
1.7
1.9
2.2
2.6
3.0
3.5
4.0
D
10
Widening, in feet, by:

+1
1.4
1.5
1.6
1.7
1.8
2.0
2.3
2.6
3.0
3.4
30
2 2
2.5
3
3.5
4
AASHO Policy, W=20 ft
Design Speed, mph
60

40
2 2 2 2 M
3
50
2 2
2.5
2.5
2 2
30
70
AASHO
W = 22 ft
Speed, mph
40
2
2
2.5
3
2
lent. In much modern alignment design no extra widening is
used, since high design speeds limit curvature to 5° or 6° and
traffic lanes are commonly 11 and 12 feet wide.
8-18. Transition to Widened Section.-Theoretically, the
full extra widening should continue for the whole length of a
circular arc. It is relatively easy to do this with spiraled
curves. When curves are not spiraled, however, it is impos-
sible to carry the full widening from the T.C. to the C.T.
without introducing undesirable kinks in the edge of the pave-
ment.
Spiraled curves may be widened by the methods shown in
Figs. 8-15 and 8-16. In Fig. 8-15 the total widening is placed
on the inside of the curve. Widening begins at the T.S.,
reaches the full amount at the S.C., and tapers off from the
C.S. to the S.T. in like manner. At intermediate points on the
transition the widenings are proportional to the distances from
the T.S. Forms for the curve at the inside edge of the roadway
are located on radial offsets from the survey center-line at
HORIZONTAL ALIGNMENT
231

Yo You
źb
źb - 1
źb
źbł
T.S.
TITTITO
=x+~+
TS.
@▪▪▪¶¶
ko
Fig. 8-15. Inside widening on spiraled curve
Ď
ко
لا
TIL
1/
S.C
Fig. 8-16. Equal lane widening on spiraled curve
Ew

JION
IS.C
5.
E21
distances equal to 6 plus the proportional amounts of the
widening.
In this method the curve of the inside edge is not a true
spiral. However, the transition is smooth except at the S.C.,
where the slight break may be remedied by eye adjustment of
the stakes or forms. The alternative would be to calculate a
separate spiral for the inside edge having a throw equal to
w plus the throw of the center-line spiral, and having a radius
232
CURVE PROBLEMS IN HIGHWAY DESIGN
R' equal to R-b-w. The length of the edge spiral is com-
puted as in the example on page 234. This procedure, pre-
ferred by some engineers, results in a spiral much longer than
that on the center-line; it also complicates the field work
somewhat. Fig. 8-17 illustrates the layout; see also Prob. 8-9.
In Fig. 8-16 the total widening is divided equally between
the inside and outside edges, and is distributed along the
transition by the proportion method used in Fig. 8-15. If
the spirals are long enough to attain the superelevation
properly, the breaks in the edges of the pavement at the S.C.
are hardly noticeable; in case they are apparent at all, they
may be rectified by eye adjustment of the forms.

žb
źb
TS.
T.S.
0
O+W
R
Xt=
x = L.√//w
=L₂y
S.C.
พ
2 Y
S.C.
Fig. 8-17. Spiraled center-line and inside edge
+
w b
The slight reverse curve in the outer edge of the pavement
near the T.S. may be avoided by starting the outside widening
at the point where the tangent produced intersects the
widened pavement, as point t in Fig. 8-16. The distance
from the T.S. to t is approximately equal to
(8-21)
Theoretically, there is a slight break at point t, but it
is imperceptible. Moreover, the small loss in widening
between t and the T.S. is negligible. For example, on a 10°
HORIZONTAL ALIGNMENT
233
curve for which L,=400 ft and w=2 ft, ≈¿=59 ft and the
widening at point t is only 0.15 ft.
When curves are spiraled, the method of Fig. 8-16 has
considerable justification, particularly in the case of a 2-lane
concrete pavement having a longitudinal joint on the survey
center-line. For one thing, staking is simplified. In addi-
tion, traffic on the outside lane is provided with the same
extra widening that is given to the inner lane, and the ten-
dency to edge across the longitudinal joint is thereby reduced.
On sharp curves requiring center striping, there is no unsightly
and confusing deviation between the striping and the center
joint.

b
źb-1
½
after which
TS
T. C.
q
I
不
​aaaaaa
R
By
▲ (in radians)
S.C
5.9977
Fig. 8-18. Inside widening on unspiraled curve
R..
Unspiraled circular curves are widened at the inside edge
of the pavement and approached by some sort of easement
curve, such as a spiral (Fig. 8–18). Theoretically, a perfectly
smooth transition will result by using a spiral which has a
throw equal to w and which terminates in a circular curve hav-
ing a radius R' equal to R-b-w. There are several methods
of determining the value of L.. The simplest is to assume that
Y = 4 times the throw (equation 5-20). Then,
L.=√24 R'o
(from equation 5-21)
(from equation 5-6)
Lis
2 R'
ہوۓ
234
CURVE PROBLEMS IN HIGHWAY DESIGN
Example.—Given: D。=10°; ž½b=12 ft; w=2 ft.
Ls=√24X558.958×2=163.8 ft, and ▲=8.395°
On a very sharp curve having a large value of ▲, the approxi-
mation Y = 40 may not give sufficient accuracy. In this case,
calculate the coefficient M (Art. 5-11) and determine ▲ from
Table XIII. Thus, in the preceding example,
M
=
R'+w
R'
Ls
560.96
558.96
From Table XIII, by interpolation, A = 8.397°. Then, from
formula 5-6, L=163.8 ft as before. (In this example the
Or, from Table XII,
two methods agree.)
=
= 1.003578
2.0
0.01221
163.8 ft
The method of Fig. 8-18 leaves a strip of ineffective pave-
ment at the outside edge near the T.C. (see Fig. 8-11). In
order to compensate for this, the outside edge may be inde-
pendently spiraled. This produces a layout in which the
survey center-line is a simple curve but the edges of the pave-
ment are true spirals. (The State of Indiana used this practice
for many years on curves having D=5° and greater.) One
method of selecting the spirals is by using the following pro-
cedure, illustrated with the aid of Fig. 8–19:
1. Choose a value for the throw o of the outside spiral.
In practice, o is usually about w.
2. Calculate L. (and ▲) for the outside spiral, as previously
described, using R' =R+b−0.
3. Calculate L, (and ▲) for the inside spiral, using a throw
equal to o+w and a radius R' = R— žb—0—w.
4. Examine the two spiral lengths and, if necessary, adjust
the value of o (and possibly of w) to give more suitable
spirals.
When the inner and outer edges are separately spiraled,
the S.C.'s of the two spirals will not be opposite each other.
Moreover, the inside spiral must be longer than that on the
outside edge, because of the greater throw (see Prob. 8–11).
HORIZONTAL ALIGNMENT
235
źb-
žb I.S.
w.
T.S.

O+W
£
T.C.
R
+
S.C.
=%
S.C.
k—b+w⋅
Fig. 8-19. True edge spirals on unspiraled curve
An objection to using a spiral for the widening transition
to unspiraled curves-aside from the fact that each combina-
tion of R, b, and w requires a different spiral-is that only one-
half the desired widening is attained at the T.C. of the circular
arc. Furthermore, if the circular curve is short, there may
not be distance enough beyond the T.C. in which to reach full
widening unless an exceedingly short spiral is used. Accord-
ingly, most State highway departments that have not yet
adopted center-line spiraling use transitions which reach full
widening on the inside edge at some relatively small fixed
distance beyond the T.C., regardless of the values of R, b, and
The length of the transition is also a fixed distance,
although some States vary the length with the degree of curve
according to a rule-of-thumb procedure. The edge transition
itself cannot be a spiral; usually it is a curve approximating a
parabola. On pavement work, the eye adjustment of the
forms required to produce a smooth curve near the point of
maximum widening is likely to be quite extensive. This
matter is treated in greater detail in Art. 8–19.
The voluminous tables worked up by some State highway
departments for staking widening transitions and supereleva-
tion runoffs on unspiraled curves-in contrast to the simple
procedures based upon Figs. 8-12, 8-15, and 8-16—are them-
selves strong arguments for adoption of center-line spiraling.
236
CURVE PROBLEMS IN HIGHWAY DESIGN
8-19. Edge Lengths. It is often necessary to determine the
curved length of the inner edge or outer edge of the pavement
or of a line a certain distance from either edge. This problem
occurs in connection with estimating the length of curb or of
guard rail and in staking an offset curve parallel to the
center line.
Unwidened circular arcs present no special difficulty. This
form of the problem is treated in detail in Art. 2–15.
In the case of a circular arc on which inside widening is
accomplished by means of a true spiral (Fig. 8–18), the length
of each spiraled edge is found from the basic spiral-length
formula, L. =2 ▲ R' (equation 5–6), ▲ being in radians and R'
being the radius of the fully widened edge, i.e., R' = R—≥b-w.
The total length of the curved inside edge between the T.S.
and S.T. of the spirals is (I+2 ▲) (R −žb−w), whereas the
length of the curved outside edge is I(R+16).
The modification of the preceding case, in which the true
spiral is replaced by a rule-of-thumb transition reaching full
widening a short distance beyond the T.C., does not admit of
any mathematically simple method of calculating edge lengths
accurately. Since high precision is rarely required, approxi-
mate methods usually suffice. Thus, in Fig. 8-20, full widen-

źb-
źb
A
a
h
T.C.
BI
Fig. 8-20
d.
R
źb
ing is attained at C, the distances a, d, h, and w being given.
The length of the curved edge AB is approximately equal to
HORIZONTAL ALIGNMENT
237
the chord distance, or
(approx.)
Also, the curve BC is approximately equal to the arc BC', or
BC = BC' = ( b-
· (R-16h) d (approx.)
Ꭱ
Curved edge lengths along an unwidened curve which is
spiraled along the center line depend on the method of aligning
the edges. If true spirals are used for the edge curves, their
lengths equal 2 ▲(R±‡b) and the total curved edge lengths
equal (1+2 ▲) (R±žb). However, the edge spirals do not
quite begin, or end, on radial lines through the T.S. and S.C.
Moreover, the pavement width is not exactly uniform between
those points. Simplicity is gained and the effects of true
spirals are closely approximated by keeping a uniform width
of pavement. That is, the edges of the pavement are on
radial offsets from the center line at distances of b(as in
Fig. 8-15, but without the widening).
formula 5-7, 8=
The lengths of the resulting edge curves between points
radially opposite the T.S. and S.C. are found as follows:
In Fig. 5-3, dl=rds, and the corresponding relations for the
inside and outside edges are
h2
AB=a+za
dl;= (r-3b) dô
and dlo = (r+b)dô
i
Consequently, dl;= dl-1bd8 and dl.=dl+3bdd. But from
2
=
5-7, 8= (2) A, from which đô
stitution, therefore,
By integration,
and
dli = dl —
2Aldl
L2
baldl
L2
1₁ = 1 - (3-4) (12)²
2
+ (32-4) (12.) ²
(8-23)
For the complete edge curve, l=Ls, li=Li, and l。=L。.
l=l+
baldl
L2
and dl。=dl+
Lis
After sub-
(8-22)
238
CURVE PROBLEMS IN HIGHWAY DESIGN
Consequently (Fig. 8-21),
and
and
in which A is in radians.
If the pavement is widened on the inside by the method
shown in Fig. 8-15, an analysis similar to the foregoing yields
L₁ =L, A(b+}w)
(8-26)
Where the widening is divided equally between the inside
and outside lanes, as in Fig. 8-16, the edge lengths are given by
Li=Ls-A(b+3w)
Lo=Ls+A(3b+} w)
b(or P)
Li=L.-1bs
Lo=L₂+&bs
C₁
8-20. Staking Offset Curve Parallel to a Spiral.-An offset
curve parallel to a spiral may be staked either by deflection
angles or by offsets (see Chapter 5). In either method it is
necessary to find the chords to be taped. To illustrate the
theory involved, Fig. 8-21 shows a center line spiral L, divided
17.5 81
TS. δι
1
C₂
82
Δ
L。
Ls
C3
83-
(8-24)
(8-25)
Fig. 8-21. Offset curve parallel to spiral
(8-27)
(8-28)

S.C.
To P.I
into three equal parts. It is required to stake the parallel
offset curve L, by setting points radially opposite the corre-
sponding points on the center line. In contrast to the analo-
gous problem at a circular curve (Art. 2-15), the chords along
the offset curve are not the same length because of the
increasing rate of curvature along the spiral and the con-
sequent increase in the values of d.
HORIZONTAL ALIGNMENT
239
If the offset curve is divided into n equal parts, for-
2
(1/2) (1)².
Ls
mula 8-23 shows that c₁ == +
N
n
2Ls
Also C2
n
2
C1,
which may be reduced to c₂ = c₁₂+2(24) (4)²
η
2
S.
T.
2
Similarly, C3=C₂+2
etc. To generalize, sub-
stitute the offset p for b. The formulas applicable to parallel
offset curves outside or inside a center-line spiral then become
()();
2
C1
=
C2=C₁±2i
C3=C₂+2i
where the increment i equals
S. C.
L±i
n
Cn=Cn-1±2i
R
•
k
b
P
N
(57.296).
At w
1-24
I
C. S. L.
+
(8-29)

S. T.
Fig. 8-22. Pavement areas at spiraled curve
The actual chords to be taped will differ slightly from the
nominal chords (given by formulas 8-29) only near the end of
a long sharp spiral. Where this is the case, the chord cor-
rections may be found with the aid of Tables II and III.
If the center-line spiral has been staked and it becomes
necessary to set parallel edge curves, the required points may
240
CURVE PROBLEMS IN HIGHWAY DESIGN
be located by offsets (Art. 5-15). Each point is at a constant
offset p from the center-line stake and also at the computed
chord distance ahead of the offset stake previously set.
8-21. Pavement Areas on Curves.-The area of pavement
along any unwidened curve equals pavement width times
length of curve. For a circular curve, A=bL, where L is the
arc length between T.C. and C.T.
At an unwidened spiraled curve (see lightly-shaded portion
of Fig. 8--22),
A=b [2L+(I −2^)R]
where I and A are in radians; or
(8-30)
100 (I-2A)
A = b[2L6+
(8-30a)
D
where I and ▲ are in degrees. In other words, the presence of
spirals does not affect the basic relation for pavement area so
long as the pavement width is uniform.
If the pavement is widened as in Fig. 8-15, the total extra
area along the inside of a curve with equal spirals consists of
a curved strip of uniform width w and two equal curved strips
of variable width. From the osculating-circle principle (Fig.
5-4), the four heavily-shaded areas in Fig. 8-22 are approxi-
mately equal. Therefore, the total extra area due to wid-
ening is
AwwI (R-6—1w)
(8-31) Approx.
where I is in radians. If slightly higher precision is needed,
a theoretically correct relation for the area of the strip of
variable width (A) may be found by expressing dA as the
difference between two sectors. Then by integration,
Avw= ½wLs—▲(}bw+}w²)
(8-32)
where A is in radians. The second term in this formula is
usually small enough to be neglected at the flat curves used on
high-speed alignment. Omission of this term indicates that
the area approximates a triangle.
8-22. Sight Distances on Horizontal Curves.-Where a
building, wall, cut slope, or other obstruction is located at the
inside of a curve, the designer must consider the possible effect
of the obstruction on the sight distance. Fig. 8-23 shows this
HORIZONTAL ALIGNMENT
241
situation for the case in which the sight line AB is shorter
than the curve. The driver's eye at A is assumed to be at
the center of the inside lane. Although the chord AB is the
actual line of sight, the stopping sight distance S is taken as
the arc AB because this is the travel distance available for
stopping in order to avoid hitting an object at B.
II

M
Fig. 8-23. Sight distance on horizontal curve
SH
m
In the following analysis, R, D, and L refer to the center
of the inside lane, not to the survey center-line. If the nota-
tion on Fig. 8-23 is used, m = R vers a. But a:D=SH:100
and R is approximately equivalent to
Hence
=
R
28.65
5,730 SHD
D
200
m=
R
vers
Cos-1
5,730
D
B
R-m
Ꭱ
L(2SH-L)
8 R
Also,
(8-34)
Fig. 8-24 is a design chart (AASHO Policy) showing the
required middle ordinates, at various degrees of curve, needed
to satisfy the stopping sight distances given in Table 8-1.
Where the obstruction is a cut slope, the criteria for height of
eye and object (Art. 8-3) can be approximated by using a
height of 2.5 feet at the point where m is measured.
In case the sight distance is longer than the curve, the
following approximate formulas may be used:
J
(8-33)
(8-35) Approx.
!
242
CURVE PROBLEMS IN HIGHWAY DESIGN
30
25
D= Degree of Curve - Center Line of Inside Lane
15
20
10
= Max D when e = 0.10
=>
I
V = Design Speed mph
V=30 S (Stopping) = 200′(Measured along & Inside Lane)
V=40 S=275
V = 50
S=350
V=60
S = 475
V = 70 S = 600
25
30
10
15
20
m = Middle Ordinate from Center Line Inside Lane to Sight Obstruction-Feet
Fig. 8-24
200
250
300
3000
35
R = Radius-Center Line of Inside Lane - Feet
Instead of using the foregoing methods, the designer may
of little value except on long flat curves.
distances given in Table 8-3. However, this application is
These formulas may also be used with the passing sight
(8-36) Approx.
and
SH
L
4 Rm
+
21
L

VERTICAL ALIGNMENT
243
prefer to scale sight distances from the plans. Since high
precision is unnecessary, the procedure is to place a straight-
edge on the survey center-line (at the station for which the
sight distance is to be determined) and tangent to the obstruc-
tion. The sight distance is then taken as the difference
in stationing between the points where the straightedge inter-
sects the center line. When scaled, it is suggested (AASHO
Policy) that SH be recorded to the nearest 50 feet when less
than 1,000 feet and to the nearest 100 feet when greater than
1,000 feet.
h1
VERTICAL ALIGNMENT
8-23. Sight Distances at Crest Vertical Curves.-At a crest
vertical curve the sight distance is considered to be the hori-
zontal projection of the line of sight for the assumed condi-
tions. Fig. 8-25 shows the situation where the sight distance S
is less than the length L of the vertical curve, or S<L. The
distances hi and h₂, representing the height of the driver's eye
G₁ =
S₁
'G₂ =
AL
h₁• 800
52
S
L
Fig. 8-25. Sight distance on vertical curve, where S<L
h₂
2
=s² : (+4) ²
2

and the height of the object, respectively, are vertical offsets
to the tangent sight line. In the general case, hi h₂, and
the sight line is not parallel to the chord joining the ends of the
parabola.
From the rule of offsets in Art. 4-2 and formula 4–2,
(In sight distance formulas, A is used as a positive number
equal to the change in gradient from G₁ to G2.) Therefore,
244
CURVE PROBLEMS IN HIGHWAY DESIGN
When S<L,
S1=
h1
Similarly,
The result obtained by substituting S for S₁+S₂ and solving
for L is the following general relation:
AS2
200(√h₁+ √h2)²
1
In the AASHO Policy the values in the case of stopping
sight distance are h₁ = 4.5 feet and h₂ = 4 inches. In the case of
passing sight distance, h₁=h₂=4.5 feet. (See Art. 8-3.) Sub-
stituting these values in formula 8-37 gives the following
practical relations:
S₂ = N
When S<L,
(non-passing conditions) L=
G1
L=
200 L hi
A
200 L h₂
A
L=
A S2
1,457
A S2
3,600
(passing conditions)
(8-39)
Figure 8-26 shows the situation where S>L. In the general
case h₁#h₂ and the sight line ad is not parallel to the chord
joining the ends of the parabola.
The problem is to find the slope of the sight line that will
make the distance ad a minimum. Let g represent the dif-
ference between the gradient of the sight line and the gradient
G₁. Then A-g will be the difference between the gradient of
V/G₂
L
2
L
(8-37)
(8-38)

·S-
Fig. 8-26. Sight distance on vertical curve, where S>L
hz
VERTICAL ALIGNMENT
245
the sight line and the gradient G2. Use is also made of the
following property of the parabola: If a tangent to the para-
bola is drawn between the main tangents, the horizontal pro-
jection of the intercept cut off on this new tangent by the main
tangents is equal to one-half the horizontal projection of the
long chord of the parabola; that is, the horizontal projection.
of bc is equal to L.
By definition, S equals the sum of the horizontal projections
of the distances ab, bc, and cd. Consequently, the general
expression for sight distance where S>L is
S=
-***
dS
dg
Solving for g gives
dS
For the sight distance to be a minimum, =0, or
dg
100 hi. L. 100 h₂
+ 2+
g 2 A-g
g
100 hi
g²
+
}
100 h2
(A—g)²
Ahi h₂-h₁ A
h₂-hi
=0
When S>L,
(non-passing conditions) L=2S-1,457
A
(A)
The result of substituting the value of g from equation B
in equation A and solving for L is the following general rela-
tion:
L=2S_3,600
A
(B)
200(√h₁+ √h₂)².
When S>L, L=2S-
(8-40)
A
For the values of h₁ and h₂ in the AASHO Policy the practi-
cal relations are:
(8-41)
(passing conditions)
(8-42)
The light solid lines in Fig. 8–27 (AASHO Policy) show
minimum lengths of crest vertical curves needed to provide
stopping sight distances at various design speeds. The value
of K, or the length of curve required to effect a 1% change in
246
CURVE PROBLEMS IN HIGHWAY DESIGN
CER 2 = 2 G
16
15
14
13
A = Algebraic Difference in Grades-Per Cent
w
12
∞
7
<O
LM
་
2
200
K = 27.5-
V=30 mph \K = 28³
V=Design Speed = 50 mph K = 80-
K= 84.1
resies
VIA.
400
V=40 mph K=
K=51.9
600
V=60 mph K = 150-
K = 155
L
Drainage Maximum K=143-
V=70 mph K = 240¬
K=247.37
800
1,000
1,200
Minimum Length of Vertical Curve-Feet
Fig. 8-27. Design controls for crest vertical curves
K===
or L= K⚫A
Calculated Value
Rounded Design Control
Drainage Maximum for Curbed Pavements
1,400
1,600
1,800
2,000

VERTICAL ALIGNMENT
247
}
A (Art. 4-2), is a simple expression of the design control. For
convenience, the theoretical values of K required to fit the
stopping distances in Table 8-1 have been rounded to integral
values; lengths of vertical curves computed from the rounded
values of K are plotted as the heavy solid lines in Fig. 8-27.
Where S>L the theoretical minimum lengths become zero
for small values of A because the sight line passes over the
crest of the vertical curve. However, good practice in design
calls for inserting a vertical curve at all changes in vertical
alignment. As an approximation of current practice, the
minimum length of vertical curve is taken as about 3 times the
design speed. This approximation is represented by the heavy
vertical lines at the lower left of Fig. 8-27.
There has been concern over the effect on the required
lengths of crest vertical curves caused by the trend toward
lower vehicles. Research suggests the advisability of reducing
the height of eye from 4.5 feet to slightly less than 4 feet.25
Such a reduction requires about a 10 percent increase in the
crest vertical curve lengths given by Fig. 8-27. However,
theoretical reductions in both stopping and passing sight dis-
tances caused by decrease in eye height are offset by improve-
ments in vehicle performance. Considering all factors, no
change in design standards for crest vertical curves appears
to be warranted.26
Drainage requirements may affect the maximum lengths of
vertical curves. If pavements are curbed, experience indi-
cates the desirability of attaining a minimum longitudinal
grade of 0.35% at a point about 50 feet from the crest. This
corresponds to a K-value of 143 feet for a change of 1 per cent
in A; the resulting line is plotted in Fig. 8-27 as the drainage
maximum. Special attention to drainage should be given for
combinations below and to the right of this line.
Some attempts have been made to introduce headlight sight
distance as a design control for stopping sight distance at crest
vertical curves, the height of headlight h₁ being taken as 2.5
feet and the height of object h₂ as 4 inches or more. Obviously
these conditions would demand much longer vertical curves,
but this requirement is considered to be unnecessary in view
of the lower running speeds used in night driving.
Although formulas 8-39 and 8-42 for passing sight distance
248
CURVE PROBLEMS IN HIGHWAY DESIGN
are contained in the AASHO Policy, there is no design chart
comparable to Fig. 8-27. (This is in contrast to the original
1940 Policy.20) Because of the high construction cost where
crest cuts are made, it is ordinarily impracticable to provide
the much longer vertical curves needed for passing.
8-24. Sight Distances at Sag Vertical Curves.-There is
no generally accepted basis for establishing the lengths of sag
vertical curves. Four criteria have been used: (1) headlight-
beam distance, (2) rider comfort, (3) drainage control, and
(4) general appearance.

2.5'
Ź
VIES
Fig. 8-28. Headlight beam distance, where S<L
S
只是​一
​އ

必
​$12.5 124.
MMMM
When S>L,
L
Fig. 8-29. Headlight beam distance, where S>L
S
Headlight-beam distance, as used by the Pennsylvania
Turnpike Commission¹, is represented by Figs. 8-28 and 8-29.
With the aid of these figures it is not difficult to derive the
practical relations, which are:
When S<L,
L=
A S2
500+3.5 S
L=2S-
+
500+3.5 S
A
(8-43)
(8-44)
VERTICAL ALIGNMENT
249
16
15
14
13
A = Algebraic Difference in Grades-Per Cent
THE RESTRICTIONS
w
12
=
G
OO
CO
LO
--
2
V = 30 mph-
200
K = 33.3
K = 35
1
V = 40 mph-
V=30 mph.
400
V = 40 mph
K = 50
X
V=50 mph
K=51.7
V=Design Speed 50 mph
S<L
K = 7C
K = 71.0
V = 60 mph´ ``K = 100
V=60 mph
V=70 mph-
K=104.33
V = 70 mph K = 140 K=138.5-
ܠܚܝ.
Drainage Maximum K=143-
K==/
or L=K.A
Headlight Sight Distance-Calculated
Headlight Sight Distance-Rounded Design Control
Comfort Control
Drainage Maximum for Curbed Pavements
1,400
1,600
600
800
1,000
1,200
L = Minimum Length of Vertical Curve-Feet
Fig. 8-30. Design controls for sag vertical curves
1,800
2,000

250
CURVE PROBLEMS IN HIGHWAY DESIGN
The light solid lines in Fig. 8-30 (AASHO Policy) show
lengths of sag vertical curves conforming to the preceding
formulas, in which values of S are the stopping distances in
Table 8-1.
When a vehicle is traversing a sag vertical curve, centrifugal
and gravitational force act in the same direction. As a result
there is some discomfort when the speed is high. The effect
is not easily evaluated, but limited attempts at its measure-
ment have led to the general conclusion (ASSHO Policy) that
riding on sag vertical curves is comfortable when the centri-
petal acceleration does not exceed 1 ft per sec². A mathe-
matical approximation of this criterion can be derived with
the aid of Fig. 8-31.
A
O
8
·L·
or L:R:
AB:OA=BC:VB
AR
V2
But
Thus, L=
100
R
and converting v to V in mph gives
A V2
46.5
L=
1
1
ic
Fig. 8-31
Because of the small gradients used in practice, the curve
AB may be taken as a circular arc and the distance VB as
approximately L. In the similar triangles AOB and CVB,
S
|

B
ALL
2002 (approximately)
1 ft per sec². Substituting for R
(8-45)
Lengths of sag vertical curves satisfying this comfort factor
are shown in Fig. 8-30 by the lines composed of short, dashes.
VERTICAL ALIGNMENT
251
The criterion for drainage is the same as on a crest vertical
curve; i.e., K should not exceed 143 feet for a change of 1 per
cent in A. This criterion is plotted in Fig. 8-30 as the drainage
maximum. It should be observed that the maximum length
for drainage on a sag curve exceeds the minimum lengths set
by the other criteria—a fortunate circumstance.
Several rule-of-thumb relationships have been used to insure
that a sag curve will have satisfactory appearance. One of
the simplest requires that L be at least 100 A. In Fig. 8-30
this corresponds to a design speed of 60 mph, based on head-
light-beam distance.
Of the four criteria for the minimum length of a sag vertical
curve, that based on headlight-beam distance appears to be
the most logical. This is the conclusion in the A ASHO Policy,
in which the K-values have been rounded off for convenience
in design, as represented by the heavy solid lines in Fig. 8-30.
8-25. Sight Distances at Interchanges.-Sight distances at
interchanges should be at least as long as the stopping dis-
tances listed in Table 8-1. When a sag vertical curve occurs
at an underpass, the overhead structure may shorten the sight
distance otherwise obtainable. However, if the length of the
sag curve conforms with that in Fig. 8-30, the structure does
not shorten the sight distance below the minimum required for
stopping. This is true even though the sight distance is
measured from a height of eye of 6 feet (truck driver) to a
height of object of 1.5 feet (tail light) and the vertical clearance
is the recommended minimum of 14 feet. The practical rela-
tions (AASIO Policy) corresponding to these conditions are:
When S<L,
A S²
L=
8,200
(8-46)
L=2S_8,200
A
When S>L,
(8-47)
In the case of dual highways it may be desirable to check
the distance available for passing at an undercrossing without
ramps. This can be done either by using the preceding
formulas or by scaling from the profile.
Limited sight distance is more likely to occur where an
interchange is located at a horizontal curve. Ordinarily, the
252
CURVE PROBLEMS IN HIGHWAY DESIGN
lateral clearance to bridge rails at an overpass or to abut-
ments at an underpass is not enough to permit use of maximum
curvature for the design speed. The formulas for SH in Art.
8-22 can be used in analyzing this situation.
DESIGN PRINCIPLES
Exact adherence to the specific controls outlined in the
foregoing articles will not guarantee attainment of the best
location. Experience, judgment, and the observance of
recognized principles of good design are also necessary. When
the geometric controls are applied, certain general principles
should also be observed.
8–26. General Controls for Horizontal Alignment.—Impor-
tant principles relating to horizontal alignment are as follows:
1. Alignment between the location control points (see Art.
9–3) should be as directional as possible. Long, flowing curves
fitted to the topography (Fig. 8-32) are better than long
tangents that slash through the terrain in an artificial manner.
2. Closely spaced short curves should be avoided. Such
unsightly kinks as broken-back and reverse curves may be
converted into more pleasing alignment in several ways. (See
Arts. 7-14 and 7-15.)
3. Small changes in direction should not be accomplished
by means of the sharpest curve permitted by the design speed.
The maximum degree of curve should only be used with large
central angles and at critical locations in general.
4. Curves, unless very flat, should be avoided on long, high
embankments.
5. Consistent alignment design should be attained (see
Art. 1-5). In case the topography requires a reduction in
design speed, the change-reflected in reduced sight distances,
increased curvature, and shorter distances between curves-
should be made gradually over a distance of several miles.
Moreover, there should be conspicuous warning signs to show
that such a change is in progress.
6. Horizontal and vertical alignment should be studied
together. Some general design controls for such study are
outlined in Art. 8-28.
254
CURVE PROBLEMS IN HIGHWAY DESIGN
8-27. General Controls for Vertical Alignment.-When
considering vertical alignment, some important principles are
the following:
1. A smooth-flowing profile with long vertical curves is
preferable to a profile with numerous breaks and short grades.
2. Care should be taken to avoid sag vertical curves on
comparatively straight horizontal alignment. These produce
“hidden dips” and are serious hazards, especially during pass-
ing maneuvers.
3. In general, long steep grades may be broken to the
advantage of traffic by placing the steepest grade at the
bottom of the ascent.
4. Steep grades should be reduced through important inter-
sections at grade, in order to minimize hazards to turning
traffic.
5. Unnatural and unsightly design should be avoided.
Among these defects are crest vertical curves on embankments
and sag vertical curves in cuts; broken-back vertical curves;
and the presence of numerous minor undulations in grade line
so located that they are visible to the driver.
6. On important highways carrying a large percentage of
commercial traffic, economic studies of motor-truck perform-
ance relative to grades should accompany the office work of
grade-line design. Some references 21,22,23 are included in the
bibliography at the end of this chapter.
8-28. Combination of Horizontal and Vertical Alignment.
Where there is a combination of horizontal and vertical curva-
ture the following principles should be kept in mind.
1. It is not essential to separate horizontal and vertical
curves; in general, the alignment is more natural and more
pleasing in appearance if they are combined-subject to the
limitations which follow in 2 and 3.
2. A change in horizontal alignment should preferably be
made at a sag vertical curve where the change in direction is
readily apparent to the driver. However, the horizontal curve
should be flat, in order to avoid the distorted appearance
caused by foreshortening.
DESIGN PRINCIPLES
255
3. If horizontal curvature at a crest vertical curve can-
not be avoided, the change in direction should precede the
change in profile.
4. Relatively small savings in cost of right-of-way or of
grading should not be an excuse for the insertion of short
sections of sub-standard design. On most locations in rural
areas the cost of these relatively permanent elements of the
highway is less than that of the pavement and other shorter-
lived appurtenances. Therefore, it is unwise to reduce the
built-in safety of some sections of a highway, and to invite
almost certain early obsolescence, for the sole purpose of
reducing the cost by a small percentage.
5. As noted in Art. 1-5, "the aim of good location should
be the attainment of consistent conditions with a proper
balance between curvature and grade." Balanced design,
everywhere consistent with the chosen design speed, is the
ideal to be constantly sought. Straight alignment obtained
at the expense of long, steep grades, or excessive curvature
inserted to follow the grade contour closely, are both poor
designs. The best design is a compromise in which safety,
economics, and aesthetics are sensibly blended.
PROBLEMS
8-1. Compute minimum stopping sight distance on level
pavement for any assigned design speed. Take values of t
and ƒ from Table 8–1.
8-2. Compute minimum passing sight distance on a two-
lane highway for any assigned design speed. Take passing
speed from Table 8--3 and interpolate in Table 8-2 for proper
values of.a, ti, t2, and dз.
8-3. Compute the side friction factor ƒ developed on the
following curves, each of which has favorable crown.
Is f
within the safe maximum recommended for curve design by
AASHO?
(a) R=5,000 ft; e=0.10; V=55 mph. Answer: 0.06; yes.
(b) Da=5°; e=0.06; V=60 mph. Answer: 0.15; no.
(c) R=1,200 ft; e=0.08; V=60 mph.
(d) Da=1°40′; e=0.08; V=60 mph.
256
CURVE PROBLEMS IN HIGHWAY DESIGN
•
8-4. If the actual coefficient of sliding friction between tires
and icy pavement is 0.05, theoretically within what range of
speeds could a vehicle be operated without sliding inward or
outward during icy conditions on the curves in Prob. 8-3?
Partial answers: (a) 61 to 106 mph; (b) 13 to 43½ mph.
8-5. Compute the stopping sight distance on straight icy
pavement when operating without chains (ƒ=0.05) at an
actual speed of 35 mph (take t from Table 8-1). Compare
answer with the range of speeds found in Prob. 8-4. What
conclusions appear to be justified (a) with regard to the super-
elevation designs in Prob. 8-3, and (b) with regard to maxi-
mum speed of operation without chains on icy pavement?
8-6. For any assigned design speed, verify:
(a) Values of R and D listed in Table 8-5.
(b) Values of R, D, and ƒ listed in Table 8-6.
(c) The fact that superelevation rates in Table 8-7 and Fig.
8-10 lie on parabolic curves.
(d) Values of L, listed in Table 8-8.
8-7. Compute design speeds at the following curves on the
Pennsylvania Turnpike consistent with values of ƒ recom-
mended by AASHO (Fig. 8-7 and Table 8-4). See Table
8-10 for the superelevation rates.
(a) Da=2°; (b) Da=3°; (c) Da=4°; (d) Da=5°. Partial
answers: (a) 78 mph; (b) 72 mph.
8-8. A typical section of the Massachusetts Turnpike is a
6-lane dual highway. On tangents, each inner lane has a 2
per cent cross slope toward the median; outer lanes, a 2 per
cent cross slope toward the shoulders. The radius of the
sharpest horizontal curve is 3,500 ft. If this section were
continued around the sharpest curve without rotating the
pavement, for what maximum speeds would the lanes be safe?
Take ƒ as recommended by AASHO in Fig. 8-7 and Table 8-4.
(Note: This curve is actually banked so that all lanes have a
2 per cent favorable cross slope.)
8-9. Compute the length of true spiral on the inside edge of
the following spiraled curves, assuming that the inside lane is
widened as in Fig. 8–17:
DESIGN PRINCIPLES
257
(a) Da=5°; b=10 ft; w=2 ft; L, on
2 ft; Ls on
(b) R=500 ft; 3b=10 ft; w=2 ft; L, on
(c) Da=13°; b=11 ft; w= 2 ft; L, on
(d) R=800 ft; b=10 ft; w=2.5 ft; Ls on
Partial answers: (a) 378.9 ft; (b) 250.1 ft.
&=300 ft.
¢=200 ft.
=210 ft.
(a) Da=5°; b=10 ft; w=2 ft.
(b) R=500 ft; b=10 ft; w=2 ft.
2 ft.
(c) Da=13°; }b=11 ft; w=2 ft.
(d) R=800 ft; b=10 ft; w=2.5 ft.
¢=230 ft.
8-10. Compute the length of true spiral on the inside edge
of the following unspiraled curves, assuming that the inside
lane is widened as in Fig. 8-18. Compare answers with those
in Prob. 8-9.
Answer: 233.3 ft.
Answer: 153.1 ft.
8-11. Compute the lengths of true spirals on the inside and
outside edges of the curves in Prob. 8–10, assuming that the
inside lane is widened as in Fig. 8-19. Use w as the throw of
the outside spiral. Compare answers with those in Prob. 8–10.
Partial answers: (a) inside 285.7 ft; outside 166.5 ft; (b) inside
187.4 ft; outside 110.5 ft.
8-12. Compute the edge lengths of the curves in Prob. 8–9
between points radially opposite the T.S. and S.C., assuming
that widening is omitted. Partial answers: (a) L¿=298.69 ft;
L=301.31 ft; (b) L;=198 ft; L,=202 ft.
8-13. Compute the inside-edge lengths of the curves in
Prob. 8-9 between points radially opposite the T.S. and S.C.,
assuming that the widening is applied as in Fig. 8-15. Partial
answers: (a) L;=298.52 ft; (b) L;=197.73 ft.
8-14. Compute the edge lengths of the curves in Prob. 8-9
between points radially opposite the T.S. and S.C., assuming
that the widening is divided equally between the lanes as in
Fig. 8–16. Partial answers: (a) L;=298.60 ft; Lo=301.40 ft;
(b) Li=197.87 ft; L,=202.13 ft.
8-15. Compute the chords needed to locate offset curves
parallel to the spirals in Prob. 8-9. (As a check, the sum of
the chords should equal the value found by formula 8-24 or
258
CURVE PROBLEMS IN HIGHWAY DESIGN
8-25, in which p is substituted for ½b.)
Spiral (a): n=6, p=60 ft outside.
Spiral (b): n=4, p=60 ft inside.
Spiral (c): n=7, p=150 ft outside.
Spiral (d): n=10, p=150 ft inside.
8-16. Compute the total pavement area in square yards
between the T.S. and S.T. of the following unwidened spiraled
curves:
(a) I=45°32′; Da=5°; b=10 ft; L=300 ft. Answer:
2,690.4 sq yd.
(b) I=51°17'; R=500 ft;
(c) I=54°46′; Da=13°;
(d) I=61°25'; R=800 ft;
b=10 ft; L=200 ft.
b=11 ft; L¸=210 ft.
b=10 ft; L=230 ft.
8-17. Compute the total extra pavement area in square
yards caused by widening the curves in Prob. 8-16 by the
method of Fig. 8-15. Use approximate formula 8-31; check
by exact method involving formula 8–32.
Answer: 200.4 sq yd approx.; 201.6 sq
Curve (a): w=2 ft.
yd exact.
Curve (c): w=2 ft.
Curve (b): w=2 ft.
Curve (d): w=2.5 ft.
8-18. Design standards for a 2-lane highway provide for
12-ft opposing traffic lanes and 8-ft shoulders. Bridge abut-
ments at underpasses are located 1 ft beyond the shoulders.
If an abutment occurs at the middle of a curve having a
center-line radius of 1,000 feet (no spirals),
Answer: 346 feet.
(a) What is SH if I equals 23°?
(b) What is SH if I equals 13°?
(c) What is the maximum recommended design speed in (a)?
(d) Would the maximum recommended design speed be
limited by the presence of the bridge abutment or by the side
friction factor developed, considering the superelevation rate
e is 0.10?
8-19. Determine the recommended minimum lengths of
crest and sag vertical curves where:
BIBLIOGRAPHY
259
(a) A=5%; V=50 mph. Answers: Crest 400 ft; sag 350 ft.
(b) A=4.75%; V = 70 mph.
8-20. At which of the four vertical curves in Prob. 8-19
would difficulty with surface drainage probably occur, assum-
ing the pavements to have edge curbs?
8-21. Compute the actual stopping sight distances at the
following vertical curves:
(a) Crest curve: L=600 ft; A=3%. Answer: 540 ft.
(b) Crest curve: L=600 ft; A =6%.
(c) Sag curve: L=600 ft; A=3%.
(d) Sag curve: L=600 ft; A=6%.
8-22. If the sag vertical curves in Prob. 8-19 occurred at
underpasses, what sight distances would be provided?
(a) Sag curve (a). Answer: 995 ft.
(b) Sag curve (b).
8-23. Principles of sight distances at vertical curves:
(a) Starting with formula A (Art. 8-23), derive formula 8-40.
(b) Verify the fact that the theoretical K-values in Figs.
8-27 and 8-30 are consistent with the rounded stopping sight
distances in Table 8-1.
(c) Derive formulas 8-43 and 8-44.
(d) Derive formulas 8-46 and 8-47.
BIBLIOGRAPHY
1. Noble, C.M., "Engineering Design of Superhigh-
ways," Proceedings, American Road Builders' Association, 1941,
pp. 183-217.
2. "A Policy on Geometric Design of Rural Highways,"
American Association of State Highway Officials, 1954.
3. Prisk, C.W., "Passing Practices on Rural Highways,"
Proceedings, Highway Research Board, 1941, pp. 366-378.
4. Leeming, J.J., "Road Curvature and Superelevation,"
Road Paper No. 7, Inst. C.E., Oct., 1942.
5. Stonex, K.A., "Car Control Factors and Their Mea-
surement," S.A.E. Journal, March, 1941.
260
CURVE PROBLEMS IN HIGHWAY DESIGN
6. Fox, M.L., "Relations Between Curvature and
Speed," Proceedings, Highway Research Board, 1937, pp. 202-
211.
7. Stonex, K.A., and Noble, C.M., "Curve Design and
Tests on the Pennsylvania Turnpike,” Proceedings, Highway
Research Board, 1940, pp. 429-451.
8. Moyer, R.A., Bulletin 120, Iowa Engineering Experi-
ment Station, 1934.
9. Moyer, R.A., and Berry, D.S., "Marking Highway
Curves with Safe Speed Indications," Proceedings, Highway
Research Board, 1940, pp. 399-428.
10. Barnett, J., "Safe Side Friction Factors and Super-
elevation Design," Proceedings, Highway Research Board, 1936,
pp. 69-76.
11. Shortt, W.H., "A Practical Method for the Improve-
ment of Existing Railway-Curves," Proceedings, Inst. C.E.,
Vol. 176, 1909, pp. 97 ff.
12. Royal-Dawson, F.G., Elements of Curve Design, E.&
F.N. Spon, Ltd., London, 1932.
13. Smirnoff, M.V., "Analytical Method of Determin-
ing the Length of Transition Spiral," Transactions, ASCE,
Vol. 116, 1951, pp. 155-185.
14. Haile, E.R., Discussion on "The Modern Express
Highway," Transactions, ASCE, Vol. 102, 1937, pp. 1091-
1099.
15. Leeming, J.J., and Black, A.N., "Road Curvature
and Superelevation: Experiments on Comfort and Driving
Practice," Journal, Inst. Municipal and County Engineers,
Vol. LXXI, No. 5, Dec., 1944, pp. 137 ff.
16. Leeming, J.J., "The General Principles of Highway
Transition Curve Design," Transactions, ASCE, Vol. 113,
1948, pp. 868-896.
17. Warren, H.A., and Hazeldine, E.R., "Experimental
Transition Curves," Journal, Inst. Municipal and County
Engineers, Vol. LXV, No. 21, March, 1939, pp. 1012 ff.
18. Welty, W.R.,. "A Method for Studying the Paths
of Motor Vehicles on Curves," Bureau of Highway Traffic,
Yale University, New Haven, Conn., 1947.
19. Bruce, A.G., and Brown, R.D., "The Trend of Highway
Design," Public Roads, Vol. 8, No. 1, March, 1927, p. 8.
BIBLIOGRAPHY
261
20. "A Policy on Sight Distance for Highways," American
Association of State Highway Officials, 1940.
21. Saal, C.C., "Hill Climbing Ability of Motor Trucks,"
Public Roads, Vol. 23, No. 3, May, 1942, pp. 33-54.
22. Taragin, A., "Effect of Length of Grade on Speed
of Motor Vehicles," Proceedings, Highway Research Board,
1945, pp. 342-353.
23. "Time and Gasoline Consumption in Motor Truck Oper-
ation," Research Report No. 9-A, Highway Research Board,
1950.
24. "Highway Curves and Test Track Design," Bulletin
149, Highway Research Board, 1957.
25. Lee, C. E., "Driver Eye Height and Related Highway-
Design Features," Proceedings, Highway Research Board, 1960,
pp. 46-60.
26. Stonex, K.A., Loutzenheiser, D.W., and Haile, E.R.Jr.,
"Driver Eye Height and Vehicle Performance in Relation to
Crest Sight Distance," Bulletin 195, Highway Research Board,
1958, pp. 1-8.
CHAPTER 9
RAILROAD SURVEYS
9-1. Foreword. The purpose of this chapter is to present
a few examples of curve and earthwork theory and surveying
procedures applied to the field of railway surveying. More
detailed descriptions of some of the subjects, particularly
track layout and maintenance (turnouts, connecting tracks,
string lining, etc.), will be found in railway surveying hand-
books, notably Railroad Curves and Earthwork, by Allen, or
Field Engineering, by Searles, Ives, and Kissam.
It is essential that Chapter 1 be restudied in connection
with the following description of survey methods and paper-
location procedure. Otherwise, much of the material in that
chapter-consisting of basic considerations more important
than the technical details to be presented-would have to be
repeated.
Fig. 9-1 shows an example of railroad location in the Rocky
Mountains near Blacktail, Montana. The pictured alignment
includes a simple 8° curve with a total central angle of 152°40′
and a total length, including two 180-foot spirals, of 2,088 feet.
On the adjoining tangents the grade is 1.80%; on the curve
it is compensated 0.04% per degree, and the actual grade is
1.48%. Shown is a 106-car freight train powered by a 5,400-
horsepower diesel locomotive at the head and a 4,050-horse-
power helper at the rear.
9-2. Reconnaissance.-Typically, the conception of a
project is followed first by a careful study of the best available
maps and then by field reconnaissance of the terrain between
the proposed termini. In the early years of railroad expansion
in the United States, it was often necessary to reconnoiter
relatively unexplored regions without the aid of maps. Direc-
tions were determined by pocket compass; elevations, by
barometer; slopes, by clinometer; and distances, by pedometer
or by timing the gait of saddle horses. Though such devices
are still useful at times, they have been made nearly obsolete
by the growing file of good maps and by improvements in
262
264
RAILROAD SURVEYS
the science of photogrammetry. In fact, it is becoming
increasingly apparent that, except for minor projects, aerial-
surveying methods may almost entirely supplant the older
methods of field reconnaissance, and perhaps of the pre-
liminary survey as well. Several particularly valuable types
of information that may be obtained more readily from aerial
photographs than from former ground methods are noted in
Chapter 12. See also Reference 7 in Art. 9–20.
9-3. Location.Controls.-The proposed railway must pass
through or near certain controlling points. Some of the
primary controls will have been fixed in the conception of the
project; others, including most of the secondary ones, will be
revealed by office studies of the maps and by whatever field
reconnaissance is needed to verify questionable points.
Typical location controls include:
Primary Controls
The termini.
Important intermediate traffic centers.
Unique mountain passes, tunnel sites, or major stream
crossings.
Secondary Controls
Minor intermediate markets or production centers.
Water courses.
Crossings of existing railroads or important highways.
Swampy areas.
Areas subject to snow or rock slides.
Areas involving costly land damages.
Topography in general, as it affects the economical attain
ment of desirable grades and curvature.
9-4. Organizing the Field Work.-Prior to starting exten-
sive field work, the locating engineer preferably should make
a field examination of the general route recommended in the
reconnaissance report. Before going into the field, he must
become acquainted with the general objectives of the project.
Moreover, he should be informed as to how much information
he may divulge to curious property owners along the route.
A great variety of useful purposes may be accomplished in
this preliminary field examination. Among these are:
RAILROAD SURVEYS
265
Arrangements for temporary office quarters, and for housing
and boarding the men.
Hiring of local help for axemen or other positions.
Permission to trespass on private property for survey pur-
poses, with whatever qualifications are imposed by the
owner.
Identification of convenient bench marks described in
Federal or State publications.
Determination of best access to the work by automobile
or other means of transportation.
Making of special notes regarding secondary controls.
Taken together, the information acquired should enable
the locating engineer to select the needed field and office
personnel and to determine the most efficient methods of per-
forming the various surveying operations.
9-5. Stadia Traverse.-In remote regions that are in-
adequately mapped and on projects for which aerial-surveying
methods are not justified, the reconnaissance may not definitely
disclose the best general route. In such cases it is wise to
run a stadia traverse along each of the possible locations.
Needed measurements are made as rapidly as possible,
consistent with the required accuracy. Transit stations are
far apart. Single deflection angles need be read no closer
than the nearest 10 minutes, though it is advisable to check
the resulting bearings by compass in order to guard against
blunders. A few intermediate shots along the traverse line
may be needed to give data for plotting a profile of the
traverse. Only enough side shots (supplemented by sketches)
need be taken to give the approximate positions and elevations
of the secondary controls.
Especially good judgment is needed in conducting this
type of stadia survey; if the survey is intrusted to inexperi-
enced personnel, it usually reverts to an unnecessarily detailed
and time-consuming topographical survey.
The resulting maps, plotted by protractor and scale, really
serve as high-grade reconnaissance; if the work is well done,
they permit the definite discard of certain routes and go far
266
RAILROAD SURVEYS
toward fixing closely the best general route to be followed in
the more precise preliminary surveys.
9-6. Transit-and-Tape Traverse.-Whenever the recon-
naissance narrows the location down to a fairly definite route,
it is good practice to run a careful transit-and-tape traverse,
called the "base line" or "P-line," joining the various loca-
tion controls. This traverse then serves not only as an accu-
rate framework for plotting the topographic details but also
as a convenient means of transferring the paper location to the
ground and checking its accuracy.
The P-line is a continuous deflection-angle traverse located
as close as convenient to the expected position of the final
location. Transit stations are marked by substantial tacked
hubs, which are driven flush, carefully referenced, and
identified by guard stakes. Deflection angles are at least
doubled, the telescope being reversed on the second angle in
order to eliminate instrumental errors. To guard against
blunders in reading angles the magnetic bearing of the forward
line should be compared with the calculated bearing before
leaving any transit station. If the survey is a long one, a
traverse bearing should be checked independently every 10
miles or so by means of a sun or star observation for azimuth.
Horizontal distances are measured with a 100-foot (or
longer) steel tape, usually being recorded to the nearest tenth
of a foot, but sometimes to hundredths. Station stakes are
set at regular 100-ft stations, at intermediate plus points
where profile breaks occur, and, possibly, at fence lines and
highways crossing the center line. It is not necessary to
center all station stakes, since they serve only for profile
leveling and cross-sectioning. The method described in Art.
2-11 is invariably close enough for the purpose. However,
chaining from pin to pin, not from stake to stake, is always
advisable.
9-7. Levels.—If the survey is over terrain permitting the
transit party to make rapid progress, it is often worth while
to start two two-man level parties at the time at which the
transit party begins its work. One, the bench level party,
carries the elevations from the nearest bench mark and sets
additional bench marks every 1,000 feet or so along the
RAILROAD SURVEYS
267
general route. It is not necessary for this party to follow
the exact path taken by the transit party; swampy areas and
steep slopes, for example, should be by-passed wherever
possible. In fact, the best plan is for the bench level party
to work ahead of the transit party along a general route
specified by the locating engineer. This party can then close
its levels back to the starting point each half-day without
falling behind the transit party.
The second level party, the profile level party, takes its
initial backsight on a bench mark established near station P
0+00 by the bench level party. It follows immediately
behind the transit party, taking rod readings to tenths of a
foot on the ground at every station stake and intermediate plus
point. Elevations are carried along through turning points
on which readings are made to hundredths. In proceeding
along the line, a check is made on bench marks previously set
by the bench level party. Thus, there is no need for the
profile level party to close its levels back; the party is always
close to the transit party, where the locating engineer fre-
quently needs the information on elevations in planning the
position of forward transit stations.
In following the dual leveling-party scheme, it is advisable
to alternate rodmen every half-day; this practice reduces
delays to the profile leveling party in searching for bench
marks established by the bench level party.
Bench-mark elevations should be adjusted as required by
the closing errors. Closures of bench levels should be less
than 0.05√M, where M is the distance between bench marks
in miles. Another criterion sometimes used is 0.01√S, where
S is the total number of set-ups in a closed circuit between
bench marks.
Bench-mark elevations as determined by the profile level
party should be compared with their adjusted values each
day. The adjusted elevations should be noted in the field
book and used on all subsequent work.
9-8. Topography by Stadia.-The method to be used for
taking topography depends on the character of the terrain,
the scale of the map, and the selected contour interval. A
scale of 1 in. =400 ft, with 10-ft contours, is about the small-
268
RAILROAD SURVEYS
est useful combination. A better general combination is
1 in. = 200 ft, with 5-ft contours; a scale of 1 in. = 100 ft, with
5-ft contours, is also popular.
Under some circumstances the stadia method of taking
topography is the most efficient one. This may be the case
in open regions permitting unobscured sights, especially if a
wide strip of topography is required. In this method, no
station stakes are set between traverse stations, and profile
leveling is unnecessary. All field work (except bench leveling)
is done by one large party. It is good practice to have a
separate computer and field draftsman in the party, in order
that the notes may be reduced and plotted as the field work
progresses. This method requires skilled personnel and care-
ful supervision by the locating engineer.
9-9. Topography by Hand Level.-Wherever a narrow
strip of accurate topography is required through a region
covered with brush or timber, the "standard" railway-survey-
ing method of taking topography with hand level, rod, and
tape is almost essential. In this method the topography
party is supplied with the ground elevation at each station
stake, as determined by the profile leveling. The locations of
contours on lines at right angles to the survey center-line
are then determined by the following method:
A perpendicular to the traverse is first established at each
station either by estimation or with the aid of a pocket com-
pass, an optical square, or a cross staff. In timber or brush
the transverse lines are kept reasonably straight by ranging
through with three flags or range poles.
The location of the first regular contour on a transverse
line is determined by hand leveling from the known ground
elevation at the station stake. A forked stick cut for a 5-ft
height of eye is a convenient support for the hand level. To
illustrate the process, assume that the center-line ground
elevation is 673.2 and that the locations of 5-ft contours are
required along rising ground to one side of the center line.
The levelman, resting the hand level in the fork of the 5-ft
stick, directs the rodman out until the hand level reads 3.2 on
the rod. This reading locates the 675-ft contour, the distance
to which, say 36 ft, is measured with a metallic tape. If notes
RAILROAD SURVEYS
269
3 6
are kept (somewhat similar to cross-section leveling, Art. 6-7),
the entry is recorded as 575. The levelman then continues
out past the rodman until he reads 10.0 on the rod, and the
distance beyond the previous point, say 42 ft, is measured.
The corresponding entry would be 80. If the distances
between contours are too great for hand-level readings, inter-
mediate readings are taken at shorter distances by the same
step-by-step process until the desired contour is reached.
With a
The hand-level method is surprisingly accurate.
little experience, levels may be carried 400 ft from the center
line and checked back with an error of less than 0.5 ft. Since
each new cross line starts with a correct center-line elevation,
there is no cumulative error.
In wooded terrain the speed in taking topography is greater
by hand leveling than by any other method giving comparable
accuracy. Only a thin gap need be cut through brush. Trees
never need be cut, for the tape and sight line may be offset
around them by eye without introducing serious error, the
scale of the map being considered.
In addition to locating contours, the topography party
locates all buildings, property lines, highways, streams, rock
outcrops, and other physical features likely to affect the loca-
tion of the railroad.
Though some engineers prefer to record notes of contour
locations (supplemented by sketches in the field book), a
method which is usually more accurate and much faster is to
plot the topography in the field at once on special field sheets.
These are usually strips of cross-section drawing paper
mounted on a topographer's sketch board about 12″x18″ in
size. The sheets are prepared in advance by drawing the
survey center-line straight and continuous through traverse
stations, provided the angles are small (possibly less than 5º).
Where the deflection angles are larger, distortion of the plotted
topography is reduced by repeating the traverse station after
leaving a 1-inch gap in the center line. A still better method
-one that eliminates all distortion at large horizontal angles—
is to cut adjoining sheets on the proper bevel through the
plotted position of the transit station and mount them on the
sketch board so that the center line is an exact reproduction
of the traverse line. The lines on the cross-section paper are
270
RAILROAD SURVEYS
thus parallel and perpendicular to the center line, thereby
being more convenient for plotting in the field. The final step
in preparing the field sheets is to record the stations and eleva-
tions along one edge.
The particular advantage of the preceding method, aside
from eliminating voluminous notes, is that contours and all
other topographic details are plotted immediately in the field,
where the faithfulness of the reproduction is readily apparent.
Moreover, field plotting to the scale that is to be used on the
subsequent map gives the topographer a clear idea of the
degree of precision needed in the field work. He is better
able to decide whether cross lines at some stations might
logically be omitted, and whether contours along additional
lines (such as on lines parallel to the center line or on ridge lines
and valley lines making an angle with the center line) are
necessary in order to complete the topography of difficult
sections.
9-10. Plotting the Preliminary Map.-If the map is to
serve most efficiently for paper-location purposes, the con-
trol traverse should be plotted by coordinates. The survey
should be tied, if convenient, to a State-plane-coordinate
system or to some other official system having coordinated
monuments; otherwise, an origin of coordinates may be
assumed arbitrarily.
Stadia topography is plotted in the usual manner by pro-
tractor angles and distances scaled from the plotted transit
stations.
Hand-level topography taken on special field sheets may
be transferred to the map by scale or dividers. However,
this is a time-consuming method. When the field sheets are
to the scale of the preliminary map-a relation strongly
recommended-a much better way is to fit each section of
field sheet to the corresponding traverse line and to transfer
the plotted topography to the map directly through thin
carbon paper, using a well-sharpened, very hard pencil or a
slightly-rounded stylus. This process is rapid, inexpensive,
and accurate; it results in the exact reproduction of the field
sheets, free from the possibility of drafting-room errors.
RAILROAD SURVEYS
271
9-11. Paper-Location Procedure. The preliminary sur-
veys and subsequent office work result in a topographic map
of a strip of territory, varying possibly from 400 to 1,000 feet.
in width, in which the ultimate location is expected to lie.
Finding a satisfactory location having suitable curves.
and grades is not usually a difficult task; it is largely a technical
process involving patience and quite a bit of routine scaling.
and calculation. But finding the best location requires some-
thing more than drafting-room technique. As stated in
Art. 1–5, “To produce a harmonious balance between curva-
ture and grade, and to do it economically, requires that the
engineer possess broad experience, mature judgment, and a
thorough knowledge of the objectives of the project.”
It is obviously impossible to write a set of rules which, if
followed, will inevitably produce the best location. The set
which follows is merely a suggested office procedure that will
be principally of value to the novice; the intangible ingredients,
skill and judgment, grow with the locator's experience.
(NOTE: In the technique to be described, splines may be
substituted for thread and curve templates.)
1. Using the preliminary map, set small pins at the termini
and also near the intermediate controls. Stretch a fine
thread around the pins.
2. Examine the terrain along the thread line, and set addi-
tional pins (angle points) where they apparently are
needed between location controls.
3. Scale the elevations at the pins and the distances between
them, thus giving the approximate grades.
4. Place transparent circular-curve templates tangent to
the thread, allowing enough room between curves for
subsequent insertion of spirals. Shift pins and change
templates until a trial alignment is obtained which fits
the alignment specifications and appears to be reasonable
in so far as gradients and probable earthwork quantities
are concerned. It may be helpful to use the grade con-
tour as a guide (see Art. 11-5).
5. Pencil the trial alignment lightly but precisely, and
station it continuously along the tangents and curves
by stepping around it with dividers.
272
RAILROAD SURVEYS
6. Plot the ground profile of the trial alignment to an
exaggerated vertical scale.
7. Using pins and thread, establish a tentative grade line
on the profile which fits the specifications for grades
and appears to produce a reasonable balance of earth-
work. The circular-curve templates may be used for
vertical curves. In estimating earthwork balance from
center cuts and fills, make approximate allowance for
the fact that the graded roadbed is wider in cuts than
on fills and that a certain percentage will be added to the
fill quantities for shrinkage (see Art. 6-14).
8. Examine the trial alignment and grade line together. Go
over the alignment station by station, visualizing the
finished roadbed if built -as- indicated. The need for
drainage structures and the probable maintenance diffi-
culties dealing with drainage, slides, and snow drifting
should be examined. Then make whatever changes will
obviously improve operating and maintenance character-
istics without changing earthwork quantities appreciably.
9. Make an earthwork estimate, using the following column
headings:
Cubic Yards
Cut
Fill+*%
(*Insert suitable shrinkage factor-possibly from 10
to 15% for purposes of estimate)
Station
Center
Height
In the station column, enter in proper order each
100-ft station, the station and plus of each high or
low point, and the station and plus of each grade point.
Scale center heights from the profile at positions midway
between the points entered in the station column;
designate the values C or F. Take earthwork quantities
from the proper table of level sections (Table XVII, Part
III). Do not forget to reduce tabulated quantities for
fractional distances or to add the shrinkage percentage to
the fill quantities.
10. Enter the sub-totals for each increment of cut and fill.
If conditions appear to warrant striving for an approxi-
mate balance of quantities along certain grading sec-
RAILROAD SURVEYS
273
tions, see how close the balance comes. Observe also
the relative sizes of adjoining cuts and fills and the
approximate distances between their centers of gravity.
For convenience, note these numerical values on the
profile.
11. Re-examine the tentative alignment and grade line in
the light of the numerical values obtained in the first
earthwork estimate. Make any minor changes in the
alignment or the grade line, or both, which will reduce
the pay quantities and improve the balance and distribu-
tion of earthwork. Do not erase the first trial; merely
use a different style of line for the revised portions. Scale
new center heights along the revised portions and deter-
mine the new quantities. Preserve all discarded tabula-
tions, but identify them and mark them "void."
12. Repeat step 11 until the location appears to be the best
one possible.
The paper location finally established by the foregoing
process should not be accepted rigidly as final; the remarks
contained in Arts. 1-11 and 1-12 are pertinent in this respect.
Though the earthwork estimate is based upon level sections,
the errors introduced by transverse slopes tend to cancel. In
a long line, level-section quantities will usually be within 5
per cent of the true values found later by cross-sectioning.
9-12. The Location Survey. The principal purpose of the
location survey is to transfer the paper location, called the
"L-line," to the ground. This may be done quite accurately
if the preliminary map is based upon a transit-tape traverse
plotted by coordinates.
The first step is to scale the coordinates of the P.I.'s of the
L-line and from them to compute the bearings of the tangents
and the distances between P.I.'s. If the map is carefully
drawn to a scale of 1 in. = 100 ft, these computed bearings and
lengths are usually reliable to the nearest minute and foot.
The bearings and lengths are useful not only in making ties
to the P-line but also in checking the location field work.
The L-line should never be run by turning the calculated
angles and measuring the calculated distances continuously
from beginning to end. Instead, each L-line tangent should
274
RAILROAD SURVEYS
be tied independently to the P-line and run to a string inter-
section with the adjacent L-line tangent, at which point the
exact central angle of the curve is then measured.
An excellent tie exists wherever an L-line tangent crosses
the P-line. The P-line stationing of the cross-
ing can be scaled close enough for the purpose if
the angle of intersection is large; otherwise, it
may be computed by coordinates. A hub at the
intersection is then located by sighting and meas-
uring from the most convenient P-line hub, after
which the L-line tangent is projected in both
directions by setting up at the intersection hub
and turning off the calculated angle.

E
())
(6)
L-line tangents may sometimes be tied to the
P-line by right-angle offsets. Another method is
to produce a P-line course for a scaled or cal-
culated distance to an intersection with the
L-line. Considerable ingenuity and field experi-
ence are needed in establishing ties rapidly and
accurately.
©
Great care should be taken to insure the
straightness of the L-line tangents. The lines
should always be produced forward through
P.O.T.'s by double centering. Obstacles on the
tangents may be by-passed by the method
described in Art. 9-13.
|
D
(
@ & C
B
(2)
After the P.I.'s have been located and the
central angles have been measured, the curves
and spirals are run in as described in Part I.
Stationing is continuous along the tangents and
curves. Some minor adjustments in alignment
are usually made during or following the staking
A
of the curves (see Art. 1-12). These adjust-
ments may involve some of the special curve |
problems described in Chapter 7.
It is a good plan to run profile levels over
the staked L-line in order to compare the actual
ground profile with the paper-location ground profile. The
degree of "fit" is a measure of the accuracy of the topog-
raphy.
Fig. 9-2
RAILROAD SURVEYS
275
If it is suspected that the earthwork estimate may be in
error, owing possibly to the prevalence of side-hill sections,
the line may be cross-sectioned by the slope-staking method
described in Art. 6-7. The more accurate earthwork quanti-
ties may then justify some slight revisions in the grade line
without necessarily changing the alignment. Further analysis
of the earthwork distribution may be made by means of a
mass diagram (see Art. 6-19).
The location survey also includes ties to property lines
and existing improvements, as well as a variety of measure-
ments needed for the design of miscellaneous structures.
9-13. By-Passing Obstacles on Tangents.-Where location
tangents (or other straight survey lines) are produced through
woods, large trees often obstruct the line. To avoid felling
them without authorization, and also to save time, the best
recourse is to by-pass them by a small-angle deflection-angle
traverse.
Sta. Defl.
Fig. 9-2 and the accompanying notes illustrate the process.
Point A is a transit station (P.O.T.) on the location tangent.
It is assumed that large trees obstruct the line beyond A.
Consequently, a small deflection angle is turned to the right
Notes for Fig. 9–2
A
B
C
Ꭰ
E
Tang.
35' R
42' L
30' L
20' R
17' R
Angle
with
Tangent
35' R 640
7' L
37' L
Dist.
zero
400
420
17' L 238.82
Products
Angle X
Dist.
+22,400
-2,800
-15,540
-4,060
Alg. Sum
zero
Offset
from
Tangent
zero
+22,400 6.5 R
+19,600 5.7 R
+4,060 1.2 R
zero
276
RAILROAD SURVEYS
and a hub B is set at any convenient distance. At B it does
not prove possible to get back on the tangent; therefore, the
auxiliary traverse is continued through convenient openings
between the trees until point D is reached, after which the
tangent is resumed at E.
It is convenient to adopt a systematic form of notes which
will indicate the distance from the tangent as well as the
measurements needed to get back on line. In the form sug-
gested, for example, the algebraic sum of the products of
distances and angles in minutes must be zero up to a point on
the tangent. Moreover, since the reciprocal of the sine of 1
minute is 3,440, the offset at any point is found by dividing
the proper algebraic sum by that constant. The sign of any
product is determined by the angle that the course makes with
the tangent. An angle to the right gives a plus product;
one to the left, minus.
In the example given, it is assumed that a deflection angle
of 20′ R at D is a convenient direction toward the tangent.
The required distance to a P.O.T. at E is, therefore, 4,060÷17
=238.82 ft. At E the deflection required to place the line
of sight on the correct tangent produced is obviously 17′ R.
If the angle between any auxiliary course and the tangent
is kept below 1 or 2 degrees, a negligible error results from
assuming that the length along the auxiliary traverse is
the same as the distance along the tangent. In the foregoing
example the true tangent distance between A and E is only
0.06 ft shorter than the traverse distance of 1,698.82 ft. This
small correction is found quickly by slide rule, if required, by
summing the products of the distances and the versines of
the angles with the tangent.
It is not usual to move station stakes back on the correct
tangent. However, if required, it may be done by eye with
the aid of the offsets in the last column of the notes.
9-14. Construction Surveys.-It is not considered necessary
in a book of this kind to give detailed examples of the great
variety of measurements required on railroad construction.
These are best covered by reference to instructions and forms
supplied by railroads to their resident engineers. Briefly,
RAILROAD SURVEYS
277
the principal survey work related to new construction includes
the following operations:
1. Re-Establishing the Final Location
Checking and referencing key points, such as P.I.'s,
T.S.'s, S.T.'s, and occasional intermediate points on
long tangents, so that they are quickly available during
construction.
Resetting enough station stakes on curves to control
clearing of the right-of-way.
Checking the bench-mark elevations and setting con-
venient new bench marks in locations in which they are
not likely to be disturbed during construction.
2. Setting Construction Stakes
Cross-sectioning the line after clearing (and just ahead
of grading operations), together with setting slope stakes
wherever needed.
Line and grade stakes (and in some cases, batter boards)
for appurtenant structures, such as buildings, bridge
piers, culverts, and trestles.
Finishing stakes for completing cuts and fills to exact
grade; center stakes for track laying; grade stakes for
ballast and final rail profile.
Stakes for borrow pits; also cross-sectioning borrow
pits after stripping.
Stakes for right-of-way fences.
3. Making Periodic Quantity Measurements
Measurements, calculations, and estimates of work done
to serve as basis of monthly payments to contractor, as
well as for progress reports to headquarters.
4. Final Measurements
Final cross-sections for calculation of total grading
pay quantities.
"As-built" measurements of all work to serve as basis for
final payment, as well as for preparation of "record"
plans.
Monumenting curve points.
278
RAILROAD SURVEYS
5. Property Surveys
Making all measurements needed for preparing legal
descriptions of easements and of land acquired by pur-
chase or through condemnation proceedings.
Setting right-of-way monuments.
9-15. Superelevation. Fundamentally, superelevation.
theory is the same on railways as on highways. Fig. 8-2
and the equations developed in Art. 8-6 are valid for both
types of operation.
The equilibrium formula for superelevation of railroad
track* is
E=0.0007 V2D
(9-1)
in which E is the superelevation, in inches, of the outer rail.
This relation corresponds to formula 8-8, and is found by
substituting E÷59.5 for e. ("Standard gage" of track is 4 ft
8 inches, but E is measured with respect to center to center
of rails, i.e., 4 ft 11½ inches.)
According to the A.R.E.A. Manual:
"If it were possible to operate all classes of traffic at the same speed
on a curve, the ideal condition for smooth riding and minimum rail
wear would be obtained by elevating for equilibrium. However,
curved track must handle several classes of traffic operating at various
speeds, which results in slow trains causing excessive wear on the inside
rail and high-speed trains causing accelerated wear on the outside rail.
"Safety and comfort limit the speed with which a passenger train.
may negotiate a curve. Any speed which gives comfortable riding on
a curve is well within the limits of safety. Experience has shown that
the conventional baggage cars, passenger coaches, diners, and Pullman
cars will ride comfortably around a curve at a speed which will require
an elevation about 3 inches higher for equilibrium. Equipment
designed with large center bearings, roll stabilizers, and outboard
swing hangers can negotiate curves comfortably at greater than 3
inches unbalanced elevation because there is less car body roll. It is
suggested that where complete passenger trains are equipped with
cars utilizing the foregoing refinements that a lean test be made on
the equipment to determine the amount of body roll. If the roll
angle is less than 1°30', experiments indicate that cars can negotiate
curves comfortably at 44 inches unbalanced elevation.
"The inner rail should preferably be maintained at grade."
*1956 Manual, American Railway Engineering Association.

279
E
TABLE 9-1
EQUILIBRIUM ELEVATION FOR VARIOUS SPEEDS ON CURVES
V=Speed, mph
Degree
10 20 30
60
65
0.71
1.06 1.26
1.48
0.87
1.42
2.12
2.52
2.96
1.29
2.13
2.63 3.18
3.78
4.44
3.50
4.24 5.04
5.92
of
Curve
De
35
0°30' 0.04 0.14 0.32 0.43
1°00' 0.07 0.28 0.63
1°30' 0.11 0.42 0.95
2°00' 0.14 0.56 1.26
2°30' 0.18 0.70 1.58
3°00' 0.21 0.84 1.89
3°30' 0.25 0.98 2.21
4°00' 0.28 1.12 2.52
5°00' 0.35 | 1.40 3.15
6°00' 0.42 1.68 3.78
7°00′ 0.49 1.96 4.41
8°00' 0.56 2.24 5.04
9°00' 0.63 2.52 | 5.67
10°00′ 0.70 2.80 6.30
11°00′ | 0.77 3.08 6.93 9.43
12°00'
0.84 | 3.36
1.72
2.84
2.14 2.80 3.54 4.38 5.29 6.30 7.39
2.57 3.36 4.25 5.25 6.35 7.56 8.87
3.00 3.92 4.96 6.13 7.41 8.82 10.35
4.48 5.67 7.00
8.47 10.08
10.59
3.43
4.29 5.60 7.09 8.75
5.15 6.72 8.51
10.50
6.00 7.84 9.92
6.86 8.96 11.34
7.72 10.08
8.58 11.20
7.56 10.29
40
0.56
1.12
1.68
2.24
45
50
0.88
1.75
55
70
75
80
1.72
1.97
2.24
2.53
3.43 3.94
4.48
5.06
5.15
5.91
6.72
7.59
6.86
7.88 8.96 10.12 11.34
8.58
9.84
11.20
10.29
11.81
85
E, in inches =0.0007 V2D
90
95
100
2.84
3.16
3.50
5.67
6.32 7.00
8.51 9.48 10.50
12.46


280
RAILROAD SURVEYS
Using these recommendations, the A.R.E.A. formula for
superelevation based on maximum speed becomes
E=0.0007 (max. V)² D-3
Formula 9-2 is analogous to equation 8-11.
Table 9-1 gives the equilibrium elevation E for various
values of Dc. The 5-mph increments represent general prac-
tice for use on speed-limit signs.
9-16. Spirals.-In contrast to the situation with regard
to the use of spirals on highways (Art. 8-13), spirals have
been used on railroad track since about 1880. (A concise
history of the use of spirals is given in Proceedings, A.R.E.A.,
Vol. 40, 1939, pp. 172–174.)
(9-2)
As a result of long years of experience in operating over
spiraled superelevated curves, American railroads almost in-
variably base minimum spiral length upon the rate of rota-
tional change. For many years it has been the recommenda-
tion of the A.R.E.A. that spiral length be based upon attain-
ing superelevation across standard-gage track at a desirable
maximum rate of 1.25 inches per second when trains are
operated at maximum speed.
or
In a committee report* on "Spirals Required for High Speed.
Operation," it was found that some American railroads operat-
ing high-speed passenger trains used values higher than 1.25
inches per second, and other railroads used lower values.
However, this value was substantially verified by modern
practice, and it was concluded that there should be no change
in the recommendation. (This report also contains an inter-
esting tabulation of spiral practice on European high-speed.
lines.)
Based upon attaining superelevation at a rate of 1.25 inches
per second, the equation for desirable minimum spiral length is
Ls=1.17 EV
Un
L= && EV
88
75
*Proceedings, A.R.E.A., Vol. 39, 1938, pp. 497-507.
(9-3)
RAILROAD SURVEYS
281
where V is the maximum train speed in mph. This equation
is the same as 8-19, except for different notation.
As in the case of highways, the superelevation is run out
uniformly over the spiral. The slight vertical curves in the
outer rail at the beginning and end of the spiral are taken
care of automatically by the flexibility of the rail.
Track spirals are staked either by deflection angles or
by offsets. The basic theory is fully covered in Chapter 5.
Art. 5-15 contains practical suggestions for applying the
offset methods.
9–17. String Lining.-In spite of ballast and rail braces,
tracks on curves tend to creep slowly out of line. This creep-
ing is due principally to the unbalanced lateral forces caused
by operation at other than equilibrium speed. Other con-
tributing factors are rapid deceleration during emergency
stops and, possibly, temperature expansion and contraction.
Track once irregularly out of line becomes progressively worse,
owing to the variable impact produced by moving trains.
The trend toward higher train speeds in both freight and
passenger operation makes it more important than ever to
maintain curved track continuously in good alignment. This
can be done either by the deflection-angle method or by string
lining. The latter method has so many obvious advantages
that it is rapidly superseding the former method.
Briefly, string lining consists in shifting the track in or out
along the circular curve until equal middle ordinates are
obtained at equal chords. Theoretically, the chord used may
be of any length; but to obtain good control on main-line
track it should be between 50 and 80 feet. Many engineers
use a 62-ft chord. This is the value recommended by the
A.R.E.A.; it is a convenient length, and also produces the
useful relation that the degree of curve is numerically equal
to the middle ordinate in inches (see equation 2–31).
Equipment consists simply of a tape, a strong fish line or a
fine wire, and a scale for measuring the middle ordinates.
Some engineers use wooden or metal templates which are held
against the rail head; the string passes through holes or slots
a fixed distance from the rail. With such devices, it is neces-
282
RAILROAD SURVEYS
sary to deduct the fixed distance from the measured middle
ordinates, or to use a special scale with an offset zero point.
The procedure involves (1) preliminary field work, (2)
calculation, and (3) track shifting in accordance with the
approved calculations.
Preliminary field work
1. Locate the T.C. by sighting along the gage side of the
outer rail. Make a keel mark on the inside of the rail
head at this point and mark it sta. O on the web.
1 a
O
-1
2e_original
Revised

-C
d
Fig. 9-3
2. Mark sta. -1 similarly, 31 feet back from sta. O on the
tangent. Then mark stations, 1, 2, 3, etc. at 31-ft
intervals along the outer rail until the last station is
beyond the end of the curve.
3. Stretch the line taut between the keel marks at stations
-1 and 1. Measure and record the middle ordinate at
sta. 0. In similar manner, stretch the cord between
RAILROAD SURVEYS
283
stations 0 and 2 and measure the middle ordinate at
sta. 1. Continue this process until the middle ordinates
become zero.
Calculation is based upon four simple rules. The first comes
from the fact that the middle ordinate m is proportional to D.
100 I
Since D=- it follows that Em ∞ I. This relation may
1
L
be expressed by the following rule:
Rule 1. For any chord length the sum of the middle
ordinates on a curve between given tangents is constant.
The other rules come from Fig. 9-3, in which the solid
line represents the outer rail of curved track badly out of
line, and the dotted line shows the correct position. (The
scale is greatly exaggerated in order to make the relations
clear.)
The offset distance between the original and final positions
of the track at any station is called the throw. Track moved
outward in revising its position is given a positive throw;
for track moved inward, the throw is negative. Thus, the
throw at sta. 1 is negative and is numerically equal to the
distance ab.
The error at any station is found from the following relation:
error=original middle ordinate minus revised. middle ordinate
(algebraically). In the sketch the error at sta. 1 is ac-bd.
It is obvious that the first throw will occur at sta. 1. Since
the middle ordinates at successive stations are practically
parallel, the throw at sta. 1 is twice the error at sta. 0; both
have negative signs. This relation may be expressed by the
following rule:
Rule 2. At the first station at which a throw occurs, the
half-throw (½ t) equals the error at the preceding station.
The throw at sta. 2 is ef (negative sign); and for all practical
purposes ef=twice cd. The length cd may be written in the
following form:
-cd=(−1 ab)+(−1 ab)+[(ab+bc)—bd]
(1) (2)
(3)
(4)
284
RAILROAD SURVEYS
Term (1) is the half-throw at sta. 2.
Term (2) is the half-throw at sta. 1.
Term (3) is the error at sta. 0.
Term (4) is the error at sta. 1.
From the foregoing relation, the following rule may be
stated:
Middle Ordinates, in.
Rule 3. The half-throw at any station equals the half-
throw at the preceding station plus the algebraic sum of
the errors up to and including the preceding station.

5
10
Original
Trial 3
15
20
Stations at 31-ft intervals
Fig. 9–4
25
30
35
In solving a string-lining problem, it is helpful to plot a
graph of the measured middle ordinates. Fig. 9-4 shows such
a graph for a spiraled curve very badly out of line. The curve
is too flat near sta. 13 and too sharp near sta. 22. The original
middle ordinates were measured to tenths of inches at stations
31 feet apart; their values (with the decimal point omitted for
simplicity) are given in Table 9-2, col. (2).
For perfect alignment the middle ordinates on the circular
curve must be constant, and those on the spirals must change
uniformly. Moreover, the half-throw at the final station
must be zero; otherwise, the forward tangent will be shifted
parallel to itself by an amount equal to the full throw.
Trial 1 is shown in Table 9-2; it approximates trial 3, shown
in Fig. 9-4. The middle ordinate for a 4.4° curve between
stations 7 and 28 was assumed to be the average of the exist-
ing middle ordinates on the circular arc.
Col. (3) contains the revised middle ordinates. The middle
ordinates on the spirals were adjusted so that the sum of the
RAILROAD SURVEYS
285
Station
(1).
101234B ONATO IN?
5
10
11
Original
ԱՆ
18
(2)
0
4
14
6 41
** 27828
FORD FOREBBE #99 88*‡‡ ;*#*
16
29
36
40
48
40
12
40
13 36
14 38
15 41
47
16 42
44
23 25 -2
19 46
20
46
21 50
47
Revised
U
42
STRING-LINING CALCULATIONS
(3)
43
(4)
0 0
0
6-2
12 +2
19-34
Error
TRIAL 1
31 +5
44 +3
44
26 40 44
22 50 44 +6
23
48
24 44 44
25 41 44
44-2-21
17 44 44 0 -21
*#*#*2 ROS
+5←
38 +3 +3
44-4 - 1
44 +6
44
44 44
44 +4 +3
44 +4
TABLE 9-2

31 24
32 15
33 14
34
9
35
0
Sum 1231 1231
Sum of
Errors
44-4
+2
44 4 -2
44-8-10
44 -6 -16
44-3 -19
66403 +2
C
44+2-19
442-17
-3
| | │
(5) (6)
0
38+5
30 34 32+2
25-1
19
12+2
6 +3
0 0
2
LO
+++
0
31∞ ∞ ∞
+6
+6
Half-
Throw
-
C
44 +3-14-116
-2
M
G
B
0
0
2
2250
-5
1++++
10
-10
-7
-8
-5
00 LO
+1
+7
+9
+7
-3
-19
-5-137
-7-142
-7-149
-156
0-158
-1-158
Revised
4 -5 -159
-3-164
0 -167
0-167
W
(7)
TRIAL 2
Error
5-1
11 +3
18-2
24-1
44
44
44
44
(8)
0
0
44-4
1
11
-8-130
-2-138 44 +6
+2-140
+2-138
-136
-38
-59
-80
-99 44 +2-11
30+6+5
+9 +5
+5+14
+9+19
44 +3 +12+28
37 +4
44-4
44 +4
44
0+12 +40
+8+52
-8
4 +4+60
-4+64
·6-10 +60
-13+50
G
Sum of
Errors
44-2
44
0
44 +2 13
(9)
44 -3
0
0
-1.
I+A!++++
44
44
44 0
O
44 +6 -2
26-2
21-6
12 +2
7 +2
0
1231
44 +4 +8
44
44 +3 -8-17
-25
+4-27
15 +37
15 +22
1+++++11 +++i
114 2
+5
39 +4 +3
33 +1
+4
1
Half-
Throw
L
(10)
0
+2
S
+8-15
+7
-6
-23
1111
2123− +CONO
-3
+4
+++
-4
+6
-2 +2
0
0
0 0
286
TABLE 9-2-(Continued)
Station
(11)
-1
0
HORED ONOG IS
1
2
3
4
6
8
10
11
7-3
13+1
18
24-1
30+6+1
35 +6
41-1
+7-13
+6
-6
44 +4 +10
9 44 +3 +13 | +10
44
+13 +23
97022 22*** ***** **
Revised
21
น
23
12 44
24
25
26
13
44
14
44
15 44
16
TRIAL 3
(12) | (13)
0 0
0
0
Error
P
44-4
111
-4
1
S
RAILROAD SURVEYS
Sum of
Errors
+9+36
44
44
+5 +45
-3 +50
-9+47
-12 +38
-14+26
0-14+12
44 +2 -12
-10-14
-7-24
-2
44 +2
44 +3
44 +6 -1-31
+6
44
44 +4
44
44
44-2
44
40 +3
35-1
(14)
C
26-2
18-3
33 12+2
34
8+
35
0
Sum 1231
2 L -
+9
44-3 +6
420
1500U NOOMN O
12 4
Half-
Throw
+2
++
(15)
+5-32
+9-27
+3
0
0
Sign
14
-
-18
-9
++++
13
+2 +2
+4
+4
0
Sta.
Revision-Trial 1 to 2

1267
Sta.
1234 LO CO
Sum
29
30
31
Sum
Net
32
33
34
Sum
Change
in
m
++11
+2
+2
22NO
-2
1
3
-1
+1+1++
123-d
Change
in
772
29
30 +2
32
34
-3
+1
Sum +1
Net
0
M
Mapa
M
1
pomoć przed pred
-6
7777 717
+1
+1
Revision-Trial 2 to 3
+1
+2
0
+6
A
0
35
Change in
tat
sta. 35
-6
-10
+9
+34
+33
-8
0
+32
+31
+30
+29
+189
111
COLÓ H CO
-
-6
Magda
Change int
at sta.
13
4
-22
+167
-68
-24
42
-66 -22 -40
+58 +14 +32
+84 +18 +45
+8 -14 -5
22
<
RAILROAD SURVEYS
287
revised middle ordinates is equal to the original sum. Rule 1
requires this relation.
The errors and their algebraic sums were next calculated;
the arithmetic was automatically checked by the zero value
at the foot of col. (5).
Then the half-throws were calculated from Rules 2 and 3
and entered in col. (6). Arrows indicate the additions.
Up to about sta. 17, it appeared that a fairly close solution
might be found on the first trial. After that, however,
the half-throws became excessive. Instead of starting over
when this happens, it is a good plan to continue col. (6) to
completion; otherwise, a large number of trials might have to
be made before finding a solution giving zero half-throw at the
final station.
In the illustrative example the result of the first trial was
modified by a method which guarantees a check on the second
trial. The method is based upon the following rule:
Rule 4. The effect upon the half-throw at any station
caused by a change in middle ordinate at any preceding
station equals the product of the change in middle ordinate
and the difference in stationing; the sign of the product is
opposite to the sign of the change in the middle ordinate.
The modification of trial 1 is shown at the top of the page
to the right of trial 3. It should be observed that the small
changes were made entirely on the spirals. The middle
ordinates were adjusted in such a way as to make their net
change zero (Rule 1) and at the same time to produce the
required change of +167 units in the half-throw at sta. 35.
The resulting half-throws are given in col. (10).
Trial 2 might be considered an acceptable solution, pro-
vided that there are no objects which might interfere with the
fairly large throws between stations 10 and 15.
There are any number of solutions which will give zero
throw at the end of the curve. The best solution is the one
having the smallest intermediate throws, consistent with
specified clearances and smooth curvature.
Trial 3 shows a second solution to the illustrative example.
In revising trial 2, an attempt was made to decrease the
throws near sta. 13 without increasing those near sta. 22 too
288
RAILROAD SURVEYS
much. The tabulation at the bottom of the page to the right
of trial 3 shows how this was done and the zero throw at the
end checked before the detailed calculations for trial 3 were
performed. The resulting half-throws are given in col. (15).
By using trial 3 instead of trial 2, the full throw is reduced
from a maximum of 12.8 inches to 10.0 inches.
It is possible to obtain further improvement in this example
by continuing the foregoing process, especially if the middle
ordinates are expressed to the nearest 0.05 inch. (This is
suggested as a profitable exercise for the student.)
String-lining problems are more complicated if it is neces-
sary to hold the track fixed at certain points, such as at
frogs, bridges, or station platforms. In such cases, zero
throws are entered at the proper stations and the middle
ordinates are adjusted so as to produce the required result.
Numerous restrictions on throws make it difficult to obtain
perfectly smooth track.
Track shifting in conformity with the throws finally
approved is controlled by setting suitable line (and, possibly,
grade) stakes.
Stout tacked line stakes are driven between the ties opposite
each station, or as close thereto as permitted by the position
of the ties. On double-track roadbeds, stakes are eliminated
by setting tacks on the ties of the parallel track.
Some engineers prefer to set the line stakes on the revised
center line. Track in the shifted position is then checked by
means of the usual track gage. Instead of being centered, the
line stakes may be set level with the base of the rail at a
distance such that, when the track is shifted, the base of the
rail will be a constant distance, e.g., 1 foot, from the tack.
On a curve requiring large throws, it may also be necessary to
set grade stakes for adjusting the rails to proper superelevation.
Track shifting was formerly done by moving track and ties
with the aid of lining bars. Lining and ballasting about 300
feet of main track was a normal day's work for a 10-man
section gang. Since about 1956, specially-designed machines
permit the same-sized gang to surface at least ten times as
much track per day.
RAILROAD SURVEYS
289
A
A' A
9-18. Spiraling Existing Curves. In earlier years of
railway surveying a variety of track realignment problems
arose in connection with spiraling existing track originally
laid out as simple curves. Since such problems are much less
common now, only one typical case will be illustrated.

EE+O
|OFF
+
02
S.C.
E
·R₁-R₂ +EE'
Fig. 9-5. Spiraling an existing curve
1
1
P.1.

Fig. 9-5 shows half of an existing circular curve, AE.
It is necessary to introduce spirals in such a way as to minimize
shifting of the track and at the same time to keep the new
and old track lengths practically the same. So doing facili-
tates shifting the track to its new position. The spiraled
half-curve is shown at A'E'. The throw EE' at the center
of the curve is usually restricted to a maximum value of 10
to 12 inches.
Obviously, the new simple curve, with radius R2, must be
somewhat sharper than the original one. The selected value
of Ls and the final value of R2 must fit the following relation:
(R1-R₂+EE') vers I= EE'+o
(9-4)
Any number of combinations of R2 and L, may be found.
A suggested procedure follows:
1. Select a trial value of D2 slightly greater than the original
degree of curve.
290
RAILROAD SURVEYS
2. Select a practical spiral length not less than that required
by equation 9-3; and calculate the resulting value of o
from the relation
2
Lis
or from Tables XI or XII.
24 R2
3. By a trial slide-rule calculation, determine the value of
EE' needed to balance equation 9-4. If EE' is greater
than the permitted maximum, go through the same
process with new values of D2 and Ls properly chosen to
bring EE' within the required limit.
4. After satisfactory values of D2 and L, have been obtained,
calculate the difference in length between the original
and revised alignments. This difference should be
figured between points common to both layouts, namely,
the T.S. and the S.T. of the new alignment.
See Prob. 9-3 for practical examples of the preceding case.
In other problems, it may be necessary to hold a certain
portion of the circular curve in its original position, such
as on a bridge, trestle, or high embankment. In this case,
it is necessary to compound the curve with slightly sharper
arcs in order to obtain the needed clearance for inserting the
spirals. As in the previous case, any number of combina-
tions of R2 and L, will fit the conditions. A practical example
of this type of problem is given in Prob. 9-4.
3. Crossings
9-19. Track Layouts.-Railway track layouts involving
surveying operations in their location are exceedingly com-
plex. Included under this heading are:
1. Turnouts
(2) Simple split-switch turnouts from straight track
(4) Turnouts from curved track
(c) Double turnouts, involving three-throw and tandem
split switches
2. Crossovers
(a) Between parallel straight tracks (see Fig. 9–6)
(b) Between parallel curved tracks
(a) Straight or curved track
(b) Combination crossings, or slip switches
RAILROAD SURVEYS
291
4. Connecting tracks from turnout to:
(a) Diverging track
(b) Another turnout, such as at wye tracks
(c) Parallel siding
5. Yard Layouts
(a) Complex combinations of the foregoing
(b) Various arrangements of ladder tracks
Each of these layouts involves a multitude of features,
including switches, frogs, guard rails, operating devices, rail
braces, and fasteners of various kinds.
-
One of the layouts most frequently used is a crossover
between parallel straight tracks, a simplified diagram of
which is shown in Fig. 9-6.

S
f
f
Fig. 9-6
f
-S
S
The layout involves two turnouts, each of which includes the
two switch rails, the frog, and the sections of curved and
straight track shown located between lines s-s and f-f. The
crossover tracks are straight between lines f-f, which are
located at the heels of the frogs.
The best general source of information concerning track
details is the portfolio of Trackwork Plans of the A.R.E.A.
These have been undergoing extensive revision since 1940.
Plans 910-41, and 911-41, covering straight split switches,
are given in Table XXV, Part III (the data, occupying two
sheets in the A.R.E.A. portfolio, have been rearranged slightly
to fit the pages of this book).
292
RAILROAD SURVEYS
The sketch on Plan No. 910-41 shows in outline form the
principal features of a straight split-switch turnout and
crossover. The data given on this plan are used in conjunc-
tion with Plan No. 911-41. Corresponding plans covering
curved split switches (not included here) are numbers 920–51
and 921-51.
The following definitions are taken from the latest A.R.E.A.
Manual:
Curved Lead-The distance between the actual point of
switch and the half-inch point of the frog measured on the
outside gage line of the turnout.
Frog-A track structure used at the intersection of two
running rails to provide support for wheels and passageways
for their flanges, thus permitting wheels on either rail to cross
the other.
Frog Angle-The angle formed by the intersecting gage
lines of a frog.
Frog Number One-half the cotangent of one-half the frog
angle, or the number of units of center line length in which the
spread is one unit.
Gage (of Track)-The distance between the gage lines,
measured at right angles thereto. (The standard gage is 4 ft
8 in.)
Guard Rail-A rail or other structure laid parallel with the
running rails of a track to prevent wheels from being derailed;
or to hold wheels in correct alinement to prevent their flanges
from striking the points of turnout or crossing frogs or the
points of switches.
Lead (Actual)-The length between the actual point of
the switch and the half-inch point of the frog measured on
the line of the parent track.
Stock Rail-A running rail against which the switch rail
operates.
Stock Rail Bend-The bend or set which must be given the
stock rail at the vertex of a switch to allow it to follow the
gage line of the turnout.
RAILROAD SURVEYS
293
Turnout—An arrangement of a switch and a frog with
closure rails, by means of which rolling stock may be diverted
from one track to another.
Turnout Number-The number corresponding to the frog
number of the frog used in the turnout.
The values most useful to the surveyor are the frog number
and the lead; these determine the distance occupied by the
turnout and the angle to whatever connection is to be made.
beyond the frog.
The field work for ordinary problems, such as locating
a turnout from straight track, follows a simple routine:
(a) Selecting the position for the frog.
(b) Choosing (tentatively) a standard combination of the
switch rail and the frog number.
(c) Measuring the lead to see where the point of switch
comes.
(d) Modifying the position of the frog (or changing the
frog number) so as to bring the switch point the proper
distance from the joints in the stock rails. (Recommended
distances are given in Plan No. 911–41.)
(e) Setting a tacked hub on the turnout opposite the actual
point of frog at a distance equal to half the gage distance from
the gage line of the main track.
(f) Laying out the turnout with the aid of the dimensions
and gage line offsets given in Plan No. 910-41.
Data for staking connections to track layouts beyond the
frog must be calculated from certain key measurements con-
trolling the individual problems. The great variety of such
problems precludes their inclusion in this textbook.
9-20. Relocations.-Though the principal trunk railroad
lines in the United States that can be justified in the foresee-
able future have already been located, relocations are con-
tinually being made in order to bring sub-standard sections
of line into conformity with modern requirements.
A typical relocation problem takes the form of flattening
a single sharp curve formerly requiring operation under a slow
order. If economic considerations do not eventually force
294
RAILROAD SURVEYS
the relocation, a serious wreck of a fast passenger train usually
does. The elimination (in 1947) of the "Gulf curve" at Little
Falls, N. Y.,* is an example.
Major relocations are invariably made for economic reasons.
The changes include (either separately or in combination)
reduction in distance, in grade, or in rate of curvature.
The objectives generally sought are: (1) increased speed, in
order to reduce running time; (2) greater riding comfort for
passengers; (3) operating savings; (4) increased train tonnage,
either as a result of grade reduction or from conservation of
momentum formerly wasted through braking at the approach
to slow-order curves.
A complete treatment of the economics of railway location
and operation is beyond the scope of a Route Surveying text-
book. Excellent analyses of these subjects will be found in
Proceedings, A.R.E.A., Vol. 39 (1938), pp. 439-560, and in
Proceedings, A.R.E.A., Vol. 45 (1944), pp. 25-44. See also
Chapter 16 in the latest A.R.E.A. Manual.
Major relocations involve surveying operations fully as
complex and interesting as those met in the original location
of trunk lines. In view of the existence of thousands of miles
of track having curvature and grades which limit economical
operation at high speeds, there is every reason to believe that
such major relocations will continue to be made for many
years. The railroads-stimulated by competition from other
forms of transportation, as well as by competition from each
other—are apparently aware of the necessity for making many
improvements in the alignment of main-line track.
References to a few examples of relocation and new location
follow (it is suggested that they be made required reading
assignments for students):
1. "The Eight-Mile Cascade Tunnel, Great Northern Rail-
way," Transactions, ASCE, Vol. 96 (1932), pp. 915–1004.
2. "Two Important Tunnels Built in 1945."
(1) The 3,015-Foot Bozeman Pass Tunnel in Montana
(Northern Pacific), Railway Age, Vol. 120, No. 19,
May 11, 1946, pp. 952–955.
*Railway Age, Vol. 122, No. 1. Jan. 4, 1947, p. 118.
RAILROAD SURVEYS
295
(2) The 2,550-Foot Tennessee Pass Tunnel in Colorado
(Denver & Rio Grande Western), Railway Age, Vol.
120, No. 21, May 25, 1946, pp. 1056–1058.
3. "Frisco Line Changes Improve Operation" (12-mile re-
location in Missouri, involving grade reduction from 2.3
to 1.27% and curvature reduction amounting to more
than 1,046 degrees), Railway Age, Vol. 121, No. 7, Aug.
17, 1946, pp. 281–284.
4. "Rock Island Completes Relocation in Iowa" (about
88 miles of relocation, involving elimination of 1,020
feet of rise and fall, and curvature reduction of 2,900
degrees), Engineering News-Record, Vol. 139, No. 22,
Nov. 27, 1947, pp. 726-730.
5. "Grade and Line Revisions Lower Operating Costs on
Missouri Pacific" (two relocations involving grade reduc-
tion from maximum of 2.45% to 1.25% compensated,
distance saving of 3 miles, and curvature reduction of
almost 900 degrees), Civil Engineering, March, 1949.
6. "Short-cut Through Missouri" (71-mile relocation on
C.B. & Q. reduces Chicago-Kansas City run by 22 miles
and passenger schedules by 5 hours. Has high design
standards, such as maximum grade of 0.8% and maxi-
mum curvature of 1°. Aerial photographs used in
planning the location). Engineering New-Record, Vol.
145, No. 19, Nov. 9, 1950, pp. 32–35.
7. "Crews Race to Complete Port, Railroad" (193-mile ore-
handling railroad built through rugged terrain in Quebec
Province. About half of line is curve track with few
curves sharper than 51°. Construction to high standards
of curvature and grade required 14 million cu yd of
excavation, half of it rock. Extensive use made of aerial
photogrammetry). Engineering News-Record, Vol. 165,
No. 24, Dec. 15, 1960, pp. 34-40.
8. "New Rails Cross Oregon Trail to Reach Mine" (76-mile
railroad spur built in Wyoming to main line criteria.
Location surveyed and designed by combining traditional
methods with use of aerial photographs and electronic
computer). Engineering News-Record, Vol. 166, No. 24,
June 15, 1961, pp. 44-46.
296
RAILROAD SURVEYS
PROBLEMS
9-1. From the notes for Fig. 9--2 compute:
(a) The true tangent distance AE by the most efficient
method.
(b) The deflection angles at D and E if distance DE were
203 ft.
(c) The distance CX and the deflection angle at X if the
tangent were resumed at X after deflecting 33′ L at station C.
9-2. Revise the string-lining problem of Art. 9-17 so as to:
(a) Obtain equal maximum positive and negative throws.
(b) Hold the track fixed at stations 17 and 18.
9-3. In the following cases of Fig. 9-5, verify the given
solutions and find another suitable combination of L, and new
D. which has a smaller difference in length, yet a throw that
does not exceed 10 inches.
(a) Given: Existing D.=5°; I=36°28′; Sta. T.C.=179+
43.27; E=5.75 inches; max. V=50 mph. Solution: New
Dc=5°30′; Ls=350 ft; throw EE' =0.36 ft; sta. T.S. 178+
01.20; revised length 0.45 ft shorter than original.
=
(b) Given: Existing D.=8°; I = 47°56′; sta. T.C.=74+81.06;
E=5 inches; max. V=40 mph. Solution: New D.-8°30′;
L=240 ft; throw EE' =0.09 ft; sta. T.S. =73+78.43; revised
length 0.50 ft shorter than original.
9-4. Given: An unspiraled curve on existing track; De=3°;
I = 27°16′; E = 2.25 inches; sta. T.C. =26+44.82; max. V = 50
mph. The central 200 feet of track is on a bridge. Verify
the following solution and find another suitable curve which,
when compounded and spiraled with values of Ls closer to
their desirable minimum, will not require shifting track on
the bridge. Solution: De of first and third arcs =3°10'; L,=
270 ft; unchanged length of 3° curve =210.21 ft; sta. T.S. =
25+28.14; revised length 0.13 ft shorter than original.
9-5. Using the standard dimensions of switch and frog
given in Plan No. 910-41, check the tabulated values of lead
and crossover distances (columns 4, 23, and 24) for a turnout
number specified by the instructor.
CHAPTER 10
HIGHWAY SURVEYS
10-1. Foreword. As a separate subject, adequate treat-
ment of Highway Surveys would require many pages of detailed
description. In this chapter, however, the subject is pre-
sented briefly in order to avoid repeating material already
covered in previous chapters. One referring directly to this
chapter for information about highway surveys should not
fail to note the discussion in Chapter 1 of the basic considera-
tions affecting route location; the details of curve and earth-
work theory given in Chapters 2, 3, 4, 5, and 6; and, in
particular, the practical treatment of curve problems in high-
way design found in Chapter 8. Much of Chapter 9 (Railroad
Surveys) applies equally to highways. Moreover, all applica-
tions of aerial photography and automation in Chapters 12 and
13 are to highway location.
Purposely omitted are samples of field notes and examples
of the various maps and profiles needed for location and con-
struction. Typical curve notes are given in Arts. 2-9 and 5-5;
notes for cross-sectioning and slope staking, in Art. 6–7. Gen-
erally, the best sources of information for practical forms of
notes are the instruction manuals issued to their engineers by
State highway departments. Highway drawings, especially
plan-profile sheets and "vicinity maps" for railroad crossings
or bridge sites, suffer so much by reduction to book-page size
as to be practically worthless for illustration. For instruction
of students, reference to a set of full-scale highway plans
(obtainable from most State highway departments) is strongly
recommended.
Continued improvements in survey methods and in stand-
ards of alignment design are due not alone to the progressive
policies of numerous State highway departments but also to
the activities of certain other organizations. Among these
are the Bureau of Public Roads (formerly the Public Roads
Administration), whose original research and whose insistence
upon high standards of design and construction in administer-
ing the provisions of the Federal-Aid Highway Act are per-
297
298
HIGHWAY SURVEYS
petual stimuli to progress; the American Association of State
Highway Officials, whose close cooperation with the B.P.R.
in developing modern "design policies" (Art. 8-1) is only one
of its many valuable activities; the American Road Builders'
Association (the oldest highway organization in the United
States), pre-eminent in highway public relations, early sponsor
of improved alignment standards, and recently active in
attracting better trained engineering personnel to the highway
profession; and the Highway Research Board, whose outstand-
ing work in basic highway research-organizing, correlating,
disseminating information—has been largely responsible for
supplanting rule-of-thumb methods with scientific design.
10-2. The Scope of a Modern Highway Project.-The
definition of Route Surveying given in Art. 1-2 suggested the
broad economic aspects of the subject. Modern highway
planning illustrates the economic relationships particularly
well, for in highway construction the taxpayer's dollar is
spent, in competition with countless other demands for public
services, upon projects which return to him only the intangible
benefits of improved transportation.
Though many phases of highway planning and design are
not properly a part of route surveying, it is felt that an illus-
tration of their interrelationship would be valuable—at least
to students. For this reason, there is included the following
outline taken directly from the Manual for Chiefs of Party
issued by the New Jersey State Highway Department.
PROCEDURES FOR PLANS PREPARATION FOR
STATE HIGHWAY PROJECTS
FIRST STEP
(Responsibility of Division of Planning and Economics)
1. Prepare brief report on purpose of highway, giving its
functions:
(a) To serve through traffic
(b) To serve local traffic predominantly
(c) To develop recreational, industrial, residential and
agricultural areas.
HIGHWAY SURVEYS
299.
2. Recommend classification of highway:
(a) Density of traffic (30th peak hour, 30 years hence)
(b) Passenger cars (P), Mixed traffic (M), Trucks (T)
(c) Design speed.
3. Recommend type of highway:
(a) Two lane
(b) Three lane (not normally recommended)
(c) Four lane undivided (not normally recommended)
(d) Four or more lanes divided (not more than four lanes
normally recommended)
(e) At grade "land service highway"
(f) At grade with major intersections separated (either
initially or eventually)
(g) Freeway
(h) Parkway.
7. Justification:
4. Recommend location of termini.
5. Recommend general area of location and reasons therefor.
6. Preliminary recommendations on traffic
Interchange locations.
(a) Traffic
(b) Economic factors
(c) Benefit-cost ratio
(d) Priority
(e) Place in comprehensive plan.
8. Review of report by Department of Design and Construc-
struction.
After approval (or modification) of above report by the
State Highway Engineer he will transmit to the Chief, Depart-
ment of Design and Construction, for progressing Surveys and
Plans. (Copies of this report shall be sent to Bureau of Public
Roads if participation in financing is expected.)
tions.
SECOND STEP
(Responsibility of Department of Design and Construction)
1. Reconnaissance surveys and proposed typical cross-sec-
300
HIGHWAY SURVEYS
2. After approval by the State Highway Engineer of Item 1,
prepare "Hearing Map" precedent to line adoption. After
line has been adopted by the State Highway Com-
missioner, design shall proceed.
3. Develop typical main roadway cross-sections (including
right-of-way width) based on satisfactory maximum
roadway capacity for the 30th peak hour, thirty years
hence. (In some cases this may be desirable, tolerable,
or maximum roadway capacity, in which case it will be
the subject of individual decision of the State Highway
Engineer with review by the State Highway Com-
missioner.)
4. Develop geometric design standards for approved design
speed.
5. Field reports on Interchange locations as outlined in
Memo of State Highway Engineer dated October 3, 1946,
shall be submitted.
6. Detailed traffic and economic data relating to Interchange
sites shall be obtained from Planning and Economics and
transmitted to field.
7. Preliminary sketch designs of Interchanges based on data
included in (6) above shall be submitted by field for
approval.
8. Submission of a general plan (generally 1″-200′) and
profile showing widths, alignment, grades, and tentative
Interchanges to State Highway Engineer for approval.
(After approval, prints of this plan shall be submitted to
Bureau of Public Roads.)
9. On approval of Interchange sketches and alignment and
grades, an accurate layout of Interchange design shall be
submitted for approval before being tied down mathe-
matically.
10. Submission of proposal for developing order and extent
of construction contracts shall be submitted to the State
Highway Engineer for approval, plus special or unusual
specification provisions.
HIGHWAY SURVEYS
301
11. Approval of construction plans and specifications by
State Highway Engineer and State Highway Com-
missioner.
In developing plans it is expected that Planning and
Economics and Department of Design and Construction will
cooperate closely and draw freely on information and tech-
niques each from the other, and that the Maintenance Division
and Parkway Engineer will be consulted freely on matters of
mutual concern.
10-3. Similarity to Railroad Surveys on New Locations.
Surveys for major rural highways on new locations may
follow identically the same procedure as those on railroad loca-
tion. For this reason, the descriptive material in Arts. 9-2
to 9-14 should also be considered as part of this chapter.
The principal differences in highway survey methods are
caused by the somewhat greater latitude with respect to pro-
file grades and especially by problems of highway interchange
design. Generally, these differences require a wider strip
of topography than on railroad location. Perhaps this is the
reason why aerial survey methods-rarely used by railroads—
are supplanting field reconnaissance, and even preliminary
surveys, on important new highway location.
examples are given in Chapter 12.)
(Some
Aside from the use of aerial survey methods, the greatest
difference in technique on new locations is in the method of
cross-sectioning, which is usually done by cross-section level-
ing rather than by slope staking (see Art. 6-7). The reason
for this is that the exact profile grades are not designed until
after the alignment is fixed. The cross-sections, plotted to
scale, aid in selecting the grades and in designing drainage
facilities.
10-4. Modifications of Railroad Methods on Relocations.
Most new highway construction is in the form of improve-
ments of existing highways. The changes usually involve
widening of pavements, increasing the number of traffic lanes,
reduction of grades, and flattening of curves to increase sight
distances and safe speeds. Frequently, most of the existing.
right-of-way can be utilized, but minor relocations or "cut-
302
HIGHWAY SURVEYS
offs" over new right-of-way are often required when sub-stand-
ard sections cannot be improved without incurring excessive
land damages. A closely related problem is the location of
by-passes around congested areas, with the accompanying
problems of grade separation and interchange design.
Surveys for relocations involve numerous modifications
of the usual railroad survey methods. A few of the common
modifications are described in the following paragraphs.
For one thing, reconnaissance is more localized. The reason
is that the need for possible relocation is usually disclosed by
accident statistics or by obvious traffic bottle-necks. Recon-
naissance is often simplified by reference to the original con-
struction plans, to tax maps, and—in increasing instances—to
aerial photographs (in New Jersey, for example, photographs
are available for the entire State and can be ordered to any
scale). Alignment closely approximating the final location
can usually be selected from a study of the reconnaissance
information.
The proposed alignment, complete with curves, is very
carefully staked, since it serves as one of the "base lines" for
design and construction. In case it is not possible to fix the
alignment in advance, the base line may follow the center
line of the old road; all drawings and designs are simplified,
however, if it follows the center line of the proposed road. In
the case of dual highways, it is advisable to use two base lines,
one at the crown line (the profile grade line) of each roadway.
Additional construction base lines are needed at interchanges;
in most cases the center lines of ramps or connections serve
best.
Ordinarily the alignment is staked at full stations and at.
half-stations. Station stakes are centered and tacked; on a
pavement each exact point is chiseled or marked by a nail
driven through a washer or a square of red cloth. Angles
are determined at least to the nearest 30 seconds; distances,
to the nearest 0.01 foot. On important surveys through
congested areas, temperature corrections are applied to the
taped distances, and tape tension is estimated carefully and
checked occasionally with a spring balance. Check chaining
or calculated traverse closures are required to have a precision
of from 1 in 5,000 to 1 in 10,000.
HIGHWAY SURVEYS
303
In comparison with surveys for new locations, those for
relocations cover a relatively narrow strip of territory. But
it is necessary to measure and record many more topographic
details. Among these are present paving, curbs, sidewalks,
trees, guard rails, drainage structures, fences, walls, property
lines, public utilities, buildings, land usage, intersecting high-
ways, streams, railroads, and-in general- all features which
might affect the grade-line design or which might have to be
removed and relocated.
Measurements for topography and right-of-way data are
customarily made by the transit party and recorded in the
field book. Details are tied to the base line by "plus and
offset," supplemented by range ties. Intersecting railroads,
pole lines, etc. are located by stationing and angle of intersec-
tion. Contours are not located in this form of topographic
survey. Some highway departments use the "field-sheet"
method described in Art. 9-9.
Cross-sectioning differs from railroad practice in that the
process of cross-section leveling is used instead of slope stak-
ing. The usual method is that given in Art. 6-7, though some
engineers prefer to run profile levels first and then to do cross-
sectioning by recording rod readings as plus or minus from
the given center-line elevations. Since no elevations are de-
termined in taking topography, the cross-sections must give
the needed information concerning elevations of existing
curbs, walls, drainage structures, utilities, and building
foundations. Other special methods are found in Art. 10–7.
Though ground cross-sections are used principally for deter-
mining earthwork end areas, they sometimes serve the purpose
of plotting contours to assist in the design of ramps and grade
separations.
10-5. Soil Surveys.-Comprehensive soil surveys, rarely
needed in railroad location, have become standard practice
among progressive State highway departments. Such surveys
are superseding the former types of soil surveys which were
often restricted to borings at bridge sites and cursory examina-
tion of the route for surface indications of snow slides or
unstable side slopes.
304
HIGHWAY SURVEYS
Modern alignment standards often require that routes
traverse topographic features formerly avoided; as a result,
long heavy fills and deep cuts-sometimes through bedrock-
are frequently necessary. Since the cost of a modern high-
way may exceed one-half million dollars per mile, possible
savings in construction and maintenance costs justify using
fairly expensive methods of determining pertinent informa-
tion about surface and subsurface soil conditions.
Soils investigations for highways continue to grow more
comprehensive in scope. This phase of highway engineering
requires close cooperation with the soil physicist, the geologist,
and even the seismologist. The modern tendency is to go
beyond merely making auger borings along the proposed route
and classifying the soil samples in the laboratory. Instead,
the past geologic history of the area is investigated. From
these studies, area soil maps are prepared which show the soil
"pattern"—land forms, types of soil deposits, swamp areas,
drainage conditions, and related information. A proposed
route traversing the region covered by an area soil map is then
the subject of a preliminary soil report which shows the rela-
tionship of the soils to the engineering considerations of align-
ment, grade, drainage, and grading and compaction processes.
The availability of materials for borrow, for subbase, or for
concrete aggregates is also indicated. The report may suggest
alignment changes; it also contains specific recommendations
regarding the extent of subsurface exploration needed to
answer detailed questions for design and construction.
In glaciated regions, erratic depths to bedrock may justify
the use of seismic methods of subsurface exploration. Mass-
achusetts, for example, has developed a procedure in which a
geologic "strip" map is prepared and the locations where
seismic studies are recommended, such as at deep cuts and
bridge sites, are shown on this map. Seismological field work
has been done under a cooperative program of the Mass-
achusetts Department of Public Works and the U.S. Geological
Survey. Extensive use was made of seismic profiles in obtain-
ing quantity estimates on the Massachusetts Turnpike.¹*
Seismic methods have become less costly with the develop-
ment of a light-weight seismograph, the sound waves for
*Superscript numbers refer to the bibliography at the end of this chapter.
HIGHWAY SURVEYS
305
which are generated by the impact of a sledge hammer on a
metal plate instead of by an explosive charge.2
Complete treatment of the subject of modern highway
soil surveys is beyond the scope of a text on Route Surveying.
For detailed information, see any modern text on Highway
Engineering. Broader aspects of the subject are described in
detail in References 3 through 7 in the bibliography.
10–6. Preparation of Plans.-Office procedure in design
and preparation of plans for a major highway project involves
a multitude of operations. Some of these are quite routine
and may be done by the sub-professional members of the
surveying crews during rainy weather; others require special-
ized training and experience.
State highway departments usually follow certain “stand-
ards" with regard to methods of design, sizes of drawing
sheets, arrangement of work, and forms for estimating quanti-
ties. No one scheme is best for all projects; a great deal
depends on the size and type of the project and on the per-
sonnel available. However, on Federal-Aid projects certain
specifications of the B.P.R. relative to size of drawing sheet
and form of layout must be followed.
The final objective of the office work is to prepare a cost
estimate and a complete set of plans showing clearly all in-
formation needed (1) by the engineers in laying out the lines
and grades to be used by the contractor in building the project,
(2) by the contractor in estimating the nature and extent of all
work to be performed, in order that he may prepare his bid,
and (3) by the legal agents to assist in preparing the right-of-
way descriptions and other data connected with land takings
and easements.
A detailed description of office design methods not only
would be too voluminous for inclusion in this book but also
would encroach upon subjects more properly treated in a
study of highway engineering. Consequently, only an outline
of conventional office routine is given to show the relation
between survey work and design. Since about 1956, the use
of automation in location and design (see Chapter 13) has led
to important changes in some of the steps described.
Supplied with the reconnaissance report and all the data
from the field surveys, the designers carry out these steps:
306
HIGHWAY SURVEYS
1. Design of typical sections: These are dimensioned
drawings showing the proposed roadway cross-sections of the
standard portions of the project. Shown are width, thick-
ness, and crown of pavements; shoulder widths and slopes;
positions of ditches, side slopes, curbs, median strips, guard
rails, and other construction details.
2. Preparation of location map: Usually done on a series
of 22"X36" Federal Aid Sheets, which show the profile as
well as the plan. Common scales are 1″-100′ horizontally;
1" 10' vertically. Plan shows survey base lines, topographic
details, and all alignment and right-of-way data. Profile
(sometimes drawn on separate sheets) shows the ground line,
grade line as finally designed, drainage structures, and esti-
mated earthwork quantities and balance points.
=
3. Plotting of cross-sections: Ground cross-sections, used
for earthwork calculation and in grade-line design, are plotted
directly from the cross-section leveling notes. A common
scale is 1'10'. Scale may be larger when end areas are to
be determined by the "strip" method (Art. 6—6); smaller
when found by calculating machine (Art. 6–4).
4. Establishing of profile grades: Grade line designed with
regard to relative importance of economy of construction,
balance of earthwork quantities, property damage, sight
distances, safety of operation, drainage and soil conditions,
aesthetics, and adaptability to future property development
and to future highway or railroad grade separations. Fre-
quent reference to plotted ground cross-sections is helpful in
design.
5. Drawing of cross-section plans: Proposed roadway
cross-sections drawn on ground cross-section sheets in con-
formity with the designed profile grades. These sections
show the pay lines for excavation. Widening and supereleva-
tion are allowed for.
6. Making of special detail drawings: Includes detail draw-
ings of all types of drainage structures; of retaining walls,
curbs, guard rails, and other appurtenances; and of compli-
cated interchanges and intersections. In connection with the
latter problems, standards often followed are those given in
the AASHO Policy on Geometric Highway Design.
HIGHWAY SURVEYS
307
7. Preparation of right-of-way plans: Property maps of all
parcels to be acquired or conveyed, showing locations, owners'
names, and ties to existing and proposed right-of-ways.
8. Estimate of quantities: Detailed estimate of quantities
of grading, paving, and other construction work, prepared
systematically with the aid of special "take-off sheets."
Summary of results, to serve as basis for engineer's cost esti-
mate and to aid contractors in preparing bids.
9. Preparation of specifications: Detailed general and
special provisions relating to proposal conditions, submission
of bids, prosecution of work, construction details, and methods
of measurement and payment.
10-7. Construction Surveys.-Generally, the types of sur-
veying operations needed on highway construction are the
same as those outlined for railroad construction in Art. 9-14;
therefore, the descriptions will not be repeated here.
In general, field layout and staking are somewhat more
complex in highway work, owing to the multiplicity of lanes
and the many ramps and intersections. Staking practices
vary with the type of highway, the nature of the terrain, the
magnitude and the cuts and fills, and the preferences of the
particular organization.
In the conventional method, tacked line stakes, marked with
station and offset, are set no more than 50 feet apart on offset
lines from the construction base lines. Their elevations are
determined and recorded for future use in setting grade stakes.
After the right-of-way has been cleared, a double line of slope
stakes or "rough grading" stakes is set at 50-foot intervals.
Finishing stakes are necessary for the final operations of side-
slope trimming, subgrade preparation, and setting of forms for
paving. After the grading has been completed, "blue-topped"
line and grade stakes are set on the subgrade near enough to
the work to permit forms to be set truly by means of a short
grade board.
In mountainous terrain, where grading is very heavy and
there are complications in the form of variable slopes and
benches, the customary method of setting construction slope
stakes is very clumsy. Instead, a "traverse method" may
be used to great advantage. Because of its specialized applica-
tion, the traverse method will not be described. (See Refer-
ence 8 for details).
viag
308
HIGHWAY SURVEYS
The use of freeway design, in which double roadways are
often at different levels separated by a median of varying
width, complicates not only the construction staking but also
the calculation of grading quantities. Good results can be
obtained by substituting a contour grading plan for the usual
voluminous set of cross-section sheets. In essence, "contour
grading" consists in superimposing contour lines of the pro-
posed construction on the existing contour map, thereby form-
ing a series of areas bounded by closed contours. The areas
are planimetered, and the volumes of the horizontal slices of
earthwork are determined by the average-end-area method.
This method is subject to further refinement and greater
accuracy if partial contour intervals are taken into account.
As a result of some time studies, it is estimated that earthwork
calculations, together with the drafting and survey operations,
can save about 40 per cent in man-hours. Even this saving
is small, however, compared with that resulting from the use
of electronic computers (see Chapter 13)
9
Warped surfaces at intersections require specially worked
out staking arrangements in order to produce smooth riding
surfaces.
Record plans of all work "as built" are worked up as con-
struction proceeds. Since pavement is usually paid for on
the basis of surface area, the final measurement of the length
of the project is somewhat greater than the horizontal survey
measurement.
10-8. Examples of Modern Practice.-Examples of good
practice in highway location and interchange design are so
numerous as to permit reference to only a few. The following
examples illustrate various types of problems:
1. "Difficult Location Problems on 476-Mile Blue Ridge
Parkway." (Shows application of railway surveying methods
to new highway location in mountainous terrain. Alignment
includes spirals and double spirals, eleven tunnels, grade
separations, and grade compensation for curvature.) Civil
Engineering, Vol. 17, No. 7, July, 1947, pp. 378 ff.
2. "North Santiam Highway Follows Difficult Route Near
Cascade Summit." (Costly 15.6-mile Oregon highway in-
volves heavy cuts and fills; stability of fills affected by possible
HIGHWAY SURVEYS
309
sudden drawdown of flood-control reservoir.) Civil Engineer-
ing, Vol. 18, No. 8, August, 1948, pp. 507 ff.
3. "Application of Coordinate Methods to Freeway Plan-
ning and Construction." (Describes precise surveying
methods used to solve complex design and right-of-way prob-
lems on freeway construction in urban areas.) California
Highways and Public Works, Nov.-Dec., 1946.
4. "Evolution of the Pennsylvania Superhighway." (Entire
issue devoted to the history, financing, design and construc-
tion of the nation's first modern toll highway.) Roads and
Streets, Vol. 82, No. 10, October, 1939.
5. "New Jersey Turnpike." (A group of articles covering
the planning, financing, design, and construction of this 118-
mile expressway.) Civil Engineering, Vol. 22, No. 1, January,
1952, pp. 1-69.
6. "New York State Thruway." (A group of articles
covering the planning, financing, and design of the first 427-
mile section of this modern toll highway.) Civil Engineering,
Vol. 23, No. 11, November, 1953, pp. 735–752.
7. "Tough Terrain Conquered by Builders of West Virginia
Turnpike." (Describes a difficult location problem in which
a modern 88-mile highway, having geometric design standards
suited to speeds of 60 mph or higher, replaced a tortuous route
107 miles long that had some grades of 9% and some curves of
50-ft radius.) Civil Engineering, Vol. 24, No. 2, February,
1954, pp. 74–80.
8. "Big Freeway Will Link Los Angeles, San Diego, Mexico."
(Describes 165-mile San Diego Freeway being constructed to
8-lane width at an ultimate cost close to one-half billion
dollars.) Engineering News-Record, Vol. 161, No. 22, Nov.
27, 1958, pp. 30 ff.
10-9. Use of Aerial Surveys.-It is noteworthy that high-
way engineers have been active in adapting aerial surveys to
ever-widening fields of usefulness in the planning, location,
and design of highways. In fact, the science has progressed
to such a point that E. T. Gawkins, commenting upon the
results of experience in New York State, wrote:10
310
HIGHWAY SURVEYS
aerial surveying . . . will in most cases obviate the need for one
or more reconnaissance surveys and all the labor required for pre-
liminary estimates of several alternate routes. Once the line has been
selected from the use of aerial surveys, field surveying can be reduced
to include only those necessary steps such as laying out of base line,
setting of stakes, determination of right-of-way limits on the ground,
and taking of sections for earthwork estimates prior to the award of the
contract-the steps that will always be required for the construction of
a highway.
The most suitable relationship between ground-survey and
aerial-survey methods has yet to be worked out. Possible
combinations of these two methods are suggested in Chapter
12. Their relative use is largely an economic question involv-
ing the size of the project, the character of the terrain, and
the availability of existing photographs to suitable scale.
Most of the United States has been photographed from the
air at least once. The work has been done by several agencies
and for a variety of purposes. Consequently, not all the
photographs are suitable for highway-location purposes.
Up-to-date information on the existence and nature of avail-
able aerial photographs may be obtained from the Map
Information Office of the United States Geological Survey.
The Bureau of Public Roads lists the six stages of highway
location as follows:11
•
First Stage-Reconnaissance survey of the entire area
between terminal points.
Second Stage-Reconnaissance survey of all feasible route
bands.
Third Stage-Preliminary survey of the best route.
Fourth Stage-Location of the highway on the ground.
Fifth Stage-Construction of the highway.
Sixth Stage-Operation and maintenance of the highway.
The earliest use of aerial photography in highway location
was as a supplement to the usual ground reconnaissance
surveys. Improvements in the art have now enabled photo-
grammetry to supplant ground methods for reconnaissance
studies of large areas.
The first stage, according to present practice, consists of
stereoscopic examination of small-scale aerial photographs
covering a broad area between the terminal points. Inter-
BIBLIOGRAPHY
311
mediate controls related to topography and land-use are
disclosed and broadly considered. The result of these studies
is the determination of all bands within the area which might
contain a feasible location for the highway.
In the second stage, large-scale aerial photographs are
taken along each of the feasible route bands. The photographs
are examined stereoscopically, as before, but their larger
scale permits the controls of topography and land-use to be
given special scrutiny. All possible route bands are compared,
after evaluating the several controls, and the best one is
chosen for more-detailed surveys. Thus, the use of aerial
photographs in two stages of reconnaissance will have dis-
closed the best route without costly ground surveys of several
alternate routes.
The third stage of location includes the making of a topo-
graphic map of the selected route band, and the projection
of a geometric location by the familiar paper-location method
(see Art. 9-11). As indicated in the quotation on page 310
photogrammetric methods are being used to an increasing
degree in studies following reconnaissance, especially in the
preparation of topographic maps. A reproduction of an
original aerial photograph, and the resulting topographic map,
are shown in Fig. 12-3. Fig. 12-4 shows a newer develop-
ment-the photo-contour map.
The fourth stage will always be done by ground surveying,
since aerial methods cannot conceivably be extended to the
processes of running in curves or setting stakes for grading and
other construction operations. Information of great value in
connection with the fifth and sixth stages can also be obtained
from aerial photographs.
In the over-all problem of route location, both photogram-
metric and ground surveying methods will be used. For
practical examples of their interrelationships the engineer
should study the up-to-date practices described in Chapter 12.
BIBLIOGRAPHY
1. Murphy, V.J., "Seismic Profiles Speed Quantity Esti-
mates for Massachusetts Turnpike," Civil Engineering, Vol.
26, No. 6, June, 1956, pp. 374-375.
312
BIBLIOGRAPHY
2. Liebenow, W.R., "Subsurface Bedrock Along Highway
Route Mapped by Seismograph," Public Works, July, 1960,
pp. 111-112.
3. "Highway-Materials Surveys," Bulletin 62, Highway
Research Board, 1952.
4. "Mapping and Subsurface Explorations for Engineering
Purposes,” Bulletin 65, Highway Research Board, 1952.
5. "Engineering Applications of Soil Surveying and Map-
ping," Bulletin 83, Highway Research Board, 1953.
6. "Air Photo and Soil Mapping Methods: Appraisal and
Application," Bulletin 180, Highway Research Board, 1958.
7. "Soil and Materials Surveys by Use of Aerial Photo-
graphs," Bulletin 213, Highway Research Board, 1959.
8. Construction Manual, State of California, Dept. of Public
Works, Div. of Highways.
9. Kane, C.V., "Contour Grading," California Highways
and Public Works, Sept.-Oct., 1952, pp. 1-5.
10. Gawkins, E.T., "Aerial Mapping Cuts Cost of Highway
Location in New York," Civil Engineering, Vol. 17, No. 2,
February, 1947, pp. 80-82.
11. Pryor, W.T., “Photogrammetry as Applied to Highway
Engineering," Photogrammetric Engineering, Vol. 17, No. 1,
March, 1951, pp. 111-125.
CHAPTER 11
SURVEYS FOR OTHER ROUTES
11-1. Foreword.-Reference to the broad definition of
transportation stated in Art. 1-2 suggests that the following
additional types of transportation may involve surveying
operations similar to those already described for railroads
and highways:
1. Transportation (transmission) of power and messages
by means of overhead tower or pole lines, or by lines in under-
ground conduits.
2. Transportation of liquids and gases through closed
conduits under pressure, such as pipe lines for water, gasoline,
oil, and natural gas; through closed conduits by means of
gravity, such as sewers and aqueducts; and through open
channels, such as canals and flumes.
3. Transportation of materials (sand, gravel, stone, or
selected borrow) to the site of large construction projects, by
means of cableways and belt conveyors.
Whenever any of the foregoing are projects of considerable
magnitude and involve termini a fairly long distance apart,
the required surveying operations may properly be included
in the term route surveying.
Special types of surveys are necessary in the case of tunnel
location and construction. These are noted briefly in Art.
11-6.
11-2. Similarity to Railroad and Highway Surveys.-Sur-
veys for all routes of transportation and communication are
similar in general respects to those described in Chapters 9
and 10 for railroads and highways. This is because all
routes have certain location controls (Art. 9-3); in fitting the
line to those controls the natural sequence of field and office
work approximates that outlined in Art. 1-8. The particular
differences that do occur are caused by requirements peculiar
to a specific type of route. The engineer acquainted with
railroad or highway surveying should have no difficulty in
313
314.
SURVEYS FOR OTHER ROUTES
adapting his knowledge to surveys for other routes, once the
uses to which the surveys are to be put are known. Surveys
for some other routes are described briefly in the succeeding
articles.
11-3. Transmission-Line Surveys.-The location of a
power transmission line is controlled less by topography than
is the location of other types of routes. Power loss due to
voltage drop is proportional to the length of the conductor;
consequently, high-tension transmission lines run as directly
as possible from generating station to substation. Changes in
direction, where required by intermediate controls, are made
at angle towers instead of along curves. A trunk telephone or
telegraph line is usually located within the right-of-way of a
highway or of a railroad, in which case the curves of the right-
of-way must be followed.
Unless aerial photographs are used, the field and office work
for transmission-line location involves, after a study of avail-
able maps, the following operations:
1. Reconnaissance for the location of intermediate features
to be avoided, such as buildings, cemeteries, extensive swamps,
stands of heavy timber, and particularly valuable improved
land; and for the location of intermediate controls fixing points
on the line, such as the most advantageous crossings of impor-
tant highways, railroads, and streams.
2. A transit-and-tape (or a stadia) traverse. The traverse
may be either a preliminary line or the final center line, the
selection depending on the difficulty of the problem. In stak-
ing the long straight sections between intermediate controls,
the deflection-angle method of by-passing obstacles on tan-
gents (described in Art. 9–13) is particularly useful. Contours
are not located; however, all topographic features and right-
of-way data are measured with respect to the traverse, as
described in Art. 10–4.
3. Levels sufficient in extent to aid in locating the towers.
These may be merely "spot" elevations or those for a complete
profile along the traverse line. On final location in difficult
terrain, it is advisable to take levels along two lines, one on
each side of the center line, in order to obtain proper conductor
clearance when spotting the positions of the towers.
SURVEYS FOR OTHER ROUTES
315
4. Office studies, including the features common to all route
location: drawing of the map, description of right-of-way ease-
ments, estimate of quantities and cost, and preparation of
specifications. A special problem in transmission-line design
is the location of the towers. This location work may be done
with the aid of special transparent templates, as described
after step 5.
X
5. Construction surveys. These are relatively simple on
transmission-line construction, since there is practically
no grading. Stakes are needed only for clearing the right-
of-way and for building tower footings. However, the sur-
veyor's assistance is also valuable in planning other details
related to construction, such as in spotting cable reels and
locating suitable dead-end and pulling points.
THETHE
Axis

Lower
-Minimum
EMEIFUR
EMELEME
Sag
GREE
Conductors
Clearance
DENEIGENENSI
deliam
T
Fig. 11-1
A convenient method of spotting tower locations on the
profile is to use a transparent template, the lower edge of
which is cut to the curve (approximately a parabola) that
will be taken by the conductor cables. Obviously the curve
must be modified to fit the scale of the profile. Two other
curves are inscribed on the template parallel to the curve of
the lower edge. The axial distance from the lower edge to
the middle curve equals the maximum cable sag for a par-
ticular span; that from the middle curve to the upper curve
equals the specified minimum ground clearance.
The template is used as shown in Fig. 11-1. First a
point X is located at a suitable position for a tower; then the
lower edge of the template is placed on this point and the
template is moved until the middle curve touches the ground
316
SURVEYS FOR OTHER ROUTES
line. The other point Y at which the lower edge of the tem-
plate intersects the ground line is the possible location of the
next tower. One template fits a considerable range of spans
with sufficient accuracy.
After the towers at certain controlling points have been
located, the location of the intermediate towers is a matter of
cut and try; the object is to cover the greatest length of line
with the least number of towers.
On important new transmission-line work, aerial photo-
graphs are now commonly used, at least for reconnaissance,
and they are sometimes used for all phases of the survey work
except final staking. The growing file of available aerial
photographs (Art. 10-9) often makes this method feasible
where it was formerly prevented by economic considerations.
11-4. Surveys for Pressure Pipe Lines and Underground
Conduits. Surveys for the location of long pressure pipe
lines are almost as simple as those for transmission lines. In
fact, the descriptions contained in steps 1 and 2 in Art. 11-3
apply also to pipe-line surveys. However, since pressure
pipe lines are usually located underground, greater attention
is paid to foundation conditions and especially to avoiding
costly rock excavation and frequent stream crossings. Acces-
sibility to power for operating booster pumping stations is also
an important intermediate control.
Grades and undulations in the profile are relatively unim-
portant, especially on small-diameter steel pipe lines; conse-
quently, detailed profile levels may be omitted and replaced
by spot elevations at proposed pumping stations and at the
high and low points along the line.
HAR
On construction, line stakes are more important than
grade stakes. In fact, grade stakes for steel pipe lines may
be needed only at pumping stations and at crossings of high-
ways, railroads, and streams. Along intervening sections,
at least in easy terrain, several sections of pipe are welded
together on the ground before being laid in the relatively uni-
form trench dug by the trenching machine.
Large reinforced-concrete pipe lines require much more
careful attention to undulations in the profile, since there is a
SURVEYS FOR OTHER ROUTES
317
practical limit to the change in direction possible at each joint,
and beyond that limit special pipe sections are necessary.
As in the case of transmission lines, right-of-way for a
pressure pipe line usually takes the form of easements for
its construction and operation.
Aerial surveys are particularly useful in the location of long
pipe lines. Only mosaics are used, since contours are not
essential. As a rule, a stereoscopic study of the photographs
will give enough information for the preliminary location.
After this the line may be "walked over" prior to deciding
upon the final location. It is possible, however, to rely upon
aerial photographs to an even greater extent.¹,2 * For example,
in building one of the longest pipe lines from the southwestern
part of the United States to the industrial middle west, no
surveyor went on the job until the sections of pipe were ready
to be laid; yet, the right-of-way agents completed much of
their work before that time.
Surveys for underground conduits containing power lines
on private right-of-way are much the same as those for pres-
sure pipe lines.
Underground communication circuits are commonly placed
in conduits located beneath the highway pavement. Access
for maintenance is by means of manholes. Coaxial cables
used for telephone, broadcast, and television circuits may be
drawn through existing conduits. However, they are also
placed directly in a shallow ploughed trench beside the high-
way. In neither case is any extensive survey work required.
11-5. Surveys for Construction at the Hydraulic Gradient.
Surveys for hydraulic construction in which flow is by gravity
require very careful attention to elevations, owing to the flat
grades used. If the flow is in an open channel, as in a canal,
the alignment may have to be circuitous in order to obtain
proper velocity and to avoid costly grading. A more direct
alignment is possible if the construction is below the ground
surface, such as in the case of a grade-line tunnel, aqueduct, or
sewer.
Surveys for surface construction may be identical with those
for railroad location, the principal modification being that
a narrower strip of topography will suffice. This condition is
*Superscript numbers refer to the bibliography at the end of this chapter.
318
SURVEYS FOR OTHER ROUTES
caused by the necessity for keeping the gradients between
relatively narrow limits.
Stations on the preliminary traverse are kept close to the
final location by setting them near the "grade contour." The
grade contour is the line on the ground (starting at a con-
trolling point) along which the grade changes at the rate best
suited to the construction. In locating the stations, it is
obviously necessary that the leveling be kept up with the
transit work. In simple irrigation-ditch construction in easy
terrain, a line may be located on the grade contour in the field
by tape and level, without need for the transit.
On a contour map the grade contour is found by starting
at a controlling point and stepping from contour to contour
with dividers set at a distance equal to the contour interval
divided by the desired rate of grade. The closer the final
alignment follows the grade contour, the lower will be the
grading quantities.
Since economy of grading is an important factor in canal
construction, careful cross-sections are taken at short inter-
vals. Construction surveys for canals are very similar to
those for highways, but all stakes must be set on offset lines.
There are cases in which aerial surveys have been used in
studies for canal location-for example, on the proposed
Florida Barge Canal. They have also been used in studies for
levees and dikes to control meandering rivers. Difficulty of
access for ground-survey parties is an important consideration
favoring the use of aerial-survey methods. However, the
small contour interval needed on maps for canal studies does
not ordinarily permit the location of contours by photo-
grammetric methods. Mosaics are useful for general studies;
but for detailed location the best method probably is to make
an accurate planimetric map from the aerial photographs and
then to add the contours by the plane-table method.
For gravity-flow structures below the ground surface, it is
entirely suitable to use railroad surveying methods, supple-
mented by adequate subsurface exploration. A most impor-
tant aspect of the office studies for such construction is to
decide whether cut-and-cover construction or tunnel con-
struction is the better. Often a combination of the two pro-
vides the most economical solution.
■
SURVEYS FOR OTHER ROUTES
319
11-6. Tunnel Surveys.-In mountainous terrain, it is
sometimes necessary to use tunnels on route alignment.
Surveying operations for locating tunnels vary greatly in
complexity. Preliminary studies are best made by using
aerial photography, especially in regions which have experi-
enced earth movements. Even detailed field studies may not
disclose old earthquake faults, but good photographs quickly
reveal them. As an example, some topographically-favorable
tunnel sites considered for Interstate Highway 70 under the
Continental Divide, west of Denver, were found to follow
major fault zones³, a fact which resulted in the choice of a
different location.¹
The final alignment of a short tunnel may be fixed by
locating a transit line on the ground directly over the tunnel.
As a rule, however, an indirect precise traverse is necessary.
In the case of subaqueous tunnels or long tunnels to be driven
through rugged mountain ranges, triangulation control must
be used. This is a subject outside the scope of route surveying.
The traverse or triangulation control provides only the
data for calculating the tunnel alignment; elevations must be
determined by careful spirit leveling between the proposed
portals.
Locating the portals, adits, and shafts by means of the
accurate control surveys is only one of the surveyor's impor-
tant tasks. His work in controlling the accuracy of the tunnel
driving is fully as important; it must be done with the highest
precision, for it cannot be verified conclusively until the head-
ings are holed through-and then it is too late to make adjust-
ments.
Surveying for tunnels driven through rock involves special-
ized operations not found in other types of route surveying.
Among these are:
1. Carrying the alignment down shafts by means of heavy
bobs damped in oil and suspended from piano wires. The
equipment also includes lateral adjusting devices for the
sheaves and scales for measuring the swing of the wires.
2. Transferring the alignment from the wires to plumb
bobs suspended from riders, or "skyhooks," mounted on scales
attached to the roof of the tunnel.
320
BIBLIOGRAPHY
3. Extending the alignment into the tunnel on "spads"
driven in plugged holes in the roof. The transitman usually
works on a suspended platform, out of the way of the muck
cars.
4. Carrying the alignment to the working face, or "painting
the heading," for locating the drill holes.
5. Transferring grade down shafts by means of weighted
tapes or by taping down elevator guides.
6. Carrying temporary grade into the tunnel by means of
inverted rod readings on the wood plugs in which the spads
are driven.
7. Cross-sectioning twice; first for locating points needing
trimming, and finally for obtaining permanent graphical
records of the sections and for computing pay yardage and
overbreak.
Several ingenious devices for cross-sectioning have been
used, such as pantographs and "sunflowers." The latter
are designed to locate breaks in the tunnel cross-section by
polar coordinates. A unique adaptation of the sunflower
was used in cross-sectioning the tunnels on the Pennsylvania
Turnpike, 5
Numerous other special procedures are necessary in survey-
ing for shield-driven tunnels and for those driven by the com-
pressed-air method.
BIBLIOGRAPHY
1. MacDonald, G.E., "Surveys and Maps for Pipelines,"
Transactions, ASCE, Vol. 121, 1956, pp. 121-134.
2. Guss, P., “Aerial Photography Aids Pipeline Location,”
Civil Engineering, Vol. 31, No. 6, June, 1961, pp. 48-51.
3. Mitcham, T.W., "Tunnel-Site Selection by Use of Aerial
Photography," Civil Engineering, Vol. 31, No. 8, August, 1961,
p. 64.
4. "Road Tunnel Will Pierce Divide," Engineering News-
Record, Vol. 167, No. 7, Aug. 17, 1961, p. 25.
5. “Evolution of the Pennsylvania Superhighway," Roads
and Streets, Vol. 82, No. 10, October, 1939, p. 77.
CHAPTER 12
AERIAL PHOTOGRAPHY IN ROUTE SURVEYING
12-1. Foreword.-The uses of aerial photographs in route
surveying justify devoting a separate chapter to aspects of the
subject not covered in textbooks on photogrammetry. Aside
from a brief review of certain definitions and mapping proc-
esses, this chapter avoids repeating the technical principles of
photogrammetric mapping as described in books on the sub-
ject.¹* Instead, the emphasis is on the special applications of
this new science to route location, and on their advantages,
limitations, and economic value. Some new developments
that involve photogrammetry appear in Chapter 13.
12–2. Definitions.-Definitions of photogrammetric terms
to be used in this chapter are given in this article. For a more
comprehensive list the engineer is referred to the Manual of
Photogrammetry of the American Society of Photogrammetry.2
Photogrammetry is the science or art of obtaining reliable
measurements by means of photography. The subject is sub-
divided into terrestrial photogrammetry and aerial photogram-
metry. In terrestrial photogrammetry the photographs are
taken from one or more ground stations; in aerial photogram-
metry, from an aeroplane in flight. Terrestrial photogram-
metry utilizes horizontal photographs or oblique photographs,
whereas aerial photogrammetry utilizes vertical photographs or
oblique photographs. Since oblique photographs are used only
for special purposes (see Art. 12-12), the aerial photographs
referred to in this chapter are considered to have been made
with the camera axis vertical, or as nearly vertical as practi-
cable in an aircraft.
The photographs used may be contact prints, made with the
negatives in contact with sensitized photographic paper; ratio
prints, the scales of which have been changed from those of
the negatives by enlargement or reduction; or stereoscopic pairs
(stereo-pairs), in which two photographs of the same area are
taken in such a manner as to afford stereoscopic vision.
*Superscript numbers refer to the bibliography at the end of this chapter.
321
322
AERIAL PHOTOGRAPHY IN ROUTE SURVEYING
An important property of a photograph is its scale, which
is the ratio of a distance on the photograph to a corresponding
distance on the ground. The scale of a photograph varies
from point to point because of displacements caused by camera
tilt and topographic relief. These displacements must be cor-
rected for, if the photograph is to be used for precise measure-
ments.
A flight strip is a succession of overlapping aerial photo-
graphs taken along a single course. The overlap is necessary
for stereoscopic examination and for the construction of
mosaics.
A mosaic is an assemblage of aerial photographs the edges
of which have been trimmed and matched to form a continu-
ous photographic representation of a portion of the earth's
surface. If the photographs are matched without reference
to ground control points, the resulting map is an uncontrolled
mosaic; if they have first been brought to a uniform scale and
fitted to ground control stations, the map is a controlled mosaic.
A planimetric map is one which shows the horizontal posi-
tions of selected natural and cultural features, whereas a
topographic map also shows relief in measurable form, usually
by contours. The term base map is used to define a large-scale
planimetric map compiled from aerial photographs. A copy
of the base map may be used for the addition of contours and
other data located by means of the plane table and/or photo-
grammetric methods.
12-3. Uncontrolled Mosaics.-In making an uncontrolled
mosaic, contact prints covering the area to be studied are
trimmed and assembled by matching like images and are then
fastened to a rigid or flexible backing. If the mosaic is for
temporary use in the field, the prints are mounted on linen or
other material that will permit the map to be rolled up. For
this purpose semi-matte prints are preferred because they take
pencil lines readily and are not scratched as easily as glossy
prints. For more permanent use, properly matched glossy
prints may be stapled or pasted to a rigid backing. If desired,
the assembly may be photographed to preserve one or more
copies of the complete map, after which the mosaic is dis-
AERIAL PHOTOGRAPHY IN ROUTE SURVEYING
323
mantled so that the contact prints may be used for stereoscopic
study.
Such a mosaic is relatively inexpensive and, though subject
to errors because of scale variations and displacements, it is
extremely valuable for reconnaissance (Art. 12–9) and mis-
cellaneous uses (Art. 12-12).
12-4. Importance of Stereoscopic Vision. Most aerial
photography is done by flying parallel flights across an area.
During the flights, photographs are taken at intervals such
that adjacent photographs will overlap approximately 30 per
cent at the sides and 60 per cent in the direction of flight.
This insures that the center (principal point) in each photo-
graph will appear in the adjacent picture taken in the line of
flight, thus providing what is called "stereoscopic overlap."
By properly orienting the overlapping photographs (stereo-
pairs) and viewing them through a stereoscope, the process
known as "stereoscopic fusion" takes place. In this process
there is a vivid mental impression of the terrain in three dimen-
sions. (A simple demonstration of stereoscopic fusion is
shown in Fig. 12-1.) In effect, two positions of the camera
lens several thousand feet apart are substituted for the
observer's eyes. In the resulting image (known as the
"stereomodel”), relative heights of hills and structures, depths
of canyons, and slopes of terrain are determinable. Used in
this way, the old principle of stereoscopic vision has become
probably the most important basic tool for studying the
manifold problems of route-location.
12-5. Controlled Mosaics.-The utility of a mosaic may
be greatly increased by bringing the photographs to a uniform
scale, correcting them for tilt, and fitting them in their correct
relative positions. This procedure requires the location of
control points by ground surveying methods.
Control stations, properly distributed over the area, are
first selected with the aid of the stereoscope. They should be
definite points easily recognized on the photographs and
accessible on the ground. Buildings, fence corners, or road
intersections usually serve this purpose. Preparatory to
planning the ground control surveys, the selected control
points are marked on each photograph by a circled prick point.
324
AERIAL PHOTOGRAPHY IN ROUTE SURVEYING
The ground control consists of suitable triangulation,
traverse, and level circuits from which the coordinates and
elevations of the control points may be computed. Ground
control is costly and should therefore be no more precise or
extensive than is required for the purpose. In research aimed
at reducing the cost of ground control for surveys of very large
areas, great progress in use of air-borne radar, and shoran
techniques, has been made.3,4,5 See also Art. 13–2.
In making the finished mosaic, the contact prints are ratioed
(brought to the same predetermined scale), rectified (corrected
for tilt), and fitted on a base board to the plotted ground con-
trol points. There are several methods of doing this, all of
which are highly technical and require special equipment.
Only the central part of each photograph is used in compiling
the mosaic, and the trimmed edges are feathered on the under-
side. In addition, prints having the same tone, or degree of
exposure, are selected. The finished mosaic then has the
appearance of a single large photograph.
12-6. Planimetric Maps.—A planimetric map, on which are
shown the accurate positions of such natural and cultural
features as watercourses, forests, highways, and buildings, may
be constructed from aerial photographs which are tied to
adequate ground control. The first step is to make a control
plot on which the ground control points are located from their
computed coordinates. Photographic control points ("pic-
ture points") must also be located on the plot to permit proper
matching of the photographs. These are points that are
clearly visible on two or more photographs but are not tied in
by the ground surveys. They are usually located on the con-
trol plot by an analytic or graphical method of radial tri-
angulation.
The positions of details may be transferred from the photo-
graphs to the plot either by a relatively simple tracing method
or by a stereographic method that employs special plotting
instruments. The tracing method yields good results only
if the photographs are relatively free from relief or tilt dis-
placements. In this method each photograph is fitted as
closely as possible to the plotted control points, and the
selected details are traced onto the plot. The positions of
AERIAL PHOTOGRAPHY IN ROUTE SURVEYING
325
details which show on two photographs may have to be
adjusted to compensate for slight differences in their traced
locations.
A more accurate map, free from the effects of tilt displace-
ment, may be constructed with the aid of complex optical and
mechanical instruments called "stereoplotters." Those most
widely used include the double-projection map compilation
plotters: Balplex, Kelsh, Multiplex, and Nistri Photocartograph;
and the more-costly optical train instruments: Galileo-Santoni
Stereocartograph, Nistri Photostereograph, Wild Autograph, and
Zeiss Stereoplanigraph.

1
X
:
O
Hold a card at right angles to the page and along
the dotted line. Closing one eye at a time,
adjust the head so that letter X (but not O) can
be seen with the left eye, and conversely with the
right eye. Then open both eyes and focus them
beyond the page. The four dots above the let-
ters will fuse and appear as two. Moreover, the
lower dot will seem to be floating in space relative
to the upper dot. To prove that all four dots are
in the image, notice that the letters X and O are
superimposed.
-
Fig. 12-1. Stereoscopic fusion
These instruments employ the principle of the floating mark.
In the stereoplanigraph and related instruments, the photo-
graphs of a stereo-pair are viewed through two movable
AERIAL PHOTOGRAPHY IN ROUTE SURVEYING
327
be made to rise or sink vertically. In planimetric mapping,
the general procedure is to operate the plotter so that the
floating mark appears to be touching the surface of the ground
(or other feature to be mapped) in the stereomodel, and then to
mark the corresponding points on the manuscript map. In
some stereoplotters the coordinates of the point viewed are
read from scales. More-complex instruments have a special
arrangement of levers, gears, and shafts which permits the
path of the floating mark to be plotted directly (Art. 12–7).
Some types of stereoplotters, particularly the optical-train
instruments, are able to extend ground control accurately
through a chain of stereomodels by a process known as bridg-
ing, which is an application of aerotriangulation. Aerotri-
angulation and the linkage of stereoplotter to electronic com-
puter are described in Art. 13-9.
Many details appearing on the aerial photographs may be
transferred to the planimetric map without commensurate
increase in cost-in contrast to the situation in ground survey-
ing. Moreover, only the relevant details need be transferred.
Thus, several different planimetric maps may be constructed
from the same photographs, the features shown on a map
depending on the use to be made of the map.
12–7. Topographic Maps by Photogrammetric Methods.
A stereoscopic plotting instrument may also be manipulated in
such a way as to measure the differences in elevation between
points on the stereo-pairs. In principle, the first step in doing
this is to adjust the instrument until the floating mark rests on
a control point of known elevation. The frame carrying the
two eyepieces is then moved in the x-and y-directions until the
mark is at the detail being measured but is apparently floating
in space above or below it. The mark is then made to rise or
sink until it is at the same elevation as the detail. This is
done by narrowing or increasing the distance between the
eyepieces. Scale readings made during the procedure can be
converted to give the x- and y-coordinates of the particular
detail viewed and its elevation with respect to the control
point. The principle can be extended to the drawing of con-
tours, thus converting the planimetric map into a complete
topographic representation.
AERIAL PHOTOGRAPHY IN ROUTE SURVEYING
329
51
per cent of the elevations tested shall be in error more than
one-half the contour interval."
That such standards are realistic and attainable was
demonstrated by their use in planning the Pennsylvania Turn-
pike extensions,6,7 which utilized photogrammetric maps made
to a scale of 1 in. =200 ft, showing planimetric details, spot
elevations, and 5-ft contours. Even more-rigid requirements
are sometimes specified. For example, topographic maps with
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AERIAL PHOTOGRAPHY IN ROUTE SURVEYING
a contour interval of 1 ft, at a scale of 1 in. =40 ft, have been
prepared for the highway commission of one eastern state.
The accuracy required was that 90 per cent of the contours be
dependable within ft.8,9 A reproduction of a portion of an
aerial photograph is shown in Fig. 12-3(a), and a photogram-
metric map of the area covered by the photograph is shown
in Fig. 12-3(b).
12-8. Orthophotography.-A new improvement over con-
ventional stereoscopic photogrammetry took place about 1956
with the development of equipment to produce the equivalent
of an orthographic photograph, i.e., a uniform-scale photo-
graph free from distortions due to tilt and relief. The U.S.
Geological Survey accomplished this by means of a machine
called the orthophotoscope.23 The original incentive came from
the acute need for geologic use of uniform-scale maps on which
the full wealth of planimetric detail is provided by photo-
graphic images.
At about the same time a different system of producing
topographic maps based on orthophotography was developed
commercially by the R.M. Towhill Corporation of Honolulu.24
The product, called the Photo-Contour Map (Fig. 12-4), is not
a mosaic with contours superimposed. Because of the method
by which the photographic perspective is rectified, it is an
orthographic projection at the same scale as the contour
plotting. The finished map is a photo copy on which contours
are shown as black or white lines, or a combination of both.
The tone of the photograph usually determines which is pre-
ferable. The Photo-Contour Map is practically self-checking.
Inaccurate work in compiling is revealed by obvious mis-
matches in contours and planimetric features. Inspection of
Fig. 12-4 shows that all contours properly track the visible
drains and roads.
In general, Photo-Contour Maps cost more than topo-
graphic maps made by conventional photogrammetric methods
(see Art. 12–11). However, where a proposed route location
passes through areas having dense planimetric detail, they
may be less costly because of saving in drafting time.
12-9. Photogrammetry for Reconnaissance.-Before mak-
ing the detailed projection of a route between selected termini,
AERIAL PHOTOGRAPHY IN ROUTE SURVEYING 331
there must be general studies dealing with intermediate con-
trols and interpretation of terrain. These studies may be
summarized in the one word reconnaissance. If the controls
usually present are examined (see Art. 9-3), they will be seen
to fall into two principal types: (1) land usage and (2)
topography. Both types of controls are recorded in great
detail on aerial photographs. In fact, it is now generally
agreed that such photographs provide the best means of mak-
ing the type of careful reconnaissance so aptly emphasized by
Wellington (Art. 1–9).
For the first stage of reconnaissance, as described in Art.
10-9, small-scale photographs of the region between the
termini will enable the designer to select the important con-
trols and locate all feasible route bands. The scale of these
photographs will depend on the distance between the termini
and the importance of the intervening terrain. In practice,
scales as small as 1 in. =2,000 ft (1:24,000) are quite suitable.
The best procedure is to lay up an uncontrolled mosaic and to
study it in conjunction with stereoscopic examination of
stereo-pairs. The result will be the selection of several bands
of terrain between the termini within each of which lies an
apparently feasible location. These bands may be from 1,800
feet to 1 mile or more in width.
The second stage of reconnaissance has for its purpose the
comparison of the route bands and the selection of the most
promising one. In some instances the choice will become
apparent during the first stage, but in more-difficult cases it
may first be necessary to study larger-scale photographs.
These may be prepared with little extra cost by enlarging the
original photographs up to a practical maximum of about four
diameters. Even when the original photographs are taken in
the summer, these enlargements are usable, since heavy foliage
does not detract seriously from their value in reconnaissance
studies.
In both stages of reconnaissance the aerial photographs
should be supplemented by other available maps. One in-
genious method is to make photographic enlargements of
U.S.G.S. sheets on a transparent-film base, to serve as an over-
lay at the same scale as the mosaic.10 Used in conjunction
with the stereo-pairs, this method adds a quantitative factor
to the study. Inaccuracies in the maps or scale distortions in
AERIAL PHOTOGRAPHY IN ROUTE SURVEYING
333
the photographs affect all route bands equally and are not
large enough to invalidate this procedure. Many of the mis-
cellaneous values disclosed by aerial photographs (Art. 12-12)
will become apparent in these studies and will also contribute
toward the selection of the best route band. Field inspection
at critical locations may be needed to resolve some difficulties.
12-10. Photogrammetry for Detailed Location Studies.-The
type of photogrammetric application used for detailed location
depends on the type of transportation route involved. It is
possible to establish the location of a transmission line or a long
pressure pipe line simply from the study of mosaics and stereo-
pairs. However, the accurate location of a highway, a rail-
road, or other route on which grades and right-of-way costs
are more important requires a complete topographic map
showing contours at not more than 5-ft intervals.
In some instances contours may be drawn by stereoscopic
plotting only on certain portions of the planimetric map, the
remaining contours being fixed by plane-table or other ground-
survey methods. This might be true, for example, in ex-
tremely flat areas or where dense ground cover prevents obtain-
ing the specified accuracy of contour location from photo-
graphs.19
Complete topographic maps, made as outlined in Art. 12-7,
vary in scale from 1 in. =200 ft, with a 5-ft or 10-ft contour
interval, to 1 in. = 40 ft, with 1-ft contours. To meet the
National Map Accuracy Standards the scale of the contact
prints cannot ordinarily exceed four times the map scale,
although the maps to a scale of 1 in. = 200 ft prepared for the
Pennsylvania Turnpike extensions.7 were made from prints
having a taking scale of about 1 in. = 1,000 ft. (The selected
route band in these examples was 1 mile wide.)
Other practical examples from states that have used photo-
grammetric maps on highway projects are:
California¹¹-Taking scale 200 ft per inch. Map scale
1 in. 50 ft, with 2-ft contours.
==
Connecticut¹²—Taking scales 500, 200, and 100 ft per inch.
Corresponding map scales 1 in. =200 ft (5-ft contours),
1 in. = 100 ft (2-ft contours), 1 in. = 40 ft (1-ft contours).
334 AERIAL PHOTOGRAPHY IN ROUTE SURVEYING
Massachusetts,12-Taking scale 600 ft per inch for rural
locations; 400 ft per inch for urban work. Map scales
1 in. =200 ft (5-ft contours) and 1 in. = 100 ft (2-ft con-
tours).
Ohio¹º-Taking scale 200 ft per inch. Map scale 1 in. =50
ft, with 2-ft contours.
Once the topographic maps are prepared, trial lines are laid
down on them with a spline, and the best route is chosen by
following the usual paper-location procedure, as outlined in
Art. 9-11.
An important feature of the paper-location study is the
estimate of grading quantities. If topographic maps made
from aerial photographs actually meet the National Map
Accuracy Standards, there is no reason why adequate quantity
estimates cannot be prepared from them. Recently, highway
engineers are finding that this is true. For example, in
Massachusetts on a 2-mile relocation the difference between
quantities computed from surveyed cross sections and from
sections plotted from aerial topographic maps was only 2.6
per cent in embankment and 3.3 per cent in excavation.13 A
similar comparison on a 7,600-ft project in Connecticut¹2
showed discrepancies of 1.4 per cent in cut and 1.2 per cent in
fill. On a 30-mile project in California"¹ the variation in the
excavation quantities was less than 2.5 per cent.
The foregoing results lead to the belief that in highway
construction we may eventually make contract payment for
excavation on the basis of quantities derived from photogram-
metric studies. A step in this direction was taken in Ohio¹0
on a 4.12-mile relocation for which a complete set of construc-
tion plans was prepared by photogrammetric studies and these
plans were used immediately for award of the construction con-
tract by the usual competitive method. Research shows that
by adjusting photogrammetric sections to field elevations
along a route center-line and taking cross sections from a
stereomodel, the resulting quantities are within limits con-
sidered satisfactory for purposes of payment.25,26 The linkage
of stereoplotter and electronic computer (Art.
further step toward this goal.
13-9) is a
AERIAL PHOTOGRAPHY IN ROUTE SURVEYING
335
12-11. Costs of Photogrammetric Mapping. In addition to
possessing many other advantages, photogrammetric maps
can often be prepared at far less cost than maps made by con-
ventional ground survey methods. The tabulation on page
336 is instructive in this regard.
In preparing the tabulation, the cost data for projects not
listed in the bibliography were obtained from commercial
organizations specializing in photogrammetric mapping. It
is important to observe that the cost per mile of photogram-
metric maps depends principally on the scale and the contour
interval. The length of the project--for given specifications
has practically no effect on unit costs. Also mapping costs
have not increased significantly over the years despite the
large increase in construction costs. This is due to better
equipment, improved techniques, and more competition among
å greater number of mapping companies. An important fac-
tor in keeping costs down is the ability of modern stereo-
plotters to extend sparse ground control by bridging.
On a cost-per-mile basis, comparable maps made by ground
survey methods will rarely cost less, and will usually cost much
more, than photogrammetric maps. However, a fairer com-
parison of costs should take into account the fact that maps
compiled from ground surveys rarely cover as wide a strip of
topography as those made photogrammetrically. On work
in Massachusetts, for example, the average cost of aerial
topographic maps was $680 per mile for a strip 6,000 ft wide;
whereas costs for ground surveys and plotting were about
$1,500 per mile for a strip only 500 ft wide.13
Regardless of whether photogrammetry produces savings in
mapping costs, it is likely to yield substantial savings in
construction costs. These savings accrue not only from time
savings but also from reduction in grading quantities and
construction difficulties because the wider strip of topography
and the astute use of stereo-pairs may result in the projec-
tion of a better location than is possible by former methods.
As an example, on a highway project in Mississippi, excavation
quantities averaged 42,872 cu yd per mile as determined from
a ground survey location, whereas another location, worked
out later with the aid of aerial photographs, was selected with
336
AERIAL PHOTOGRAPHY IN ROUTE SURVEYING
Year
1948
1950
1960
1950
1952
1961
1944
1944
1947
1947
1950
1952
1953
1954
1954
1960
1961
Conn.
Conn.
R.I.
State
Conn.
Mass.
N.Mex.
Cost Data on Photogrammetric Maps for Highway-Route Location
Cost per Mile
Cost per Acre
40-ft Scale-1-ft Contours-1,000 ft Wide
$25
16
14
N.Y.-N.J.
Mass.
Pa.
Pa.
Conn.
Mass.
Pa.
Pa.
N.Y.
Wisc.
N.Y.
Mileage
Specification:
4
3.8
7.5
7.7
Specification: 100-ft Scale-2-ft Contours-Mile Wide
$1,020
1,033
1,130
15
4.6
31
20.5
140
68
Specification: 200-ft Scale-5-ft Contours-1 Mile
$ 570
629
586
825
7.1
$3,000
1,950
2,587
60
150
27
100
10.5
24
610
958
953
922
820
$3.19
7.10
1,199
979
23.
Special Notes
Ref. No. 8, p. 341
Ref. No. 8, p. 341
50-ft scale; 1,500 ft wide
$0.79
0.86
0.92
1.29
0.96
1.50
1.49
1.44
1.28
1.88
1.53
Ref. No. 8, p. 341
Strip 1,200 ft wide
Strip 400 ft wide
Wide
6,000-ft strip. Ref. 21, p. 342
6,000-ft strip. Ref. 22, p. 342
Ref. No. 6, p. 341
Ref. No. 6, p. 341
Ref. No. 8, p. 341
AERIAL PHOTOGRAPHY IN ROUTE SURVEYING
337
excavation quantities averaging only 27,536 cu yd per mile.¹4
Similar comparisons on highway projects in Central America
and Alaska showed phenomenal savings in construction costs
in addition to improvements in alignment and gradients.15
The comments made in Art. 1-13 relative to costs and
ultimate accuracies are especially pertinent in aerial photog-
raphy and mapping. Photographs should be taken at a scale
suited to resolving the smallest detail that must actually be
shown on the map; otherwise additional field surveys will be
needed and costs will mount. The topographic map itself
should be drawn with the smallest horizontal scale and largest
contour interval that will serve the requirements. Greater
accuracy and larger scales than are needed are wasteful and
costly.
12-12. Miscellaneous Uses of Aerial Photographs.-The
technical utility and economic advantages of aerial photo-
graphs, as described in the preceding articles, carry enough
weight to justify their employment on all important route-
location projects. However, it is worth emphasizing that
aerial photographs possess other inherent values which may
result as by-products from their primary uses. Often, these
miscellaneous uses will require little additional cost or effort.
The auxiliary uses of aerial photographs are many and
varied. They may be found in any stage of route location,
from the preliminary planning to studies made after the route
is in operation. The following valuable uses have been re-
ported:
For Preliminary Planning
Interpretation of terrain. Drainage patterns. Soil
types. Land use as affecting costs of right-of-way.
Location and extent of wooded areas, swamps, rock
outcrops, snow-slide and erosion scars, and borrow
material and granular deposits for use in construction.
For Detailed Studies
Size of drainage areas for culvert determination. Plan-
ning of interchanges, using oblique photographs.
Large-scale site maps for bridges, intersections, and
338
AERIAL PHOTOGRAPHY IN ROUTE SURVEYING
other detailed studies, prepared from one pair of
photographs with a stereoscopic plotting instrument.¹0-
For Construction
Determination of best means of access. Type of clearing.
Possible effect of terrain and climatic conditions on
choice of construction equipment. Progress reports,
using series of oblique photographs taken in sequence
along route at convenient time intervals."
After Construction
Traffic studies, including traffic counts, speeds, and
densities, congestion, railroad grade-crossing elimina-
tion, efficiency of existing parking facilities, and location
of new parking areas. Road-inventory studies, includ-
ing changes in use of abutting land and pavement-con-
dition surveys.1
16
General
Public-relations purposes in general,¹ such as illustrations
at public hearings and legislative reviews pertaining
to proposed route location and land takings, using
oblique photographs that are easier for the layman to
understand (see the Frontispiece).
12-13. Limitations of Photogrammetry.-Photogrammetry
for purposes of route location is not without its limitations.
Clarity in photographs requires good atmospheric conditions—
freedom from clouds, mist, smoke, or severe haze. In some
parts of the world such ideal conditions may occur only one
or two days in the year.18
The interrrelationship between aerial photographs and
ground control requires careful planning as to their timing. A
hasty decision to take aerial photographs, solely in the belief
that they always save time, invites inefficiency and may raise
costs.
It is virtually impossible to determine precise elevations
from photographs taken when there is heavy snow cover or
dense vegetation. Consequently, for contour mapping the
photographs should be taken when the ground is bare and the
trees are defoliated. Regions having dense forests of different
AERIAL PHOTOGRAPHY IN ROUTE SURVEYING
339
types and heights of evergreens obviously present a difficult
problem. Reference 19 offers a possible solution.
Photogrammetric methods should not usually be used for
maps requiring elevations with an error of less than 1 ft. For
example, contour mapping by photogrammetry is questionable
for canal location in areas when contours are widely spaced
and where high precision is needed. In such circumstances
is would be better to draw a planimetric map by photogram-
metric methods and then add contours and spot elevations by
appropriate ground surveying methods.
If the area to be mapped is small or if only a narrow strip
of topography is needed, photogrammetry may then be re-
stricted by economics rather than by technical limitations.
Finally, projects such as highway relocation require addi-
tional data about details that can only be obtained by ground
methods. Among these are: precise location of hydrants,
water gates, and property-line markers; utility-pole numbers;
type, critical elevation, and house number of buildings; size
and invert elevation of culverts; and all the information regard-
ing existing subsurface installations that are needed for design
drawings and construction planning.
12-14. Summary of Advantages of Aerial Photography in
Route Surveying.-To summarize the information in this
chapter, it is apparent that the salient advantages of aerial
photography in route surveying are:
1. The larger area and wider route bands covered by the
photographs give greater flexibility in route location
and practically insure that no better location has been
overlooked.
2. Practically all the studies preceding construction can be
made without encroaching on private property or arous-
ing premature fears in regard to the extent of property
damage. Land speculation is thereby reduced.20 More-
over, the eventual acquiring of right-of-way is expedited
because property owners can see clearly on the photo-
graphs the effects of the takings.
3. On a large project the elapsed time between starting the
survey work and construction can be greatly shortened.
Should weather conditions provide only a short field
340 AERIAL PHOTOGRAPHY IN ROUTE SURVEYING
season, ground control and photography can be done
then, the map compilation being left for indoor work dur-
ing the winter months. In contrast to plane-table work,
the stereoscopic plotting of contours is independent of
weather and can be done on a day-and-night schedule if
necessary.
4. Over-all survey and mapping costs may be considerably
less than by ground methods.
5. Maps made by photogrammetric methods possess a
more uniform accuracy than is usually found in those
compiled from ground surveys.
6. What was formerly thought to be the ultimate goal of
aerial photography-the compilation of detailed topo-
graphic maps-is being extended to even more useful
purposes, such as quantity estimates and complete con-
struction plans.
7. Photogrammetry is of direct value in reducing the short-
age of engineering services by releasing engineering
personnel from routine survey work for more-advanced
employment in design.
8. Aerial photographs have many auxiliary uses and con-
tain more information about a variety of significant fea-
tures than the engineer can obtain by ground methods
except at greatly increased cost.
BIBLIOGRAPHY
1. Moffitt, F.H., Photogrammetry, Scranton: International
Textbook Company, 1959.
2. Manual of Photogrammetry, 2d ed., American Society of
Photogrammetry, Washington, D.C., 1952.
3. Hoover, C.R., "Development of an Optical Radar
System for Surveying Purposes," Transactions, American
Geophysical Union, Vol. 31, No. 4, August, 1950, pp. 518-524.
4. Ross, J.E.R., "Shoran, Application to Geodetic Tri-
angulation," The Canadian Surveyor, Vol. 10, No. 3, Jan-
uary, 1950, pp. 9-18.
5. Aslakson, C.I., "The Importance of Shoran Surveying,"
Transactions, ASCE, Vol. 120, 1955, pp. 225-234.
BIBLIOGRAPHY
341
6. Williams, F.J., "Photogrammetry Locates 208 Miles of
Pennsylvania Turnpike Extensions," Civil Engineering, Vol.
20, No. 12, December, 1950, pp. 761-763.
7. Gilbert, G.B., "Photogrammetry Aids the Pennsyl-
vania Turnpike,” Proceedings ASCE, Vol. 80, Separate No.
577, December, 1954.
8. Hooper, C.J., "Photogrammetry and Its Uses in High-
way Planning and Design," Photogrammetric Engineering, Vol.
XVII, No. 1, March, 1951, pp. 133-137.
9. Quinn, A.O., "Photogrammetry Aids Highway Engin-
eers,” Photogrammetric Engineering, Vol. XVIII, No. 5, De-
cember, 1952, pp. 787-790.
10. Meyer, R. W., "Aerial Photography Streamlines Ohio's
Highway Program," Photogrammetric Engineering, Vol. XIX,
No. 5, December, 1953, pp. 771-776.
·
11. Telford, E.T., "Photogrammetry as Applied to High-
way Engineering," Photogrammetric Engineering, Vol. XVII,
No. 1, March, 1951, pp. 175-180.
12. Perkins, E.T., "Use of Aerial Surveys in Highway
Location," Proceedings of Conference on Modern Highways,
Massachusetts Institute of Technology, June, 1953, pp.
70-77.
13. Houdlette, E.C., "Photogrammetry as Applied to
Highway Engineering in Massachusetts," Photogrammetric
Engineering, Vol. XVII, No. 1, March, 1951, pp. 138-143.
14. Brown, I. W., "Photogrammetry as Applied To High-
way Engineering in Mississippi," Photogrammetric Engineer-
ing, Vol. XVII, No. 1, March, 1951, pp. 151-160.
15. “Photogrammetry and Aerial Surveys,” Bulletin 157,
Highway Research Board, 1957, p. 59.
16. McMaster, H.M., and Legault, A.R., "Pavement Con-
dition Surveys by Means of Aerial Photographs," Univ. of
Nebraska, Eng. Exp. Sta., Bulletin No. 1, April, 1952.
17. Johnson, D.S., "Photogrammetry in Highway Plan-
ning," Bulletin 228, Highway Research Board, 1959, pp. 12–20.
18. Cottrell, C.M., Panel: "Engineering Applications of
Photogrammetry," Photogrammetric Engineering, Vol. XX,
No. 3, June, 1954, pp. 516-520.
19. Sternberg, I., "Aerial Mapping in Areas of Heavy
342 AERIAL PHOTOGRAPHY IN ROUTE SURVEYING
Ground Cover," Bulletin 199, Highway Research Board, 1958,
pp. 44-48.
20. Winsor, D.E., "Survey Security Through Photogram-
metry," Bulletin 199, Highway Research Board, 1958, pp. 14-23.
21. Nelson, S., "Aerial Surveys Expedite Highway Plan-
ning," 1944 Group Meeting Book, American Association of
State Highway Officials.
22. Houdlette, E.C., "Massachusetts Adapts Aerial Photo-
graphy to Highway Location," Civil Engineering, Vol. 17,
No. 2, February, 1947, pp. 85 ff.
23. Southard, R.B., Jr., “Orthophotography-Its Tech-
niques and Applications," Photogrammetric Engineering, Vol.
XXIV, No. 3, June, 1958, pp. 443-451.
24. Mahan, R.O., "The Photo-Contour Map," Photogram-
metric Engineering, Vol. XXIV, No. 3, June, 1958, pp. 451-457.
25. Funk, L.L., "Terrain Data for Earthwork Quantities,"
Bulletin 228, Highway Research Board, 1959, pp. 49-65.
26. Funk, L.L., "Adjustment of Photogrammetric Surveys,"
Bulletin 228, Highway Research Board, 1959, pp. 12-20.
CHAPTER 13
AUTOMATION IN LOCATION AND DESIGN
13-1. Foreword.-Automation is entering every aspect of
transportation. Electronic controls have already been ap-
plied to road-building equipment, notably, graders and pavers.
Automatic train control has proved its worth, as has radio
control of highway traffic signals. The present decade will
bring practical applications of startling new developments
in highway operating controls, such as induction radio to
transmit instructions to the driver of a moving vehicle, induc-
tion loops under the road surface to detect the passage of a
vehicle and actuate warning signals to trailing vehicles, and
the complete electronic guidance of streams of traffic on im-
portant arteries. Despite the importance of these subjects,
this chapter is restricted to automation in the field and office
aspects of route location.
The use of automation in location and design is so new that
most of the applications to be described date since publication
of the second edition of this book in 1956. These applications
are largely in the office procedures of mapping and design.
Yet some revolutionary improvements in surveying field work
have also occurred-especially in distance measurement.
13-2. Automation in Field Measurements.-A major break-
through in distance measurement has occurred with the intro-
duction of accurate portable devices that depend on wave
mechanics and electronics. Their development was stimu-
lated by the successful application to geodetic surveying of
Shoran¹*, a blind-bombing instrument used in World War II.
Without Shoran, for example, the islands adjoining the missile
test range southeast of Cape Canaveral would remain geodeti-
cally isolated.
In distance measurement using light waves, an instrument
called the geodimeter transmits a beam of light through crossed
polaroids and a Kerr cell, which modulates the light at a
*Superscript numbers refer to the bibliography at end of this chapter.
343
344
AUTOMATION IN LOCATION AND DESIGN
definite frequency between 10 and 30 mc (megacycles) per
second. A passive mirror or bank of prisms ("slave unit") at
the remote station reflects the light back to a phototube
multiplier in the geodimeter. Here a detector instantaneously
compares the phase of the returning light with that of the beam
just leaving. The phase detector is then moved to zero
("nulled") through manual control, thus bringing the phases
into agreement and yielding readings which, with the velocity
of light known, may be converted to the distance between
the stations.
Geodimeters come in several models, the heaviest of which
weighs over 200 pounds. The accuracy of these non-portable
models is such that they can replace conventional first-order
triangulation with trilateration (measurement of the sides of
a network of triangles). The maximum effective distance is
limited only by attainment of "line-of-sight" conditions; this
approximates 40 to 50 miles except under unusual circum-
stances.
A lighter model geodimeter weighing 58 pounds is well
adapted to route-surveying projects requiring photogram-
metric ground control by precise traverse or trilateration.
This model (Fig. 13-1)² can measure distances up to 20 miles
long with a maximum error of about 4 inches. An even
lighter model (35 pounds) is available for short-range work-
50 feet to 5 miles-on which cheap expendable reflectors may
be used as unmanned receiving stations. This model can
replace taping on most route layout, boundary line surveys,
and traversing in general.
The principal disadvantage of the geodimeter is its depend-
ence on fairly good visibility. The light beam will not pene-
trate dense fog or haze. Measurements over fairly long lines
must be made at night.
Instruments for measuring distances using microwaves have
reached a high degree of efficiency since 1957. These devices,
like geodimeters, are also phase-comparison systems; but their
operating principle is based on the instantaneous sampling of
the phase of independent crystal oscillators located at each
end of the line being measured.
In the tellurometer (Fig. 13-2) a carrier microwave radiated
at the master unit at about 3,000 mc per sec is modulated to
AUTOMATION IN LOCATION AND DESIGN
347
frequencies of 10 mc per sec and others. The remote unit
receives, analyzes, and retransmits the pulses to the master
unit, where the phase shift is portrayed on a cathode-ray
tube and measured with the aid of a calibrated graticule over
the tube. The distance between the units is derived from the
phase shift and the known velocity of radio waves corrected
for humidity and atmospheric pressure.
The master and remote units of the original tellurometer
each weighed 57 pounds including built-in radio-telephone
system, vibrator supply unit, carrying case, and tripod. Fig.
13-2(a) shows a master unit of the original model. A newer
version called "Micro-Distancer" [Fig. 13–2(b)] is lighter and
has the advantage that master and remote units are inter-
changeable. Other tellurometer systems ("Aero-Dist," "Hy-
dro-Dist," "Terra-Fix") are available for special commercial
and military uses.
-
The electrotape³ is a light-weight, all-transistorized measur-
ing system that employs the same basic principle as the
tellurometer. The units are interchangeable and are mounted
on standard surveyor's tripods. Round-trip transmission
times come from a direct-reading counter instead of a cathode-
ray tube.
In contrast to the geodimeter, measuring systems using
microwave transmission are practically independent of the
weather. They are effective day or night, even during fog or
light rain. Like the geodimeter, they require an optical line
of sight and they measure slope distances. Therefore, vertical
angles must also be measured with a theodolite so that slope
distances may be reduced to horizontal.
All measuring systems using light waves or radio waves have
an accuracy which is a certain percentage of the distance plus
or minus a fixed amount. Depending on the type and model
of instrument, the fixed amount (according to manufacturers'
claims) ranges from 1 centimeter to 2 inches; the percentage,
from 1 to 3 millionths of the distance measured. Unbiased
tests2,4,5,6 substantially verify these claims and also indicate
survey cost savings of from 40 to 70 per cent over conven-
tional precise traverse or triangulation. For all but very
short lines, the accuracy attained is far better than that
required on most route-location surveys.
348
AUTOMATION IN LOCATION AND DESIGN
The accurate determination of ground profile by electronics
is inherently more difficult than slope-distance measurement.
However, research on radar altimetry has resulted in an air-
borne profile recorder capable of determining elevations to
+10 feet from an altitude of 45,000 feet. A truck-mounted
elevation meter" with a revolution counter on a fifth wheel
and a pendulum cutting an electromagnetic field has been
designed so that electrical signals generated by counter and
pendulum are combined by an electronic integrator to produce
a continuous record of the road profile; the accuracy is satis-
factory for fourth-order vertical control. Further refinement
in such devices is inevitable.
13-3. Terminology in Automation.-Automation is break-
ing down the traditional distinction between field work and
office work. As an example, making rod readings in cross
sectioning and recording them in a field book is considered to
be field work. But is notekeeping field or office work when
the readings are recorded on Mark Sense cards (Art. 13-6)
which are to be run through automatic office machines?
Among the common terms used in automation are data
procurement, data processing, data reduction, data transmission,
and data presentation. These expressions do not involve bas-
ically new operations for the route surveyor or designer.
Measurements of distances taped in the field and of xyz
coordinates of points read from a stereomodel as the output of
a stereoplotter are both forms of data procurement. Com-
putations made in the field by slide rule and in the office by
electronic computer are merely different forms of data proc-
essing. The stereomat (Art. 13-8), an electronic plotting
instrument, is a data-reduction system; but so is the drafts-
man's hand and contour pen. Mailing a typewritten record
of earthwork quantities from field to office serves the same
purpose in data transmission as sending the record over a
telephone line in the form of binary-coded electrical pulses.
Cross sections are a form of data presentation, whether plotted
in the field directly from rod readings or stripped from a
photogrammetric map and drawn by an automatic line plotter.
In route location and design, the electronic computer serves
as a bridge linking these applications of automation into an
integrated system (Art. 13–9). For this reason it is advisable
to review the fundamentals of modern computers.
AUTOMATION IN LOCATION AND DESIGN
349
13-4. Fundamentals of Modern Computers.-It would be
inconsistent with the title of this book to cover intricate
details of the design, circuitry, and programming of modern
computers; nor is this essential. Most engineers and science-
motivated high-school students know, at least in a general
way, what computers do and how they do it. The purpose of
this article is to review the fundamentals of computers for
readers unfamiliar with them so that the special applications
described later in the chapter will be better understood.
Engineering data may be presented in analog (graphical)
form or in digital (numerical) form. Plotted cross sections
and contour maps are examples of analog form; rod readings
and distances, of digital form. Engineers have long used
mechanical types of analog and digital computers such as the
planimeter (analog) and desk calculator (digital). But the
application of electronic computers to route-location problems
dates only since about 1955.
The modern analog computer is especially suited to the
analysis of flow in networks, e.g., water or gas pipe systems
and electrical circuits; it has also been applied to traffic
engineering problems. But all subsequent references to com-
puters in this book are to the more widely used electronic
digital computer.
The stored-program computer has five basic components:
input, memory (storage), arithmetic, control, and output.
The input unit takes information (problem data and program
instructions) from punched cards. punched tape, magnetic
tape, or a special typewriter and places it in memory. All
items of information must be stored in primary or auxiliary
memory before the computer can solve a problem without
interruption. The arithmetic unit usually performs only four
arithmetic operations: addition, subtraction, multiplication,
and division. (Multiplication may be done by repeated addi-
tion and shifting of the decimal point; division, by repeated
subtraction.) Another function of the arithmetic unit is to
assist in making decisions by comparing two numbers (through
subtraction) and then following either of two prescribed
courses of action depending on which number is the larger.
The control unit decodes the program instructions stored in
memory and directs signals to the other units based on those
instructions and on the results of arithmetic operations. The
350
AUTOMATION IN LOCATION AND DESIGN
output unit records numerical answers and any other specified
data taken from memory, employing one of the forms used for
input or even a graphical display.
In most computers the internal computations are done in
the binary system rather than the decimal-number system.
There are several ways of coding binary variables to represent
one decimal digit. In straight binary, for example, the deci-
mal number 18.625 is written as 10010.101, each digit in the
binary number being the coefficient of a power of 2. Thus
(18.625)10 = (10010.101)2, which equals 1·2ª+0·2³+0•2²+1•2¹+
0·2º+1·2¬¹+0·2-2+1·2˜³. Evaluating these terms gives 16+
2+0.5+0.125, or 18.625. The reason for using binary num-
bers lies in the ease with which they can be represented by
cards, tape, or circuitry. In the computer something is either
present or absent-a particular place on a punched card or
tape either has a hole or does not; a switch or vacuum tube is
either on or off; or a magnetic memory core is magnetized in
one direction or the other at a given point. The digit () is
used to symbolize one condition; the digit 1, the other. These
binary digits are called bits.
It is not essential that the computer user or programmer be
concerned with further details of the binary system. Num-
erical data are "read" into memory by the input device as
decimal numbers. Built-in circuitry converts the numbers to
binary form for computation and back to decimal form for
output.
The medium-size stored-program computer used in most
route-location problems has an internal memory unit capable
of storing from 1,000 to 8,000 combinations of bits, with each
combination (word) in a separate location (address). A prob-
lem to be solved by the computer is broken down to a definite
sequence of elementary steps called the program. Each step
is a coded number which is also stored in memory at a par-
ticular address.
Since the computer can only add, subtract, multiply and
divide, all required algebraic and trigonometric operations.
must be converted to arithmetic. If a problem contains many
special operations, such as finding the square root of a number
or evaluating a function of an angle by means of a power series,
too many addresses in the high-speed memory unit may be
AUTOMATION IN LOCATION AND DESIGN
351
used up. For this reason some of these operations may be
supplied as subroutines in auxiliary memory, usually in the
form of slower but more economical magnetic tape.
Some newer computers are designed to accept input instruc-
tions in a code based on common English terms-and even to
translate simple formulas (e.g., Fortran) into machine lan-
guage for internal processing. Programming is thus simplified.
Because of its electronic circuitry, a computer operates at
very high speed. The over-all time for solving a given prob-
lem depends on the efficiency of the program and the design
of the computer, particularly with respect to access time
(to memory) and to the type of input and output units.
Actual internal operations proceed at rates as high as millions
of bits per minute. Reference 8 gives further non-technical
information about computers.
The advantages and uses of computers in route location and
design are outlined in the next article. Before considering
these applications, however, it is worth remembering that the
computer, though a powerful tool for the engineer, is not his
master. Though future computers may be developed to
function like the human brain, those presently used in high-
way-engineering applications were characterized sensibly as
follows by the Chief of the Electronics Branch of the Bureau
of Public Roads: 9
"The electronic computer is only a calculating machine. The
thinking involved in the solution of an engineering problem must
continue to be done by the engineer. Although the computer can
call upon its memory for information and can make simple logical
decisions, it can do these things only when explicitly directed to do
them by means of instructions in the program. There is no
computer instruction which says, 'Use your own judgment.' The
computer cannot exercise judgment, it cannot reason or accum-
ulate experience, and it cannot create. It will do exactly what it
is directed to do-no more and no less."
13-5. Computer Programs in Route Location. In 1955,
the Arizona State Highway Department and the California
Division of Highways pioneered in adapting electronic data
processing to traverse and earthwork computations. Stirred
by the significance of these applications, regional conferences
on electronics in highway engineering were held in widely
separated States for several years (starting in 1956) under the
joint sponsorship of the American Association of State High-
way Officials, the Bureau of Public Roads, and the host high-
M
352
AUTOMATION IN LOCATION AND DESIGN
way department. No more effective action could have been
taken to stimulate interest among highway engineers, admin-
istrators, and computer manufacturers toward exchange of
ideas and development of computer programs. The con-
ference reports* are invaluable sources of reference on com-
puter applications in all phases of highway operation.
9
Alignment and earthwork computations are repetitive in
nature; they consume countless man-hours that the engineer
could use to better advantage on more professional work.
This characteristic of a lengthy numerical problem-frequent
Occurrence is what makes the use of the computer attractive
and economical. Once a program has been prepared, its high
development cost can be spread over repeated applications.
Computers yield extraordinary savings in computation time.
"For example, in computing the volumes of earth to be
moved in constructing a proposed highway, about 80 to 90
man-hours per mile are required using the traditional method.
In the electronic-computer method, using an intermediate
size computer, only about 15 minutes per mile of computer
time is required. Additional time is, of course, required to
punch the input data on cards or tape, but even including
this, the time consumed is only about one-thirtieth of that
required by the traditional method. Also, because the com-
puter results are obtained entirely by arithmetic computation,
errors and inaccuracies in plotting, measuring, and tran-
scribing data are eliminated."
"In other types of problems, even more impressive savings
are attained. The computation of the dimensions and eleva-
tions needed for laying out, fabricating, and constructing
multispan curved bridges with various skewed piers and
abutments is tedious and time consuming. In one project
of this kind with fourteen spans, a complete solution was
obtained on an intermediate size computer in 11 minutes.
Based on similar problems done previously, it was estimated
that it would have taken 22 man-days to solve the problem
using a desk calculator. This is a time ratio of about one
to a thousand."
*For information on these reports write to Secretary of Committee,
Division of Development, Office of Operations, Bureau of Public Roads,
Washington 25, D.C.
AUTOMATION IN LOCATION AND DESIGN
353
More significant than saving in computation time is the
potential upgrading in the quality of the end result. Tradi-
tional approximations and rules of thumb can be replaced by
more exhaustive treatments based on exact theory. The
computer can handle solutions of special alignment problems
by analytic geometry (Art. 7-19) with great speed and ease.
Refinements in design formerly avoided to simplify computa-
tions become practicable-such as more careful fitting of the
grade line to terrain by use of unequal-tangent vertical curves
(Art. 4-6). As a result of evaluating more possible routes by
use of the computer, the savings in the cost of building a
project—and particularly in the capitalized saving in annual
operation and maintenance costs-can far exceed the mone-
tary value of saving in computation time.
When first applied to route location, the digital computer
merely replaced the desk calculator in performing familiar
routine computations. Among these were: (1) calculation of
a closed traverse including error of closure, balancing, coordi-
nates of stations, and area enclosed; (2) determination of any
two items of missing data in an open traverse; and (3) calcula-
tion of earthwork quantities from cross sections by any of the
methods described in Arts. 6-8 to 6-10.
As the spectacular time savings became known, computer
programs were developed for all phases of horizontal and
vertical alignment-straight and curved, for structural design,
traffic problems, soils analyses, and administrative procedures.
And the use of computers mushroomed. In 1960, for example,
the California Division of Highways processed each month an
average of 125,000 traverse courses and over 300 miles of
earthwork calculations alone.10 By 1961, 49 of the 53 member
departments of AASHO were using computers in highway
work.11
The financial burden of program development has been
eased by the formation of "user groups" composed of highway
departments and consulting firms using the same make and
model of computer. However, as of 1961, at least twelve
different computers were in use on highway work, each having
its own coding system and terminology. In order to over-
come difficulties in exchange of programs, the Bureau of
Public Roads has a Computer Program Library that serves as
354
AUTOMATION IN LOCATION AND DESIGN
an agency for the receipt and distribution of programs used in
all fields of highway operation. Perhaps the most valuable
feature of this service is the conversion of programs to a
"library form" consisting of common English and mathe-
matical terms and standardized flow-charting symbols. As a
result, any BPR program can be readily coded for use with
any computer. A library memorandum which catalogues all
programs on file and all those available in library form is
distributed periodically. (For information on these memo-
randa, see address in footnote on page 352.) Memorandum
No. 8 dated November, 1960 gives the details of 30 programs
in library form and lists 349 other programs on file. Abstracts
of four of the library-form programs follow:
PROGRAM BPR SU-2
Traverse Computation-developed in Cali-
fornia Division of Highways
This program is designed to perform the trigonometric computations
encountered in surveys, or in any problem involving the relations
between the sides and angles of triangles. It will deduce from
given elements, other required elements, calculate area and produce
an orderly tabulation of courses with their computed or known
factors. Provision is made for some interdependence of traverses,
allowing data from specified courses to be stored for later use.
Included in the program is a routine for the solution of horizontal
curve data.
PROGRAM BPR E-2 Earthwork Quantities-developed in Missouri
Highway Department
This is a design program which is used for both undivided and
divided roadways. In addition, it computes rock excavation
quantities separately from earth excavation quantities. The
program has 7 segments:
1. Station sequence check
2. Conversion of rod readings to elevations
3. Computation of template points and corrections for super-
elevation and variable width
4. Slope selection and slope stake coordinates
5. Total areas
6. Earth and rock areas
7. Shrinkage, volumes and cumulative volumes
PROGRAM BPR HA-2
Horizontal Alignment--developed as Digital
Terrain Model* Systems by Massachusetts
Institute of Technology under sponsorship
of Massachusetts Department of Public
Works in cooperation with the Bureau of
Public Roads
*See Art. 13-9 for explanation of the Digital Terrain Model System.
AUTOMATION IN LOCATION AND DESIGN
355
Part 1-Basic Horizontal Alignment Program computes centerline
geometry of alignment including the station of each PI, TC,
and CT; external angle between tangents and tangent dis-
tance for each curve; azimuth of each tangent; and center-
line station, baseline offset, ground elevation at centerline
and skew angle for each terrain cross section.
Part 2-Even Station Interpolation Program computes ground ele-
vations at equal intervals (either full stations or plusses)
for plotting a ground profile.
Part 3-Parallel Offset Alignment Program computes geometry of
two parallel offset lines as well as the centerline and relates
it to DTM terrain data.
Part 4-Special Alignment Geometry Program computes centerline
geometry only without reference to DTM terrain data.
Vertical Alignment-developed as in Pro-
gram HA-2
PROGRAM BPR VA-1
Part 1-Basic Vertical Alignment Program computes profile geome-
try including the station and elevation of each end of vertical
curves, the gradient for each tangent, and the profile grade
elevation at each station where a cross section was taken.
Part 2-Profile Geometry Program computes profile grade elevations
at constant interval stations (either full stations or plusses).
Successful computer programs are not yet in their final
form. Further improvements are inevitable. A promising
new approach is represented by COGO12 (derived from
Coordinate Geometry). COGO is a programming system
which frees the engineer from having to force a solution to an
established form. It gives him full control over the form of
input/output and the sequence of mathematical operations.
Moreover, a COGO program is written quickly on any sheet
of paper, used once, and then discarded. No costly program
investment is involved.
13-6. Recording Field Data.-The worst bottleneck in the
efficient use of the computer is in preparing input data. It
takes far longer to punch cards or tape with readings from
field notebooks than it does for the computer to process them
and perform the computations programmed. Moreover, mis-
takes arising from illegible figures and from human lapses in
transcribing data are difficult to avoid.
Methods of reducing these effects show great promise. The
Arizona and Texas¹³ highway departments eliminate hand-
written cross section field notes by substituting IBM Mark
Sense Cards. These cards are imprinted with rows and
356
AUTOMATION IN LOCATION AND DESIGN
columns of small ellipses identified by the digits 0 through 9.
In the field, the station and plus, leveling readings, and cross
section notes are recorded by marking the proper ellipses with
a special graphite pencil. The cards are collated in the office
and the marks are read by an interpreting machine as the
cards pass under metal brushes. Electrical contacts made
through the graphite cause holes corresponding to the field
readings to be punched in another deck of cards. These
cards are processed through a printer to furnish the field
office with a typed record of the cross section data; they also
serve as input to the computer for use in the preliminary or
final earthwork programs. The Remington Rand Optical
Scanning Punch¹¹ operates in much the same way.
Several highway departments have used other devices for
recording field data. These include a pocket-size card punch,
a portable imprinter for tape, and field dictating equipment.15
13-7. Data Transmission.-High-speed transmission of in-
ter-office data in highway work takes several forms. Teletype
has been used, but because of its limitations in speed and
format it is being superseded by methods employing elec-
tronics, microwave radio, and optics.
Commercial systems are available whereby signals in binary
code are shaped for transmission over regular telephone lines
at speeds of thousands of bits per second and reshaped at a
remote receiver into the original form transmitted. Digital
data for input to these transceiver systems may be punched
cards, punched tape, or magnetic tape, with output in any of
these media or in hard copy readout. Such a system will
probably be adopted by most very large highway organizations
having outlying offices and a centralized computer complex.
Microwave radio is used in somewhat similar data-trans-
mission systems, and has the advantage of being free from
line failures during storms.
Analog as well as digital data can be transmitted by a
facsimile printing system that employs optical-electronic
scanning.¹4 In this process the document, map, or photo-
graph to be reproduced is scanned by a spot of light generated
by a cathode-ray tube. The light is reflected to a photo-
multiplier and converted to electrical pulses for transmission
over coaxial cable or microwave relay system. At the re-
358
AUTOMATION IN LOCATION AND DESIGN
to continuous line representations such as profiles or cross
sections.
A further step toward automation in plotting is represented
by an electronically-activated instrument formerly called
"Auscor" (Automatic Scanning Correlator)16, an improved
form of which was patented in 1961 and is marketed under
the name Stereomat. This device is fitted to a double-projec-

D
balan
F
(A) Aerial Photos
Negatives
Diapositives
B
C
Stereomodel
Measuring
System
Terrain Data
(xyz coords. of points)
Visual
Display
Computer Programs
For example, HA-2,
VA-1, E-2 (Art. 13-5)
Printed Copy
Detailed align-
ment, profile,
coordinates of
Computer Input E
(Cords or tape)
cross sections,
earthwork quan-
tities, etc.,
e. g., HA-2, VA-1,
E-2 (Art. 13-5)
Computer
Memory
Computer
Arithmetic
Computer
Output
Electrical
Signals
On-Line
Plotter
G
H
Field
Surveys
Topographic
Map
Preliminary Studies
of Trial Lines
(Involving right of way,
alignment, grades, drain-
age, earthwork, etc.)
Controlling
Dimensional Data
For detailed study.
of trial and final
horizontal and vert-
ical alignments
Punched
Cards or Tape
1
Off-Line K
Plotter
Drawings
(Plan, pro-
file, mass
diagram,
x - sections)
Fig. 13-4. Flow diagrams in highway design
2
4
O
L
3
AUTOMATION IN LOCATION AND DESIGN
359
tion stereoplotter and achieves automation in the operations of
clearing parallax and guiding the floating mark so as to
produce profiles or contours as well as in determining spot
elevations.
One of the photogrammetric engineer's goals is a completely
automated, integrated mapping system. That its attainment
is in sight is supported by a progress report¹ on an electron-
ically-controlled system in which stereoplotter, coordinato-
graph, and orthophotoscope (Art. 12-8) are integrated.
13-9. Integration of Photogrammetry and Computer.-
Before photogrammetry and electronics were adapted to route
location, the usual procedure in design followed the sequence
traced through Fig. 13-4 by 1-2-3-4-H-1-H-I-L (see Chap-
ters 9 and 10). The computations (H) were done with the
aid of logarithms, desk calculator, and planimeter; drawings
(L) were plotted manually. With the advent of automation,
however, several systems were developed for mechanizing
these operations.
In California an economical method of obtaining digital
terrain data for determining highway earthwork quantities is
based on key punching for computer use directly from strips
of map manuscript.18 This is a special example of semi-
automation.
The Digital Terrain Model (DTM) System, 19 developed non-
commercially at the Massachusetts Institute of Technology,
is a prototype of a system which effects the maximum inte-
gration of photogrammetry and computer in highway work.
Other functionally-similar systems differ from DTM princi-
pally in the design and arrangement of the components and
in the extent to which automation is used. These include
unique systems assembled for their own use by the Ohio
Department of Highways, by commercial mapping agencies,
and by consulting engineering firms, e.g., Photronix, Inc. No
one of these automated systems is portrayed perfectly by the
oversimplified diagram in Fig. 13-4. The best approximation
to the DTM System is the sequence: A-B-2; B or 2-C-D-E;
E, F, and 4-G-H-I and/or J-K-L.
A digital terrain model is a statistical representation of the
ground surface within a route band by the xyz coordinates of a
large number of points. When stored on computer input
AUTOMATION IN LOCATION AND DESIGN
361
the horizontal guide bar (cross section). Gear trains convert
shaft positions to a digital coded output for storage in the
control chassis. When the floating mark is on the ground at a
desired point, the plotter operator steps on a foot-treadle
switch. This action (by the controls exercised in the para-
meter board, chassis, and programmer assembly) causes the.
station number and the yz coordinates to be typed on hard
copy and punched on paper tape in a standard code. An
automated stereoplotter such as the Stereomat would elim-
inate manual operation of the plotter.16
Many engineers feel that there is no good substitute for a
topographic map on which the designer can lay his trial lines
and visualize many other matters pertaining to his design.
However, automatic readout of terrain data directly from a
stereomodel theoretically eliminates the need for a map. In
test comparisons on a highway project in Pennsylvania, 20
controlling dimensional data for trial lines were taken success-
fully from banks of multiplex models rather than from a
topographic map. Obviously this matter needs thorough
study from a fresh, unbiased viewpoint. Even if a map
is used, extraction of terrain data from the model can be done
independently and filed in the computing center ready for
use whenever the map has been compiled.
The computer has almost entirely supplanted the tradi-
tional cross section and planimeter method of determining
grading quantities. In States where cross sections are still
included in the construction plans,21 they may be drawn by an
off-line plotter (K) of the type shown in Fig. 13-3. However,
a survey22 shows a definite trend toward digital cross sections
in lieu of drawings. Instead of a large number of drawn
cross sections, a few sheets of numerical data (I) supply even
more information in a form felt to be superior by both engi-
neers and contractors.23 In California where this is done, it
is the practice to supply a few sample cross sections in the
contract plans merely for pictorial purposes.24
The terrain data (D) may be taken along a particular trial
alignment as in the traditional method of design. This pro-
cedure requires repeating the readout operation for each new
alignment, but it has advantages where topography limits the
line to only a few logical trials. By contrast, in the DTM
362
AUTOMATION IN LOCATION AND DESIGN
System all terrain data in the route band are taken from a
generalized baseline and stored at one time; readings do not
have to be retaken with each change in alignment. This
contributes to the ease of carrying out the premise on which
the DTM System is built, i.c., that the best use of computer
time is for studying a large number of alignments and profiles.
However, the fact that DTM output data are for skewed
sections (with respect to the route alignment) creates some
problems. These are discussed in a series of searching evalu-
ation studies in the report* that includes Reference 20.
Fig. 13-4 does not portray the details of the computer
phase. With the medium-size computer used by most State
Highway Departments, it is necessary to run appropriate data
through the computer in a series of passes. Eventually some
States may join in integrating these smaller computers with
regionally-located and financed super-computers each having
memory storage large enough to handle all programs on im-
portant projects in one pass. If line representations of cross
sections are to be continued as part of the construction plans,
the slower off-line plotter may be replaced by a cathode-ray
tube and camera assembly connected on-line with the com-
puter in such a way that cross sections are portrayed and
photographed rapidly and automatically. This was done to a
limited degree in Ohio.25
A special problem involving the integration of photogram-
metry and computer is that of analytic aerotriangulation.
Aerotriangulation uses measurements between points on aerial
photographs for the purpose of determining the corresponding
ground positions of the points. The process, if done analyti-
cally, is more complex than geodetic triangulation because the
lines joining the ground points and aerial camera position
must be treated as a system of rays in three dimensional space.
In route surveying, instrumental aerotriangulation has been
used for many years to extend control through a chain of
stereomodels by a process called "bridging." With the im-
provement in stereoplotters, bridging serves to reduce the
amount of costly ground control. An excellent example of
the advantages of bridging is described in Reference 21.
Because of the complexity of the mathematical relationships
*For further information, see footnote on page 352.
BIBLIOGRAPHY
363
in aerotriangulation, most of the early attempts to accomplish
the process analytically involved approximations, and all were
time-consuming.26 With the advent of the computer, how-
ever, analytic aerotriangulation can now be done accurately
and rapidly. The U. S. Coast and Geodetic Survey devised a
system in which the positions of points on aerial photographs
are read to the nearest micron by a precise comparator. The
measurements are transmitted to an electronic readout device
and simultaneously punched on computer input material.
Finally, the computer processes the data through a program
built around the correct mathematical formulas and delivers
the positions and elevations of the points.
BIBLIOGRAPHY
1. Aslakson, C.I., "The Importance of Shoran Surveying,"
Transactions, ASCE, Vol. 120, 1955, pp. 225-234.
2. Alexander, I.H., and Goldin, A.K., "Sierra Highway-
Geodimeter Speeds Relocation Survey," California Highways
and Public Works, May-June, 1960, pp. 16 ff.
3. Hempel, C.B., "Electrotape-A Surveyor's Electronic
Eyes," Surveying and Mapping, Vol. XXI, No. 1, March,
1961, pp. 85-88.
4. Demuth, H.P., "Tellurometer Traverse Surveys," Tech-
nical Bulletin No. 2, U.S. Coast and Geodetic Survey, March,
1958.
5. Jones, E.J., "An Electronic Break-Through in Survey-
ing," Civil Engineering, Vol. 29, No. 7, July, 1959, pp. 479-481.
6. "Electronic Surveying-1960 Developments," Bulletin
258, Highway Research Board.
7. Moore, R.H., "What's New in Surveying Instruments,”
Civil Engineering, Vol. 29, No. 8, August, 1959, pp. 550-553.
8. Merritt, F.S., "What You Should Know About Com-
puters," Engineering News-Record, Vol. 164, No. 15, April 14,
1960, pp. 39-63.
9. Schureman, L.R., 'The Use of Electronic Computers in
Highway Engineering,” Miller-Smith Lecture Series, Rutgers
University, April 20, 1959.
364
AUTOMATION IN LOCATION AND DESIGN
10. Reynolds, F.M., "Data Processing," California High-
ways and Public Works, March-April, 1960, pp. 39-42.
11. Radzikowski, H.A., "Advanced Techniques in the High-
way Departments," American Highways, Vol. XL, No. 2,
April, 1961, pp. 11 ff.
12. Miller, C.L., "COGO-A Computer Programming Sys-
tem for Civil Engineering Problems," Civil Engineering Sys-
tems Laboratory, Massachusetts Institute of Technology, August
15, 1961.
13. Dingwall, J.C., "Mark Sensing-Field Recording of
Earthwork Cross Sections," American Highways, Vol.
XXXVIII, No. 2, April, 1959, pp. 9 ff.
14. Schureman, L.R., "Optical-Electronic Scanning-
Planes, Lines and Print," American Highways, Vol. XL, No. 2,
April, 1961, pp. 12 ff.
15. Nielsen, T.R., "Data Transcription from Maps and
Field by Use of Dictating Equipment," American Highways,
Vol. XXXVIII, No. 2, April, 1959, p. 10.
16. "Oscar' Can Cut Road Costs By Millions," Engineering
News-Record, Vol. 160, No. 25, June 19, 1958, pp. 23-24.
17. Boyajean, J., "The Implementation of the Integrated
Mapping System," Photogrammetric Engineering, Vol. XXVII,
No. 1, March, 1961, pp. 55-60.
18. Funk, L.L., "From Map to Computer," Bulletin 283,
Highway Research Board, 1960, pp. 49-55.
19. Miller, C.L., and LaFlamme, R.A., "The Digital Ter-
rain Model—Theory and Application," Photogrammetric En-
gineering, Vol. XXIV, No. 3, June, 1958, pp. 433-442.
20. Richardson, E.C., "An Evaluation of the DTM System
of Electronic Computer Programs for Highway Location and
Design," Report of Electronics Committee, AASHO, San
Francisco, Calif., December 4-5, 1958, pp. 62-70.
21. "Eleven Miles of Interstate Designed in 16 Weeks,"
Engineering News-Record, Vol. 161, No. 13, Sept. 25, 1958,
pp. 42-48.
BIBLIOGRAPHY
365
22. Huffine, W.B., "Highway Department Usage of Ad-
vanced Techniques," American Highways, Vol. XL, No. 2,
April, 1961, pp. 14 ff.
23. Schlitt, H.G., "Our Experience on the Use of Digital
Cross-Section Presentations," Report of Electronics Committee,
AASHO, Boston, Mass., October 15-16, 1959, pp. 15-17.
24. Funk, L.L., "Applications of Photogrammetry to the
Location and Design of Freeways in California," Bulletin 157,
Highway Research Board, 1957, pp. 26-38.
25. "Ohio Mechanizes Highway Design," Engineering
News-Record, Vol. 158, No. 11, March 14, 1957, pp. 37 ff.
26. McNair, A.J., "General Review of Analytical Aero-
triangulation," Photogrammetric Engineering, Vol. XXIII, No.
3, June, 1957, pp. 573-582.
PART III
TABLES
367
I.
II.
III.
LIST OF TABLES
Radii, Deflections, Offsets, etc....
Lengths of Arcs and True Chords.
XVII-A.
Correction Coefficients for Subchords. Chord
Definition of D..
IV. Correction Coefficients for Subchords. Arc
Definition of D………..
381
V.
Even-Radius Curves. Deflections and Chords 382
VI. Lengths of Circular Arcs; Radius 1....... 384
VII. Minutes and Seconds in Decimals of a Degree 385
VIII. Tangents and Externals for a 1° Curve.
386
409 '
IX. Corrections for Tangents. Chord Def. of D..
X. Corrections for Externals. Chord Def. of D..
XI. Selected Spirals.
409
411
XII. Spiral Functions for Ls
= 1
420
XIII. Coefficients for Curve with Equal Spirals.... 429
XIV. Tangents and Externals for Unit Double-
Spiral Curve....
XV. Deflection Angles for 10-Chord Spiral..
XVI. Coefficients of a₁ for Deflections to any Chord
Point on Spiral. .
XVI-C. Corrections to Table XVI for Large Deflec-
tions..
•
XVII. Level Sections..
•
Corrections to Table XVII for Transverse
Ground Slopes..
*****
XVII-B. Corrections to Table XVII for Transverse
Ground Slopes.
•
•
368
380
•
•
381
430
432
447
458
460
XVIII. Triangular Prisms. Cubic Yards per 50 feet 448
XIX. Cubic Yards per 100-Foot Station.
XX. Natural Trigonometric Functions.
XXI. Logarithms of Numbers...
XXI-A. 7-Place Logarithms of Numbers from 1 to 100 575
XXI-B. 7-Place Logarithms of Useful Constants..
551
575
576
XXII. Logarithmic Sin, Cos, Tan, and Cot...
XXIII. Stadia Reductions....
622
630
XXIV. Turnout and Crossover Data.
XXV. Trigonometric Formulas..
637
XXVI. Squares, Cubes, Square Roots, Cube Roots,
and Reciprocals..
438
439
440
447
639
368
TABLE I.-RADII, DEFLECTIONS, OFFSETS, ETC.
Chord Definition of Degree of
Curve*
(Dc)
MATHEMATICAL RELATIONS
1. sin
sin D.=
50
R
1002
R
•
Arc Definition of Degree of Curve*
(Da)
MATHEMATICAL RELATIONS
exactly 1. Da=
5,729.58
Ꭱ
exactly
2. C.O. =200 sin D. 2. C.O.=4 R sin² Da
exactly
exactly
3. T.O. = C........ exactly 3. T.O.C.O...... exactly
4. M.O. = C.O…………. approx.
C.O.....
4. M.O.
.-c.......
C.O………. approx.
8
Notes
1) For values of Da less than 12°, the C.O. (not tabulated)
may be taken equal to the tabulated C.O. for the correspond-
ing value of De. Exact values of C.O. are tabulated for both
definitions of D, where D equals or exceeds 12°.
2) For values of D. and Da less than 12°, relation 4 will
give M.O. without perceptible error. Beyond D.-12°, the
exact values of M.O. are tabulated.
is numerically the
M.O.
3) For any value of D, the ratio
C.O.
same for both definitions of D. Thus, if Dɑ were 52°, the
11.54
M.O. equals
(84.69), which is 11.17.
87.67
a
4) To obtain c.o., t, and m.o. for a chord or arc shorter than
100 ft, multiply C.O., T.O., and M.O. by (chord or are
100
* See Arts. 2-4 and 2–14.
369
TABLE I.-RADII, DEFLECTIONS, OFFSETS, ETC.

DEGREE
OF
CURVE
D
0° 0'
I
GOVOG EWN.
51
6'
י7
81
9'
10'
יוד
12'
131
141
151
16'
י7ו
181
19'
20'
211
22'
231
241
25'
261
27'
28'
291
30'
311
32'
331
341
351
36'
371
381
391
40'
411
421
·431
44'
45'
46'
471
48'
49'
50!
511
52'
531
54'
55'
56'
57'
58'
59'
DEFL.
PER FT
OF STA.
(MIN)
0.005
0.01
0.015
C.02
0.025
0.03
0.035
0.04
0.045
C.05
0.055
0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
C.I
C.105
0.11
0.115
0.12
0.125
0.13
0.135
-0.14
0.145
0.15
0.155
0.16
0.165
0.17
0.175
0.18
0.185
0.19
0.195
0.2
0.205
0.21
0.215
0.22
0.225
C.23
0.235
0.24
0.245
0.25
0.255
C.26
0.265
0.27
0.275
0.28
0.285
0.29
0.295
RADIUS
R
CHORD DEFINITION
LOG R
Infinite
343775.
171887.
114592.
Infin.
5.536274
5.235244
5.059153
85943.7 4.934214
68754.9 4.837304
57295.8 4.758123
49110.7 4.691176
42971.8 4.633184
38197.2 4.582031
34377.5 4.536274
31252.3
4.494881
4.457093
4.422331
28647.8
26444.2
24555.4 4.39C146
19098.6 4.281002
18093.4
4.257521
17188.8 4.235244
16370.2 4.214055
15626.1 4.193852
14946.8 4.174547
14324.0
4.156064
13751.C 4.138335
13222.1
4.121302
12732.4
4.104911
12277.7 4.089117
11854.3 4.073877
11459.2
11089.6
10743.0
10417.5
10111.1
22918.3 4.360183 C.44
21485.9
4.332154
0.47
20222.1 4.305825
0.49
4.059154
4.044914
4.031125
4.017762
4.004797
9822.18 3.992208
9549.34 3.979973
9291.25 3.968074
9046.75 3.956493
8814.78 3.945212
8594.42 3.934216
8384.80 3.923493
8185.16 3.913027
7994.81 3.902808
7813.11 3.892824
7639.49 3.883065
7473.42 3.873519
7314.41 3.864179
7162.03 3.855036
7015.87 3.846082
6875.55 3.837308
6740.74 3.828708
6611.12 3.820275
6486.38 3.812002
6366.26 3.803885
C.O.
1 STA.
6250.51 3.795916
6138.90 3.788091
6031.20 3.780404
5927.22 3.772851
5826.76 3.765427
C.03
0.06
0.09
0.12
C.15
0.17
0.20
C.23
0.26
C.29
0.32
0.35
0.38
0.41
0.52
0.55
0.58
0.61
0.64
0.67
0.70
0.73
0.76
0.79
0.81
0.84
0.87
0.90
C.93
0.96
0.99
1.02
1.05
1.07
….[ ]
1.13
1.16
1.19
1.21
1.25
1.28
1.31
1.34
1.37
1.39
1.43
1.45
1.48
1.51
1.54
1.57
1.60
1.63
1.66
1.69
1.72
ARC DEFINITION
RADIUS
R
Infinite
343775.
171887.
114592.
Infin.
5.536274
5.235244
5.059153
85943.7 4.934214
68754.9
57295.8
49110.7
42971.8
38197.2
22918.3
21485.9
20222.0
34377.5
4.536274
31252.2
4.494881
28647.8 4.457093
26444.2
4.422331
4.390146
24555.3
19098.6
18093.4
17188.7
16370.2
15626.1
14946.7
14323.9
LOG R
13751.C
13222.1
12732.4
12277.7
11854.3
4.837304
4.758123
4.691176
4.633184
4.582031
A
4.360183
4.332154
4.305825
4.281001
4.257520
4.235244
4.214055
4.193851
4.174546
4.156063
4.138334
4.121300
4.104910
4.089116
4.073876
||459.2
4.059153
11089.5 4.044912
10743.0
10417.4
10111.0
4.031124
4.017760
4.004795
9822.13
3.992206
9549.29
3.979971
3.968072
9291.21
9046.70 3.956490
8814.73 3.945209
8594.37 3.934214
8384.75 3.923490
8185.11 3.913025
7994.76 3.902805
7813.06 3.892821
7639.44 3.883061
7473.36 3.873516
7314.35 3.864176
7161.97 3.855033
7015.81 3.846078
6875.49 3.837304
6740.68 3.828704
6611.05 3.820270
6486.31 3.811998
6366.20 3.803880
6250.45 3.79591|
6138.83 3.788086
6031.14 3.780399
5927.15 3.772846
5826.69 3.765422
370
TABLE I.-RADII, DEFLECTIONS, OFFSETS, ETC.

DEGREE
OF
CURVE
D
1° 0'
|
21
31
41
51
6'
י7
81
91
10'
111
12'
131
14'
151
16'
171
181
19'
201
211
22'
231
241
251
26'
271
281
291
30'
311
32'
331
341
351
361
371
38'
391
40'
411
42'
431
441
451
46'
471
481
491
50'
511
521
531
541
551
56'
571
581
59'
DEFL.
PER FT
OF STA.
(MIN)
0.3
0.305
0.31
0.315
0.32
0.325
0.33
0.335
0.34
0.345
0.35
0.355
0.36
0.365
0.37
0.375
0.38
0.385
0.39
0.395
0.4
0.405
0.41
0.415
0.42
0.425
C.43
0.435
0.44
0.445
0.45
0.455
0.46
0.465
0.47
0.49
0.495
0.5
0.505
0.51
0.515
0.52
RADIUS
R
CHORD DEFINITION
LOG R
3.758128
5729.65
5635.72
3.750950
5544.83 3.743888
5456.82
3.736939
5371.56 3.730100
0.575
0.58
0.585
0.59
0.595
5288.92
3.723367
5208.79
3.716737
5131.05 3.710206
5055.59 3.703772
4982.33 3.697432
4583.75
3.661221
4523.44
3.655469
4464.70
3.649792
4407.46 3.644189
4351.67 3.638656
4911.15
3.691183
2.03
4841.98
3.685023
2.07
4774.74
3.678949 2.09
4709.33 3.672959 2.12
4645.69 3.667051
2.15
4044.51
3997.49
3951.54
3906.64
3862.74
0.475
3618.80 3.558564
3581.10
0.48
0.485 3544.19 3.549517
3508.02
3472.59
3819.83
3777.85
3736.79 3.572499
3696.61 3.567804
3657.29 3.563160
4297.28
2.33
3.633194
4244.23 3.627799
2.35
4192.47 3.622470
4141.96 3.617206
2.39
2.41
4092.66 3.612005 2.44
3.606866
3.601787
3.596766
3.591803
3.586896
0.55
3125.36
0.555 3097.20
0.56
3069.55
0.565 3042.39
0.57
3015.71
3.582044
3.577245
3437.87
3403.83
3370.46
3337.74
3305.65 3.519257
2989.48
2963.72
2938.39
2913.49
2889.01
C.O.
1 STA.
0.525
3274.17 3.515101
3243.29 3.510985
0.53
0.535 3212.98 3.506908
0.54
3183.23
3154.03
3.502868
3.498866
0.545
1.75
1.77
1.80
1.83
1.86
1.89
1.92
1.95
1.98
2.01
2.18
2.21
2.24
2.27
2.30
2.47
2.50
2.53
2.56
2.59
2.76
3.554017 2.79
2.82
3.545063 2.85
3.540654
2.88
3.536289 2.91
3.531968
2.94
3.527690
2.97
3.523453 3.00
3.03
ARC DEFINITION
3.494900
3.20
3.490970 3.23
3.487075
3.26
3.29
3.483215
3.479389
3.32
RADIUS
R
3.475596
3.35
3.471836
3.37
3.468109
3.40
3.464413 3.43
3.460749 3.46
5729.58 3.758123
5635.65 3.750944
5544.75
3.743882
5456.74
3.736933
5371.48
3.730094
5288.84
3.723360
5208.71
3.716730
5130.97
3.710199
5055.51 3.703765
4982.24
3.697425
4911.07
3.691176
4841.90
3.685015
4774.65 3.678941
4709.24
4645.60
4583.66
4523.35
4464.61
4407.37
4351.58
4297.18
4244.13
4192.37
4141.86
4092.56
4044.41
3997.38
3951.43
3906.53
3862.64
LOG R
2.62
3819.71
3.582031
2.65
3777.74 3.577232
2.68
3736.68
3.572486
3.567791
2.71 3696.50
2.73
3657.18
3.563146
3618.68
3580.99
3544.07
3507.91
3472.47
3.05
3274.04
3.08 3243.16
3.11
3212.85
3183.10
3.14
3.17
3153.90
3.672951
3.667042
3.661213
3.655460
3.649783
3.644179
3.638647
3.633184
3.627789
3.62246C
3.617196
3.611995
3.606855
3.601775
3.596755
3.591791
3.586884
3.558550
3.554003
3.549502
3.545048
3.540638
3437.75
3.536274
3403.71
3.531952
3370.34 3.527673
3337.62
3.523437
3305.53
3.519241
3.515085
3.510968
3.506890
3.502850
3.498847
3125.22 3.494881
3097.07
3.490951
3069.42
3.487056
3042.25
3.483195
3015.57
3.479369
2989.34
3.475576
2963.58 3.471816
2938.25
3.468087
2913.34
3.464392
2888.86
3.460727
371
TABLE I.-RADII, DEFLECTIONS, OFFSETS, ETC.

DEGREE
OF
CURVE
D
2° 0'
ין
21
31
4'
5'
6'
י7
8'
9'
10'
||'
121
131
141
151
161
171
18'
19'
201
211
22'
231
241
25'
26'
27'
28'
291
30'
311
321
331
341
351
36'
371
381
391
40'
411
42'
431
44'
45'
46'
471
48'
49'
50'
51'
52'
53'
54'
55'
561
57'
58'
591
DEFL.
PER FT
OF STA.
(MIN)
C.6
C.605
0.61
C.615
C.62
0.625
0.63
C.635
0.64
C.645
0.65
0.655
0.66
C.665
C.67
0.675
0.68
0.685
0.69
0.695
C.7
0.705
0.71
0.715
0.72
0.725
0.73
0.735
0.74
0.745
0.75
0.755
0.76
0.765
0.77
0.775
0.78
C.785
0.79
0.795
0.8
C.805
0.81
0.815
0.82
C.85
0.855
RADIUS
R
0.86
0.865
0.87
CHORD DEFINITION
LOG R
3.457115
2864.93
2841.26 3.453511
2817.97 3.449937
3.446392
3.442876
2795.06
2772.53
2750.35
3.439388
2728.52 3.435928
27C7.C4 3.432495
2685.89
3.429089
2665.08 3.42571C
2644.58
2624.39
26C4.51
3.422356
3.419029
3.415727
2584.93 3.412449
2565.65 3.409197
2546.64 3.405968
2527.92 3.402763
2509.47 3.399582
2491.29 3.396424
2473.37 3.393289
2455.70 3.390176
2438.29
2421.12
2404.19
2387.50
2371.04
3.374938
2354.80 3.371954
2338.78 3.368990
2322.98
2307.39
3.387085
3.384016
3.38C969
3.377943
2261.86
2247.08
2232.49
2292.0I
3.360217
2276.84 3.357332
3.354466
3.351618
3.348789
2218.09
2203.87
3.366046
3.363122
3.345797
3.343187
2189.84
3.340412
2175.98 3.337655
2162.30
3.334916
0.825 2083.68 3.318832
0.83
2071.13 3.316208
2058.73 3.313600
0.835
C.84
2046.48 3.311008
2034.37 3.308431
0.845
2148.79 3.332193
2135.44 3.329488
2122.26 3.326799
2109.24
3.324127
2096.39 3.321471
2C22.41
2010.59 3.303323
1998.90
3.300791
1987.35 3.298274
1975.93 3.295771
0.875 1964.64
3.293283
0.88
1953.48
3.290809
0.885
1942.44 3.288349
0.89
1931.53
1920.75
0.895
C.O.
1 STA.
3.285902
3.283470
3.49
3.52
3.55
3.58
3.61
3.64
3.66
3.69
3.72
3.75
3.78
3.81
3.84
3.87
3.90
3.93
3.96
3.98
4.01
4.04
4.10
4.13
4.16
4.19
4.22
4.25
4.28
4.30
4.33
4.36
4.39
4.42
4.45
4.48
4.51
4.54
4.57
4.60
4.62
4.65
4.68
3.305869 4.94
4.71
4.74
4.77
4.07 2455.53
2438.12
2420.95
2404.C2
4.80
4.83
4.86
4.89
4.92
4.97
5.00
5.03
5.06
ARC DEFINITION
5.09
5.12
5.15
5.18
5.21
RADIUS
R
2864.79
2841.11
3.457093
3.453488
2817.83 3.449914
2794.92
3.446369
2772.38
3.442852
2750.20
3.439364
2728.37
3.435903
2706.89 3.43247C
2685.74
3.429064
2664.92
3.425684
2644.42
2624.23
2604.35
2584.77
2565.48
2546.48
2527.75
2509.30
2491.12
2473.20
2370.86
2354.62
2338.60
2322.80
2307.21
2291.83
2276.65
2261.68
2246.89
2232.30
3.380938
2387.32 3.377911
2217.90
2203.68
2189.65
2175.79
2162.IC
2148.59
2135.25
2122.07
2109.05
2096.19
LOG R
2083.48
2070.93
2058.53
2046.28
2034.17
2022.20
2010.38
1998.69
1987.14
1975.72
3.422330
3.419002
3.415700
3.412422
3.409169
3.405940
3.402735
3.399553
3.396395
3.393259
3.390145
3.387055
3.383985
3.374905
3.371921
3.368956
3.366012
3.363087
3.360183
3.357297
3.354430
3.351582
3.348753
3.345942
3.343149
3.340374
3.337617
3.334877
3.332154
3.329448
3.326759
3.324086
3.321430
3.318790
3.316166
3.313557
3.310964
3.308387
3.305825
3.303278
3.300745
3.298228
3.295725
1964.43
3.293236
1953.27 3.290761
1942.23
3.288301
1931.32
3.285854
1920.53 3.283421
372
TABLE I.-RADII, DEFLECTIONS, OFFSETS, ETC.
DEGREE
OF
CURVE
D
3° 01
INDE DOгöä ö -~мE
21
31
Ц
51
61
71
8'
91
101
121
131
141
151
161
171
181
191
201
211
221
231
241
251
26'
271
28'
29'
30'
311
321
331
341
351
361
371
381
391
40¹
41
42'
431
441
45'
46'
47'
481
491
501
51'
521
53'
541
55'
56'
57'
58'
59'
DEFL.
PER FT
OF STA.
(MIN)
0.9
0.905
0.91
0.915
0.92
0.925
0.93
0.935
0.94
0.945
0.95
0.955
C.96
0.965
0.97
C.975
C.98
C.985
C.99
C.995
1.0
1.005
1.01
1.C15
1.02
1.025
1.03
1.035
1.C4
1.C45
1.05
1.055
1.06
1.C65
1.07
1.075
1.ca
1.085
1.09
1.095
I
1.105
1.11
1.115
1.12
1.125
1.13
1.135
1.14
1.145
1.15
1.155
1.16
1.165
1.17
1.175
1.18
1.185
1.19
1.195
RADIUS
R
>
CHORD DEFINITION
LOG R
1910.08
1899.53
1889.09
1878.77
1868.56
1858.47
1848.48
1838.59 3.264486
1828.82 3.262170
1819.14 3.259667
1763.18
1754.19
1745.29
1736.48
1727.75
1809.57 3.257576
18CC.IC 3.255296
1790.73 3.253029
1781.45 3.250774
1772.27 3.248530
3.281051
3.278646
3.276253
3.273874
3.271508
1677.20
1669.06
1661.CO
1653.CI
1645.11
3.269155
3.266814
1599.21
1591.81
1584.48
1577.21
1570.01
3.246297
3.244077
3.241867
3.239669
3.237481
3.235305
3.233140
1475.71
1469.41
1637.28
3.214122
1629.52
3.212060
1621.84
3.210007
1614.22
3.207964
1606.68 3.205930
3.224584
3.222472
3.220369
3.218277
3.216195
1719.12
5.82
1710.57
5.85
1702.1C 3.230985 5.88
1693.72 3.228841
5.90
1685.42
3.226707
5.93
3.203906
3.201892
3.199886
1528.16 3.184169
1521.40
1514.17
1508.06
3.178419
1501.48 3.176519
C.O.
1 STA.
3.182244
3.180327
1494.95 3.174627
1488.48 3.172744
1482.07 3.170868
3.169001
3.167142
5.24
5.26
5.29
5.32
5.35
5.38
5.41
5.44
5.47
5.50
1463.16
3.165291
1456.96 3.163447
1450.81
5.53
5.56
5.58
5.61
5.64
5.67
5.7C
5.73
5.76
5.79
3.197890 6.34
3.195903
6.37
1562.88
1555.81
3.193925 6.40
3.191956
6.43
1548.80 3.189996
6.46
1541.86 3.188045
6.49
1534.98 3.186103
6.51
5.96
5.99
6.02
6.05
6.08
6.11
6.14
6.17
6.19
6.22
6.25
6.28
6.31
6.54
6.57
6.60
6.63
6.66
6.69
6.72
6.75
6.78
6.81
6.83
6.86
3.161612 6.89
1444.72 3.159784 6.92
1438.68 3.157963
6.95
ARC DEFINITION
RADIUS.
R
1909.86
1899.31
1888.87
1878.55
1868.34
1809.34
1799.87
1790.49
1781.22
1772.03
1762.95
1753.95
1745.C5
1736.24
1727.51
1718.87
171C.32
1701.85
1693.47
1685.17
1858.24 3.269102
1848.25
3.266761
1838.37
3.264432
1828.59
1818.91
1676.95
1668.81
1660.75
1652.76
1644.85
1637.02
1629.26
1621.58
1613.96
1606.42
1598.95
1591.55
1584.21
1576.95
1569.75
1562.61
1555.54
7

1527.89
1521.13
1514.43
1507.78
1501.20
1494.67
1488.20
1481.79
1475.43
1469.12
LOG R
1462.87
1456.67
1450.53
1444.43
1438.39
3.281001
3.278595
3.276203
3.273823
3.271456
3.262116
3.259812
3.257520
3.255240
3.252973
3.250716
3.248472
3.246239
3.244018
3.241808
3.239609
3.237421
3.235244
3.233078
3.230922
3.228778
3.226644
3.224520
3.222407
3.220303
3.218210
3.216128
3.193851
3.191881
1548.53
3.189921
1541.59 3.187969
1534.71
3.186026
3.214C55
3.211991
3.209938
3.207894
3.205860
3.203835
3.201820
3.199814
3.197817
3.195830
3.184091
3.182165
3.180248
3.178339
3.176438
3.174546
3.172661
3.170786
3.168918
3.167058
3.165206
3.163362
3.161526
3.159697
3.157876
373
TABLE I.-RADII, DEFLECTIONS, OFFSETS, ETC.

DEGREE
OF
CURVE
D
4° 0'
I
2'
31
4
51
6'
7'
8'
91
101
111
121
131
141
15'
16'
171
18'
19'
20'
211
22'
231
24'
251
26'
27'
28'
291
30'
311
321
331
341
351
361
371
381
391
40'
41
42'
431
44
451
46'
471
481
49'
50'
511
521
531
541
55'
561
571
581
591
DEFL.
PER FT
OF STA.
(MIN)
1.2
1.205
1.21
1.215
1.22
1.225
1.23
1.235
1.24
1.245
1.25
1.255
1.26
1.265
1.27
1.275
1.28
1.285
1.29
1.295
1.3
1.305
1.31
1.315
1.32
1.325
1.33
1.335
1.34
1.345
1.35
1.355
1.36
1.365
1.37
1.375
1.38
1.385
1.39
1.395
1.4
1.405
1.41
1.415
1.42
1.425
1.43
1.435
1.44
1.445
1.45
1.455
1.46
1.465
1.47
1.475
1.48
1.485
1.49
1.495
RADIUS
R
CHORD DEFINITION
LOG R
1432.69 3.156151
1426.74 3.154346
1420.85 3.152548
1415.01
3.150758
1409.21
3.148975
1403.46
1397.76
1392.1C
1386.49 3.141916
1380.92
3.140170
1375.40
3.138430
1369.92
3.136697
1364.49
3.134971
1359.10 3.133251
1353.75 3.131539
3.147200
7.13
3.145431
7.15
3.143670 7.18
7.21
7.24
1348.45 3.129833
1343.18
3.128134
1337.96
3.126442
1332.77 3.124756
1327.63
1273.57
1268.87
1264.21
1259.58
1254.98
1322.53
1317.46
1312.43
3.118078
1307.45
3.116424
1302.50 3.114777
3.121404
3.119738
1297.58
3.113136
1292.71
3.111501
1287.87
3.109872
1283.07 3.108249
1278.30
3.106632
7.42
7.45
7.47
7.50
3.123077 7.53
3.105022
3.103417
3.101818
3.100225
3.098638
1165.70
1161.76
C.O.
1 STA.
1228.11
3.089236
1223.74
3.087689
1219.40
3.086147
1215.09
3.084610
1210.82 3.083079
6.98
7.01
7.04
7.07
7.10
1206.57 3.081553
1202.36 3.080033
1198.17 3.078518
1194.01 3.077008
1189.88 3.075504
7.27
7.30
7.33
7.36
7.39
7.56
7.59
7.62
7.65
7.68
1250.42
3.097057
8.00
1245.89 3.095481
8.03
1241.40 3.093912 8.06
1236.94 3.092347
8.08
1232.51
3.090789
8.11
7.71
7.74
7.76
7.79
7.82
7.85
7.88
7.91
7.94
7.97
8.14
8.17
8.20
8.23
8.26
8.29
8.32
8.35
8.38
8.40
1185.78 3.074005
8.43
1181.71 3.072511 8.46
1177.66 3.071022 8.49
1173.65 3.069538 8.52
1169.66
3.068059
8.55
3.066585
8.58
3.065116
8.61
157.85 3.063653 8.64
1153.97
3.062194 8.67
1150.11 3.C6C740 8.69
ARC DEFINITION
RADIUS
R
1432.39 3.156063
1426.45 3.154257
1420.56
3.152458
1414.71
3.150667
1408.91 3.148884
1403.16
1397.46
1391.80
1386.19
1380.62
1375.10
1369.62
1364.19
1358.79
1353.44
1322.21
1317.14
1312.12
1307.13
1302.18
1348.14
3.129734
1342.87 3.128034
1337.64
1332.46
1327.32
1273.24
1268.54
LOG R
.1263.88
1259.25
1254.65
1250.09
1245.56
3.147108
3.145339
3.143577
3.141822
3.140074
1241.06
1236.60
1232.17
3.138334
3.13660C
3.134873
3.133153
3.131440
1297.26
3.113028
1292.39
3.111392
3.109762
1287.55
1282.74 3.108139
1277.97 3.106521
3.126341
3.124654
3.122974
3.121300
3.119633
3.117972
3.116318
3.114669
3.104910
3.103304
3.101705
3.100111
3.098523
3.096941
3.095365
3.093974
3.092229
3.090670
1227.77
3.089116
1223.40
3.087566
1219.06 3.086025
1214.75 3.084487
1210.47
3.082955
1206.23 3.081429
1202.01
3.079908
1197.82
3.078392
1193.66
3.076881
1189.53
3.075376
1185.43 3.073876
1181.36 3.072381
1177.31
3.070891
1173.29
3.069406
1169.30
3.067927
1165.34 3.066452
1161.40
3.064982
1157.49
3.063517
1153.61
3.062057
3.060603
149.75
374
TABLE I.-RADII, DEFLECTIONS, OFFSETS, ETC.

DEGREE
OF
CURVE
D
ONE DOгää ö-~ SONG 8
5° 0'
2'
31
4
5
6'
י7
81
91
101
יו!
121
131
141
151
16'
י7ו
181
19'
20'
211
22'
23'
241
251
26'
271
28'
29'
30'
311
32'
331
341
351
361
371
381
391
40'
411
42'
431
441
451
461
47.'
48'
49'
50'
51'
52'
531
54'
551
56'
571
58'
59'
DEFL.
PER FT
OF STA.
(MIN)
1.5
1.505
1.51
1.515
1.52
1.525
1.53
1.535
1.54
1.545
1.55
1.555
1.56
1.565
1.57
1.575
1.58
1.585
1.59
1.595
1.6
1.605
1.61
1.615
1.62
1.625
1.63
1.635
1.64
1.645
1.65
1.655
1.66
1.665
1.67
1.675
1.68
1.685
1.69
1.695
1.7
1.705
1.71
1.715
1.72
1.725
1.73
1.735
1.74
1.745
1.75
1.755
1.76
1.765
1.77
1.775
1.78
1.785
1.79
1.795
RADIUS
R
CHORD DEFINITION
LOG R
1146.28 3.059290
1142.47 3.057846
1138.69
1134.94
1131.21
1127.50
1123.82
1120.16
1116.52
1112.91
1091.73
1088.28
1084.85
1081.44
IC78.05
1109.33
3.045059
1105.76 3.043662
1102.22 3.042268
1098.70 3.040880
1095.20 3.039495
1074.68
1071.34
1068.01
1064.71
1061.43
1058.16
1054.92
1051.70
1048.49
1045.31
3.056407
3.054972
3.053542
1011.51
1008.55
1005.60
1002.67
3.035368
3.034C02
3.C32639
3.052116
8.87
1127.13
3.050696 8.90 1123.45
3.049280
8.93 1119.79
3.047868
8.96
1116.15
3.046462
8.99
1112.54
3.031281
3.029927
3.028577
3.027231
3.C25890
3.024552
3.C23219
3.021890
3.020565
3.019244
3.017927
1042.14
1039.00
3.016614
IC35.87 3.015305
1032.76 3.C13999
1029.67 3.C12698
C.O.
1 STA.
3.038115
1091.35
9.16
3.036740 9.19 1087.89
9.22 1084.46
9.25 1081.05
9.28 1077.66
3.007532
3.006250
3.004972
3.003698
3.002427
3.001160
8.72
1145.92
8.75 1142.11
8.78 1138.33
8.81 1134.57
8.84
1130.84
9.01
1108.95
9.04 1105.38
9.07 1101.84
9.10 IC98.32
9.13 IC94.82
9.45
9.48
9.51
1026.60 3.011401
9.74
1023.55 3.010107 9.77
1020.51
3.008818 9.80
1017.49
1014.50
9.31 IC74.30
9.33 1070.95
9.36
1067.62
9.39 1064.32
9.42
1061.03
9.54
9.57
ARC DEFINITION
RADIUS
R
9.60
1041.74
9.62 IC38.59
9.65
IC35.47
9.68 1C32.36
9.71 IC29.27
9.97
999.762 2.999897 10.00
996.867 2.998637
10.03
993.988 2.997381 10.06
991.126 2.996129 IC.09
988.28C 2.994880
985.451 2.993635
1026.19
1023.14
1020.10
9.83 1017.08
9.86 IC14.08
IC.12
10.15
9.89 1011.10
9.92 I008.14
9.94
IC05.19
1002.26
982.638 2.992393 10.18
979.840 2.991155 10.21
977.060 2.989921
10.23
10.26
974.294 2.988690
971.544 2.987463 IC.29
1057.77
IC54.52
968.810 2.986239 10.32
966.091 2.985018 10.35
963.387 2.983801 IC.38
960.698 2.982587 IC.41
958.025 2.981377 10.44
1051.30
1 с48.09
1044.91
999.345
996.448
993.568
990.705
987.858
985.028
982.21.3
979.415
976.632
973.866
971.115
968.379
965.659
962.954
LOG R
3.059153
3.057707
3.056267
3.054831
3.053400
3.051974
3.050553
3.049135
3.C47723
3.046315
3.044912
3.043514
3.042119
3.040729
3.039343
3.037963
3.036587
3.035215
3.033847
3.032483
3.031124
3.029769
3.028418
3.027071
3.025729
3.024390
3.023056
3.021726
3.020400
3.019078
3.017760
3.016446
3.015136
3.013829
3.012527
3.011229
3.009935
3.008644
3.007357
3.006074
3.004795
3.003520
3.002248
3.000980
2.999715
2.998455
2.997198
2.995944
2.994695
2.993448
2.992206
2.990967
2.989731
2.988499
2.987271
2.986045
2.984824
2.983606
960.264 2.982391
957.590 2.981179
375
TABLE I.-RADII, DEFLECTIONS, OFFSETS, ETC.

DEGREE
OF
CURVE
D
6° 0'
2¹
4
6'
8'
10'
12'
141
16'
181
201
22'
241
26'
28'
30'
321
341
36'
38'
40'
42'
44'
461
48'
50'
52'
541
56'
58'
7° 0'
21
41
6
81
IC'
12'
14
DEFL.
PER FT
OF STA.
(MIN)
1.8
1.81
1.82
1.83
1.84
1.85
1.86
1.87
1.88
1.89
40'
421
441
46'
48'
2.0
2.C1
2.02
2.03
2.04
1.9
905.131
1.91
900.397
1.92 895.712
1.93
891.076
1.94
886.488
1.95
881.946
1.96
877.451
1.97
873.002
1.98
868.598
1.99 864.238
2.05
2.06
2.1
2.11
2.12
2.13
2.14
2.15
2.16
2.17
16' 2.18
18'
2.19
RADIUS
R
20'
2.2
22'
2.21
241 2.22
26' 2.23
28' 2.24
CHORD DEFINITION
LOG R
838.972
834.904
2.07 830.876
955.366
950.093
944.877
939.719
934.616 2.970633
929.569
924.576
919.637
914.750
909.915
2.08
826.886
2.09 822.934
2.3
2.31
2.32
2.33
2.34
50'
2.35
52' 2.36
54' 2.37
56' 2.38
58' 2.39
819.020
815.144
811.303
807.499
803.731
301
2.25
764.489
32'
2.26
761.112
341 2.27
757.764
36' 2.28 754.445
38' 2.29 751.155
10.47
2.980170
2.977766
10.53
2.975375
10.58
2.972998 10.64
10.7C
859.922 2.934459 11.63
855.648 2.932295
11.69
2.930142 11.75
851.417
847.228
2.928000 11.80
843.080
2.925869 11.86
2.968282 10.76
2.965943 10.82
2.963616
2.961303
2.959001
C.O.
1 STA.
2.956711
11.05
2.954434
|1.11
2.952168 11.16
2.949915 11.22
2.947673 11.28
2.945442 11.34
11.40
2.943223
2.941015
2.938819
11.45
11.51
2.936633
11.57
10.27
10.93
IC.99
2.923747
2.921637
2.919536
11.92
11.98
12.04
2.917446
12.09
2.915365 12.15
781.840
778.307
774.806
771.336 2.887244
767.897
2.913295
12.21
2.911234 12.27
2.909183
12.33
2.907142
12.38
2.905111
12.44
799.997
2.903089
12.50
2.901076 12.56
796.299
792.634
2.899073 12.62
789.003
2.897079 12.67
785.405 2.895094 12.73
2.893118 12.79
2.891151 12.85
2.889193 12.91
12.96
2.885303 13.02
2.883371 13.08
2.881448 13.14
2.879534 13.20
2.877627 13.25
2.875730 13.31
747.894
744.661 2.871959
741.456
738.279
735.129
2.873840 13.37
13.43
2.870086 13.49
2.868221 13.55
2.866363
13.60
732.005 2.864514
13.66
728.909 2.862673 13.72
725.838 2.860840 13.78
722.793
719.774
2.859014 13.84
2.857196 13.89
ARC DEFINITION
RADIUS
R
954.930
949.654
944.436 2.975173
939.275 2.972793
934.170 2.970426
929.121
924.126
919.184
914.294
909.457
881.474
876.976
872.525
868.118
863.756
859.437
855.161
850.927
846.736
842.585
904.670
2.956490
899.934 2.954211
895.247
2.951943
2.949687
890.608
886.017
2.947442
838.475
834.405
830.374
826.381
822.427
818.511
814.632
810.789
806.983
803.212
799.476
795.775
792.108
788.474
784.874
781.306
777.771
774.267
770.795
767.354
763.944
760.563
757.213
753.892
750.600
LOG R
747.336
744.101
740.894
737.714
734.561
2.979971
2.977565
731.436
728.336
725.263
722.216
719.194
2.968072
2.965731
2.963402
2.961086
2.958783
2.945209
2.942988
2.940778
2.938579
2.936391
2.934214
2.932048
2.929892
2.927748
2.925616
2.923490
2.921377
2.919274
2.917181
2.915098
2.913025
2.910961
2.908908
2.906864
2.904830
2.902805
2.900790
2.898784
2.896787
2.894800
2.892821
2.890852
2.888891
2.886939
2.884996
2.883061
2.881135
2.879218
2.877309
2.875408
2.873516
2.871631
2.869756
2.867888
2.866028
2.864176
2.862332
2.860496
2.858667
2.856846
376
TABLE I.-RADII, DEFLECTIONS, OFFSETS, ETC.

DEGREE
OF
CURVE
D
8° 0'
2'
4'
6'
8'
10'
121
141
16'
18'
20'
22'
241
26'
281
30'
321
341
36'
38'
50'
52'
541
561
58'
6'
8'
DEFL.
PER FT
OF STA.
(MIN)
2.4
2.41
2.42
2.43
2.44
20'
221
2.45
2.46
2.47
40'
2.6
42'
2.61
441
2.62
46' 2.63
481
2.64
241
26'
28'
2.48
2.49
2.55
2.56
2.57
2.58
2.59
9° 0' 2.7
21
2.71
4'
2.72
10'
2.75
12' 2.76
[4¹ 2.77
16'
2.78
18'
2.5
688.156
685.419
2.51
2.52 682.704
2.53 680.010
2.54
677.338
2.73
2.74
2.8
2.81
2.82
2.83
2.84
30'
2.85
32' 2.86
341
2.87
RADIUS
R
36' 2.88
381
2.89
CHORD DEFINITION
LOG. R
2.65 649.274
2.66
646.838
2.67
644.420
2.68
642.021
2.69
639.639
40'
2.9
42'
2.91
ЦЦІ 2.92
46' 2.93
481 2.94
716.779
713.810
50'
2.95
521 2.96
541 2.97
56' 2.98
58'
2.99
710.865
707.945
705.048
702.175
699.326
696.499
693.696
690.914
674.686
672.056
669.446
666.856
664.286
621.203
618.974
2.79 616.760
661.736
659.205
656.694
654.202
651.729
637.275
634.928
632.599
630.286
627.991
625.712
623.450
614.563
612.380
610.214
608.062
605.926
603.805
601.698
599.607
597.530
595.467
593.419
591.384
589.364
587.357
585.364
583.385
581.419
579.466
577.526
575.599
2.855385
2.853583
2.851787
14.07
2.849999 14.13
2.848219 14.18
C.O.
1 STA.
2.846446 14.24
2.844679 14.30
2.842921 14.36
2.841169 14.42
2.839424
14.47
13.95
14.01
2.837687 14.53
2.835956 14.59
2.834232 14.65
2.832515 14.71
2.830805 14.76
2.829102 14.82
2.827405 14.88
2.825715
14.94
15.00
2.824032
2.822355 15.05
2.820685
15.11
2.819021 15.17
2.817363 15.23
2.815712 15.29
2.814067
15.34
2.812428
15.40
2.810796
15.46
2.809169
15.52
2.807549
15.58
2.805935 15.63
2.804327 15.69
2.802724
15.75
2.801128
15.81
2.799538
15.87
2.797953 15.92
2.788566
2.787021
2.777867
2.776360
2.774858
16.27
16.33
2.785482 16.39
2.783948
16.45
2.782420 16.50
2.780897
16.56
2.779379 16.62
16.68
16.74
16.79
2.773361
2.771870
2.770383 16.97
2.768902 17.03
2.767426 17.08
ARC DEFINITION
16.85
16.91
RADIUS
R
2.765955 17.14
2.764489
17.20
17.26
2.763028
2.761572 17.32
2.760120
17.37
2.796374 15.98
625.045
2.794801
16.04
622.780
16.10 620.532
2.793234
2.791673
2.790117
716.197
2.855033
713.226 2.853227
710.278 2.851428
707.355 2.849638
704.456 2.847854
701.581 2.846078
698.729 2.844309
695.900 2.842547
693.094
690.311
687.549
684.810
682.093
679.397
676.722
16.16 618.300
16.21
616.084
661.105
658.572
656.059
653.564
651.088
648.632
646.193
643.773
641.371
638.986
636.620
634.271
631.939
674.068
2.828704
671.435 2.827004
668.822
666.23C
663.658
629.624
627.326
LOG R
613.883
611.699
609.530
607.376
605.237
603.114
601.005
598.911
596.831
594.766
592.715
590.678
588.655
586.646
584.651
2.840792
2.839045
2.837304
2.835570
2.833843
2.832123
2.830410
2.825311
2.823624
2.821944
2.820270
2.818603
2.816941
2.815288
2.814640
2.811998
2.810362
2.808732
2.807109
2.805492
2.803880
2.802275
2.800675
2.799081
2.797494
2.795911
2.794335
2.792764
2.791199
2.789640
2.788086
2.786538
2.784995
2.783457
2.781926
2.780399
2.778878
2.777362
2.775851
2.774346
2.772846
2.771351
2.769861
2.768376
2:766897
582.669 2.765422
580.700 2.763952
578.7452.762487
576.803 2.761028
574.874 2.759573
377
TABLE I-RADII, DEFLECTIONS, OFFSETS, ETC.

DEGREE
OF
CURVE
D
10° 0'
2'
41
6'
81
10'
12¹
141
16'
181
201
221
241
26'
28'
3C'
321
341
361
38'
40'
42'
441
461
48'
50'
52'
541
56'
58'
11° 0¹
21
41
6'
81
10'
12'
141
16'
18'
20'
22'
241
26'
28'
30'
321
341
361
38'
40'
42'
441
DEFL.
PER FT
OF STA.
(MIN)
3.0
3.CI
3.02
3.03
3.C4
3.05
3.06
3.07
3.08
3.09
3.1
3.11
3.12
3.13
3.14
3.15
3.16
3.2
3.21
3.22
3.23
3.24
3.25
3.26
3.27
3.28
3.29
3.3
3.31
3.32
3.33
3.34
3.35
3.36
3.37
3.38
3.39
546.438
544.714
3.17 543.CCI
3.18
3.19
3.4
3.41
3.42
3.43
3.44
3.45
3.46
3.47
3.48
3.49
RADIUS
R
3.5
3.51
3.52
46' 3.53
48'
3.54
CHORD DEFINITION
LOG R
573.686
571.784
569.896
568.C2C
566.156
564.305
562.466
560.638
558.823
557.019 2.745870
555.227
553.447
551.678
549.920
548.174
541.298
539.606
536.253
534.593
532.943
531.303
529.673
528.053
526.443
524.843
523.252
521.671
520.ICC
518.539
2.758674
2.757232
2.755796
17.55
2.754364 17.61
2.752937 17.66
516.986
515.443
C.O.
1 STA.
2.744471
2.743076
17.43
17.49
2.751514 17.72
2.750096
17.78
2.748683
17.84
2.747274 17.89
17.95
537.924 2.730721 18.59
18.65
2.729370
2.728023
18.71
2.726681
2.725342
18.CI
18.07
2.741686 18.13
2.74C30C
18.18
2.738918
18.24
2.737541 18.30
2.736169
2.734800
18.36
18.42
2.733436 18.47
2.732C77 18.53
18.76
18.82
2.724CC8 18.88
2.722677 18.94
2.721351 19.00
2.72CC29
19.05
2.718711
19.11
2.717397
19.17
2.716087 19.23
2.714781
19.28
2.713479 19.34
2.712181 19.40
513.909
512.385 2.709596
2.71C887 19.46
19.52
510.869 2.70831C 19.57
509.363 2.707027 19.63
5C7.865 2.705748
19.69
506.376
2.704473
19.75
504.896 2.703202
19.81
503.425 2.701934 19.86
501.962
2.700671 19.92
500.507 2.69941C 19.98
499.061
2.698154 20.04
2.6969C1 20.10
497.624
496.195
2.695652 20.15
494.774 2.694407 20.21
493.361 2.693165 20.27
491.956 2.691926 20.33
490.559 2.690692 20.38
489.171 2.689460
20.44
487.790 2.688233 20.50
486.417
2.687C08 20.56
5C1
3.55 485.051 2.685788 20.62
3.56 483.694
521
2.684570 20.67
482.344
541 3.57
2.683357 20.73
561 3.58 481.CCI 2.682146 20.79
58! 3.59 479.666 2.680939 20.85
ARC DEFINITION
RADIUS
R
572.958
571.054
569.163
567.285
565.419
559.894
558.076
556.270
554.475
552.692
550.921
563.565
2.750944
561.723 2.749522
549.161
547.412
545.674
543.947
542.231
54C.526
538.832
528.884
527.262
525.649
524.047
522.454
520.871
519.297
517.733
516.178
514.633
513.097
511.569
510.051
5C8.542
507.C42
LOG R
498.224
496.784
495.353
493.929
492.514
2.758123
2.756667
491.107
489.708
488.316
2.755237
2.753801
2.75237C
537.148
2.73C094
535.475 2.728739
533.812
2.727388
532.159 2.726041
530.516
2.724699
486.933
485.557
2.748106
2.746693
2.745285
2.743882
2.742483
2.741089
2.7397CC
2.738314
2.736933
2.735557
2.734185
2.732817
2.731453
2.723360
2.722026
2.720696
2.719370
2.718048
2.71673C
2.715417
2.714106
2.712799
2.711497
5C5.551
504.068
502.595
501.129 2.699950
499.672
2.698685
2.710199
2.708905
2.707614
2.706327
2.705044
2.703765
2.702490
2.701218
2.697425
2.696168
2.694914
2.693665
2.692418
2.691176
2.689937
2.688701
2.687469
2.686241
484.19C
2.685015
482.830 2.683974
481.477
2.682576
480.132 2.681361
478.795
2.680149
378
DEGREE
OF
CURVE
D
12° 0'
101
20'
30'
401
50'
13° 0'
10'
201
301
401
50'
14° 0'
10'
201
301
40'
50'
15° 0'
10¹
20'
30'
40'
50'
16° 0'
10'
201
30'
40'
501
17° 0'
101
201
301
40'
50'
18° 0'
10'
201
30'
40'
50'
19° 0'
10'
201
30'
40'
50'
20° 0'
10'
20'
30'
401
50'
21° 01
101
20'
30'
40'
50'
TABLE I.-RADII, DEFLECTIONS, OFFSETS, ETC.

DEFL.
PER FT
OF STA.
(MIN)
RADIUS
R
CHORD DEFINITION
LOG R
3.6
3.65
20.91
21.19
2.62
2.66
3.7
478.339 2.679735
471.810 2.673767
465.459 2.667881 21.48 2.69
459.276 2.662074 21.77 2.73
453.259 2.656345 22.06
3.85 447.395 2.650691❘ 22.35
3.75
3.8
2.77
2.80
3.9
3.95
4.0
4.05
4.1
420.233 2.623490 23.80
4.15 415.194 2.618251 24.09
4.8
4.85
4.9
4.95
5.0
5.05
C.O.
1 STA.
4.2
4.25
4.3
410.275 2.613075 24.37
405.473 2.607962 24.66
400.782 2.602908 24.95
396.200 2.597914 25.24
391.722 2.592978 25.53
4.45 387.345 2.588097 25.82
4.35
4.4
4.5
4.55
4.6
383.065 2.583272 26.11
378.880 2.578501 26.39
374.786 2.573783 26.68
4.65 370.780 2.569116 26.97
366.859 2.564500| 27.26
363.022 2.559933 27.55
4.7
4.75
441.684 2.645111 22.64
2.84
2.88
436.117 2.639603 22.93
430.690 2.634164 23.22
2.91
425.396 2.628794 23.51 2.95
6.0
6.05
6.1
6.15
6.2
335.0132.525062
5.4
319.623 2.504638 | 31.29
5.45 316.715 2.500668 31.57
313.860 2.496736 31.86
5.5
5.55 31.056 2.492839 32.15
5.6
308.303 2.488978 32.44
5.65 305.599 2.485152 32.72
M.O.
1 STA.
359.265 2.555415 27.83 3.50
355.585 2.550944 28.12 3.53
351.98 2.546519 28.41 3.57
348.450 2.542140 28.70 3.61
344.990 2.537806 28.99 3.64
341.598 2.533516| 29.27 3.68
338.273 2.529268 29.56 3.72
29.85 3.75
30.14 3.79
2.516774 30.42 3.82
325.604 2.512690 | 30.71
5.35 322.585 2.508645 31.00
5.1
5.15
5.2
331.816 2.52C898
5.25
328.689
5.3
5.7
5.75
5.8
5.85
5.9
295.247 2.470186 33.87
292.770 2.466526 34.16
5.95 290.334 2.462897
2.462897 34.44
287.939 2.459300 34.73
285.583 2.455733 35.02
283.267 2.452195 35.30
280.988 2.448688 35.59
278.746 2.445209 35.87
6.25 276.541 2.441759 36.16
2.99
3.02
3.06
3.10
3.13
3.17
3.20
3.24
3.28
3.31
3.35
3.39
3.42
3.46
3.86
3.90
302.943 2.481361 33.01 4.16
300.333 2.477603 33.30 4.19
297.768 2.473878 33.58 4.23
3.94
3.97
4.01
4.04
4.08
4.12
4.26
4.30
4.34
4.37
4.41
4.45
4.48
4.52
4.56
6.3
274.370 2.438337 36.45
4.59
6.35 272.234 2.434943 36.73 4.63
6.4 270.132 2.431576
6.45 268.062
266.024
6.55 264.018
2.428235
2.428235
37.02 4.67
37.30 4.70
2.424921
2.424921 | 37.59 4.74
6.5
2.421633 37.88
4.78
RADIUS
R
ARC DEFINITION
LOG R
C.O.
1 STA.
477.465 2.678941 20.87
470.924 2.672951 | 21.15
464.560 2.667042 21.44
458.366 2.661213 21.73
452.335 2.655460 | 22.02
446.461 2.649783 | 22.30
440.737 2.644179 22.59
435.158 2.638647 22.88
429.718 2.633184 23.16
424.413 2.627789 | 23.45
419.237 2.622460 23.74
414.186 2.617196 24.03
409.256 2.611995 24.31
404.441 2.606855 24.60
399.738 2.601775 24.89
395.143 2.596755 25.17
390.653 2.591791 25.46
386.264 2.586884 | 25.74
381.972 2.582031 26.03
377.774 2.577232 26.32
373.668 2.572486 26.60
369.650 2.567791 26.89
365.718 2.563146 27.17
361.868 2.558550 27.46
358.099 2.554003 27.74
354.407 2.549502 28.03
350.790 2.545048 28.3!
347.247 2.540638 28.60
343.775 2.536274 28.89
340.371 2.531952 29.16
337.034 2.527673 29.45
333.762 2.523437 29.73
330.553 2.519241 30.02
327.404 2.515085 30.31
324.316
321.285
2.510968❘ 30.59
2.506890 30.87
318.310
2.502850 31.16
315.390 2.498847 31.44
312.522 2.494881 31.72
309.707 2.490951 32.00
306.942 2.487056 32.29
304.225 2.483195 32.58
301.557
298.934
2.479369 32.86
2.475576 33.14
296.357 2.471816 33.42
293.825 2.468087 33.71
291.334 2.464392 33.99
289.886 2.460727 34.27
281.783
279.492
277.238
275.020
286.479 2.457093 34.55
284.111 2.453488 34.83
2.449914 35.12
2.446369 35.40
2.442852 | 35.68
·
2.439364 35.96
272.837 2.435903 36.24
270.689 2.432470 36.52
268.574 2.429064 36.80
266.492 2.425684 37.08
264.442 2.422330 37.37
262.423 2.419002 37.65
379
DEGREE
OF
CURVE
D
22° 0'
ICI
201
301
40'
5C'
23° C'
ICI
201
301
40'
50'
24° CI
10'
201
30'
40'
50'
25° O'
ICI
2C'
301
40'
50'
26° 0'
301
27° 0'
30'
28° C'
30'
29° 0'
30'
30° 0'
32° O'
TABLE I.-RADII, DEFLECTIONS, OFFSETS, ETC.

DEFL.
PER FT
OF STA.
(MIN)
6.7
6.75
6.8
6.85
6.9
6.95
7.0
7.05
7.1
7.15
6.6
262.042 2.418371 38.16
6.65 260.098 2.415134 38.45
258.180 2.411922 38.73
256:292 2.408734 39.C2
254.431 2.405571 39.30
252.599 2.402431 | 39.59
7.2
7.25
7.3
7.35
RADIUS
R
CHORD DEFINITION
LOG R
C.O.
1 STA.
8.4
8.55
250.793 2.399315 39.87
249.013 2.396222 | 40.16
247.258 2.393151 40.44
245.529 2.390103 | 40.73
243.825 2.387077 41.CI
242.144 2.384074 41.30
7.5 231.011 | 2.363633 | 43.29
7.55 229.506 2.360794 | 43.57
7.6 228.020 2.357974 43.86
7.65 226.555 2.355173
NOTE:
238.853 2.37813C
240.487 2.381091 41.58 5.26
41.87 5.29
2.375190 42.15 5.33
2.372270 | 42.44 5.37
234.C84 2.369371 42.72
7.45 232.537 2.366492 43.00
237.241
235.652
7.4
5.40
5.44
7.8 222.271 2.346882 44.99
7.95 218.150 2.338755 45.84
8.1 214.183 2.330785 46.69
8.25 210.362 2.322967❘ 47.54
M.O.
1 STA.
4.81
4.85
4.89
4.92
4.96
5.00
7.7 225.1C8 2.352391 44.42 5.62
7.75 223.68C 2.349627 44.71
5.C4
5.07
5.11
5.14
5.18
5.22
5.48
5.51
5.55
44.14 5.59
2C6.678 2.315295 48.38
203.125 2.307764 | 49.23
5.92
6.03
6.14
6.25
6.36
6.47
6.58
7.03
7.47
7.92
8.7
199.696 2.300370 50.08
8.85 196.385 2.293108 | 50.92
9.C
193.185 2.285974 51.76
181.398 2.258632 55.13
9.6
34° 01
10.2
36° 0'
10.8
8.37
9.04
38° 0' 11.4
41° 0' 12.3
44° C' 13.2
48° 0' 14.4
52° O' 15.6
9.72
171.015 2.233035 | 58.47
161.803 2.208988 61.80
153.578 2.186328 | 65.11
142.773 2.154645 70.04
133.473 2.125395 74.92
122.930 2.089657 81.35 10.63
14.058 2.057128
2.057128 87.67 11.54
104.787 2.020307 95.43 12.7C
94.354 1.974760 106.0 14.34
85.C65 1.929751 117.6 16.25
76.213 1.882027 131.2 18.69
67.817 1.831339 147.5 22.00
59.284 1.772941 168.7 27.43
57° 0'
17.1
64° 0' 19.2
72° C¹ 21.6
82° 0'
24.6
95° 0' 28.5
115° 0' 34.5
RADIUS
R
ARC DEFINITION
260.435
258.477
256.548
254.648
LOG R
5.66
5.70 220.368 2.343149 44.60
5.81 216.21C 2.334877 45.43
2.415700 37.93
2.412422 38.21
2.409169 | 38.49
2.405940 38.77
252.775 2.402735 39.05
250.93C 2.399553
39.33
249.112 2.396395 39.61
247.320 2.393259 39.89
245.553 2.390145 40.16
243.812 2.387C55 40.44
242.095 2.383985 40.72
240.402 2.380938 41.00
C.O.
1 STA.
238.732 2.377911 41.28
237.C86 2.374905 41.56
235.462 2.371921 41.84
233.860 2.368956 | 42.11
232.280 2.366012 | 42.39
230.721 2.363087 42.67
229.183 2.360183 42.95
227.665 | 2.357279 | 43.23
226.168 2.354430 43.50
224.689 2.351582 43.78
44.06
223.230 2.348753
221.790 2.345942 44.33
204.628
2C1.C38
212.207 2.326759 | 46.26
2.318790 47.08
208.348
2.310964 47.91
2.303278 48.72
2.295725
2.295725 49.55
197.572
194.223
2.288301
50.36
190.986 | 2.281001
51.18
179.049 2.252973 54.42
168.517 2.226644 | 57.62
159.155 2.201820 60.79
150.778 2.178339 63.93
139.746 2.145339 68.55
130.218 2.114669 | 73.09
119.366 2.076881 | 78.98
110.184 2.C42119 84.69
100.519 2.002248 91.55
89.525 1.951943 100.6
79.577 1.90079C 110.0
69.873 1.844309 120.3
1.780399 131.1
49.822 1.697425 141.8
60.311
The odd values of D between 30° and 115° are those
whose arc-definition radil vary approximately from
190 feet to 50 feet by 10-ft intervals. For curves
having exactly these radii, see Table V.
380
O
ooooo
12345 6NOGO
ooooo
| °
2°
4°
5°
70
8°
9°
10°
OOOOO
11°
12°
13°
14°
15°
16°
17°
18°
19°
20°
21°
22°
23°
24°
25°
26°
27°
28°
29°
30°
32°
34°
36°
38°
41°
44°
48°
52°
57°
64°
72°
82°
95°
115°
TABLE II.-LENGTHS OF ARCS AND TRUE CHORDS

ARC FOR
1 STA.
100.001
100.005
100.011
100.020
100.032
100.046
100.062
100.081
100.103
100.127
100.154
100.183
100.215
100.249
100.286
100.326
100.368
100.412
100.460
100.510
100.562
100.617
100.675
ICO.735
100.798
100.863
100.931
101.002
101.075
101.152
101.312
101.482
101.664
101.857
102.166
102.500
102.986
103.516
104.246
105.394
106.896
109.073
112.445
118.992
CHORD DEFINITION OF D
1/10 STA.
10
10
10
10
10
10.01
10.01
10.01
10.01
10.01
10.02
10.02
10.02
10.02
10.03
10.03
10.04
10.04
10.04
10.05
10.06
10.06
10.07
10.07
10.08
10.08
10.09
10.10
10.11
10.11
10.13
10.15
10.16
10.18
10.21
10.25
10.30
10.35
10.42
10.53
10.68
10.90
11.23
11.88
TRUE CHORDS
1/4 STA.
5555
25
25
25
25
25.01
25.01
25.02
25.02
25.02
25.03
-25.04
25.04
25.05
25.06
25.07
25.08
25.09
25.10
25.11
25.12
25.13
25.14
25.16
25.17
25.19
25.20
25.22
25.23
25.25
25.27
25.31
25.35
25.39
25.43
25.51
25.59
25.70
25.82
25.98
26.26
26.61
27.12
27.91
29.44
1/2 STA.
50
50
50
50.01
50.01
50.02
50.02
50.03
50.04
50.05
50.06
50.07
50.08
50.09
50.11
50.12
50.14
50.16
50.17
50.19
50.21
50.23
50.25
50.27
50.30
50.32
50.35
50.38
50.40
50.43
50.49
· 50.56
·
50.62
50.69
50.81
50.94
51.12
51.31
51.59
52.01
52.57
53.38
54.63
57.03
1/4 STA.
25
~~~~~ ~~~~~ ~~~~~
25
25
25
25
25
25
25
25
25
25
25
25
25
ARC DEFINITION OF D
25
25
25
24.99
24.99
24.99
24.99
24.99
24.99
24.99
24.99
24.99
24.99
24.98
24.98
24.98
24.98,
24.98
24.97
24.97
24.97
24.96
24.95
24.95
24.94
24.92
24.90
24.87
24.82
24.74
TRUE CHORDS
1/2 STA.
unuun ung
50
50
50
50
50
49.99
49.99
49.98
49.98
49.98
49.97
49.97
49.96
49.96
49.95
49.95
49.94
49.94
49.93
49.92
49.92
49.91
49.90
49.89
49.88
49.88
49.87
49.86
49.84
49.82
49.79
49.77
49.73
49.69
49.63
49.57
49.49
49.35
49.18
48.94
48.58
47.93
1 STA.
100
100
99.99
99.98
99.97
99.95
99.94
99.92
99.90
99.88
99.85
99.82
99.79
99.75
99.72
99.68
99.63
99.59
99.54
99.49
99.44
99.39
99.33
99.27
99.21
99.14
99.08
99.01
98.94
98.86
98.71
98.54
98.36
98.18
97.88
97.56
97.10
96.60
95.93
94.88
93.55
91.68
88.93
84.04
Chord Definition of D
For degrees of curve not listed obtain excess arc per station approximately
by interpolation, or exactly to 3 decimal places (up to D = 15°) from:
Excess = 0.00127 D2
Arc Definition of D
For degrees of curve not listed obtain chord deficiency per station approxi-
mately by interpolation, or exactly to 2 decimal places (up to D = 30°) from:
Deficiency = 0.00127 D2
381
TABLES III & IV.-CORRECTION COEFFICIENTS FOR SUBCHORDS
Chord Definition of D
The following table may be
used to obtain true lengths of
subchords not listed in Table
II.
Arc Definition of D
The following table may be
used to obtain true lengths of
subchords not listed in Table
II.
For any degree of curve the
small correction to be added
to the nominal length in order
to obtain the true length is
almost a constant percentage
of the excess of arc for 1 sta-
tion on a curve of that degree.
The maximum correction is
required for a nominal sub-
chord about 57.5 feet long.
Nominal
Subchord
5
10
HOONER IGUA AKUN NAUG
20
30
35
60
65
70
75
80
85
90
95
100
Ratio of Chord
Correction to
Excess Arc
per Station
.050
.099
.147
.192
.234
.273
.307
.336
.359
.375
.383
.384
.375
.357
.328
.288
.236
.171
.093
0
HIDD
EXAMPLE. Given: a 20°
curve. Required: true sub-
chord for 75-ft nominal
length.
-
Solution: Excess arc per
sta.=0.510 (from Table II).
Corr.=0.510X0.328 0.17.
Add corr. to nominal length
giving true subchord=75.17.
For subchords not listed,
interpolate for correction co-
efficients.
For any degree of curve the
small correction to be sub-
tracted from an arc length in
order to obtain the true chord
length is almost a constant
percentage of the chord defi-
ciency for a 100-foot arc on a
curve of that degree. These
percentages vary approxi-
mately as the cubes of the arc
lengths.
Length
of Arc
5
PO42 ROG FOBO BR858
20
25
30
35
40
50
55
60
75
80
90
95
100
Ratio of Chord
Correction to
Chord Deficiency
for 100-ft Arc
.0001
.0010
.0034
.0080
.016
.027
.043
.064
.091
.125
.166
.216
.275
.343
.422
.512
.614
.729
.857
1
EXAMPLE. Given: a 25°
curve. Required: true sub-
chord for 75-ft arc.
Solution: Chord def. per
sta.=0.79 (from Table II).
Corr.=0.79×0.422=0.33.
Subtract corr. from arc giv-
ing true subchord=74.67.
For arcs not listed, note
that correction coefficients
are proportional to the cubes
of the arcs.
382
RADIUS
FT
80
90
ICO
110
50
60
70 24.555
150
16C
170
180
190
200
225
250
275
3C0
TABLE V.-EVEN-RADIUS CURVES. DEFLECTIONS
AND CHORDS

DEFL.
MIN PER
FT OF
ARC
17.189 2°51.89'
15.626 2°36.26'
120 14.324 2°23.241
130
140
65C
700
750
800
850
900
950
1000
1050
IICC
1200
1300
1400
1500
1600
34.377
5°43.78'
28.648 4°46.48'
4°05.55¹
10 FT
21.486 3°34.86'
19.099 3°10.99'
DEFLECTIONS FOR ARCS OF
325
350
375 4.584
400
425
13.222 2°12.22'
12.278
2°02.78'
450 3.820
475
500
550
600
11.459 1°54.59'
10.743
1°47.43'
10.111
1°41.11
9.549
9.047
1°35.49'
1°30.471
25 FT
3°34.86'
8.594
7.639 3°10.991
6.875 2°51.89'
6.250 2°36.26'
5.73C 2°23.24'
5.289 2°12.221
4.911 2°02.78'
1°54.59'
4.297 1°47.43!
4.044 1°41.111
1°35.49!
3.619 1°30.47'
3.438 1°25.94'
3.235 1°18.13'
2.865 1°11.62'
2.644
1°06.11
2.455
1°01.39'
2.292 0°57.30'
2.149 0°53.71'
2.022 0°50.56'
1.910 0°47.75'
1.809 0°45.23'
1.719 0°42.97'
1.637 0°40.93'
1.563 0°39.071
1.432 0°35.811
1.322 0°33.06'
1.228
0°30.69'
1.146 0°28.65'
1.074 0°26.86'
1.011 0°25.28'
1700
1800 0.955 0°23.87'
1900 0.905 0°22.62'
2000 0.859 0°21.49'
2100
0.819 0°20.46'
2300
2200 0.781 0°19.53'
0.747 0°18.68'
0.716 C°17.911
2400
2500 0.688 0°17.19'
2600 0.661 0°16.53'
25 FT
14°19.441
11°56.20'
10°13.88'
8°57.15'
7°57.46'
7°09.72¹
6°30.65'
5°58.10'
5°30.55'
5°06.94'
4°46.48'
4°28.58'
4°12.77'
3°58.73¹
3°46.17'
50 FT
7°09.72'
6°21.97'
5°43.78'
5°12.52'
4°46.48
4°24.441
4°05.55'
3°49.18'
3°34.86'
3°22.221
3°10.99'
3°00.93'
2°51.891
2°36.26'
2°23.241
2°12.22'
2°02.78'
1°54.59'
1°47.43'
1°41.11'
1°35.49'
1°30.47'
1°25.94'
1°21.85'
1°18.131
1°11.62'
1°C6.111
!°CT.391
0°57.30'
0°53.72'
0°50.56'
C°47.75'
C°45.23'
C°42.97'
0°40.93'
0°39.C7'
0°37.37!
0°35.811
0°34.38'
0°33.06'
100 FT
57°17.75'
47°44.79
40°55.53'
35°48.591
31°49.86'
28°38.87'
26°02.61'
23°52.39'
22°02.21
20°27.77'
19°C5.92'
17°54.301
16°51.IC'
15°54.93'
15°C4.67'
14°19.441
12°43.941
11°27.55'
10°25.05'
9°32.96'
8°48.88
8° | | . || '
7°38.37!
7°09.72'
6°44.441
6°21.97'
6°C1.87'
5°43.771
5°12.52'
4°46.48'
4°24.44
4°05.55'
3°49.181
3°34.86'
3°22.221
3°10.99'
3°00.931
2°51.89'
2°43.70¹
2°36.26'
2°23.241
2°12.22¹
2°C2.78'
1°54.59'
1°47.431
1°41.111
1°35.49'
1°30.471
1°25.94'
1°21.851
1°18.13'
1°14.731
1°11.62'
1°08.75'
1°06.11
10 FT
9.98 24.74
24.82
24.87
9.99
9.99
CHORDS FOR ARCS OF
IC 24.90
24.92
10
IC
ãããoо UOUOO OOUOU
10
IC
10
24.93
95.89
24.95 96.59
97.13
IC 24.96 97.55
24.96
97.88
10
10
IC
10
IC
25 FT
24.98
24.99
•~~~~
24.99
25
25
25
~~~~~
25
50 FT
49.87 98.96
49.90 99.18
99.34
49.93 99.45
24.99
49.92
24.99
24.99 49.94 99.54
25
annnn
25
25
~~~~~
25
25
nnn⌁N
25
25
~~~~~
25
25
VANNY
25
25
~~~~~
25
25
VANNY
25
25
25
~~~~~
V⌁⌁⌁
25
25
25
25
~~~~~
25
⌁⌁⌁⌁⌁
25
25
25
25
25 FT
⌁⌁⌁⌁n
25
25
25
25
25
24.96
24.97
24.97
24.97
24.98
24.98
49.95
49.96
49.96
49.97
49.97
49.97
49.98
49.98
49.98
49.99
49.99
10000 00000 0.00 0.00 0.00
49.99
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
5C
50
5C
50
50
5C
100 FT
5C
84.15
88.82
91.73
50
93.62
94.94
98.16
98.38
98.57
98.72
98.85
99.61
99.66
99.70
99.74
99.77
99.79
99.82
99.83
99.86
99.89
99.90
99.92
99.93
99.93
99.94
99.95
99.95
99.96
99.96
99.96
99.97
99.97
99.98
99.98
99.98
99.99
99.99
ICO
ICC
IOC
ICC
100
IOC
ICC
ICO
383
RADIUS
FT
2700
2800
2900
3CCC
3100
3200
3300
3400
3500
3600
3700
3800
3900
4000
4100
4500
5000
5500
6000
6500
7000
7500
8000
9000
10000
NOTES:
TABLE V.-EVEN-RADIUS CURVES. DEFLECTIONS
AND CHORDS

DEFL.
MIN PER
FT OF
ARC
C.637
C.614
0.593
C.573
0.555
25 FT
DEFLECTIONS FOR ARCS OF
0° 15.911
0°15.35'
0°14.82
0°14.321
0°13.86'
C.537 0°13.431
C.521 0°13.C2'
0.506 0°12.64'
C.491 0°12.28¹
C.477
C. 191
C. 172
0°11.941
C.465 0°11.611
C.452 0°11.311
0.441
0°11.02'
C.430 C°IC.741
0.419 0°IC.48¹
C.382 C°09.55'
0.344 C°C8.59'
C.313
0°07.81'
0.286 0°07.16'
0.264
C°06.61'
C.246 0°C6.141
C.229 C°C5.73'
0.215
C°C5.371
C°C4.771
C°C4.30'
50 FT
0°31.83'
0°3C.69'
0°29.641
C°28.65'
0°27.72¹
C°26.86'
0°26.04'
0°25.28'
C°24.56!
0°23.87'
C°23.23'
0°22.62'
C°22.04'
0°21.49'
0°2C.96'
0°19.10'
0°17.19'
0°15.63'
0°14.32'
0°13.22'
0°12.28¹
0°11.46'
0°IC.74'
0°09.551
0°C8.59'
100 FT
=
1°03.66'
T°CI.39'
0°59.27!
0°57.30'
C°55.451
C°53.711
0°52.09'
0°50.55'
0°49.11'
0°47.75'
0°46.46'
C°45.23'
0°44.C71
0°42.97'
0°41.92'
C°38.20'
0°34.38'
C°31.25'
C°28.65'
C°26.441
C°24.56'
0°22.92'
0°21.49'
0°19.10'
C°17.19'
CHORDS FOR ARCS OF
25 FT
~~~~~ ~~~~~ ~~~~~ ~~~~~ ~~~~~
25
слепя спел сполосся слося слоя спепел слоя слепелся сл
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
25
50 FT
× Arc Length
50
50
88888 88888
50
50
50
50
50
50
50
50
50
50
ggggཤྩ ggggg ggg
50
50
50
50
50
50
50
50
50
50
50
50
50
100 FT
100
100
ICO
ICC
ICC
ICC
ICC
ICC
IOC
ICC
ICC
ICC
ICC
ICC
ICC
Degree of Curve (arc definition) = twice the deflection for a 100-ft arc.
Deflections for even-radius curves or arcs not listed may be computed from:
1718.873
1718 R
Defl. (in minutes)
ICC
ICC
100
IOC
ICC
100
ICC
ICO
ICO
100
Chords not listed may be obtained by interpolation or computed from:
Chord = 2R sin Defl.
Total length of curve may be determined by use of Table VI. See Art. 5-||
for selection of spirals for even-radius curves.
384
DEG
1234
augue bajuд 55555 55 wwwww wwwwww~~~~ ~~~ Ñ-Ō 600 voc
10
11
12
13
14
15
17
18
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
39
40
41
43
45
47
50
53
60
ெ
TABLE VI.-LENGTHS OF CIRCULAR ARCS; RADIUS=1
(Degrees, Minutes, and Seconds to Radians)

LENGTH
0.017 45 329
.034 90 659
.052 35 988
.069 81 317
0.087 26 646
.104 71 976
.122 17 305
.139 62 634
.157 07 963
0.174 53 293
.191 98 622
.209 43 951
.226 89 280
.244 34 610
0.261 79 939
.279 25 268
.296 70 597
.314 15 927
.331 61 256
0.349 06 585
.366 51 914
.383 97 244
.401 42 573
.418 87 902
0.436 33 231
453 78 561
.471 23 890
.488 69 219
.506 14 548
0.523 59 878
.541 05 207
.558 50 536
.575 95 865
.593 41 195
0.610 86 524
.628 31 853
.645 77 182
.663 22 512
.680 67 841
DEG
61
62
63
64
0.959 93 109
0.977 38 438
0.994 83 767
1.012 29 097
1.029 74 426
1.047 19 755
65
66
67
68
69
PENNE POROº 8.*** 88... a.dk. Kukan code 2008
92
95
96
97
98
0.698 13 170
.715 58 499
.733 03 829
100
101
102
.750 49 158 103
.767 94 487 104
0.785 39 816 105
.802 85 146 106
.820 30 475
.837 75 804
107
.855 21 133 109
0.872 66 463
.890 11 792
.907 57 121
...925 02 450
.942 47 780
110
111
112
113
114
115
116
117
118
119
120
LENGTH
1.064 65 084
.082 10 414
.099 55 743
.117 01 072
1.134 46 401
.151 91 731
.169 37 060
.186 82 389
.204 27 718
1.221 73 048
.239 18 377
.256 63 706
.274 09 035
.291 54 365
1.308 99 694
.326 45 023
.343 90 352
.361 35 682
.378 81 011
MIN
-23ª
2
3
4
1.832 59 571
.850 04 901
.867 50 230
56
KONOM ONMI MONOR ONE ON ONE FOR CE 9599 ANKZ KUNANO
.884 95 559
5
6
7
8
9
10
11
12
13
14
15
16
17
18
1.396 26 340
.413 71 669
.431 16 999
.448 62 328
.466 07 657 24
19
20
21
22
1.483 52 986 25
.500 98 316
.518 43 645
.535 88 974 28
.553 34 303 29
23
26
1.570 79 633 30
27
.588 24 962 31
.605 70 291 32
.623 15 620 33
.640 60 950 34
1.658 06 279 35
.675 51 608 36
.692 96 937 37
.710 42 267 38
.727 87 596 39
1.745 32 925 40
.762 78 254
.780 23 584
41
42
.797 68 913 43
.815 14 242
44
45
46
47
48
.902 40 888 49
1.919 86 218 50
.937 31 547 51
.954 76 876
.972 22 205
.989 67 535 54
52
53
2.007 12 864 55
.024 58 193 56
.042 03 522
57
58
.059 48 852
.076 94 181 59
.094 39 510 60
LENGTH
.000 29 089
0 58 178
0 87 266
1 16 355
.001 45 444
I 74 533
2 03 622
2 32 711
2 61 799
.002 90 888
3 19 977
3 49 066
3 78 155
4 07 243
.004 36 332
4 65 421
4 94 510
SEC
.011 63 553
11 92 642
12 21 730
12 50 819
12 79 908
.013 08 997
13 38 086
13 67 175
13 96 263
14 25 352
.014 54 441
14 83 530
15 12 619
15 41 708
15 70 796
1231
UK DULU EESES £5 wwwwwwwwww ~~~~~ ~~ GOVOG FŵÑ=Ō 600 vo
.015 99 885
16 28 974
16 58 063
16 87 152
17 16 240
17 45 329
10
5 23 599 18
5 52 688
15
.005 81 776 20
6 10 865 21
6 39 954
22
24
6 69 043 23
6 98 132
.007 27 221
7 56 309
7 85 398
8 14 487
8 43 576
.008 72 665
9 01 753
9 30 842
9 59 931
9 89 020
17
25
26
27
28
29
.010 18 109 35
10 47 198
10 76 286 37
II 05 375
11 34 464
38
39
40
30
31
32
33
34
44
47
48
50
52
53
54
55
56
57
տ
58
59
60
LENGTH
.000 00 485
00 970
01 454
01 939
.000 02 424
02 909
03 394
03 879
04 363
.000 04 848
05 333
05 818
06 303
06 787
.000 07 272
07 757
08 242
08 727
09 211
.000 09 696
10 181
10 666
11 151
11 636
.000 12 120
12 605
13 090
13 575
14 060
.000 14 544
15 029
15 514
15 999
16 484
.000 16 969
17 453
17 938
18 423
18 908
.000 19 393
19 877
20 362
20 847
21 332
.000 21 817
22 301
22 786
23 271
23 756
.000 24 241
24 726
25 210
25 695
26 180
.000 26 665
27 150
27 634
28 119
28 604
29 089
385
MIN
FELLE DELU 56565 55 wwww wwww XNK EN OG FUN-0 6000 VOG IWN-0
8
10
14
15
16
17
18
19
20
21
22
~~ ~~~~~ MMMMM MMMMM
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
And
TABLE VII.-MINUTES AND SECONDS IN DECIMALS
OF A DEGREE

0
.0000
.0167
.0333
.C500
.0667
.0833
.ICCO
.1167
.1333
.1500
.1667
.1833
.2000
.2167
.2333
.25CC
.2667
.2833
.3000
.3167
.3333
.350C
3667
.3833
.4000
.4167
.4333
.4500
.4667
4833
.5000
.5167
.5333
.5500
.5667
.5833
.6000
.6167
.6333
.6500
.6667
.6833
.7000.
.7167
.7333
.7500
.7667
.7833
.8000
.8167
.8333
.8500
.8667
.8833
.9000,
.9167
.9333
.9500
.9667
.9833
10
.0028
.C194
.0361
.C528
.0694
.086 I
.1028
.1194
.1361
.1528
.1694
.1861
.2028
.2194
.2361
.2528
.2694
.2861
.3028
.3194
.3361
.3528
.3694
.3861
.4028
.5028
.5194
.5361
.5528
.5694
.5861
.6028
.6194
.6361
.6528
.6694
.6861
.7028
.7194
.7361
.4194
.4361
.4528
.4542
.4694 .4708
.4861
.4875
.7528
.7694
.7861
.8028
.8194
.8361
.8528
.8694
.886I
.9028
15
-.9194
.936)
.9528
.9694
.9861
.0042
.0208
.C375
.0542
.0708
.C875
.1042
.1208
.1375
.1542
.1708
.1875
.2042
.2208
.2375
.2542
.2708
.2875
.3042
.3208
.3375
.3542
.3708
.3875
.4042
.4208
4375
.5042
.5208
.5375
.5542
.5708
.5875
.6042
.6208
.6375
.6542
.6708
.6875
.7042
.7208
.7375
.7542
7708
.7875
.8042
.8208
.8375
.8542
.8708
.8875
.9042
.9208
.9375
.9542
.9708
.9875
20
SECONDS
.0056
.C222
.0389
.C556
.C722
.C889
.1056
.1222
.1389
.1556
.1722
.1889
.2C56
.2222
.2389
.2556
.2722
.2889
.3056
.3222
.3389
.3556
.3722
.3889
.4056
.4222
4389
.4556
.4722
.4889
.5056
.5222
.5389
.5556
.5722
.5889
.6056
.6222
.6389
.6556
.6722
.6889
.7056
.7222
.7389
.7556
.7722
.7889
.8056
.8222
.8389
.8556
.8722
.8889
.9056
.9222
.9389
.9556
.9722
.9889
30
.0083
.0250
.C417
.C583
.075C
.C917
.IC83
.1250
.1417
.1583
.175C
.1917
.2083
.225C
.2417
.2583
.2750
.2917
.3083
.3250
.3417
.3583
.3750
.3917
.4083
.4250
.4417
.4583
.4750
.4917
.5083
.5250
.5417
.5583
.575C
.5917
.6083
.6250
.6417
.6583
.6750
.6917
.7083
.7250
.7417
.7583
.775C
.7917
.8083
.8250
.8417
.8583
.8750
.8917
.9083
.9250
.9417
.9583
.9750
.9917
40
.CIII
.0278
.C444
.0611
.0778
.C944
| | | | *
.1278
.1444
.1611
.1778
.1944
.2111
.2278
.2444
.2611
.2778
.2944
.3111
.3278
.3444
.3611
.3778
.3944
.4111
.4278
.4444
.4611
.4778
.4944
.5111
.5278
.5444
.5611
.5778
.5944
.6111
.6278
.6444
.6611
.6778
.6944
.7111
.7278
.7444
.7611
.7778
.7944
.8111
.8278
.8444
.8611
.8778
.8944
.9111
.9278
.9444
.9611
.9778
.9944
45
.C125
.C292
.0458
.0625
.C792
.0958
.1125
.1292
.1458
.1625
.1792
.1958
.2125
.2292
.2458
.2625
.2792
.2958
.3125
.3292
3458
.3625
.3792
.3958
.4125
.4292
.4458
.4625
.4792
.4958
.5125
.5292
.5458
.5625
.5792
.5958
.6125
.6292
.6458
.6625
.6792
.6958
.7125
.7292
.7458
.7625
.7792
.7958
.8125
.8292
.8458
.8625
.8792
.8958
.9125
.9292
9458
.9625
.9792
.9958
50
.C139
.0306
.C472
.0639
.0806
.C972
1139
.1306
.1472
.1639
.1806
.1972
.2139
.2306
.2472
.2639
.2806
.2972
.3139
.3306
3472
.3639
3806
.3972
.4139
.4306
.4472
.4639
.4806
4972
.5139
.53C6
.5472
.5639
.5806
.5972
.6139
.6306
.6472
.6639
.6806
.6972
.7139
.7306
.7472
.7639
.7806
.7972
.8139
.83C6
.8472
.8639
.8806
.8972
.9139
.9306
.9472
.9639
.9806
.9972
386
TABLE VIII. TANGENTS AND EXTERNALS FOR A 1° CURVE
(Chord or Arc Definition)
Explanation
Chord Definition of Degree of Curve
(De)
MATHEMATICAL RELATIONS
1.
50
sin D. = 200
R
la. De
2.
=
2a. T-
=
3a. E
5,730
R
T1°
3. ER exsec
=
T = R tan¦ I……..exactly 2.
Dc
E₁º
Dc
•
1
醾
​•
•
exactly 1.
approx.
Arc Definition of Degree of Curve
(Da)
approx. la. Da
MATHEMATICAL RELATIONS
· approx.
...exactly 3.
T=3,919.5X
E=1,212.3X
Da
2a. T
3
26
3
26
3a. E
=
*
5,729.58
R
5,730
Ꭱ
T = R tan I....exactly
=
.exactly
Table VIII gives values of T and E for a 1° curve having
various central angles. The values may be considered correct
for both the chord definition and the arc definition of D, since
the difference in R for D= 1° is only seven units in the sixth
significant figure. To find T and E for any given values of I
and D, use relations 2a and 3a, as in the following example:
Given: I=68°45', D=8°40′. (Note: To
avoid lack of precision in calculation, use D as
EXAMPLE.
2 26
3'
instead
of 8.666 ...)
For Arc Definition of D,
T1°
Da
E1°
·
E = R exsec I...exactly
...exactly
Da
..exactly
452.25
·
= 139.88
approx.
..exactly
For Chord Definition of D,
T= Arc-definition value plus correction of 0.43 found
from Table IX by interpolation; or T=452.68
E-Arc-definition value plus correction of 0.14 found
from Table X by interpolation; or E=140.02
387
D
TABLE VIII.-TANGENTS AND EXTERNALS FOR A 1° CURVE
(Chord or Arc Definition)

T
0-231 DONO ONE ON M2 KO700 27~m7 ❤❤♪❤8 22 400 850HZ KONAA
6
8
9
10
12
13
14
20
21
22
23
24
15 12.5 C.C
25
16 13.3
C.C
17 14.2 O.C
18 15.C 0.0
19 15.8 0.0
27
28
29
30
36
20.8 C.C
26 21.7 0.0
22.5 C.O
23.3 C.C
24.2 0.1
37
38
39
40
41
42
43
T
25.0 C.I
31 25.8 0.1
32
33
34
44
1 = 0°
35 29.2
52
E
0.0
C.C
0.8
0.0
1.7 0.0
0.0
2.5
3.3
C.C
53
4.2 C.O
5.0 0.0
5.8 0.0
6.7
0.0
7.5 0.0
54
8.3 C.C
9.2 0.0
10.0 0.0
10.8 0.0
11.7 0.0
55
16.7 0.0
17.5 0.0
18.3 C.0
19.2 0.0
20.0 0.0
26.7 0.1
27.5
0.1
28.3
C. I
45 37.5 C.I
46 38.3 0.1
47 39.2 0.1
48 40.0 C.I
49 40.8 0.1
41.7 0.2
51 42.5 0.2
43.3 C.2
44.2 0.2
45.0 0.2
0.1
30.0 C.I
30.8 C.I
31.7 C.I
32.5 0.1
33.3 0.1
34.2 C.1
35.0 0.1
35.8 C.I
36.7 C.I
45.8 0.2
56 46.7 C.2
57 47.5 0.2
58 48.3 0.2
59 49.2 C.2
T
1 = 1°
50.0
50.8
51.7
52.5
53.3
LUJ
58.3
59.2
E
C.2
C.2
C.2
C.2
0.2
54.2 C.3
55.0 0.3
55.8 C.3
56.7 C.3
57.5 0.3
0.3
0.3
60.C
0.3
60.8 0.3
61.7 0.3
62.5 C.3
63.3 C.4
64.2 0.4
65.C 0.4
65.8 0.4
66.7 0.4
67.5 C.4
68.3
C.4
69.2
C.4
70.0
C.4
70.8 0.4
71.7
0.4
72.5
C.5
73.3 C.5
74.2 C.5
75.0 C.5
75.8 0.5
76.7 0.5
77.5 0.5
78.3 0.5
1 = 2°
T
100.0
0.9
ICC.8 C.9
ICI.7 0.9
102.5 0.9
103.3 C.9
104.2
105.0
IC5.8
IC6.7
107.5
112.5
113.3
114.2
115.0
115.9
116.7
117.5
E
IC8.3
109.2
110.0
110.8 1.1
1|1.7 1.1
118.4
119.2
120.0
C.9
1.0
1.0
1.0
1.0
129.2
130.0
130.9
131.7
132.5
1.0
1.C
1. I
I. J
1.I
1.2
1.2
1.2
1.2
1.2
1.2
1.3
120.9
1.3
121.7
1.3
122.5
1.3
123.4 1.3
124.2 1.3
125.C 1.4
125.9 1.4
126.7 1.4
127.5 1.4
128.4 1.4
79.2 0.5
80.0 C.6
80.8 C.6
81.7 0.6
82.5
133.4
0.6 134.2
135.C
C.6
83.3 C.6
84.2
85.C C.6
85.8 C.6
86.7 C.7
87.5 0.7
88.3 0.7
89.2 0.7
90.0 C.7
90.8 0.7
91.7 C.7 141.7
1.8
92.5 0.7 142.5 1.8
93.3 0.8 143.4 1.8
94.2 0.8 144.2 1.8
95.C 0.8 145.C 1.8
95.8 0.8 145.9 1.9
96.7 0.8 146.7 1.9
97.5 C.8 147.5 1.9
98.3 C.8 148.4 1.9
99.2 0.9 149.2 1.9
135.9
136.7
137.5 1.7
138.4 1.7
139.2
140.C
140.9
1.5
1.5
1.5
1.5
1.5
1.6
1.6
1.6
1.6
1.6
1 = 3°
T
E
150.0
150.9
151.7
2.C
152.5 2.0
153.4
2.1
2.0
2.0
154.2
155.C
155.9
156.7 2.1
157.5
2.2
2.1
2.1
2.1
158.4
2.2
159.2 2.2
160.0 2.2
160.9 2.3
161.7 2.3
162.5
2.3
163.4
2.3
164.2
2.4
165.0 2.4
165.9 2.4
166.7 2.4
167.6 2.4
168.4 2.5
169.2 2.5
170.1 2.5
170.9
2.5
171.7
2.6
172.6
2.6
173.4 2.6
174.2 2.6
175.1 2.7
175.9 2.7
176.7 2.7
177.6 2.8
178.4 2.8
179.2 2.8
180.1 2.8
180.9 2.9
181.7
182.6 2.9
183.4 2.9
184.2
3.0
185.1 3.0
185.9
186.7 3.C
3.0
| = 4
187.6 3.1
188.4 3.1
1.7 189.2
1.7
1.7
3.1
190.1 3.2
190.9
3.2
T
2CC. I
200.9
201.8 3.6
202.6 3.6
203.4 3.6
204.3
205. I
205.9
206.8
207.6
208.4
209.3
210.1
210.9
211.8
212.6
213.4
214.3
215.1
215.9
216.8
217.6
218.4
219.3
220. I
220.9
221.8
222.6
223.4
224.3
E
225. I
226.0
226.8
227.6
228.5
3.5
3.5
3.7
3.7
3.7 6
0-23A
1231
MONDO O-NE DONOR OU202 20700 870M7 MOM08 2007 2000 OMZ KONAA
5
3.7 8
3.8 9
7
3.8 10
3.8
3.9 12
3.9 13
3.9 14
3.9 15
4.0 16
4.0 17
4.0 18
4.1 19
4.1
4.1 21
4.3
4.2 22
4.2 23
4.2
24
4.3 25
4.3 26
4.4 28
4.4 29
4.4 30
4.5 31
4.5 32
4.5 33
229.3
230.I
231.C
2.9 231.8 4.7 38
4.6 34
4.6 35
4.6 36
4.7 37
232.6 4.7 39
233.5 4.8 40
234.3 4.8 41
235.1
4.8 42
236.0 4.9 43
191.7 3.2
192.6 3.2
193.4 3.3
194.2 3.3
195.1
244.3 5.2
3.3 245.2 5.2
195.9 3.3
196.7 3.4
197.6 3.4
198.4 3.4
199.2
3.5
236.8 4.9 44
237.6 4.9 45
238.5 5.0 46
239.3 5.C
47
246.C 5.3
246.8
247.7 5.3
240.1 5.0 48
241.C 5.1 49
241.8
5.1
50
242.6 5.151
243.5 5.2 52
53
54
55
5.356
57
248.5 5.4 58
249.3 5.4 59
388
TABLE VIII.-TANGENTS AND EXTERNALS FOR A 1° CURVE
(Chord or Arc Definition)

T
01234 FOTOD 0-231 567
5
0 250.2 5.5
7
9
254.3 5.6
6 255.2 5.7
256.C 5.7
5.8
257.7 5.8
12
T
8 256.8
| =
22222 207~~~ 87~37 68788 222 700 85NOT BONGO
21
5°
251.0
28
29
30
E
300.3
5.5
3C1.I
251.8
5.5 301.9
252.7 5.6 302.8
253.5
5.6
303.6
15 262.7 6.0
16 263.5 6.1
17 264.4 6.1
18
265.2 6.1
19 266.0 6.2
308.6
258.5 5.8
259.3 5.9
309.5
260.2 5.9 310.3
261.0 5.9
311.1
312.0
14 261.9 6.0
48
T
275.2 6.6
276.1 6.6
276.9 6.7
32
33 277.7
34 278.6 6.8
35 279.4 6.8
36 280.2 6.9
37 281.1 6.9
38 281.9 6.9
-39 282.7 7.0
1 = 6°
20 266.9 6.2
267.7 6.3
268.5 6.3
23 269.4 6.3
317.0 8.8
317.8 8.8
318.7 8.9
319.5 8.9
24 270.2 6.4 320.3 8.9
25 271.0 6.4 321.2 9.0
26 271.9 6.4 322.0 9.0
272.7 6.5 322.8 9.1
273.5 6.5 323.7 9.1
274.4 6.6 324.5 9.2
E
7.9
7.9
8.0
8.0
8.0
8.1
304.5
305.3 8.1
306.1 8.2
307.0 8.2
307.8 8.3
55
296. I
7.6
56
296.9
7.7
57 297.8 7.7
298.6 7.8
58
59 299.4
8.3
8.4
8.4
8.4
8.5
312.8 8.5
313.7 8.6
314.5 8.6
315.3 8.7
316.2 8.7
325.4
326.2
327.0
9.3
6.7 327.9 9.4
328.7 9.4
329.5 9.5
330.4 9.5
331.2 9.6
332.0 9.6
332.9 9.7
40
283.6 7.0
333.7 9.7
42
43 286. I
41 284.4 7.1 334.6 9.8
285.2 7.1 335.4 9.8
7.1 336.2 9.9
7.2 337.1 9.9
45 287.7 7.2
288.6 7.3
44
286.9
337.9 10.0
338.7 10.0
46
47
289.4 7.3 339.6 10.I
340.4 10.1
290.3 7.3
49 291.1 7.4 341.2 10.2
9.2
9.3
50 291.9 7.4 342.1 10.2
51 292.8 7.5 342.9 10.3
293.6 7.5 343.7 10.3
52
53 294.4 7.6
344.6 10.4
345.4 10.4
54
295.3
7.6
T
1 = 7°
350.4
351.3
10.7
10.8
352.1 10.8
352.9 10.9
353.8 11.0
354.6
355.5
346.3
E
357.1
358.0
11.0
II.C
1. I
11.1
11.2
11.5
11.5
363.C
363.8
364.7 11.6
365.5 11.6
366.3 11.7
367.2 11.8
368.0 11.8
368.8 11.9
369.7 11.9
37C.5 12.0
371.4 12.0
372.2 12.1
373.0 12.1
373.9 12.2
374.7 12.2
375.5 12.3
376.4 12.3
377.2 12.4
378.1 12.5
378.9 12.5
379.7 12.6
380.6 12.6
381.4 12.7
382.2 12.7
383.1 12.8
383.9 12.8
384.7 12.9
385.6
386.4
13.0
13.0
387.3
13.1
388. I
388.9
13.1
13.2
389.8 13.2
390.6 13.3
391.4 13.4
392.3
13.4
393.1 13.5
394.0
394.8
395.6 13.6
13.5
13.6
346.3 10.5 396.5 13.7
347.1 10.5 397.3 13.8
347.9 10.6 398.1 13.8
348.8 10.6 399.0 13.9
7.8 349.6 10.7 399.8 13.9
· 1 = 8°
T
E
358.8
11.2
359.6
11.3
409.0 14.6
409.9 14.6
11.3 410.7 14.7
361.3 11.4 411.5 14.8
362.2
412.4 14.8
360.5
11.4
400.7
14.C.
401.5
14.1
402.3
14.1
403.2 14.2
404.0 14.2
404.8 14.3
405.7 14.3
406.5 14.4
407.4 14.5
408.2 14.5
413.2 14.9
414.1 14.9
414.9 15.0
415.7 15.1
416.6 15.1
417.4 15.2
418.2 15.2
419.1 15.3
419.9 15.4
420.8 15.4
421.6 15.5
422.4 15.6
423.3 15.6
424.1
424.9
15.7
15.7
425.8 15.8
426.6 15.9
427.5 15.9
428.3 16.0
429.1 16.0
430.0 16.1
430.8 16.2
431.7 16.2
432.5 16.3
433.3 16.4
434.2 16.4
435.C 16.5
435.8 16.6
436.7 16.6
437.5 16.7
438.4 16.7
439.2 16.8
440.0 16.9
440.9
441.7 17.0
17.0
442.5 17.1
443.4 17.1
444.2 17.2
445.1 17.3
445.9 17.3
446.7 17.4
447.6 17.5
448.4 17.5
449.3 17.6
450.1 17.7
1 = 9°
T
E
450.9 17.7
451.8 17.8
452.6 17.8
453.4 17.9
454.3
18.0
455.I
18.0
456.0 18.1 6
456.8 18.2
457.6 18.2 8
458.5 18.3 9
01234
DO700 0-2MI BORDO 2022 20702 87~87 HOMMA GENRE OF OZNOZ HONO.
459.3 18.4 10
460.2 18.4
18.5 12
461.0
461.8
462.7
5
18.6 13
18.7 14
18.7 15
463.5
464.4
18.8 16
465.2 18.9 17
466.0
466.9
18.9 18
19.0 19
19.1
467.7
468.5 19.1 21
469.4
470.2
19.2 22
19.3 23
471.1 19.3 24
471.9 19.4❘ 25
472.7 19.5 26
473.6
474.4
475.3
19.5 27
19.6 28
19.7 29
19.7 30
476.1
476.9
19.8 31
477.8 19.9 32
478.6 20.0 33
479.5 20.0 34
480.3 20.1 35
481.1 20.2 36
482.0 20.2 37
482.8 20.3 38
483.6 20.4 39
484.5 20.4 40
486.2 20.6
485.3 20.5 41
42
487.0 20.7 43
489.5 20.9
487.8 20.7 44
488.7 20.8 45
46
490.4 20.9 47
491.2 21.0 48
492.C 21.1 49
492.9 21.2 50
493.721.2 51
494.6
21.3
52
495.4 21.4 53
496.2
21.5 54
497.1 21.5 55
497.9 21.6 56
498.8 21.7❘ 57
499.6 21.7 58
500.4 21.8 59
389
TABLE VIII.-TANGENTS AND EXTERNALS FOR A 1° CURVE
(Chord or Arc Definition)

0-231 VOTO O-NMI DONDO COME DO700 87~ma ❤8588 2222 9999 upofu bubon
667
0 501.3 21.9
22.0
503.0 22.0
22.1
22.2
1 502.
2
3
503.8
4504.6
5 5C5.5 22.3
15
6 506.3 22.3
7 507.2 22.4
8 508.0 22.5
9 508.8 22.6
12
10 509.7 22.6
11 510.5 22.7
511.4 22.8
13 512.2 22.8
513.0 22.9
513.9 23.C
14
16 514.7
17
18
20
515.6 23.1
23.2
516.4
19 517.2
517.2 23.3
25
| = 10°
518.1
21 518.9 23.5
519.8 23.5
23 520.6 23.6
24 521.4 23.7
22
T
30
31
23.8
26 523.1 23.8
524.C 23.9
524.8 24.0
29 525.6 24.1
27
28
37
38
39
40
E
441
42
35 530.7 24.5
36 531.5 24.6
532.4 24.7
533.2 24.8
534.0 24.8
44
522.3
43537.4
54
55
32
526.5 24.1 576.9 29.0
527.3 24.2 577.8 29.I
528.2 24.3 578.6 29.1
33 529.0 24.4 579.5 29.2
529.8 24.4 580.3 29.3
34
I = 11°
45 539.1 25.3
46 539.9 25.4
540.8 25.5
47
48 541.6 25.5
542.5 25.6
49
T
50 543.3 25.7
51 544.1 25.8
52 545.0 25.9
545.8 25.9
546.7 26.0
53
551.7
552.5
553.4
554.2
555. I
555.9
556.7
557.6
558.4
559.3
564.3 27.7
23.1 565.2 27.8
566.0 27.9
566.8 28.0
567.7 28.1
E
26.5
26.6
26.6
26.7
26.7
26.8
23.4 568.5 28.1
569.4 28.2
570.2 28.3
571.1 28.4
571.9 28.5
26.9
27.0
27.1 ||
27.2
27.2
27.2
560. I 27.3
561.0 27.4
561.8 27.5
562.6 27.6
563.5 27.6
572.7 28.6
573.6 28.6
574.4 28.7
575.3 28.8
576.1 28.9
534.9 24.9 585.4 29.8
535.7 25.0 586.2 29.9
536.6 25.1 587.C 30.0
25.1 587.9 30.1
538.2 25.2 588.7 30.2
581.2 29.4
582.0 29.5
582.8 29.6
583.7 29.7
584.5 29.7
547.5 26.1
548.3 26.2
56
57 549.2 26.3
58 550.C 26.3
59 550.9 26.4 6C1.4 31.5
T
| = 12°
E
602.2
31.6
31.6
603.1
603.9 31.7
604.7 31.8
605.6 31.9
606.4 32.0
6C7.3 32.Į
608.1 32.2
609.0 32.3
6C9.8 32.4
610.6 32.4
611.5 | 32.5
612.3 32.6
613.2 32.7
614.0 32.8
614.9 32.9
615.7 33.0
616.5 33.1
617.4 33.2
618.2 33.3
619.1 33.3
619.9 33.4
620.8 33.5
621.6 33.6
622.4 33.7
623.3 33.8
624.1 33.9
625.0 34.0
625.8 34.1
626.7 34.2
627.5 34.3
628.3 34.4
629.2 34.4
630.0 34.5
630.9 34.6
598.0 31.1 648.6
598.8 31.2
649.4
599.7 31.3 650.3
60C.5 31.4 651.1 36.9
652.C 37.0
T
6=
| = 13°
E
Les
652.8 37.I
653.7 37.2
654.5 37.3
655.3 | 37.4
656.2 37.5
657.0 37.6
657.9 37.6
658.7 37.7
659.6 37.8
660.4 37.9
661.3 38.0
662.1 38.1
662.9 38.2
663.8 38.3
664.6 38.4
665.5 38.5
666.3 38.6
667.2 38.7
668.0 38.8
668.9 38.9
631.7 34.7
632.6 34.8
633.4 34.9
634.2 35.0
635.1 35.1
635.9 35.2
636.8 35.3
637.6 | 35.4
638.5 35.5
639.3 35.6
640.2 35.7
41.6
41.7
41.8
690.8 41.5
641.0 35.7 691.7
641.8 35.8 || 692.5
642.7 35.9 693.4
643.5 36.C 694.2 41.9
644.4 36.1 695.I 42.0
645.2 36.2 || 695.9
695.9 42.1
646.1 36.3
646.9 36.4
597.2 31.0 647.7 36.5
589.6 30.3
590.4 30.3
591.3 30.4
592.1 30.5
592.9 30.6
593.8 30.7
594.6 30.8
595.5 30.9
596.3 30.9
696.7 42.2
697.6 42.3
698.4 42.4
669.7 39.0
670.5 39.I
671.4 39.2
672.2 39.3
673.1 39.4
673.9 39.5
674.8 39.6
675.6 39.7
676.5 | 39.8
677.3 39.9
678.1 40.0
679.C 4C.I
679.8 40.2
680.7 40.3
681.5 40.4
682.4
683.2
684. I
684.9
685.8 40.9
4C.5
40.6
40.7
4C.8
686.6 41.C
687.4 41.1
688.3 41.2
689.I 41.3
690.0 41.4
36.6 699.3 42.5
36.7 7CC.1 42.6
36.8 7C1.C 42.7
701.8 42.8
7C2.7 42.9
1 = 14°
T
7C3.5 43.0
704.4 43.1
705.2 43.2
706.1 43.3
7C6.9 43.4 4
707.7 43.5 5
7C8.6 43.7 6
709.4 43.8 7
710.3 43.9 8
711.1 44.0 9
712.0 44.1 10
712.8
44.2 "
713.7 44.3 12
714.5 44.4 13
715.4 44.5 14
720.4
721.3
E
716.2 44.6 15
717.1
717.9
44.7 16
718.7
719.6
724.7
725.5
726.4
727.2
T
44.8 17
44.9 18
0-231 BOT∞ OUNCE CONDO ONME KONDO OMMma ❤❤m88 22 202 85AMZ KAKO.
45.C 19
45.1 20
45.2 21
722.1 45.3 22
723.0
45.4 23
723.8 45.5 24
45.6 25
45.8 26
45.9 27
46.0 28
728.1 46.1 29
728.9 46.2 30
729.8 46.3 31
730.6 46.4 32
731.4 46.5 33
732.3 46.6 34
733.1 46.7 35
734.0 46.8 36
734.8 46.9 37
735.7 47.0 38
736.5 47.1 39
737.4 47.3 40
738.2 47.4 41
739.1 47.5 42
739.9 47.6
43
740.8 47.7
42.2
741.6 47.8 45
742.5
47.9 46
743.3 48.0
44
744.2 48.1 48
745.C
747.5 48.6
47
745.8 48.3 50
749.2 48.8
746.7 48.5 51
52
748.4 48.7 53
54
750.1 48.9 55
750.9 49.C
56
751.8 49.1 | 57
752.6 49.2 | 58
753.5 49.3
59
390
TABLE VIII.-TANGENTS AND EXTERNALS FOR A 1° CURVE
(Chord or Arc Definition)

1
E
754.3
49.4
755.2 49.6
756.0 49.7
3 756.9 49.8
4 757.7 49.9
0123A KO789
6
5 758.6 50.0
759.4 50.I
760.3 50.2
761.1 50.3
762.0 50.4
~37
13
10 762.8 50.6
11 763.7 50.7
764.5 50.8
765.3 50.9
766.2 51.0
12
14
17
18
2002 DONDº 870M2 28788 2
15 767.0 51.1
16 767.9 51.2
768.7 51.3
769.6 51.5
19 770.4 51.6
25
26
1 = 15°
771.3 51.7
21 772.1 51.8
773.0 51.9
23 773.8 52.0
24 774.7 52.1
27
28
29
T
--
40
41
30 779.8 52.8
31 780.6 52.9
781.5 53.0
33 782.3 53.2
34
32
783.2 53.3
788.3 54.0
789.1 54.1
42 790.0 54.2
43 790.8 54.3
4444
791.7 54.4
g་ྒུ ཀྱྰཀྨཤྩགgདྩོཆོ
55
775.5 52.2
776.4 52.4
777.2 52.5
770. i 52.6
778.9 52.7
45 792.5
54.6
46
793.4 54.7
47 794.2
54.8
54.9
48
795. I
795.9
49
55.0
796.8 55. I
51797.6
55.2
55.4
798.5
53 799.3 55.5
54 800.2 55.6
56
826.5 59.3
827.4 59.4
828.2 59.5
829.1 59.7
829.9 59.8
830.8 59.9
831.6 60.0
832.5 | 60.2
833.3 60.3
834.2 60.4
835.0 60.5
35 784.0 53.4
37
38
36 784.9 53.5 835.9 60.7
785.7 53.6 836.7 60.8
786.6 53.7 837.6 60.9
39 787.4 53.9 838.4 61.0
839.3 61.1
840. I 61.3
841.0 61.4
841.8 61.5
842.7 61.6
801.0 55.7
801.9
802.7
55.8
56.0
57
58 803.6
59 804.4
| = 16°
56. I
56.2
T
805.2
806. I
806.9
807.8
808.6
E
809.5 56.9
810.3 57.0
811.2 57.1
812.0 57.3
812.9 57.4
813.7 57.5
814.6 57.6
815.5 57.7
816.3 57.9
817.2
58.0
822.3
823. I
824.0
824.8
825.7
58. I
58.2
818.0
818.9
819.7 58.3
820.6 58.5
821.4 58.6
56.3 856.3 63.6
56.4 857.2 63.8
56.5 858.0 63.9
56.7 858.9 64.0
56.8 859.7 64.1
860.6 64.3
861.4 64.4
862.3 64.5
863.1
64.6
864.0
64.8
58.7
58.8
58.9
59. I
59.2
843.5 61.8
844.4 61.9
845.2 62.0
846.1 62.1
846.9 62.3
| = 17°
T
E
864.8 64.9
865.7 65.0
866.5 65.2
867.4 65.3
868.2 65.4
65.5
869. I
869.9 65.7
870.8 65.8
871.6 65.9
872.5 66. I
873.3 66.2
874.2 66.3
875.1 66.4
875.9 66.6
876.8 66.7
877.6 66.8
878.5 67.0
879.3 67.1
880.2 57.2
881.0 67.3
881.9 67.5
882.7 67.6
883.6 67.7
884.4 67.9
885.3 68.0
886. 68.I
887.0 68.3
887.9 68.4
888.7 68.5
889.6 68.6
890.4 68.8
891.3 68.9
892. I 69.0
893.0 69.2
893.8 69.3
894.7 69.4
895.5 69.6
896.4 69.7
897.2 69.8
898.1
70.0
70. I
70.2
847.8 62.4 898.9
848.6 62.5 899.8
849.5 62.6 900.7 70.4
850.3 62.8 901.5 70.5
851.2 62.9 902.4 70.6
852.0 63.0 .903.2 70.8
852.9 63.1 904.1 70.9
853.7 63.3 904.9 71.0
854.6 63.4 905.8 71.2
855.5 63.5 906.6 71.3
1 = 18°
T
E
907.5 71.4
908.3 71.6
909.2 | 71.7
910. I 71.8
910.9
72.0
911.8 72.1
912.6 72.2
913.5 72.4
914.3 72.5
915.2 72.6
916.0 72.8
916.9 72.9
917.7 73.0
918.6 73.2
919.5 73.3
920.3 73.4
921.2 73.6
922.0 73.7
922.9 73.8
923.7 74.0
924.674.1
925.4 74.3
926.3 74.4
927.I 74.5
928.0 74.7
928.9 74.8
929.7 74.9
930.6 75.1
931.4 75.2
932.3
75.4
75.5
933. I
934.0
75.6
934.8 75.8
935.7 75.9
936.6 76.0
937.4 76.2
938.3 76.3
939.1 76.5
940.0 76.6
940.8 76.7
941.7 76.9
942.5 77.0
943.4 77.1
944.3 77.3
945. I
77.4
946.0 77.6
946.8 77.7
947.7 77.8
948.5 78.0
949.4 78.1
950.2 78.3
951.1 78.4
952.0 78.5
952.8 78.7
953.7
78.8
954.5 79.0
955.4 79.1
956.2
957. I
958.0
79.2
79.4
79.5
1 = 19°
T
958.8 79.7
959.7 79.8
960.5 80.0
961.4 80.1
962.2
80.2
E
963. I
964.0
964.8 80.7
80.4
80.5
976.0
976.8
977.7
978.5
979.4
965.7 80.8 8
966.5 80.9 9
967.4
81.1 10
968.2
81.2
969. I
970.0
81.5 13
970.8 81.7 14
1
971.7
81.8 15
972.5 82.0 16
973.4
974.2
82.1 17
82.2 18
82.4 19
975. I
0-23+ DONO IN ON 222 00702 87♡♡7 60m88 20 29500 8508Z HANAA
55
5
81.4 12
82.5 20
82.7
82.7 21
82.8 22
83.0 23
83.1 24
83.2 25
980.2
981.1
982.0
982.8 83.7 28
83.4 26
83.5 27
983.7 83.8 29
984.5 84.0 30
985.4 84.1 31
986.3 84.3 32
987.1
988.0
84.4 33
84.6 34
988.8
989.7
84.8 36
990.5 85.0 37
991.4 85.1 38
992.3 85.3 39
993.1 85.4 40
994.0 85.6 41
994.8 85.7 42
84.7 35
995.7 85.9 43
87.3
996.5 86.0 44
87.5
997.4 86.2 45
998.3 86.346
999. I
1000.0
86.5
47
86.6 48
1000.8 86.8 49
1001.7 86.9
1002.6 87.1 51
1003.4
1004.3
87.2
52
53
54
1005. 1
88.2
1006.0
87.6
55
1006.9 87.8 56
1007.7
87.9
57
1008.6 88. I 58
1009.4
59
391
TABLE VIII.—TANGENTS AND EXTERNALS FOR A 1º CURVE
(Chord or Arc Definition)

1
01234
2
3
4
56789 0-234
10
12
13
15
16
17
18
19
66-66 185 55555 £55±5 wwwww wwwww N8280 281
27
14 1022.3
36
37
38
39
20
1027.5
21
1028.3
22
1029.2
23 1030.I
1030.9
24
41
T
25
1031.8
26 1032.6
1033.5
1034.4
29 1035.2
42
1 = 20°
1010.3
1011.2
1012.0
1012.9
1013.7
54
1014.6
89.1
1015.4 89.3
89.4
1017.2 89.6
1018.0
89.7
1016.3
55
56
1018.9
1019.8
1020.6
1021.5
58
1023.2
1024.0
1024.9
1025.8
1026.6
1044.7
1045.6
1046.4
43
1047.3
44 1048.1
45
1036. I
1037.0
1037.8
1038.7
1040.4
1041.3
1042.1
1043.0
1043.8
49
50 1053.3
1054.2
1055.0
53 1055.9
1056.8
1049.0
1049.9
1050:7
1051.6
1052.4
92.9
93.1
93.2
93.4
1039.5 93.5
E
88.4
88.5
88.7
88.8
89.0
1057.6
1058.5
57 1059.3
1.060.2
1061.I
89.9
90.0
90.2
90.3
90.5
90.6
90.8
90.9
91.1
91.2
91.4
91.6
91.7
91.9
92.0
92.2
92.3
92.5
92.6
92.8
93.7
93.8
94.0
94.2
94.3
94.5
94.6
94.8
94.9
95.1
95.2
95.4
95.5
95.7
95.9
96.0
96.2
96.3
96.5
96.6
96.8
97.0
97.1
97.3
97.4
1 = 21°
T
1061.9
1062.8
1063.7
1064.5
1065.4
1066.2
1067.1
1068.0
1068.8
1069.7
1070.6
1071.4
1072.3
1073. I
1074.0
1074.9
1075.7
1076.6
1077.5
1078.3
1083.5
1084.4
1085.2
1086. I
1086.9
1087.8
1088.7
1089.5
1090.4
1091.3
1093.9
1094.7
1095.6
1079.2
100.7
1080.0 100.9
1080.9
101.1
1081.8
101.2
1082.6 101.4
1100.8
1101.6
1102.5
་་་
1103.4
1104.2
97.6
97.7
97.9
98.1
98.2
1105. I
1105.9
1106.8
1107.7
1108.5
98.4
98.5
98.7
98.8
99.0
99.2
99.3
99.5
99.6
99.8
1109.4
1110.3
100.0
100. I
100.3
100.4
100.6
1092.1 103.2
1093.0 103.3
103.5
101.6
101.7
101.9
102.0
102.2
1096.4 104.0
1097.3 104.1
1098.2 104.3
1099.0 104.5
1099.9
104.6
102.3
102.5
102.7
102.8
103.0
103.6
103.8
104.8
104.9
105. I
105.3
105.4
105.6
105.8
105.9
106. I
106.3
106.4
106.6
111.1 106.7
1112.0 106.9
1112.9 107.1
| = 22°
T
E
1113.7
107.2
1114.6 107.4
1115.5
107.6
1116.3
107.7
1117.2
107.9
1118.1
1118.9
1119.8
1120.7
1121.5
1122.4
1123.2
1124.1
1125.0
1125.8
1131.0
1131.9
1132.8
1133.6
1134.5
1126.7
109.7
1127.6
109.9
1128.4 110.1
1129.3 110.2
1130.2
110.4
108.1
108.2
108.4
108.6
108.7
108.9
109.1
109.2
109.4
109.6
1145.8
1146.6
1147.5
1135.4
111.4
1136.2 111.6
1137.1 111.8
1138.0
11.9
1138.8
112.1
1148.4
1149.2
1150. I
1151.0
1151.8
110.6
110.7
1139.7
112.3
1140.6
112.4
1141.4 112.6
1142.3
112.8
112.9
1143.2
110.9
III.
[11.1
111.2
1144.0 113.1
1144.9 113.3
113.4
1159.6
1160.5
113.6
113.8
113.9
114.1
114.3
114.5
114.6
1152.7 114.8
1153.6 115.0
1154.4 115.1
1155.3 115.3
1156.2 115.5
1157.0 115.7
1157.9 115.8
1158.8
116.0
116.2
116.3
1161.4
116.5
1162.2 116.7
1163.1
1164.0
116.9
117.0
1164.8 117.2
1 = 23°
T
E
1165.7
117.4
1166.6
117.6
1167.4
117.7
1168.3 117.9
1169.2 118.1
1170.1
118.3
1170.9 118.4
1171.8 118.6
1172.7 118.8
1173.5 118.9
1174.4
1175.3
1176.1
1177.0
1177.9
1183.1
1183.9
1184.8
1185.7
1186.6
0
I
·237
119.1 10
119.3
119.5 12
6678σ
5
119.6 13
119.8 14
1178.7 120.0 15
1179.6 120.2 16
1180.5
1181.3
1182.2
9
120.3 17
120.5 18
120.7
-NMI BOMO UN D700 8~~~
120.9 20
121.0 21
121.2 22
1200.5 124.4
1201.3 124.6
1202.2 124.8
1203.1 124.9
1203.9 125.1
121.4 23
121.6 24
121.8 25
121.9 26
1187.4
1188.3
1189.2 122.1 27
1190.0
122.3 28
122.5
I 190.9
1209.2
1210.0
1210.9 126.6
1191.8 122.6 30
1192.6 122.8 31
1193.5
1194.4
I 195.2
123.0 32
123.2
123.3
1211.8 126.7
123.5
1196.1
1197.0
1198.9
123.7 36
123.9
I 198.7 124.1
1199.6 124.2
1217.0 127.8
80 38588 FIRE CONDO OUNCE
33
34
1204.8 125.3
1205.7 125.5
1206.6 125.7
1207.4 125.8 48
1208.3
126.0 49
40
42
126.2 50
126.4
51
46
LOLOLO LO LO LO LO L
6678 σ
52
53
1212.6 126.9 54
1213.5 127.1 55
1214.4
127.3
1215.3 127.5 57
56
1216.1 127.6 58
59
392
T
TABLE VIII.-TANGENTS AND EXTERNALS FOR A 1° CURVE
(Chord or Arc Definition)

567
0-2MH BOTH OUNCE ON ONE DONO CUMA KOMMA 220 250g unõbõ hubog
नं
3
Ц
5
6
7
8
9
10
15
16
17
18
12
13
14 1230.1
22
25
26
19 1234.4 131.5
27
20 1235.3 131.7
28
21 1236.2 131.8
1237.0
132.0
30
23 1237.9 132.2
32
24 1238.8 132.4
1239.7
1240.5
132.6
132.8
1241.4 132.9
1242.3 133.1
29 1243.2 133.3
36
1 = 24°
37
T
E
1217.9
128.0
1218.7 128.2
1219.6 128.4
128.5
128.7
1220.5
1221.4
38
1222.2 128.9
1244.0 133.5
31 1244.9 133.7
39
1223.1 129.1
1224.0
1224.8
40
42
1225.7 129.6
33 1246.6 134.1
134.2
34 1247.5
43
44
1226.6
129.8
130.0
1227.5
130.2
1228.3
1229.2 130.4
130.6
35 1248.4
1249.3
1250. I
1251.0
1251.9
129.3
129.5
45
1230.9 130.7
1231.8 130.9
1232.7 131.1
1233.6 131.3
46
47
གླ846n
58
1252.8 135.4
41 1253.6 135.5
1254.5 135.7
1255.4 135.9
1256.3 136.1
1245.8 133.9
1257.1 136.3
1258.0 136.5
1258.9 136.7
1259.7 136.9
48
49 1260.6 137.0
1261.5 137.2
50
51 1262.4 137.4
52 1263.2 137.6
53 1264.1 137.8
1265.0 138.0
54
134.4
134.6
134.8
135.0
135.2
55 1265.9 138.2
56 1266.7 138.4
1267.6 138.5
1268.5 138.7
57
59 1269.4 138.9
1 = 25°
T
1270.2
1271.1
1272.0
1272.9
1273.7
1274.6
1275.5
1276.4
1277.2
1278.1
1279.0
141.0
1279.9 141.2
1280.7 141.4
1281.6 141.6
1282.5 141.8
1285.1
1286.0
1286.9
1283.4 142.0
1284.2
142.2
142.4
142.5
142.7
1287.7
1288.6
1289.5
1290.4
E
139.1
139.3
139.5
139.7
139.9
140.1
140.3
140.4
140.6
140.8
142.9
143.1
143.3
143.5
1291.2 143.7
1294.7
1295.6
1292.1
143.9
1293.0
144.1
1293.9 144.3
1296.5
1297.4
1298.2
1299. I
1300.0
144.5
144.7
144.9
145.0
145.2
145.4
145.6
1300.9 145.8
1301.7 146.0
1302.6 146.2
1303.5
146.4
1304.4 146.6
1305.3
146.8
1306.1 147.0
1307.0 147.2
1307.9 147.4
1308.8 147.6
1309.6 147.8
1310.5 148.0
1311.4 148.2
1312.3 148.4
1313.1 148.6
1314.0 148.7
1314.9 148.9
1315.8 149.1
1316.7 149.3
1317.5 149.5
1318.4 149.7
1319.3 149.9
1320.2 150.1
1321.0
1321.9
150.3
150.5
1 = 26°
T
1322.8
1323.7
1324.5
1325.4
1326.3
1327.2
1328.1
1328.9
1329.8
1330.7
1331.6
1332.5
1333.3
1334.2
1335.1
1336.0
1336.8
1337.7
1338.6
1339.5
1340.4
1341.2
1342.1
1343.0
1343.9
E
150.7
150.9
151.1
151.3
151.5
151.7
151.9
152.1
152.3
152.5
152.7
152.9
153.1
153.3
153.5
1356.2
1357.1
153.7
153.9
154. I
154.3
154.5
154.7
154.9
155. I
155.3
155.5
1344.8
155.7
1345.6
155.9
1346.5
156.1
1347.4 156.3
1348.3
156.5
1349.2
156.7
1350.0 156.9
1350.9
157.1
1351.8 157.3
1352.7 157.5
1353.6 157.7
1354.4 157.9
1355.3 158.1
158.3
158.5
1358.7
1358.8
1359.7
159.1
1360.6 159.3
1361.5
159.5
158.7
158.9
1362.4 159.7
1363.2 159.9
1364.1 160.I
1365.0
1365.9
160.4
160.6
1366.8 160.8
1367.6
161.0
1368.5
161.2
1369.4
161.4
1370.3 161.6
1371.2 161.8
1372.0 162.0
1372.9 162.2
1373.8 162.4
1374.7 162.6
| = 27°
T
E
1375.6 162.8
1376.4 163.0
1377.3 163.2
1378.2 163.4
1379.1 163.6
1381.7 164.2
1380.0
163.8
1380.9 164.0 6
1382.6 164.5 8
1388.8
1389.7
1390.6
1391.4
1392.3
1383.5 164.7 9
0-23-
1384.4 164.9 10
1385.3 165.1
1386.1
165.3 12
1387.0
1387.9
4
165.5 13
165.7 14
BO700 0-2MI DONOR OU222 KON≈≈ 87~♡7 68588 2722E OF 85087 KANO8
LO
165.9
166. I
166.3
166.5 18
1393.2
1394.1
1395.0 167.4
1395.9
1396.7
166.7 19
167.0 20
15
16
167.2 21
17
167.6 23
167.8 24
168.0 25
1397.6
1398.5 168.2 26
1399.4 168.4 27
1400.3 168.6 28
1401.2 168.8 29
169.0 30
1402.0
1402.9
1403.8
169.5 32
1404.7 169.7 33
1405.6
169.9 34
169.3 31
171.6
1406.5
170.1 35
1407.3 170.3 36
1408.2
170.5
1409.1 170.7 38
1422.4 173.9
1423.3
1410.0 170.9 39
1410.9
1411.8
1412.6
1413.5 171.8 43
1414.4 172.0 44
1415.3 172.2
1416.2 172.4
1417.1 172.6
1417.9
172.8
1418.8 173.1
1424.1
1425.0
1425.9
1426.8
1427.7 175.2
171.2 40
37
171.4 41
174.8
175.0
42
45
46
47
1419.7 173.3
1420.6 173.5 51
1421.5 173.7 52
48
49
174.1 54
53
174.3 55
174.6 56
57
58
59
393
1
0-231 DONO DINE KONCO ~~~
TABLE VIII.—TANGENTS AND EXTERNALS FOR A 1º
(Chord or Arc Definition)
4
5
6
7
8
9
10
11
12
13
1437.4
177.6
1438.3 177.8
1439.2 178.0
1440.1
178.2
14 1441.0 178.4
33
122 20~~2 2~~m~ komm E NO COME DOOR
24
20 1446.3
21
22
23
32
40
41
42
15 1441.8 178.6
16 1442.7 178.9
17 1443.6 179.1
1444.5 179.3
19 1445.4 179.5
18
44
45
46
47
48
49
55L55
T
1 = 28°
25 1450.7 180.8
26
1451.6 181.0
27 1452.5 181.2
1453.4 181.5
1454.3 181.7
1455.1 181.9
1456.0 182.1
1456.9 182.3
1457.8 182.5
34 1458.7
182.8
55
1428.6
1429.4
1430.3
1431.2
1432.1
56
57
LAS
58
E
59
175.4
175.6
175.8
176.1
176.3
1433.0
176.5
1433.9 176.7
1434.8
176.9
1435.6
177.1
1436.5
177.3
179.7
1447.2 180.0
1448.1 180.2
1448.9 180.4
1449.8 180.6
1472.9 186.3
51 1473.8 186.5
52
1474.7 186.7
53 1475.6 187.0
54 1476.5 187.2
1459.6
183.0
183.2
1460.5
1461.4 183.4
1462.2 183.6
1463.1 183.9
1464.0 184.1
1464.9 184.3
1465.8 184.5
1466.7 184.7
1467.6 185.0
1468.5 185.2
1469.3 185.4
1470.2 185.6
1471.1 185.8
1472.0 186.1
1477.3 187.4
1478.2 187.6
1479.1 187.8
1480.0 188.1
1480.9 188.3
1 = 29°
T
E
1481.8 188.5
1482.7
188.7
189.0
1483.6
1484.5
189.2
1485.3
189.4
1486.2
189.6
1487.1 189.8
1488.0
190.1
1488.9
1489.8
1490.7
190.7
1491.6
191.0
1492.5
191.2
1493.4
191.4
1494.2 191.6
190.3
190.5
1495.1 191.9
1496.0
1496.9
1497.8
1498.7
1499.6
1500.5
1501.4
1502.3
1503.1
1504.0
1504.9
1505.8
192.1
192.3
192.5
192.8
193.0
193.2
193.4
193.7
193.9
194.1
194.3
194.6
1506.7 194.8
1507.6 195.0
195.2
1508.5
1509.4
195.5
1510.3
195.7
1511.2
195.9
1512.1 196.2
1512.9
196.4
1513.8 196.6
1514.7 196.8
1515.6 197.1
1516.5 197.3
1517.4 197.5
1518.3 197.8
1519.2 198.0
1520.1
198.2
1521.0
198.4
1521.9 198.7
1522.8 198.9
1523.6 199.1
1524.5
199.4
199.6
1525.4
1526.3
199.8
1527.2
200. I
1528.1 200.3
1529.0 200.5
1529.9 200.7
1530.8 201.0
1531.7 201.2
1532.6 201.4
1533.5 201.7
1534.4
201.9
1 = 30°
T
E
1535.3 202.1
1536.1 202.4
1537.0❘ 202.6
1537.9 202.8
1538.8
203.0
1539.7
1540.6
1541.5
1542.4
1543.3
1544.2
1545.1
1546.0
1546.9
1547.8
1548.7
1549.6
1550.4
1551.3
1552.2
1557.6
1558.5
1559.4
1560.3
1561.2
1562.1
1563.0
203.3
203.5
1563.9
1564.8
1565.7
203.7
204.0
204.2
1553.1
206.8
1554.0
207.0
1554.9 207.2
1555.8 207.5
1556.7
207.7
204.4
204.7
204.9
205.1
205.4
205.6
205.8
206.1
206.3
206.5
207.9
208.2
208.4
208.7
208.9
209.1
209.4
209.6
209.8
210. I
1566.6
210.3
1567.5
210.5
1568.4
210.8
1569.2
211.0
1570.1 211.2
1571.0 211.5
1571.9 211.7
1572.8 212.0
1573.7
1574.6
212.2
212.4
1575.5 212.7
1576.4 212.9
1577.3 213.1
1578.2 213.4
1579.1 213.6
1580.0 213.9
1580.9 214.1
1581.8 214.3
1582.7 214.6
1583.6 214.8
1584.5 215.1
1585.4 215.3
1586.3 215.5
215.8
216.0
1587.2
1588. I
1 = 31°
CURVE

T
E
1589.0 216.2
1589.9 216.5
1590.8 216.7
1591.7 217.0
1592.6 217.2
1593.5 217.5
1594.4 217.7
1595.3 217.9
1596.2 218.2
1597.1
218.4
1598.0
1598.8
1599.7
1600.6
1601.5
1
OI234 5O700 0-231
1616.8
1617.7
1618.6
1619.5
0
6
8
9
218.7 10
218.9
1615.9 223.5
223.8
224.0
224.2
224.5
11
219.1 12
219.4 13
219.6 14
1602.4
1603.3
1604.2
220.1 16
220.4 17
220.6 18
1605.1
1606.0 220.8 19
219.9 15
1606.9 221.1 20
1607.8 221.3 21
1608.7 221.6
1609.6 221.8
1610.5 222.1
56
22222
6666 65 55555 555 wwwww wwwww ~~ ~~~ ~~
23
1611.4
222.3
1612.3 222.5
1613.2 222.8 27
1614.1 223.0
1615.0 223.3
24
25
26
28
29
1620.4
224.7
35
1621.3 225.0 36
1622.2 225.2
1623.1 225.5 38
1624.0
225.7
39
37
40
41
1624.9 226.0
1625.8 226.2
1626.7 226.5 42
1627.6 226.7
1628.5 226.9
43
44
45
49
1629.4 227.2
1630.3 227.4 46
1631.2 227.7
1632.1 227.9 48
1633.0 228.2
1633.9 228.4
1634.8 228.7
1635.7 228.9 52
1636.6 229.2
1637.5 229.4
1638.4 229.7
1639.3 229.9 56
1640.2 230.2 57
1641.1 230.4 58
1642.0 230.7 59
53
54
55
394
01234
TABLE VIII.-TANGENTS AND EXTERNALS FOR A 1° CURVE
(Chord or Arç Definition)

2
4
66789 0-2M‡
5
10
11
1652.0
1652.9
1653.8
13
1654.7
14 1655.6
12
DOMOO 87♡♡
17
15 1656.5
16
1657.4
1658.3
1659.2
1660. I
22222 222
20
1661.0
21 1661.9
24
26
235.9
236.2
22 1662.8 236.4
27
ოოო
23 1663.7 236.7
1664.6 236.9
33
25 1665.5 237.2
35
36
40
41
1 = 32°
42
43
44
T
E
1643.0 230.9
1643.9 231.1
1644.8 231.4
1645.7 231.6
1646.6
231.9
45
46
1647.5
232.1
1648.4 232.4
1649.3
1650.2
1651.1
30 1670.0 238.4
1670.9
238.7
32 1671.8 238.9
47
48
49
100 85087 HONOR
55 LO
1672.8 239.2
34 1673.7 239.4
51
1674.6 239.7
1675.5 239.9
37 1676.4 240.2
38
39
1677.3 240.5
1678.2 240.7
1679.1 241.0
1680.0 241.2
1680.9 241.5
1681.8 241.7
1682.7 242.0
52
53
232.6
232.9
233. I
233.4
233.6
233.9
234. I
234.4
55
56
234.6
234.9
57
235. I
235.4
235.6
1666.4 237.4
1667.3 237.7
1668.2 237.9
1669.1 238.2
1688. I
1689.0
1690.0
243.5
243.8
244.0
1690.9 244.3
54 1691.8 244.5
1692.7
1693.6
1694.5
58 1695.4
1683.6 242.2
1684.5 242.5
1685.4 242.8
1686.3 243.0
1687.2 243.3
244.8
245. I
245.3
245.6
59 1696.3 245.8
1 = 33°
T
1697.2
1698. I
1699.0
1699.9
1700.8
1701.7
247.4
1702.6
247.6
1703.5 247.9
1704.5 248.I
1705.4 248.4
1706.3 248.7
1707.2
248.9
1708.1
249.2
1709.0 249.4
1709.9 249.7
E
246.1
246.3
246.6
246.9
247. I
1710.8 250.0
1711.7 250.2
1712.6 250.5
1713.5 250.7
1714.4 251.0
1715.3 251.3
1716.3
251.5
1717.2
1718. I
1719.0
1719.9
1720.8
1721.7
1722.6
1723.5
251.8
252.0
252.3
1738.1
1739.0
1739.9
1740.8
1741.7
252.6
252.8
253. I
253.3
253.6
1724.4
253.9
1725.3 254.I
1726.2
254.4
254.7
1727.2
1728.1
254.9
1729.0 255.2
1729.9 255.4
1730.8 255.7
1731.7 256.0
1732.6 256.2
1733.5 256.5
1734.4 256.8
1735.3 257.0
1736.3 257.3
1737.2 257.6
257.8
258. I
258.3
258.6
258.9
1742.6
259. I
1743.5
259.4
1744.4 259.7
1745.4 259.9
1746.3 260.2
1747.2 260.5
1748.1 260.7
1749.0 261.0
1749.9 261.3
1750.8 261.5
| = 34°
T
1751.7
1752.6
1753.6
1754.5
1755.4
1756.3
1757.2
1758.1
1759.0
1759.9
1760.8
1761.8
1765.4
1766.3
1767.2
1768. I
1769.1
E
261.8
262. I
1762.7 265.0
265.3
1763.6
1764.5
265.5
262.3
262.6
262.9
1777.3
1778.2
263. I
263.4
263.7
263.9
264.4
264.5
264.7
1770.0
267.2
1770.9
267.4
1771.8
267.7
1772.7 268.0
1773.6 268.2
265.8
266. I
266.4
266.6
266.9
1774.5 268.5
1775.4
268.8
1776.4
269. I
1797.4
1798.3
1799.2
1800. I
1801.1
269.3
269.6
1779.1 269.9
1780.0 270.1
1780.9 270.4
1781.8 270.7
1782.8
270.9
271.2
1783.7
1784.6
271.5
1785.5 271.8
272.0
1786.4
1787.3
272.3
1788.2 272.6
1789.2 272.9
1790.1 273.1
1791.0 273.4
1791.9 273.7
1792.8 273.9
1793.7 274.2
1794.6 274.5
1795.6 274.8
1796.5
275.0
275.3
275.6
275.9
276.1
276.4
1802.0 276.7
1802.9 277.0
1803.8 277.2
1804.7 277.5
1805.6 277.8
| = 35°
T
E
1806.6 278.1
1807.5 278.3
1808.4 278.6
1809.3 278.9
1810.2 279.2 4
1811.1 279.4
1812.1 279.7
1813.0 280.0
1813.9 280.3
1814.8 280.5
1815.7
1816.6
1817.6
1818.5
1819.4
1820.3
1821.2
1822.1
1823. 1
1824.0
1832.2
1833.2
280.8 10
0123A
281.I "
1838.7
1839.6
1840.5
1841.4
1842.4
56789 0-23A
281.4 12
1829.5
285.0
1830.4 285.3
1831.3
285.6
285.8
286. I
281.7 13
281.9 14
282.2 15
282.5 16
282.8 17
283.0 18
283.3
19
1824.9
283.6 20
1825.8 283.9 21
1826.7 284.2
1827.6
284.4
1828.6 284.7 24
DOMOD 222 2
1843.3
1844.2
289.2
289.5
1845.1 289.8
1846.0 290.0
1847.0 290.3
1847.9 290.6
1848.8 290.9
1849.7 291.2
1850.6 291.5
1851.5 291.7
1852.5
292.0
1853.4 292.3
1854.3
292.6
1855.2
292.9
1856.2
293.2
6 O EEEEE E55=5 wwwww wwwww ~~28 EN
23
25
26
1834.1 286.4
1835.0 266.7
1835.9 287.0 32
27
29
1836.8 287.2 33
1837.8 287.5
34
30
31
287.8 35
288. I 36
288.4
37
288.6
288.9
38
39
40
41
42
44
45
53
1857.1 293.4 55
1858.0
1858.9 294.0
1859.8 294.3
1860.8
O O
293.7 56
58
294.6 59
395
TABLE VIII.-TANGENTS AND EXTERNALS FOR A 1° CURVE
(Chord or Arc Definition)

T
0 1861.7
1862.6
2 1863.5
3 1864.4
4 1865.4
O-NMH BOTOK O-NET DOO ON 2 HO700 87~~~ ~8~8. O 00 8500 HONOR
5
1866.3
1867.2
7 1868. I
6
8
9
16
10 1870.9 297.7
1871.8 298.0
12
13
14 1874.6 298.9
1872.7 298.3
1873.7 298.6
1876.4
17 1877.4
15 1875.5 299.2
299.4
299.7
18 1878.3 300.0
300.3
20
19 1879.2
21
22
23
24
27
28
29
30
26
25 1884.7 302.0
1885.7 302.3
1886.6 302.6
1887.5 302.9
1888.4 303.2
1889.4 303.5
1890.3 303.8
1891.2 304.1
1892.1 304.3
1893.1 304.6
1894.0 304.9
36 1894.9 305.2
1895.8 305.5
38 1896.7 305.8
31
32
33
34
35
1 = 36°
37
40
1869.0
1870.0
41
39 1897.7 306.1
47
48
49
E
294.9
295. I
295.4
295.7
296.0
306.7
42 1900.4 307.0
43 1901.4
44
296.3
296.6
296.9
297.1
297.4
300.6
1880. I
1881.0
300.9
1882.0
301.2
1882.9
301.5
1883.8 301.7
51
45
1903.2
46 1904. I
1898.6 306.4
1899.5
307.2
1902.3 307.5
307.8
308.1
308.4
308.7
1906.9 309.0
1905.1
1906.0
1907.9 309.3
1908.8 309.6
52 1909.7 309.9
1910.6 310.2
53
54 1911.6 310.5
55 1912.5 310.8
56 1913.4 311.0
57 1914.3 311.3
58 1915.3 311.6
59 1916.2 311.9
| = 37°
T
E
312.2
1917.1
1918.0
312.5
1919.0 312.8
1919.9 313.1
1920.8 313.4
1921.7
313.7
1922.7 314.0
1923.6 314.3
1924.5 314.6
1925.5
314.9
315.2
1926.4
1927.3
315.5
315.8
1928.2
1929.2 316.1
1930.1 316.4
1931.0 316.7
1932.0 316.9
1932.9 317.2
317.5
1933.8
1934.7
317.8
1935.7 318.1
1936.6 318.4
1937.5 318.7
1938.5
319.0
1939.4 319.3
1940.3 319.6
1941.2 319.9
1942.2 320.2
1943.1 320.5
1944.0 320.8
1945.0
321.1
1945.9 321.4
1946.8 321.7
1947.7 322.0
1948.7 322.3
1949.6 322.6
1950.5 322.9
1951.5 323.2
1952.4 323.5
1953.3
323.8
1954.3
1955.2
1956. I
1957.0
1958.0
324. I
324.4
324.7
325.0
325.3
1958.9 325.6
1959.8 325.9
1960.8 326.2
1961.7 326.5
1962.6 326.8
1963.6 327.1
1964.5 327.4
1965.4 327.7
1966.4 328.0
1967.3 328.3
1968.2
1969.1
1970. I
1971.0
1971.9
329.2
329.5
329.8
1 = 38°
T
1972.9
1973.8
1974.7
1975.7
1976.6
1977.5
1978.5
1979.4
1980.3
1981.3
1982.2
1983.1
1984.
1985.0
1985.9
1986.9
334.7
1987.8 335.0
E
330. I
330.5
330.8
331.1
331.4
1988.7 335.3
1989.7
335.6
1990.6
335.9
2000.9
2001.8
2002.8
2003.7
2004.6
331.7
332.0
332.3
332.6
332.9
1991.5
336.2
1992.5
336.6
1993.4 336.9
1994.3
337.2
1995.3
337.5
333.2
333.5
333.8
334.1
334.4
1996.2
1997.1
1998.1
338.4
1999.0 338.7
1999.9 339.0
2014.9
2015.9
2016.8
2017.7
2018.7
2019.6
2020.5
2021.5
2022.4
2023.4
328.6 2024.3
328.9
2025.2
2026.2
337.8
338. I
2005.6 340.9
2006.5 341.2
2007.4 341.5
2008.4
341.8
2009.3
342.1
2010.2
342.4
2011.2
342.7
2012.1 343.0
2013.0 343.3
2014.0 343.7
339.3
339.6
339.9
340.2
340.6
344.0
344.3
344.6
344.9
345.2
345.5
345.8
346. I
346.5
346.8
347.1
347.4
347.7
2027. I 348.0
2028.0 348.3
| = 39°
T
E
2029.0 348.6
2029.9 349.0
2030.9 349.3
2031.8 349.6 3
2032.7 349.9
2035.5 350.8
2033.7 350.2
2034.6 350.5 6
1
2047.8
2048.7
2036.5 351.2 8
2037.4 351.5 9
2038.4 351.8 10
2039.3
352.I
2040.2 352.4 12
2041.2
2042.1
2052.5
2053.4
2054.3
0-NMI DO700 O-NMI CON UNA PO708 87~MA HOMOM DE NO AGAMZ KANO.
==
232
2
2055.3
2056.2
2057.2
2058. I
2059.0
2060.0
2060.9
5
2043.1
353.4 15
2044.0 353.7 16
2044.9
354.0 17
2045.9
2046.8
354.3 18
354.6 19
2066.6
2067.5
352.7 13
353.0 14
2049.6
2050.6 355.9 23
2051.5
354.9 20
355.3 21
355.6 22
356.2 24
356.5 25
356.8 26
357.2 27
357.5 28
357.8
29
358. I
358.4
358.7 32
30
31
359.1 33
359.4 34
2061.9
2062.8
2063.7
360.3 37
2064.7 360.7 38
2065.6
361.0 39
359.7 35
360.0 36
361.3 40
361.6
41
2068.5 361.9 42
2069.4 362.3 43
2070.3 362.6
44
45
46
362.9
2071.3
2072.2
363.2
2073.2 363.5
2074. I
363.9 48
2075.0 364.2
49
47
2076.0
2076.9 364.8 51
364.5 50
2077.9 365.1
52
2078.8 365.5 53
2079.8 365.8
54
2080.7 366.1 55
2081.6 366.4 56
2082.6
366.8
57
2083.5 367.1 58
2084.5 367.4 59
396
↑
TABLE VIII.—TANGENTS AND EXTERNALS FOR A 1° CURVE
(Chord or Arc Definition)

01234
2222
DO700 0-2MI DONDO OUNCE DONO0 OZ~M7 KOMMA DE 2700 8OOOZ KONAO
01234
6678
2089.2
5 2090. I
6
2091.1
2092.0
2093.0
8
369.3
369.7
370.0
370.3
9 2093.9 370.6
12
13
10 2094.9 371.0
2095.8
2096.8
371.3
371.6
2097.7 371.9
17
14 2098.6 372.3
15 2099.6 372.6
16 2100.5 372.9
2101.5 373.2
21
18 2102.4 373.6
22
19 2103.4 373.9
20 2104.3
T
1 = 40°
E
2085.4 367.7
2086.4 368.0
2087.3 368.4
2088.3
23 2107.2
27
24 2108.1
25 2109.0
375.8
26 2110.0 376.2
376.5
376.8
377.1
32
28 2111.9
29 2112.8
36
30 2113.8 377.5
374.2
2105.3 374.5
2106.2
374.9
31 2114.7 377.8
37
33 2116.6 378.5
40
368.7
369.0
2110.9
34 2117.6 378.8
41
35 2118.5 379.1
42
43
2119.5
2120.4
38 2121.4 380. I
380.4
39 2122.3
46
49
2115.7 378.1
2123.3 380.8
2124.2 381.1
2125.1 381.4
2126.1 381.8
44 2127.0 382.1
375.2
375.5
50
45 2128.0
382.4
2128.9 382.7
2129.9 383.I
47
48 2130.8 383.4
2131.8 383.7
52 2134.6
379.4
379.8
2132.7 384.1
51 2133.7 384.4
384.8
385. I
385.4
53 2135.6
54 2136.5
55 2137.5 385.7
56 2138.4 386.1
57 2139.4 386.4
58 2140.3
59 2141.3
386.7
387.0
| = 41°
T
2142.2
2143.2
2144.1
2145. I
2146.0
2147.0
2147.9
2148.9
2149.8
2150.8
2151.7
2152.7
2153.6
2154.6
2155.5
2156.5
2157.4
2158.4
2159.3
2160.3
2161.2
2162.2
2163.2
2164.1
2165. I
2166.0
2167.0
2167.9
2168.9
2169.8
2170.8
2171.7
2172.7
2173.6
2174.6
2175.5
2176.5
2177.4
2178.4
2179.4
་་
387.4
387.7
388.0
388.4
388.7
389.0
389.4
389.7
390.0
390.4
390.7
391.0
391.4
391.7
392. I
392.4
392.7
393. I
393.4
393.7
394. I
394.4
394.7
395. I
395.4
395.7
396. I
396.1!
396.8
397. I
397.4
397.8
398. I
398.4
398.8
399. I
399.5
399.8
400. I
400.5
2180.3
400.8
2181.3
401.2
2182.2 401.5
2183.2
2184.1
401.8
402.2
2185.1 402.5
2186.0 402.9
2187.0 403.2
2187.9 403.5
2188.9
403.9
2189.9 404.2
2190.8 404.6
2191.8 404.9
2192.7 405.2
2193.7 405.6
2194.6 405.9
2195.6
406.3
2196.5 406.6
2197.5 407.0
2198.5
407.3
| = 42°
T
2199.4
2200.4
2201.3
2202.3
408.3
408.7
2203.2 409.0
2204.2
2205. I
2206. I
2207. I
2208.0
2209.0
411.1
2209.9
411.4
2210.9 411.8
2211.8 412.1
2212.8
412.5
E
407.6
408.0
2213.8
412.8
2214.7
413.1
2215.7
413.5
2216.6 413.8
2217.6
414.2
2223.3
2224.3
2225.3
2226.2
2227.2
2218.6
414.5
2219.5 414.9
2220.5 415.2
2221.4
415.6
2222.4
415.9
2228. I
2229. I
2230. I
2231.0
2232.0
409.4
409.7
410.0
410.4
410.7
2232.9
2233.9
2234.9
2235.8
2236.8
416.3
416.6
416 9
2252.2
2253. I
2254. I'
2255.0
2256.0
417.3
417.6
418.0
418.3
418.7
419.0
419.4
419.7
420. I
420.4
420.8
421.1
2237.7
421.5
2238.7 421.8
2239.7 422.2
2240.6 422.5
2241.6
422.9
2242.5
423.2
2243.5
423.6
2244.5
423.9
2245.4 424.3
2246.4 424.6
2247.3 425.0
2248.3 425.3
2249.3
425.7
426.0
2250.2
2251.2
426.4
426.7
427.1
427.4
427.8
428. I
1 = 43°
T
E
2257.0
2257.9
428.5
428.9
2258.9 429.2
2259.9 429.6
2260.8 429.9
2261.8 430.3
2262.7 430.6
2263.7 431.1
2264.7 431.3
2265.6 431.7
0
I
234 BO7∞ 0-234 S
2266.6 432.0 10
2273.4
2274.3 434.9
2275.3
435.2
2276.2
2277.2
2278.2
2279.1
2280. I
8
2267.6
432.4
2268.5 432.8 12
2269.5 433.1 13
2270.5 433.5 14
9
2271.4
2272.4 434.2 16
433.8 15
434.5 17
435.6
435.9 21
0 2002 DON¤* 8.~♡~ ❤❤❤❤. 2 00 85087 KOKOO
436.3
436.7 23
43.7.0 24
2281.1
437.4
437.7
2282.0
2283.0 438.1 27
447. 1
447.4
447.8
25
26
2284.0 438.4 28
2284.9 438.8 29
2285.9
439.2
2286.9
439.5 31
2287.8
439.9 32
440.2
2288.8
2289.8 440.6 34
2290.7 441.0 35
2291.7 441.3 36
2292.7 441.7 37
2293.6 442.0
2294.6 442.4 39
2295.6
442.7 40
2296.5 443.1 41
2297.5 443.5 42
2298.5 443.8 43
2299.4 444.2 44
2300.4 444.6 45
2301.4 444.9
46
2302.3 445.3 47
2303.3 445.6 48
2304.3 446.0 49
2305.2 446.4
2306.2
446.7
2307.2
2308. I
2309.1
52
54
2310.1 448.2 55
2311.1 448.5 56
2312.0
2313.0
448.9
449.3
2314.0 449.6 59
57
397
TABLE VIII.-TANGENTS AND EXTERNALS FOR A 1° CURVE
(Chord or Arc Definition)

T
0 2314.9
01234 VOT∞0 0-2M7 DO789
2
3
42318.8
5
7
8
T=
2319.8
451.8
6 2320.7
452.2
2321.7
452.5
2322.7
452.9
9 2323.7 453.3
2315.9
2316.9
2317.8
12
10 2324.6
453.6
11 2325.6 454.0
15
17
44°
2326.6
13 2327.5
14 2328.5 455.1
16 2330.5
E
450.0
450.3
450.7
451.1
451.4
2329.5 455.4
455.8
2331.4 456.2
18 2332.4 456.5
19 2333.4 456.9
25 2339.2
26 2340.2
27 2341.1
28
2342. |
29 2343.1
454.4
454.7
20 2334.3 457.3
21 2335.3 457.6
458.0
22 2336.3
23 2337.3
242338.2
458.4
458.7
50 2363.5
51 2364.5
52 2365.5
53 2366.5
54 2367.4
459.I
459.5
459.8
460.2
460.6
30 2344.1
31 2345.0
32 2346.0
33 2347.0 462.1
34 2348.0
462.4
460.9
461.3
461.7
35 2348.9 462.8
36 2349.9 463.2
37 2350.9 463.5
38 2351.8 463.9
39 2352.8 464.3
40 2353.8 464.6
41 2354.8 465.0
422355.7 465.4
43 2356.7 465.8
442357.7
466. I
45 2358.7 466.5
46 2359.6 466.9
47 2360.6 467.2
48 2361.6 467.6
49 2362.6
468.0
468.4
468.7
469. I
469.5
469.8
55 2368.4 470.2
56 2369.4 470.6
57 2370.4 471.0
471.3
58 2371.3
59 2372.3 471.7
| = 45°
T
2373.3
2374.3
2375.3
2376.2
2377.2
2378.2
2379.2
2380.I
2381.1
2382.1
2383. I
2384.0
2385.0
2386.0
2387.0
E
472.1
472.5
472.8
473.2
473.6
2402.6
2403.6
2404.6
2405.6
2406.6
474.0
474.3
474.7
475. I
475.4
475.8
476.2
477.7
2388.0
2388.9
478.1
2389.9
478.5
2390.9
478.8
2391.9 479.2
476.6
477.0
477.3
2392.8 479.6
2393.8 480.0
2394.8
480.3
2395.8
480.7
2396.8
481.1
2397.7
481.5
2398.7 481.9
2399.7 482.2
2400.7 482.6
2401.7
483.0
2427.2
2428.2
2429.1
2430. I
2431.1
483.4
483.8
484. I
484.5
484.9
2407.5
485.3
2408.5 485.7
2409.5 486.0
2410.5 486.4
2411.5 486.8
2412.4
2413.4
2414.4 487.9
2415.4 488.3
2416.4 488.7
487.2
487.6
2417.4 489.1
2418.3 489.5
2419.3 489.8
2420.3 490.2
2421.3 490.6
2422.3 491.0
2423.2
2424.2
2425.2
2426.2
491.4
491.7
492. I
492.5
492.9
493.3
493.7
494. I
494.4
1 = 46°
T
2432.1
2433. I
2434.1
2435.0
2436.0
2437.0
496.7
2438.0
497.1
2439.0 497.5
2440.0
497.9
498.3
2440.9
2441.9 498.7
2442.9 499.1
2443.9
499.4
2444.9 499.8
2445.9 500.2
2446.9
2447.8
2448.8
2449.8
2450.8
2451.8
2452.8
2453.8
2454.7
2455.7
E
494.8
495.2
495.6
496.0
496.4
2459.7
2460.7
2456.7
504.5
2457.7 504.9
2458.7
505.3
2466.6
2467.6
2468.6
2469.6
2470.5
500.6
501.0
501.4
501.8
502.2
2471.5
2472.5
2473.5
2474.5
2475.5
502.5
502.9
503.3
503.7
504.I
2461.7
506.4
2462.6 506.8
2463.6 507.2
2464.6
507.6
2465.6
508.0
505.6
506.0
508.4
508.8
509.2
509.5
509.9
510.3
510.7
511.1
511.5
511.9
2476.5 512.3
2477.5 512.7
2478.5
513.1
513.5
2479.4
2480.4
513.9
2481.4 514.3
2482.4 514.6
2483.4 515.0
2484.4 515.4
2485.4 515.8
2486.4 516.2
2487.4 516.6
2488.4
517.0
2489.3
517.4
2490.3
517.8
1 = 47°
T
E
2491.3
518.2
2492.3 518.6
2493.3 519.0
2494.3 519.4
2495.3 519.8
2496.3
520.2
2497.3
520.6
2498.3
521.0
2499.3 521.4
2500.2 521.8
2506.2
2507.2
2508.2
2509.2
2510.2
2511.2
2512.2
2513.2
2514.1
2515. I
2516. I
2517.1
2518. I
2519.1
2520. I
2521.I
2522.1
2523. I
2524.I
2525. I
2526. I
2527. I
2528. I
2529. I
2530. I
2531.I
2532.I
2533. I
2534.1
2535.0
2501.2 522.2 10
2502.2
522.6 11
2503.2
2504.2
2505.2
I
0-234
DONOR ONE DONOR 7222 20702 A.~♡~ ❤❤M88 279 28500 8OOOZ KANAA
555555
5
6
7
8
526. I
526.5
526.9
527.3
527.7
9
523.0 12
523.4 13
523.7 14
524.1 15
524.5 16
534.1
534.6
535.0
535.4
535.8
524.9 17
525.3 18
525.7
19
20
21
23
24
528. I
528.5
528.9
529.3
529.7 29
25
26
28
'530.1 30
530.5 31
530.9
531.3
531.7 34
32
33
532.1
532.5 36
532.9
533.3
533.7 39
35
37
38
40
41
42
43
44
2536.0 536.2 45
2537.0 536.6 46
2538.0
537.0
47
2539.0 537.4 48
2540.0 537.8 49
2541.0 538.2 50
2542.0 538.6 51
2543.0 539.0
52
2544.0 539.4 53
2545.0
539.8 54
2546.0 540.2
2547.0
2548.0
540.6
541.0
2549.0
541.4
2550.0
541.8
55
56
57
58
59
398
01234 5O789
2
3
TABLE VIII.-TANGENTS AND EXTERNALS FOR A 1° CURVE
(Chord or Arc Definition)

42555.0
2556.0
6 2557.0
7 2558.0
8 2559.0
9 2560.0
NMI D
10
2561.0
546.3
11 2562.0 546.7
12 2563.0 547.1
13 2564.0
547.5
547.8
14 2565.0
15
2566.0
2567.0
17 2568.0
18 2569.0
19
2570.0
16
1 = 48°
222222
T
E
2551.0
542.2
542.6
2552.0
2553.0
543.0
2554.0 543.5
543.9
22 20700 800
20 2571.0 550.4
21 2572.0 550.8
22 2573.0 551.2
23
2574.0 551.6
24 2575.0
552.0
30 2581.0
31
25 2576.0 552.4
26 2577.0 552.9
27 2578.0 553.3
28 2579.0 553.7
2580.0 554.I
554.5
554.9
555.3
33 2584.0 555.7
34 2585.0 556.2
2582.0
32 2583.0
6600 20000 55555 £5555
42
43
44
35 2586.0 556.6
2587.0
557.0
36
37
2588.0 557.4
38 2589.0 557.8
39
2590.0
558.2
40
47
544.3
544.7
545. I
545.5
545.9
45 2596.1
46
2597. I
2598. I
2599. I
48
49 2600.1
548.3
548.8
549.2
549.6
550.0
558.6
2591.I
2592.1
559.0
2593. I 559.5
559.9
2594. I
2595.1
560.3
50 2601.I
51
2602. I
52 2603.1
53 2604.1
54 2605.1
55 2606. I
2607.1
56
57 2608. I
58 2609.1
59 2610.1
560.7
561.1
561.5
561.9
562.4
564.9
565.3
565.7
1 = 49°
566. I
566.5
T
2611.2
2612.2
2613.2
2614.2
2615.2
2616.2
2617.2
2618.2
2619.2
2620.2
E
566.9
567.4
567.8
568.2
568.6
2621.2 571.I
2622.2
571.5
2623.2
572.0
2624.2
572.4
2625.3
572.8
569.0
569.4
569.9
570.3
570.7
2626.3 573.2
2627.3
573.6
2628.3 574.1
2629.3
574.5
2630.3
574.9
2641.4
2642.4
2643.4
2644.4
2645.4
2631.3 575.3
2632.3
575.7
2633.3 576.2
2634.3
576.6
577.0
2635.3
2636.4 577.4
2637.4 577.9
2638.4
2639.4
578.3
578.7
2640.4
579. I
2651.5
2652.5
2653.5
2654.6
2655.6
579.5
580.0
580.4
580.8
581.2
581.7
2646.5
2647.5
582. I
2648.5
582.5
2649.5
582.9
2650.5 583.4
583.8
584.2
584.6
585. I
585.5
562.8
563.2
2661.6 588.0
2662.7 588.5
563.6 2663.7 588.9
564.0 2664.7 589.3
2665.7 589.8
2666.7 590.2
2667.7 590.6
564.4
2668.7 591.0
2669.8 591.5
2670.8 591.9
2656.6 585.9
2657.6
586.3
2658.6 586.8
2659.6 587.2
2660.6 587.6
T
1 = 50°
2671.8
2672.8
2673.8
2674.8
2675.8
2681.9
2682.9
2684.0
2676.9
594.5
2677.9 594.9
2678.9
595.3
2679.9 595.8
2680.9
596.2
2685.0
2686.0
2687.0
2688.0
2689.0
2690. I
2691. I
2692.1
2693. I
2694.1
2695.2
2696.2
2697.2
2698.2
E
592.3
592.8
593.2
593.6
594.0
596.6
597. I
597.5
597.9
598.3
2712.5
2713.5
2714.5
2715.5
2716.6
598.8
599.2
599.6
600. I
600.5
603. I
603.5
2699.2
604.0
2700.2 604.4
2701.3
604.8
2702.3
605.3
2703.3
605.7
2704.3 ❘ 606.1
2705.3 606.6
2706.4
607.0
2707.4
607.4
2708.4
607.9
2709.4
608.3
2710.4 608.8
2711.5
609.2
609.6
610. I
610.5
610.9
611.4
2717.6
611.8
2718.6 612.2
2719.6 612.7
2720.6 613.1
2721.7 613.6
2722.7
2723.7
614.0
614.4
2724.7 614.9
2725.7 615.3
2726.8 615.8
2727.8 616.2
2728.8
2729.8
616.6
617.1
2730.9 617.5
2731.9 618.0
T
E
2732.9 618.4
2733.9 618.8
2734.9 619.3
2736.0 619.7
2737.0 620.2
1 = 51°
***
2738.0 620.6
2739.0 621.0
2740.1
621.5
621.9
2741.1
2742. I
622.4
2743. I
2744.2
2745.2
2746.2
2747.2
2748.3
2749.3
2750.3
2751.3
2752.4
2758.5
2759.5
2760.6
2761.6
2762.6
2763.7
2764.7
2765.7
2766.7
2767.8
622.8
623.3
623.7
2773.9
2775.0
2776.0
2777.0
2778.1
2779. I
2780.I
2781.1
600.9 2753.4 627.2 20
601.4
627.7
2754.4
2755.4
21
601.8
602.2
2756.5
602.7
2757.5
122 2
0-23I DONO DINA DOO ~Z KO700 A.~m~ ❤omo R 9700 850MZ BONAA
ოოოო ოოო
6
7
8
9
624. I 13
10
624.6 14
628. I
628.6
629.0
12
625.0 15
625.4 16
625.9 17
626.4 18
626.8 19
634.4
636.2
636.6
637. I
637.5
638.0
22
23
629.5
629.9
630.4 27
24
25
630.8 28
26
631.2 29
631.7
632. I
632.6
633.0
633.5 34
30
2768.8
633.9 35
2769.8
2770.8
634.8
36
37
2771.9 635.3 38
2772.9 635.7
39
40
41
31
32
33
42
43
44
638.4 45
638.9
46
639.3 47
2782.2 639.8 48
2783.2 640.2
49
2784.2
640.7
2785.3 641.1
2786.3 641.6
2787.3 642.0 53
2788.4 642.5 54
51
52
2789.4 642.9 55
2790.4 643.4 56
2791.4
643.8
57
2792.5 644.3 58
2793.5 644.7
59
399
TABLE VIII.—TANGENTS AND EXTERNALS FOR A 1° CURVE
(Chord or Arc Definition)

1 = 52°
T
0 2794.5
I 2795.6
2 2796.6
3 2797.6
24
2798.7
5 2799.7
6 2800.7
22
15
16
17
18
19
7 2801.8 648.3
8
2802.8
648.8
9 2803.8
649.2
E
645.2
645.6
646. I
10 2804.9
649.7
11 2805.9 650.2
12
2806.9❘ 650.6
2808.0 651.1
14 2809.0 651.5
13
646.5
647.0
647.4
647.9
2810.0 652.0
2811.1
652.4
2812.1
652.9
2813. I
653.3
2814.2
653.8
20
2815.2
654.3
21 2816.2
654.7
22 2817.3
655.2
23 2818.3 655.6
24 2819.3 656.I
25 2820.4
656.5
26 2821.4 657.0
27 2822.4 657.5
28 2823.5 657.9
29 2824.5 658.4
37 2832.8
38 2833.8
39
30 2825.6 658.8
31 2826.6
659.3
32 2827.6
659.7
33 2828.7
660.2
34 2829.7 660.7
35 2830.7 661.1
36 2831.8 661.6
662.0
662.5
2834.9 663.0
50 2846.3
51 2847.3
52 2848.4
53 | 2849.4
542850.5
663.4
2837.0 663.9
42 2838.0 664.3
40 2835.9
41
43 2839.0
44 2840.I
664.8
665.3
45 2841.I 665.7
46 2842.|
47 2843.2
48 2844.2
49 2845.3
666.2
666.6
667. I
667.6
668.0
668.5
669.0
669.4
669.9
55 2851.5 670.3
56 2852.5 670.8
57 2853.6
671.3
58 2854.6 671.7
59 2855.7 | 672.2
| = 53°
T
2856.7
2857.7
2858.8
2859.8
2860.9
2861.9
2862.9
2864.0
2865.0
2866. I
2872.3
2873.4
2874.4
E
672.7
673.1
673.6
674. I
674.5
675.0
675.5
2867.1
677.3
2868.2
677.8
2869.2 678.3
2870.2
678.7
2871.3
679.2
675.9
676.4
676.9
679.7
680. I
680.6
2875.5 681.1
2876.5 681.5
2888.0
2889.0
2890. I
2891.1
2892.2
2877.5 682.0
2878.6 682.5
2879.6
682.9
683.4
2880.7
2881.7 683.9
2882.8 684.3
2883.8 684.8
2884.8 685.3
2885.9 685.7
2886.9 686.2
686.7
667.2
687.6
688. I
688.6
2893.2
689.0
2894.3 689.5
2895.3
690.0
2896.3
690.5
2897.4
690.9
2898.4 691.4
2899.5
691.9
2900.5 692.3
2901.6
692.8
2902.6 693.3
2903.7 693.8
2904.7 694.2
2905.8 694.7
2906.8 695.2
2907.9 695.7
2908.9 696. I
2910.0
2911.0
696.6
697.I
2912.1 697.6
2913.1 698.0
2914.2 698.5
2915.2 699.0
2916.3 699.5
2917.3 699.9
2918.4 700.4
| = 54°
T
2919.4
2920.5
2921.5
2922.6
2923.6
2924.7
2925.7
2926.8
2927.8
2928.9
2929.9
2931.0
2932.0
2933. I
2934.1
2935.2
2936.2
2942.5
2943.6
2944.6
708. I
708.6
2937.3
709.0
2938.3
709.5
2939.4 710.0
2948.9
2949.9
2951.0
2952.0
2940.4 710.5
2941.5 710.9
711.4
711.9
712.4
2953. I
2954. I
2955.2
་་
700.9
701.4
701.8
702.3
702.8
2945.7
712.9
2946.7
713.4
2947.8 713.8
714.3
714.8
703.3
703.8
704.2
704.7
705.2
2966.8
2967.9
2968.9
705.7
706.1
706.6
2970.0
2971.0
707.1
707.6
2972. I
2973. I
2974.2
2975.3
2976.3
2956.2
717.8
2957.3
718.2
2958.3
718.7
2959.4 719.1
2960.5
719.6
715.3
715.6
716.2
2961.5
720. I
2962.6
720.6
721.1
2963.6
2964.7
721.6
2965.7 722.1
716.7
717.2
722.5
723.0
723.5
724.0
724.5
725.0
725.5
725.9
726.4
726.9
2977.4
727.4
2978.4 727.9
2979.5 728.4
2980.5 728.9
2981.6 729.4
| = 55°
E
729.9
730.3
2984.8 730.8
2985.8 731.3
2986.9
731.8
T
2982.7
2983.7
2988.0
2989.0
2990: I
2991.I
2992.2
2998.6
2999.6
3003.9
3004.9
3006.0
3007.1
3008. I
3009.2
3010.3
3011.3
3012.4
3013.5
3014.5
3015.6
3016.6
3017.7
3018.8
3019.8
3020.9
732.3
732.8
3022.0
3023.0
3024. I
733.3
2993.3
2994.3
735.3
2995.4 735.7 12
2996.5
2997.5
733.8
734.3
1
3000.7 738.2 17
3001.8 738.7 18
3002.8
739.2 19
0-23=
734.8 10
DO700 0-2I DONOR 2222 20702 07~M7 ❤❤MMM DEINE 2700 8887 KADAR
3041.2
3042.2
3043.3
3044.4
3045.4
5
6
736.2 13
8
736.7 14
737.2 15
737.7 16
743.1
739.7 20
740.2 21
740.7
741.2 23
741.7
24
742.1 25
742.6 26
743.6 28
744.1 29
744.6
30
745. I
745.6
745.1 33
3032.6 753.1
746.6 34
3033.7 753.6
31
747.1 35
32
747.6 36
748.1 37
748.6 38
749.1
39
3025.2
749.6
3026.2
750. I
3027.3
750.6
3028.4 751.1 43
757.1
757.6
3029.4 751.6 44
40
3030.5 752.1 45
41
3031.6 752.6 46
42
47
3034.8 754.1 49
48
3035.8 754.6 50
3036.9 755.I 51
3038.0
755.6
52
3039.0
756.1 53
756.6
3040. I
54
55
56
758. I
758.6
759.1 59
57
58
400
TABLE VIII.-TANGENTS AND EXTERNALS FOR A 1° CURVE
(Chord or Arc Definition)

0-23=
6678σ O-2M4 50
5
3048.6
3049.7
4 3050.8
9
II
12
13
14
10 3057.2
3058.3
3059.3
15
16
17
18
19
22222 007°° 67087 COMM DIE DOOD OTOMY HONO
21
23
24
T
| = 56°
20 3067.9
3069.0
3070. I
3071.1
3072.2
38
39
3046.5
3047.6
42
3051.9
3052.9
3054.0
43
44
3055. I
3056.1
47
48
49
3060.4
3061.5
3062.6
3063.6
3064.7
3065.8
3066.8
55
56
57
3089.4
40
41❘ 3090.5
E
759.6
760. I
25 3073.3 772.2
26 3074.4 772.7
27
28
3075.4 773.2
3076.5 773.7
29 3077.6 774.2
760.6
761.1
761.6
762. I
762.6
763. I
763.6
764.1
30
31
32 3080.8 775.7
764.6
765. I
765.6
766. I
766.6
33
3081.9 776.3
34 3082.9 776.8
767.1
767.6
768. I
35
3084.0 777.3
36 3085.1 777.8
37 3086.2 778.3
3087.2 778.8
3088.3 779.3
768.6
769.2
769.7
770.2
770.7
771.2
771.7
58
59 3109.9
45 3094.8 782.4
3095.9 782.9
46
3096.9
3098.0
3099. I
779.8
780.3
3091.6 780.9
3092.6. 781.4
3093.7 781.9
50
31.00.2 784.9
51 3101.2 785.5
3078.7 774.7 3143.4
3079.7 775.2
3144.5
3145.6
3146.6
3147.7
52 3102.3 786.0
53 3103.4 786.5
54 3104.5 787.0
| = 57°
T
E
3110.9 790.1
3112.0
790.6
791.1
3113.1
3114.2 791.6
3115.3 792.1
3116.3
792.7
3117.4 793.2
3118.5 793.7
3119.6 794.2
3120.7 794.7
3121.7 795.2
3122.8 795.8
3123.9 796.3
3125.0 796.8
3126.1 797.3
3127.2 797.8
798.3
3128.2
3129:3
798.9
3130.4
799.4
3131.5 799.9
3132.6 800.4
3133.6 800.9
3134.7
801.5
3135.8 802.0
3136.9 802.5
3138.0 803.0
3139.1
803.5
3140.1
3141.2
3142.3
804.1
804.6
805. I
805.6
806. I
806.7
807.2
807.7
3148.8 808.2
3149.9 808.8
3151.0 809.3
3152. 809.8
3153.2 810.3
3159.7 813.5
3160.8 814.0
783.4 3161.8 814.5
3162.9 815.0
783.9
784.4
3164.0
815.6
3165. I
3166.2
3167.3
3168.4
3169.5
3154.2 810.9
3155.3 811.4
3156.4 811.9
3157.5 812.4
3158.6 812.9
816. I
816.6
817.2
817.7
818.2
3105.6 787.5
3170.6 818.7
3106.6 788.0 3171.6 819.3
3107.7 788.5
3108.8 789.1
789.6
3172.7 819.8
3173.8 820.3
3174.9 820.8
|=
T
E
3176.0 821.4
3177.1 821.9
3178.2 822.4
3179.3 823.0
3180.4
823.5
3181.4
3182.5
3183.6
3184.7
3185.8
58°
3195.6
3196.7
826.7
3186.9
3188.0
827.2
3189.1 827.7
3190.2
828.3
3191.3
828.8
824.0
824.5
3192.4
829.3
3193.5 829.9
3194.5
830.4
830.9
831.4
3203.3
3204.4
825.1
825.6
826. I
3197.8 832.0
3198.9
3200.0
3201.I
3202.2
832.5
833.0
833.6
834.1
834.6
835.2
3205.5 835.7
3206.6 836.2
3207.7 836.8
3208.8 837.3
3209.9 837.8
3210.9 838.4
3212.0 838.9
3213.1
839.5
3214.2 840.0
3215.3 840.5
3216.4 841.1
3217.5 841.6
3218.6 842.1
3231.8
3232.9
3234.0
3235. I
3219.7
842.7
843.2
3220.8
3221.9
843.7
3223.0 844.3
3224.1 844.8
3225.2
845.4
3226.3
3227.4
845.9
846.4
3228.5 847.0
3229.6 847.5
3230.7 848.1
848.6
849.1
849.7
850.2
3236.2
850.8
3237.3 851.3
3238.4 851.8
3239.5 852.4
3240.6 852.9
1 = 59°
T
3241.7
3242.8
3243.9
3245.0
3246.1
3249.4 857.3
3252.7
3253.8
3247.2
3248.3 856.7 6
3254.9
3256.0
3257.1
3258.2
3259.3
3260.4
3261.5
3262.6
E
853.5
854.0
854.5
855. I
855:6
3263.7
3264.8
3265.9
3267.0
3268. I
3269.2
3270.3
1
3250.5 857.8 8
3251.6
858.3
3271.4
3272.6
3273.7
01234
856.2 5
KONOK DIEMI DONOR ONE DONOR 87~M7 HOM08 2007 20000 850MZ HONOR
7
9
858.9
859.4
860.0 12
10
£60.5 13
861.1 14
861.6 15
862.2 16
862.7 17
863.2 18
863.8 19
864.3 20
864.9 21
865.4 22
866.0
866.5
23
24
867. I
867.6
958.2 27
25
26
868.7 28
869.3 29
869.8 30
3274.8
3275.9
3277.0
3278.1 871.5 33
870.4 31
870.9 32
3279.2 872.0 34
3280.3 872.6 35
3281.4 873.1 36
3282.5
873.7 37
3283.6 874.2 38
3284.7 874.8 39
3285.8
875.3 40
3286.9 875.9 441
3288.0 876.4 42
3289.2
3290.3
877.0 43
877.5 44
878. I 45
3291.4
3292.5
3293.6 879.2 47
3294.7 879.7 48
3295.8 880.3 49
3296.9 880.8
3298.0 881.4 51
3299.1
3300.2
881.9
882.5
3301.4
878.6 46
52
53
883. I 54
3302.5
3303.6 884.2
3304.7 884.7
3305.8
885.3
3306.9 885.8 59
883.6 55
56
57
58
401
↑
TABLE VIII.-TANGENTS AND EXTERNALS FOR A 1° CURVE
(Chord or Arc Definition)

0-234 BOTO0 0-2MT DO
0 3308.0
3309. I
3310.2
3311.3
3312.5
5
6
7
8
9
10
12
15
16
13 3322.5
17
893.6
14 3323.6 894.2
- ONE DO700 8.
18
2222
3324.7 894.7
3325.8 895.3
3326.9 895.9
3328.0 896.4
19 3329.2 897.0
ოოო ოო
27
28
29
20
21
3330.3 897.5
3331.4 898.1
3332.5 898.7
22
23
3333.6 899.2
24 3334.7 899.8
T
| = 60°
34
25
3335.9 900.3
26 3337.0 900.9
3338.1 901.5
33
36
3313.6
3314.7
3315.8
890.3
3316.9
890.8
3318.0 891.4
30 3341.4 903.2
3342.6 903.7
32 3343.7 904.3
E5558
3319.1 891.9
3320.2 892.5
3321.4
893.1
33 3344.8 904.8
3345.9 905.4
35 3347.0 906.0
3348.1 906.5
37 3349.3 907.1
38 3350.4
907.7
908.2
39 3351.5
40 3352.6
41 3353.7
60000 20000 55555
E
886.4
886.9
887.5
828. I
888.6
42 3354.8
45
889.2
889.7
43 3356.0
44 3357.1
47
3358.2 911.6
3359.3 912.2
3360.4 912.8
48 3361.6 913.3
49
3362.7
913.9
58
59
50 3363.8
51 3364.9
52 3366.0
53 3367.2
54 3368.3
55 3369.4
56 3370.5
3403.1
934.5
3404.3 935.0
3405.4
935.6
3339.2 902.0 3406.5
936.2
3340.3 902.6
3407.7
936.8
908.8
909.4
909.9
910.5
911.1
917.3
917.9
57 3371.7 918.4
3372.8 919.0
3373.9 919.6
914.5
915.0
915.6
916.2
916.7
T
| = 61°
3375.0
3376.1
3377.3
3378.4
3379.5
3380.6
923.0
3381.8 923.6
3382.9
924.1
3384.9
924.7
3385.1
925.3
3386.3
3387.4
3388.5
3389.6
3390.8
3397.5
3398.6
3399.8
3391.9
928.7
3393.0
929.3
3394.1 929.9
3395.3
930.4
3396.4
931.0
3400.9
3402.0
E
920.1
920.7
921.3
921.9
922.4
925.8
926.4
927.0
927.6
928. I
931.6
932.2
932.7
933.3
933.9
937.3
3408.8
3409.9
3411.0
937.9
938.5
3412.2 939.1
3413.3
939.7
940.2
3414.4
3415.6
940.8
3416.7 941.4
3417.8 942.0
3418.9
942.5
3431.4
3432.5
3433.7
3420. I
3421.2
3422.3
3423.5
944.9
3424.6 945.4
3434.8
3435.9
3425.7 946.0
3426.9 946.6
3428.0 947.2
3429.1 947.8
3430.3 948.3
948.9
949.5
950. I
950.7
951.3
1 = 62°
3437.1 951.8
3438.2
952.4
3439.3 953.0
3440.5 953.6
3441.6 954.2
T
3442.7
3443.9
3445.0
3446. I
3447.3
3448.4
3449.5
3450.7
3451.8
3452.9
3454.1
3455.2
3456.3
3457.5
3458.6
3459.8
3460.9
3462.0
3465.4
3466.6
3467.7
953.5
964.1
964.7
3463.2 965.3
3464.3
965.9
3474.6
3475.7
943.1
943.7 3489.4
944.3
3490.5
3491.7
3492.8
3476.8
3478.0
3479.1
3480.3
3481.4
3468.9 958.2
3470.0
968.8
E
954.8
955.3
955.9
956.5
957. I
.:3471.1
3472.3
3473.4 970.6
957.7
958.3
958.8
959.4
960.0
3494.0
3495. I
3496.2
3497.4
3498.5
960.6
961.2
3499.7
3500.8
3502.0
3503. I
3504.3
961.8
962.4
963.0
3505.4
3506.6
3507.7
3508.8
3510.8
966.5
967.I
967.7
975.3
3482.5
3483.7
975.9
3484.8 976.5
3486.0 977.1
3487.1 977.7
3488.2 978.3
978.9
979.5
980. I
980.7
969.4
970.0
971.2
971.8
972.4
973.0
973.6
974.2
974.8
981.3
981.9
982.5
983.1
983.7
984.3
984.9
985.5
986. I
986.7
987.3
987.9
988.4
989.0
989.6
1 = 63°
T
E
3511.1
990.2
3512.3
990.8
3513.4 991.4
3514.6
3515.7
3516.9
3518.0
3519.2
3520.3
3521.5
3522.6
3523.8
3524.9
3526. I
3527.2
3528.4
3529.5
3530.7
3531.8
3533.0
3534.1
3535.3
3536.4
3537.6
3538.7
3539.9
3541.0
3542.2
3543.3
3544.5
3545.6
3546.8
3547.9
3549. I
3550.2
3551.4
3552.5
3553.7
3554.8
3556.0
3557.2
3558.3
3559.5
3560.6
3561.8
3562.9
3564.1
3565.2
3566.4
3567.5
3568.7
3569.9
3571.0
3572.2
3573.3
3574.5
3575.6
3576.8
3578.0
3579.1
1
01237
992.0
992.6 4
993.2 5
993.8 6
994.4 7
995.0 8
995.6 9
996.2 10
996.8 11
997.5 12
998. I
13
998.7 14
999.3 15
999.9 16
ICCO.5 17
1001. 18
1001.7 19
1002.3 20
1002.9 21
1003.5 22
1004. 23
1004.7 24
1005.3 25
1005.9 26
1006.5 27
1007.1 28
1007.7 29
1008.3 30
1008.9 31
1009.5 32
1010.1 33
1010.7 34
1011.3 35
1012.0 36
1012.6 37
1013.2 38
1013.8 39
1014.4 40
1015.0 41
1015.6 42
1016.2 43
10.6.81 44
1017.4 45
1018. 1 46
1018.7 47
1019.3 48
1019.9 49
1020.5 50
1021.1 51
1021.7 52
1022.3 53
1022.9 54
1023.5 55
1024.2 56
1024.8 57
1025.4 58
1026.0 59
402
TABLE VIII.-TANGENTS AND EXTERNALS FOR A 1° CURVE
(Chord or Arc Definition)

0-NMI MONO DINA DONO ONE LO700 07037 0OMMA DIN 99700 8508Z KANAA
2
3 3583.8 1028.5
3584.9 1029.1
7
5 3586.1
6 3587.2
3588.4
8 3589.6
9 3590.7
||
12
13
10 3591.9
15
143596.5
17
1035.9
16 3598.8 1036.5
22
18 3601.2
25
1037.1
1037.8
19 3602.3 1038.4
27
20 3603.5 1039.0
1 = 64°
T
E
T
3650.2
3580.3 1026.6
3581.4 1027.2 3651.4
3582.6 1027.8 3652.5
3653.7
3654.9
3673.7
21 3604.7 1139.6 3674.8
23 3607.0 1040.9
24 3608.2 1041.5
26 3610.5
35
37
38
29 3614.0
40
3593.0
3594.2
3595.4 1034.7
3597.7
الا
3600.0
28 3612.8 1044.0
1044.6
43
45
1045.2
31 3616.3 1045.8
46
47
48
49
51
52
1029.7
1030.3
1030.9
1031.6
1032.2
32
33
34 3619.8 1047.7
54
1032.8
1033.4
1034.0
3661.9
3663. I
3664.3
3665.4
1035.3 3666.6
3611.6
3609.3 1042.1
42 3629.I
56
3605.8 1040.2 3676.0
3677.2
3678.4
59
3615.1
3617.5
3621.0 1048.3 3691.3
1048.9 3692.5
36 3622.|
3623.3 1049.5 3693.7
3624.5 1050.2 3694.9
393625.6 1050.8 3696.0
1042.7
1043.3
1046.4
3617.6 1047.1
3626.8 1051.4 3697.2
3628.0 1052.0 3698.4
1052.7 3699.6
3630.3 1053.3 3700.8
44 3631.5 1053.9 3702.0
53.3642.0
1 = 65°
3656. I
3657.2
3658.4
3659.6
3660.7
3667.8
3669.0
3670. I
3671.3
3672.5
3679.5
3680.7
3681.9
3683. I
3684.3
3685.4
3686.6
3687.8
3689.0
3690. I
3705.5
3706.7
3707.9
55 3644.3 1060.8 3715.0
3645.5 1061.4 3716.1
58 3647.8 1062.7 3718.5
1063.3 3719.7
3649.0
E
T
1063.9
3720.9
3722.1
1064.5
1065.2
1065.8
3723.2
3724.4
1066.4 3725.6
1067.0
1067.7
1068.3
1068.9
1069.6
1073.4
1074.0
1074.7
1075.3
1076.0
1070.2
1070.8
1071.5 3735.1
1072.1
3736.3
1072.8
3737.5
1076.6
1077.2
1077.9
1078.5
1079. I
1079.7
1080.4
1081.0
1081.6
1082.3
1082.9
1083.5
3632.6
3633.8
3635.0
3636. I
3637.3
3638.5
1057.7
3709.0
1095.7
3639.7 1058.3 3710.2 1096.3
3640.8 1058.9 3711.4
1097.0
1059.6 3712.6 1097.6
3643.2 1060.2 3713.8
1098.3
1054.5 3703.1 1092.5
1055.2 3704.3 1093.1
1055.8
1056.4
1093.8
1094.4
1057. I
1095. I
1086. I
1086.7
1087.4
1088.0
1088.7
1098.9
1099.6
57 3646.7 1062.0 3717.3 1100.2
1089.3
1089.9
1090.6
1091.2
1091.9
| = 66°
3729.2
3730.4
3731.6
3726.8
1105.4
3728.0 1106.0
1106.7
1107.3
1108.0
3732.7
3733.9
3742.2
3743.4
3744.6
3745.8
3747.0
1084.2 3758.9
1084.8
3760.1
1085.5
3761.3
3748.2
3749.4
3738.7
1!11.8
3739.9
1112.5
3741.1 1113.1
3750.6
3751.8
3752.9
3754. I
3755.3
3756.5
3757.7
3762.5
3763.7
3764.9
3766. I
3767.3
3780.4
3781.6
E
1102.2
1102.8
3782.8
3784.0
3785.2
1103.5
1104.1
1104.8
3786.4
3787.6
3788.8
1100.9 3790.0
1101.5 3791.2
1108.6
1109.2
1113.8
|||4.4
115.1
1115.8
1116.4
[17. |
1117.7
3805.6
1109.9 3806.8
1110.5
3808.0
|||1.2
3809.2
118.4
1119.1
!!!9.7
1120.4
1121.0
1128.2
3768.5
3769.6
1128.9
3770.8 1129.5
3772.0 1130.2
3773.2 1130.8
3774.4 1131.5
3775.6 1132.2
3776.8 1132.8
3778.0 1133.5
3779.2
1121.7
1122.3
1123.0
1123.6
1124.3
1125.6
1126.2
1 = 67°
T
E
3792.4 1141.4
3793.6 1142.1
3794.8 1142.7
3796.0 1143.4
3797.2
1144.0
1134.8
1135.5
3798.4
3799.6
3800.8
3802.0
3803.2
1138.1
1138.8
1139.4
1140. I
1140.7
3804.4 1148.0
1148.7
3810.4
3811.6
3812.8
3814.0
3815.2
3816.4
3817.6
3818.8
3820.0
3821.2
3822.4
3823.6
1124.9 3834.5
3835.7
3836.9
1126.9 3838.1
1127.5 3839.3
3824 8
3826.0
3827.2
3828.4
3829.6
3830.8
3832.0
3833.3
3840.5
3841.7
3842.9
3844.1
3845.3
3846.5
3847.7
1134.1 3851.4
3849.0
3850.2
3852.6
3853.8
1136.1 3855.0
1136.8
3856.2
3857.4
1137.4
3858.6
3859.8
3861.1
3862.3
3863.5
1144.7 5
1145.4
1146.0
1146.7 8
1147.3 9
0123=
1149.3
1150.0
1150.7 14
667000-231 LOT OF 20
1151.3 15
1152.0 16
1152.7 17
1153.4 18
1154.0 19
1154.7
20
1155.421
1156.0 22
1156.7
23
157.3 24
1158.7
1158.0 25
26
1159.3 27
1160.0 28
1160.6 29
1161.3 30
1162.0 31
1162.7 32
1163.3 33
1164.0 34
1164.7 35
1165.4 36
1166.1 37
1166.7 38
1167.4 39
1168.1 40
1168.8 41
1169.4 42
1170. I 43
1170.8 44
1171.4 45
1172. 46
1172.8 47
1173.5 48
1174.49
1174.8 50
1175.5 51
1176.2 52
1176.8 53
1177.5 54
1178.2 55
1178.9 56
1179.6 57
1180.2 58
180.9 59
403
1
TABLE VIII.—TANGENTS AND EXTERNALS FOR A 1° CURVE
(Chord or Arc Definition)

01234
KONOK DINMI CON ~~2 20700 07087 667MM DIE 1792 mon
9
8 3874.4
3864.7 1181.6
3865.9 1182.3
3867.1 1183.0
3868.3 1183.6
3869.6
5 3870.8
6 3872.0
3944.0
3945.2
7 3873.2 1186.4 3946.5
1187.0 3947.7
3948.9
3875.6 1187.7
12
15
16
17
18
19
10 3876.8 1188.4 3950.2
|| 3878.0 1189.1 3951.4
3879.3 1189.8 3952.6
13 3880.5 1190.4 3953.9
14 3881.7 1191.1
3955. I
23
22 3891.4
25
26
24 3893.9
29
T
20 3889.0
1195.2
21 3890.2 1195.9
1 = 68°
3882.9
3884. I
3885.3
3886.6
3887.8
35
3892.6
38
39
E
....
T
3937.9
3939. I
3940.3
3941.6
1184.3 3942.8
3895. I
3896.3
3897.5
40
3907.3
36 3908.5
41
1198.6
1199.3
1200.0
28 3898.7 1200.6 3972.3
3900.0 1201.3 3973.6
30 3901.2 1202.0
42
1185.0
1185.7
31 3902.4 1202.7
32 3903.6 1203.4
1191.8
1192.5
1193.2
1193.8
1194.5
33 3904.8 1204.1
55555
1196.6
1197.2
197.9
1 = 69°
3974.8
1243.7
3976.0
1244.4
3977.3
1245.1
3978.5
1245.8
34 3906.1 1204.8 3979.7 1246.5
3956.3
3957.5
3958.8
3960.0
3961.2
1205.4
1206. I
1206.8
1207.5
43
44 3918.3 1211.7
45 3919.5 1212.3
46
3962.5
3963.7
3981.0 1247.2
3982.2 1248.0
3983.4 1248.7
3984.7 1249.4
37 3909.7
3910.9
3912.2 1208.2 3985.9 1250.1
E
1222.7
1223.4
1224.1
1224.8
1225.5
1226.2
1226.9
1227.6
1228.3
1229.0
3913.4 1208.9 3987.2
3914.6 1209.6 3988.4
3915.8 1210.3
3989.6
3917.1 1211.0 3990.9
1229.7
1230.4
1231.1
1231.8
1232.5
3968.6 1240.2
3969.9
1240.9
3971.1 1241.6
1236.7
1237.4
3964.9
1238.1 4039.3
3966.2
1238.8
4040.6
3967.4 1239.5
4041.8
1242.3
1243.0
1233.2
4030.6
1233.9
4031.8
1234.6
4033.1
1235.3 4034.3
1236.0
4035.6
T
1250.8
1251.5
1252.2
1252.9
3992.1 1253.6
1 = 70°
E
4011.9
1265.0
4013.2 1265.7
4014.4
1266.4
4015.7
1267.1
4016.9
1267.8
1254.3
1255.1
1255.8
4018.2
4019.4
4020.6
4021.9
4023. I
4024.4
4025.6
4026.9
4028. I
4029.4
4036.8
4038.1
4043.1
4044.3
4045.6
4046.8
4048.1
4049.3
4050.6
4051.8
4053.I
4054.3
4055.6
4056.8
4058. I
4059.3
4060.6
4061.8
4063.1
4064.3
4065.6
4066.8
4068.1
4069.3
4070.6
4071.9
4073. I
4074.4
4075.6
4076.9
4078.1
4079.4
1275.7
1276.4
1277.1
1277.9
1278.6
1268.5
4093.2
1269.3 4094.5
1311.9 5
1312.6 6
1313.4 7
1270.0
4095.7
1270.7
4097.0 1314.1 8
1271.4
4098.2
1314.9 9
1272.1
1272.8
1273.5
1274.3
4103.3
1275.0 4104.5
1282.9
1283.6
1284.3
1285.1
1285.8
3993.3
3920.7 1213.0 3994.6
3921.9 1213.7
47
3995.8
48 3923.2 1214.4 3997.1 1256.5
3924.4 1215.1
1257.2
3998.3
3999.5
1215.8
1257.9
50 3925.6
51 3926.8 1216.5
52 3928.1 1217.2
53 3929.3
54 3930.5
55 3931.7 1219.2 4005.7
56 3933.0 1219.9
57 3934.2 1220.6 4008.2 1262.9 4083.1
4000.8 1258.6
4002.0 1259.3
1217.9 4003.3 1260.0
1218.6 4004.5 1260.7
1261.4
4007.0 1262.2
4080.6 1304.5
4081.9
1305.3
58 3935.4 1221.3
59 3936.7 1222.0
4009.5 1263.6
4010.7 1264.3
1306.0
4084.4 1306.7
4085.7 1307.5
| = 71°
T
E
4086.9
4088.2
1308.2
1308.9
4089.4 1309.7
4090.7 1310.4
4091.9 1311.2
1279.3
1280.0
1280.7
1281.5
4115.9
1282.2 4117.2
1293.7
1294.3
1295. I
1295.8
1296.5
4099.5
4100.8
4102.0
1297.2
1298.0
1298.7
1299.4
1300.2
4105.8
4107.1
4108.3
4109.6
4110.9
4112.1
4113.4
4114.6
1286.5
4124.8
1287.2
4126.0
1287.9
4127.3
1288.6
4128.6
1289.3 4129.8
1290.0 4131.1
1290.8
1291.5
4132.4
4133.6
4134.9
1292.2
1292.9
4136.2
4118.4
4119.7
4121.0
4122.2
4123.5
4137.4
4138.7
4140.0
4141.2
4142.5
4143.8
4145.0
4146.3
4147.6
4148.8
1300.9
4150.1
1301.6 4151.4
1302.4
4152.7
4153.9
1303.1
1303.8
4155.2
0-23A
4156.5
4157.7
4159.0
4160.3
4161.6
1315.5 10
1316.3 ||
1317.1 12
1317.8 13
1318.5 14
1319.2 15
1320.0 16
1320.7 17
1321.4 18
1322.2 19
1322.9 20
1323.6 21
1324.4 22
1325. 23
1325.9 24
1326.6 25
1327.3 26
1328.1 27
1328.8 28
1329.6 29
1330.3 30
1331.0 31
1331.8 32
1332.5 33
1333.3 34
1334.0 35
1334.7 36
1335.5 37
1336.2 38
1337.0 39
1337.7 40
1338.4 41
1339.2 42
1339.9 43
1340.744
1341.4 45
1342. 46
1342.9 47
1343.6 48
1344.4 49
1345.1 50
1345.8 51
1346.6 52
1347.3 53
1348.1 54
1348.8 55
1349.6 56
1350.3 57
1351. 58
1351.859
404
TABLE VIII.-TANGENTS AND EXTERNALS FOR A 1° CURVE
(Chord or Arc Definition)

1
0123➡
5678 σ
9
10
11
123I BOMB ONE O❤7°º 8~~ma
14
21
22
15 4182.0 1363.8
16 4183.2 1364.6
17 4184.5 1365.3
ოო
19 4187.1 1366.8
27
20 4188.4
4189.6
32
23
24 4193.5
33
1=72°
T
E
4162.8 1352.6
4164.1 1353.3
4165.4 1354.1
4166.7 1354.8
4167.9 1355.6
== w w wy
£5555
39
42
4169.2 1356.3 4246.2
1357.1 4247.5
1357.8
4170.5
4171.8
4173.0
4174.3
1358.6 4250.0
1359.3
4251.3
35 4207.6
36 4208.8
37 4210.1
ཝཟ ཟ
4252.6
4253.9
4175.6 1360.1
4176.9 1360.8
4178.1 1361.6 4255.2
4179.4 1362.3 4256.5
4180.7 1363.1 4257.8
1367.6
1368.4
4190.9 1369. I
4192.2
25 4194.8 1371.4 4272.0
26 4196.0 1.372.2 4273.3
4197.3 1372.9 4274.6
45
28 4198.6 1373.7 4275.9
29 4199.9 1374.4
4277.2
4185.8 1366.1 4263.0
4264.3
30 4201.2 1375.2 4278.5
31 4202.4 1376.0
4279.8
4281.1
4203.7 1376.7
4205.0 1377.5
4282.4
34 4206.3 1378.2
4283.7
40 4214.0
1555
-234
41 4215.3
4216.5
51
43 4217.8
44 4219.1
1369.9
1370.6
1 = 73°
T
4239.7
4241.0
4242.3
4243.6
4244.9
1379.0
1379.8
1380.5
52
53
54 4232.0
4259.1
4260.4
4261.7
1383.6
1384.3
1401.8
1402.6
4248.8 1403.4
1385. I
1385.8
4211.4 1381.3 4288.9
4212.7
1382.0
4290.2
4285.0
4286.3
4287.6
1382.8 4291.5
4292.8
4294.1
4295.4
4296.7
50 4226.8 1390.4 4304.6
E
1398.0
1398.8
1399.5
1400.3
1401.1
4228.1 1391.2 4305.9
4229.4 1391.9 4307.2
4230.7
1404.2
1404.9
4265.6 1413.5
4266.9 1414.3
4268.2 1415.0 4346.4
1405.7
1406.5
1407.3
1408.0
1408.8
55
56 4234.6 1395.0 4312.4
57 4235.9 1395.7
4313.7
58
4237.1 1396.5 4315.0
59 4238.4 1397.2 4316.3
1409.6
1410.4
1411.2
1411.9
1412.7
4269.5 1415.8 4347.7
4270.7
1416.6
4349.0
4220.4
1386.6
4298.0 1432.9
46 4221.7 1387.4 4299.3 1433.7
47 4223.0 1388.1 4300.6
1434.5
1435.2
48 4224.3 1388.9 4301.9
49 4225.5 1389.6 4303.2
1436.0
1417.3
1418.1
1418.9
1419.7
1420.4
1421.2
1422.0
1422.8
1423.5
1424.3
1425.1
1425.9
1426.7
1427.4
1428.2
1429.0
1429.8
1430.6
1436.8
1437.6
1438.4
1392.7 4308.5 1439.1
1393.4 4309.8 1439.9
| = 74°
T
4317.6
4318.9
4320.2
4321.5
4322.8
4233.3 1394.2 4311.1 1440.7
1441.5
1442.3
1443.0
1443.8
4324.1
4325.4
4326.8
4328.1
4329.4
4330.7
4332.0
4333.3
4334.6
4335.9
4337.2
4338.5
4339.9
4343.8
4345.1
4350.4
4351.7
4353.0
4354.3
4355.6
4370. I
4371.4
4372.7
1431.3
4374.1
1432.1 4375.4
4341.2 1458.8
4342.5 1459.6
4366.1
4367.5
4368.8
4376.7
4378.0
4379.3
4380.6
4382.0
E
T
E
1444.6
1445.4
4396.5 1492.4
4397.8 1493.2
1446.2
4399.2 1494.0
1447.0 4400.5 1494.8
1447.8 4401.8 1495.6
4383.3
4384.6
4395.9
4387.3
4388.6
1448.5 4403.1
1449.3
1450.1
1450.9
1451.7
1452.5
1453.3
4389.9
4391.2
4392.5
4393.9
4395.2
1454.1
1454.9
1455.7
1456.4
1457.2
1458.0
4363.5
4364.8 1473.2
1474.0
1474.8
1475.6
1460.4
1461.2
1462.0
1462.8
1463.6
| = 75°
4356.9
1468.4 4436.4
4358.2 1469.2 4437.7
4359.6
1470.0
4360.9
1470.8
4362.2
1471.6
1476.4
1477.2
1478.0
1478.8
1479.6
4404.5
4405.8 1498.1
4407.1
4408.4
1480.4
1481.2
1482.0
1482.8
1483.6
4409.8
4411.1
4412.4
4413.8
4415.1
1464.4 4429.7
1465.2 4431.0
1466.0
4432.4
1466.8 4433.7
1467.6 4435.0
4416.4
4417.7
4419.1
4420.4
4421.7
4423.1
4424.4
4425.7
4427.0
4428.4
1472.4 4443.0
4439.0
4440.4
4441.7
4444.4
4445.7
4447.0
4448.4
4449.7
4451.1
4452.4
4453.7
4455. I
4456.4
4457.7
4459. I
4460.4
4461.7
1496.4
1497.3 6
1499.7
1498.9 8
01234
I 500.5
1501.3
1502.1 12
DO700 0-2MI MOZOM OU DON°2 2~~♡~ ❤OM88 GENRE CONDO IGNAZ KANH8
5
1502.9 13
9
1503.7 14
1504.5 15
10
1505.4 16
1506.2 17
1507.0 18
1507.8 19
1508.6 20
1509.4 21
1510.2 22
1511.0 23
1511.8 24
1512.6 25
1513.5 26
1514.3 27
1515. 28
1515.9 29
1516.7 30
1517.5 31
1518.3 32
1519.2 33
1520.0 34
1520.8 35
1521.6 36
1522.4 37
1526.5
1523.3 38
1524. 39
1524.9 40
1525.7 41
1530.6
1527.4 43
1528.2 44
42
1529.0 45
1488.4 4469.8
1489.2
1490.0 4472.5 1538.9
1490.8
1491.6
1529.8 46
1537.2
1531.5 48
47
1532.3 49
1484.4 4463.1 1533.1 50
1485.2
4464.4 1533.9 51
1486.0 4465.8 1534.8
52
1486.8 4467.1 1535.6 53
1487.6
4468.4 1536.4 54
55
4471.1 1538.1 56
57
4473.8 1539.7 58
4475.2 1540.6 59
405
1
TABLE VIII.—TANGENTS AND EXTERNALS FOR A 1° curve
(Chord or Arc Definition)

1 = 76°
T
E
T
0 4476.5 1541.4 4557.6
1542.2 4558.9
I
4477.8
2
4479.2
1543.1
4560.3
3
1543.9
4561.7
4480.5
4481.9
1544.7
4563.0
5 4483.2
6
4484.6
7
4485.9
1547.2
8
4487.2 1548.0
9 4488.6
1548.9
10 4489.9
11 4491.3
12 4492.6
13 4494.0
144495.3
154496.7
16
4498.0
4499.4
17
18
4500.7
19 4502.0
1545.5
1546.4
25 4510.I
26 4511.5
27 4512.8
4514.2
29 4515.5
28
30 4516.9
31 4518.2
32 4519.6
334520.9
34 4522.3
1549.7
1550.5
1551.4
20 4503.4
1558.0
4504.7 1558.8
22 4506.1 1559.7
21
23 4507.4
1560.5
1561.3
24 4508.8
4571.2
4572.6
4573.9
1552.2 4575.3
1553.0 4576.6
| = 77°
1553.8 4578.0
1554.7 4579.4
1555.5
4580.7
1556.3
4582.1
1557.2
4583.5
4564.4
4565.7
4567. I
4568.5
4569.8
1570.5
1571.3
1562.1 4591.7
1563.0 4593.I
1563.8
4594.4
1564.6 4595.8
1565.5 4597.2
1572.2
1573.0
1573.9
་
4605.4
4606.8
4608. I
4609.5
4610.9
T
1591.6 4639.8
1592.4 4641.2
1593.3
4642.5
1594.1
4643.9
1595.0
4645.3
1595.8 4646.7
1596.7
4648.1
1597.5
4649.4
1598.4
4650.8
1799.2
4652.2
1600. I
1600.9
1601.8
1602.6
1603.5
4584.8 1608.6
4586.2 1609.4
4587.6
1610.3
4589.0
4590.3
1604.3
1605.2
1606.0
1606.9
1607.7
1566.3 4598.5
1567.1 4599.9
1568.0
4601.3
4602.6
1568.8
1569.7
4604.0 1620.5
1611.1
1612.0
1612.8
1613.7
1614.5
1615.4
1616.2
35 4523.7
36 4525.0
37 4526.4
384527.7
39 | 4529.1
4612.2
1625.7
1574.7
1575.5
4613.6 1626.6
40 4530.4
41 4531.8
424533.1 1576.4 4615.0 1627.4
43 4534.5 1577.2 4616.4 1628.3
444535.8 1578.1
45 4537.2 1578.9 4619.1
46 4538.6 1579.7 4620.5
474539.9 1580.6 4621.9
48 4541.3 1581.4 4623.2
49 4542.6 1582.3 4624.6
1 = 78°
1630.0
1630.9
1631.8
1632.7
1633.5
4653.6
4655.0
50 4544.0 1583.1
4626.0 1634.4
51 4545.3
4627.4
1635.3
1583.9
1584.8
1636. I
4628.8
1585.6 4630.I 1637.0
1586.5 4631.5 1637.8
52 4546.7
53 4548.I
544549.4
554550.8
564552.1
57 4553.5
58 4554.9
594556.2
4656.4
4657.7
4659.1
1587.3 4632.9 1638.7
1588.2 4634.3 1639.6
1589.0
1589.9
4635.6 1640.4
4637.0
1641.3
1590.7
4638.4
1642. I
4660.5
4661.9
1617.1 4681.3
1618.0
4682.7
1618.8
1619.7
4663.3
4664.7
4666.1
4667.4
4668.8
4670.2
4671.6
4673.0
1621.4 4688.3
1622.3 4689.7
1623.1 4691.1
1624.0 4692.4
1624.8 4693.8
4674.4
4675.8
4677.2
4678.5
4679.9
4695.2
4696.6
4698.0
4699.4
4617.7 1629.2 4700.8
4684.1
4685.5
4686.9
4702.2
4703.6
4705.0
4706.4
4707.8
4709.2
4710.6
4712.0
4713.4
4714.8
4716.2
4717.6
4719.0
4720.4
4721.8
E
1643.0
1643.9
1644.7
1645.6
1646.5
1647.3
1648.2
1649.1
1650.0
1650.8
1651.7
1652.6
1653.5
1654.3
1655.2
1656. I
1657.0
1657.9
1658.7
1659.6
1664.8
1665.7
1666.6
1667.5
1668.3
1669.2
1670. I
1671.0
1671.9
1672.8
1660.5
4751.2
1661.4
4752.6
1662.2
4754.0
1663.1 4755.4
1664.0
4756.8
1682.5
1683.4
1684.3
| = 79°
1685. I
1686.0
T
E
4723.2 1695.8
4724.6 1696.7
4726.0 1697.6
4727.4 1698.5
4728.8 1699.4
1691.3
1692.2
4737.2
4738.6
4740.0
4741.4
4742.8
1693. I
1694.0
1694.9
4744.2
4745.6
4747.0
4748.4
4749.8
4730.2 1700.2
4731.6 1701.1
4733.0 1702.0
4734.4 1702.9 8
4735.8
4758.3
4759.7
4761.1
4772.4
1673.6
1674.5
4773.8
1675.4 4775.2
1676.3 4776.6
1677.2 .4778.0
4762.5
4763.9
4765.3
4766.7
4768.1
4769.5
4770.9
1
4786.5
4787.9
4789.3
4790.7
4792.1
0-231 567BK 0-2MA DOT6K
4
4800.7
4802.1
4803.5
4804.9
4806.3
1703.8 9
1704.7 10
1705.6
1706.5 12
1707.4 13
1708.3 14
1709.2 15
1710.1 16
1711.0 17
1711.9 18
1712.8 19
1713.7 20
22222 222
1678. I
4779.4
1679.0 4780.8
4782.2
4783.7 1734.4 43
1731.7 40
1732.6 41
1733.5 42
1679.9
1680.7
1681.6
4785.1 1735.3 44
1736.2 45
1737.246
1738.1 47
1739.0 48
1739.9 49
1714.6 21
1715.5
22
1716.4 23
1717.3 24
1718.2 25
1719.1 26
1720.0 27
1720.9 28
1721.8 29
1722.7 30
1723.6 31
1724.5 32
1725.4 33
1726.3 34
1686.9
4793.6 1740.8 50
1687.8 4795.0 1741.751
1688.7
4796.4 1742.6 52
1689.6
4797.8
1743.5 53
1744.4 54
1690.5
4799.2
1727.2 35
1728.1 36
1729.0 37
1729.9 38
1730.8 39
1745.355
1746.3 56
1747.2 57
1748.1 | 58
1749.0 59
406
1
TABLE VIII.-TANGENTS AND EXTERNALS FOR A 1° CURVE
(Chord or Arc Definition)

01234
KONOM ONMI DONDO ONE D0700 8.~M7 KOMM. 22 0 508
7
5 4814.9 1754.4
1755.4
6 4816.3
1756.3
1757.2
1758. I
8
9
10
12
19
4824.8 1760.8
1761.8
14 4827.7 1762.7
21
13 4826.2
22
1763.6
15 4829.1
16 4830.5 1764.5
17
4831.9 1765.4
24
25
| = 80°
T
27
E
4807.7 1749.9 4893.6
4809.2 1750.8 4895.0
4810.6 1751.7 4896.5
4812.0 1752.6 4897.9
1753.5
4813.4
4899.4
28
29
4817.7
4819.1
4820.5
18 4833.4 1766.4 4919.6
4921.0
31
4822.0
4823.4
32
20 4836.2 1768.2 4922.5
34
1759.0
1759.9
4834.8 1767.3
23 4840.5 1771.0 4926.8
51
| = 81°
T
30 4850.5
4900.8
4902.2
4903.7
4905. I
4906.6
26 4844.8 1773.7 4931.2
1774.6 4932.6
1846.2
4847.6
4849. I
4934.1
1775.6
1776.5
4935.5
4908.0
4909.5
4910.9
4912.4
4913.8
4915.2
4916.7
4837.6 1769.1 4923.9
33 4854.8 1780.2
4839.1 1770.0 4925.4
35
4857.7
36 4859. I
37
4860.5
38 4862.0
4841.9 1771.9 4928.3
4843.4 1772.8 4929.7
1819.4
1820.3
4918.1 1821.3
1822.2
1823.2
50 4879.2 1796.0
4880.6 1796.9
52 4882.1 1797.9
1777.4
4937.0
4851.9 1778.3 4938.4
4853.4 1779.3 4939.9
4941.3
4856.2 1781.1 4942.8
1782.0
1783.0
1783.9
1784.8
39 4863.4 1785.8 4950. I
4944.2
4945.7
40
4864.8 1786.7 4951.5
41 4866.3 1787.6 4953.0
4867.7 1788.6 4954.4
42
43 4869.1 1789.5 4955.9
44 4870.6 1790.4 4957.3
45 4872.0 1791.3 4958.8
46 4873.4 1792.3 4960.3
47 4874.9 1793.2 4961.7
48 4876.3 1794.1 4963.2
49 4877.8 1795.1 4964.6
4947.2
4948.6
4966. I
4967.6
4969.0
E
1805.3
1806.2
1807.2
53 4883.5 1798.8 4970.5
54 4884.9 1799.7 4971.9
1800.6
4886.4
4973.4
55
1802.5 4976.3
56 4887.8 1801.6 4974.9
57 4889.3
58 4890.7 1803.4 4977.8
59 4892. I
1804.4
1808. I
1809. I
1814.7
1815.6
1816.6
1817.5
1818.5
1810.0 4988.0
1810.9
4989.5
4991.0
1811.9
1812.8
4992.4
1813.8
4993.9
1824.1
1825.0
1826.0
1826.9
1827.9
1833.6
1834.5
1835.5
1836.4
1837.4
1838.3
1839.3
1840.2
1841.2
1842.1
1843.1
1844.0
1845.0
1845.9
1846.9
1 = 82°
T
4980.7
4982.2
4983.6
1852.6
1853.6
1854.5
1855.5
1856.4
4985.1
4986.6
1857.4
1858.4
1859.3
1860.3
4979.2 1861.2
1828.8
5017.4
1829.8 5018.9
1830.7
5020.3
1831.7
1832.6
4995.4
4996.8
4998.3
4999.8
5001.2
5002.7
5004.2
5005.6
5007. I
5008.6
5010.0
5011.5
5013.0
5021.8
5023.3
5024.8
5026.2
5027.7
5029.2
5030.7
5032.1
5033.6
5035. I
5036.6
5038. I
1847.8
5046.9
1848.8
5048.4
1849.7 5049.9
1850.7 5051.4
1851.6 5052.8
5039.5
5041.0
5042.5
5044.0
5045.4
E
1862.2
1863.2
1864. I
1865. I
1866.0
1867.0
1868.0
1868.9
1869.9
1870.8
1871.8
1872.8
1873.7
1874.7
1875.7
1876.6
1877.6
1878.6
1881.5
5099.0
1882.5 5100.4
1883.4
5101.9
5014.5 1884.4
5103.4
5015.9
1885.4
5104.9
1891.2
1892.2
1893.1
1894. I
1895.1
1 = 83°
1896.0
1897.0
1898.0
1899.0
1900.0
T
5069.2
5070.7
1900.9
1901.9
1902.9
1903.8
i 904.8
5072.1
5073.6
5075. I
5094.5
1879.6 5096.0
1880.5 5097.5
5076.6
5078. I
5079.6
5081.1
5082.6
5084.0
5085.5
5087.0
5088.5
5090.0
1886.3
5106.4
1887.3 5107.9
1888.3
5109.4
1889.3
5110.9
1890.2
5112.4
5061.7
1915.6
5063.2
5064.7
1916.6
1917.6
5066.2 1918.5
5067.7 1919.5
5091.5
5093.0
E
1920.5
1921.5
1922.5
1923.5
1924.5
1925.4
1926.4
1927.4
1928.4
1929.4
1930.4
1931.4
1932.4 12
1933.4 13
1934.4 14
10
1935.3 15
1936.3 16
1937.3 17
1938.3 18
1939.3
1940.3 20
60700 0-2MI DONDO OU DONº 8.
1941.3 21
1942.3 22
1943.3 23
1944.3 24
1945.3 25
1946.3 26
1
1947.3 27
1948.3 28
1949.3 29
0123A
1950.3 30
5113.9
5115.4
1951.3 31
5116.9 1952.3 32
5118.4 1953.3 33
5119.9 1954.3 34
5142.4
5143.9
5055.8 1911.7 5145.4
5054.3 1910.7
5057-3 1912.7 5146.9 1972.3
5058.8
1913.6
1914.6
5060.3
5121.4 1955.2 35
5122.9 1956.2 36
5124.4 1957.2
5125.9
1958.2 38
5127.4 1959.2 39
COLOR 20000 55555 £5555 wwwww
1960.2❘ 40
1961.241
5128.9
5130.4
5131.9 1962.2
5133.4 1963.2 43
5134.9
1964.2 44
37
5136.4
1905.8
1906.8
1965.2 45
5137.9 1966.3 46
1907.8 5139.4
1967.3
47
1908.7 5140.9 1968.3 48
1909.7
1969.3 49
42
1970.3
50
1971.3 51
52
5148.4 1973.3 53
5150.0
1974.3 54
5151.5 1975.3 55
5153.0 1976.4 56
5154.5 1977.4 57
5156.0 1978.4
5157.5
1979.4 59
58
407
TABLE VIII. TANGENTS AND EXTERNALS FOR A 1º CURVE
(Chord or Arc Definition)
T
222 2
0-2MT DONCK OUNCE ON OMA KONDO OM~m7 ❤❤♪❤. O 00 85885 6858
E
T
0 5159.0 1980.4 5250.3
5160.5 1981.4
1
5251.8
5253.3
5254.9
5256.4
2 5162.0 1982.4
3 5163.5 1983.4
5165.0 1984.4
5 5166.6
65168.1
75169.6
8 5171.1
9 5172.6
105174.1
II 5175.6
125177.1
13 5178.7
145180.2
21 5190.8
24 5195.3
2001.6 5282.5
22 5192.3 2002.6 5284.1
27
23 5193.8 2003.7 5285.6
2004.7
5287.2
28
T
5265.6
5267.I
1990.5
1991.5
1992.5 5268.7
1993.5 5270.2
1994.5 5271.8
1995.5 5273.3
15
2057.3
5181.7
16 5183.2 1996.6 5274.8 2058.0
1997.6 5276.4 2059.4
1998.6 5277.9
5279.5
17 5184.7
185186.2
19 5187.7 1999.6
2060.4
2061.5
20 5189.3 2000.6 5281.0
| = 84°
25 5196.8 2005.7
5288.7
26 5198.4 2006.7 5290.3
30
32
M

29 5202.9 2009.8 5294.9
5204.5
315206.0
5207.5
33 5209.0
40
2010.8 5296.4
2011.8 5298.0
2012.9 5299.5
2014.0 5301.1
34 5210.5 2014.9 5302.6
38 5216.6
1985.4 5257.9
1986.5 5259.5
1987.5 5261.0
1988.5
1989.5
5262.5
5264. I
48
35 5212.1 2015.9 5304.2
36 5213.6 2017.0 5305.7
37
5215.1
49
1 = 85°
39 5218.2 2020. I
50
51
5201.4 2008.8 5293.3
52
2067.7
2068.8
5199.9 2007.7 5291.8 2069.8
2070.9
2071.9
41
42
5222.7 2023.2
43 5224.3 2024.2
44 5225.8 2025.2
5219.7
45 5227.3 2026.2
5228.8 2027.3
5230.4 2028.3
5231.9
46
47
57
2021.1
5221.2 2022. I
E
2041.7
2042.7
2043.8
2044.8
2045.9
2078.2
2079.3
2018.0 5307.3 2080.3
2019.0
5308.8 2081.4
5310.4
2082.4
2029.3 5324.3
5233.4 2030.4 5325.9
2052. I
2053. I
2054.2
2055.2
2056.3
5234.9 2031.4 5327.4
5236.5 2032.4 5329.0
5238.0 2033.5 5330.5
54 5241.1 2035.5 5333.6
2046.9
2110.0
2047.9
2111.1
2049.0 5353.9 2112.1
2050.0
5355.5 2113.2
2051.1
5357.5
2114.2
2062.5
2063.5
2064.6
2065.6
2066.7
2073.0
2074.0
2075. I
2076. I
2077.2
5311.9
5313.5
5315.0
5316.6 2086.7
5318.1 2087.7
2094. I
2095.2
2096.2
53 5239.5 2034.5 5332.1 2097.3
2098.3
5319.7 2088.8
5321.2 2089.9
5322.8 2090.9
2092.0
2093.0
2083.5
2084.6
55 5242.6 2036.5 5335.2 2099.4
5244.1 2037.6 5336.8 2100.5
5245.7 2038.6 5338.3 2101.5
58 5247.2 2039.6 5339.9 2102.6
1 = 86°
T
5343.0
5344.5
5346. I
5347.7
5349.2
59 5248.7 2040.7 5341.4 2103.6
5350.8
5352.3
5358.6
5360.1
5361.7
5405.6
5407.2
2085.6 5408.8
5410.4
5412.0
2115.3
2116.4
2117.4
5363.3 2118.5
5364.8
2119.6
5366.4
5368.0
5369.5
5371.1
5372.7
5374.2
5375.8
5377.4
5378.9
5380.5
5382.1
5383.6
5385.2
5386.8
5388.3
5389.9
5391.5
5393.1
5394.6
5396.2
5397.8
5399.3
5400.9
5402.5
5404. I
E
2104.7
2105.8
2106.8
2107.9
2108.9
2120.6
2121.7
2122.8
2123.9
2124.9
5429.3
5430.9
5432.5
5434.1
5435.7
2126.0
2127.1
2128.1
2129.2
2130.3
2131.3
2132.4
2133.5
2134.6
2135.6
2139.9
2141.0
2142.1
2143.2
2144.3
5413.5
5415.1
5416.7
5418.3 2156.2
5419.8
2157.3
| = 87°
T
5437.2 2169.2
5438.8 2170.3
5440.4 2171.4
5442.0 2172.5
5443.6 2173.6
2152.9
2154.0
2155.1
5421.4 2158.4
5423.0
2159.5
5424.6 2160.6
5426.2
5427.7
2161.6
2162.7
5445.2 2174.7
5446.7 2175.8
5448.3 2176.9
5449.9
2178.0
5451.5
5453. I
5454.7
5456.3
5457.9
5459.5
2136.7 5484.9
2137.8 5486.5
2138.9 5488.1
5489.7
5491.3
5461.0
5462.6
5464.2
5465.8
5467.4
5469.0
5470.6
5472.2
5473.8
5475.4
5492.9
5494.5
5496.1
2145.3 5497.7
2146.4
5477.0
5478.6
5480.2
5481.8
5483.4
་་
01234
2179.1 9
5678∞
2180.2 10
2181.3 "1
2182.4 12
2183.5 13
22
IZMI MODE. ONE HO700 87087 68788 25222 99700 BDOBJ
2184.6 14
2185.6 15
2186.7 16
2187.8 17
2188.9 18
2190.0 19
2191.1 20
2192.2 21
2193.3 22
2194.4
2195.5
23
2196.6 25
24
2197.8 26
2198.9 27
2200.0 28
2201.1 29
2202.2 30
2203.3 31
2204.4 32
2205.5 33
2206.6 34
2207.7 35
2208.8 36
2209.9 37
2147.5
5500.9 2213.2 40
2148.6 5502.5 2214.3 41
2149.7 5504.1 2215.4
42
2150.8 5505.7 2216.5 43
2151.9
5507.3 2217.6 44
5509.0
2218.7 45
5510.6 2219.9 46
5512.2
2221.0
2222.1
5513.8
5515.4
2211.0 38
2224.3
5499.3 2212.1 39
5520.2 2226.5
47
2223.249
48
5517.0
5518.6 2225.4 51
50
52
5521.8 2227.7 53
5523.4 2228.8 54
2163.8 5525.0 2229.9 55
2164.9 5526.6 2231.0 56
2166.0
2167.0
2168.1
5528.2 2232.1 57
5529.8 2233.3 58
5531.4
2234.4 59
408
TABLE VIII.-TANGENTS AND EXTERNALS FOR A 1° CURVE
(Chord or Arc Definition)

INN NN
56
0-NMI BONO OLEME DONE ONE DONDO CO~~7 ❤om. I DO 885
2
3
4
5
6
7
8
9
10
12
13
14
16
17
19
2252.3
2253.5
2254.6
18 5562.1 2255.7
5563.7 2256.9
20
15 5557.3
25
26
27
28
32
33
34
5565.4 2258.0
21 5567.0 2259.1
5568.6 2260.3 5666.7
22
23
5570.2 2261.4
5668.3
24 5571.8 2262.5 5670.0
30 5581.6 2269.3
35
5573.5
2263.6 5671.6
2264.8
5673.3
5575. I
5576.7
2265.9
5674.9
5578.3
2267.0 5676.6
29 5580.0 2268.2 5678.2
37
T
--|
40
41
1 = 88°
2274.9
2276. I
2277.2
38 5594.6 2278.3
42
E
5533.1 2235.5
5534.7 2236.6
5536.3 2237.7
5537.9
2238.9
5539.5
2240.0
5589.7
36 5591.3
5593.0
2241.1
5541.1
5542.7
2242.2
5544.3 2243.3
5546.0 2244.5
5547.6 2245.6
5679.9
31 5583.2 2270.4 5681.5
5584.8
5586.5
5588. I
2271.6
2272.7
2273.8
5549.2 2246.7
5550.8
5552.4
2247.8
2249.0
5554.0
5555.7
2250. I
2251.2
45
39 5596.2 2279.5
47
5558.9
5560.5
48
49
43 5602.7 2284.0
51
44 5604.4 2285.2
53
46 5607.6 2287.4
5597.9 2280.6
5599.5 2281.7
5601.1 2282.9
1 = 89°
T
5630.5
5632.2
5633.8
5635.4
5637. I
5638.7
5640.3
5642.0
5643.6
5645.3
5646.9
5648.6
5650.2
5651.8
5653.5
5655. I
5656.8
5658.4
5660.I
5661.7
5663.4
5665.0
5696.4
5698. I
5699.7
5701.4
5703.0
5606.0 2286.3 5704.7
E
2303.5
2304.6
2305.8
2306.9
2308. I
50 5614.2 2292.0 5713.0
5615.8 2293.1
52 5617.4 2294.3
2309.2
2310.4
2311.5
2312.7
2313.8
2315.0
2316.2
2317.3
2318.5
2319.6
2320.8
2322.0
2323. I
2324.3
2325.4
54 5620.7 2296.6 5719.7
2326.6
2327.8
2328.9
2330. I
2331.2
5683.2 2340.5
5684.8 2341.7
5686.5
2342.8
2332.4
2333.6
2334.7
2335.9
2337.0
2338.2
2339.4
2355.6
5706.4 2356.8
2358.0
2359.2
2349.8
2351.0
2352.1
2353.3
2354.5
5609.3 2288.6 5708.0
5610.9 2289.7 5709.7
5612.5 2290.9 5711.3 2360.3
2361.5
5714.7 2362.7
5716.3 2363.9
5619.1 2295.4 5718.0 2365.0
2366.2
55 5622.3 2297.7 5721.3
56 5624.0 2298.9
5723.0
5724.7
57
5625.6 2300.0
58 5627.2 2301.2 5726.3 2370.9
59 5628.9 2302.3
5728.0 2372.1
| = 90°
5688. I
2344.0
5689.8
2345.2
5691.4
2346.3 5791.7
5693.1 2347.5 5793.3
5694.8
2348.6 5795.0
2367.4
2368.6
2369.8
T
5729.7
5731.3
E
2373.3
2374.5
5733.0 2375.7
5734.7
2376.8
5736.3
2378.0
5738.0
5739.7
5741.3
5743.0
5744.7
5746.3
5748.0
5749.7
5751.4
5753.0
5754.7
5756.4
5758. I
5759.7
5761.4
5763. I
5764.8
5766.4
5768. I
5769.8
5771.5
5773.2
5774.9
5776.5
5778.2
5779.9
5781.6
5783.2
5784.9
5786.6
5788.3
5790.0
5796.7
5798.4
5800. I
5801.8
5803.5
5805. 1
5806.8
5808.5
5810.2
5811.9
5813.6
5815.3
5817.0
5818.7
5820.4
5822.1
5823.8
5825.4
5827.1
5828.8
2385. I
2386.3
2387.5
2388.7
2389.9
2391.0
2392.2
2393.4
2394.6
2395.8
2397.0
2398.2
2399.4
2400.6
2401.8
2379.2 5839.0
2380.4
5840.7
2381.6
5842.4
2382.7
5844. I
2383.9
5845.8
2402.9
2404. I
2405.3
2406.5
2407.7
2408.9
2410. I
2411.3
2412.5
2413.7
| = 91°
2426.9
2428.1
2429.3
2430.5
2431.7
T
5830.5
5832.2
E
2444.9
2446. I
2447.3
2448.6
5837.3 2449.8
2438.9
2440. I
2441.3
5833.9
5835.6
2442.5
2443.7
5852.6
5854.3
5847.5
5849.2 2458.3
5850.9
5856.0
5857.7
5859.4
5861.1
5862.9
5864.6
5866.3
5868.0
5869.7
5871.4
2414.9
5890.2
2416.1 5891.9
2417.3
5893.6
5895.4
2418.5
2419.7 5897.1
5873.1
5874.8
5876.5
5878.2
5879.9
2420.9 5898.8
5900.5
2422.1
2423.3
5902.2
2424.5 5903.9
2425.7
5905.7
5881.7
5883.4
5885. I
5886.8
5888.5
2451.0
2452.2
2453.4
2454.7
2455.9
5907.4
5909. I
5910.8
5912.5
5914.3
0-23♬
6678σ
9
2457.1 10
IZMI KONO ME DON** 8~~87 KHM88 27 700 85087 KANOO
2459.5 12
2460.8 13
2462.0 14
2463.2 15
2464.4 16
2465.6 17
2466.9 18
2468. I 19
2469.3 20
2470.5 21
2471.7 22
2473.0 23
2474.2 24
2475.4 25
2476.6 26
2477.8 27
2479. 28
I
2480.3 29
2481.5 30
2482.7 31
2484.0 32
2485.2 33
2486.4 34
2487.6 35
2488.9 36
2490. 37
2491.3 38
2492.6 39
2493.8 40
2495.0 41
2496.3 42
2497.5 43
2498.7 44
2499.9 45
2501.246
2502.4 47
2503.6 48
2432.9
5916.0
2434.1 5917.7
2507.3 51
2435.3 5919.4 2508.6 52
2436.5
5921.2 2509.8 53
5922.9
2437.7
2511.1 54
2504.9 49
2506. 50
5924.6
2512.3 55
5926.3 2513.5 56
5928.0
5929.8
2514.8 57
5931.5 2517.3 59
2516.0 58
409
I
5°
10°
2° 30 4°
5°
.00 .01 .01.01 .02 .02
.01 .01 .02 .03 .03.04
15° .01 .02 .03.04 .05.06
20°
.02.04.05
.05.06.08
.01
25° .02
30° .02
35°
40°
I
TABLE IX. —CORRECTIONS FOR TANGENTS*
(Chord Definition of D)
After dividing T,°
(Table VIII) by D add the correction.
с

.10
.03 .05.07 .08
.04.06
.06.08 .10 .12
.02.04.07.09
.II .14
03.05 .08.11
.13 .16
.13
.17
45° .03 .06 .09 .12
50° .03 .07 .10 .14
55° .04 .07 .11 .16
60° .04 .08
65° .04 .09
70° .05 .10 15 .21
75°.05.11 .17 .23
80° .06 .12 .18 .25
10°
]。
85° .06 .13.20 .27
90° .07 .14 .22 .30
95° .08 .15 .24 .33
100°❘ .08 .17 .26 .36
105° .09 .18.29
110° .10 .20 .31
115° 11 .22
.24
120°
.12
95°
100°
105°
110°
.14.19
·
1°
FOT
.35
38
•
Degree of Curve (Dc)
6°
.05 .10
115° .06 .12
120° .07 .14
.15.18
.17 .20
.19.23
.21 .25
.23.28
.25 .31
.28 .33
.30 .37
.33.40
.36 .44
.39 .48
.43.52
8° 10°
.03
.03
.05
.06
.08
.10 .12
.10
.13
.15
.13
.16
.18
.21
.24
.27
.30
.34
6°
.37
.41
.45
.49
.53
.58
.63
.69
.16
.20
.23
.26
.30
.34
.38
.42
.46
.51
.56
.61
.03 .07 .11 .14
.17 .21 .28
.04.08 .12 .17 .20 .24 .32
.04.09 .14 .19 .23.28 .37
Degree of Curve (D)
2° 3° 4° 5°
8° 10° 12°
.00 .00 .00 .00 .00 .00 .00 .00 .00
20° .00 .00 .00 .00 .01.01.01 .01 .01
30° .00 .00 .01.01 .01.02 .02.03.03
40° .00 .01.01.02.02.03 .04 .05 .06
50° .01 .01.02.03 .04.05 .06 .08 .09
60° .01 .02 .03.05
.05.06.07
.07.09 .11 .14
12°
.04
.04
.08 .09
.19
.23
.28
.32
.36
.41
65° .01.03.04.06 .07 .08 .11 .14 .16
70° .02.03.05.07 .08 .10 .13 .16 .19
75° .02 .04.06 .08 .09 .|| .15 .19
80°
.02.04 .07 .09 .11
85°
.13
.03 .05 .08 .11 .13 .16
.03 .06 .09 .12 .15 .18 .24
.18 .22
.21
.23
.27
.26 .31
.30 .36
90°
.45
.50
.56
.61
.67
.73
•
14°
.13
.18
* See page 386 for explanation and example.
TABLE X.-CORRECTIONS FOR EXTERNALS*
(Chord Definition of D)
.23 .26
.27 .31
.32
.37
.42
.48
.53
.59
After dividing E,° (Table VIII) by De add the correction.
с
16°
.05
.10
.15
.21
.65
.74
.71
.82
.78
.89
.86 .98
.67
.73 .87
.80 .93 1.07 85°
1.02 1.17 90°
.79 .95 1.11 1.27 95°
.87 1.04 1.21 1.39
.39 .47 .57 .76 .95 1.14
1.33 1.52 105°
.43 .51.62 .83 1.04 1.25 1.46 1.67 110°
.47 .57 .68 .91
1.14 1.37 1.60 1.83 115°
.52.62.76 | 1.01 1.26 1.51 1.76
120°
2.02
.61
.67
.16 .22 .27 .32 .43 .54 .65 .. 76
.19 .26 .31
.22 .30 .36
* See page 386 for explanation and example.
25°
30°
.37 35°
.42
40°
I
.48 45°
.54 50°
5°
16°
10°
15°
20°
I
55°
60°
65°
70°
75°
80°
100°

14°
.00 .00 10°
.02 .02 20°
.04 .04 30°
.07 .07 40°
.12
.18
.!|
50°
.16
60°
.19 .22
.22 .26
.27
.30
.31
.36 80°
.36
.42 85°
.48 90°
.42
65°
70°
75°
.35
.40
42 .49
.56 95°
.49 .57 .65
.47 .56 .65
100°
.75
105°
.87
110°
.38.50 .63 .75 .88 1.00 115°
.58.73 .87 1.02 1.17 120°
.44
410
TABLES XI & XII.—SPIRAL TABLES-GENERAL EXPLANATION
D
PARTIAL NOTATION
(See Chapter 5 for complete theory and notation)
Degree of central circular curve (Da=arc definition of
D; De-chord definition of D).
Length of spiral curve, in feet.
Central angle of the spiral, or spiral angle.
Ls
Δ
X, Y Coordinates (abscissa and ordinate) of the S.C. referred
to the T.S. as origin and to the initial tangent as X-axis.
Xo, o Coordinates (abscissa and ordinate) of the offset T.C.,
which is the point where a tangent to the circular curve
produced backward becomes parallel to the tangent at
the T.S.
L.T. "Long tangent" of the spiral.
S.T. "Short tangent" of the spiral.
L.C. "Long chord" of the spiral.
Table XI. This table gives spiral parts for various selected
values of D and Ls. It may be used for both chord definition and
arc definition of D. The definition of D affects only the co-
ordinates of the offset T.C.; correct values of these coordinates
may be obtained by observing the following:
1. When arc definition of D is used, select values of o and Xo
from sub-columns headed Da.
2. When chord definition of D is used, subtract the correc-
tions in the columns headed from the arc-definition
values of o and Xo.
Table XII. This table may be used to obtain spiral parts
for any combination of D and Ls up to A=45. Proceed as
follows:
1. When arc definition of D is used, enter table with given
value of ▲ (interpolating if necessary) and multiply the
tabulated coefficients by the value of L.
2. When chord definition of D is used, calculate spiral parts as
above. Then, since only the coordinates of the offset
T.C. are affected by the definition of D, correct the calcu-
lated values of o and Xo by subtracting the products of D
and the coefficients in the columns headed by an asterisk.
Example. D=12; L,=320; A=19.2. For arc definition,
8.90
o=320X.02781
Xo=320X.49813=159.40
For chord definition,
=
0 = 8.90-12X.0041
Xo=159.40-12X.0240=159.11
=
8.85
411
L.
100
125
150
200
250
0 37.5
300
0 45.0
350
0 52.5
400 1 00.0
700
800
900
1000
450 1 07.5
500 1 15.0
550 1 22.5
600
1 30.0
100
125
150
200
Δ
500
550
600
0°15.0'
0 18.8
0 22.5
0 30.0
700
800
900
1000
250
0 50.0
300 1 00.0
350 1 10.0
400 1 20.0
100
125
150
200
450 1 30.0
1 40.0
1 50.0
2 00.0
1 45.0
2 00.0
2 15.0
2 30.0
0°20.0'
0 25.0
0 30.0
0 40.0
700
800
900
1000
2 20.0
2 40.0
3 00.0
3 20.0
0°25.0'
0 31.3
0 37.5
0 50.0
250 1 02.5
300 1 15.0
350
400
450
1 52.5
500 2 05.0
550 2 17.5
600
2 30.0
TABLE XI.—SELECTED SPIRALS
Xo | X Y L.T. |
1
|
D=0° 30'
100.00 0.14 66.67
125.00 0.23 83.33
100.00
150.00
0.33
200.00
0.58 133.33
0
3 45.0
4 10.0
0.03
0.06
0.08
0.15
0.23
0.31
0.44
0.58
0.74
0.91
1.10
1.31
1.78
2.33
2.94
3.64
0.05
0.08
0.11
0.20
50.00
62.50
75.00
100.00
2.38
3.10
3.93
4.85
125.00
150.00
0.06
0.09
0.14
0.24
175.00
200.00
225.00
250.00
274.99
299.99
0.30
125.00
0.44
150.00
0.59 175.00
0.78
200.00
50.00
62.50
75.00
100.00
0.98
225.00
1.21
249.99
1.47 274.99
1.75
299.99
349.99
699.93
399.98
799.90
449.98
899.86
11.78
499.97 999.81 14.54
1 27.5 0.74 175.00
1 40.0
0.97
199.99
0.38 125.00
0.55 150.00
250.00
299.99
349.99
399.99
224.99
1.23
1.51 249.99
1.83 274.99
2.18
299.98
449.98
499.98
549.97
599.96
4.91 449.93
6.06 499.91
125.00
150.00
200.00
0.91 166.67
1.31
200.00
1.78 233.34
2.33 266.67
D=0° 40'
249.99
299.99
349.99
399.98
2.95
300.01
3.64 333.34
4.40 366.68
5.24
400.01
100.00 0.18
0.30
0.44
0.78
449.97
499.96
549.95
599.93
7.13
9.31
349.98
699.88
399.97
799.83
12.41
449.96 899.75
15.70
499.94 999.66 19.39
449.95
499.93
549.91
599.89
466.69
533.37
600.05
666.73
1.21 166.67
1.75
200.00
2.38 233.34
3.10
266.67
3.93
300.01
4.85 333.35
5.87
366.69
6.98
400.03
D=0° 50'
50.00
100.00 0.24
62.50 125.00 0.38
75.00 150.00 0.55 100.00
100.00
200.00
0.97 133.33
9.50 466.71
66.67
83.33
249.99
166.67
200.00
349.98 2.97 233.34
1.52
299.99 2.18
399.97
3.88
266.68
100.00
66.67 33.33
83.33 41.67 125.00
100.00
50.00 150.00
200.00
133.33
66.67
4.91 300.02
6.06 333.36
7.33
8.73
2 55.0
2.97
349.97 699.82 11.88
466.73
3 20.0 3.88 399.95 799.73 15.51 +533.43
899.63 19.63 600.14
999.47 24.23 666.85
S.T.
366.70
400.04
I
L.C.
33.33 100.00
41.67 125.00
50.00 150.00
66.67 200.00
83.33
100.00
116.67 350.00
133.34 399.99
250.00
300.00
233.37 699.95
533.39 266.72 799.92
600.09
666.79
300.08 899.89
333.44 999.85
150.01 449.99
166.67 499.99
183.34 549.99
200.01 599.98
233.35
699.97
266.70 799.96
300.05 899.94
333.39
999.92
83.33 250.00
100.00
300.00
116.67
349.99
133.34
399.99
150.01
449.99
166.68
499.98
183.35 549.98
200.02 599.97
33.33
100.00
41.67 125.00
50.00
150.00
66.67 200.00
83.34 250.00
100.00 299.99
116.67 349.99
133.34 399.99
150.02 449.98
166.69
499.97
183.36 549.96
200.04
599.95
233.39 699.92
266.75 799.88
300.12
899.83
333.50
999.77
412
L8
100
125
150
200
250
300
350
400
450
500
550
600
700
800
900
1000
100
125
150
200
250
300
350
400
450
500
550
600
700
800
900
1000
100
125
150
200
250
300
350
400
450
500
A
550
600
700
800
900
0°30.0'
0 37.5
0 45.0
1 00.0
1 15.0
1 30.0
1 45.0
2 00.0
2 15.0
2 30.0
2 45.0
3 00.0
3 30.0
4 00.0
4 30.0
5 00.0
0°37.5'
0 46.9
0 56.2
1 15.0
1 33.8
1 52.5
2 11.3
2 30.0
2 48.8
3 07.5
3 26.3
3 45.0
4 22.5
5 00.0
5 37.5
6 15.0
0°45.0'
0 56.2
1 07.5
1 30.0
1 52.5
2 15.0
2 37.5
3 00.0
3 22.5
3 45.0
4 07.5
4 30.0
0
5 15.0
6 00.0
6 45.0
TABLE XI.-SELECTED SPIRALS
| L.T.
0.07
0.11
0.16
0.29
1.47
1.82
2.20
2.62
Xo
Da
0.45
125.00
0.65
150.00
0.89 174.99
1.16
199.99
50.00
62.50
75.00
100.00
224.99
249.98
274.98
299.97
0.09
50.00
0.14
62.50
0.20
75.00
0.36 | 100.00
0.11
0.17
0.25
0.57
124.99
149.99
0.82
1.11 174.99
1.45 | 199.99
50.00
62.50
75.00
0.44 100.00
0.68 124.99
•
• •
3.56
349.94
4.65 399.94
.01
5.89 449.91 .01
7.27
499.87 .01
•
· -
· •
*
•
•
•
·
·
•
· •
• •
• •
•
*
•
1.84 224.98
2.27 249.97
2.75 274.97 .01
3.27
299.96 .01
•
4.45 349.93 .01
5.82 | 399.90
7.36 | 449.86
9.08 | 499.80
.01
.01
.01
·
• ·
D=1° 15'
X
D=1°
0.98 149.99
1.33 174.99
1.74 199.98
2.20 | 224.97 .01
2.72 249.96 .01
•
•
66.67
100.00 .0.29
125.00 0.45 83.33
150.00 0.65 100.00
199.99 1.16 133.34
249.99 1.82 166.67
299.98 2.62 200.01
349.97
399.95
3.56 233.34
4.65266.68
Y
449.93 5.89 300.02
499.91 7.27 333.37
549.87 8.80 366.71
599.84 10.47 | 400.06
699.74 | 14.25 | 466.76
466.76
799.61 18.61 533.47
899.44 23.55 | 600.19
999.24 29.07 | 666.93
666.93
100.00 0.36 66.67 Į
125.00 0.57 83.33
150.00 0.82 100.00
199.99 1.45 133.34
249.98 2.27 166.67
299.97 3.27 200.01
349.95 4.45 233.35
399.92 5.82 266.69
449.89
499.85
549.80
599.74 |
D=1° 30'
7.36 300.04
9.09 333.39
11.00
11.00 366.74
13.09 | 400.09
13.09 400.09
100.00 0.44
125.00
0.68
149.99
0.98
199.99
249.97
66.67
83.33
100.00
1.75
133.34
2.73 166.68
S.T.
299.95
349.93
3.93 | 200.02
5.34233.36
399.89 6.98 266.71
449.85 8.83 300.06
499.78 10.90 333.41
L.C.
33.33 | 100.00
41.67 125.00
50.00 150.00
66.67 200.00
249.99
83.34
100.01 299.99
116.68 349.99
133.35 | 399.98
150.02 449.97
166.70 499.96
183.37 549.94
200.05 599.93
233.42
233.42
466.81
699.59 17.81
799.39 23.26
233.46 699.82
533.54 266.86 799.73
899.13 29.43 600.30 300.28 899.62
998.80 | 36.33 | 667.08
333.71 999.48
699.89
799.82
266.79
300.18 | 899.76
333.58
333.58 | 999.66
33.33 100:00
41.67 125.00
50.00 150.00
66.67 200.00
83.34 244.99
100.01 299.99
116.68 349.98
133.36 399.97
150.03 | 449.95
166.71 499.93
183.40
549.91
599.89
200.08
183.42 549.87
200.12 | 599.84
3.30 274.95 .01
3.92 | 299.94 .01
5.35 | 349.90 .01
6.98 399.86 .01
8.83 449.79 .01
549.71 13.19 366.77
599.63 15.70 400.13
699.41 21.36 466.88 233.52 699.74
799.12 27.90 | 533.64 | 266.94 | 799.61
898.75 | 35.31 | 600.44 | 300.40 | 899.44
Subtract from tabulated value for Da when chord definition of D is used.
33.33
100.00
41.67 125.00
50.00 150.00
66.67
199.99
83.34 249.99
100.01 299.98
116.69 | 349.97
133.37 | 399.95
150.05 | 449.93
166.74 499.90
413
Ls
100
125
150
200
250
300
350
400
450
500
550
600
700
800
900
100
125
150
200
250
300
350
400
450
500
550
600
700
800
100
125
150
200
250
300
350
400
450
500
Δ
0°52.5'
1 05.6
1 18.8
1 45.0
2 11.3
2 37.5
3 03.8
3 30.0
3 56.3
4 22.5
4 48.8
5 15.0
6 07.5
7 00.0
7 52.5
1°00.0'
1 15.0
1 30.0
2 00.0
2 30.0
3 00.0
3 30.0
4 00.0
4 30.0
5 00.0
5 30.0
6 00.0
7 00.0
8 00.0
1°07.5'
1 24.4
1 41.2
2 15.0
2 48.8
3 22.5
3 56.3
4 30.0
5 03.8
5 37.5
550
6 11.3
600 6 45.0
700 7 52.5
800 9 00.0
0
TABLE XI.—SELECTED SPIRALS
1.15
1.56
2.03
2.57
3.18
Χρ
0.13 50.00
0.20
62.50
0.29
75.00
0.51
100.00
0.80 | 124.99
Da
0.15
0.23
0.33
0.58
0.91
X
D=1° 45'
100.00
125.00
0.51 | 66.67
0.80
83.33
149.99 1.15 100.00
199.98 2.36 133.34
249.96 3.18 166.68
*
|
149.99 .01 299.94
174.98 .01 349.90
199.98 .01 399.85
224.96 .01 449.79
249.95 .01
499.71
3.85 274.93 .01
4.58
299.92 .01
6.24
349.87 .01
399.80 .02
8.14
10.30
449.71 .02
•
D=2°
549.61
599.50
699.20
798.81
898.31
0.16
50.00
0.26
62.50
0.37
74.99
0.65 99.99 .01
1.02 124.99 .01
4.94 274.90
5.89 | 299.86
8.02
10.46
50.00
62.50
75.00
•
124.99 0.91
149.99
100.00 | .01 199.98
124.99 .01 249.95
•
Y
4.58 200.02
6.23 | 233.37
8.14 266.72
10.30
300.07
333.43
12.72
L.T.
15.39 366.80
18.32 | 400.18
24.92 | 466.95
32.54 | 533.75
41.18 600.60
100.00 0.58 66.67
83.34
1.31 100.00
2.33 133.34
3.64 166.68
1.30 | 149.99 .01
1.78 174.98.01
2.32 199.97 .01
2.94 224.96.01
3.64249.94 .01
4.40 | 274.92 | .01 549.49 17.59 366.84
5.23 | 299.89 .02 | 599.34 | 20.93 | 400.23
7.13 | 349.82 .02 698.96 28.48 467.03
|
9.30 399.74 .02 | 798.44 37.18 | 533.88
D=2° 15'
1.47 149.98 .01 299.90
2.00 174.97 .01 349.83
2.62 199.96 .01 399.75
3.31
4.09
299.92 5.24 200.03
349.87 7.13 233.38
399.80 9.30 266.73
449.72 11.78 | 300.10
499.62 | 14.54 | 333.47
100.00 0.65
124.99 1.02
149.99 1.47
199.97
249.94
66.67
83.34
100.00
133.34
2.62
4.09 166.69
5.89 | 200.04
8.02 | 233.39
10.47 266.75
224.94 .01 449.65 13.24 300.12
249.92 .02
16.35 333.50
499.52
.02
549.36
19.78 366.89
.02 599.17 23.54 | 400.29
349.78 | .03 | 698.69 | 32.03 | 467.13
399.67
399.67 | .03 798.03 | 41.82 | 534.02
S.T. L.C.
33.33 100.00
41.67 125.00
50.00 150.00
66.67 199.99
83.35 | 249.98
100.02
116.70
299.97
349.96
133.38 399.93
150.07 | 449.91
166.76 | 499.87
183.46
549.83
599.78
200.16
233.59
699.65
267.05 799.47
300.55
300.55 | 899.25
33.33 100.00
41.67 125.00
50.00 150.00
66.67 199.99
83.35
249.98
100.03 299.96
116.71 349.94
133.40 399.91
150.09 | 449.88
166.79 499.83
183.50 549.77
200.21 | 599.71
233.67 699.54
267.16 799.30
33.33 | 100.00
41.67 125.00
50.00 | 149.99
66.68 199.99
83.35 249.97
100.03 299.95
116.72 349.93
133.41 399.89
150.11 449.84
166.82 499.79
183.54 549.72
200.26 599.63
233.75 699.42
267.30 799.12
* Subtract from tabulated value for Da when chord definition of D is used.
A.
Eque ار
tem
to by
totul (bungi duitur ton Ch
23
L.
100
125
150
200
250
300
350
400
450
500
550
600
700
800
100
125
150
200
250
300
350
400
450
500
550
600
700
100
125
150
200
250
300
350
400
450
500
Δ
1°15.0'
1 33.8
1 52.5
2 30.0
3 07.5
3 45.0
4 22.5
5 00.0
5 37.5
6 15.0
6 52.5
7 30.0
8 45.0
10 00.0
But berat 414 A
1°22.5'
1 43.2
2 03.8
2 45.0
3 26.3
4 07.5
4 48.8
5 30.0
6 11.3
6 52.5
7 33.8
8 15.0
9 37.5
1°30.0'
1 52.5
2 15.0
3 00.0
3 45.0
4 30.0
5 15.0
6 00.0
6 45.0
7 30.0
TABLE XI.—SELECTED SPIRALS

0.18
0.28
0.41
0.73
1.13
1.63
2.22
2.91
3.68
4.54
0.20
0.31
0.45
0.80
1.25
1.80
2.45
3.20
4.05
5.00
6.04
7.20
9.80
0.87
1.36
Xo
1.96
2.67
3.49
Da
X
D=2° 30'
100.00
0.72
66.67
33.33 100.00
83.34
124.99 1.14
149.98
41.67 125.00
50.01 149.99
75.00 .01
100.01
1.64
2.91
99.99 .01
199.96
133.35 66.68 199.98
124.99 .01 249.93
4.54 166.69 83.36 249.97
5.49
21.98
26.15
274.87.02 | 549.21
6.55 | 299.83 | .02 | 598.97
8.90 349.73.03 | 698.37
698.37 35.57
399.59 | .03 | 797.57 | 46.44
11.62
D=2° 45'
99.99
124.99
149.98
199.95
249.91
4.42
5.46
50.00
62.50
•
149.98.01
299.87
174.97 | .01 349.80
199.95 .02 | 399.70
224.93 | .02 | 449.56
249.90 | .02 | 499.40
50.00
62.50 .01
75.00 .01
99.99 .01
124.98 | .01
0.22 50.00 | .01
99.99
0.34
62.50 .01
124.99
0.49
75.00 .01 149.98
99.99 .01 199.95
124.98.01 | 249.89
149.97 .01 299.85
174.96 | .02 | 349.75
199.94 | .02 | 399.63
224.91.02 | 449.47
249.88.02 | 499.28
499.28
274.84 .03 549.03
299.79 .03 598.76
349.67 | .03 | 698.02
698.02
D=3°
149.97 .02 299.81
174.95 .02 349.71
199.93 .02 | 399.56
224.89 .03 449.37
249.86 | .03 | 499.14
6.60 274.81 .03
7.85 | 299.75 | .03
|
10.68 349.61 .04
548.86
548.86
Y
598.52
598.52
6.54
8.91
697.66
697.66
11.63
14.72
18.16
24.17
28.76
39.12
0.87
1.36
1.96
3.49
5.45
L.T.
0.80
1.25
1.80
3.20 133.35
5.00
166.70
7.85
10.68
13.95
17.65
21.79
26.36
31.36
42.66
200.04 100.04 299.94
349.91
233.40 116.73
266.77
266.77 | 133.43
133.43
399.86
300.15 150.14
449.81
333.54 | 166.86
499.73
366.94 183.59
400.36
400.36 | 200.33
467.24 | 233.86
534.18 267.45
|
7.20
299.93
349.89
9.79
12.79
266.80 133.45 399.84
300.19
19.98 | 333.58
333.58
16.19
150.17 449.77
166.90 | 499.68
S.T. L.C.
66.67 33.34 100.00
83.34 41.67 125.00
100.01
50.01 149.99
66.68 199.98
83.36
249.96
200.06 | 100.05
233.42 116.75
367.00
400.43
467.36
66.67
83.34
549.65
599.54
699.27
798.92
100.01
133.35
166.70
183.64
549.58
200.40 599.45
233,96 | 699.13
33.34 | 100.00
41.67 124.99
50.01 149.99
66.68 199.98
83.37 249.95
550
8 15.0
600
9 00.0
367.06 183.70 549.49
|
400.52 | 200.47 | 599.34
467.49 234.08 | 698.96
700
10 30.0
* Subtract from tabulated value for Da when chord definition of D is used.
200.06
100.06 299.92
233.44 116.76 | 349.87
266.82 | 133.47 | 399.80
300.22 150.20 | 449.72
333.63 166.94 499.62
415
La
100
125
150
200
250
300
350
400
450
500
L8
100
125
150
200
250
Δ
1°45.0'
2 11.3
550 9 37.5
600 10 30.0
700 12 15.0
100
125
150
200
250
2 37.5
3 30.0
4 22.5
300
350
5 15.0
6 07.5
7 00.0
7 52.5
8 45.0
300
6 00.0
350 7 00.0
400
8 00.0
450
9 00.0
500 10 00.0
A
2°00.0'
2 30.0
3 00.0
4 00.0
5 00.0
550 11 00.0
600 12 00.0
650 13 00.0
2°15.0'
2 48.8
3 22.5
4 30.0
5 37.5
6 45.0
7 52.5
400 9 00.0
450 10 07.5
500 11 15.0
+
0
X
D=3° 30'
0.25 50.00 .01 99.99
0.40
62.50 | .01
124.98
0.57
74.99 .01
149.97
1.02
99.99
.02 199.93
1.59 | 124.98 | .02 | 249.85
TABLE XI.-SELECTED SPIRALS

2.29
3.12
4.07
5.15 | 224.86.03
6.36
249.80.04
0
7.69 274.74.04
9.16 299.66 .05
12.45 | 349.46
.05
Da *
0.29
0.45
0.65
1.16
1.82
2.62
3.56
4.65
5.89
7.26
0.33
0.51
0.74
1.31
2.04
...
2.94
4.01
5.23
………
• •
• • •
Xo
Do I *
•
8.79 .01 274.66 .06 547.98
|
10.46 | .01 299.56.06 | 597.37
.01 299.56 .06 597.37
12.27.01 324.44.07 646.66
| |
► ··
•
149.96 .02 299.75 9.16 200.09
174.93.03 | 349.60 12.46 233.47
199.90.03 399.40 | 16.27 266.88
449.15 | 20.59 | 300.30
498.84 | 25.41 333.74
....
Xo
Do I *
X
50.00.01 99.99
62.50 .01 | 124.98
74.99 .02 | 149.96
99.98 .02 | 199.90
124.97 .03
249.81
149.95.03
174.91.04
Y
L.T.
D=4° 30'
50.00 | .01 | 99.98
62.50 .02 124.97
74.99 .02 149.95
99.98 .03 | 199.88
124.96.03 249.76
|
1.02
1.59
2.29
100.01
4.07
133.36
6.36 166.72
66.67
83.34
"
548.45 | 30.73 | 367.21 | 183.83 | 549.31
597.99 36.56 400.71 | 200.64 599.10
696.80 | 49.72 | 467.79 | 234.35 | 698.58
D=4°
Y
1.16
100.01
1.82
2.62
4.65 133.37
7.27
166.73
299.67 10.46
349.48 14.24
200.11 | 100.10 | 299.85
233.52 116.83 | 349.77
199.87 .04 399.22 18.59 266.94 133.58 399.65
224.82.05 448.89 23.52 300.39 | 150.35 | 449.50
|
249.74 .05 498.48 29.02 333.87 167.15 499.32
|
|
183.98 549.10
|
200.84 598.83
|
217.74 648.51
|
S.T. L.C.
33.34 | 100.00
41.67 124.99
149.99
50.01
66.69 199.97
83.38 249.94
L.T. S.T. L.C.
66.67 33.34
83.34
99.99
41.67 124.99
50.01 149.98
66.70 199.96
83.39 249.92
367.38
400.92
48.98 434.51
35.11
41.75
299.89
100.08
116.97 349.82
133.52 | 399.73
150.27 | 449.63
167.04 499.48

1.31 66.67
2.04
2.94
83.34
100.02
5.23 | 133.38
8.18 166.75
149.93 .04 299.58 11.77 200.15
100.13 | 299.81
174.89 .04 | 349.34 | 16.02 | 233.56 116.88 349.71
199.84.05 | 399.02 | 20.91 267.01 | 133.65 | 399.56
6.62.01 224.77 .06 448.60 26.45 300.49 150.45 449.37
8.17.01 249.68 .06 498.08 | 32.64 | 334.01 | 167.28 | 499.14
33.34 99.99
41.67 124.99
50.02 | 149.98
66.71 199.95
83.41 249.89
550 12 22.5 9.88 .01 274.57 .07
600 13 30.0 11.76 .01 299.45.08
|
650 14 37.5 13.79 .01 | 324.30 .08
547.44 39.47 | 367.57 | 184.15
184.15
596.68 46.94 401.17 | 201.07
| |
645.78 55.05 434.82 218.02 648.12
548.86
598.52
*Subtract from tabulated value for Dɑ when chord definition of D is used..
416
I
Ls
100
125
150
2°30.0'
3 07.5
3 45.0
2005 00.0
Δ
250 6 15.0
300 7 30.0
350 8 45.0
400 10 00.0
450 11 15.0
500 12 30.0
550 13 45.0
600 15 00.0
100
125
150
200
250
300
2°45.0'
3 26.3
4 07.5
5 30.0
6 52.5
8 15.0
350 9 37.5
400 11 00.0
450 12 22.5
500 13 45.0
550 15 07.5
600 16 30.0
100
3°00.0'
125
3 45.0
150 4 30.0
200 6 00.0
250 7 30.0
300 9 00.0
350 10 30.0
400 12 00.0
450 13 30.0
500 15 00.0
100
3°15.0'
125
4 03.8
150 4 52.5
200 6 30.0
250
8 07.5
Da
0
0.36
0.57
0.82
1.45
0.40
0.62
0.90
1.60
TABLE XI. SELECTED SPIRALS
Xo
0.44
0.68
0.98
1.74
2.73
0.47
0.74
1.06
1.89
2.95
•
2:27
3.27
124.95 .04 249.70
149.91 .05 299.49
4.45
174.86 .06 349.18
5.81 .01 199.80
199.80 .06 398.78
.07
7.36 .01| 224.71
9.08.01 249.60.08
10.98 .01 274.47 .09
13.06
• •
·
D
•
* Da
•
•
..
·
•
·
8.09.01 224.65 .09
9.98 | |
.01 249.52.10
12.07 .01 274.36.10
14.36 .02 | 299.17 .11
•
-
*
►
•
D=5°
50.00 | .02 99.98
62.4902 124.96
74.99 .02 149.94
99.97.03 199.85
449.23
448.27
448.27 29.37 | 300.61 | 150.55
150.55
497.62 | 36.24 | 334.17 | 167.43
167.43
546.84 43.82 367.78 | 184.36
| |
498.94
548.59
.01 299.32.09 595.90 52.10 401.45 201.32 | 598.17
| |
|
D=5° 30′
50.00 .02 99.98
62.49 .02 124.96
74.99 .03 149.92
99.97 .04 199.82
2.50
124.94.05 | 249.64 9.99 166.79 83.45 249.84
3.60
149.90 .06 299.38 14.38 200.22
| | | 100.20
100.20 299.72
4.90 .01 174.84
174.84 .07 349.01 19.56 | 233.68 | 116.98 | 349,56
|
6.39 .01 199.76 .08 398.53 25.53 267.18 133.80 399.34
|
| | |
X
•
D=6°
Y
L.T.
50.00.02 99.97
62.49.03
.03 124.95
74.98 .03 149.91
99.96 .05 | 199.78
124.93 .06 249.57
1.45
66.67
2.27
83.35
3.27 | 100.02
5.81 133.39
D=6° 30'
49.99 .03 99.97
62.49.03
.03124.94
74.98.04
.04 149.89
99.96 .05
.05 199.74
124.91 .07 249.50
9.08 166.77 83.43 249.87
13.07 200.18 100.16 | 299.77
17.79 | 233.62 | 116.93 | 349.64
23.22 267.09 133.72 399.46
| |
1.60
2.50
66.67
83.35
3.60 100.03
6.40 133.40
S.T. L.C.
33.34 99.99
41.68 124.98
50.02 149.97
66.72 199.93
447.90 32.29 300.74 | 150.67 | 449.07
| |
497.13 39.83 334.35 167.59 498.72
| |
546.16 48.15 368.02 | 184.56 | 548.30
|
595.04 57.26 401.75 | 201.59 | 597.79
| |
3.92 .01
5.34 01
6.97 .01
149.88
149.88 .07
174.80
174.80 .08
299.26 15.68 200.26 100.24 299.67
| |
348.83 21.33 233.75 117.04 | 349.48
199.71 .09 398.25 27.84 267.28 133.89 399.22
8.82 .01| 224.59 | .10|447.51 | 35.20 | 300.88 | 150.80 | 448.89
10.88 .01 | 249.43 .11 496.58 | 43.42 | 334.54 | 167.76 | 498.48
1.74
2.73
66.68
83.35
3.93 | 100.03
6.98 | 133.41
10.90 166.82
|
33.34 99.99
41.68 124.98
50.02 149.97
66.73 199.92
1.89 66.68
2.95 83.36
4.25 | 100.04
7.56 133.42
11.80 166.84
33.34 99.99
41.68 124.98
50.03 149.96
66.74
83.47
199.90
249.81
33.34 99.99
41.69 124.97
50.04 149.95
66.75 | 199.89
83.49 249.78
300 9 45.0
350 11 22.5
4.25 | .01 | 149.85
149.85 .08 299.13 16.98 200.30 100.28 | 299.61
174.77 | .09 | 348.62 | 23.10 | 233.82 | 117.11 | 349.39
5.78.01
400 13 00.0
7.55.01
|
.02
450 14 37.5 9.55
500 16 15.0 11.78 .02
199.66
199.66.11 397.94 30.14
224.51.12 447.08 38.11
| |
249.33.13 495.99 47.00
| |
267.39 133.99 | 399.08
|
301.03 150.94 | 448.70
|
334.75 167.96 498.22
|
* Subtract from tabulated value for Da when chord definition of D is used.
417
L8
Δ
100
125
3°30.0'
4 22.5
150 5 15.0
200 7 00.0
250 8 45.0
300 10 30.0
350 12 15.0
400 14 00.0 |
450 15 45.0
50017 30.0
4°00.0'
100
125
5 00.0
150 6 00.0
200 8 00.0
250 10 00.0
300 12 00.0
350 14 00.0
400 16 00.0
45018 00.0
500 20 00.0
100 4°30.0'
125 5 37.5
150 6 45.0
200 9 00.0
250 11 15.0
30013 30.0
350 15 45.0
400 18 00.0
45020 15.0
500 22 30.0
100 5°00.0'
125 6 15.0
150 7 30.0
200 10 00.0
250 12 30.0
TABLE XI.-SELECTED SPIRALS

Da |
Xo
* Da l * X
4.58 .01
6.23.01
8.13 .02
10.28 .02
12.68 .02
D=7°
0.51
0.79
49.99 .03 99.96 2.04 66.68
62.49.04 | 124.93 3.18 83.36
74.98.05 | 149.87 4.58 100.04
99.95.06 | 199.70 8.14 133.44
3.18 .01 124.90 .08 249.42 | 12.70 166.87
1.15
2.04
0.58
0.91
1.31
2.33 .01
3.63 .01
·
0.65
1.02
1.47
2.62.01
4.09 .01
Y
0.73
1.14
1.64 .01
2.91 .01
4.54 .02
··
L.T.
2.33 66.68
3.63
83.37
49.99 | .04 99.95
62.48.05 | 124.90
74.97 .06 149.83
99.93.08 | 199.61
100.06
5.23
9.30 133.47
124.87.10 | 249.24 | 14.51 | 166.93
49.99 .05 99.94 2.62 66.69
62.48 .06 | 124.88 4.09 83.38
74.96.08 149.79 5.88 100.07
99.92 .10 | 199.51
199.51 | 10.45 | 133.51
124.84 .13 249.04 16.32 167.00
|
| |
D=10°
49.99 .06 99.92
62.48 .08 124.85
74.96 .10 149.74
99.90 .13
124.80.16
S.T.
|
149.83.09 298.99 18.28 200.35 100.32 299.55
| | |
174.73.11 | 348.40 | 24.86 233.89 117.18 349.29
|
199.60.12 397.62 | 32.44 | 267.51 | 134.10 398.94
224.43
224.43.14 | |
.14 446.61 | 41.01 301.20 151.09 448.49
|
249.22.15 495.36 | 50.56 | 334.98 168.16 | 497.93
|
D=8°
5.23.01
7.11.02
9.28.02
11.74 .03
149.78.12 298.69 20.88 200.46 100.42 | 299.42
174.65.14 347.92 | 28.38 | 234.07 117.33 | 349.07
| |
199.48 .16 | 396.89 | 37.03 | 267.76 | 134.33 398.62:
224.26.18 445.58 46.79 | 301.57 151.42 | 448.03
14.48 .04 | 248.99 | .20 | 493.94 | 57.68 | 335.49 | 168.63 497.30
D=9°
L.C.

1
33.35 99.98
41.69 124.97
50.04 149.94
66.76 199.87
83.52 249.74
2.91 66.69
4.54 83.39
6.54 100.09
199.39 11.61 133.55
| |
248.81
248.81 18.12 | 167.08
|
5.88 .02 149.72 .15 298.34 | 23.47200.58 100.53 299.26
|
7.99 .02 174.56.18 347.36 31.90 234.26 | 117.51 | 348.83
| |
| |
10.43 .03 | 199.34 .20 396.07 41.59 268.06 134.60 398.25
13.19.04 224.07.23 | 444.41 | 52.54 301.99 151.81 447.51
| | |
16.27.05 248.72.25 492.34 64.73 | 336.07 169.15 496.58
|
|
|
33.35 99.98
41.70 124.96
50.05 149.93
66.79 199.83
83.58 249.66.
33.35 99.97
41.71 124.95.
50.07 149.91
66.82 199.78
83.64 249.57
33.36 99.97
41.71 124.93
50.08 149.89
66.86 199.73
83.71 249.47
297.95 26.05 200.72 100.66 299.09
| | | |
346.75 35.40 | 234.48 117.71 348.55
| | |
300 15 00.0
350 17 30.0
400 20 00.0
450 22 30.0
500 25 00.0
6.53 .02 149.66.19
8.88 .03 174.46 .22
|
11.58 | |
.04 199.19.25
14.64 .06 223.85.28
| |
18.06.07 248.42.31
395.15 46.14 268.39 134.90 397.84
| | | |
443.11 58.26 302.46 | 152.24 | 446.92
|
490.56 71.74 336.72 169.75 495.78
| | | |
|
*Subtract from tabulated value for Da when chord definition of D is used.
418
Ls
Δ
4°24.0'
80
100
5 30.0
125 6 52.5
150 8 15.0
200 11 00.0
250 13 45.0
300 16 30.0
35019 15.0
400 22 00.0
500 27 30.0
4°48.0'
80
100
6 00.0
125 7 30.0
150 9 00.0
200 12 00.0
250 15 00.0
300 18 00.0
350 21 00.0
400 24 00.0
500 30 00.0
80
5°12.0'
100
6 30.0
125 8 07.5
150 9 45.0
200 13 00.0
250 16 15.0
300 19 30.0
350 22 45.0
400 26 00.0
500 32 30.0
80 5°36.0'
100 7 00.0
125 8 45.0
150 10 30.0
200 14 00.0
Da
250 17 30.0
300 21 00.0
350 24 30.0
400 28 00.0
500 35 00.0
0
TABLE XI.—SELECTED SPIRALS
*
0.51
0.80
·
1.25 .01
1.80 .01
3.20 .01
• · • •
4.99 .02
7.18 .03
9.76 .04
12.73 .06
19.84 .09
0.56
0.87
1.36 .01
1.96 .01
3.49.02
··
5.44.03
.03
7.82.04
· ··
0.60
0.94 .01
1.48 .01
2.13.01
3.77 .02
Χο
Da l *
5.89 .04
8.47.05
X
0.64
1.02 .01
1.59 .01
2.29 .02
4.06 .03
D=11°
39.99.06 79.95
49.98 .08 99.91
62.47 | .10 | 124.82
74.95.12 | 149.69
99.88 .15 199.26
D=12°
39.99 | .07 79.94
49.98 .09 99.89
62.46.11 124.79
74.94.14 | 149.63
99.85.18 199.12
Y
L.T.
D=13°
39.99.09 79.93
49.98 .11 99.87
62.45.13 124.75
74.93 .16 149.57
99.83.21 198.97
2.05 53.35
3.20
66.70
5.00
83.40
7.19
12.77
19.92
124.76.19 248.56
167.17
149.59.23 297.52 28.63 | 200.88
174.34 .26 346.07 38.88 234.73
199.02 .30 394.14 50.66 268.76
248.09 .37 488.60
337.45
78.69
100.11
133.59
2.23 53.35
3.49 66.70
5.45 83.41
7.84 100.13
13.92 | 133.64
124.72.23 248.29 21.71 167.27
| |
83.88
149.51 .27 | 297.05 | 31.19 | 201.04 | 100.95
100.95
10.64 .06 | 174.22.31 | 345.33 | 42.35 | 235.00 | 118.18
118.18
13.88.08 198.84.36
198.84 .36 393.04 55.16 269.16
| | | 135.60
135.60
21.60.12 247.73.44 486.46 85.57 338.25 171.14
| |
|
S.T.
L.C.
26.68 | 79.98
33.36 99.96
41.72 124.92
50.10 149.86
66.90 199.67
39.99.10 79.93
49.98.12 99.85
62.45.16 124.71
74.92 .19 | 149.50
99.80.25 198.81 | 16.22 | 133.75
83.79 249.36
100.80 298.90
117.94 | 348.25
135.23 | 397.39
170.41 494.90
26.68 79.98
33.37 99.95
41.74 124.90
50.12 149.84
66.95 199.61
124.67 .27 248.00 23.50 167.37 83.98 249.11
149.42 .32 296.54 33.75 201.23 101.12 | 298.46
11.52 .07 174.09 .37 344.52 45.80 235.29
| | | 118.45
118.45 347.55
15.01.10 198.63.42 391.84 59.62 | 269.60
|
| 136.00
136.00 396.35
23.36.15 |
.15 247.34 | .51 | 484.15 92.39 339.13 | 171.95 492.89
|
D=14°
249.24
298.69
347.91
396.89
493.93
2.42 53.36 26.69 79.97
3.78 66.71 33.37 99.94
5.90 83.42 41.75 124.89
8.49 100.15 50.14 149.81
15.07 133.69
|
67.00 199.54
2.56 53.36 26.69 79.97
4.07 66.72 33.38 99.93
6.35 83.44 41.76 124.87
9.14 100.18 50.16 149.78
67.05 199.47
124.61.31
247.68 25.28 167.49
84.08 248.96
6.34 .05
|
9.12 .07 149.33 .37 296.00 36.30 201.43 101.30
| | |
12.39 .09 173.94 .42 343.65
|
16.15.12 198.42 .48 390.55
|
25.12.18 246.92.59 481.66
298.21
347.16
49.24 235.61 118.74
| |
64.06 270.08 136.44 395.77
| |
99.13 340.09 172.83 491.76
| |
|
*Subtract from tabulated value for Da when chord definition of D is used.
419
Ls
Δ
80 6°00.0'
100 7 30.0
125 9 22.5
150 11 15.0
200 15 00.0
250 18 45.0
300 22 30.0
350 26 15.0
400 30 00.0
500 37 30.0
80 6°24.0'
100 8 00.0
125 10 00.0
150 12 00.0
200 16 00.0
80 7°12.0'
100 9 00.0
125 11 15.0
150 13 30.0
200 18 00.0
250 22 30.0
300 27 00.0
350 31 30.0
400 36 00.0
500 45 00.0
TABLE XI.-SELECTED SPIRALS
60 6°00.0'
80 8 00.0
100 10 00.0
125 12 30.0
150 15 00.0
0
Da | *
0.70 .01
1.09.01
1.70 .01
2.45 .02
4.35 .04
6.79.06
9.76 .08
13.26 .11
17.28 .15
26.86 .22
250 20 00.0
300 24 00.0
350 28 00.0
400 32 00.0
500 40 00.0 | 28.59 .27
0.74 1.01
1.16 .01
1.82 .02
2.61 .03
4.64 .05
0.84 1.01
1.31 .02
2.04.03
2.94 .04
5.22 .06
8.14 10
11.69.14
.14
15.86.19
20.65 | .25
32.02 .38
Χο
0.52 | .01
0.93 .01
1.45.02
2.27 .03
3.26.05
Da
*
X
D=15°
7.24 .07 124.50 .40 246.97
10.41.10 149.13.48 294.78
|
14.13.14 | 173.62 | .55 341.73
|
18.41 .18 197.94
197.94.62 | 387.70
246.00 .75 476.18
| |
D = 18°
124.56.35 247.34
|
149.23.42 295.41
173.78 .48 342.72
198.18.55 389.17
|
246.48 .67 479.00
|
D=16°
39.99.11 79.91 2.79 53.36 26.69 79.96
49.97.14 99.83 4.36 66.73 33.39 99.92
62.44.18 | 124.67 6.80 83.45 41.77 124.85
74.90.21 149.42 9.79 100.20 50.18 149.74
99.77 .28 | 198.63 17.37 133.82 67.11 199.39
39.98.13 79.90
49.97.16 99.80
62.44.20 | 124.62
74.89 .24| 149.34
99.74 .32 | 198.45
39.98.15 79.87
49.96.21 99.75
62.42 .26 | 124.52
74.86 .31 | 149.17
99.67.41 | 198.04
Y
124.36.50
148.90.60
173.25.69
197.40 .77 384.50
244.95 .93 470.02
|
D=20°
L.T. S.T. L.C.
29.99 .15 59.93
39.97 .20 79.84
49.95 25 99.70
62.40.31 | 124.41
74.83 .38 148.98
|
27.06 167.61
38.84 201.64
52.65 235.95
68.46 270.60
105.79 341.14
2.98] 53.37
4.65 66.74
7.26 83.47
10.44 100.23
26.70 79.96
33.40 99.91
41.79 124.83
50.21 149.71
18.51 133.88 67.17 199.31
246.17 32.36 168.03
293.41 46.38 202.38
339.57 62.77 237.14
81.44 272.40
125.24 344.78
84.19 248.81
101.49 297.95
119.05 | 346.75
136.92 395.15
|
173.78 490.55
|
28.84 167.74 84.31 248.65
41.37 201.87 101.70 297.67
56.05 236.32 119.39 346.30.
72.82 271.16 137.43 394.48
112.36 342.26 174.81 489.25
3.481 53.38
5.23 66.75
8.16 83.50
11.73 100.29
20.80 134.03
26.71 79.94
33.41 99.89
41.82 124.79
50.27 | 149.63
67.30 199.12
84.58 248.29
102.16| 297.05
120.13 345.32
|
138.56 | 393.03
177.12 486.43
20.02 59.97
79.93
99.86
2.09 | 40.02
3.72
53.39 26.72
5.80 66.77
9.06 83.54
13.03 100.36
|
33.43
41.86 124.74
50.33 149.54
200 20 00.0
250 25 00.0
300 30 00.0
350 35 00.0
400 40 00.0
5.79.09 99.60 .50 197.58 23.07 134.19
9.03.14 124.21 .62 245.28 35.87 | 168.36
12.96 | |
.20 148.64 .73 291.88 51.34 202.95
17.58 .27 172.85 .84 337.16 69.39 238.06
|
22.87.34 196.80.94 380.94 89.89 273.81
| | |
67.45 198.92
84.87 247.89
102.69 296.36
|
120.98 344.23
139.85 391.40
.34
* Subtract from tabulated value for Da when chord definition of D is used.
420
Δ
0.0
.I
.2
~33
.3
.4
0.5
.6
.7
.8
.9
1.0
.1
.2
.3
.4
1.5
.6
.7
.8
.9
IME 5
.3
.4
2.0.00291
.1
.2
2.5
.6
.7
.8
.9
~~+ Dorog 0-234
.4
3.5
.6
.7
.8
.9
Da
.00000
015
029
044
058
.00073
.2
088
102
117
131
3.0.00435
. I
.2
.3
.3
.00146
161
175
190
204
.4
.00219
233
248
262
277
305
320
334
349
.00363
377
392
406
421
4.0.00581
450
464
479
493
.00508
523
537
552
566
596
610
625
639
4.5
.6
.7
.8
.9
5.0 .00727
.00654
669
683
698
712
TABLE XII. SPIRAL FUNCTIONS FOR LS=1

⭑
ဗွီ8888 §8888 ဗြိ8888 ဗွီ8888 ဗြိ8867 gမံ ဂ္ယီဝ
.0000
.0000
.0000
00
.0001
.0000 |.49999
.0001
01
01
01
ខ្ញុំ 88 ធ្នូ8888 g8888
.0001
02
.0002
Do
.50000.0000
000
000
000
000
.0002
Xo
.50000
000
000
000
000
999
998
998
998
858 88822
998
997
997
997
01
03
04
05
.0006
08
.49999.0013
999
999
999
999
09
10
14
15
17
18
.0019
20
DONNE
22
23
24
.49998.0025
22223
27
28
29
31
.49997.0032
996
996
996
996
33
34
36
37
OM
JŌ.
.49995.0038
995
994
994
994
39
41
42
43
.49994.0045
993
993
993
992
46
Anuo nuuko 855
.49992.0051
991
991
991
990
.49990.0057
989
989
988
988
0003 ..49987
62
0064
X
Y
1.00000.00000
1.00000
1.00000
.99999
.99999
058
116
175
233
.99999.00291
999
998
998
997
349
407
465
524
.99997.00582
996
995
995
994
640
698
756
814
.99993.00873
992
991
989
.99988.01163
987
985
984
982
931
989
990.01047
105
222
280
338
396
.99981.01454
979
978
976
975
512
571
629
687
.99973.01745
971
969
967
965
803
861
919
978
.99963.02036
961
958
956
953
094
152
210
268
.99951.02326
948
946
943
941
384
443
501
559
L.T.
.66667
667
667
667
667
*
.66668
668
668
669
669
.66667 |.33334
667 334
668
334
668
334
668
334
.66669
669
670
670
671
.66671
671
672
672
673
.66676
677
678
678
679
.66680
681
681
682
683
S. T.
.66684
685
686
686
687
.99938.02617
935 675
.66688
689
733
690
932
930
791
691
927
849
692
.99924.02907 1.66693
.33333 1.00000
333.00000
1.00000
1.00000
1.00000
334
334
334
.33334
335
335
335
335
.66673.33339
674
675
675
676
.33336
336
336
337
337
.33337
338
338
339
339
340
340
341
341
.33342
343
343
344
345
.33345
346
347
347
348
.33349
350
350
351
352
.33353
354
355
L.C.
356
357
.33358
1.00000
1.00000
.99999
999
999
.99999
998
998
998
997
99997
997
996
996
995
.99995
994
993
993
992
.99992
991
990
990
989
.99988
987
986
985
984
.99983
982
981
980
979
.99978
977
976
975
974
.99973
971
970
969
967
.99966
When chord
* Multiply functions (except * values) by the given value of Ls.
definition of degree of curve (De) is used, correct the calculated values of
o or Xo by subtracting the product of De and the values. See page 410 for
example.
421
1°
5.0
.I
.2
NME
.3
.4
5.5.00800
.6
.7
.8
.9
*
6.0
.I
.2
234
.3
.4
.6
.7
.8
.9
6.5.00945
.I
.2
.3
.4
7.5
.6
.7
.8
.9
8.0
.I
234 567BG
.2
.3
.4
8.5
.6
.7
.8
.9
.I
23=
7.0.01018 .0005
.2
.3
.4
Da
.00727
742
756
771
785
5678 M
814
829
843
858
.8
.9
10.0
.00872
887
901
916
930
9.0.01308
960
974
989
.01003
033
047
062
076
.01091
105
120
134
149
.01163
178
192
207
221
9.5 .01381
.6
.7
.01236
250
265
279
294
TABLE XII.-SPIRAL FUNCTIONS FOR L:=1

323
337
352
366
395
410
424
439
.01453
*
.0003
g838 83ဝတီတီ ဗွီ388 ဗွီ8888 g8888 ဗွီ ဒွ588 ဗွီဒီ888 g882-
.0004
.0006
Da
.0003.49985.0070
984
984
Xo
.0009
.0010
10
10
||
||
0011
.49987.0064
987
986
986
985
983
983
.0007 .49967
966
965
965
964
⭑
.49982.0076
981
981
980
979
.0008 .49963
65
66
67
69
.49979.0083
978
977
976
976
962
961
961
960
71
73
74
75
958
957
956
955
78
79
.49954
953
952
951
950
.49949
80
81
.49975.0089
974
973
973
972
84
85
86
.49971
970
969
969
968 .0100
ww
88
90
91
93
94
.0095
97
98
99
.0102
DE BON FOND ONIO
03
04
05
07
.49959.0114
.0108
09
10
12
13
16
17
18
19
.0120
22
23
24
26
.0127
X
.99924
921
914
911
918.03023
.99908
904
901
897
894
886
882
879
875
.99871
867
863
859
.99829
824
819
815
810
Y
800
795
790
785
.99780
775
.02907.66693
965
855.04010
082
140
764
759
.99890.03488.66705 .33368
546
369
604
371
662
372
720
373
748
742
737
731
372
430
.03198.66699
256
314
719
713
708
702
.99696
.03778
836
894
952
.99851 .04068.66719
846
842
838
833
126
184
L.T.
242
300
.04358
416
474
694
696
697
698
532
590
700
701
703
704
04937
995
706
708
709
710
.66712
713
715
716
717
720
722
724
725
.66727
728
730
732
733
.99805.04648.66735 .33395
706
397
764
399
822
400
879
402
737
738
740
742
.66744
745
747
749
751
S. T.
755
757
759
761
.33358
359
360
361
362
765
767
.33363
364
365
366
367
769
771
770.05053
…
169
.99754.05227.66753 .33412
285
414
342
416
400
417
458
419
.33374
376
377
378
380
.33381
382
384
385
386
.33388
389
391
392
394
.99725.05516.66763 .33421
574
632
690
747
.05805.66773
.33403
405
407
409
410
423
425
427
428
33430
L.C.
.99966
965
963
962
961
.99959
958
956
954
953
.99951
950
948
946
944
.99943
941
939
937
936
.99934
932
930
928
926
99924
922
920
918
916
.99913
911
909
907
904
.99902
900
897
895
893
.99890
888
885
883
880
.99878
875
873
870
867
.99865
When chord
Multiply functions (except * values) by the given value of Ls.
definition of degree of curve (De) is used, correct the calculated values of
o or Xo by subtracting the product of Dc and the values. See page 410 for
example.
422
10.0
1
.2
IMI SO760 0
.3
.4
10.5
.6
.7
.8
.9
11.0
. I
.2
22 567∞a
.3
.4
11.5
.6
.7
.8
.9
12.0
.│
.2
.3
.4
12.5
.6
.7
.8
.9
13.0
:|
.2
.3
H BOT∞0
.4
13.5
.6
.7
.9
.I
.2
.3
.4
567
Da
.01453
468
482
.6
497
511
.7
TABLE XII.-SPIRAL FUNCTIONS FOR LS=1

.01526
540
555
569
584
.01598
613
627
642
656
.01671
685
700
714
729
.8 .02003
018
.01743
757
772
786
801
.01815
829
844
858
873
14.0.02032
.01887
902
916
931
945
046
061
14.5.02104
075
090
118
133
147
162
★
.8
.9
15.0 .02176
.001 1
IL
~~~ ~~MMM
12
12
12
.0012
12
13
13
13
.0013
14
14
14
14
5555 O
.0015
15
.01960.0020
974
989
15
15
16
.0016
16
16
17
17
.0017
18
18
18
18
.0019
19
19
20
20
20
21
21
21
.0022
22
22
23
23
Xo
Da
.49949
948
947
946
945
.49944
943
942
94-1
940
.49939
938
937
935
934
.49933
932
931
929
928
.49927
926
924
923
922
.49921
919
918
917
915
.49914
913
911
910
909
.49908
906
905
904
902
*
.0127
28
29
2006 CXFUN DE555 £55 wwwww wwnn
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.0133
34
36
37
38
.0139
43
44
.0146
48
51
.0152
55555
53
54
56
57
.0158
59
63
.0164
65
67
68
69
.0170
72
73
74
75
.49901.0177
900
898
897
895
78
79
80
82
.0023
.0183
.49894
892
24
84
24
891
85
24
889
86
25
888
88
.0025 .49886 .0189
X
.99696
690
684
677
671
.99632
625
618
612
605
.99665.06094
658
652
645
639
547
540
532
.99525
517
509
502
494
.99486
478
470
462
454
.06671
.99598
591
729
584
787
576
844
569
902
.99562 .06959
555.07017
.99446
438
430
421
413
Y
.99405
396
387
379
370
99362
353
344
335
326
.99317
.05805.66773
863
921
978
.06036
152
210
267
325
556
614
.06383.66796
440
498
075
132
190
.07248
305
363
420
478
L.T.
.07823
880
938
995
.08053
776
778
780
782
.66784
787
789
791
794
168
225
282
340
798
801
803
806
.66808
811
813
816
818
.66821
823
826
828
83!
.66834
836
839
.07535.66847
593
650
708
765
842
845
850
853
856
859
.66862
865
868
871
874
S.T.
880
883
886
889
.33430
432
434
436
438
.33440
442
444
447
449
.33451
453
455
457
460
.33462
464
466
469
471
.33473
476
478
480
483
33485
488
490
493
495
.33498
500
503
505
508
.08110.66877 .33524
527
530
532
535
.33511
513
516
519
521
T
08397
.66892
455
895
512
898
569
901
627
904
08684 66908 .33553
.33538
541
544
547
550
L.C.
.99865
862
859
856
854
99851
848
845
842
839
.99836
833
830
827
824
.99821
818
815
812
808
.99805
802
799
795
792
.99789
785
782
778
775
.99771
768
764
761
757
.99753
750
746
742
739
.99735
731
727
723
720
99716
712
708
704
700
.99696
When chord
* Multiply functions (except * values) by the given value of Ls.
definition of degree of curve (Dc) is used, correct the calculated values of
o or Xo by subtracting the product of Dc and the values. See page 410 for
example.
*
423
4°
15.0
.2
.3
NAI Đi
.4
15.5.02248
.6
.7
.8
.9
16.0
.1
.2
.3
.4
16.5
.6
.7
.8
.9
17.0
.I
.2
*
.3
VMI S
.4
.6
.7
.8
.9
18.0
.I
.2
.3
4
•
18.5
.6
.7
∞ a
.8
.9
19.0
I
.2
VMI S6
.3
.4
19.5
Da
.02176
190
205
219
234
.6
.7
.8
.9
20.0
262
277
291
306
TABLE XII.-SPIRAL FUNCTIONS FOR L.=1

.02320
335
349
364
378
.02393
407
422
436
451
.02465
479
494
508
522
17.5.02537 | .0034
551
565
579
594
.02608
622
637
651
666
.02680
694
709
723
738
.02752
766
781
795
810
*
.02824
838
853
867
882
.02896
.0025
25
26
26
26
27
27
28
28
.0028
29
02228
29
.0026.49878 | .0195
29
30
.0030
30
31
31
32
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32
33
33
33
34
35
35
55
35
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36
37
37
37
12
.0032.49854 | .0213
.0040
40
41
41
41
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Do
.0042
42
Xo
43
43
44
.0044
.49886.0189
884
883
881
880
876
875
865
864
873
872 .0200
.49870.0201
868
867
.0038.49827
38
39
39
39
*
*
852
850
849
847
.49845
843
841
840
838
.49862.0207
860
859
857
856
.49836
834
832
830
828
90
91
93
94
*ឥតខ្ញុំ ផ្លូវ88 គ្ន8---
825
823
821
819
.49808
806
804
04
35678
15
16
17
18
.0220
21
22
23
24
.0226
27
28
29
30
.49817.0238
815
813
811
809
.0232
33
34
35
36
39
40
41
42
.0244
45
46
47
48
802
800
49798 .0250
X
308
299
289
280
.99317.08684.66908
741
799
856
913
911
914
918
921
.99271.08970 | .66924
261 .09028
928
252
085
931
242
142
934
233
200
938
213
203
194
184
164
154
143
133
.99123
113
102
092
Y
.99223.09257.66941 .33583
314
586
371
589
428
593
485
596
060
050
039
029
.99018
007
.98996
985
974
.98962
951
940
929
917
.99174.09543.66959 .33599
600
602
657
606
714
609
771
612
.98906
894
883
871
860
.98848
836
824
812
800
.98788
.09828
885
942
999
081.10056
.9907.10113.66995
170
L.T.
284
341
.10398
455
512
569
626
945
948
952
955
.10967
.11024
962
966
970
973
227.67003
081
138
194
.66977
981
984
988
992
999
007
011
.10683.67035
740
797
854
910
.67015
019
023
027
031
039
043
047
051
.67055
059
063
067
072
.11251.67076
308
364
421
478
.11535
S. T.
080
084
.33553
556
559
089
093
.67097
561
564
.33567
571
574
577
580
.33615
619
622
626
629
.33632
636
639
643
646
.33650
654
657
661
664
.33668
672
675
679
683
.33687
690
694
698
702
.33706
709
713
717
721
.33725
L.C.
.99696
692
688
683
679
.99675
671
667
662
658
.99654
649
645
641
636
.99632
627
623
618
614
.99609
605
600
595
591
.99586
581
576
572
567
99562
557
552
547
542
.99537
532
527
522
517
.99512
507
502
497
491
.99486
481
476
470
465
.99460
When chord
Multiply functions (except * values) by the given value of Ls.
definition of degree of curve (De) is used, correct the calculated values of
o or Xo by subtracting the product of Dc and the values. See page 410 for
example.
津
​424
4°
20.0
. I
.2
.3
233 5
.4
20.5
.6
.7
.8
.9
21.0
./
.2
IMI SOT∞G O-~Mª 5678
.3
.4
21.5
.6
.7
·8
.9
22.0
.1
.2
.3
.4
22.5
.6
.7
.8
.9
MI NO7∞
.3
.4
23.5
.6
.7
.8
23.0.03326
.I
.2
.9
24.0
.1
.2
.3
.4
567
24.5
.6
.7
Do
.02896
910
925
939
954
.8
.9
25.0
.02968
982
997
.03011
026
.03111
125
140
154
169
.03183
197
211
226
240
.03254
268
283
297
312
340
354
369
383
.03469
483
497
512
526
TABLE XII.-SPIRAL FUNCTIONS FOR Ls=1

⭑
.03040.0048
054
068
083
097
.03540
554
568
583
597
.03611
.0044
44
45
C0555 55556 5&6EE
45
46
.0046
47
47
48
48
49
49
An gaana
53
.0051.49766
51
52
5555555
57
58
.0053.49755
54
54
98888
.03397.0061
411
426
440
455
60
60
Xo
Do
.49798
796
794
791
789
.0056 .49745
743
740
738
735
61
62
62
63
.49787
785
783
781
779
.0063
64
64
.49777
775
773
770
768
.0058.49733
59
59
65
65
764
762
759
757
753
751
749
747
•
731
728
726
723
.49721
719
716
714
711
49709
707
704
702
699
.0066 .49697
66
694
67
692
67
689
68
687
.0068 .49684
*
.0250
51
52
53
54
.0256
57
58
59
60
.0262
63
3338 GOREN PAPER 2.0¤ã
64
.0267
.0285
86
88
89
90
.0291
92
93
95
96
.0297
98
99
.0300
02
.0303
04
05
06
07
.0309
X
.98788
776
764
752
740
.98728
715
703
690
.0273 .98536
523
509
496
482
.98601
588
575
562
549
678.12044
.98665 .12101
652
639
627
614
.98469
455
442
428
415
.98401
387
· 373
359
345
.98331
316
302
288
274
.98260
245
231
216
202
Y
.98187
172
157
143
128
.98113
.11535
591
648
705
761
.11818
875
931
988
157
214
270
327
.12383
439
496
552
609
.12665
721
777
834
890
.12946
.13002
059
115
172
.13508
564
621
677
733
L.T.
845
901
957
.14012
.67097
102
106
|||
115
.14068
124
180
236
292
.14348
.67119
124
128
133
137
.67142
147
151
156
161
.67165
170
175
180
184
.13228.67238
284
340
396
452
.67189
194
199
204
208
.67213
218
223
228
233
243
248
254
259
.67264
269
274
280
285
S.T.
.67317
322
328
333
339
67344
.33725
729
733
737
741
.33745
749
753
758
762
.33766
770
774
779
783
.33787
791
796
800
804
.33809
813
818
822
826
.33831
835
840
844
849
.33854
858
863
868
872
.33877
882
886
891
896
906
910
915
920
L.C.
.33925
930
935
940
945
.33950
.99460
454
449
443
438
.99432
427
421
415
410
.99404
399
393
387
381
99376
370
364
358
352
.13789.67290.33901 .99222
295
216
301
209
306
203
311
196
.99346
340
334
328
322
.99316
310
304
298
292
.99286
279
273
267
261
.99254
248
242
235
229
.99190
183
177
170
163
.99157
* Multiply functions (except * values) by the given value of Ls. When chord
definition of degree of curve (De) is used, correct the calculated values of
o or Xo by subtracting the product of Dc and the * values. See page 410 for
example.
425
1°
25.0
.1
.2
~M 5O7∞.
.3
.huf
25.5
.6
.7
.8
.9
26.0
.1
.2
.3
.4
5678σ O
.6
.7
.8
26.5.03825
.9
27.0
.I
233 56
.2
.3
.4
27.5
.6
.7
.8
.9
28.0
.I
.2
.3
.4
28.5
.6
.7
.8
.9
o
29.0
.1
233
.2
.3
.4
5678
29.5
.6
.7
Da
.03611
625
640
654
669
.8
.9
30.0
.03683
697
711
725
739
.03753
767
782
796
811
839
853
868
882
910
924
939
953
.03967
981
995
.04009
023
.04108
122
136
151
165
.04179
193
207
222
236
TABLE XII.—SPIRAL FUNCTIONS FOR Ls=1

⭑
.04250
264
278
293
307
.04321
.0068
69
69
70
71
.0071
72
72
73
73
ZIPPO FRO
.0074
74
75
76
.03896 .0080
76
.0077
77
78
78
79
.04037 .0085
051
86
065
87
080
094
80
81
81
82
.0082
83
84
84
85
88888
87
88
.0088
89
90
90
91
.0092
92
93
93
94
.0095
95
96
97
97
Xo
Da
.49684
681
679
676
674
.49671
668
666
663
661
.49646
643
640
638
635
.49632
629
626
624
621
.49618
615
613
610
608
.49658.0320
656
653
651
648
.49590
587
584
582
579
•
.49576
573
570
567
564
⭑
.0309
10
11
12
13
.0314
=
15
17
18
19
2~~~~
21
22
23
25
.0326
27
28
29
30
.0331
33
34
.49605.0343
602
599
596
593
*
35
36
.0337
38
39
40
42
44
45
46
47
.0348
997nn kunni göðbó
.0354
.49561
558
555
552
549
64
.0098 49546 .0365
.0359
62
X
.98038
022
007
.97991
976
.98113.14348 | .67344
098 404 350
083
459
355
068
515
361
053
571
366
945
929
913
898
.97882
866
850
834
818
.97721
704
688
671
655
Y
.97802.15461
786
770
753
737
621
604
588
571
.97960.14905.67400 .34001
961
.15016
072
128
.97554
537
520
503
486
.14627
682
738
794
849
.97469
452
434
417
399
.97382
364
346
329
311
.97293
.15183
239
294
350
405
516
571
627
682
.15738
793
848
903
959
L.T.
180
235
.16290
345
400
455
510
.67372.33975
378
383
389
395
406
412
418
424
.67430
435
441
447
453
.67459
465
471
477
483
.67490
496
502
508
514
.97638.16014.67520 .34111
069
116
124
122
128
134
527
533
539
546
.67552
558
565
571
578
S.T.
.16565.67584
620
675
730
785
.33950
955
960
965
970
591
597
604
610
.16840
.67617
895
623
630
950
.17005
637
060
643
17114 .67650
981
986
991
996
007
012
017
023
.34028
033
039
044
049
.34055
060
066
071
077
.34083
088
094
099
105
.34139
145
151
157
163
34169
165
171
177
193
.34199
205
211
217
223
.34229
L.C.
.99157
150
143
136
129
.99123
116
109
102
095
.99088
081
074
067
060
99053
046
038
031
024
.99017
009
002
.98995
987
.98980
973
965
958
950
.98943
935
928
920
913
98905
897
890
882
874
98866
859
851
843
835
.98827
819
811
803
795
.98787
* Multiply functions (except values) by the given value of Ls. When chord
definition of degree of curve (De) is used, correct the calculated values of
o or Xo by subtracting the product of Dc and the values. See page 410 for
example.
426
4°
30.0.04321
./
.2
.3
.4
233 56
30.5
.6
.7
.8
.9
31.0
..
.2
0123I SOT
.3
. Ц
31.5
.6
*
.7
.8
.9
32.0
. |
.2
.3
.4
32.5
I 56
.6
.7
.8
·9
33.0
. |
.2
234 56
.3
.4
33.5
•
.6
.7
.8
.9
34.0
.1
.2
.3
•
UMI 56
Do
.4
TABLE XII.-SPIRAL FUNCTIONS FOR Ls=1

335
349
363
377
.04391
405
419
434
448
.04462
476
490
504
518
.04532
546
560
574
588
.04602
616
630
645
659
.04673
687
701
715
729
.04743
757
771
785
799
.04813
827
841
855
869
.04883
897
911
925
939
04953
967
981
995
*
.0098
98
99
.0100
88 ㅇㅇㅇ ​00880 88
00
.0101
02
03
04
.0104
05
06
06
07
09
10
10
1110*
-~~~# I5OON
12
12
13
14
.0108.49500
15
16
16
17
.01 18
18
19
20
21
.0121
22
23
23
24
.0125
26
26
27
28
34.5
.6
.7
.8
.9 .05009
35.0 .05023 .0132
.0128
29
30
31
31
Xo
Da
.49546
543
540
•
537
534
49531
528
525
522
519
.49516
513
510
506
503
497
494
490
487
.49484
481
478
475
472
466
462
459
455
49452
449
445
442
438
.49435
432
429
425
422
49419
415
412
408
405
.49401
398
395
*
.0114.49469 | .0392
391
388
49385
.0365
66
67
600 INNER FOR
68
69
.0371
72
73
74
75
.0376
77
78
79
80
.0381
83
84
85
86
.0387
88
89
90
91
93
94
95
97
.0398
99
.0400
01
02
.0403
04
05
06
07
.0408
09
10
[1
235
12
.0413
15
16
17
18
.0419
X
.97293
275
257
239
221
.97203
185
167
148
130
057
038
.97112.17661
094
075
907
887
868
849
Y
.97020.17934
001
.96982.18043
963
944
.96733
713
694
674
655
.17114
169
224
279
333
.17388
443
498
552
607
.96536
516
496
475
455
96435
414
716
770
825
879
.96830.18478
811
791
772
752
394
373
353
.96332
988
097
152
.96635.19019
615
595
576
556
532
586
640
694
.18748
803
857
911
965
073
127
181
234
.96926.18206.67790 .34356
260
363
315
370
369
376
424
383
.19288
342
396
450
504
.19557
611
665
718
772
.19826
L.T.
.67650
657
664
*
670
677
.67684
691
698
705
712
.67719
726
733
740
747
.67754
761
768
775
783
797
804
812
819
.67826
834
841
849
856
.67863
871
878
886
894
.67901
909
916
924
932
.67939
947
955
963
971
.67979
987
994
S.T.
.68002
010
.68018
.34229
235
241
248
254
.34260
266
273
279
285
.34292
298
304
311
317
.34324
330
337
343
350
.34390
397
403
410
417
.34424
431
438
444
451
.34458
465
472
479
486
.34493
500
508
515
522
.34529
536
544
551
558
34565
L.C.
.98787
779
771
763
755
.98746
738
730
722
713
.98705
697
688
680
672
.98663
655
646
638
629
.98621
612
603
595
586
.98577
569
560
551
542
.98534
525
516
507
498
.98489
480
471
462
453
.98444
435
425
416
407
.98398
389
379
Multiply functions (except
values) by the given value of Ls.
When chord
definition of degree of curve (De) is used, correct the calculated values of
values. See page 410 for
o or Xo by subtracting the product of Dc and the
example.
370
361
.98351
427
4°
35.0
.1
.2
.3
.4
35.5
5678σ o
.6
.7
.8
.9
36.0
.I
.2
.3
234
.4
36.5
.6
.7
.8
.9
37.0
·/
0-234 5
.2
.3
.4
37.5
.6
·
.7
.8
.9
38.0
.1
.2
.3
4
38.5
.6
.7
.8
.9
·23=
.2
.3
.4
39.5
.6
.7
.8
∞ a
.9
40.0
Da
.05023
037
051
065
079
TABLE XII.—SPIRAL FUNCTIONS FOR L,=1

177
191
204
218
.05232
246
260
273
287
.05301
39.0.05579
.I
.05093.0136
107
121
135
149
315
329
343
357
.05371
385
399
413
427
.05441
455
469
482
496
.05510
524
538
551
565
⭑
.05163.0139.49349
593
607
620
634
.0132
33
33
.05648
662
676
690
704
.05718
Ew wwwww wwww
34
35
36
37
38
39
40
41
42
42
.0143
44
45
45
46
.0147
48
49
49
50
.0151
52
52
53
54
.0155
56
56
57
58
.0159
59
60
61
62
.0163
63
64
65
66
.0167
68
68
Xo
69
70
.0171
Da
.49385
381
378
374
371
345
342
338
335
.49331
327
324
320
317
.49367.0424
363
360
356
353
.49313
309
306
302
299
.49295
291
287
284
280
..49256
252
249
245
242
⭑
.49238
234
230
226
222
.0419
2022 2020
21
23
25
26
27
28
.0429
30
31
32
33
.0434
35
36
37
38
.0439
40
41
42
43
.49276.0449
272
268
264
260
•
.0444
45
46
47
48
50
51
52
53
.0454
55
56
57
58
.0459
60
61
62
63
.49218.0464
214
210
207
203
49199
65
66
67
68
0469
X
.96332
311
291
270
.96229
208
187
250.20040
.20094
147
201
254
307
166
145
.96017
.95996
.96124.20361
103
081
060
038
974
953
931
.95800
778
756
734
712
.95690
668
645
623
601
.95578
556
533
Y
.95910.20893
888
866
844
822
511
488
.19826
879
933
987
.95466
443
420
397
374
.95351
328
305
281
258
.95235
414
467
521
574
.20627
680
734
787
840
946
999
.21052
105
.21158
211
264
317
370
.21423
475
528
581
634
.21686
739
792
844
897
.21949
.22002
054
107
159
.22212
264
316
369
421
.22473
L.T.
.68018
026
034
042
051
.68059
067
075
083
092
.68100
108
116
125
133
.68141
150
158
167
175
.68184
192
201
210
218
.68227
236
244
253
262
.68271
279
288
297
306
.68315
324
333
342
351
.68360
369
379
388
397
S.T.
34565
573
580
587
595
.34602
610
617
625
632
.34640
647
655
663
670
34678
686
693
701
709
.34717
725
732
740
748
.34756
764
772
780
788
.34796
804
812
820
829
.34837
845
853
861
870
.34878
886
895
903
911
.34920
928
937
945
954
.68406
415
424
434
443
68452 .34962
L.C.
.98351
342
333
323
314
.98304
295
285
276
266
.98257
247
237
227
218
.98208
198
188
179
169
.98159
149
139
129
119
.98109
099
089
079
069
.98059
049
038
028
018
.98008
.97997
987
977
967
.97956
946
935
925
914
.97904
893
883
872
861
.97851
values) by the given value of Ls.
When chord
* Multiply functions (except
definition of degree of curve (De) is used, correct the calculated values.of
o or Xo by subtracting the product of Dc and the * values. See page 410 for
example.
428
4°
40.0
.I
.2
.3
.4
40.5
.6
.7
.8
.9
41.0
.1
.2
.3
.4
4 567
41.5
.6
.7
.8
.9
42.0
} •
.2
.3
.4
Our
42.5
·6
.7
.8
.9
43.0
.]
.2
.3
.4
43.5
.6
.7
.8
.9
44.0
.I
.2
~34 5 6
.3
.4
44.5
•
.6
7
.8
.9
45.0
Do
.05718
732
745
759
772
.05786
800
814
827
841
.05855
869
883
896
910
.05924
938
952
965
979
.05993
.06007
020
034
047
075
089
102
116
212
226
239
253
TABLE XII. SPIRAL FUNCTIONS FOR Ls=1

.06267
281
294
308
321
*
.06335
.0171
72
72
349
362
376
389
06403
ZZZZZ ZZZñ
73
74
.0175
76
77
77
78
.0179
80
81
82
82
.0183
84
85
86
87
.0196
06130
144
97
157
98
171
99
184 .0200
.0187
88
89
90
91
33883 8588
Do
04
Xo
.0205
.49199.0469
195
191
187
183
.49179
175
171
167
163
.49138
134
130
126
122
.06061.0192.49097.0493
93
93
94
95
.49159.0479
155
151
146
142
105
101
093
088
084
079
⭑
.49118.0488
114
110
70
71
72
73
.06198.0200.49054 | .0503
.0474
75
76
77
78
028
024
019
015
.0209.49011
10
80
81
82
83
007
003
.0484
85
86
87
88
11
12 .48998
13
994
.0214 .48990
.49075
.0498
071
99
067 .0500
062
058
89
90
91
92
94
95
96
97
050
045
041
036
.49032 .0507
að 855 * § 8 =
.0512
13
14
14
15
.0516
X
.95235
212
188
165
141
95118
094
071
047
023
95000
.94976
952
928
904
.94880
856
832
807
783
.94759
734
710
685
661
.94636
612
587
562
538
.94388
363
337
312
287
Y
.94262
236
211
185
160
.22473
526
578
630
682
.22734
786
838
890
942
.22994
.23046
098
150
202
.23254
306
358
409
461
.23513
564
616
667
719
.23771
822
874
.94513.24028
488
463
438
413
925
976
079
130
182
233
.24284
335
387
438
489
.24540
591
642
693
744
.94134
108
082
057
031
94005 .25049
.24795
846
896
947
998
L.T.
.68452
462
471
481
490
.68500
509
519
528
538
.68547
557
567
577
586
.68596
606
616
626
635
68645
655
665
675
685
.68695
706
716
726
736
.68746
756
767
777
787
.68798
808
818
829
839
.68850
860
871
882
892
.68903
914
924
935
946
.68957
S.T.
.34962
971
980
988
997
.35006
014
023
032
041
.35049
058
067
076
085
.35094
103
| 12
121
130
.35139
148
158
167
176
.35185
194
204
213
222
.35232
241
250
260
269
.35279
288
298
307
317
.35327
336
346
356
365
35375
385
395
405
415
.35424
L.C.
.9785!
840
829
819
808
.97797
786
775
765
754
.97743
732
721
710
699
.97688
677
666
655
643
.97632
621
610
599
587
.97576
565
553
542
531
.97519
508
496
485
473
.97462
450
438
427
415
.97404
392
380
368
357
.97345
333
321
309
297
.97285
* Multiply functions (except * values) by the given value of Ls. When chord
definition of degree of curve (De) is used, correct the calculated values of
o or Xo by subtracting the product of Dc and the values. See page 410 for
example.
*
429
Δ
TABLE XIII.—COEFFICIENTS FOR CURVE WITH
EQUAL SPIRALS*

508508 508508 508508 508508 508508 508508 508508 508508 508508 508588
1 00
20
40
2 00
20
40
3 00
20
4.00
20
40
5 00
20
40
6 00
20
40
7 00
20
40
8 00
20
40
9.00
20
40
10 00
20
40
11 00
20
40
12 00
20
40
13 00
20
40
14 00
20
40
15.00
20
40
16.00
20
40
17 00
20
40
18 00
20
40
19.00
20
40
20 00
20
40
N
0.034906
0.046540
0.058173
0.069805
0.081435
0.093064
0.104691
0.116316
0.127938
0.139558
0.151175
0.162789
0.174400
0.186007
0.197611
0.209210
0.220805
0.232396
0.243982
0.255563
0.267139
0.278709
0.290274
0.301832
0.313358
0.324931
0.336471
0.348004
0.359530
0.371049
0.382560
0.394063
0.405558
0.417046
0.428524
0.439994
0.451455
0.462907
0.474350
0.485783
0.497206
0.508619
0.520022
0.531414
0.542795
0.554166
0.565525
0.576873
0.588209
0.599534
0.610846
0.622146
0.633433
0.644708
0.655969
0.667217
0.678452
0.689673
0.700881
0.712074
P
0.000203
0.000361
0.000564
0.000812
0.001106
0.001 444
0.001827
0.002256
0.002729
0.003248
0.003812
0.004421
0.005074
0.005773
0.006517
0.007305
0.008139
0.009017
0.009940
0.010908
0.011921
0.012979
0.014081
0.015229
0.016420
0.017657
0.018938
0.020264
0.021634
0.023049
0.024508
0.026011
0.027559
0.029152
0.030788
0.032469
0.034194
0.035963
0.037777
0.039634
0.041535
0.043480
0.045469
0.047502
0.049579
0.051699
0.053863
0.056071
0.058322
0.060616
0.062954
0.065335
0.067759
0.070227
0.072737
0.075291
0.077888
0.080527
0.083209
0.085934
M
1.000051
1.000090
1.000141
1.000203
1.000277
1.000361
1.000457
1.000564
1.000682
1.000812
1.000953
1.001106
1.001269
1.001444
1.001630
1.001827
1.002036
1.002255
1.002486
1.002728
1.002982
1.003247
1.003523
1.003811
1.004108
1.004419
1.004739
1.005072
1.005415
1.005770
1.006135
1.006511
1.006900
1.007300
1.007710
1.008131
1.008564
1.009008
1.009464
1.009930
1.010407
1.010895
1.011395
1.01 1906
1.012428
1.012961
1.013505
1.014061
1.014627
1.015204
1.015792
1.016391
1.017002
1.017624
1.018256
1.018899
1.019554
1.020220
1.020896
1.021584
S
0.017454
0.023271
0.029088
0.034905
0.040722
0.046539
0.052355
0.058171
0.063986
0.069802
0.075616
0.081430
0.087244
0.093057
0.098870
0.104682
0.110492
0.116306
0.122113
0.127921
0.133729
0.139536
0.145342
0.151146
0.156951
0.162753
0.168555
0.174356
0.180155
0.185954
0.191751
0.197546
0.203340
0.209134
0.214925
0.220715
0.226504
0.232291
0.238077
0.243861
0.249643
0.255424
0.261203
0.266980
0.272755
0.278529
0.284300
0.290070
0.295837
0.301604
0.307367
0.313129
0.318888
0.324646
0.330401
0.336154
0.341904
0.347653
0.353400
0.359143
430
1°
- P
FORON MINUN AMOIN MIDON NOO
4
6
8
10
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
TABLE XIV.-TANGENTS AND EXTERNALS FOR
UNIT DOUBLE-SPIRAL CURVE

1.00178.02918
1.00257.03507
Ts
Es
Ts
1.00028 .01164 42.5 1.03394
1.00064 .01747 43.0 1.03479
1.00114 .02332 43.5 1.03566
44.0 1.03653
44.5
1.00794
1.00873
1.00955
1.01042
1°
1.00302 .03802
1.03922 .14191
1.00402
1.00458
45.0
1.00350.04098 45.5
.04396 46.0 1.04015 .14370 .4
.04696 46.5 1.04109
1.00518.04992 47.0 1.04204
.14550
.14730
1.00581 .05292 47.5 1.04301
1.00648 .05593 48.0 1.04399
1.00719.05895 48.5 1.04498
.06198 49.0 1.04598
.06502 49.5 1.04701
.06808 50.0
.07115
1.01 132 .07424
1.01226 .07734
1.01324.08045
30.0 1.01644 .08990
30.5 1.01700 ..09149
31.0 1.01758 .09309
31.5 1.01817
1.01817.09469
32.0 1.01877 .09630
.2
.4
27.5 1.01375 .08201 51.0
28.0 1.01427 .08358
1.01480 .08515
28.5
29.0
29.5 1.01588 .08831
.6
.8
.2
.4
1.01533.08674 .6
.8
37.5 1.02609 .11432
38.0 1.02682 .11599
38.5
1.02756 .11767
39.0 1.02832 .11936
39.5
1.02909 .12105
52.0
.2
.4
.6
.8
53.0
32.5 1.01938 .09791
33.0 1.02000 .09952 .2
33.5 1.02064 .10114 .4
34.0 1.02128 .10277 .6
34.5 1.02194 .10440 .8
35.01.02260 .1060454.0
35.5 1.02327 .10768
36.0 1.02396 .10933
36.5 1.02466 .11099
37.0 1.02537 .11265
.2
.4
Es
Ts
Es
.13135 57.0 1.06399 .18536
.13309 .2 1.06449 .18617
.13484 .4 1.06499 .18698
.13660
1.03742 .13836
.6 1.06550 .18778
!.06600 .18859
.8
55.0
.2
.4
.6
.8
40.0
1.02987 .12275 56.0
40.5 1.03066 .12446
41.0 1.03146 .12617
41.5
42.0
1.03227 .12789
1.03310 .12962
.2
.4
.6
.8
1.03831.14013 58.0
1.04929 .16055
1.04971 .16130
1.05014.16206
1.05057 .16281
1.05100 .16356
1.05143 .16432
1°
.14912 59.0
.15094
.15277
.15460 .6
.15645
.8
1.05318.16736
1.05362.16813
1.05407.16889
1.05452.16966
∞ ONICO ON
1.05913
1.05961
1.04804.15831|| 60.0
1.04845.15905 .2
1.04887.15980
.2
.6
1.06009.17899
1.06057.17979
1.06105.18058
.8
1.06250.18296
.2
1.06299 .18376
1.06349 .18456
.4
68 ON=6∞
1.05186.16508
1.05230.16584 62.0
1.05274 .16660 .2
.4
.4
.6
.8
61.0
.2
.4
.6
1.05680 .17352 64.0
1.05726 .17430
1.05772 .17508
.6 1.05819 .17586
.8 1.05866
.17664
.8
.6
1.05497 .17043
1.05542 .17120
1.05588 .17197 .6
1.05634
.17275 .8
.8
∞ O
63.0
.2
.17742 65.0
.17820
.4
.6
.8
∞∞ ONE
1.07953 .20947
1.08010
.21034
.2 1.08068 .21120
.4 1.08126 .21207
1.08184
1.08243
.2
1.06153 .18137 66.0
1.06201 .18217
1.06651.18940
1.06754.19103
1.06703 .19021
ONJO∞
1.06806 .19184
1.06857 .19266
1.06909 .19348
1.06962 .19431
1.07015 .19513
1.07068 | .19596
1.07121 .19679
.4
1.07174
1.07228
1.07282.19929
1.07391.20097
1.07446.20181
1.07501.20266
.19762
.19846
1.07337 .20013
1.07556 .20350
1.07612 .20435
1.07668 .20519
1.07724 .20604
1.07781 | .20690
1.07838 .20775
1.07895.20861
.4 1.08724 .22086
1.08302 .21468
.21555
1.08361
1.08421
.21643
1.08481 .21731
1.08541 .21820
.21294
.21381
1.08602 .21908
1.08663
.21997
.6
.8 1.08847 .22265
1.08785.22175
.2 1.08971
1.08909 .22355
.22445
.6 1.09097
.8 1.09160
1.09034.22535
.22625
.22716
431
1°
TABLE XIV.-TANGENTS AND EXTERNALS FOR
UNIT DOUBLE-SPIRAL CURVE

67.0
.2 1.09287
1.09351
1.09416
1.09481
.4
600 ON‡60
.6
.8
68.0
.2
.4
69.0
.2
.4
.6
.23544
1.09744
.8 1.09810 .23637
.6
.8
70.0
.2
.6
.8
∞ O
.4 1.10352
1.10421
1.10491 .24585
71.0
.6
000
.8
72.0
.2
.6
.8
Ts
Es
1.09223 .22807 75.0
.22898
.22990
.23082
.23174
73.0
.2
1.09546.23266
1.09612.23358
1.09678.23451
.8
∞ ON468
74.0
.2
.4
1.10078 .24013
1.10146 .24108
.4 1.11062
.6
.8
1.11281 .25660
1.11355 .25760
.4 1.11429 .25860
.6 1.11504
.25960
.26060
1.11579
1.09876 .23731 77.0
1.09943.23825 .2
1.10010.23919 .4
.6
.8
.24203 78.0
1.10214
1.10283 .24298
.24393
.24489
1.10561 .24681 79.0 1.13661
.2 1.10632
.24777
.24874
.4
1.10703
1.10774.24971
1.10854
1.10917
1.10989
.25069
.25167
.25265
.25363
1.11135 .25462
1.11208 .25561
1°
1.11654 .26161
1.11730.26262
1.11806 .26363
11882.26465
1.11959 | .26567
.2
.4
.6
.8
·
76.0
.2
.4
.6
.8
.2
.4
.6
.8
80.0
.2
44
•
Ts
Es
1.12036 .26669 83.0
1.12124 | .26772 .2
1.12192 | .26875
1.12270 .26978
1.12348
.4
.6
.27082
.8
.27186
.27290
.27394
1.12667 .27499
1.12747 .27604
.28786
.2
1.13746 .28896
·
4 1.13832 .29006
.6 1.13918 .29116
1.14005 .29226
.8
81.0
.2
1.12427
1.12507
1.12587
ON:
1.12828 .27710
1.12909 .27816
.29337
.29449
1.14267
.29561
.6
.29673
1.14356
.8 1.14445 .29785
82.0
1.12991.27923
1.14092
1.14179
1°
84.0
.2
.
•
.4
.2
.4 1.15173 .30699
.6 1.15266 .30815
.8 1.15359 .30931
.6
.8
1.13074 .28030
1.13157 .28137
1.13240 .28244 86.0
1.13323 .28352 .2
1.13407
.28460
.4
1.13491
.28568
.6
1.13576
.28677
.8
87.0
.2
.4
85.0
0226
.2
.4
.6
.8
.6
.8
88.0
.2
4
•
1.14535 .29898 89.0
1.14625 .30011
.4 1.14715
•
.30125
.30239
.6 1.14805
.8 1.14896 .30353
Ú∞ ONIO∞ ONIO∞
.8
.2
.4
.6
1.18174 .34373
.6.1.18282
.34503
.34633
1.18391
1.18500
1.18500 .34764
.34895
1.14988 .30468 90.0
1.15080 .30583
.2
.4
Ts
Es
1.15453 .31048
1.15547 .31165
1.15642 .31283
1.15738 .31401
1.15834 .31520
1.15930 .31639
1.16027 .31759
1.16124 .31879
1.16222 .31999
1.16320 .32120
.6
1.16418 .32241
1.16517 .32363
1.16617 .32485
1.16717 .32607
1.16818 .32730
.8
1.16919.32854
1.17021.32978
1.17123
1.17226
1.17329
.33478
1.17433
1.17537
1.17642 .33732
.33605
1.17748.33859
1.17854.33987
.8 1.18942
.33102
.33227
.33352
1.17960.34115
1.18067.34244
I.18609
1.18719.35027
1.18830.35159
.35292
1.19054 35425
.35559
1.19167
1.19394
.35828
•
1.19280.35693
1.19508.35963
NOTES: (1) For any given I and selected spiral length, Ts and Es
may be found by
for the double-spiral, or all-transitional, curve
(2) If
multiplying the tabulated values by the length of spiral.
the given I does not correspond
a tabulated value, Ts and Es
may be found by simple interpolation. (3) Deflection angles for
staking the curve may be computed, as for any spiral, by using
Δ
4 = 1/1/
432
A
2
6
10
/
O
O /
/
0.0 0 00.00 0 00.00 00.0 10 00.0 0 00.0 0 00.0 10 00.0 0 00.0 |0 00.0 10 00.0
.1 00.02 00.1 00.2 00.3 00.5 00.7 01.0 01.3 01.6 02.0
00.04 00.2
00.06 00.2
00.08
02.0
00.4
00.5
00.6
01.0
01.0 01.4
01.5 02.2
02.0 02.9
02.6 03.2
04.9
06.5 08.0
04.0
06.0
02.9
03.8
00.3
00.7
01.3
03.9
05.1
öävää Awniŏ OONGG AWN
.6
0.5 0 00.10 0 00.4
234 56789
.4
1.0 0 00.20 0 00.8
00.22 00.9
00.24 01.0
0 01.8 0 03.2 0 05.0 0 07.2 0 09.8
02.0
05.5 07.9 10.8
06.0
03.5
02.2
03.8
08.6
11.8
04.2 06.5
09.4
12.7
00.26 01.0 02.3
.4 00.28 01.1
07.0
10.1
13.7
1234
2.0 0 00.40 0 01.6 0 03.6 0 06.4 0 10.0 0 14.4
.1
00.42 01.7 03.8 06.7
10.5 15.1
07.0
11.0
15.8
00.44 01.8 04.0
00.46 01.8 04.1
00.48 01.9
07.4
11.5
16.6
04.3
07.7
12.0
17.3
TABLE XV.—DEFLECTION ANGLES FOR 10-CHORD SPIRAL
1
3
5
9
1

00.32 01.3
1.5 0 00.30 0 01.2 0 02.7 10 04.8 0 07.5 0 10.8 0 14.7 0 19.2
02.9 05.1 08.0 11.5 15.7 20.5
03.1 05.4
12.2
03.2 05.8
13.0
01.5 03.4 06.1 09.5
00.34
00.36
00.38
08.5
09.0
16.7 21.8
17.6 23.0
18.6 24.3
13.7
.1
23+
0 00.9
00.12 00.5 01.1
00.14 00.6 01.3
00.16 00.6 01.4
00.18 00.7 01.6
.4
.7
.8
4
2.5 0 00.50 0 02.0 0 04.5 0 08.0 0 12.5 0 18.0 0 24.5
.6 00.52 02.1 04.7 08.3 13.0
.7 00.54
02.2 04.9
02.2 05.0
08.6
09.0
.8
00.56
.9 00.58 02.3 05.2
09.3
.9
01.4
01.4
O-NET MONO!!
0 01.6 0 02.5 0 03.6 0 04.9 0 06.4 0 08.1 0 10.0
01.9 03.0 04.3 05.9 07.7
09.7
12.0
02.2
03.5
05.0
06.9
09.0
11.3
02.6
04.0
05.8
10.2
13.0
07.8
06.5 08.8
02.9
04.5
11.5
14.6
02.5 04.5
.3
7
.4
5.5
8
5.0 0 01.00 0 04.0 0 09.0 0 16.0 10 25.0
.1 01.02 04.1 09.2 16.3 25.5
.2 01.04 04.2 09.4 16.6
01.06 04.2 09.5 17.0
01.08 04.3 09.7 17.3
0 19.6
20.6
3.0 0 00.60 0 02.4 0 05.4 0 09.6 0 15.0 0 21.6 0 29.4 0 38.4 0 48.6 1 00.0
00.62 02.5 05.6 09.9 15.5 22.3 30.4 39.7 50.2 02.0
00.64 02.6 05.8 10.2 16.0 23.0 31.4 41.0 51.8 04.0
00.66 02.6 05.9 10.6 16.5 23.8 32.3 42.2
00.68 02.7 06.1 10.9 17.0 24.5 33.3 43.5
53.5
06.0
55.1
08.0
0 12.8
14.1
15.4
16.6
17.9
0 25.6
26.9
21.6 28.2
22.5
23.5
3.5
0 00.70 0 02.8 0 06.3 0 11.2 0 17.5 0 25.2 0 34.3 0 44.8 0 56.7 1 10.0
00.72 02.9 06.5 11.5 18.0
.6
25.9
35.3 46.1
58.3 12.0
.7 00.74 03.0
26.6
06.7 11.8 18.5
00.76 03.0 06.8 12.2 19.0
00.78 03.1 07.0 12.5
.8
36.3
27.4 37.2
19.5 28.1 38.2
14.0
16.0
.9
18.0
4.0 0 00.80 0 03.2 0 07.20 12.8 0 20.0 0 28.8 10 39.2
0 51.2 1 04.8 1 20.0
00.82 03.3 07.4 13.1 20.5 29.5 40.2 52.5 06.4 22.0
.2 00.84 03.4 07.6 13.4 21.0 30.2 41.2 53.8 08.0 24.0
00.86 03.4 07.7 13.8 21.5 31.0 42.1 55.0
00.88 03.5 07.9 14.1 22.0 31.7 43.1
09.7
26.0
56.3 11.3
28.0
29.4
30.7
0 16.2 0 20.0
17.8
22.0
19.4
24.0
21.1
26.0
22.7 28.0
4.5 0 00.90 10 03.6 10 08.1 0 14.4 10 22.5 0 32.4 0 44.1 0 57.61 12.9
.6 00.92 03.7 08.3 14.7 23.0 33.1 45.1 58.9 14.5
00.94
23.5
33.8
03.8 08.5 15.0
03.8 08.6 15.4
00.98 03.9 08.8 15.7 24.5
00.96
16.1
17.8
46.1 1 00.2
34.6 47.0 01.4
35.3 48.0 02.7
24.0
19.4
0 24.3 0 30.0
25.9 32.0
27.5
34.0
29.2
36.0
30.8
38.0
0 32.0
18.7 25.5 33.3
13.5 19.4 26.5 34.6
14.0 20.2 27.4 35.8
14.5 20.9 28.4 37.1 47.0 58.0
0 40.5 0 50.0
42.1 52.0
43.7
45.4
54.0
56.0
14.0.
16.0
18.0
0 40.0
0 32.4
34.0
42.0
35.6
44.0
37.3
46.0
38.9 48.0
47.4 59.9
48.6 1 01.6
49.9 03.2
26.0 37.4 51.0 06.6 24.2
26.5 38.2 51.9 07.8 25.9
27.0 38.9 52.9 09.1 27.5
1 30.0
32.0
34.0
36.0
38.0
0 36.0 0 49.0 1 04.0 1 21.0 1 40.0
36.7 50.0 05.3 22.6
42.0
44.0
46.0
48.0
54.9 11.7 30.7 52.0
.6
.7 01.14
0 01.10 0 04.4 0 09.9 0 17.6 0 27.5 0 39.6 0 53.91 10.41 29.1 1 50.0
01.12 04.5 10.1 17.9 28.0 40.3
28.5 41.0
29.0 41.8
29.5 42.5
54.0
.8
01.16
04.6 10.3 18.2
04.6 10.4 18.6
04.7 10.6 18.9
55.9 13.0
56.8 14.2
57.8 15.5
32.3
34.0
35.6
56.0
.9 01.18
58.0
433
TABLE XV.-DEFLECTION ANGLES FOR 10-CHORD SPIRAL
1
3
6
9
O
O /
G /
6.0 0 01.20 0 04.8 0 10.8 0 19.2 0 30.0 0 43.2 0 58.8 1 16.8 1 37.2 2 00.0
.1 01.22 04.9 11.0 19.5
.2 01.24
.3
30.5
05.0
19.8
31.0
43.9 59.8
44.6 1 00.8
45.4 01.7
18.1 38.8 02.0
19.4 40.4 04.0
42.1 06.0
20.2
31.5
01.26 05.0
.4 01.28 05.1
20.6
20.5
32.0
46.1
02.7
21.9
43.7 08.0
Δ
O
2
11.2
11.3
11.5
4
5
/
7
8
6.50 01.30 10 05.2 0 11.7
0 20.8
1 03.7 1 23.2
04.7
.6 01.32 05.3 11.9 21.1
24.5
0 32.5 0 46.8
33.0 47.5
.7 01.34 05.4 12.1 21.4 33.5 48.2
05.7
.8 01.36 05.4 12.2 21.8 34.0 49.0 06.6
.9 01.38 05.5 12.4 22.1 34.5 49.7 07.6 28.3 51.8 18.0
25.8
27.0
7.00 01.40 0 05.6 0 12.6 0 22.4 10 35.0 10 50.4 1 08.6 1 29.6 1 53.4 2 20.0
.1 01.42 05.7 12.8 22.7 35.5 51.1 09.6 30.9 55.0 22.0
.2 01.44 05.8 13.0 23.0 36.0 51.8 10.6 32.2
.3 01.46 05.8 13.1 23.4 36.5 52.6
.4 0.1.48 05.9 13.3 23.7 37.0 53.3 12.5
24.0
11.5
33.4
26.0
34.7
28.0
.1
.2
.3
41.0 59.0
41.5 59.8
42.01 00.5
8.00 01.60 0 06.4 0 14.4 0 25.6 0 40.0 0 57.6 1 18.4 1 42.4
01.62 06.5 14.6 25.9 40.5 58.3 19.4 43.7
01.64 06.6 14.8 26.2
01.66 06.6 14.9 26.6
.4 01.68 06.7 15.1 26.9
8.50 01.70 10 06.8 0 15.3 0 27.2
.6 01.72 06.9 15.5 27.5
.7) 01.74 07.0 15.7 27.8
.8 01.76 07.0 15.8 28.2
.9 01.78 07.1 16.0 28.5 44.5
0 42.5 1 01.2 1 23.3
43.0 01.9
24.3
43.5 02.6
25.3
44.0
03.4
04.1
1 45.3 2 10.0
46.9 12.0
48.5 14.0
50.2 16.0
7.50 01.50 0 06.00 13.5 0 24.0 0 37.5 0 54.0 1 13.5 1 36.0 2 01.5
.6 01.52 06.1 13.7 24.3 38.0 54.7 14.5
06.2
.8 01.56 06.2
.9 01.58 06.3 14.2 25.3
37.3 03.1
38.6
.7 01.54
04.7
13.9 24.6
14.0 25.0
38.5
39.0
55.4 15.5
56.2 16.4 39.8 06.4
17.4 41.1 08.0
39.5
56.9
I
56.6
58.3
59.9
10
2 40.0
2 09.6
11.2
42.0
12.8
44.0
20.4 45.0
21.3 46.2 14.5 46.0
47.5 16.1 48.0
22.3
51.4
26.2
52.6
27.2 53.9 24.2
2 30.0
32.0
34.0
36.0
38.0
1 48.8 2 17.7 |2 50.0
50.1 19.3 52.0
20.9 54.0
22.6
56.0
58.0
9.00 01.80 0 07.2 0 16.2 0 28.8 0 45.0 1 04.8 1 28.2 1 55.2 2 25.8 3 00.0
.1 01.82 07.3 16.4 29.1
45.5 05.5
29.2
56.5 27.4 02.0
.2 01.84 07.4
29.4
46.0
06.2
30.2
57.8
16.6
16.7
29.8
46.5
.3 01.86 07.4
.4 01.88 07.5 16.9 30.1
07.0
31.1 59.0
32.1 2 00.3
47.0
07.7
29.0
30.7
32.3
04.0
06.0
08.0
09.1
34.1 02.9
9.50 01.90 0 07.6 0 17.1 0 30.4 0 47.5 1 08.4 1 33.1 2 01.6
.6 01.92 07.7 17.3 30.7 48.0
.7 01.94 07.8 17.5 31.0 48.5
01.96 07.8 17.6 31.4 49.0
.9 01.98 07.9 17.8 31.7 49.5 11.3
09.8
2 33.9 3 10.0
35.5 12.0
35.1 04.2 37.1 14.0
36.0 05.4 38.8 16.0
37.0 06.7 40.4 18.0
.8
10.6
12.7
10.00 02.00 0 08.0 0 18.0 0 32.0 0 50.0 1 12.0 1 38.0 2 08.0 2 42.0 3 20.0
.1 02.02 08.1 18.2 32.3 50.5
02.04 08.2 18.4 32.6
02.06 08.2 18.5 33.0
.4 02.08 08.3 18.7 33.3 52.0 14.9
.2
51.0
39.0 09.3 43.6 21.9
13.4 40.0
14.2 40.9
.3
51.5
10.6 45.2 23.9
11.8 46.8 25.9
41.9 13.1 48.5 27.9
10.5 0 02.10 0 08.4 0 18.9 0 33.6 0 52.5 1 15.6 1 42.9 2 14.4 2 50.1 3 29.9
.6 02.12 08.5 19.1 33.9 53.0 16.3 43.9 15.7 51.7 31.9
.7 02.14 08.6 19.3 34.2 53.5 17.0 44.9 17.0
.8 02.16 08.6 19.4 34.6 54.0 17.8 45.8 18.2
.9 02.18 08.7 19.6 34.9 54.5 18.5 46.8 19.5
53.3
54.9
33.9
35.9
56.6
37.9
.2 02.24 09.0 20.2
11.00 02.20 0 08.8 0 19.8 0 35.2 0 55.0 1 19.2 1 47.8 2 20.8 2 58.2
.1 02.22 08.9 20.0 35.5 55.5 19.9 48.8 22.1 59.8
35.8 56.0 20.6 49.8
.3 02.26 09.0 20.3 36.2 56.5 21.4 50.7
.4 02.28 09.1 20.5 36.5 57.0 22.1
11.50 02.30 0 09.2 0 20.7 0 36.8 0 57.5 1 22.8 1 52.7 2 27.2 3 06.3 3 49.9
.6 02.32 09.3 20.9 37.1 58.0 23.5 53.7 28.5 07.9 51.9
23.4 3 01.4
24.6 03.0
25.9 04.7
51.7
.7 02.34
09.4
37.4 58.5 24.2
21.1
21.2
09.5 53.9
11.1 55.9
.8 02.36
09.4
09.5 21.4
37.8 59.0
38.1 59.5
54.7 29.7
55.6 31.0
32.3 12.7
25.0
25.7
.9 02.38
56.6
57.9
3 39.9
41.9
43.9
45.9
47.9
434
TABLE XV.-DEFLECTION ANGLES FOR 10-CHORD SPIRAL
1
2
3
4
5
6
8
9
O /
/
O
/
12.00 02.40 0 09.6 0 21.6 0 38.4 1 00.0 1 26.4 1 57.6 2 33.6 3 14.4 3 59.9
.1 02.42 09.7
21.8
38.7 00.5 27.1 58.6
34.9
16.0 4 01.9
.2
09.8
22.0
39.0
01.0
36.1
17.6 03.9
.3
02.44
02.46 09.8
.4 02.48 09.9
22.1
39.4
01.5
37.4
19.2
05.9
22.3
39.7
02.0
38.7
20.8
07.9
Δ
O
12.50 02.50 0 10.0 0 22.5
.6 02.52 10.1 22.7
.7 02.54 10.2 22.9
.8 02.56 10.2 23.0
.9 02.58 10.3 23.2
13.00 02.60 0 10.4 0 23.4
.1 02.62 10.5 23.6
.2 02.64 10.6
.3 02.66 10.6
.4 02.68 10.7 24.1
23.8
23.9
0 40.0
40.3
40.6
41.0
41.3
42.6
42.9
1 02.5
03.0
03.5
04.0
04.5
48.6
49.0
49.3
7
0 41.6 1 05.0 1 33.6 2 07.4 2 46.4
41.9 05.5 34.3 08.4
35.0 09.3
47.7
42.2
48.9
35.8 10.3
50.2
36.5 11:3
51.5
06.0
06.5
07.0
ܡ
27.8 59.5
28.6 2 00.5
29.3 01.5
1 30.0 2 02.5
30.7 03.5
2 40.0
41.3
04.4 42.5
05.4
32.9 06.4
31.4
32.2
.7 03.34 13.4 30.1
.8
53.4
03.36 13.4 30.2 53.8
.9 03.38 13.5 30.4 54.1
25.7
43.8 27.3
45.1
28.9
14.5 0 02.90 0 11.6 0 26.1 0 46.4 1 12.5 1 44.4 2 22.1 3 05.6
23.1 06.8
24.0 08.1
.6 02.92 11.7 26.3 46.7
13.0 45.1
13.5
45.8
.8
.7 02.94 11.8 26.5 47.0
11.8 26.6 47.4
26.8 47.7
14.0
46.6
25.0
02.96
02.98 11.9
.9
14.5
47.3 26.0
3 22.5
24.1
13.50 02.70 0 10.8 0 24.3 0 43.2 1 07.5 1 37.2 2 12.3 2 52.8 13 38.6 4 29.9
.6 02.72 10.9 24.5 43.5 08.0 37.9 13.3 54.0 40.3
08.5 .38.6
55.3 41.9
09.0 39.4
56.6 43.5
57.9 45.1
31.9
33.9
.7 02.74 11.0
24.7
43.8
14.2
15.2
24.8
44.2
35.9
44.5
09.5 40.1
16.2
37.9
.8 02.76 11.0
.9 02.78 11.1 25.0
14.00 02.80 0 11.2 0 25.2 0 44.8
.1 02.82 11.3 25.4 45.1
.2 02.84 11.4 25.6 45.4
11.4
.3 02.86
25.7 45.8
25.9 46.1
.4 02.88
11.5
1 10.0 1 40.8 2 17.2 2 59.2
10.5 41.5 18.2 3 00.4
11.0
19.1 01.7
11.5 43.0 20.1 03.0
12.0 43.7 21.1 04.3
42.2
10
3 30.5 4 19.9
32.2 21.9
33.8
23.9
35.4
259
37.0
27.9
4 09.9
11.9
13.9
15.9
17.9
51.6
53.2
3 46.7 4 39.9
48.4 41.9
50.0
43.9
45.9
47.9
15.0 0 03.00 0 12.0 0 27.0 0 48.0 1 15.0 1 48.0 2 27.0 3 12.0 4 02.9 4 59.8
.1 03.02 12.1 27.2 48.3 15.5 48.7 28.0 13.2 04.5 5 01.8
.2 03.04 12.2 27.4
16.0 49.4 28.9 14.5 06.1 03.8
.3 03.06 12.2 27.5
16.5 50.2 29.9 15.8 07.8
.4 03.08 12.3 27.7
17.0 50.9 30.9
05.8
17.1
09.4
07.8
3 54.8 4 49.9
56.4
51.9
58.1
09.4 59.7
10.7 4 01.3
15.50 03.10 0 12.40 27.9 0 49.61 17.5 1 51.6 2 31.9 3 18.4 4 11.0 5 09.8
.6 03.12 12.5 28.1 49.9 18.0 52.3 32.9 19.6 12.6 11.8
.7 03.14 12.6 28.3 50.2 18.5 53.0 33.8 20.9 14.2 13.8
.8 02.16 12.6 28.4 50.6 19.0 53.7 34.8 22.2 15.9 15.8
.9 03.18 12.7 28.6 50.9 19.5 54.5 35.8 23.5 17.5 17.8
53.9
55.8
57.8
16.00 03.20 0 12.8 0 28.8 0 51.2 1 20.0 1 55.2 2 36.8 3 24.8 4 19.1 5 19.8
03.22 12.9 29.0 51.5 20.5 55.9 37.8 26.0 20.7 21.8
.2 03.24 13.0 29.2 51.8 21.0 56.6 38.7
23.8
.3 03.26 13.0 29.3 52.2 21.5 57.3 39.7
25.8
.4 03.28 13.1 29.5 52.5 22.0 58.1 40.7 29.9 25.6
27.3 22.3
28.6 23.9
27.8
16.50 03.30 0 13.2 0 29.7 0 52.8 1 22.5 1 58.8 2 41.7 3 31.1 4 27.2 5 29.8
.6 03.32 13.3 29.9 53.1 23.0 59.5
23.5 2 00.2
24.0 00.9
24.5 01.7
42.7 32.4 28.8 31.8
43.6 33.7 30.4 33.8
44.6 35.0 32.0 35.8
45.6 36.3 33.6 37.8
17.00 03.40 0 13.6 0 30.6 0 54.4 1 25.0 2 02.4 2 46.6 3 37.5 4 35.3 5 39.8
.1 03.42 13.7 30.8 54.7 25.5 03.1 47.5 38.8 36.9 41.8
.2 03.44 13.8 31.0 55.0
26.0 03.8 48.5 40.1 38.5 43.7
03.46 13.8 31.1 55.4 26.5 04.5 49.5 41.4 40.1
45.7
.4 03.48 13.9 31.3 55.7 27.0 05.3 50.5 42.7 41.7 47.7
.3
46.6
48.2
49.0 49.8
17.50 03.50 0 14.0 0 31.5 0 56.0 1 27.5 2 06.0 2 51.5 3 43.9 4 43.4 5 49.7
.6 03.52 14.1 31.7 56.3 28.0 06.7
52.4 45.2 45.0 51.7
.7 03.54 14.2 31.9
28.5
07.4
53.4 46.5
53.7
56.6
57.0
.8
29.0
08.1
54.4 47.8
55.7
03.56 14.2 32.0
.9 03.58 14.3 32.2 57.3
29.5
08.9
55.4
5.77
435
Δ
TABLE XV.—DEFLECTION ANGLES FOR 10-CHORD SPIRAL
1
3
9
O
2
4
5
O
/
/
O /
32.6
57.9
51.6
18.00 03.60 0 14.4 0 32.4 0 57.6 1 30.0 2 09.6 2 56.4 3 50.3 4 51.5 5 59.7
.1 03.62 14.5
03.64
03.66 14.6
.4 03.68 14.7
.2
14.6
32.8
58.2
30.5 10.3 57.3
11.0 58.3
11.7 59.3
31.0
52.9
53.16 01.7
54.7 03.7
54.2 56.3 05.7
55.4 57.9
07.7
.3
32.9
58.6
31.5
33.1 58.9
32.0
12.5 3 00.3
1 32.5
33.0
6
/
7
14.6
33.5
34.0
34.5 16.1
15.3
41.0
41.5
8
2 13.2 3 01.3
13.9
3 56.7 4 59.5
02.2 58.0 5 01.2
03.2 59.3
02.8
04.2 4 00.6
05.2 01.8
23.00 04.60 0 18.4 0 41.4
.1 04.62 18.5 41.6
.2 04.64 18.6 41.8
.3
04.66 18.6 41.9
.4 04.68 18.7 42.1
18.5 0 03.70 0 14.8
0 33.3 0 59.2
.6 03.72 14.9 33.5 59.5
5 07.6 6 19.7
09.2 21.7
36.0
18.2
10.9
23.7
.7 03.74 15.0 33.7 59.8
.8 03.76
15.0 33.8 1 00.2
.9 03.78 15.1 34.0 00.5
19.00 03.80 0 15.2 0 34.2 1 00.8 1 35.0 2 16.8 3 06.2 4 03.1
.1 03.82 15.3 34.4 01.1 35.5 17.5 07.1 04.4
.2 03.84 15.4 34.6
08.1 05.7
.3 03.86 15.4 34.7
09.1 06.9 12.5
.4 03.88 15.5 34.9 02.1
10.1 08.2 14.1 27.6
19.5 0 03.90 0 15.6 0 35.11 02.4 1 37.5 2 20.4 3 11.14 09.5 5 15.7 6 29.6
.6 03.92 15.7 35.3 02.7 38.0 21.1 12.0 10.8 17.3
.7 03.94 15.8
35.5
.8 03.96 15.8 35.6
.9 03.98 15.9 35.8 03.7 39.5 23.3
01.4
01.8
18.9
25.6
36.5
37.0 19.7
31.6
33.6
03.0
38.5
21.8
03.4
39.0
22.5
13.0 12.1
14.0 13.3
15.0 14.6
18.9
20.6
22.2 37.6
35.6
10
6 09.7
11.7
13.7
15.7
04.4
06.0 17.7
20.0 0 04.00 0 16.0 0 36.0 1 04.0 1 40.0 2 24.0 3 16.04 15.9 5 23.8 6 39.6
.1 04.02 16.1 36.2 04.3 40.5 24.7 16.9 17.2 25.4
41.6
43.6
18.5
27.0
.2 04.04 16.2
.3 04.06 16.2
.4 04.08 16.3 36.7 05.3 42.0
36.4
36.5
04.6
05.0
25.4 17.9
26.1 18.9 19.7
26.9 19.9 21.0 30.3 47.6
28.6
45.6
20.50 04.10 0 16.4 0 36.9 1 05.6 1 42.5 2 27.6 3 20.9 4 22.35 31.9 6 49.6
.6 04.12 16.5 37.1 05.9 43.0 28.3 21.8 23.6 33.5 51.6
.7 04.14 16.6
53.6
.8 04.16 16.6
55.6
.9 04.18 16.7 37.6 06.9 44.5 30.5
37.3 06.2
37.4 06.6
43.5 29.0
44.0 29.7
22.8 24.8 35.1
23.8 26.1
36.7
24.8 27.4 38.3 57.5
21.00 04.20 0 16.8 0 37.8 1 07.2 1 45.0 2 31.2 3 25.8 4 28.7 5 40.0 6 59.5
31.9 26.7 30.0
.1 04.22 16.9 38.0 07.5 45.5
.2 04.24 17.0 38.2 07.8
46.0
.3 04.26 17.0 38.3 08.2 46.5
.4 04.28 17.1 38.5 08.5 47.0 34.1 29.7 33.8
32.6 27.7
33.4 28.7
31.2
32.5
41.6 7 01.5
43.2 03.5
44.8 05.5
46.4 07.5
21.50 04.30 0 17.2 0 38.7 1 08.8 1 47.5 2 34.8 3 30.7 4 35.1 5 48.0 7 09.5
.6 04.32 17.3 38.9 09.1 48.0 35.5 31.6 36.4 49.7 11.5
39.1 09.4 48.5 36.2 32.6 37.6 51.3
04.36 17.4 39.2 09.8 49.0
37.0 33.6 38.9 52.9
.9 04.38 17.5 39.4 10.1 49.5 37.7 34.6 40.2 54.5
.7 04.34 17.4
13.5
15.5
.8
17.5
22.00 04.40 0 17.6 0 39.6 1 10.4 1 50.0 2 38.4 3 35.5 4 41.5 5 56.1
.1 04.42 17.7 39.8 10.7 50.5 39.1 36.5 42.7 58.7
.2 04.44 17.8 40.0 11.0 51.0 39.8 37.5 44.0 59.3
.3 04.46 17.8 40.1 11.4 51.5
40.6 38.5 45.3 6 01.0
.4 04.48 17.9 40.3 11.7 52.0 41.3 39.5 46.6 02.6
22.50 04.50 0 18.0 0 40.5 1 12.0 1 52.5 2 42.0 3 40.5 4 47.9 6 04.2 7 29.4
.6 04.52 18.1 40.7 12.3 53.0 42.7 41.4 49.1 05.8
.7 04.54 18.2 40.9 12.6
53.5 43.4 42.4 50.4 07.4
.8 04.56 18.2 41.0 13.0 54.0 44.2 43.4 51.7 09.0
.9 04.58 18.3 41.2 13.3 54.5 44.9 44.4 53.0 10.7 37.4
31.4
33.4
35.4
7 19.5
21.5
23.5
25.5
27.5
23.50 04.70 0 18.8 0 42.3 1 15.2 1 57.5 2 49.2 3 50.2 5 00.6 16 20.4
.8
.6 04.72 18.9 42.5 15.5 58.0 49.9 51.2 01.9 22.0
.7 04.74 19.0 42.7 15.8 58.5 50.6 52.2
03.2 23.6
53.2 04.5 25.2
54.2 05.7
04.76 19.0 42.8
16.2
16.5
59.0 51.4
59.5 52.1
.9
04.78 19.1 43.0
26.8
1 13.6 1 55.0 2 45.6 3 45.31 54.2 6 12.3 7 39.4
13.9 55.5 46.3 46.3 55.5 13.9 41.4
14.2 56.0 47.0 47.3 56.8
14.6 56.5 47.8 48.3 58.1
15.5
43.4
17.1
45.4
14.9 57.0 48.5 49.2 59.4 18.7
47.4
7 49.4
51.4
53.3
55.3
57.3
436
24.0
.1
.2
.3
.4
24.5
.6
.7
.8
.9
25.0
.1
.2
.3
.4
25.5
.6
.7
.8
.9
26.0
.1
.2
.3
.4
26.5
.6
.7
.8
.9
27.0
.2
.4
.6
.8
28.0
.2
.4
.6
.8
29.0
.2
.4
.6
.8
30.0
.2
.4
.6
.8
31.0
.2
.4
.6
•
32.0
.2
.4
.6
.8
TABLE XV.-DEFLECTION ANGLES FOR 10-CHORD SPIRAL

1
04.84
04.86
04.88
0 04.90
04.92
04.94
04.96
04.98
0 04.80 0 19.2 0 43.21 16.8
04.82
19.3
43.4
17.1
19.4
43.6
17.4
19.4
43.7
17.8
19.5
43.9 18.1
0 05.00
05.02
05.04
05.06
05.08
0 05.10
05.12
05.14
05.16
05.18
05.22
05.24
05.26
05.28
2
0 05.30
05.32
05.34
05.36
0 19.6
19.7
19.8
19.8
19.9
0 20.0
20.1
20.2
20.2
20.3
0 05.80
05.84
05.88
05.92
05.96
0 20.4
20.5
20.6
20.6
20.7
0 21.2
21.3
21.3
21.4
05.38 21.5
0 05.400 21.6
05.44 21.8
05.48 21.9
05.52
22.1
05.56
22.2
0 05.200 20.8
0 20.80 46.8
0 46.8 1 23.2
20.9 47.0
23.5
21.0
47.2
23.8
21.0
47.3
24.2
21.1
47.5 24.5
0 05.60
05.64
0 22.4
22.6
05.68 22.7
05.72 22.9
05.76 23.0
0 23.2
23.4
23.5
23.7
23.8
3
0 24.0
0 06.00
06.04 24.2
06.08 24.3
06.12
24.5
24.6
06.16
4
0 06.40 0 25.6
06.44 25.8
06.48 25.9
06.52 26.1
06.56 26.2
0 44.11 18.4.
44.3
18.7
44.5
19.0
44.6
19.4
44.8 19.7
0 45.01 20.0
45.2 20.3
45.4 20.6
45.5 21.0
45.7 21.3
0 45.91 21.6
46.1 21.9
46.3
22.2
46.4
22.6
46.6 22.9
0 47.71 24.8
47.9 25.1
48.1
25.4
48.2
25.8
48.4 26.1
0 48.6 1 26.4
49.0 27.0
49.3 27.7
49.7 28.3
50.0
29.0
0 50.41 29.6
50.8 30.2
51.1 30.9
51.5 31.5
51.8 32.2
0 52.21 32.8
52.6 33.4
52.9 34.1
53.3
34.7
35.4
53.6
5
2 00.0 2 52.8
00.5
53.5
01.0
54.2
55.0
55.7
01.5
02.0
2 02.5
03.3
03.5
04.0
04.5
2 07.5
08.0
08.5
09.0
09.5
2 05.0 | 3 00.0
05.5 00.7
06.0 01.4
06.5
02.1
07.0
02.9
2 10.0
10.5
11,0
11.5
12.0
2 12.5
13.0
13.5
14.0
14.5
6
2 56.4
57.1
57.8
58.6
59.3
2 20.0
21.0
3 03.6
04.3
05.0
05.8
06.5
0 54.0 1 36.0 2 30.0
54.4 36.6 31.0
54.7 37.3 320
55.1 37.9 33.0
55.4 38.6 34.0
3 07.2
07.9
08.6
09.3
10.1
2 15.0 | 3 14.4
16.0 15.8
17.3
17.0
18.0
18.7
19.0
20.1
3 10.8
11.5
12.2
13.0
13.7
3 21.5
23.0
22.0 24.4
23.0 25.9
24.0 27.3
2 25.0 3 28.8
26.0 30.2
27.0 31.6
28.0 33.1
29.0
34.5
3 35.9
37.4
38.8
40.3
41.7
0 06.20
0 24.8 0 55.81 39.2 2 35.0 3 43.1
06.24 25.0 56.2 39.8 36.0 44.6
06.28 25.1 56.5 40.5 37.0 46.0
06.32 25.3 56.9 41.1 38.0 47.5
06.36 25.4 57.2 41.8 39.0 48.9
0 57.6 1 42.4 2 40.0 3 50.3
58.0 43.0 41.0 51.8
58.3 43.7 42.0 53.2
58.7 44.3
59.0 45.0
43.0
54.7
44.0
56.1
7
3 55.1
56.1
57.1
58.1
59.0
4 00.0
01.0
02.0
03.0
03.9
8
5 07.0
08.3
09.6
11.8
12.8
13.7
10.9
12.1
5 13.4
14.7
16.0
17.2
18.5
4 04.9 5 19.8
05.9 21.1
06.9 22.4
07.9 23.6
08.8 24.9
4 09.8 5 26.2
10.8 27.5
28.7
30.0
31.3
4 14.7 5 32.6
15.7 33.9
16.7
35.1
17.6
36.4
18.6 37.7
4 19.6 5 39.0
20.6 40.2
21.6
41.5
22.5
42.8
23.5
44.1
4 24.5 5 45.4
26.5 47.9
28.4 50.5
30.4 53.0
32.3 55.6
4 34.3 5 58.1
36.2 6 00.7
38.2
03.2
40,2
05.8
42.1
08.3
4 44.1 6 10.8
46.0 13.4
48.0
16.0
49.9
18.5
51.9
21.0
4 53.8
6 23.6
26.2
55.8
57.8
28.8
59.7 } 31.3
5 01.7 33.9
5 03.66 36.4
05.6
39.0
07.5 41.5
09.5 44.0
11.5 46.6
5 13.4 6 49.2
15.4 51.7
54.2
17.3
19.3 56.8
21.2 59.3
9
6 28.4
30.1
31.7
33.3
34.9
6 36.5
38.1
39.7
41.3
43.0
6 44.6
46.2
47.8
49.4
51.0
6 52.7
54.3
55.9
57.5
59.1
7 00.7
02.3
04.0
05.6
07.2
7 08.8
10.4
12.0
13.7
15.3
7 16.9
20.1
23.3
26.6
29.8
7 33.0
36.3
39.5
42.7
45.8
7 49.1
52.3
55.6
58.8
8 02.0
8 05.3
08.5
11.7
15.0
18.2
8 21.4
24.6
27.8
31.0
34.3
8 37.5
40.7
43.9
47.1
50.4
10
7 59.3
8 01.3
03.3
05.3
07.3
8 09.3
11.3
13.3
15.3
17.2
8 19.2
21.2
23.2
25.2
27.2
8 29.2
31.2
33.2
35.2
37.1
8 39.1
41.1
43.1
45.1
47.1
8 49.1
51.1
53.1
55.1
57.0
8 59.0
9 03.0
06.9
10.9
14.9
9 18.9
22.9
26.8
30.8
34.7
9 38.7
42.7
46.7
50.7
54.6
9 58.6
10 02.6
06.5
10.5
14.5
10 18.5
22.5
26.4
30.4
34.3
10 38.3
42.3
46.2
50.2
54.2
437
Δ
O
33.00 06.00
.2
06.64
.4
06.68
.6
06.72
.8
06.76
35.0
34.0
0 06.80
.2
06.84
.4
06.88
.6
06.92
.8 06.96
0 07.00
.2 07.04
.4 07.08
.6
07.12
.8
07.16
37.0
.2
TABLE XV.—DEFLECTION ANGLES FOR 10-CHORD SPIRAL

36.0 0 07.20
.2
07.24
.4
07.28
.6
07.32
.8
07.36
1
0 07.40
07.44
.4 07.48
.6
07.52
.8
07.56
.6
.8
38.0
0 07.60
07.64
.2
.4 07.68
39.0
.2
0 07.80
07.84
.4 07.88
.6
.8
41.0
.2
07.72
07.76
40.00 08.00
.2 08.04
.4 08.08
.6
08.12
08.16
.8
07.92
07.96
0 08.20
08.24
.4 08.28
.6
.8
44.0
.2
.4
.6
.8
45.0
08.32
08.36
2
0 08.40
08.44
08.48
0 26.4 0 59.4
26.6 59.8
26.7 1 00.1
26.9
00.5
27.0
00.8
0 27.2
27.4
27.5
27.7
1 01.2
01.6
01.9
02.3
27.8 02.6
0 28.0 1 03.0
28.2
03.4
03.7
28.3
28.5
28.6
0 28.8
29.0
29.1
29.3
29.4
3
0 30.4
30.6
30.7
30.9
31.0
0 29.6 1 06.6
29.8
07.0
29.9
07.3
30.1 07.7
30.2
08.0
42.0
.2
.4
.6 08.52 34.1
.8
08.56
34.2
04.1
04.4
1 04.8
05.2
05.5
05.9
06.2
0 31.2 1 10.2
31.4 10.6
31.5
10.9
31.7
11.3
31.8
11.6
0 33.6
33.8
33.9
1 08.4
08.8
09.1
09.5
09.8
0 32.0 1 12.0
32.2 12.4
32.3 12.7
32.5 13.1
32.6 13.4
0 32.8 1 13.8
33.0 14.2
14.5
33.3 14.9
33.4 15.2
33.1
1
16.0
16.3
4
0 08.60
0 34.4
43.0
.2
34.6
08.64
.4 08.68 34.7 18.1
.6
08.72
08.76
34.9
35.0
18.5
18.8
.8
1 45.6
46.2
46.9
47.5
48.2
1 48.8
49.4
50.1
50.7
51.4
53.3
53.9
54.6
1 55.2
55.8
56.5
57.1
57.8
1 58.4
59.0
5
2 45.0
46.0
3 57.5
59.0
47.0 4 00.4
48.0
01.8
49.0
03.3
2 50.0
51.0
52.0
53.0
16.7
17.0 17.0
1 52.0 2 55.04 11.9 | 5 42.8
5 42.8
52.6 56.0 13.3 44.7
57.0
58.0
59.0
3 05.0
06.0
59.7
07.0
08.0
2 00.3
01.0 09.0
6
3 00.0
01.0
02.0
03.0
04.0
1
4 04.75 33.0
06.2 34.9
07.6
36.9
09.0
38.8
54.0 10.5 40.8
201.6 3 10.0
02.2 11.0
12.0
02.9.
03.5
04.2
4 19.1
20.5
22.0
23.4
24.8
4 26.3
27.7
29.2
30.6
32.0
4 33.5
34.9
36.3
13.0 37.8
14.0 39.2
7
14.8 46.7
16.2
48.6
17.7
50.6
2 04.8 3 15.04 40.7
05.4 16.0 42.1
06.1
17.0
43.5
06.7
45.0
18.0
07.4 19.0
46.4
5 23.2
25.1
27.1
29.1
31.0
2 11.2 3 24.9 4 55,0
11.8 25.9
56.5
12.5 26.9 57.9
13.1 27.9 59.4
13.8 28.9 5 00.8
5 52.5
54.5
56.5
58.4
6 00.4
8
7 01.9
04.5
07.0
09.5
12.1
6 21.9
23.8
25.8
27.7
29.7
7 14.7
17.2
19.7
22.4
24.9
7 27.4
29.9
32.6
35.1
37.6
7 40.2
42.7
45.2
47.8
50.3
6 02.3
04.3
7 52.9
55.5
58.0
08.2 8 00.6
06.2
10.1 03.1
6 12.18 05.6
14.0 08.2
16.0
10.7
18.0
13.3
19.9
15.8
2 08.03 20.0 4 47.9 6 31.6 8 31.1
08.6 20.9 49.3 33.6 33.7
09.3 21.9 50.7 35.5
09.9 22.9 52.2 37.5
10.6 23.9 53.6 39.5
36.2
38.8
41.3
2 14.43 29.9 5 02.2
15.0 30.9 03.7
15.7 31.9 05.1
16.3 32.9 06.5
08.0
33.9
1 17.4 2 17.6 3 34.9 5 09.4
17.8
35.9 10.9
36.9 12.3
19.5 37.9 13.7 06.7 16.9
15.2 08.7 19.5
18.2
700.9 9 09.3
02.9 11.9
04.8 14.4
18.9
20.2 38.9
8 18.4
20.9
23.5
26.0
28.6
6 41.4
8 43.9
43.4
46.4
45.3
49.0
47.3 51.5
49.2 54.0
6 51.18 56.5
53.1 59.1
55.19 01,7
57.0 04.2
59.0 06.8
0 08.80
0 35.2
35.4
5 16.67 10.79 22.0
18.0 12.6 24.6
08.84
08.88
19.5
14.6
27.1
1 19.2 2 20.83 39.9
19.6 21.4 40.9
35.5 19.9 22.1 41.9
08.92 35.7 20.3 22.7 42.9 20.9
08.96 35.8 20.6 23.4 43.9 22.4
0 09.00 0 36.0 1 21.0 2 24.0 3 44.9 5 23.8
16.5
29.7
18.5
32.2
7 20.4
9 34.8
9
8 53.6
56.8
9 00.0
03.2
06.5
9 09.7
12.9
16.1
19.4
22.6
9 25.8
29.0
32.3
35.5
38.7
9 41.9
45.1
48.3
51.6
54.8
9 58.0
10 01.2
04.4
07.6
10.9
10 14.1
17.3
20.5
23.7
26.9
10 46.2
49.4
52.7
55.9
59.1
11 02.3
05.5
08.7
11.9
15.1
10
11 34.4
37.6
40.8
44.0
47.2
10 58.1
11 02.1
06.1
10.0
14.0
11 50.4
53.6
56.8
12 00.0
03.3
12 06.5
11 18.0
21.9
25.9
29.9
33.8
11 37.8
41.7
45.7
49.7
53.6
10 30.1 12 56.9
33.4 13 00.9
36.6 04.8
08.8
39.8
43.0
12.7
11 57.6
12 01.6
05.5
09.5
13.4
12 17.4
21.3
25.3
29.3
33.2
12 37.1
41.1
45.0
49.0
52.9
13 16.7
20.6
24.6
28.5
32.4
11 18.3 13 56.1
21.6 14 00.1
24.8 04.0
08.0
28.0
31.2
11.9
13 36.4
40.4
44.3
48.3
52.2
14 15.8
18.8
23.7
27.7
31.6
14 35.5
39.5
43.4
47.4
51.3
14 55.2
438

DEFLECTION
TO CHORD
POINT NO.
T.S. = 0
-2315 67∞σo
8
9
10
||
-2315
12
13
14
15
16
17
18
19
20
NOTES:
TABLE XVI.-COEFFICIENTS OF a: FOR DEFLECTIONS TO ANY CHORD POINT ON SPIRAL⭑
0
9
16
25
36
49
64
81
100
-
NOVORO OH
4
40
108
8507
322-
~70-
M DIODON HERON AND
2
16
27
3
18
14
10
22
40 36
55 52
72
91
112
70
90
112
4
32
27
20
28
45
5
││
0
13 0
0000 00041 OERSESS
64
6
0
19
40
85 76 63
108
88
133 126
7
72
65
56
45
32
54
17 38
DOO NENO 08.
6593
68
72
115
100
160 154 144 130
TRANSIT AT CHORD POINT NUMBER
8
9 10 T
12 13 14 15
128 162
450
434
119
108
476
162
200| 242 288 338 392
152 189 230 275 324 377
140 176 216 260 308 360 416
95 126 161 200 243 290 341 396 455
80 110 144 182 224
224 270
270 320 374 432
63 92 125
203 248 297 350 407
140❘ 180
180❘ 224 272
116 155 198 245
128 170 216
140 185
152
99
68
108
20 44
00DKOM FRONT
72
104
23 50 81
0
26
56
0
29
25
52
28
0
90
62
32
121 130 135 136
58
31
0
144
154 160 162
90 64 34
112
145
124
99
169 180 187 190 189 184 175 162
196 208 216 | 220 220 216| 208 196 180 160 136
225 238 247 252 253 250 243 232 217 198
70
108
256
270| 280 286 288 286
289 304 315 322 325 324
280
319
364 364 360
175 148
270 256 238 216 190 160
310| 297 280 259 | 234 205
352 340 324 304 280 252
361 378 391 400 405 406 403 396 385 370 351 328 | 301
400 418 432 442 448 450 448 442 432 418 400 378 352
324 340 352 360
See examples on next page and theory in Art. 5-7.
35 74
38
0
37
76
O
40
117 82
16 17 18 19 20
512
578 648
800
495
560
629
722
702 779
680 | 756
540
608
518
585
656 731
494 560 630 704
468
468, 533 602 675
0
43
324
296
380 440 504
351410 473
266 320 378 440
234
234 287 344 405
200 252 308
368
117 164 215
270
329
392
459
80 126 176 230 288 350
416
41 86 135 188 245 306 371
44 92 144 200 260 324
O 47 98 153 212 275
125 07858
126 88 46
172 135 94 49
220 184 144 100
270
322
50
0
52
104
53
572 644
540 611
506 | 576
470❘ 539
432 500
235 196 153 106
288 250 208 162 112
162 224
110 171
0 56 116
55
0 59
0
58
DEFLECTION
TO CHORD
POINT NO.
T.S. = 0
-2345 67000 -2MIS Ø7600
8
9
10
||
12
13
14
15
16
17
18
19
20
439
TABLE XVI.-C.-CORRECTIONS TO TABLE XVI
FOR LARGE DEFLECTIONS (MINUTES)
15°
68ម
25
32
3333444444
34
36
38
40
41
42
43
44
45
Ratio of 1s
0.6
to Ls
0.7
0.02
0.01
0.02 0.05
0.04
0.10
0.5
0.01
0.01
0.02
0.07
0.03
0.08
0.03
0.10
0.04
0.11
0.29
0.04 0.13 0.34
0.05
0.06
0.06
0.06
0.07
0.16
0.20
0.24
0.16
0.17
0.18
0.88
0.94
1.01
0.19
0.49
1.09
0.21
0.53
1.17
0.07 0.22 0.56 1.25
"
0.8
0.04
0.11
0.21
0.39
0.42
0.46
-
0.37
0.45
0.54
0.64
0.75
0.9
0.09
0.22
0.43
0.74
0.90
1.09
1.29
1.52
1.78
1.91
2.06
*1, length of spiral sighted over
S
2.21
2.37
2.54
1.0
0.17
0.41
0.81
1.40
1.70
2.05
2.43
2.86
3.34
3.60
3.87
4.16
4.46
4.77
Table XVI provides a rapid method of obtain-
ing deflections to locate any chord point of a spiral
from a transit set up at any other point. It is
valid for any number of equally-spaced chord
points up to twenty, provided that a, is always de-
fined as the deflection from the tangent at T.S._to
the first chord point. Calculate a₁ from the rela-
204 (in degrees)
tion a₁ (in minutes)
where
n = the number of equal chords in the spiral,
n2
For large deflections and long sights the results obtained by
using Table XVI should be altered by applying small correc-
tions found by interpolation from Table XVI-C. For sights
toward the S.C., these corrections are subtracted; for sights toward
the T.S., they are added. The source of these corrections is
explained in Appendix A.
Given: A spiral with L. = 150 ft and ▲=6°,
EXAMPLE 1.
to be staked as a 3-chord spiral.
20 X 6
9
13.33'
Deflections from the tangent at the T.S. are found by multi-
plying 13.33′ by the coefficients 1, 4, 9. Local tangent at S.C.
is found by turning off deflection of 13.33 × 18=4°00' from
backsight to T.S. No corrections are required.
a1=
EXAMPLE 2. Given: A spiral with Ls=800 ft and ▲ =32°,
to be staked as a 20-chord spiral.
20X32
a1
400
= 1.6'
""
Deflections from the tangent at the T.S. are found by multi-
plying 1.6' by coefficients 0 to 400, found in the column headed
"Transit at 0. Corrections vary from -0.5' for chord point
16 to 1.7', for the sight to the S.C. If chord points 13 to
S.C. are located from a set-up at point 12, deflections from
local tangent are found by multiplying 1.6' by coefficients 37
to 352. The corrections are negligible, the maximum value
being +0.08′ for the backsight to the T.S.
G

440
HEIGHT
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
5www wwww NNNNN ~~~~~ 60 GG FIWW~ ~-
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20
21
22
23
24
25
26
27
28
29
30
32
33
34
35
36
37
38
39
40
BASE
14
26
53
80
107
135
163
193
222
252
282
313
344
376
408
441
474
508
542
576
611
647
682
719
756
793
831
869
907
947
986
1026
1067
1108
1149
1191
1233
1276
1319
1363
1407
1497
1589
1682
1778
1875
1974
2075
2178
2282
TABLE XVII.—LEVEL SECTIONS
BASE
16
3307
3431
3556
30
60
91
122
154
186
219
252
285.
319
354
389
424
460
497
533
571
608
647
685
724
764
804
844
885
926
969
1010
1054
1096
1141
1184
1230
1274
1321
1366
1413
1460
1508
1556
1653
1752
1853
1956
2060
2166
2274
2384
2496
2389
2611
2497 2726
2607
2719
2833
2949
3067 3333
3186
3460
3589
CUBIC YARDS PER 100-FT STATION
BASE
18
3719
3852
34
68
102
137
172
208
245
281
319
356
395
433
472
512
552
593
634
675
717
759
802
845
889
933
978
1023
1069
1115
1161
1208
1256
1304
1352
1401
1450
1500
1550
1601
1652
1704
1808
2833
2956
2844 3081
2964 3208
3085
3337
2245
2359
2475
2593
2712
3208 3468
3600
3734
3870
4008
4148
BASE
20
37
75
113
152
191
231
271
311
352
394
435
478
521
564
608
652
696
742
787
833
880
927
974
1022
1071
1119
1169
1219
1269
1352
1964
1915 2078
2023
2194
2133
2311
1319
1371
1422
1474
1527
1580
1633
1687
1742
1797
2431
2552
2675
2800
2927
BASE
24
45
90
135
181
3727
3867
4008
4152
4297
4444
228
275
322
370
419
468
517
567
617
668
719
770
822
875
928
981
1035
1090
1144
1200
1256
1312
1369
1426
1484
1542
1600
1659
1719
1779
1839
1900
1961
2023
2085
2148
2275
2404
2534
2667
2801
2937
3075
3215
3356
3056
3186
3319 3793
BASE
30
56
112
169
226
4245
4400
4556
4715
4875
5037
284
342
400
459
519
579
639
700
761
823
885
948
1011
1075
1139
1204
1269
1334
1400
1467
1534
1601
1669
1737
1806
1875
1945
2015
2085
2157
2228
2300
2593
2742
2893
3045
3200
3357
3514
3675
3837
4001
3500
4167
3645 4334
4504
3453 3942 4675
3589
4093
4848
BASE
40
74
149
224
300
5742
5926
376
453
530
607
685.
764
843
922
1002
1083
1163
1245
1326
1408
1491
1574
1658
1742
1826
1911
2373 3058
2445
3149
2519
3241
1996
2082
2169
2256
2343
2431
2519
2607
2696
2786
2876
2967
3333
3520
3707
3897
4089
4282
4478
4675
4874
5075
5278
5482
5689
5897
6108
5023
6320
5200 6533
5379
6749
5559
6967
7186
7407
1/4:1

PER FT OF
ADDED BASE
1.85
3.70
5.56
7.41
9.26
11.11
12.96
14.81
16.67
18.52
20.37
22.22
24.07
25.93
27.78
29.63
31.48
33.33
35.18
37.04
38.89
40.74
42.59
44.44
46.30
48.15
50.00
51.85
53.70
55.56
57.41
59.26
61.11
62.96
64.82
66.67
68.52
70.37
72.22
74.07
77.78
81.48
85.19
88.89
92.59
96.30
100.00
103.70
107.41
|||||
114.81
118.52
122.22
125.93
129.63
133.33
137.04
140.74
144.44
148.15
441
HEIGHT
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20
O-23I DON∞0
21
22
23
24
25
26
27
28
29
30
31
32
33
34
MMM
35
36
37
38
39
40
BASE
14
26
54
82
111
141
172
204
237
271
306
341
378
415
454
493
533
575
617
660
704
749
794
841
889
937
987
1037
1089
1141
194
1249
1304
1360
1417
1475
1533
1593
1654
1715
1778
1906
2037
2172
2311
2454
2600
2750
2904
3061
3222
3387
3556
3728
3904
4083
4267
4454
4644
TABLE XVII.—LEVEL SECTIONS
BASE
16
30
61
93
126
160
194
230
267
304
343
382
422
463
506
549
593
638
683
730
778
826
876
926
978
1030
1083
1138
1193
1248
1306
1363
1422
1482
1543
1604
1667
1730
1794
1860
1926
2061
2200
2343
2489
2639
2793
2950
3111
3276
3444
3617
3793
3972
4156
4343
4533
4728
4926
4839
5128
5037 5333
BASE
18
CUBIC YARDS PER 100-FT STATION
34
69
104
141
178
217
256
296
338
380
423
467
512
557
604
652
700
750
800
852
904
957
1012
1067
1123
1180
1238
1296
1356
1417
1478
1541
1604
1669
1734
1800
1867
1935
2004
2074
2217
2363
2513
2667
2824
2985
3150
3319
3491
3667
3846
4030
4217
4407
4602
4800
5002
5207
5417
5630
BASE
20
38
76
115
156
197
239
282
326
371
417
463
511
560
609
660
711
763
817
871
926
982
1039
1097
1156
1215
1276
1338
1400
1463
1528
1593
1659
1726
1794
1863
1933
2004
2076
2149
2222
2372
2526
2683
2844
3009
3178
3350
3526
3706
3889
4076
4267
4461
4659
4861
5067
5276
5489
5706
5926
BASE
24
45
91
138
185
234
283
334
385
438
491
545
600
656
713
771
830
889
950
1012
1074
1138
1202
1267
1333
1400
1469
1538
1607
1678
1750
1823
1896
1971
2046
2123
2200
2278
2357
2438
2519
2683
2852
3024
3200
3380
3563
3750
3941
4135
4333
4535
4741
4950
5163
5380
5600
5824
6052
6283
6519
BASE
30
56
113
171
230
289
350
412
474
538
602
667
733
800
869
938
1007
1078
1150
1223
1296
1371
1446
1523
1600
1678
1757
1838
1919
2000
2083
2167
2252
2338
2424
2512
2600
2689
2780
2871
2963
3115
3341
3535
3733
3935
4141
4350
4563
4780
5000
5224
5452
5683
5919
6157
6400
6646
6896
7150
7407
BASE
40
75
150
226
304
382
461
541
622
704
787
871
956
1041
1128
1215
1304
1393
1483
1575
1667
1760
1854
1949
2044
2141
2239
2338
2437
2538
2639
2741
2844
2949
3054
3160
3267
3375
3483
3593
3704
3928
4156
4387
4622
4861
5104
5350
5600
5854
6111
6372
6637
6906
7178
7454
7733
8017
8304
8594
8889
1/2:1

PER FT OF
ADDED BASE
1.85
3.70
5.56
7.41
9.26
| | . ||
12.96
14.81
16.67
18.52
20.37
22.22
24.07
25.93
27.78
29.63
31.48
33.33
35.18
37.04
38.89
40.74
42.59
44.44
46.30
48.15
50.00
51.85
53.70
55.56
57.41
59.26
61.11
62.96
64.82
66.67
68.52
70.37
72.22
74.07
77.78
81.48
85.19
88.89
92.59
96.30
100.00
103.70
107.41
|| | . ||
114.81
118.52
122.22
125.93
129.63
133.33
137.04
140.74
144.44
148.15
442
HEIGHT
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
22222222
20
21
22 22222 222 Mongo
ƆMM MMMM
567
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
BASE
14
27
250
56
81
119
153
189
227
267
308
352
397
444
494
544
597
652
708
767
827
889
953
1019
1086
156
1227
1300
1375
1452
1531
1611
1694
1778
1864
1952
TABLE XVII.—LEVEL SECTIONS
BASE
16
31
63
97
133
171
211
253
4889
5167
5452
5744
6044
296
342
389
438
489
542
596
653
711
771
833
897
963
1031
1100
1171
1244
1319
1396
1475
1556
1638
1722
1808
1896
1986
2078
2042
2171
2133
2267
2227 2364
2322
2463
2419 2564
3611
3852 4044
4100
4300
4356
4563
4619
4833
CUBIC YARDS PER 100-FT STATION
5111
5396
5689
5989
6296
BASE
18
6352
6611
6667 6933
6989 7263
7319
7656
7600
7944
8000 8296
34
70
108
148
190
233
279
326
375
426
479
533
590
648
708
770
834
900
968
1037
1108
1181
1256
1333
2519
2667
2722
2878
2933 3096
3259
3152
3322
3493
3378 3556 3733
1412
1493
1575
1659
1745
1833
1923
2015
2108
2204
2301
2400
2501
2604
2708
3796 3981
4237
4500
4770
5048
2815
3033
5333
5626
5926
6233
6548
6870
7200
7537
7881
8233
8593
BASE
20
38
78
119
163
208
256
305
356
408
463
519
578
638
700
764
830
897
967
1038
1111
1186
1263
1342
1422
| 505
1589
1675
1763
1853
1944
2038
2133
2231
2330
2431
2533
2638
2744
2853
2963
3189
3422
3663
3911
4167
4430
4700
4978
5263
5556
5856
6163
6478
6800
7130
7467
7811
8163
8522
8889
BASE
24
45
93
142
193
245
300
356
415
475
537
601
667
734
804
875
948
1023
1100
1179
1259
1342
1426
1512
1600
1690
1781
1875
1970
2068
2167
2268
2370
2475
2581
2690
2800
2912
3026
3142
3259
3500
3748
4004
4267
4537
4815
5100
5393
5693
6000
6315
6637
6967
7304
7648
8000
8359
8726
9100
9481
BASE
30
56
115
175
237
301
-367
434
504
575
648
723
800
879
959
1042
1126
1212
1300
1390
1481
1575
1670
1768
1867
1968
2070
2175
2281
2390
2500
2612
2726
2842
2959
3079
3200
3323
3448
3575
3704
3967
4237
4515
4800
6667
7004
BASE
40
8426
8800
9181
75
152
231
311
9570
9967
10370
394
478
564
652
742
833
927
1022
1119
1219
1319
1422
1527
1633
1742
1852
1964
2078
2194
2311
2431
2552
2675
2800
2927
3056
3186
3319
3453
3589
3727
3867
4008
4152
4297
5093
6019
5393 6356
5700
6015
6337
4444
4744
5052
5367
5689
7778
8152
7348 8533
7700
8922
8059
9319
6700
7052
7411
9722
10133
10552
10978
[1411
11852
1:1

PER FT OF
ADDED BASE
1.85
3.70
5.56
7.41
9.26
11.11
12.96
14.81
16.67
18.52
20.37
22.22
24.07
25.93
27.78
29.63
31.48
33.33
35.18
37.04
38.89
40.74
42.59
44.44
46.30
48.15
50.00
51.85
53.70
55.56
57.41
59.26
61.11
62.96
64.82
66.67
68.52
70.37
72.22
74.07
77.78
81.48
85.19
88.89
92.59
96.30
100.00
103.70
107.41
|||.||
114.81
118.52
122.22
125.93
129.63
133.33
137.04
140.74
144.44
148.15
443
HEIGHT
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
NMMIT DOWN N'occio 22222 20700 E COM
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20
21
23
26
28
29
30
31
32
33
im MMMMM=
34
35
36
37
38
39
40
BASE
14
25
27
57
90
126
164
206
250
296
346
398
453
511
572
635
701
770
841
917
994
1074
1157
1243
1331
1422
1516
1613
1713
1815
1920
2028
2138
2252
2368
2487
2609
2733
2861
2991
3124
4769
5104
5450
5807
6176
TABLE XVII.-LEVEL SECTIONS
7761
8185
BASE
16
8620
9067
9524
9993
31
65
101
141
183
228
275
326
379
435
494
556
620
687
757
830
905
983
1064
1148
1235
1324
1416
1511
1609
1709
1813
1919
2027
2139
2253
2370
2490
2613
3259
3407
3539 3694
3830
3993
4131
4302
4444
4622
2738
2867
6556
6946
7348 7585
8006
8437
4954
5296
5650
6015
6391
CUBIC YARDS PER 100-FT STATION
BASE
18
8880
9333
9798
10274
10761
10472
10963 11259
35
72
113
156
201
250
301
356
413
472
535
600
668
739
813
889
968
1050
1135
1222
1313
1406
1501
1600
2868
3000
2998
3135
3131
3272
3268 3413
1701
1806
1913
2022
2135
2250
2368
2489
2613
2739
4472
4800
5139
5489
5850
6222
6606
6778
7000
7176 7406
7822
8250
8689
9139
9600
10072
10556
BASE
20
| 1050
11556
38
80
124
170
220
272
327
385
446
509
575
644
716
791
868
948
1031
1117
1205
1296
1390
1487
3556
3704
3850 4006
4156
4319
4643
4978
1587
1689
1794
1902
2013
2126
2242
2361
2483
2607
2735
2865
2998
3133
3272
3413
3557
5324
5681
6050
6430
6820
7222
7635
8059
8494
8941
BASE
24
46
94
146
200
257
317
379
444
513
583
657
733
813
894
979
1067
1157
1250
1346
1444
1546
1650
1757
1867
2213
2333
2457
2583
2713
2844
2979
3117
3257
3400
3546
3694
3846
4000
4317
4644
4983
5333
5694
6067
6450
6844
7250
1979
2257
2094 2383
2513
7667
8094
BASE
30
57
117
179
244
9398
9917
9867
10400
10346
10894
10837 11400
313
383
457
533
613
694
779
867
957
1050
1146
1244
1346
1450
1557
1667
1779
1894
2013
2133
2644
2779
2917
3057
3200
3346
3494
3646
3800
3957
4117
4279
4444
4783
5133
5494
5867
6250
6644
7050
7467
7894
8333
8783
9244
9717
8533
8983
9444 10200
10694
I 1200
BASE
40
75
154
235
319
405
494
587
681
779
880
983
1089
1198
1309
1424
1541
1661
1783
1909
2037
2168
2302
2438
2578
2720
2865
3013
3163
3316
3472
3631
3793
3957
4124
4294
4467
4642
4820
5001
5185
5561
5948
6346
6756
7176
7607
8050
8504
8969
9444
9931
10430
10939
11459
11991
12533
13087
11717
12244
13652
11339 11917 12783
14228
11852 12444 13333 14815
112:1

PER FT OF
ADDED BASE
1.85
3.70
5.56
7.41
9.26
11.11
12.96
14.81
16.67
18.52
20.37
22.22
24.07
25.93
27.78
29.63
31.48
33.33
35.18
37.04
38.89
40.74
42.59
44.44
46.30
48.15
50.00
51.85
53.70
55.56
57.41
59.26
61.11
62.96
64.82
66.67
68.52
70.37
72.22
74.07
77.78
81.48
85.19
88.89
92.59
96.30
100.00
103.70
107.41
||| . ||
114.81
118.52
122.22
125.93
129.63
133.33
137.04
140.74
144.44
148.15
444
HEIGHT
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20
20
21
22
23
24
~~~~~ MMMMM MMMMMI
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
BASE
14
28
59
94
133
176
222
272
326
383
444
509
578
650
726
806
889
976
1067
1161
1259
1361
1467
1576
1689
1806
1926
2050
2178
2309
2444
2583
2726
2872
3022
3176
3333
3494
3659
3828
4000
4356
4730
5111
5511
TABLE XVII.—LEVEL SECTIONS
BASE
16
31
67
106
148
194
244
298
356
417
481
550
622
698
778
861
948
1039
1133
1231
1333
1439
1548
1661
1778
1898
2022
2150
2556
2698
2844
2994
3148
3306
3467
3631
3800
3972
4148
4511
4889
5281
5689
5926
6111
6356 6548
6800
7000
7467
7259
7733 7948
CUBIC YARDS PER 100-FT STATION
8444
8956
9481
BASE
18
35
74
1991
2119
2250
2281
2385
2417 2524
117
163
213
267
324
385
450
519
591
667
746
830
917
1007
1102
1200
1302
1407
1517
1630
1746
1867
2667
2813
2963
3117
3274
3435
3600
4296
4667
5052
5452
5867
6296
6741
7200
7674
8163
8222
8667
8726
9185
9244
9719
9778 10022 10267
10326
10578 10830
BASE
20
10889
11467
I 2059 12333 12607
12667 12948
13230
13289 13578 13867
39
81
128
178
231
289
350
415
483
556
631
711
794
881
972
1067
1165
3565
3733
3769 3906
3941
4081
4117
4261
1267
1372
1481
1594
1711
1831
1956
2083
2215
2350
2469
2631
2778
2928
3081
3239
3400
4444
4822
5215
5622
6044
6481
6933
7400
7881
8378
8889
9415
9956
10511
11081
11148 11407 11667
11733 12000 12267
12881
13511
14156
13926 14222 14519 14815
BASE
24
46
96
150
207
269
333
402
474
550
630
714
800
891
985
1083
1185
1291
1400
1513
1630
1750
1874
2002
2133
2696
2846
3000
3157
3319
3483
3652
3824
4000
4741
5133
5541
5963
6400
2269
2546
2407 2696
2550
2850
3007
3169
6852
7319
7800
8296
8807
BASE
30
9333
9874
10430
11000
I 1585
57
119
183
252
12185
12800
13430
14074
14733
324
400
480
563
650
741
835
933
1035
1141
1250
1363
1 480
1600
1724
4180 4591
4363
4785
4550
4983
3333
3502
3674
3850
4030
4213
4400
5185
5600
6030
6474
6933
1852
2222
1983
2372
2119
2526
2257 2683
2400 2844
7407
7896
8400
8919
9452
BASE
40
12963
13600
14252
14919
76
156
239
326
15600
15407 16296
417
511
609
711
817
926
1039
| 156
1276
1400
1528
1659
1794
1933
2076
3009
3178
3350
3526
3706
3889
4076
4267
4461
4659
4861
5067
5276
5489
5706
5926
6378
6844
7326
7822
10000
|||||
10563 11711
11141 12326
11733
12956
12341
13600
8333
8859
9400
9956
10526
1 4259
14933
| 5622
16326
17044
17778
2:1

PER FT OF
ADDED BASE
1.85
3.70
5.56
7.41
9.26
11.11
12.96
14.81
16.67
18.52
20.37
22.22
24.07
25.93
27.78
29.63
31.48
33.33
35.18
37.04
38.89
40.74
42.59
44.44
46.30
48.15
50.00
51.85
53.70
55.56
57.41
59.26
61.11
62.96
64.82
66.67
68.52
70.37
72.22
74.07
77.78
81.48
85.19
88.89
92.59
96.30
100.00
103.70
107.41
||| . ||
114.81
118.52
122.22
125.93
129.63
133.33
137.04
140.74
144.44
148.15
445
HEIGHT
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
76000 ~~~~~ ~~~~~ MMMMM MMMMM=
17.5
18.0
18.5
19.0
19.5
20
21
22
wwwww N
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
BASE
14
29
63
103
148
199
256
318
385
458
537
621
711
806
907
1014
1126
1244
1367
1495
1630
1769
1915
2066
2222
2384
2552
2725
2904
3088
4310
4533
4762
4996
5236
TABLE XVII.-LEVEL SECTIONS
BASE
16
8241
8859
32
70
114
163
218
278
344
415
492
574
662
756
855
959
1069
1185
1306
1433
1566
1704
1847
1996
2150
2311
3278
3389
3473
3588
3674 3793
3881
4003
4093
4219
2477
2648
2825
3007
3195
4440
4667
4899
5137
5381
5481
5630
5989 6144
6519
6681
7241
7070
7644
7822
CUBIC YARDS PER 100-FT STATION
9500
10163 10370
10848 11063
BASE
18
36
78
125
178
236
300
369
444
525
611
703
800
903
1011
1125
1244
1369
1500
1636
1778
1925
2078
2236
2400
2569
2744
2925
3111
3303
3500
3703
3911
4569
4800
5036
5278
5525
5778
6300
6844
7411
8000
8426
9052
9700 9900
10578
11278
8611
9244
BASE
20
40
85
136
193
255
322
395
474
558
648
744
844
951
1063
1181
1304
1432
1567
1706
1852
2003
2159
4125
4247
4344 4470
2321
2489
2662
2841
3025
3215
3410
3611
3818
4030
4699
4933
5173
5419
5669
5926
6456
7007
7581
8178
8796
9437
10100
10785
11493
BASE
24
12285 12515
13037 13274
13811 14056
14607 14859 15111 15363
47
100
158
222
292
367
447
533
625
722
825
933
1047
1167
1292
1422
1558
1700
1847
2000
2158
2322
2492
2667
2847
3033
3225
3422
3625
3833
4047
4267
4492
4722
4958
5200
5447
5700
5958
6222
6767
7333
7922
8533
11556 11778 12000 12222 12667
12744
13511
12974 13433
13748 14222
14300 14544 15033
15426 15685
16267 16533
17130 17404
15867
15944 16204 16722
16800 17067 17600
17678
17952 18500
18015 18296 18578 18859 19422
18922 19211 19500 19789
19852 20148 20444
20367
20741 21333
BASE
30
58
122
192
267
347
433
525
622
725
833
947
1067
1192
1322
1458
1600
1747
1900
2058
2222
2392
2567
2747
2933
3125
3322
3525
3733
3947
4167
4392
4622
4858
5100
5347
5600
5858
6122
6392
6667
7233
7822
8433
9067
9167
9722
9822 10400
10500 11100
11200
11922
13333
14122
BASE
40
77
159
247
341
14933
15767
440
544
655
770
892
1019
1151
1289
1432
1581
1736
1896
2062
2233
2410
2593
2781
2974
3173
3378
3588
3804
4025
4252
4484
4722
4966
5215
5469
5730
5995
6267
6544
6826
7114
7407
8011
8637
9285
9956
11822 12859
12567
13641
10648
11363
12100
14444
15270
16119
16989
16622 17881
18796
17500
18400
19733
19322 20693
20267
21674
22678
22222 23704
21233
3:1

PER FT OF
ADDED BASE
1.85
3.70
5.56
7.41
9.26
11.11
12.96
14.81
16.67
18.52
20.37
22.22
24.07
25.93
27.78
29.63
31.48
33.33
35.18
37.04
38.89
40.74
42.59
44.44
46.30
48.15
50.00
51.85
53.70
55.56
57.41
59.26
61.11
62.96
64.82
66.67
68.52
70.37
72.22
74.07
77.78
81.48
85.19
88.89
92.59
96.30
100.00
103.70
107.41
||1.11
114.81
118.52
122.22
125.93
129.63
133.33
137.04
140.74
144.44
148.15
446
The coefficients in the tables on the opposite page provide a
rapid method of correcting the level-section quantities of
Table XVII for transverse ground slopes.
.b.

or
From the figure,
Similarly,
21
which reduces to
$
b
di==+sc+
2
dr=
dr
b
2 + sc
1-
b
SG
100
+sc
SG
100
1+-
C
sd G
100
G
100
dr.
d.G
A = AL + 1 ( +sc) ( 100 ) − 1 ( +) (100)
-
-sc
212
A = AL+260 (2 +sc) (dı−d,)
2002
From equation 6-3, the area of the level section is AL=
c(b+cs). On a transverse slope, the level-section area is in-
creased by one triangular area and decreased by another, as
shown by the cross-hatched areas in the figure. The resulting
net area A is
(1)
33
(+2) (0)
[ ]
1002
-s2
G
(2)
(3)
When equations 1 and 2 are substituted in equation 3, the
net area A reduces to
A=AL+(c+;
Therefore,
2
Cr Cu yd per 100 ft = level-section yardage+ c(c+1) 3
(4)
447
BASE
b
14
16
18
20
24
30
40
50
60
GROUND
SLOPE
G
TABLE XVII -A. VALUES OF b/2s

where
:1
28
32
36
40
48
60
80
100
120
2:1
14
16
18
20
24
30
28988
40
50
60
SIDE SLOPE RATIO, s
1:1
7
8
9
10
12
15
20
25
30
12:1
4.67
5.33
6
6.67
8
10
SIDE SLOPE RATIO, $
4:1
2:1
1:1
1:1
5%
0.0001
0.0012
0.0093
0.0313
10%
0.0006 0.0046
0.0374
0.1278
15%
0.0013 0.0105
0.0852
0.2962
20%
0.0023 0.0187
0.1543
0.5494
25% 0.0036 0.0294
0.2469
0.9090
30% 0.0052 0.0426
0.3663 1.411
35%
0.0071 0.0585
0.5170
2.113
40%
0.0093 0.0771
0.7054
3.125
45%
0.0119 0.0988
0.9405
4.650
50%
0.0147 0.1234
1.235
7.143
60%
2.083
0.0213 0.1831
70% 0.0292 0.2585 3.557
80% 0.0386 0.3526 6.584
90% 0.0494 0.4702 15.79
100% 0.0617 0.6172
13.3
16.7
20
TABLE XVII.-B. VALUES OF C
23.68
100
C-(199) [(0) --]
с
27
1002
s2
=
2:1
3.50
4
4.50
5
6
7.5
10
12.5
15
2:1
0.0748
0.3086
0.7325
1.411
2.469
4.167
7.117
13.17
31.58
3:1
2.33
2.67
3
3.33
4
5
6.67
8.33
10

3:1
0.2558
1.099
3.821
6.250
14.29
47.37
Example. Given: a cross section on a uniform transverse
ground slope G of 20%. Other data are: base b=30 ft; center
cut or fill c=15 ft; side slope ratio s=
=1:1.
From Table XVII,
Level-section quantity =2917 cu yd per sta.
From Tables XVII-A and XVII.-B,
Correction for slope
(15+10)2(0.5494)
343 cu yd per sta.
Yardage corrected for slope=3260 cu yd per sta.
448
HEIGHT
OR
WIDTH
0.1
0.09
.2 0.19
0.28
0.37
.3
=
.4
0.5
.6
.7
.8
.9
1.0
.I
•~34 567
.2
.3
.4
1.5
.6
.7
.8
.9
2.0
.2
.3
E SO780
.4
2.5
.6
.7
1.85
.I 1.94
2.04
2.13
.8
.9
3.0
. I
.2
.3
= 56 |
.4
3.5
.6
.7
4.0
.I
234
.2
.3
.4
TABLE XVIII.-TRIANGULAR PRISMS. CUBIC YARDS
1
0.46
0.56
0.65
0.74
0.83
•234 56769
0.93
1.02
5.5
.6
.7
.8
.9
1.11
3.24
3.33
3.43
.8 3.52
.9
3.61
1.20
1.30
1.39
1.48
1.57
1.67
1.76
2.22
2.31
2.41
2.50
2.59
2.69
2.78
2.87
2.96
3.06
3.15
4.5
.6
.7
.8 4.44
.9 4.54
3.70
3.80
3.89
3.98
4.07
4.17
4.26
4.35
2
0.19
0.37
0.56
0.74
0.93
1.11
1.30
1.48
1.67
1.85
2.04
2.22
2.41
2.59
2.78
2.96
3.15
3.33
3.52
4.63
4.81
5.00
5.19
5.37
3
5.56
5.74
5.93
6.11
6.30
0.28
0.56
0.83
1.11
1.39
1.67
1.94
2.22
2.50
2.78
3.06
3.33
3.61
3.89
3.70
5.56
3.89
5.83
4.07
6.11
4.26 6.39
4.44
6.67
4.17
4.44
4.72
5.00
5.28
8.61
8.89
9.17
9.44
6.48
9.72
6.67 10.00
6.85
7.04 10.56
7.22
10.83
7.41 11.11
7.59 11.39
7.78 11.67
7.96 11.94
8.15 12.22
5.0 4.63
.I 4.72
.2 4.81
.3
9.26 13.89
9.44 14.17
9.63 14.44
4.91
9.81 14.72
.4 5.00 10.00 15.00
PER 50 FEET

8.33 12.50
8.52 12.78
8.70
13.06
8.89 13.33
9.07
13.61
WIDTH OR HEIGHT
0.37
0.74
1.11
1.48
6.94
9.26
7.22
9.63
7.50
10.00
7.78 10.37
8.00
10.74
1.85
2.22
2.59
2.96
3.33
8.33 11.11
11.48
11.85
12.22
12.59
3.70
4.07
4.44
4.81
5.19
5.56
5.93
6.30
6.67
7.04
7.41
7.78
8.15
8.52
8.89
12.96
13.33
10.28 13.70
14.07
14.44
14.81
15.19
15.56
15.93
16.30
18.52
18.89
19.26
19.63
20.00
5.09 10.19 15.28
5.19 10.37 15.56
5.28 10.56 15.83 21.11
20.37
20.74
5.37 10.74 16.11 21.48
5.46 10.93 16.39 21.85
5
0.46
0.93
1.39
1.85
2.31
2.78
3.24
3.70
4.17
4.63
5.09
5.56
6.02
6.48
6.94
7.41
7.87
8.33
8.80
16.67
20.83
17.04 21.30
17.41
21.76
17.78
18.15
18.52
19.98
19.44
19.91
20.37
6
0.56
1.11
1.67
2.22
9.26
9.72
10.19
12.22
10.65
12.78
|.|| 13.33
22.22
22.69
2.78
3.33
3.89
23.15
23.61
24.07
4.44
5.00
24.54
25.00
5.56
6.11
6.67
7.22
7.78
16.20 19.44
16.67 20.00
17.13 20.56
17.59
18.06
25.46
25.93
26.39
26.85
27.31
7
0.65
1.30
11.57 13.89
16.20
12.04
14.44
16.85
12.50
15.00
17.50
12.96 15.56 18.15
13.113
!6.!!
18.80
1.94
2.59
8.33
8.89
9.44
10.00 11.67
10.56
12.31
13.89
16.67 19.44
14.35
17.22
20.09
14.81
17.78
20.74
15.28 18.33
21.39
15.74 18.89
22.04
3.24
3.89
4.54
5.19
5.83
6.48
7.13
7.78
8.43
9.07
11.11 12.96
11.67
9.72
10.37
11.02
26.11
26.67
27.22
13.61
14.26
14.91
15.56
22.69
23.33
23.98
21.11 24.63
21.67
25.28
22.22
25.93
22.78 26.57
23.33
23.89
24.44
27.22
27.87
28.52
25.00
29.17
25.56 29.81
30.46
8
0.74
1.48
30.56
35.65
31.11 36.30
31.67
2.22
2.96
3.70
4.44
5.19
5.93
6.67
7.41
8.15
8.89
9.63
10.37
||.||
11.85
12.59
13.33
14.07
18.52
19.26
20.00
20.74
21.48
22.22
22.96
23.70
24.44
25.19
25.93
26.67
27.41
28.15
28.89
29.63
30.37
14.81 16.67
15.56
17.50
16.30
18.33
17.04
19.17
17.78 20.00
31.11
31.85
32.59
33.33
34.07
34.81
31.11 35.56
31.76 36.30
38.52
39.26
40.00
9
40.74
41.48
36.94 42.22
0.83
1.67
2.50
3.33
32.22 37.59 42.96
32.78
38.24
43.70
4.17
5.00
5.83
6.67
7.50
8.33
9.17
10.00
10.83
11.67
12.50
13.33
14.17
15.00
15.83
20.83
21.67
22.50
23.33
24.17
25.00
25.83
26.67
27.50
28.33
29.17
30.00
30.83
31.67
32.50
27.78 32.41 37.04 41.67
28.33 33.06 37.78
42.50
28.89 33.70
43.33
29.44
34.35
44.17
30.00
35.00
45.00
33.33
34.17
35.00
35.83
36.67
37.50
38.33
39.17
40.00
40.83
45.83
46.67
47.50
48.33
49.17
449
HEIGHT
OR
WIDTH
0-23=
6.0
. I
.2
.3
.4
5678 σ
6.5
.6
.7
.8
.9
7.0
.2
.3
= 56
4
7.5
6.48
. Į 6.57
6.67
6.76
6.85
.6
234
.3
8.5
.6
.7
.I
.2
.3
~MI 56
.4
9.5
TABLE XVIII.-TRIANGULAR PRISMS. CUBIC YARDS
1
6.94
13.89 20.83
.7
21.67
7.04 14.07 21.11
7.13 14.26 21.39
.8 7.22 14.44
.9 7.31 14.63
8.0 7.41 14.81
7.50 15.00
21.94
.2 7.59 15.19
.6
·
5.56
5.65
11.11
11.30
5.74
11.48
5.83 11.67
5.93
11.85
6.02
6.11
6.20
6.30
6.39
22.22
22.50
22.78
7.69 15.37 23.06
.4 7.78 15.56 23.33
~MI NO76a
.3
.7 8.98
2
9.0 8.33
8.43
8.52
8.61 17.22 25.83
8.70
17.41
26.11
.8 9.07
9.17
8.80
8.89
.8
12.04
12.22
12.41
12.59
12.78
3
16.67
16.94
9.54
18.06
18.33
18.61
18.89
19.17
12.96
19.44
13.15 19.72
13.33
20.00
13.52 20.28
13.70
20.56
17.22 22.96
17.50
17.78
7.87 15.74
15.93
7.96
8.06
16.11
23.61 31.48
23.89 31.85
24.17
16.30 ! 24.44
16.48
32.22
.8 8.15
32.59
.9 8.24
24.72
32.96
.9
10.0
9.26 18.52
27.78
.I 9.35 18.70 28.06
16.67 25.00
16.85 25.28
17.04 25.56
.2 9.44 18.89 28.33
.9 10.09 20.19
.4 9.63 19.26 28.89
PER 50 FEET

4
WIDTH OR HEIGHT
22.22
22.59
23.33
23.70
20.37 30.56
20.56
20.74
20.93
21.11
24.07
24.44
24.81
25.19
25.56
25.93
26.30
26.67
27.04
27.41
27.78
28.15
28.52
28.89
29.26
17.59 26.39 35.19
17.78 26.67 35.56
17.96 26.94 35.93
18.15 27.22 36.30
18.33 27.50
36.67
29.63
30.00
30.37
30.74
31.11
10.5
38.52
19.44 29.17 38.89
.6 9.81 19.63 29.44 39.26
9.72
.7
9.91
19.81 29.72
39.63
10.00 20.00 30.00
40.00
30.28
40.37
40.74
30.83 41.11
31.11 41.48
31.39 41.85
31.67 42.22
6
27.78 33.33
28.24
33.89
5
28.70
29.17
29.63
30.09
30.56
31.02
31.48
31.94
32.41
32.87
33.33
33.80
34.26
34.72
35.19
35.65
36.11
36.57
37.04
37.50
37.96
38.43
38.89
33.33 41.67
33.70
42.13
34.07
34.44
34.81
37.04
46.30
37.41
46.76
37.78
47.22
19.07 28.61 38.15 47.69
40.28
40.74
41.20
42.59
43.06
43.52
43.98
44.44
44.91
45.37
45.83
34.44
35.00
35.56
39.35
47.22
39.81 47.78
36.11
36.67
37.22
37.78
38.33
50.93
51.39
51.85
38.89
39.44
40.00
40.56
41.11
44.44
45.00
45.56
46.11
46.67
48.33
48.89
49.44
41.67
48.61
42.22 49.26
42.78
49.91
43.33 50.56
43.89
51.20
50.00
50.56
51.1
51.67
52.22
52.78
53.33
53.89
54.44
55.00
48.15
48.61 58.33
49.07 58.89
49.54
50.00
50.46
7
11.0
10.19
.1
10.28
.2
10.37
.3 10.46
10.56
10.65 21.30 31.94 42.59
53.24
63.89
.4
11.5
.6 10.74 21.48 32.22 42.96
.7 10.83 21.67 32.50 43.33 54.17 65.00
.8 10.93 21.85 32.78 43.70 54.63
.9 11.02 22.04 33.06 44.07 55.09
53.70
64.44
65.56
66.11
38.89
39.54
40.19
40.83
41.48 47.41
42.13
42.78
43.43
44.07
44.72
61.11
61.67
62.22
52.31 62.78
52.78
63.33
45.37
46.02
46.67
8
44.44
45.19
45.93
51.67
46.67 52.50
53.33
51.85
52.59
53.33
48.15 54.17
48.89 55.00
49.63 55.83
50.37
56.67
51.11
57.50
51.85
59.26
52.50 60.00
53.15
60.74
53.80 61.48
54.44
62.22
71.30
71.94
47.31 54.07
47.96 54.81 61.67
55.56
56.30
57.04
57.78
58.52
72.59
73.24
73.89
55.09
62.96
55.74 63.70
56.39
64.44
57.04
65.19
57.69
55.56
64.81 74.07
56.11
65.46
74.81
56.67 66.11
75.56
57.22 66.76
76.30
57.78
67.41
77.04
9
61.57
62.22
62.87
63.52 72.59
64.17 73.33
50.00
50.83
71.11
71.85
58.33
59.17
60.00
60.83
70.83
71.67
72.50
73.33
65.93 74.17
58.33
66.67 75.00
58.98
67.41
75.83
59.63 68.15 76.67
60.28 68.89 77.50
60.93 69.63
78.33
81.48
82.22
82.96
83.70
84.44
62.50
63.33
64.17
65.00
65.83
66.67
67.50
68.33
70.37 79.17
80.00
80.83
69.17
70.00
68.06
77.78
68.70 78.52
59.44 69.35 79.26 89.17
60.00
70.00
80.00
90.00
60.56
70.65
80.74
90.83
81.67
82.50
83.33
84.17
85.00
85.83
86.67
87.50
88.33
91.67
92.50
93.33
94.17
95.00
74.54
85.19
95.83
75.19 85.93
96.67
97.50
75.83 86.67
76.48 87.41 98.33
77.13 88.15 99.17
450
HEIGHT
OR
WIDTH
12.0
.1
.2
.3
.4
12.5
.6
.7
.8
.9
13.0
234 SU
.3
.4
12.04
.1 12.13
13.5
.6
.7
.8
.9
14.0
.2 12.22 24.44
JMI
.3
15.5
.6
.7
TABLE XVIII.-TRIANGULAR PRISMS. CUBIC YARDS
1
.8
.9
| | . ||
11.20
11.30
11.39
11.48
16.5
2
233 567∞a
22.22
22.41
11.57 23.15
11.67
23.33
11.76
23.52
11.85
23.70
11.94
23.89
12.50
12.59
12.69
12.78
12.87
12.96
25.93
.I 13.06 26.11
.2
.3
.4
22.59
33.89
22.78
34.17
22.96 34.44
24.07
24.26
3
33.33
33.61
34.72
35.00
35.28
35.56
35.83
12.31 24.63
12.41 24.81 37.22
14.5
13.43
.6
13.52
.7
13.61
.8 13.70 27.41
.9 13.80
15.0
13.89 27.78 41.67
.1 13.98 27.96 41.94
14.07 28.15
14.17 28.33 42.50
.2
.4 14.26 28.52 42.78
25.00
37.50
25.19 37.78
25.37
25.56
25.74
38.89
39.17
13.15 26.30 39.44
13.24 26.48
39.72
26.67 40.00
13.33
26.85
40.28
27.04
40.56
27.22 40.83
41.11
27.59 41.39
PER 50 FEET

4
50.00
50.37
38.06
50.74
38.33 51.11
38.61
51.48
14.35 28.70 43.06
14.44
14.54
28.89 43.33
29.07 43.61
29.26 43.89
14.63
14.72
29.44 44.17
16.0
.J
.2 15.00 30.00 45.00
.3 15.09 30.19
45.28
45.56
.4
15.19 30.37
WIDTH OR HEIGHT
44.44
44.81
45.19
45.56
45.93
36.11
48.15
60.19
36.39
48.52 60.65
36.67
48.89
61.11
36.94 49.26
61.57
49.63
62.04
46.30
46.67
47.04
47.41
47.78
51.85
52.22
52.59
52.96
53.33
55.56
55.93
42.22 56.30
56.67
57.04
53.70
54.07
54.44
54.81
55.19
14.81 29.63 44.44 59.26
14.91 29.81 44.72
59.63
60.00
60.37
60.74
15.28
.6 15.37
.7
15.46 30.93 46.39
.8 15.56 31.11 46.67 62.22
.9 15.65
31.30 46.94
62.59
30.56
45.83 61.11
30.74 46.11 61.48
61.85
17.0
62.96
15.74 31.48 47.22
.1 15.83 31.67 47.50 63.33
.2 15.93 31.85 47.78 63.70
16.02
.3
.4
17.5
.6
32.04 48.06 64.07
16.11 32.22 48.33 64.44
16.20 32.41 48.61 64.81
16.30 32.59 48.89 65.19
16.39 32.78 49.17 65.56
16.48 32.96 49.44 65.93
.9 16.57 33.15 49.72 66.30
.7
.8
5
55.56
56.02
56.48
56.94
57.41
57.87
58.33
58.70
59.26
59.72
62.50
62.96
63.43
63.89
64.35
57.41 71.76
57.78
58.15
58.52
58.89
72.22
72.69
73.15
73.61
74.07
74.54
64.81
77.78
65.28
78.33
65.74
78.89
66.20 79.44
66.67
80.00
67.13
80.56
67.59
81.11
68.06 81.67
68.52
82.22
68.98
82.78
76.39
76.85
69.44
83.33
69.91 83.89
70.37 84.44
70.83 85.00
71.30
85.56
77.31
77.78
78.24
6
78.70
79.17
79.63
80.09
80.56
66.67 77.78
67.22
67.78
68.33
68.89
69.44
70.00
70.56
81.02
81.48
81.94
71.11
71.67
82.41
82.87
75.00
75.56
76.11
76.67
77.22
7
72.22
84.26
72.78 84.91
73.33
85.56
73.89
86.20
74.44
86.85
78.43
79.07
79.72
80.37
81.02
81.67
82.31
82.96
83.61
88.89 103.70
89.44 104.35
90.00 105.00
75.00
75.46 90.56 105.65
75.93
87.50
88.15
8
86.11
100.46
86.67
101.11
87.22 101.76
87.78
102.41
88.33 103.06
9
88.89 100.00
89.63
100.83
90.37
101.67
91.11 102.50
91.85 103.33
92.59 104.17
93.33 105.00
94.07 105.83
112.50
113.33
88.80
101.48 114.17
89.44 102.22 115.00
90.09 102.96
115.83
94.81 106.67
95.56 107.50
96.30 108.33
97.04 109.17
97.78 110.00
98.52 110.83
99.26
111.67
100.00
100.74
116.67
90.74 103.70
91.39 104.44 117.50
105.19 118.33
92.04
92.69 105.93 119.17
93.33
106.67
120.00
93.98 107.41 120.83
94.63 108.15
121.67
95.28
122.50
108.89
95.93 109.63
123.33
96.57 110.37 124.17
97.22 |||.||
97.87 111.85
98.52 112.59 126.67
99.17 113.33 127.50
99.81 114.07 128.33
125.00
125.83
114.81 129.17
115.56 130.00
116.30 130.83
117.04 131.67
117.78 132.50
118.52 133.33
119.26
134.17
120.00
135.00
120.74
135.83
91.11 106.30 121.48 136.67
91.67 106.94 122.22 137.50
92.22 107.59 122.96 138.33
92.78 108.24 123.70 139.17
93.33 108.89 124.44 140.00
93.89 109.54 125.19
140.83
94.44 110.19 125.93
141.67
95.00 110.83 126.67 142.50
111.48 127.41 143.33
95.56
96.11 112.13 128.15 144.17
96.67 112.78 128.89 145.00
97.22 113.43 129.63 145.83
97.78 114.07 130.37 146.67
98.33 114.72 131.11 147.50
98.89
115.37 131.85 148.33
99.44 116.02 132.59 149.17
451
HEIGHT
OR
WIDTH
18.0
|
.2 16.85
16.94
17.04
.3
M=
.4
18.5
.9
19.0
17.13
.6 17.22
.7
.8
123=
•
.2
.3
.9
20.0
.1
17.59 35.19
.I 17.69 35.37
17.78
19.5
.6
.7
.8 18.33
18.43
234 567∞a
.3
TABLE XVIII.—TRIANGULAR PRISMS. CUBIC YARDS
1
17.87 35.74
.4 17.96 35.93
20.5
.8
.2 18.70
21.0
.1
.2
0-23I SO7∞a
16.67 33.33
16.76
17.31
17.41
17.50
[
18.06
18.15
18.24
21.5
2
18.52
55.56
37.04
18.61 37.22 55.83
37.41 56.11
37.59 56.39
.4 18.89 37.78 56.67
234
50.00
33.52 50.28
33.70
50.56
33.89
50.83
34.07 51.11
18.80
34.26
34.44
34.63
34.81
35.00
18.98 37.96 56.94
.6 19.07 38.15 57.22
19.17 38.33
19.26 38.52 57.78
58.06
.7
.9 19.35 38.70
52.78
53.06
35.56 53.33
53.61
53.89
36.11
36.30
3
23.5 21.76
.6 21.85
.7 21.94
.8 22.04
.9 22.13
51.39
51.67
51.94
52.22
52.50
.3 19.72 39.44
36.48
54.72
36.67
55.00
36.85 55.28
.4 19.81 39.63
54.17
54.44
19.63 39.26 58.89
PER 50 FEET

59.17
59.44
WIDTH OR HEIGHT
66.67
67.04
43.52 65.28
43.70 65.56
43.89 65.83
44.07 66.11
44.26 66.39
67.41
67.78
68.15
19.44 38.89 58.33
77.78
19.54 39.07 58.61 78.15
78.52
68.52
68.89
69.26
69.63
70.00
70.37
70.74
71.11
71.48
71.85
75.93
76.30
57.50 76.67
77.04
77.41
72.22
72.59
72.96
73.33
73.70
5
78.89
79.26
83.33 100.00
83.80
84.26
84.72
85.19
6
85.65
102.78
86.11 103.33
86.57 103.89
104.44
87.50 105.00
87.04
87.96 105.56
88.43 106.11
88.89 106.67
89.35
107.22
89.81
107.78
74.07
92.59 |||||
74.44
93.06
[11.67
74.81 93.52
75.19
93.98
75.56
94.44
100.56 117.31
101.11 | 117.96
101.67 118.61
102.22 119.26
97.22
97.69
98.15
109.44
110.00
110.56
94.91 113.89
95.37
114.44
115.00
95.83
96.30
115.56
96.76 116.11
136.11 155.56 175.00
136.76
156.30 175.83
137.41
138.06
19.91 39.81 59.72 79.63
.6 20.00 40.00 60.00
.7
180.83
181.67
157.04 176.67
157.78 177.50
138.70 158.52 178.33
99.54 119.44 139.35 159.26 179.17
80.00 100.00 120.00 140.00 160.00 180.00
20.09 40.19 60.28 80.37 100.46 120.56 140.65 160.74
.8 20.19 40.37 60.56 80.74 100.93 121.11 141.30 161.48
.9 20.28 40.56 60.83 81.11 101.39 121.67 141.94 162.22
22.0 20.37 40.74 61.11 81.48 101.85
20.46 40.93 61.39 81.85 102.31
.2 20.56 41.11 61.67 82.22 102.78
.3 20.65 41.30 61.94 82.59 103.24
.4 20.74 41.48 62.22 82.96 103.70
41.67 62.50 83.33
41.85 62.78 83.70
182.50
104.17
104.63
84.07 105.09
84.44 105.56
84.81 106.02
22.5 20.83
.6 20.93
.7 21.02 42.04 63.06
.8 21.11 42.22 63.33
.9 21.20 42.41 63.61
23.0 21.30 42.59 63.89 85.19 106.48
.1 21.39 42.78 64.17 85.56 106.94
.2 21.48 42.96 64.44 85.93 107.41
86.30 107.87
86.67 108.33
.3
21.57
43.15 64.72
.4 21.67 43.33 65.00
87.04 108.80
87.41 109.26
87.78 109.72
88.15 110.19
88.52 110.65
112.22
112.78
113.33
11.6.67
117.22
117.78
98.61 118.33
99.07 118.89
7
90.28
108.33 126.39 144.44
162.50
90.74 108.89 127.04 145.19
163.33
91.20
127.69 145.93
91.67
164.17
128.33 146.67 165.00
128.98 147.41 165.83
92.13
116.67 133.33 150.00
134.07 150.83
134.81 151.67
135.56 152.50
136.30 153.33
123.15
123.80
124.44
125.09
125.74
119.91 137.04 154.17
120.56 137.78 155.00
121.20 138.52 155.83
121.85 139.26 156.67
122.50 140.00 157.50
8
9
132.87
133.52
134.17
134.81
135.46
129.63 148.15 166.67
130.28 148.89 167.50
130.93 149.63 168.33
131.57 150.37 169.17
132.22 151. 170.00
140.74 158.33
141.48 159.17
142.22 160.00
142.96 160.83
143.70
161.67
126.11 147.13
126.67 147.78
127.22 148.43
127.78
128.33
128.89
129.44
130.00 151.67
151.85 170.83
152.59 171.67
153.33 172.50
154.07 173.33
154.81 174.17
184.17
122.22 142.59 162.96 183.33
122.78 143.24 163.70
123.33 143.89 164.44
123.89
185.00
144.54 165.19 185.83
124.44 145.19 165.93
186.67
125.00 145.83 166.67 187.50
125.56 146.48
3/
167.41 188.33
168.15 189.17
168.89 190.00
169.63 190.83
149.07
170.37 191.67
149.72 171.11 192.50
150.37
151.02 172.59
171.85 193.33
194.17
195.00
173.33
130.56 152.31 174.07
131.11 152.96 174.81
131.67 153.61 175.56
132.22 154.26 176.30 198.33
132.78 154.91 177.04 199.17
195.83
196.67
197.50
452
HEIGHT
OR
WIDTH
24.0
.I
24.5
22.31
.2 22.41
.3 22.50
.4 22.59
25.5
22.69
.6 22.78
.7 22.87
.8 22.96
.9 23.06
23.15
23.24
25.0
.1
.2 23.33
.3 23.43
.4 23.52
23.61
.6 23.70
.7
23.80
.8 23.89
.9
26.0
.3
.4
26.5
.6
.7
.8
.9
24.07
. 1 24.17
.2
24.26
TABLE XVIII.-TRIANGULAR PRISMS. CUBIC YARDS
1
34
22.22 44.44
44.63
44.81
45.00
45.19
28.5
· 6
.>
29.0
· /
~MI 5o7
29.5
28.0 25.93
.6
2
47.78
23.98 47.96
24.81
24.91
48.15
48.33
48.52
24.35 48.70
24.44
51.85
.I 26.02 52.04
.2 26.11 52.22
26.20 52.41
.3
.4
47.22
47.41
47.59
46.30 69.44
46.48
69.72
46.67
70.00
46.85
70.28
47.04 70.56
24.54
49.07
24.63
49.26
24.72 49.44
49.63
49.81
3
66.67
66.94
67.22
67.50
67.78
45.37
68.06
45.56 68.33
45.74
45.93
68.61 91.48
68.89
46.11 69.17
91.85
92.22
70.83
71.11
72.22
72.50
72.78
73.06
48.89 73.33
.9 27.69
•
71.39
71.67
71.94
PER 50 FEET

4
74.44
74.72
WIDTH OR HEIGHT
88.89
89.26
89.63
90.00
90.37
90.74 113.43
91.11 113.89
114.35
114.81
115.28
96.30
96.67
97.04
50.00
75.00
100.00
125.00
27.0 25.00
.1 25.09
.2 25.19 50.37
50.19 75.28
100.37
125.46
100.74
125.93
126.85
75.56
.3 25.28 50.56 75.83 101.11 126.39
.4 25.37 50.74 76.11 101.48
27.5 25.46 50.93 76.39 101.85
.6 25.56 51.11 76.67 102.22
.7 25.65 51.30
.8 25.74 51.48
.9 | 25.83
76.94 102.59
77.22
102.96
103.33
51.67
77.50
97.41
97.78
73.61 98.15
73.89
98.52
74.17
98.89
99.26
99.63
92.59 115.74
92.96 11.6.20
93.33
116.67
93.70 117.13
94.07
117.59
77.78
103.70
78.06 104.07
78.33 104.44
78.61 104.81
26.30 52.59 78.89 105.19
5
94.44
118.06
94.81 118.52
95.19 118.98
95.56 119.44
95.93 119.91
||| . ||
111.57
112.04
27.31 54.63 81.94 109.26
133.33 155.56 177.78 200.00
133.89 156.20 178.52 200.83
134.44 156.85 179.26 201.67
112.50 135.00 157.50 180.00 202.50
112.96 135.56 158.15 180.74 203.33
120.37
120.83
121.30
121.76
122.22
122.69
123.15
123.61
124.07
124.54
6
129.63
130.09
136.57
27.41 54.81 82.22 109.63 137.04
138.89
139.44
140.00
26.39
26.48
26.57
.8 26.67
.9
26.76
26.85
80.56 107.41
134.26
161.11
53.70
26.94 53.89 80.83 107.78
.2 27.04 54.07 81.11 108.15
134.72
161.67
135.19
162.22
.3
54.26 81.39 108.52
135.65
162.78
27.13
.4 27.22 54.44 81.67 108.89 136.11 163.33
.8 27.59 55.19 82.78 110.37 137.96
55.37 83.06 110.74 138.43
136.11 158.80 181.48 204.17
136.67 159.44 182.22 205.00
137.22 160.09 182.96 205.83
137.78
160.74
183.70 206.67
184.44 207.50
138.33
161.39
140.56
141.11 164.63
144.44
145.00
145.56
146.||
146.67
127.31 152.78
127.78 153.33
128.24
153.89
128.70
154.44
129.17
155.00
155.56
156.11
156.67
130.56
131.02
157.22
131.48 157.78
150.00
150.56
151.11
151.67
152.22
52.78 79.17 105.56 131.94 158.33
52.96
79.44 105.93 132.41
53.15 79.72 106.30 132.87
53.33 80.00 106.67
53.52
133.33
80.28 107.04
133.80
7
147.22
171.76
147.78 172.41
148.33
173.06
148.89
173.70
174.35
149.44
141.67 165.28 188.89 212.50
142.22 165.93 189.63 213.33
142.78 166.57 190.37 214.17
143.33 167.22
143.89
191.11 215.00
167.87 191.85 215.83
162.04
162.69
163.33
163.98
158.89
159.44
160.00
160.56
.7 27.50 55.00 82.50 110.00 137.50 165.00
165.56
166.11
8
9
185.19 208.33
185.93 209.17
186.67 210.00
187.41 210.83
188.15 211.67
192.59 216.67
168.52
169.17
193.33 217.50
169.81
194.07 218.33
170.46 194.81 219.17
171.11
195.56 220.00
196.30 220.83
197.04 221.67
197.78 222.50
198.52 223.33
199.26 224.17
175.00 200.00 225.00
175.65 200.74 225.83
176.30 | 201.48 226.67
176.94 202.22 227.50
177.59 202.96 228.33
178.24 203.70 229.17
178.89 204.44 230.00
179.54 205.19 230.83
180.19 205.93 231.67
180.83 206.67 232.50
181.48 207.41 233.33
182.13 208.15 234.17
182.78 208.89 235.00
183.43 209.63 235.83
184.07 210.37 236.67
184.72 211.11 237.50
185.37 211.85 238.33
186.02 212.59 239.17
186.67 213.33 240.00
187.31 214.07 240.83
187.96 214.81 241.67
188.61 215.56 242.50
189.26 216.30
243.33
189.91 217.04 244.17
190.56 217.78
245.00
245.83
163.89 191.20
164.44 191.85
219.26 246.67
220.00 | 247.50
192.50
193.15 220.74 248.33
193.80 221.48
249.17
218.52
218.52
453
HEIGHT
OR
WIDTH
.2
30.0 27.78
.127.87
27.96
28.06
.4 28.15
JMI
.3
30.5
0123-
.6 28.33
.7
28.43
.8 28.52
.9
28.61
31.0
.I
TABLE XVIII.—TRIANGULAR PRISMS. CUBIC YARDS
1
33.5
32.5 30.09
.6 30.19
.7 30.28
.8 30.37
.930.46
28.24 56.48
84.72
85.00
56.67
56.85
85.28
57.04 85.56
57.22
85.83
31.5 29.17 58.33
.6 29.26
.7 29.35
.8 29.44
.9 | 29.54
58.52
58.70
32.0 29.63
.129.72
.229.81
.3 29.91
.430.00
33.0 30.56
.I 30.65
.2 30.74
.3 30.83
.430.93
2
31.02
.6 31.11
.731.20
.831.30
.931.39
0-23=
55.56
55.74
55.93
56.11
56.30
28.70
86.11 114.81
143.52
28.80
86.39 115.19
143.98
.2
28.89
86.67
115.56
|44.44
.3 28.98 57.96 86.94 115.93 144.91
.4 29.07 58.15 87.22 116.30
145.37
34.0 31.48
.1 31.57
.231.67
.331.76
.4 31.85
57.41
57.59
57.78
3
58.89
59.07
PER 50 FEET

4
WIDTH OR HEIGHT
5
138.89 166.67
139.35
83.33 111.11
83.61 111.48
83.89 111.85
84.17 112.22 140.28
84.44 112.59
167.22
167.78
139.81
168.33
168.89
140.74
112.96 141.20
113.33 141.67
113.70 142.13
114.07
142.59
114.44
143.06
87.50 116.67
87.78 117.04
88.06 117.41
88.33 117.78
88.61 118.15
145.83
146.30
146.76
147.22
147.69
6
123.70154.63
35.5
32.87 65.74 98.61 131.48 164.35
.6 32.96 65.93 98.89 131.85 164.81
.7 33.06 66.11 99.17 132.22 165.28
.8 33.15
66.30 99.44 132.59 165.74
33.24 66.48 99.72 132.96
.9
166.20
7
169.44
197.69
170.00
198.33
170.56
198.98
171.11 199.63
171.67
200.28
90.28
|
243.70
92.22 122.96
59.26 88.89 118.52 148.15 177.78 207.41
59.44 89.17 118.89 148.61 178.33 208.06
59.63 89.44 119.26 149.07 178.89 208.70
59.81 89.72 119.63 149.54 179.44 209.35
60.00 90.00 120.00 150.00 180.00 210.00
60.19
120.37 150.46 180.56 210.65 240.74 270.83
60.37 90.56 120.74 150.93 181.11 211.30 241.48 271.67
60.56 90.83 121.11 151.39 181.67 211.94 242.22 272.50
60.74 91.11 121.48 151.85 182.22 212.59 242.96 273.33
60.93 91.39 121.85 152.31 182.78 213.24
274.17
61.11 91.67 122.22 152.78 183.33 213.89 244.44 275.00
61.30 91.94 122.59 153.24 183.89 214.54 245.19 275.83
61.48
153.70 184.44 215.19 245.93 | 276.67
61.67 92.50 123.33 154.17 185.00 215.83 246.67 277.50
61.85 92.78
185.56 216.48 247.41 278.33
62.04 93.06 124.07 155.09 186.11 217.13 248.15 279.17
62.22 93.33 124.44 155.56 186.67 217.78 248.89 280.00
62.41 93.61 124.81 156.02 187.22 218.43 249.63 280.83
62.59 93.89 125.19 156.48 187.78 219.07 250.37 281.67
62.78 94.17 125.56 156.94 188.33 219.72 251.11 282.50
62.96 94.44 125.93 157.41 188.89 220.37 251.85 283.33
63.15 94.72 126.30 157.87 189.44 221.02 252.59 | 284.17
63.33 95.00 126.67 158.33 190.00 221.67 253.33 285.00
63.52 95.28 127.04 158.80 190.56 222.31 254.07 285.83
63.70 95.56 127.41 159.26 191.|| 222.96 254.81 286.67
34.5 31.94 63.89 95.83 127.78 159.72
.6 32.04 64.07 96.11 128.15 160.19
.7 32.13 64.26 96.39 128.52 160.65
.8 32.22 64.44 96.67 128.89 161.11
.9 32.31 64.63 96.94 129.26 161.57
35.0 32.41 64.81 97.22 129.63 162.04
.I 32.50 65.00 97.50 130.00 162.50
194.44
195.00
162.96 195.56
163.43 | 196.11
.2
.3
.4
32.59 65.19 97.78 130.37
32.69❘ 65.37 98.06 130.74
32.78 65.56 98.33 131.11
163.89 196.67
8
194.44 222.22 250.00
195.09 222.96 250.83
195.74 223.70 251.67
196.39 224.44 252.50
197.04
225.19 253.33
175.00 204.17
175.56 204.81
176.II 205.46
176.67
177.22
197.22
197.78
9
225.93 | 254.17
226.67
226.67 255.00
172.22
200.93 229.63 258.33
172.78
201.57 230.37
259.17
173.33
202.22 | 231.11
260.00
173.89
202.87 231.85
260.83
174.44 203.52 232.59 261.67
227.41 | 255.83
228.15 | 256.67
228.89 | 257.50
262.50
233.33
234.07
263.33
234.81 264.17
206.II 235.56 265.00
206.76
236.30 | 265.83
237.04 266.67
237.78 267.50
238.52 | 268.33
239.26 | 269.17
240.00 270.00
191.67 223.61 255.56 287.50
192.22 224.26 256.30 288.33
192.78 224.91 257.04 289.17
193.33 225.56 257.78 290.00
193.89 226.20 258.52 290.83
226.85 259.26 | 291.67
227.50
228.15
228.80
229.44
260.00 292.50
260.74 | 293.33
261.48 | 294.17
262.22 295.00
230.09
230.74
262.96 | 295.83
263.70 | 296.67
231.39 264.44 297.50
298.33
299.17
198.33
198.89 232.04 265.19
199.44
232.69 265.93
454
HEIGHT
OR
WIDTH
36.0
. I
~MJ 5678σ
·2
.3
36.5
.6
.7
.8
33.33 66.67 100.00
33.43 66.85 100.28
33.52 67.04 100.56
67.22 100.83
33.61
.4 33.70 67.41 101.11 134.81 168.52
.9
233 5
.3
.4
38.5
35.19 70.37 105.56
.I 35.28 70.56 105.83
.2
.6
.7
.8
89
137.04
137.41
37.0 34.26 68.52 102.78
.I 34.35 68.70 103.06
.2 34.44 68.89 103.33
.3 34.54 69.07 103.61 138.15
137.78
.4 34.63
69.26
138.52
37.5
.6
103.89
34.72 69.44 104.17
34.81 69.63 104.44 139.26
.7 34.91 69.81 104.72 139.63
.8 35.00
140.00
.9 35.09
140.37
38.0
70.00 105.00
70.19 105.28
.9
39.0
40.0
TABLE XVIII.—TRIANGULAR PRISMS. CUBIC YARDS
1
.I 36.20
.2
36.30
36.39
.3
.4
36.48
∞σ 0-23=
.9
2
.2 37.22
.3
37.31
37.41
.4
3
39.5 36.57
.6
36.67
.7 36.76
.8 36.85
.9 36.94
37.04 74.07 ||1.11
.I 37.13
∞ a
40.5 37.50
.6 37.59
.7 37.69
.8
37.78
.8
.9
PER 50 FEET

33.80
33.80 67.59 101.39 | 135.19 | 168.98 202.78 236.57
33.89 67.78 101.67 135.56 169.44 203.33
33.98 67.96 101.94 135.93 169.91 | 203.89
34.07 68.15 102.22 136.30 170.37 204.44
34.17 68.33 102.50
170.83
136.67
WIDTH OR HEIGHT
140.74
141.11
35.37 70.74 106.|| 141.48
141.85
142.22
35.46 70.93 106.39
35.56 71.11 106.67
35.65 71.30 106.94
35.74 71.48 107.22
35.83 71.67 107.50 143.33
35.93 71.85 107.78 143.70
36.02 72.04 108.06 144.07
142.59
142.96
41.5 38.43
.6 38.52
.7 38.61
6
133.33 166.67 200.00 233.33
133.70 167.13 200.56| 233.98
134.07
201.11 | 234.63
134.44
201.67 235.28
202.22 | 235.93
167.59
168.06
36.11 72.22 108.33
72.41 108.61
72.59 108.89 145.19
72.78 109.17 145.56
72.96 109.44 145.93
144.44
144.81
73.15 109.72 146.30
73.33 110.00 146.67
73.52 110.28 147.04
73.70 110.56 147.41
73.89 110.83 147.78
75.00 112.50 150.00
75.19 112.78 150.37
75.37 113.06 150.74
75.56 113.33
151.11
37.87 75.74 113.61 151.48
37.96 75.93 113.89
151.85
41.0
.I
38.06 76.11 114.17
152.22
152.59
.2
38.15
76.30 114.44
.3 38.24 76.48 114.72 152.96
.4 38.33 76.67 115.00 153.33
138.89 173.61 208.33
174.07 208.89
174.54 209.44
175.00 210.00
175.46 210.56
148.15
74.26 111.39 148.52
74.44 III.67 148.89
74.63 ||1.94 149.26
74.81 112.22 149.63
5
76.85 115.28 153.70
77.04 115.56 154.07
77.22 115.83 154.44
38.70 77.41 116.11 154.81
38.80
77.59 116.39 155.19
171.30
171.76
172.22
172.69
173.15
175.93
176.39
7
178.24 213.89
178.70 214.44
179.17 215.00
179.63 215.56
180.09 | 216.11
270.37 304.17
237.22 | 271.11 305.00
237.87 271.85 305.83
238.52 272.59 | 306.67
205.00 239.17 | 273.33 307.50
205.56 | 239.81 274.07
206.11 240.46
206.67 | 241.||
207.22 241.76
207.78 242.41
211.11
211.67
211.67 246.94
176.85
212.22 247.59
177.31 212.78 248.24
177.78 213.33
248.89
218.33
218.89
187.50
187.96
188.43
188.89
189.35
180.56 216.67 252.78
181.02 217.22 | 253.43
181.48 217.78 254.07
181.94
182.41
8
266.67 300.00
267.41 300.83
268.15 301.67
268.89 302.50
269.63 303.33
246.30 281.48
189.81
227.78
190.28 | 228.33
190.74 228.89
191.20 229.44
191.67 230.00
192.13 230.56
192.59 231.||
193.06
193.52
193.98
243.06
243.70
244.35
277.78 312.50
278.52 | 313.33
279.26 | 314.17
245.00 280.00 315.00
245.65 | 280.74
280.74
315.83
185.19 222.22 259.26
185.65 222.78❘ 259.91
186. 223.33 | 260.56
186.57 223.89 261.20
187.04 224.44 | 261.85
249.54 285.19
250.19 | 285.93
250.83 286.67
251.48 287.41
252.13 288.15
9
274.81
275.56 310.00
276.30 310.83
277.04 311.67
182.87
183.33
219.44 256.02 292.59
220.00 | 256.67 293.33
183.80 220.56 257.31 | 294.07
316.67
282.22 317.50
282.96 | 318.33
283.70 319.17
284.44
320.00
308.33
309.17
231.67 270.28
232.22 | 270.93
232.78 | 271.57
254.72 291.11 327.50
255.37 | 291.85
328.33
288.89
325.00
289.63
325.83
290.37 326.67
331.67
184.26 | 221.1|| 257.96 | 294.81
184.72 221.67 258.61 295.56 332.50
225.00 262.50
225.56 | 263.15| 300.74
226.11 | 263.80 | 301.48
226.67 | 264.44 302.22
227.22 | 265.09
302.96
320.83
321.67
322.50
323.33
324.17
329.17
330.00
330.83
296.30 | 333.33
297.04 334.17
297.78 335.00
298.52 335.83
299.26 | 336.67
300.00 337.50
338.33
339.17
340.00
340.83
303.70
341.67
265.74
266.39
304.44 342.50
267.04 305.19 343.33
267.69 305.93
305.93
268.33 306.67
344.17
345.00
268.98 307.41 345.83
269.63 308.15 346.67
308.89 347.50
309.63 348.33
310.37 349.17
455
HEIGHT
OR
WIDTH
42.0
38.89
38.98
.2 39.07
.I
.3
39.17
39.26
.4
43.0
39.81
.1 39.91
.2 40.00
.3 40.09
.4
40.19
42.5
39.35
78.70 118.06
.6 39.44 78.89 118.33
.7 39.54 79.07
.8 39.63 79.26
118.61
118.89
.9 39.72
79.44
119.17
43.5 40.28
.6 40.37
.7 40.46
.8 40.56
.9 40.65
44.0
234 56
.3
44.5
.6
TABLE XVIII.—TRIANGULAR PRISMS. CUBIC YARDS
1
45.0 41.67
.I 41.76
.2
41.20 82.41 123.61
41.30
7.41.39
.8 41.48
.941.57
46.0
.3 41.94
.4
42.04
MI SO7∞g o-NKI SO
45.5 42.13
.6 42.22
.742.31
.8 42.41
.9 42.50
81.48 122.22
40.74
.1 40.83 81.67 122.50
.2 40.93
81.85 122.78
82.04 123.06
41.02
.441.11
82.22 123.33 164.44 205.56
42.59
.I 42.69
.2 42.78
.3
.4
46.5
2
47.0
3
155.93
77.78 116.67 155.56
77.96 116.94
78.15 117.22
78.33 117.50 156.67
78.52 117.78 157.04
156.30
47.5
PER 50 FEET

41.85 83.70 125.56
WIDTH OR HEIGHT
157.41
157.78
158.15
158.52
158.89
278.70
79.63 119.44
79.81 119.72
279.35
280.00
159.26 199.07 238.89
159.63 199.54 239.44
80.00 120.00 160.00 200.00 240.00
80.19 120.28 160.37 200.46 240.56 280.65
80.37 120.56 160.74 200.93 241.11 281.30
80.56 120.83 161.11
80.74 121.11 161.48
80.93 121.39
81.11 121.67
81.30 121.94
6
194.44 233.33 272.22
194.91 233.89 272.87
195.37 234.44 273.52
195.83 235.00 274.17
196.30 235.56 | 274.81
196.76 236.11 275.46
197.22 | 236.67 | 276.11
197.69 237.22 276.76
198.15 237.78 277.41
198.61 238.33 278.06
5
83.33 125.00 166.67
83.52 125.28 167.04
162.96
203.70
163.33
204.17
163.70 204.63
164.07
205.09
167.41
83.89 125.83 167.78
84.07 126.11 168.15
247.22
164.81 206.02
82.59 123.89 165.19 206.48 247.78 289.07
82.78 124.17 165.56 206.94 248.33 289.72
82.96 124.44 165.93 207.41 248.89
83.15 124.72 166.30 207.87
249.44
84.26 126.39 168.52
210.65
84.44 126.67 168.89 211.11
84.63 126.94 169.26
84.81 127.22 169.63
85.00 127.50 170.00
85.19 127.78 170.37 212.96
85.37 128.06 170.74 213.43
85.56 128.33 171.11 213.89
42.87 85.74 128.61 171.48 214.35
42.96 85.93 128.89 171.85 214.81
201.39 241.67 281.94 322.22 362.50
201.85 242.22 282.59 322.96 363.33
161.85 202.31 242.78 283.24 | 323.70 364.17
162.22 | 202.78 243.33 283.89 324.44 365.00
162.59 203.24 243.89 284.54 | 325.19 | 365.83
43.06 86.11 129.17 172.22 215.28
.6 43.15 86.30 129.44 172.59 215.74
.7
43.24 86.48 129.72 172.96
.8 43.33 86.67 130.00 173.33
.9 43.43 86.85 130.28 173.70
43.52 87.04 130.56 174.07
.1 43.61 87.22 130.83 174.44 218.06
.2 43.70
.3 43.80
.4
7
87.41 131.11 174.81 218.52
87.59 131.39 175.19
218.98
43.89 87.78 131.67 175.56 219.44
219.44
43.98 87.96 131.94 175.93 219.91
.6 44.07 88.15 132.22 176.30 220.37
.7 44.17 88.33 132.50 176.67 220.83
.8 44.26 88.52 132.78
.9 44.35 88.70 133.06
177.04 221.30
|
177.41 221.76
216.20 259.44
216.67 | 260.00
217.13 | 260.56
290.37
291.02
208.33
208.80
250.00 291.67 333.33 375.00
250.56 292.31 334.07 375.83
209.26 251.11 292.96 334.81 376.67
209.72 | 251.67 293.61 335.56 377.50
210.19
252.22 | 294.26
336.30 378.33
252.78
252.78 294.91
244.44285.19 | 325.93 | 366.67
245.00 285.83 326.67 367.50
245.56 286.48 | 327.41368.33
246.11 287.13 | 328.15 369.17
287.78 328.89 | 370.00
288.43
246.67
8
337.04 379.17
253.33 295.56 337.78 380.00
211.57 253.89
253.89 296.20
212.04| 254.44 296.85 339.26
212.50 255.00 .297.50 | 340.00
338.52 380.83
217.59 261.11
9
311.11 350.00
311.85 | 350.83
312.59 351.67
313.33 352.50
314.07 353.33
314.81 354.17
315.56 355.00
316.30 355.83
317.04 356.67
317.78 357.50
304.63
261.67 305.28
262.22 | 305.93
262.78
306.57
307.22
263.33
263.33
318.52 | 358.33
319.26 | 359.17
320.00 | 360.00
320.74 360.83
321.48 361.67
263.89
307.87
264.44 | 308.52
265.00 | 309.17
265.56 309.81
266.11 310.46
329.63 | 370.83
330.37 371.67
331.11 372.50
331.85 373.33
332.59 | 374.17
383.33
384.17
255.56 298.15 340.74
256.11 | 298.80 341.48
256.67 | 299.44 342.22 385.00
257.22 300.09 342.96 385.83
257.78 300.74 343.70 386.67
258.33 301.39 344.44 387.50
258.89 302.04 345.19 388.33
302.69 345.93 389.17
303.33
303.98
346.67 390.00
347.41 390.83
381.67
382.50
348.15 | 391.67
348.89 | 392.50
349.63 393.33
350.37 394.17
351.11 | 395.00
351.85 | 395.83
352.59 396.67
353.33 397.50
354.07 398.33
354.81 399.17
456
HEIGHT
OR
WIDTH
48.044.44
.1
.2
.3
.4
44.54
44.63
44.72
44.81
48.5 44.91
.645.00
.7 45.09
.8
45.19
45.28
.9
49.0
50.0 46.30
.! 46.39
.246.48
.3 46.57
.4 46.67
.7
.8
.9
50.5 46.76
.646.85
46.94
47.04
47.13
56 |
TABLE XVIII.-TRIANGULAR PRISMS. CUBIC YARDS
1
51.5
45.37
.1 45.46
.2 45.56
.3 45.65
.4 45.74
49.5
.6
45.83
45.93
.7
46.02
.8
46.11 92.22 138.33
.9 46.20 92.41 138.61
51.0
47.22
.1
47.31
.2 47.41
.3 47.50
.4
.6
.7
52.0
. I
.2
53.5
5
2
3
6
88.89 133.33 177.78 222.22 266.67
89.07 133.61 178.15 222.69 267.22
89.26 133.89 178.52 223.15 267.78
89.44 134.17 178.89 223.61 268.33
89.63 134.44 179.26 224.07 268.89
179.63 224.54 269.44
180.00 225.00 270.00
180.37 225.46 270.56
180.74 225.93 271.11
135.83 181.11 226.39 271.67
89.81 134.72
90.00 135.00
90.19 135.28
90.37 135.56
90.56
90.74
90.93
91.11 136.67
91.30 136.94
91.48 137.22
49.07
49.17
91.67 137.50
91.85 137.78
92.04 138.06
49.54
.6 49.63
.7 49.72
.8 49.81
.9 49.91
47.59 95.19 142.78
PER 50 FEET

.8 47.96 95.93 143.89 191.85
.9 48.06 96.11 144.17 192.22
52.5 48.61 97.22 145.83
.6 48.70 97.41 146.11
.7 48.80 97.59 146.39
.8 48.89 97.78 146.67
.9 48.98 97.96 146.94
98.15 147.22
4
WIDTH OR HEIGHT
240.28288.33
48.15 96.30 144.44 192.59 240.74 | 288.89
48.24 96.48 144.72 192.96 241.20 289.44
48.33 96.67 145.00 193.33 241.67 290.00
.3
48.43 96.85 145.28 193.70 242.13
.4 48.52 97.04 145.56
290.56
291.11
194.07
242.59
136.11 | 181.48
226.85 272.22 317.59 362.96 408.33
136.39 181.85 227.31 272.78 318.24 363.70 409.17
182.22 | 227.78 273.33 318.89 364.44 410.00
182.59 228.24 273.89 319.54 365.19 410.83
182.96 228.70 274.44 320.19 365.93 411.67
183.33 229.17
183.70 229.63
184.07
184.44
230.09
230.56
231.02
184.81
53.0
.I
.2
.3 49.35 98.70 148.06 197.41
.4 49.44 98.89
5
283.33 330.56 377.78
283.89 | 331.20 | 378.52
284.44 331.85 379.26
285.00
190.37 237.96 285.56
47.69 95.37 143.06 190.74 238.43 286.||
47.78 95.56 143.33 191.11 238.89
332.50
333.15
380.00
380.74
333.80 381.48
286.67 334.44 382.22
47.87 95.74 143.61 191.48 239.35 287.22
239.81 287.78
7
49.26 98.52 147.78 197.04
8
92.59 138.89 185.19 231.48 277.78
92.78 139.17 185.56 231.94 278.33
92.96 139.44 185.93 232.41 278.89
93.15 139.72 186.30 232.87 279.44
93.33 140.00 186.67 233.33 280.00
93.52 140.28 187.04 233.80 280.56 327.31 374.07 420.83
93.70 140.56 187.41 234.26 281.11 327.96 374.81 421.67
93.89 140.83 187.78 234.72 281.67 328.61 375.56 422.50
94.07 141.11 188.15 235.19 282.22 329.26 376.30
94.26 141.39 188.52 235.65 282.78
94.44 141.67 188.89 236.11
94.63 141.94 189.26 236.57
94.81 142.22 189.63 237.04
95.00 142.50 190.00 237.50
423.33
424.17
329.91 377.04
9
311.11 355.56 400.00
311.76 356.30 400.83
312.41 357.04 401.67
313.06 357.78
357.78 402.50
313.70 358.52 403.33
196.30 245.37
98.33 147.50 196.67
314.35 359.26 404.17
315.00 360.00
360.00 405.00
315.65 360.74 405.83
316.30 361.48 406.67
316.94 352.22
352.22 407.50
275.00 320.83
275.56 321.48
276.11 | 322.13
366.67 412.50
367.41 413.33
368.15 414.17
276.67 322.78 368.89 415.00
277.22
323.43 369.63
415.83
324.07 370.37 416.67
324.72 371.11 417.50
325.37 371.85 418.33
326.02 372.59 419.17
326.67 373.33 420.00
425.00
425.83
194.44 243.06
243.06
291.67
340.28 388.89 437.50
194.81 243.52 292.22 340.93 389.63 438.33
195.19 243.98 | 292.78 341.57 390.37
195.56 244.44 293.33 | 342.22 | 391.11
195.93 244.91 293.89 342.87 391.85
426.67
427.50
428.33
431.67
432.50
429.17
430.00
335.09 382.96 430.83
335.74 383.70
336.39 384.44
337.04 385.19
337.69 | 385.93
338.33 386.67
338.98 287.41
339.63
433.33
434.17
435.00
435.83
388.15 436.67
439.17
440.00
440.83
294.44 343.52 392.59 441.67
245.83 295.00 344.17 393.33 422.50
246.30 295.56 344.81 394.07 443.33
246.76 296.II 345.46 394.81 444.17
247.22 296.67 346.11 395.56 445.00
148.33 197.78
99.07 148.61 | 198.15 247.69 | 297.22 346.76 396.30 445.83
99.26 148.89 198.52 248.15 297.78 347.41 397.04 446.67
99.44 149.17 198.89 248.61 298.33
99.63 149.44 199.26 249.07 298.89
99.81 149.72 199.63 249.54 299.44
348.06 397.78 447.50
348.70 398.52 448.33
349.35 399.26 449.17
457
HEIGHT
OR
WIDTH
233 SON
54.5
233 56789
55.5
.6
.8
0-23-
56.0
5678 σ
56.5
.6
.7
54.0
50.00 100.00 150.00 200.00
. I 50.09 100.19 150.28
.2 50.19 100.37 150.56
.3 50.28 100.56 150.83
.4 50.37 100.74 151.11
400.00 | 450.00
400.74 450.83
401.48 | 451.67
453.33
.6
202.22 252.78
404.44
250.00 300.00 350.00
200.37 250.46 300.56 350.65
200.74 | 250.93 | 301.1| 351.30
201.11 251.39 301.67 351.94 402.22 452.50
201.48251.85 | 302.22 | 352.59 | 402.96
50.46 100.93 151.39 201.85 252.31 302.78 353.24 403.70
50.56 101.11 151.67
303.33 | 353.89
.7 50.65 101.30 151.94 202.59 | 253.24 303.89 354.54
50.74 101.48 152.22 202.96 253.70 | 304.44 | 355.19
50.83 101.67 152.50
50.93 101.85 152.78
.I 51.02 102.04 153.06 204.07
.2 51.11 102.22
.3 51.20 102.41
.4 51.30 102.59 153.89 205.19
405.19
355.19 405.93
405.93
.8
.9
203.33
254.17 305.00 | 355.83406.67
55.0
203.70
203.70
153.33 204.44
.8
.9
51.39 102.78
51.48 102.96
.751.57 103.15
51.67 103.33 155.00 206.67 258.33
.9 51.76 103.52 155.28 207.04
258.80
57.0
.I
.2
~34 5678
51.85 103.70 155.56 207.41
.151.94 103.89 155.83 207.78
52.04 104.07 156.11 208.15
.3 52.13 104.26 156.39 208.52
.4 52.22 104.44 156.67 208.89
.2
.3
57.5
.6
.8
.9
58.0
.I
.2
~34 5678σ o
52.96 105.93 158.89 211.85
53.06 106.11 159.17 212.22
.4 53.15 106.30 159.44 212.59
.3
.9
TABLE XVIII.—TRIANGULAR PRISMS. CUBIC YARDS
1
59.0
2
. I
3
233 561
.3
.4
PER 50 FEET

4
59.5
WIDTH OR HEIGHT
153.61204.81
53.24 106.48 159.72 212.96 266.20
53.33 106.67 160.00 213.33 | 266.67
.753.43 106.85 160.28 | 213.70 | 267.13|320.56
53.52 107.04 160.56 214.07 | 267.59 321.11
53.61 107.22 160.83
214.44 268.06 | 321.67
209.26
52.31 104.63 156.94
52.41 104.81 157.22 209.63
52.50 105.00 157.50
52.59 105.19 157.78
52.69 105.37 158.06
5
210.00
210.37
210.37
210.74
254.63 | 305.56 356.48 407.41 458.33
255.09 | 306.11 357.13 408.15 459.17
255.56 306.67 357.78 408.89 460.00
256.02 307.22 358.43 409.63 460.83
256.48 307.78 359.07 410.37 | 461.67
154.17 205.56 256.94
411.11 462.50
154.44 205.93 | 257.41
411.85 463.33
154.72 206.30 | 257.87
412.59 464.17
413.33 465.00
414.07 465.83
6
52.78 105.56 158.33 211.11
263.89
52.87 105.74 158.61 211.48 264.35
264.81
265.28
265.74
308.33 | 359.72
308.89 360.37
309.44 | 361.02
310.00 | 361.67
310.56 362.31
259.26 311.11 | 362.96
259.72 | 311.67 363.61
260.19 | 312.22 | 364.26
260.65 | 312.78 | 364.91
261.|| 313.33 365.56
7
8
261.57 313.89 | 366.20 | 418.52
252.04 314.44 366.85
262.50 315.00 | 367.50
262.96 | 315.56 368.15
263.43 316.11 368.80
319.44
320.00
53.70 107.41 161.11 214.81
483.33
484.17
215.19
215.56
215.56
485.00
53.80 107.59 161.39
53.89 107.78 161.67
53.98 107.96 161.94 215.93 269.91 323.89 377.87
.4 54.07 108.15 162.22
431.85 485.83
58.5
.6
.7
325.00 379.17 433.33 487.50
325.56 379.81 434.07 488.33
271.76 326.11 | 380.46
272.22 | 326.67 | 381.11 435.56
272.69 | 327.22 | 381.76436.30
434.81
489.17
490.00
54.54
490.83
54.63 109.26 163.89
491.67
216.30 | 270.37 324.44 378.52 432.59 486.67
54.17 108.33 162.50 216.67 270.83
54.26 108.52 162.78 217.04 271.30
54.35 108.70 163.06 217.41
.8 54.44 108.89 163.33 217.78
109.07 163.61 218.15
218.52
54.72 109.44 164.17 218.89
.254.81 109.63 164.44219.26
54.91 109.81 164.72 219.63
329.44 384.35
55.00 110.00 165.00 220.00
330.00 | 385.00
55.09 110.19 165.28 220.37 275.46 330.56 385.65
55.19 110.37 165.56 220.74
386.30
110.56 165.83 221.11 276.39 331.67 386.94
110.74 166.11 221.48 276.85
110.93 !66.39 221.85 277.31
492.50
383.70 438.52 493.33
274.07 328.89
274.54
439.26 494.17
440.00 495.00
275.00
.6
.7 55.28
.855.37
.955.46
440.74 495.83
441.48 496.67
442.22 497.50
332.22 387.59 442.96 498.33
332.78 388.24 443.70 499.17
273.61328.33
9
| 414.81 | 466.67
415.56 467.50
416.30
417.04
417.78
316.67
317.22
369.44 422.22 475.00
370.09
370.09 422.96 475.83
317.78 | 370.74
318.33 | 371.39
423.70 476.67
424.44
318.89 | 372.04 425.19
477.50
478.33
275.93331.11
268.52 322.22
268.98 322.78
375.93 429.63
376.57 | 430.37
269.44 323.33 377.22 431.11
454.17
455.00
455.83
456.67
457.50
470.83
419.26 471.67
420.00 | 472.50
420.74 473.33
421.48 | 474.17
372.69 | 425.93
373.33 426.67
373.98 427.41
374.63 428.15
375.28 428.89
273.15 327.78 382.41 437.04
383.06 437.78
468.33
469.17
470.00
479.17
480.00
480.83
481.67
482.50
458
TABLE XIX.-CUBIC YARDS PER 100-FOOT STATION

YD
↓
ዐ ← በጋ
9.3
11.1
13.0
0 100
0.0
1.9
2
3.7
5.6 3
7.4 4
0-NMI NONOOD ON MONDO ONE ON ONE HUN ER NOG KONK
ZAKA KOJOM 78858 ONZ~~ ZPPFO 28.** 188.8 88.** 12. 28000
∞ a
5
6
14.8 8
7
16.7 9
22.2 12
46.3
48.1
11
27.8 15
77.8
37.0 20
18.5 10 64
20.4
54
55
56
25
26
58
55.6 30
57.4
59.3 32
61.1 33
63.0 34
57 111
112
69
29.6 16 70
31.5 17
33.3 18
35.2 19
71
72
73
60
61
66
24.1 13 67 121
25.9 14
59 113
62
63
81.5 44
83.3 45
65
24 78
68
74
50.0 27 81
51.9 28
53.7 29
79
128 182 236
290
38.9 21 75 129 183 237 291
40.7 22
42.6 23
44.4
76 130 184 238
77 131 185 239
132 186 240
83
84
64.8 35 89
66.7 36
68.5 37
91
70.4 38 92
72.2 39 93
74.1
40 94
41
75.9
42
200| 300 400 500 600 700 800 900
|| 1
87
108 162 216 270 324
109 163 217 271 325
110 164 218
165 219
166 | 220
272 326
273 327
274 328
90
SUM OF END AREAS IN SQ FT
378
379
380
381
382
114
115
167 221
168 222
169 223
116 170 224
117 171 225
138 192
31 85 139 193
118 172 226
119 173 227
120 174
228
98
175
122 176
133 187 241 295
80 134 188 242 296
99
86 140 194
123 177 231
124 178 232
125 179 233
126 180
127 181
229
230
0 100
95 149 203 257
79.6 43 97 151 205 | 259
152 206 260
96 150 204 258
329
275
276 330
277
278
279
234 288
235
289
280
334
281 335
282
336
283
337
284
338
285
286
349
350
135 189 243 297 351
136 190 244 298
137 191
245
299
339
340
287 341
342
343
246
247
248
302
141 195 249 303
142 196 250 304
48 102 156 210
153 207 | 261
154 208 262
209
155
143 197 251
144
198 252
145 199 253
146 200 254 308
147 201 255 309
148 202 256
346
292
293 347
294 348
383
384
331 385
332
386
333
387
300
301
344
345
310
311
312
313
314
85.2 46
87.0 47 101
88.9
263 317
264 | 318
90.7 49 103 157 211 265 319
352
353
388
389
390
391
392
432
433
434
435
436
437
491
438 492
439 493
440
494
441 495
442
443
444
445
446
486
487
488
489
490
393
394
395 449
396 450
397
92.6
50 104 158 212
94.4 51 105 159
213
266 320 374 428
267 321 375 429
96.3 52 106 160 214 268 322 376 430
98.1
107 161 215 269 | 323 | 377 431
200 300 400 500 600 700
53
496
497
498
499
500
447 501
448 502
503
504
451 505
398 452 506
399 453 507
400 454 508
401 455 509
402 456 510
305 359 413 467 521
306 360
414 468 522
307 361 415 469 523
362 416 470 524
363 417 471 525
403 457 511
404 458 512
459
513
405
406
460 514
407 461 515
354
408 462 516
355 409 463 517
356 410 464 518
357 411 465 519
412 466 520
358
364
418 472 526
365 419 473 527
366
420 474 528
367
421 475 529
368 422 476 530
315 369 423 477 531
316 370 424 478 532
371 425 479 533
372 426 480 534
373 427 481 535
1000
540 594
541
595
542
596
543
544
545
546
547
548
549
552
553
554
1100
550
604
551 605
597
598
599
600
601
602
603
555
609
556 610
557 611
558 612
559 613
606
607
608
614
560
561 615
562 616
563
617
564
618
565
566
567
568 622
569
623
619
620
621
624
570
571 625
572
573
574
536
482
483
537
484 538
592
485 539 593
800
900 1000
590
591
575
629
576
630
577 631
578 632
579
633
626
627
628
580
634
581 635
582 636
583 637
584
638
585 639
586 640
587 641
588 642
589 643
644
645
646
647
1100
459
1200
1300 1400
648 702 756
649 703 757
650 704
758
651
705
759
652
706 760
653 707
761
654 708 762
6.55 709
656 710
657 711
763
764
765
658 712
659 713
660 714
661
662
715
716
663 717
771
664 718 772
665 719 773
666 720
667 721
668 722
669 723
670 724
671 725
672 726
766
767
768
769
770
678 732
679 733
680 734
681 735
682 736
FROM SUM OF END AREAS

673 727 781
674 728 782
675 729 783
676 730 784
677 731 785
1500 1600 1700 1800 1900 2000
SUM OF END AREAS IN SQ FT
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
774 828
775
829
830
776
777 831
778 832
779 833
780
834
835
836
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
837 891
838 892
839 893
786 840 894
787 841 895
788 842
789 843
790 844
683 737 791 845
684 738 792 846
685 739 793 847
686 740 794 848 902
687 741 795 849
903
918 972 1026
919
973
920
921
922
923
924
925
926
927
688 742 796 850 904
689 743 797 851 905
690 744 798 852
691 745 799 853
692 746 800 854
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
896
897
898 952
951
948
949
950
899
900
953
954
901 955
956
957
1080
1027 108-1
1028 1082
1029 1083
1030
1084
974
975
976
977
1031
978 1032 1086
979 1033
1087
1088
980
1034
981
1035 1089
1036
1037
1038
982
983
984
985 1039
986
1040
987
988
989 1043
1044
990
991 1045
992
993
994
995 1049
996
1050
1041
1042 1096
1002❘ 1056
1003❘ 1057
1004❘ 1058
1005 1059
1006
1060
1085 1139
9.3
1140
||.|
1141
13.0
1142
14.8
1143 16.7
2100 Cu
YD
↓
997 1051
998 1052
999 1053
1000❘ 1054
1108
1001 1055 1109
1134
1135
1136
1137
1138
1090 1144
18.5
1091 1145 20.4
1092❘ 1146 22.2
1093 1147 24.1
1094 1148 25.9
1046 1100 1154 37.0
1047 1101 1155 38.9
1048
1102 1156 40.7
1103 1157 42.6
1104 1158 44.4
958
1012
959 1013
906 960 1014
907 961 1015
908 962 1016
963 1017
964 1018
693 747 801 855 909
694 748 802 856
910
911
695 749 803 857
965 1019
696 750 804 858 912 966 1020
697 751 805 859 913 967 1021
698 752 806 860 914 968 1022 1076
699 753 807 861 915 969 1023 1077
700 754 808 862 916 970 1024
1078
1079
863 917
701 755 809
1200 1300 1400
971❘ 1025
1500 1600 1700
1800
1900
0.0
1.9
3.7
5.6
7.4
1095 1149 27.8
1150 29.6
1097 1151 31.5
1098
1099
1152 33.3
1153 35.2
1105
1106
1107
1159 46.3
1160 48.1
1161
50.0
1162
51.9
1163
53.7
1110
||||
1112
1165
1164 55.6
57.4
1166 59.3
1113 1167 61.1
1114 1168 63.0
1061 1115 1169
1008 1062 1116 1170
1007
64.8
66.7
1009❘ 1063| 1117
1010 1064 1118
1065 1119
1171 68.5
1172 70.4
1173 72.2
1011
1066 1120 1174 74.1
1067 1121 1175 75.9
1068 1122 1176 77.8
1069 1123 1177 79.6
1070 1124 1178 81.5
1073
1074
1075
1071 1125 1179 83.3
1072 1126 1180 85.2
1127 1181 87.0
1128 1182 88.9
1129 1183 90.7
1130 1184 92.6
1131 1185 94.4
1132 1186 96.3
1133 1187 98.1
2000 2100
460
TABLE XX.-NATURAL SINES, COSINES,
Explanation
This table is adapted to the rapid solution of route-survey-
ing problems by computing machine. Its special advantages.
are: (1) six significant figures; (2) most-frequently used func-
tions in one table; (3) "streamline" interpolation for multiples
of 10" and 15"; (4) chances for mistakes reduced by elimina-
tion of column headings at bottom of page and by use of only
one degree per page.
Precise angle and distance measurements in route surveying
justify one more figure than in the usual 5-place table, but
hardly justify 7-place tables.
In a large number of curve formulas the sine and versine
of the same angle appear. Other combinations are sin-tan-cos
and sin-cos-tan-vers. The appearance of these functions in
the same table should save time. The omission of cotangents
is of little consequence since they are rarely used in route sur-
veying; if needed, they are equivalent to the tangent of the
complement.
G
The most serious defect of most tables of natural functions
is the absence of a convenient aid in interpolating for seconds.
This defect is remedied in Table XX, largely through the use
of sets of corrections for multiples of 10" and 15". These
multiples are the ones most frequently needed when single or
double angles are turned with a 20", 30", or 1' instrument.
Example. Find the tangent of 28° 18′ 45″. Answer:
0.538444+281=0.538725.
The foregoing is the quickest method and is sufficiently
precise for most purposes. However, because the sets of
corrections are exact at the middle of the indicated range,
slightly greater precision would be obtained in this example
by adding a correction of 282, since the 45" correction at
28° 30′ is 283. This method gives 0.538726. Many com-
puters would prefer to subtract the 15" correction of 94 from
the tangent of 28° 19′ (an excellent method), giving also
0.538726. Use of these more precise methods is recommended
for finding exsecants of angles between 45° and 60°, and for
tangents and exsecants of angles above 70°.
Obviously, corrections for any number of seconds could be
obtained speedily, if the field work justifies, by combining
the tabulated corrections and shifting the decimal point.
461
0
Zero
1.000291
234
4
EETE O)
5.001454
1745
2036
2327
2618
6
7
8
9
10.002909
U
01234 56
12
13
14
16
222
22222
234
21
20.005818
22
23
24
15.004363 30 145.999990
4654 40 194
17 4945 45 218
18
523650 242
19
5527
26
27
25.007272
7563
7854
8145
8436
28
66L6G QUººº 55555 £55±5 wwwww wwwww Nig
29
32
33
34
30.008726
9017
9308
9599
9890
36
37
38
39
35.010181
0472
0763
1054
1344
41
42
SINE
43
44
40.011635
0582
0873
1164
46
47
48
49
51
3200 " Corr
3491 10 48
73
378215
4072 20 97
52
6109
6400
6690
6981
45.013090 30 145
3380 40 194
3671 45 218
3962 50 242
4253
53
54
50.014544
4835
5126
5416
5707
57
58
59
VERSINES, EXSECANTS, AND TANGENTS
55 015998
56
CORR.
FOR
SEC.
+
1926
" Corr.
221710 48
250815 73
2799 20 97
6289
6580
6871
7162
60.017452
COSINE
One
One
One
One
.999999
.999999
9998
9998
9997
By
9997 Inspec-
tion
.999996
9995
9994
9993
9992
9989
9988
9986
9985
.999983
9981
9980
9978
9976
.999974
9971
9969
9967
9964
999962
9959
9957
9954
9951
|.999948
9945
9942
9939
9936
.999932
9929
9925
9922
9918
.999914
9910
9906
9902
9898
.999894
9890
9886
9881
9877
CORR.
FOR
SEC.
.999872
9867
9862
9858
9853
999848
0°

VERSINE
Zero
Zero
Zero
Zero
EXSEC
.000001.00000!
Zero
Zero
Zero
Zero
100000 || 100000*
0005
0006
0007
0008
0002
0002
0003 By
0002
0003
0003 0003 Inspec-
tion
.000004.000004
0001
0005
0006
0007
0008
.0000 10.000010
0011
0012
0014
0015
0061
0064
0011
0012
0014
0015
.000017.000017
0019
0020
0022
0024
0019
0020
0022
0024
.000026.000026
0029
0031
0033
0036
0029
0031
0033
0036
.000038.000038
0041
0041
0043
0043
0046
0046
0049 0049
.000052.000052
0055 0055
0058
0058
0061
0064
.000068.000068
0071
0075
0078
0082
0071
0075
0078
0082
.000086.000086
0090 0090
0094
0094
0098
0098
0102
0102
.000106.000106
0110
0110
0114
0114
0119
0119
0123 0123
.000128.000128
0133
0138
0142
0147
0133
0138
0142
0147
CORR.
FOR
SEC.
+
000152.000152
CORR.
FOR
SEC.
+
11
Corr.
10 48
15 73
20 97
30 145
40 194
45 218
50 242
"Corr.
48
10
15 73
20 97
30 145
40 194
145 218
50 242
TANGENT
Zero
000291
0582
0873
01234
.001454
2
3
1164 4
1745 6
2036 7
2327 8
2618 9
002909 10
3200 11
3491 12
3782 13
407214
004363 15
4654 16
4945 17
5236 18
5527 19
.005818 20
6109 21
6400 22
6690 23
6981 24
.007272 25
7563 26
7854 27
8145 28
8436 29
.008727 30
9018 31
9309 32
9600 33
989034
.010181 35
047236
0763 37
1054 38
1345 39
.011636 40
1927 41
2218 42
250943
2800 44
.01309145
3382 46
3673 47
3964 48
4254 49
.014545 50
483651
5127 52
5418 53
5709 54
016000 55
629156
6582 57
6873 58
7164 59
.017455 60
462
0.017452
7743
8034
8325
8616
1234
5.018907
9197
9488
9779
9.020070
10.020361
681OG
6
7
8
01234 56
11
12
13
14
16
17
18
19
15.02181530 145
2106 40 194
22222
234
21
20.023269
22
23
24
227** 8.ma
26
28
29
25.024723
5014
5305
5595
5886
31
32
33
34
30.026177
6468
6758
7049
7340
36
37
38
39
55±5 88!
35.027631
7922
8212
8503
8794
41
42
43
66966 20ººº 55555
SINE
46
40.029085
9376 " Corr.
9666 10 48
995715 73
44.030248 20 97
47
48
49
45.030538 30 145
0829 40 194
1120 45 218
1411 50 242
1701
51
0652" Corr.
0942 10 48
1233 15 73
1524 20 97
52
53
54
2396 45 218
2687 50 242
2978
3560
3851
4141
4432
50.031992
2283
2574
2864
3155
56
57
58
59
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
55.033446
3737
4027
4318
4609
60.034899
COSINE
.999848
9843
9837
9832
9827
.999821
9816
9810
9804
9799
.999793
9787" Corr.
978110
977415
9768 20
.99976230
9756 40
9749 45
9743 50
9736
.999729
9722
9716
9709
9702
.999694
9687
9680
9672
9665
.999657
9650
9642
9634
9626
.999618
9610
9602
9594
9585
CORR.
FOR
SEC.
.999488
9479
9469
9460
9450
I
.999440
9431
9421
9411
9401
999391
22 MI55
2
3
4
5
" Corr.
.999577
9568
9560 10
955115
9542 20 3
I
2
.999534 30 4
9525 40 6
9516 45 7
9507 50 7
9497
+
1°

VERSINE
.000152.000152
0157 0157
0163
0163
0168
0168
0173 0173
.000179.000179
0184 0184
0190
0190
0196
0196
0201
0201
EXSEC
.000207.000207
0213
0219
0226
0232
0244
0251
0257
C264
11
0213 Corr.
0219 10 1
0226 15 2
0232 20 2
.000238.000238 30 3
0244 40 4
025145
0257 50
5
5
0264
.000271.000271
0278 0278
0284
0284
0291
0298
0291
0299
.000306.000306
0313
0320
0328
0335
0449
0458
0313
0320
0328
0335
.000343.000343
0350
0358
0366
0374
0350
0358
0366
0374
.000382.000382
0390 0390
0398
0398
0406
0406
0415 0415
.000423.000423
0432
0440
0432
" Corr.
0440 10 1
0449 15 2
0458 20 3
.000466.000467|30
CORR.
FOR
SEC.
+
4
0476 40 6
0485 45 7
0494 50 7
0475
0484
0493
0503 0503
000512.000512
0521
0531
0540
0550
0521
0531
0540
0550
000559.000560
0569 0570
0579
0579
0589
0589
0599
0599
000609.000609
CORR.
FOR
SEC.
+
" Corr.
10
49
15 73
20 97
30 146
40 194
45 218
50 243
" Corr.
10 49
15 73
20 97
30 146
40 194
45 218
50 243
TANGENT
.017455
7746
8037
01234 56789
8328
8619 4
.018910
9201
7
9492
9783 8
.020074
5
.020365 10
0656 11
0947 12
1238 13
1529 14
.021820 15
21| 1| 16
2402 17
2693 18
2984 19
.023275 20
3566 21
3857 22
4148 23
4439 24
.024730 25
5022 26
531327
5604 28
5895 29
.026186 30
6477 31
6768 32
7059 33
7350 34
.027641 35
7932 36
8224 37
8515 38
8806 39
.029097 40
9388 41
967942
9970 43
.030262 44
.030553 45
0844 46
1135 47
1426 48
1717 49
032009 50
2300 51
2591 52
2882 53
3173 54
.033465 55
3756 56
404757
4338 58
4629 59
.034921 60
463
1
0.034900
5190
5481
5772
6062
01234 KON7∞Q
5.036353
6
8
9
01234
10.037806
||
12
13
14
6678 σ
16
17
19
IEEE KON~~ ~~27 MOMMA DIRE BONO OGNOZ KANOO O
21
22
20.040713
1004
1294
1585
1876
23
24
26
27
25.042166
2457
2748
3038
3329
28
29
15.039260 30 145.999229 30
9550 40 194
9841 45 218
18.040132 50 242
0422
31
32
33
34
30.043619
3910
4201
4491
4782
36
37
35.045072
5363
5654
5944
6235
38
39
SINE
41
42
43
44
40.046525
46
6644
6934
7225
7516
47
48
49
45.047978
51
8097 " Corr.
8388 10 48
8678 15 73
8969 20 97
53
54
56
57
50.049431
9721
52.050012
0302
0593
58
59
VERSINES, EXSECANTS, AND TANGENTS
2°

CORR.
FOR
SEC.
+
6816
"Corr.
7106 10 48
7397 15 73
7688 20 97
55.050884
1174
1464
1755
2046
60.052336
30 145
8269 40 194
8559 45 218
8850 50 242
9140
COSINE
.999391
9381
9370
9360
9350
.999339
9328
9318
9307
9296
.999285
9274 " Corr
9263 10 2
9252 15 3
9240 20 4
6
921840
8
9206 45 9
9194 50 10
9183
.999171
9159
9147
9135
9123
.999|||
9098
9086
9073
906 1
.999048
9036
9023
9010
8997
.998984
8971
8957
8944
8931
-
998917
8904
CORR.
FOR
SEC.
8890 10 2
8876 15 3
8862 20 5
998778
8763
8749
8734
8719
.998848 30 7
8834 40 9
8820 45 10
.998705
8690
8675
8660
8645
998630
" Corr.
8806 50 12
8792
+
VERSINE
000609.000610
0619
C630
0640
0650
0672
199000199000*
EXSEC
0682
0693
0620
0630
0640
0651
0683
0694
0704 0704
.000771
0782
0794
0806
0817
0672
.000715.000715
0726
0737
0748
0760
0726 " Corr.
0738 10 2
0749 15 3
0760 20 4
.000829.000830
0841
0853
0865
0877
.000772 30 6
0783 40 8
0795 45 9
0806 50 10
0818
1166
1180
1194
1208
0842
0854
0866
0878
.000889.000890
0902
0914
0927
0939
0902
0915
0927
0940
.000952.000953
0964 0965
0977
0978
0990
0991
1003
10041
.001016 .001017
1029
1030
1043
1044
1056
1057
1069
1070
.001083.001084
1096
1110
1124
1138
1098 " Corr.
CORR.
FOR
SEC.
+
1125 15
" Corr.
1111 10 2 10 49
15 73
20 97
.001152.001153 30
1167 40
1139 20 5
.001222.001224
1237
1251
1266
1281
1238
1253
1268
1282
.001295.001297
1310
1325
9
118145 10
1195 50 12
1210
1312
1327
1342
1357
~MI MOON
1340
1355
.001370.001372
3
CORR.
FOR
SEC.
+
" Corr.
10
49
15 73
20 97
30 146
40 194
45 219
50 243
7 30 146
40 194
45 219
50 243
TANGENT
1
.034921 0
5212
5503 2
5794 3
6086 4
5
.036377
6668 6
6960 7
7251 8
7542 9
.037834 10
8125 11
8416 12
8707 13
8999 14
.039290 15
9581 16
9873 17
.040164 18
0456 19
.040747 20
1038 21
1330 22
1621 23
1912 24
.042204 25
2495 26
2787 27
3078 28
3370 29
.043661 30
3952 31
4244 32
4535 33
4827 34
.0451 18 35
5410 36
570137
5993 38
6284 39
.046576 40
6867 41
715942
7450 43
7742 44
.048033 45
8325 46
861747
8908 48
9200 49
.049491 50
9783 51
.050075 52
0366 53
0658 54
050950 55
1241 56
153357
182458
2116 59
.052408 60
464
0.052336
2626
2917
3207
3498
234
5.053788
6678σ
9
||
10.055241
234
12
13
14
16
17
18
19
2~222 2~~~~ ..~♡~ 66588 II 500 85007 HONG
21
23
24
7854
20.058145
8435
8726
9016
9306
26
28
29
25.059597
9887
27.060178
0468
0758
31
15.056693 30 145.998392 30 8
6983 40 194
8375 40 |
32
33
34
30.061048
1339
1629
1920
2210
36
38
SINE
35.062500
2790
3081
3371
3661
41
42
43
44
4079
4369
4660
4950
46
47
48
40.063952
4242
" Corr.
4532 10 48
482315 73
5113 20 97
49
5531 " Corr.
582210 48
611215 73
6402 20 97
51
45.065403 30 145
5693 40 194
5984 45 218
6274 50 242
6564
52
7274 45 218
7564 50 242
50.066854
7145
7435
53 7725
54
8015
56
57
58
59
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
55.068306
8596
8886
9176
9466
60.069756
COSINE
998630
8614
8599
8584
8568
.998552
8537
8521
8505
8489
.998473
8457 Corr.
8441110 3
8424 15 4
8408 20 5
998308
8291
8274
8257
8240
8358 45 12
8342 50 14
8325
.998222
8205
8188
8170
8152
.998135
8117
8099
8081
8063
998045
8027
8008
7990
7972
CORR.
FOR
SEC.
11
.997763
7743
7724
7704
7684
.997664
7644
7624
7604
7584
1.997564
+
|.997953
11
7934 Corr.
7916 10 3
789715 5
7878 20 6
.997859 30 9
7840 40 13
7821 45 14
7802 50 16
7782
3°

VERSINE
.001370.001372
1386
1401
1416
1432
EXSEC
.001448.001450
1463
1479
1495
1511
1388
1403
1418
1434
1625
1642
1658
1675
1466
1481
.001527.001.529
1.543
| 559
1576
1592
1497
1513
.001608.001611130
1545 "Corr.
156210 3
157815 4
1594 20 5
8
162840 11
1644 45 12
1661 50 14
1678
.001692.001695
1709
1726
1743
1760
2160
2179
2198
2218
1712
1729
1746
1763
.001778.001781
1795
1812
1830
1848
1798
1816
1833
1851
.001865.001869
1883
1901
1919
1937
1887
1904
1922
1941
.001955.001959
1973
1977
1992
1996
2010
2014
2028 2033
.002047.002051
2066
2084
2103
2122
"
3
2070 Corr.
2089 10
210815 5
2127 20 6
.002141.002146 30 9
2165 40 13
2184 45 14
2203 50 16
2223
CORR.
FOR
SEC.
+
.002237.002242
2257
2276
2296
2316
2262
2282
2301
2321
.002336.002341
2356
2376
2396
2416
.002436.002442
2361
2381
2401
2422
CORR.
FOR
SEC.
+
"Corr.
10 49
15 73
20 97
"Corr.
49
10
15 73
20 97
TANGENT
30 146
140 195
145 219
50 243
.052408
2700
2991
3283
3575
.053866
4158
4450
4742
5033
01234 56789
8
30 146.056784 15
40 195 7076 16
45 219
7368 17
50 243
7660 18
7952 19
.055325 10
5617 11
5909 12
6200 13
6492 14
.058243 20
8535 21
8827 22
911923
941 24
.059703 25
9995 26
.060287 27
0579 28
087 29
.061163 30
145531
1747 32
2039 33
2331 34
062623 35
291536
3207 37
3499 38
3791 39
.064083 40
4375 41
4667 42
4959 43
5251 44
.065544 45
5836 46
6128 47
6420 48
6712 49
.067004 50
7297 51
7589 52
788 53
8173 54
.068465 55
8758 56
9050 57
9342 58
9634 59
.069927 60
465
0.069756
0-234 567BD 0-234 DO
1.070047
8
5.071207
9
11
12
10.072658
13
14
16
22222 222~~ ~~~~~
21
15.074108 30 145
4399 40 193
23
24
17 4689 45 218
18
19
20.075559
5849
6139
6429
6719
26
27
28
29
25.077009
7299
7589
7879
8169
31
32
33
34
SINE
30.078459
8749
9039
37
38
39
55555
0337
0627
0917
41
42
43
44
1497
1788
2078
2368
35.079909
36.080199
51
52
40.081359
3815 ༄གླགླུ
2948
"Corr.
3238 10 48
3528 15 73
3818 20 97
53
54
4979 50 242
5269
56
57
59
9329
9619
46
47
48
49 3968
50.084258
45.082808 30 145
3098 40 193
3388 45 217
3678 50 242
0489
0779
1069
VERSINES, EXSECANTS, AND TANGEnts
4°

CORR.
FOR
SEC.
+
11
1939 10
2228 15
2518 20
1649 Corr
48
72
97
55.085707
5997
6286
6576
6866
60.087156
4547
4837
5127
5417
COSINE
.997564
7544
7523
7503
7482
997462
7441
7420
7399
7378
.997357
4
7336 " Corr.
7314 10
729315 5
7272 20 7
997250 30 11
7229 40 14
1.997141
7119
7097
7075
7053
7207 45 16
7185 50 18
7163
.997030
7008
6985
6963
6940
.996917
6894
6872
6848
6825
996802
6779
6756
6732
6708
CORR.
FOR
SEC.
.996685
666 I
" Corr.
6637 10 4
6614 15 6
6590 20 8
1.996444
6420
6395
457
.996566 30 12
6541 40 16
6517 45 18
6493 50 20
6468
6370
6345
.996320
6295
6270
6245
6220
|.996195
+
VERSINE
.002436.002442
2456
2477
2497
2518
.002538.002545
2559
2580
2601
2622
2771
2793
2815
2837
EXSEC
002643.002650
2664
2686
2707
2728
2462
2483
2504
2524
2671 " Corr.
2693 10 4
271415 5
2736 20 7
.002750.002757 30 11
2779 40 14
2801 45 16
2823 50 18
2845
3152
3175
2566
2587
.002859.002867
2881
2903
2925
2947
2608
2629
.002970.002979
2992
3015
3037
3060
3459
3483
3507
3532
2889
2911
.003083.003092
3106
3128
2934
2956
3630
3655
3001
3024
3046
3069
.003198.003208
3221
3244
3268
3292
3115
3138
3162
3185
.003315.003326
3339 3350
3363
3386
3410
3232
3255
3279
3302
.003434.003446 30 12
347140 16
3495 45 18
3520 50 20
3544
.003556.003569
3580
3605
337410
3398 15 6
3422 20 8
CORR.
FOR
SEC.
+
3593
3618
3643
3668
.003680 1.003693
3705
3730
3755
3769
3780
3794
.003805 003820
" Corr.
4
3718
3744
CORR.
FOR
SEC.
+
11
Corr.
10 49
15 73
20 97
30 146
40 195
45 219
150 244
" Corr.
10
49
15 73
20 98
30 146
40 195
45 220
50 244
TANGENT
.069927 0
.070219 I
0512 2
0804 3
1096 44
.071388
1681 6
1973
2266 8
2558 9
.072850 10
314311
3435 12
3728 13
4020 14
.074313 15
4605 16
4898 17
5190 18
5483 19
.075776 20
6068 21
636122
6653 23
6946 24
.077238 25
75326
7824 27
8116 28
8409 29
078702 30
8994 31
9287 32
9580 33
9873 34
.08016535
0458 36
0751 37
1044 38
1336 39
.081629 40
1922 41
221542
2508 43
2801 44
.083094 45
3387 46
3679 47
3972 48
4265 49
.084558 50
4851 51
5144 52
5437 53
5730 54
.086023 55
631656
6609 57
690258
7196 59
.087489 60
466
0.087156
01234 6O76&
8
9
5.088605
11
234 567B9
12
10.090053
13
14
16
17
18
19
2222 7ºº 87287 COMO DE 9 85887 68788 8
21
23
24
20.092950
3240
3529
3819
4108
26
27
28
29
15.091502 30 145.99580530 13
577840 18
1791 40 193
2081 45 217
5752 45 20
2371 50 241
5725 50 22
2660
5698
25.094398
4688
4977
5267
5556
31
32
33
34
30.095846
6135
6425
6714
7004
36
37
38
39
41
42
43
44
SINE
35.097293
7583
7872
8162
8451
46
47
48
7446
7735
49
8025
8315
51
8894
9184
9474
9764
52
40.098741
9030 " Corr.
9320 10 48
9609 15 72
9899 20 96
45.100188 30 145
0478 40 193
0767 45 217
1056 50 241
1346
53
54
50.101635
1924
2214
2503
2792
56
57
0343 "Corr.
0633 10 48
0922 15 72
121220 97
58
59
55.103082
3371
3660
3950
4239
60.104528
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
COSINE
.996195
6169
6144
6118
6093
996067
6041
6015
5989
5963
11
5937
5911 Corr.
5884 10 4
5858 15 7
5832 20 9
995671
5644
5616
5589
5562
.995534
5507
5480
5452
5424
.995396
5368
5340
5312
5284
.995256
5227
5199
5170
5142
CORR.
FOR
SEC.
.995113
5084
" Corr.
5056 10 5
5027 15 7
4998 20 10
994968 30 15
4939 40 20
4910 45 22
488150 24
4851
994822
4792
4762
4733
4703
+
994673
4643
4613
4582
4552
.994522
5°

VERSINE
.003805.003820
3831
3856
3882
3907
EXSEC
4222
4248
4275
4302
3845
3871
.003933.003949
3959 3975
3985
4001
4011
4037
3897
3923
4411
4438
4027
4053
4063
4089
4133 10 4
4116
4142 4159 15 7
4168 4186 20 9
.004195||.004213 30 13
4240 40 18
4267 45 20
4294 50 22
4321
4080
4106
.004329.004348
4356
4384
4375
4403
4430
4458
.004466.004486
4493
4520
4548
4576
4513
4541
4569
4597
.004604.004625
4632
4660
4688
4716
4653
4682
4710
4738
.004744.004767
4773 4796
4801
4824
4853
4830
4858
4882
.004887.004911
4916
4944
4973
5002
.005178.005205
5208
5238
5267
5297
CORR.
FOR
SEC.
+
4940
4969 10
5
4998 15 7
5028 20 10
.005032.005057 30 15
5061
5090
5086 40 20
511645 22
5146 50 24
5175
5119
5149
5235
5265
5295
5325
005327.005356
5357 5386
5387
5416
5418
5447
5448
5478
.005478.005508
" Corr.
" Corr.
CORR.
FOR
SEC.
+
"Corr.
10 49
15 73
20 98
30 147
40 196
45 220
50 244
" Corr.
10
49
15 73
20 98
30 147
40 196
45 220
50 245
TANGENT
.087489 0
7782 I
8075 2
8368 3
8661 4
.088954
5
7
9248 6
9541
9834 8
.090127
9
0421 10
0714 11
1007 12
1300 13
1594 14
.091887 15
2180 16
2474 17
2767 18
3061 19
.093354 20
364721
3941 22
4234 23
452824
.09482125
511526
5408 27
5702 28
5996 29
.096289 30
6583 31
6876 32
7170 33
746434
.097757 35
8051 36
8345 37
8638 38
8932 39
099226 40
952041
981342
.100107 43
040144
.10069545
0989 46
1282 47
1576 48
1870 49
.102164 50
2458 51
2752 52
3046 53
334054
.103634 55
3928 56
4222 57
451658
4810 59
.105104 60
467
1
0.104528
4818
5107
01234 567∞σ
9
5.105975
TI
-234
12
10.107421
13
14
16
17
18
22222
-234
21
22
23
24
15.108867 30 145
9156 40 193
9445 45 217
9734 50 241
19.110023
20.110313
2272 8~~♡~ ❤❤❤
26
28
29
32
25.11758
2047
2336
2625
2914
37
SINE
41
42
43
44
5396
5686
30.113203
3492
3781
46
47
g ggg g 5ངྒོ
6264
6553
33 4070
34
4359
48
6842
7132
35.114648
4937
5226
5515
5804
49
56
57
" Corr.
48
7710
799910
828815 72
8578 20 96
0602
0891
40.116093
6382
" Corr.
667110 48
6960 15 72
7248 20 96
45.117537 30 144
7826 40 193
8115 45 217
8404 50 241
8693
58
59
1180
1469
50.1.18982
9270
51
52
9559
53 9848
54.120137
55.120426
VERSINES, EXSECANTS, AND TANGENTS
CORR.
FOR
SEC.
+
0714
1003
1292
1581
60.121869
COSINE
.994522
4491
4461
4430
4400
.994369
4338
4307
4276
4245
.994214
4182
4151 TO 5
4120 15 8
4088 20 ||
.993897
3865
3833
3800
3768
.994056 30 16
4025 40 21
3993 45 24
3961 50 26
3929
.993736
3703
3670
3638
3605
.993572
3539
3506
3473
3440
.993406
3373
3339
3306
3272
CORR.
FOR
SEC.
" Corr.
.993238
3204
" Corr.
3171 10 6
3137 15 9
3103 20 |
.992896
2862
2827
993069 30 17
3034 40 23
3000 45 26
2966 50 29
2931
2792
2757
992722
2687
2652
2617
2582
1.992546
6°

VERSINE
EXSEC
.005478.005508
5509 5539
5539
5570
5570
5601
5600
5632
.005631.005663
5662
5693
5724
5755
5694
5726
5757
5788
.005786.005820
5818
5849
5880
5912
5852 Corr.
588310 5
5915 15 8
5947 20 11
.005944.005979 30 16
5975 60 40 21
6044 45 24
607650 27
6108
6007
6039
6071
.006103.006140
6135 6173
6167
6206
6238
6200
6232
6271
.006264.006304
6297
6330
6362
6395
6527
6560
6337
6370
6403
6436
.006428.006470
6461
6494
6503
6537
6570
6604
.006594.006638
6627
6661
6694
6728
6671
6705
6739
6774
.006762.006808
6796
6829
6863
6897
.007104.007154
7138
7173
7208
7243
.006931.006980 30 17
6966 7015 40 23
7049 45 26
7084 50 29
7000
7034
7069
7119
7190
7225
7260
7296
.007278.007331
7313
7348
7383
7418
.007454.007510
CORR.
FOR
SEC.
+
11
6842 "Corr." Corr.
6876 10 6 10 49
69115 9
6945 20 12
15 74
20 98
7367
7402
7438
7474
CORR.
FOR
SEC.
+
" Corr.
10 49
15 74
20 98
30 147
40 196
45 221
50 245
130 147
40 197
45 221
50 246
TANGENT
.105104
5398
5692
5987
6281
.106575
6869
01234 56789
7163 7
7458
7752 9
.108046 10
8340 11
8635 12
8929 13
922314
.109518 15
981216
.110107 17
0401 18
0696 19
. I 10990 20
1284 21
1579 22
1873 23
216824
.112462 25
2757 26
3052 27
3346 28
3641 29
.113936 30
423031
4525 32
482033
511434
.115409 35
5704 36
5999 37
6294 38
6588 39
.11688340
717841
7473 42
7768 43
8063 44
.118358 45
865346
8948 47
9243 48
9538 49
.19833 50
.1201285ĺ
0423 52
0718 53
101354
.121308 55
1604 56
1899 57
219458
2489 59
.122785 60
468
0.121869
2158
2447
01234 BONBO
6
7
5.123313
8
9
0123⇒
10.124756
11
12
13
14
5045 "Corr.
5333 TO 48
562215 72
5910 20
96
15.126199 30 144
648840 192
17 6776 45 216
16
7065 50 241
7353
0 20 207** 87♡♡
21
22
23
24
20.127642
7930
8219
8507
8796
26
28
25.129084
9372
9661
9949
31
32
29.130238
30.130526
33
34
F
41
42
43
44
SINE
35.131968
36 2256
37
2545
38
2833
39
3121
46
47
40.133410
48
2736
3024-
g gགླ$ ༄གྲུཤྩg 5ཆོ
49
3602
3890
4179
4467
51
45.134851 30 144
5139 40 192
5427 45 216
5716 50 240
6004
52
53
54
56
0815
1103
50.136292
6580
6868
7156
7444
57
1391
1680
58
59
55.137733
8021
8309
8597
8885
3698 " Corr.
398610 48
4274 15 72
4563 20 96
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
60.139173
COSINE
.992546
2511
2475
2439
2404
||.992368
2332
2296
2260
2224
.992187
#1
2151 Corr.
2115 10
2078 15
2042 20 12
.991820
1783
1746
1709
1671
.992005 30 18
1968 40 25
1931 45 28
1894 50 31
1857
.991634
1596
1558
1521
1483
.991445
1407
1369
1331
1292
.991254
1216
1177
1138
1100
CORR.
FOR
SEC.
.991061
1022
0983 10 7
094415 10
0905 20 13
990669
0629
0589
0549
0510
loon
990469
0429
0389
.990866 30 20
0827 40 26
0787 45 29
0748 50 33
0708
0349
0308
990268
+
"Corr.
7༠

VERSINE
.007454.007510
7489 7546
7525
7582
7561
7618
7596 7654
EXSEC
.007632.007691
7668
7704
7740
7776
7922
7958
.007813.007874
7849
7885
7727
7764
7800
7837
.008180.008247
8217
8254
8291
8329
.007995.008060 30 19
8032 8097 40 25
8134 45 28
8172 50 31
8069
8106
8143
8209
7911 Corr.
794810 6
7985 15 9
8022 20 12
.008366.008437
8404
8442
8479
8517
9173
9213
8285
8323
8361
8399
9252
9292
8475
8514
.008555.008629
8593
8631
8669
8708
8552
8590
.008746.008823
8784
8823
8862
8900
8668
8706
8745
8784
8862
8902
8941
8980
.008939.009020
8978
9017
9059
" Corr.
9099 10 7
9056 913915 10
9095 9178 20 13
.009134.009218 30 20
9258 40 27
9298 45 30
9339 50 333
9379
.009331.009419
9371
9411
9451
9490
#1
"
9460
9500
9541
9582
CORR.
FOR
SEC.
+
.009531.009622
9571
9611
9651
9692
.009732.009828
9663
9704
9745
9786
CORR.
FOR
SEC.
+
Corr.
10 49
15 74
20
99
30 148
40 197
45 222
50 246
•
TANGENT
122785
3080
3375
3670 3
3966
.124261
4557
4852
5147
0123➡
5678σ
5443 9
.12573810
603411
6329 12
6625 13
69201 14
.127216 15
7512 16
7807 17
8103 18
8399 19
128694 20
8990 21
9286 22
9582 23
9877 24
.130173 25
0469 26
0765 27
1061 28
1357 29
.131652|30
1948 31
2244 32
2540 33
2836 34
.133132 35
3428 36
3725 37
4021 38
4317 39
.134613 40
" Corr.
4909 41
10 49 520542
15 74 5502 43
20 99 5798 44
30 148.136094 | 45
40 198 6390 46
45 222
50 247
6687 47
6983 48
7279 49
.137576 50
7872 51
816852
8465 53
8762 54
.139058 55
9354 56
965157
9948 58
.140244 59
140541 60
469
0.139173
9461
9749
3.140037
0325
1234 6O7Bσ
5.140613
0901
1189
1477
1765
8
9
10.142053
0-234 DONO
||
12
13
14
16
17
18
22 22
15.143493 30 144
3780 40 192
4068 45 216
435650 240
4644
19
21
22
20.144932
5220
5508
23
24
26
27
28
29
25.146371
8 6686 UNDO 55555 £5555 wwwww wwwww
36
30.147809
8097
8385
8672
8960
37
39
35.149248
9535
9823
38.15011|
0398
SINE
42
43
44
40.150686
46
47
48
49
2341 " Corr.
2629 10
48
291715 72
3205 20 96
51
5795
6083
52
53
54
45.152123 30 144
2411 40 192
2698 45 216
2986 50 240
3273
6658
6946
7234
7522
56
50.15356|
3848
4136
4423
4710
57
58
59
55.154998
5285
5572
5860
6147
VERSINES, EXSECANTS, AND TANGENTS
8°

CORR.
FOR
SEC.
+
#
0973 Corr.
1261 TO 48
1548 15 72
1836 20 96
60.156434
COSINE
------
.990268
0228
0187
0146
0106
.990065
0024
.989983
9942
9900
.989859
9818 " Corr.
9776 10 7
973515 10
9693 20 14
989651 30 21
9610 40 28
9568 45 31
9526 50 35
9484
.989442
9399
9357
9315
9272
.989230
9187
9144
9102
9059
.989016
8973
8930
8886
8843
.988800
8756
8713
8669
8626
.988582
CORR.
FOR
SEC.
n
8538 Corr.
8494 10 7
8450 15 |
8406 20 15
.988362 30 22
8317 40 29
8273 45 33
8228 50 37
8184
.988139
8094
8050
8005
7960
.987915
7870
7824
7779
7734
.987688
+
VERSINE
.009732.009828
9772
9813
9854
9894
EXSEC
9869
9910
.009935.010035
9976
.010017
0058
0100
9952
9993
0898
0941
0077
0119
0161
0203
.010141.010245
0182
0287
"Corr.
0224 0329 10 7
0265 0372 15 11
0307 041420 14
.010349.010457 30 21
0390 050040 28
0542 45 32
058550 36
0432
0474
0516 0628
.010558.010671
0601
0643
0685
0728
.010770.010888
0813
0856
0714
0757
0801
0844
1683
1727
1772
1816
0931
0975
1018
1062
.010984.011106
1027
1070
1114
1157
| 150
1194
1238
1283
.011200.011327
1244
1287
1331
1374
1372
1416
1461
1505
.011418.011550
1462
1506
1550
1594
.011638.011776 30 23
1821 40 30
1866 45 34
1912 50 38
1958
.011861.012003
1906
1950
1995
2040
CORR.
FOR
SEC.
+
2049
2095
2141
2187
"
1595 Corr." Corr.
1640 10 8
168515 1
1730 20 15
10 50
15 74
20 99
.012085.012233
2130
2176
2221
2266
.012312.012465
2279
2326
2372
2418
CORR.
FOR
SEC.
+
" Corr.
10 50
15 74
20 99
30 149
40 198
45 223
50 248
30 149
40 199
45 223
50 248
TANGENT 1
140541
I
0838
1134 2
1431 3
1728 4
.142024
5
2321 6
2618 7
2915 8
3212 9
.143508 10
3805 1
4102 12
439913
4696 14
.144993 15
5290 16
5587 17
5884 18
6181 19
.146478 20
6776 21
7073 22
7370 23
7667 24
.147964 25
8262 26
8559 27
8856 28
9154 29
.149451 30
9748 31
.150046 32
0343 33
0641 34
.150938 35
1236|36
1533 37
1831 38
2128 39
.152426 40
2724 41
3022 42
331943
3617 44
.15391545
4212 46
4510 47
480848
5106 49
.155404 50
570251
6000 52
6298 53
659654
.156894 55
719256
7490 57
7788 58
8086 59
.158384 60
470
0.156434
1234 5O769
5.157871
16
17
18
19
2222
10.159307
21
22
23
24
11
││
9594 Corr.
12 988 10 48
13.160168 15 72
14 0456 20 96
15.160743 30 144
103040 191
131745 215
1604 50 239
1891
20.162178
2465
2752
3039
3326
26
27
28
29
SINE
wwwww wwwww
6722
7009
7296
7584
25.163613
3900
4187
4474
4761
41
42
43
4444
8158
8445
g gགླ་ྒུ༄གླུཤྩgČཆོ
8732
9020
35.166482
6769
7056
7342
7629
52
53
40.167916
54
56
57
58
165048
5334
5621
5908
6195
50.170783
1069
1356
1642
1929
45.169350 30 143
9636 40 191
9923 45 215
46
47
48.170210 50 239
49 0496
8203
8489 10
" Corr.
48
8776 15 72
9063 20 96
55.172216
2502
2789
3075
3362
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
60.173648
6255
COSINE
987688
7643
7597
7551
7506
987460
7414
7368
7322
7275
.987229
7183
7136 10 8
7090 15 12
7043 20 16
|.986996 30 23
6950 40 31
6903 45 35
6856 50 39
6809
986762
6714
6667
6620
6572
.986525
6477
6429
6382
6334
.986286
6238
6189
6141
6093
.986044
5996
5948
5899
5850
985309
5259
5209
CORR.
FOR
SEC.
5159
5109
.985801
5752
" Corr.
5704 10 8
565415 12
5605 20 16
985556 30 25
5507 40 33
5457 45 37
5408 50 41
5358
985059
5009
4959
4909
4858
.984808
+
"Corr.
9°

VERSINE
.012312.012465
2357
2403
2449
2494
EXSEC
.012540.012700
2586 2747
2632
2794
2678
2841
2725 2889
2512
2559
2606
2652
.012771.012936
"
2817
2864
2910
2984 Corr.
3031 10 8
307915 12
2957 3127 20 16
.013004 ||.013175 30 24
3050 3223 40 32
3271 45 36
3097
3144
3319 50 40
3191
3368
.013238.013416
3286
3333
3380
3428
4004
4052
4101
4150
.013475.013660
3523
3571
3618
3666
3465
3513
4493
4543
4592
4642
3562
3611
.013714.013905
3762
3811
3859
3907
4841
4891
3708
3757
3807
3856
|.013956 .014153
4203
4253
4303
4353
3954
4004
4054
4103
.014199.014403
4248
4296
4346
4395
.014444.014656 30 25
4706 40 34
4757 45 38
4808 50 42
4859
CORR.
FOR
SEC.
+
|.014691 ||.014910
4741
4962
4791
5013
5064
5116
4454
"Corr.
4504 10 8
455415 13
4605 20 17
.014941.015167
4991
5219
5041
5271
5091
5323
5142
5375
.015192 .015427
CORR.
FOR
SEC.
+
"Corr.
10 50
15 75
20 100
" Corr.
10 50
15 75
20 100
TANGENT
.158384
8683
30 150
40 200
45 225
50 250
8981
9279
9577
.159876
.160174
0472
0771
1069
01234 56789
30 149.162860 15
40 199
3159 16
145 224
3458 17
50 249
3756 18
4055 19
161368 10
1666 11
1965 12
2263 13
2562 14
.164354 20
4652 21
495 22
5250 23
5549 24
.165848 25
6147 26
6446 27
6745 28
7044 29
167343 30
7642 31
7941 32
824033
8539 34
.168838 35
913736
9437 37
9736 38
.170035 39
170334 40
063441
0933 42
1232 43
153244
.171831 45
2131 46
2430 47
2730 48
303049
.173329 | 50
3629 51
3928 52
4228 53
4528 54
.17482855
512856
5427 57
5727 58
6027 59
176327 60
471
:
0.173648
01234 567∞M OI231
8
5.175080
9
II
12
10.176512
13
14
16
5878 ❤
17
18
19
~~
15.177944 30 143
8230 40 191
8516 45 215
8802 50 239
9088
AUGE N°° 8.~♡~ 6888 23 99509 85õõõ kubog 8
21
22
24
20.179375
9661
9947
23.180233
0519
25.180805
1091
1377
1664
1950
26
27
29
31
32
34
30.182236
2522
2808
3094
3380
36
37
38
35.183665
3951
4237
SINE
41
42
44
3935
4221
4508
4794
46
5367
5653
5940
6226
40.185095
49
52
53
54
11
6798 Corr.
7085 10 48
737115 72
7657 20 95
56
57
4523
4809
45.18652430 143
681040 190
7096 45 214
738 50 238
7667
50.187953
8238
8524
8810
9095
55.189381
9667
9952
58.190238
0523
VERSINES, EXSECANTS, AND TANGENTS
10°

CORR.
FOR
SEC.
+
5381 " Corr.
5667 TO 48
595215 71
6238 20 95
60.190809
COSINE
.984808
4757
4707
4656
4605
984554
4503
4452
4401
4350
.984298
11
4247 Corr.
4196 10 9
4144 15 13
4092 20 17
.984041 30 26
3989 40 35
3937 45 39
3885 50 43
3833
.983781
3729
3676
3624
3572
.983519
3466
3414
3361
3308
983255
3202
3149
3096
3042
.982989
2935
2882
2828
2774
982721
CORR.
FOR
SEC.
2667" Corr.
2613 10 9
2559 15 14
2505 20 18
.982450 30 27
2396 40 36
2342 45 41
2287 50 45
2233
982178
2123
2069
2014
1959
.981904
1848
1793
+
1738
1683
981627
VERSINE
.015192.015427
5243
5293
5344
5395
EXSEC
.015446.015688
5497
5548
5599
5650
5479
5531
5583
5636
6011
6063
6115
6167
.015702.015952
5753
5804
5856
5908
5741
5793
5846
5899
6005 " Corr.
6058 10 9
615 13
6165 20 18
.015959.016218 30 27
6272 40 36
6325 45 40
6379 50 45
6433
.016219.016486
6271
6324
6376
6649
6428 6703
7172
7226
6540
6595
.016481.016757
6534
6586
6639
6692
6812
6866
6921
6976
.016745.017030
6798
6851
6904
6958
7085
7140
7195
7250
.017011.017306
7065
7118
7361
7416
7472
7528
017279.017583
7333
7387
7441
7495
7639
" Corr.
7695 10 9
7751 15 14
7807 20 19
.017550.017863 30 28
7604
7658
7919 40 37
7976 45 42
803250 47
7713
7767
8089
.017822.018145
7877
7931
7986
8041
CORR.
FOR
SEC.
+
8202
8259
8316
8373
.018096.018430
8152
8487
8207
8544
8262
8602
8317
8659
.018373 .018717
CORR.
FOR
SEC.
+
Corr.
10 50
15 75
20 100
30 150
40 200
45 225
50 250
" Corr.
10
50
15 75
20 100
30 151
40 201
45 226
50 251
TANGENT 1
.176327
6630
6927
7227
7527
.177827
0
1231
5
8127 6
8427 7
8727 8
9028 9
.179328 10
9628 11
9928 12
.180229 13
0529 14
.180830 15
1130 16
1430 17
1731 18
2031 19
. 182332 20
2632 21
2933 22
3234 23
3534 24
183835 25
4136 26
4436 27
4737 28
5038 29
. 185339 30
5640 31
5941 32
624233
6543 34
.186844 35
7145 36
7446 37
7747 38
8048 39
.18835040
865141
8952 42
9253 43
9555 44
.189856 45
.190157 46
0459 47
0760 48
1062 49
.191363 50
1665 51
1966 52
2268 53
2570 54
.192871 55
3173 56
3475 57
3777 58
4078 59
.194380 60
472
0.190809
0-23A BONBO
4
6
9
5.192236
||
12
13
14
10.193664
DONOR OU~~2 20702 87♡♡7 20788 222
17
18
" Corr.
3949
423410 48
4520 15 71
4805 20 95
15.195090 30 143
5376 40 190
5661 45 214
5946 50 238
6231
22
21
22
20.196517
6802
7087
7372
7657
23
24
26
28
29
25.197942
8228
8513
8798
9083
31
32
34
30.199368
9653
9938
33.200223
0508
36
37
38
35.200793
1078
1363
1648
1933
42
SINE
43
44
1094
1380
40.202218
46
47
48
49
1666
1951
85087 KONGO 8
2522
2807
3093
3378
51
52
53
54
56
50.205066
57
59
2502
" Corr.
278710 47
307215 71
3357 20 95
55.206489
4496 50 237
4781
5350
5635
5920
6204
TABLE XX.-NATURAL SINES, COSINEs,
CORR.
FOR
SEC.
+
45.203642 30 142.979046 30
3926 40 190
421 45 214
6773
7058
58 7343
7627
60.207912
COSINE
.981627
1572
1516
1460
1404
980500
0443
0386
0329
0271
.981349
1293
1237
1180
π
1124
.981068
1012 Corr.
0955 10 9
0899 15 14
0842 20 19
.980785 30 28
0728 40 38
0672 45 43
0615 50 47
0558
.980214
0156
0098
0040
.979983
.979925
9867
9809
9750
9692
.979634
9575
9517
9458
9399
-
CORR.
FOR
SEC.
978748
8689
8629
8569
8509
+
.979341
9282
" Corr.
922310 10
9164 15 15
9105 20 20
978449
8389
8329
8268
8208
.978148
8986 40 40
8927 45 44
8867 50 49
8808
11°

VERSINE
.018373.018717
8428
8484
8540
8596
8820
8876
.018651.019006
8707
8763
9045
9101
9158
EXSEC
.018932.019297
9272
9328
9385
9442
8774
8832
8890
8948
8988 9356 Corr.
9415 10 10
9473 15 15
9532 20 20
.019215.019591 30
9650 40
9709 45
976950
9828
9671
9729
9064
9122
9180
9239
.019500.019887
9557 9947
9614.020006
0483
0542
0601
.019786.020186
9844
9902
9960
.020017
0066
0126
1073
1133
| 192
.020075.020487
0133
0191
0250
0308
0246
0306
0366
0426
.020366.020790
0425 0851
0912
0973
1034
0547
0608
0668
0729
.020659.021095
0718
0777
0836
0895
30.020954.021403 30 31
1014
1465 40 41
CORR.
FOR
SEC.
+
1157
" Corr.
121810 10
15
1341 | 20 21
1280 15
.021252.021713
1311
1371
1431
1491
29
39
44
49
1527 45 46
1589 50 51
1651
1776
1838
1900
1963
.021551.022026
1611
2088
1671
2151
1732
2214
1792
2277
.021852 .022341
CORR.
FOR
SEC.
+
π
10 50
15 76
20 101
Corr.
30 151
40 202
45 227
50 252
" Corr.
10
51
15 76
20 101
30 152
40 202
45 228
50 253
TANGENT
.194380
4682
4984
5286
5588
1
01234
5678✪
.195890
6192
6494
6796
7099 9
8
.197401 10
7703 11
8005 12
8308 13
8610 14
.198912 15
9215 16
9517 17
9820 18
.200122 19
.200425 20
0727 21
1030 22
1333 23
1635 24
.201938 25
2241 26
2544 27
2846 28
3149 29
.203452 30
3755 31
405832
436133
4664 34
.204967 35
5270136
5574 37
5877 38
6180 39
.206483 40
6787 41
7090 42
7393 43
7697 44
.208000 45
8304 46
860747
8911 48
921449
.209518 50
9822 51
.210126152
0429 53
0733 54
.211037 55
134156
1645 57
1949 58
2252 59
.212557 60
473
0.207912
8196
8481
0-23=
4
10
66789 0-2MT DON
5.209334
9619
9903
8.210187
0472
||
10.210756
12
13
14
16
17
18
19
15.212178 30 142
2462 40 189
2746 45 213
3030 50 237
3315
~~~~~ ~~~~~ ..~87 68
21
20.213599
3883
4167
4451
4735
22
23
24
34 50
26
27
25.215019
28
29
32
33
34
36
37
38
39
30.216440
6724
7008
བྲཱཎྜཱམྦྷཏྠ
SINE
42
35.217859
8143
8427
8711
8995
44
8765
9050
46
47
48
49
40.219279
11
9562 Corr.
9846 10 47
43.220130 15 71
0414 20 95
51
234
OKZ KONA8 8
" Corr.
1040
1325 10 47
1609 15 71
1893 20 95
52
53
5304
5588
5872
6156
45.220697 30 142
0981 40 189
1265 45 213
1548 50 236
1832
50.222116
2399
2683
2967
3250
54
7292
7575
56
57
58
59
55.223534
VERSINES, EXSECANTS, AND TANGENTS
12°

CORR.
FOR
SEC.
+
3817
4101
4384
4668
60.224951
COSINE
.978148
8087
8026
7966
7905
976922
6859
6797
.977844
7783
7722
7661
TT
7600
.977539
7477 Corr.
7416 10 10
7354 15 15
7293 20 21
.977231 30 31
7169 40 41
7108 45 46
7046 50 51
6984
6735
6672
.976610
6547
6484
6422
6359
.976296
6233
6170
6107
6044
.975980
5917
5853
5790
5726
CORR.
FOR
SEC.
.975662
5598
5534 10
547115
5406 20
975020
4956
4891
4826
4761
.975342 30
5278 40
5214 45
5149 50
5085
.974696
4631
4566
4501
4436
974370
+
" Corr.
||
16
21
32
43
48
53
VERSINE
EXSEC
.021852.022341
1913
1974
2034
2095
2404
2467
2531
2594
.022156.022658
2217
2278
2339
2400
2722
· 2785
2849
2913
.022461.022977
2523
2584
2646
2707
" Corr.
3042
3106 10
3170 15 16
3235 20 21
.022769.023299 30
2831 336440
342945
3494 50
2892
2954
3016
3559
.023078.023624
3141
3203
3265
3328
3689
3754
3820
3885
.023390.023950
3453
3516
3578
3641
4722
4786
4851
4915
4016
4082
4148
4214
.023704.024280
3767 4346
3830
4412
3893
4478
3956
4544
.024020.02461|
4083
4147
4210
4274
4678
4744
4811
4878
.024338.024945
4402
4466
4529
4594
5012
" Corr.
5079 10 ||
5146 15 17
5214 20 22
22
.024658.025281 30 34
5349 40 45
5416 45 51
5484 50 56
5552
.024980.025620
5044
5109
5174
5239
5688
5756
5824
5892
CORR.
FOR
SEC.
+
.025304.025961
5369
5434
5499
5564
025630.026304
6029
6098
6166
6235
32
43
49
54
CORR.
FOR
SEC.
+
" Corr.
10 51
15 76
20 102
30 152
40 203
45 228
50 254
" Corr.
10
51
15 76
20 102
30 153
40 204
45 229
50 255
TANGENT
1
.212557 0
2861
3165
3469
3773
.214077
01234
2
5
4381 6
4686 7
4990 8
5294 9
.215599 10
5903 1
6208 12
6512 13
6817 14
.217121 15
7426 16
7731 17
8035 18
8340 19
.218645 20
8950 21
9254 22
9559 23
9864 24
.220169 25
0474 26
0779 27
1084 28
1390 29
.221695 30
2000 31
2305 32
2610 33
2916 34
.223221 35
3526 36
3832 37
4137 38
4443 39
.224748 40
5054 41
5360 42
5665 43
597 44
.226277 45
6583 46
6888 47
719448
7500 49
.227806 50
8112 51
8418 52
8724 53
9031 54
.229337 55
9643 56
9949 57
.230256 58
0562 59
.230868 60
474
0.224951
01234
5678 O
5.226368
6
9
11
12
234
10.227784
13
14
16
17
19
22222
21
23
24
15.229200 30 142
948440 189
9767 45 212
18.230050 50 236
0333
20.230616
0899
1182
1465
1748
26
27
28
29
866466 100º 55555 £55±5 wwwww wwwww
25.232031
2314
2597
2880
3102
36
30.233445
3728
4011
4294
4577
37
38
39
35.234859
5142
5425
5708
5990
41
SINE
42
43
5234
5518
5801
6085
40.236273
46
6651
6935
7218
7501
47
48
49
51
52
8068
" Corr.
835110 47
863415 71
8917 20 94
53
45.237686 30 141
7968 40 188
8251 45 212
8534 50 235
8816
56
57
50.239098
9381
9663
9946
58
59
54.240228
55.240510
6556
" Corr.
6838 10 47
71215 71
7403 20 94
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
0793
1075
1357
1640
60.241922
COSINE
.974370
4305
4239
4173
4108
974042
3976
3910
3844
3778
.973045
2978
2910
2843
2776
973712
3645 " Corr.
357910
11
3512 15 17
3446 20 22
973379 30 33
3312 40 44
3246 45 50
3179 50 56
3112
.972708
2641
2573
2506
2438
.972370
2302
2234
2166
2098
.972029
1961
1893
1824
1755
CORR.
FOR
SEC.
.971342 30
.971687
1618
" Corr.
1549 10 12
1480 15 17
1411 20 23
970995
0926
0856
0786
0716
+
970647
0577
0506
1273 40 46
1204 45 52
1134 50 58
1065
0436
0366
970296
13°

VERSINE
.025630.026304
5695 6373
5761
6442
5827
6511
5892
6581
EXSEC
.025958.026650
6024
6090
6156
6222
6719
6789
6859
6928
.026288.026998
6355
7068 " Corr.
7138 10
12
6421
6488
7208 15 18
6554 7278 20 23
23
.026621.027349 30 35
7419 40 47
7490 45 53
6821 7560 50 59
6888 7631
6688
6754
026955.027702
7022
7090
7157
7224
.027292.028057
7359
7427
17494
7562
8176
8245
7773
7844
7915
7986
.027630.028415
7698
7766
7834
7902
8129
8200
8272
8343
9214
9284
.027971.028776
8039
8107
8487
8559
8631
8703
.028313.029138
8382
8451
8520
8589
35.028658.029503 30 37
8727
8796
8866
8935 9797
9577 40 49
9650 45 55
9724 50 61
8848
8920
8993
9066
.029005.029871
9074 9945
9144.0300 1.9
0093
0167
CORR.
FOR
SEC.
+
9211
" Corr.
9284 10 12
9357 15 18
9430 20 24
029353.030241
9423
9494
9564
9634
.029704.030614
0315
0390
0464
0539
CORR.
FOR
SEC.
+
"Corr.
10
51
15 77
20 102
30 153
40 205
45 230
50 256
"Corr.
10 51
15 77
20 103
30 154
40 206
45 231
50 257
TANGENT
.230868
1175
1481
1788
2094
.232401
2707
1
0-231 Borg 0-234 567
5
6
7
30141
3321 8
3627 9
.233934 10
424111
4548 12
4855 13
5162 14
.235469 15
5776 16
6083 17
6390 18
6697 19
.237004 20
731221
7619 22
7926 23
8234 24
.23854 25
8848 26
9156 27
9464 28
9771 29
.240079 30
0386 31
0694 32
100233
1310 34
.24161835
1926 36
2233 37
2541 38
2849 39
243158 40
3466 41
3774 42
4082 43
4390 44
.244698 45
500746
5315 47
5624 48
5932 49
.246240 50
6549 51
6858 52
7166 53
747554
.247784 55
8092 56
840157
8710 58
9019 59
249328 60
475
0.241922
2204
2486
2768
3051
I2222 222
1231
0-231 DON00 0-2M² DONO ONEZ PON** OZ~m7 ❤❤♪❤. O 00 85887 HONOR 8
4
5.243333
3615
3897
4179
4461
6
8
9
10.244743
12
13
14
16
17
18
15.246153 30 141
6435 40 188
671745 211
6999 50 235
7281
19
21
20.247563
7844
8126
8408
8690
22
23
24
26
25.248972
27
28
31
32
29.250098
30.250380
0662
0943
1225
1506
33
34
36
37
35.251788
38
39
41
42
43
44
46
47
SINE
40.253195
48
49
51
52
45.254602 30 141
4883 40 188
5164 45 211
5446 50 234
5727
53
5025 " Corr.
5307 10 47
5589 15 70
5871 20 94
54
50.256008
9253
9535
9817
56
57
58
2069
2351
2632
2914
59
55.257414
" Corr.
3477
3758 0 47
4039 15 70
4321 20 94
6289
6570
6852
7133
VERSINES, EXSECANTS, AND TANGENTS
14°

CORR.
FOR
SEC.
+
7695
7976
8257
8538
60.258819
COSINE
.970296
0225
0155
0084
0014
.969943
9872
9801
9730
9659
.969588
9517 " Corr.
9445 10 12
9374 15 18
9302 20 24
.969231 30
9159 40
968872
8800
8728
8656
8583
.968511
8438
8366
8293
8220
9088 45 54
9016 50 60
8944
.968148
8075
8002
7929
7856
.967782
7709
7636
7562
7489
CORR.
FOR
SEC.
.967415
+
967046 30
7342 " Corr.
7268 10
7194 15
7120 20
966675
6600
6526
6451
6376
36
48
.966301
6226
6151
6076
600 I
965926
6972 40 49
6898 45 56
6823 50 62
6749
DREW
VERSINE
.029704.030614
9775
9845
9916
9986
EXSEC
.030057.030989|
0128
0199
0270
0341
0841
0912
0688
0763
0838
0913
0984
1056
.030412.031366
0483
0555
0626
0698
1064
1139
1215
1290
.030769.031746 30 38
1822 40 51
1898 45 57
1975 50 64
2052
1442
"Corr.
1518 10 13
1594 15 19
1670 20 25
.031128.032128
1200
1272
1344
1417
3028
3102
3177
3251
2205
2282
2359
2436
.031489.032513
1562
1634
1707
1780
2590
2668
2745
2823
.031852.032900
1925
1998
2071
2144
2978
3056
3134
3212
.032218.033290
2291
2364
2438
2511
3368
3447
3525
3604
.032585.033682
2658
2732
2806
2880
Corr.
3761
3840 10 13
3919 15 20
3998 20 26
.032954.034077 30 40
4156 40 53
53
4236 45 59
4315 50 66
4395
.033325.034474
3400
3474
3549
3624
4554
4634
4714
4794
CORR.
FOR
SEC.
+
033699.034874
3774
3849
3924
3999
034074.035276
=
4954
5035
5115
5196
CORR.
FOR
SEC.
+
" Corr.
10 52
15 77
20 103
30 155
40 206
45 232
50 258
" Corr.
10 52
15 78
20 104
30 156
40 207
45 233
50 259
1
TANGENT
.249328 0
9637 T
9946 2
.250255 3
0564 4
.250873
5
1183 6
1492 7
1801 8
21119
.252420|10
2729 1
3039 12
3348 13
3658 14
.253968 15
4277 16
4587 17
4897 18
5207 19
.255516 20
5826 21
6136 22
6446 23
6756 24
.257066 25
7377 26
7687 27
7997 28
8307 29
.258618 30
8928 31
9238 32
9549 33
9859 34
.260170 35
0480 36
079 i 37
1102 38
1413 39
.26172340
2034 41
2345 42
2656 43
2967 44
263278 45
3589 46
390047
4211 48
4523 49
.264834 50
5145 51
5457 52
5768 53
6079 54
266391 55
6702 56
7014157
7326 58
7637 59
.26794960
476
·
0.258819
9100
9381
O-234 SOTHO 0-237 56
6
5.260224
8
9
12
13
:0.261628
14
16
17
18
19
21
22
23
24
22
***** 8.♡♡~ 68788 GENRE BU90 85087 KAKAO 8
15.263031-130 140
3312 40 187
3592 45 210
3873 50 234
4154
20.264434
4715
4995
5276
5556
25.265837
6117
27 6397
26
6678
28
29
6958
31
32
33
34
30.267238
7519
7799
8079
8359
36
37
38
39
35.268640
8920
9200
9480
9760
41
SINE
42
43
444
9662
9943
40.270040
46
0504
0785
1066
1347
47
51
1908 " Corr.
2189 10 47
2470 15 70
2751 20 94
45.271440
53
54
56
58
59
50.272840
3120
52 3400
48 2280 150 233
49
2560
0320 " Corr.
060010 47
0880 15 70
1160 20 93
30 140
1720 40 187
2000 45 210
55.274239
4519
57 4798
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
3679
3959
5078
5358
60 .275637
COSINE
.965926
5850
5775
5700
5624
.965548
5473
5397
5321
5245
.964404
4327
4250
4173
4095
.965169
13
5093 "Corr.
501610
494015
4864 20
.964787 30
19
25
38
4711 40 51
4634 45 57
4557 50 64
4481
.964018
3941
3863
3786
3708
.963630
3553
3475
3397
3319
.963241
3163
3084
3006
2928
CORR.
FOR
SEC.
962849
2770
" Corr.
269210 13
2613 15 20
2534 20 26
+
.962455 30
40
2376 40 53
2297 45 59
2218 50 66
2139
.962059
1980
1900
1821
1741
.961662
1582
1502
1422
1342
.961262
15°

VERSINE
EXSEC
.034074.035276
4150
4225
4300
4376
5357
5438
5519
5600
.034452.035681
4527
4603
4679
4755
5762
5844
5925
6006
.034831.036088
4907
4984
5060
5136
5827
5905
6170 Corr.
6252 10 14
6334 15 21
6416 20 27
.035213.036498 30 41
658040 55
6662 45 62
5443 6745 50 68
5519 6828
5289
5366
.035596.036910
5673 6993
5750
7076
7159
7242
.035982.037325
6059 7408
6137
7492
7575
6214
6292
7658
.036370.037742
6447
6525
6603
6681
7826
7910
7994
8078
.036759.038162
6837
6916
6994
7072
8246
8331
8415
8500
.037151.038584
7230
7308
7387
7466
.037545.039009 30 43
9095 40 57
9180 45 64
9266 50 71
9351
7624
7703
7782
786 I
.037941.039437
8020
8100
8179
8259 9781
8669 "Corr.
875410 14
8839 15 21
8924 20 28
9523
9608
9694
CORR.
FOR
SEC.
+
11
.038338.039867
8418 9953
8498 1.040040
8578
8658
.038738.040299
0126
0213
CORR.
FOR
SEC.
+
" Corr.
10 52
15 78
20 104
30 156
40 208
45 234
50 260
TANGENT
"Corr.
10 52
15 79
20 105
1
.267949
8261
8573
8885
9197 4
.269509
9821
0-23♬
6678σ
5
.270133 7
0445
0757
.271069 10
1382 11
169412
2006 13
231914
9
.27263 15
2944 16
3256 17
3569 18
3882 19
.274194 20
4507 21
4820 22
5133 23
5446 24
.275759 25
6072 26
6385 27
6698 28
701 29
.277324 30
7638 31
7951 32
8265 33
8578 34
.278892 35
9205 36
951937
9832 38
.280146 39
.280460 40
0774 41
1087 42
1401 43
1715 44
30 157.282029 45
40 209
45 235
50 262
2343 46
2657 47
2972 48
3286 49
.283600 50
391451
4229 52
454353
4858 54
.285172 55
5487 56
58057
611658
6431 59
.28674560
477
0.275637
5917
6197
6476
6756
0-231 DO700 O-NMI DONOR OU~~2 2070° 8~~67 4688 FINE B
~
ოოო
5.277035
7315
7594
7874
8153
6
8
9
10.278432
12
13
14
17
18
19
21
22
20.281225
23
24
26
28
29
15.279829 30 140.960050 30 41
16.280108 40 186.959968 40 54
9887 45 61
9805 50 68
9724
25.282620
31
32
33
34
30.284015
36
37
38
39
41
SINE
35.285410
42
43
44
46
47
48
g gg ༄གཤྩགཤྩ 65
8712 " Corr.
8991 10 47
9270 15 70
9550 20 93
40.286803
038845 209
0667 50 233
0946
1504
1783
2062
2342
51
2900
3178
3458
3736
56
58
59
4294
4573
4852
5131
45.288196
5688
5967
6246
6525
49 9310
50.289589
9867
52.290146
53 0424
0702
54
7082 " Corr.
7360 10 46
7639 15 70
7918 20 93
VERSINES, EXSECANTS, AND TANGENTS
16°

CORR.
FOR
SEC.
+
30 139
8475 40 186
8753 45 209
9032 50 232
55.290980
1259
1537
1815
2094
60.292372
COSINE
961262
1182
1101
1021
0940
960860
0779
0698
0618
0537
960456
0375 " Corr.
0294 10 14
0212 15 20
0131 20 27
.959642
9560
9478
9396
9314
.959232
9150
9067
8985
8902
958820
8737
8654
8572
8489
958406
8323
8239
8156
8073
957990
CORR.
FOR
SEC.
7906 " Corr.
7822 10 14
7739 15 21
7655 20 28
.957151
7067
6982
+
.957571 30 42
7488 40 56
7404 45 63
7320 50 70
7235
6898
6814
.956729
6644
6560
6475
6390
.956305
VERSINE
EXSEC
038738.040299
8818
8899
8979
9060
0386
0473
0560
0647
.039140.040735
9221
9302
9382
9463
.040032
0113
0195
0276
0822
0909
0997
1084
.039544.041172
9625
9706
9788
9869
.039950.04161330
.040358.042055
0440 2144
0522
2233
0604
2322
0686
2412
1260 " Corr.
1348 10 15
143615 22
1524120 29
44
1701 40 59
1789 45 66 45 237
1878 50 7450 263
1967
30 158
40 210
.040768.042501
0850
0933
1015
1098
2590
2680
2769
2859
.041180.042949
1263
1346
1428
1511
3039
3129
3219
3309
.041594.043400
1677
1761
1844
1927
3490
3580
3671
3762
.042010.043853
2094
2178
2261
3944
" Corr.
403510 15
412615 23
2345 4217 20 30
.042429.044309 30 46
2512 4400 40 61
4492 45 69
4583 50
2596
2680
2765 4675
.042849.044767
2933
3018
3102
3186
CORR.
FOR
SEC.
+
4859
4951
5043
5136
.043271.045228
3356
5321
3440
5413
3525
5506
3610
5599
.043695 .045692
CORR.
FOR
SEC.
10 53
15 79
20 105
" Corr.
10 53
15 79
20 106
TANGENT
.286745
7060
7375
7690
8005
30 159
140 212
45 238
7650 264
.288320
8950
9266 8
9581 9
289896 10
"Corr..29021111
0
1234 BO7Bσ
·
5
8635 6
0527 12
0842 13
115814
|.29147315
1789 16
2105 17
2420 18
2736 19
293052 20
3368121
3684 22
4000 23
4316 24
.294632 25
4948 26
5264 27
5581 28
5897 29
.296214 30
653031
6846 32
7163 33
7480 34
.297796 35
8113 36
8430 37
8746 38
9063 39
.299380 40
9697 41
.30001442
0332 43
064944
30096645
128346
1600 47
1918 48
223549
.302553 50
287051
3188 52
3506 53
3823 54
.30414155
4459 56
4777 57
509558
5413 59
.30573160
478
0.292372
2650
2928
3206
3484
01234
56789 0-234
5.293762
12
13
14
10.295152
16
17
18
19
~~~~~
232
5430
" Corr.
570810 46
5986 15 69
6264 20 93
15.296542 30 139
6819 40 185
7097 45 208
7375 50 231
7653
21
20.297930
22
23
24
222
26
25.299318
27
29
ggggg 55555 £55±5 wwwww wwwww
9596
9873
28.300151
0428
32
33
30.300706
0983
1261
1538
1815
36
37
SINE
38
39
35.302093
2370
2647
2924
3202
42
4040
4318
43
44
4596
4874
40.303479
46
47
48
49
8208
8486
8763
9041
52
53
54
45.304864 30 138
514140 185
5418 45 208
5695 50 231
5972
56
57
58
59
50.306249
6526
6803
7080
7357
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
3756
" Corr.
4033 10 46
4310 15 69
4587 20 92
55.307633
7910
8187
8464
8740
60.309017
COSINE
956305
6220
6134
6049
5964
.955878
5793
5707
5622
5536
954588
4501
4414
.955450
5364 " Corr.
5278 10 14
519215 22
5106 20 29
955020 30 43
4934 40 58
4847 45 65
4761 50 72
4674
4327
4240
.954153
4066
3979
3892
3804
.953717
3629
3542
3454
3366
.953279
3191
3103
3015
2926
.952838
2750
CORR.
FOR
SEC.
"Corr.
2662 10 15
2573 15 22
2484 20 30
|.952396 30
230740
2218 45
212950
2040
.951951
1862
1773
1684
1594
+
.951505
1415
1326
1236
1146
951056
44
59
67
74
17°

VERSINE
043695.045692
3780
3866
3951
4036
EXSEC
.044122.046158
4207
4293
4378
4464
5066
5153
5239
5326
5785
5878
5971
6065
.044550.046627
4636
4722
4808
4894
5673
5760
6252
6345
6439
6533
" Corr.
10 53
6721
" Corr.
681510 16
6910 15 24 15 80
7004 20 32 20 106
.044980.047099 30 47 30 159
7193 40 63 40 213
7288 45 71 45 239
7383 50 79 50 266
7478
.045412.047573
5499
5586
7668
7763
7859
7954
.045847.048050
5934
6021
6108
6196
7427
7516
8145
8241
8337
8433
.046283.048529
6371
6458
6546
6634
8625
8722
8818
8915
046721 .049011
6809
9108
6897
9205
6985
9302
7074 9399
.047162.049496
7250
7338
9593 " Corr.
9691 10 16
978815 24
9886 20 33
.047604 .049984 30
7693.050082 40
7782 0179 45
0277 50
7871
7960
0376
.048049.050474
8138
8227
8316
8406
CORR.
FOR
SEC.
+
0572
0671
0769
0868
.048495 .050967
8585
1066
8674
1165
8764
1264
8854
1363
.048944 .051462
22QE UNO
49
65
73
82
CORR.
FOR
SEC.
+
11
Corr.
10 53
15 80
20 107
30 160
40 214
45 241
||50 267
TANGENT
.305731
6049
6367
1
.307322
7640
0123➡
2
3
6685
7003 4
5678σ
7959
8277 8
8596 9
.30891410
9233 II
9552 12
9870 13
.310189 14
.310508 15
0827 16
146 17
1465 18
1784 19
.312104 20
2423 21
2742 22
3062 23
3381 24
.313700 25
4020 26
4340 27
4659 28
4979 29
.315299 30
5619 31
5938 32
6258 33
6578 34
.316899|35
721936
7539 37
7859 38
817939
31850040
882041
9141 42
9461 43
9782 44
.320102 45
0423 46
0744 47
1065 48
1386 49
.321707 50
2028 51
2349 52
2670 53
2991 54
.323312 55
363456
3955 57
4277 58
4598 59
324920 60
479
0.309017
9294
9570
9847
4.310123
5.310400
0676
0953
1231
6
789
||
12
13
14
10.311782
NMI DONDO 00002 2007°2 2~~~~ H☹♪88 ===
17
18
19
21
22
261|15 69
2888 20 92
15.313164 30 138
3440 40 184
3716 45 207
3992 50 230
4269
20.314545
4821
5097
23
24
26
29
SINE
25.315925
6201
6477
6753
7029
31
33
1.229
1506
30.317305
7580
7856
37
38
42
43
44
5373
5649
35.318684
8959
9235
9511
9786
46
47
48
49
40.320062
51
52
53
54
8132
8408
2059 " Corr.
2335 10 46.949972 10 15
23
30
VERSINES, EXSECANTS, AND TANGENTS
18°

CORR.
FOR
SEC.
+
45.321440 30 138
1715 40 184
1990 45 207
2266 50 229
2541
50.322816
3092
3367
3642
3917
0337
" Corr.
061310 46
088815 69
1164 20 92
55.324193
56
4468
57
4743
58
5018
59 5293
60.325568
COSINE
951056
0967
0877
0786
0696
950606
0516
0425
0335
0244
.950154
0063 " Corr.
988115
9790 20
.949243
9151
9060
8968
8876
949699 30 46
9608 40 61
9517 45 68
9426 50 76
9334
948784
8692
8600
8508
8416
.948324
8231
8139
8046
7954
CORR.
FOR
SEC.
947861
7768
7676
7583
7490
.947397
+
7304
721010
711715
7024 20
.946462
6368
6274
6180
6085
.945991
5897
5802
5708
5613
945519
COMO COL
" Corr.
946930 30
6837 40
6743 45
6649 50 78
6556
FORP
47
62
VERSINE
.048944.051462
9033
9123
9214
9304
EXSEC
.049394 .051960
9484
2060
9575
2160
9665
2261
9756
2361
1562
1661
1761
1861
.049846.052461
9937
.050028
0119
0210
0392
0483
0574
0666
2562 " Corr.
2662 10
276315
2864 20
.050757.053471
0849
0940
1032
1124
.050301.052965 30 51
3066 40 67
3167 45 76
3269 50 84
3370
3573
3675
3776
3878
051216.053980
1308
1400
1492
1584
3163
3257
3351
3444
4083
4185
4287
4390
.051676.054492
1769
1861
1954
2046
4595
4698
4801
4904
.052139.055007
2232
2324
2417
2510
5110
5213
5317
5420
.052603.055524
2696
2790
2883
2976
.053070.056044 30
5628
573210
583615
5940 20
053538.056567
3632
3726
3820
3915
CORR.
FOR
SEC.
+
6672
6777
6882
6987
.054009.057092
4103
4198
7198
7303
7409
7515
2055600
52
6148 40 70
6253 45 78
6358 50 87
6462
4292
4387
.054481.057621
17
34
CORR.
FOR
SEC.
+
" Corr.
" Corr.
17 10 54
26 15 81
35
3520 108
" Corr.
10 54
15 81
20 107
30 161
40 215
145 242
50 269
30 162
40 216
45 243
50 270
TANGENT 1
324920
5241
5563
5885
6207
.326528
6850
7172
7494
7816
0
1
232 56789
328139 10
846111
8783 12
91C6 13
9428 14
.32975015
330073 16
0396 17
0718 18
1041 19
.331364 20
1687 21
2010 22
2333 23
2656 24
332979 25
330226
3625 27
3948 28
4272 29
.334595| 30
491931
5242 32
5566 33
589034
.33621335
653736
686 37
7185 38
750939
.337833 40
8157 41
8481 42
8806 43
913044
.339454 45
9779 46
.340103 47
0428 48
0752 49
.341077 50
1402 51
1727 52
2052 53
2376 54
.342702 55
3027 56
3352 57
3677 58
400259
.344328 60
480
0.325568
01234 5O7BQ 0-234 5
6
8
5.326943
9
12
13
14
16
10.328317
18
19
27222 207°° 87~~~ 68788 2-RE
23
24
26
15.329691|30 137
9965 40 183
17.330240 45 206
0514 50 229
0789
28
29
20.331063
1338
1612
1887
2161
31
32
25.332436
2710
2984
3258
3533
33
34
36
30.333807
4081
4355
4629
4903
37
38
39
41
35.335178
5452
5726
6000
6274
42
SINE
43
44
5843
6118
46
47
48
6393
6668
40.336548
90 85087 KONAR 8
49
7218
7493
7768
8042
51
52
54
8592" Corr.
8867 10
46
914115 69
9416 20 92
45.337917 30 137
8190 40 183
846445 205
8738 50 228
9012
56
57
58
50.339285
59
9559
9832
53.340106
0380
55.340653
" Corr.
46
6821
709510
736915 68
7643 20 91
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
0926
1200
1473
1747
60.342020
•
COSINE
945519
5424
5329
5234
5139
945044
4949
4854
4758
4663
944568
4472
" Corr.
437610 16
428115 24
4185 20 32
.944089 30
.943608
3512
3416
3319
3223
.943126
3029
2932
2836
2739
3993 40 64
3897 45 72
3801 50 80
3705
.942642
2544
2447
2350
2252
.942155
2058
1960
CORR.
FOR
SEC.
1862
1764
+
|.941176|30
.941666
1569
" Corr.
147010 16
1372 15 25
1274 20 33
.940684
0585
0486
0387
0288
.940189
0090
939991
9891
9792
.939693
49
1078 40 66
0979 45 74
088 50 82
0782
19°

VERSINE
.054481.057621
4576 7727
4671
7833
7939
4766
4861
8045
.054956.058152
5051
5146
5242
5337
EXSEC
.055432.058686
5528
5624
5719
5815
48.055911.059222 30
6007
6103
6199
6295
8258
8365
8472
8579
8793 " Corr.
8900 10 18
9007 15 27
9115 20 36
.056392.059762
6488
6584
6777
933040
9438 45
954550
9653
6681.060086
9870
9978
0195
.056874.060304
6971
7068
7164
7261
8922
9021
9119
9218
.057358.060849
7456
7553
7650
7748
9613
9712
0412
0521
0630
0740
.057845.061397
7942
8040
8138
8236
0958
1068
1177
1287
.058334.061947
8431
8530
8628
8726
1506
1616
1726
1837
CORR.
FOR
SEC.
+
.058824.062500 30 56
2612 40 74
2723 45 83
2834 50 93
2945
.059316.063057
9415 3168
9514
3280
3392
3504
2058
" Corr.
2168|10 19
2279 15 28
2390 20 37
.059811.063616
9910
.060009
0109
0208
060307.064178
3728
3840
3953
4065
CORR.
FOR
SEC.
+
" Corr.
10 54
15 82
20 109
TANGENT
" Corr.
1
54
30 163
72
40 218
.349216 15
9542 16
81 45 245 9868 17
90 50 272.350195 18
0522 19
.350848 20
10 55
15 82
20 109
.344328 0
4653 T
4978 2
5304 3
5630 4
.345955 5
6281 6
6607 7
6933 8
7259 9
.347585 10
79|||||
8237 12
8563 13
888914
117521
1502 22
1829 23
2156 24
.352483 25
2810 26
3137 27
3464 28
3791 29
.35411930
4446 31
477332
510133
5429 34
357396 40
7724 41
8052 42
838043
8708 44
.35903745
9365 46
9694 47
50 274.36002248
30 164
40 219
45 246
.355756 35
6084 36
6412 37
6740 38
7068 39
0351 49
360680 50
1008 51
1337 52
1666 53
1995 54
362324 55
2653 56
2982 57
3312 58
3641 59
363970 60
481
0.342020
2294
2567
2840
3113
0-23⇒
4
5.343386
3660
3933
4206
4479
5678 ∞
9
10.344752
"
12
234
13
14
16
17
18
19
22222
15.346117 30 136
639040 182
666345 205
6936 50 227
7208
20.347481
21
7754
8027
8299
8572
25.348845
9117
9390
9662
9935
23
24
2070 87087 ❤❤♪❤. DER NO ONE
26
28
29
31
30.350207
0480
0752
32
SINE
33
34
36
37
38
39
35.351569
1842
2114
2386
2658
41
42
5025 " Corr.
5298 10 45
557115 68
5844 20 91
43
4444
40.352931
46
47
48
49
1025
1297
51
VERSINES, EXSECANTS, AND TANGENTS
20°

CORR.
FOR
SEC.
+
45.354291 30 136
4563 40 181
4835 45 204
5107 50 227
5379
50.355651
5923
52
6194
53 6466
54
6738
3203
" Corr.
3475 10 45
3747 15 68
4019 20 91
55.357010
56
7281
57
7553
58 7825
59 8096
60.358368
COSINE
.939693
9593
9494
9394
9294
939194
9094
8994
8894
8794
.937687
7586
7485
7383
7282
.938694
8593 "Corr.
8493 10 17
839215 25
8292 20 34
.938191 30 50
8091 40 67
7990 45 76
7889 50 84
7788
.937181
7079
6977
6876
6774
.936672
6570
6468
6366
6264
.936162
6060
5957
5855
5752
.935650
CORR.
FOR
SEC.
5547
" Corr.
544410 17
534115 26
5238 20 34
.934619
4515
4412
+
.935135 30 52
5032 40 69
4929 45 77
4826 50 86
4722
4308
4204
.934101
3997
3893
3789
3685
.933580
VERSINE
.060307.064178
0407
0506
0606
0706
EXSEC
4290
4403
4516
4629
.060806.064742
0906
1006
I 106
1206
4856
4969
5083
5196
.061306.065310
1407 5424 Corr.
1507 5538 10 19
1608
1708
5652 15 29
5766 20 38
.061809.065881 | 30
1909 5995 40
6110 45
57
76
86
6224 50 95
2010
2111
2212
6339
.062313.066454
2414
2515
2617
2718
6569
6684
6799
6915
.062819.067030
2921
3023
3124
3226
4968
5071
5174
5278
7146
7262
7377
7493
.063328.067609
3430
3532
3634
3736
7726
7842
7958
8075
.063838.068191
3940
4043
4145
4248
8308
8425
8542
8659
064350.068776
4453
4556
4659
4762
.064865.069364 30 59
9482 40 79
9600 45 88
9718 50 98
9836
" Corr.
20
8894
901110
912915 29
9246 20 39
.065381.069955
5485.070073
CORR.
FOR
SEC.
+
5588 0192
5692
0311
5796
0430
065899.070548
6003
6107
6211
6315
.066420.071145
0668
0787
0906
1025
CORR.
FOR
SEC.
+
"Corr.
10 55
15 83
20 10
30 165
40 220
45 248
50 275
" Corr.
10 55
15 83
20 111
30 166
40 222
45 250
50 277
TANGENT
.363970
4300
1
.365618
5948
0
46291 2
4959 3
5288
=2Ò# 56789
6278
6608 8
6938 9
.367268 10
7598 11
7928 $2
825913
8589| ₪4
368920 15
9250 16
9581 17
991|| 18
.370242 19
.370573 20
0904 21
1235 22
1566) 23
1897 24
.372228 25
2559 26
2890 27
3222 28
3553 29
.373885 30
4216 31
454832
4880 33
521234
.37554335
5875 36
6207 37
6539 38
6872 39
.37720440
7536 41
7868 42
820143
8534 44
.37886645
9199 46
9532 47
9864 48
.380197 49
.380530 50
0863 51
I 196 52
1530 53
1863 54
.38219655
2530 56
2863 57
3197 58
353059
383864 60
482
0.358368
1234 6O7BO
5.359725
9997
7.360268
8
9
||
12
10.361082
23
13
14
22222 2
16
17
18
19
1353 " Corr.
1625 10 45
189615 68
2167 20 90
15.362438 30 136
2709 40 181
2980 45 203
3251 50 226
3522
21
20.363793
~MA
22
23
24
26
27
28
29
wwwww
25.365148
32
33
34
SINE
36
37
38
39
8640
8911
9182
9454
30.366501
6772
7042
7313
7584
0540
0811
46
47
48
49
35.367854
8125
8395
བྲཱ་ྒུ ཀྱྰགཀླg55f;
51
4064
4335
4606
4877
52
53
54
5418
5689
5960
6231
40.369206
41 9476 " Corr.
42 9747 10 45
43.370017|15 68
0287 20 90
44
56
45.370557 30 135
0828 40 180
1098 45 203
1368 50 225
1638
57
58
59
50.371908
2178
2448
2718
2988
8665
8936
55.373258
3528
3797
4067
4337
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
60.374607
COSINE
.933580
3476
3372
3267
3163
.933058
2954
2849
2744
2639
.931480
1374
1268
.932534
2429 " Corr.
2324 10 18
221915 26
2113 20 35
|.932008 30 53
1902 40 70
1797 45 79
1691 50 88
1586
1162
1056
930950
0843
0737
0631
0524
.930418
0311
0204
0097
.929990
.929884
9776
9669
9562
9455
.929348
9240
CORR.
FOR
SEC.
+
"Corr.
9133 10 18
9025 15 27
8917 20 36
|.928810 30 54
8702 40 72
928270
8161
8053
7945
7836
8594 45 81
8486 50 90
8378
.927728
7619
7510
7402
7293
.927184
.21°

VERSINE
.066420.071145
6524
6628
6733
6837
7256
7361
.066942.071744
7046
7151
7571
7676
7781
7887
EXSEC
1865 " Corr.
1985 10 20
2106 15 30
2226 20 40
.067466.072347 30 60
2468 40 81
2589 45 91
2710 50 101
2831
1265
1384
1504
1624
.067992.072952
8098 3074
8203 3195
3317
8309
8414 3439
.068520.073561
8626
8732
8838
8944
0438
0545
069050.074172
9157
9263
9369
9476
3683
3805
3927
4050
.070116.075404
0224
0331
1514
1622
1839
1947
2055
2164
4295 Corr.
"Corr.
441710 21 10 56
454015 31 15 84
4663 20 41 20 112
.069582.074786 30 62 30 168
9689 491040 82 40 224
9796 5033 45 92 45 252
9903 5156 50 103 ||50 280
.070010
5280
.070652.076024
0760 6148
0867
6273
6397
6522
5527
5651
5775
5900
0975
1083
.071190.076647
1298
1406
CORR.
FOR
SEC.
+
#
6772 " Corr.
6897 10 21
7022 15 31
7148 20 42
.071730.077273 30 63
7399 40 84
7525 45 94
7650 50 105
7776
.072272.077902
2381
2490
2598
2707
.072816.078535
8029
8155
8282
8408
CORR.
FOR
SEC.
+
" Corr.
10 56
15 84
20 112
30 167
40 223
45 251
50 279
TANGENT 1
.383864
4198
4532
" Corr.
10 56
15 84
20 113
01234
4866
5200 4
.385534
5
5868
6202
6536
6871 9
.387205 10
75401
7874 12
8209 13
8544 14
5678✪
.388879 15
9214 16
9549 17
9884 18
.390219 19
.390554 20
0889 21
1225 22
1560 23
1896 24
392231 25
2567 26
2903 27
3239 28
3574 29
393910 30
424631
4583 32
4919 33
5255 34
.395592 35
5928 36
6264 37
6601 38
6938 39
.397275 40
761141
7948 42
8285 43
862244
398960 45
9297 46
9634 47
9972 48
400309 49
30 169.400646 | 50
40 225 0984 51
45 253 1322 52
||50 281 1660 53
1997 | 54
.402335 55
267356
301257
3350 58
3688 59
404026 60
483
0.374607
01234
5678σ
||
12
13
14
~~~~~ 2222
5.375955
6224
6494
6763
7033
22222
10.377302
7571
" Corr.
784110 45
8110 15 67
8379 20 90
15.378649 30 135
16 891840 180
9187 45 202
9456 50 224
9725
17
18
19
20.379994
21.380263
23
24
26
27
28
25.381339
29
31
32
33
34
SINE
36
37
38
39
নননন
4876
5146
5416
5685
30.382683
2952
3221
3490
3758
41
42
43
44
35.384027
4295
4564
46
47
48
49
0532
080 I
1070
40.385369
51
52
53
54
1608
1877
2146
2415
56
57
59
4832
5101
45.38671130 134
6979 40 179
7247 45 201
VERSINES, EXSECANTS, AND TANgents
22°

50.388052
8320
8588
8856
9124
CORR.
FOR
SEC.
+
5638 " Corr.
5906 10 45
617415 67
6443 20 89
55.389392
9660
9928
58.390196
0463
60.390731
7516 50 224
7784
COSINE
.927184
7075
6966
6857
6747
.926638
6529
6419
6310
6200
.924989
4878
4768
4657
4546
926090
5980 " Corr.
587 10 18
576 15 28
565 20 37
.92554030 55
5430 40 73
532045 83
5210 50 92
5099
.924435
4324
4213
4102
3991
923880
3768
3657
3545
3434
923322
3210
3098
2986
2874
.922762
CORR.
FOR
SEC.
"Corr.
2650
2538 10 19
2426 15 28
2313 20 37
+
.922201 30 56
2088 40 75
.921638
1525
1412
1976 45 84
1863 50 94
1750
1299
1185
.921072
0959
0846
0732
0618
.920505
VERSINE
.072816.078535
2925
3034
3143
3253
EXSEC
073362.079170
3471
3581
3690
3800
4239
4349
8662
8788
8916
9043
9298 " Corr.
9425 10 21
9553 15 32
9680 20
43
.073910.079808 30 64
9936 40 85
4129.08006545 96
0193 50 107
4020
0321
.074460.080450
4570
4680
4790
4901
.07501.081094
5122
5232
5343
5454
6232
6343
6455
6566
0578
0707
0836
0965
.075565.081742
5676
5787
5898
6009
1223
1353
1482
1612
.076120.082392 30
8475
8588
8701
8815
1872 " Corr.
2002 10
22
2132 15 33
2262 20
.076678.083046
6790
6902
7014
7126
3177
3308
3440
3571
.077238.083702
7350
7462
7574
7687
6530 170
87 40 227
9845 256
2784 50 109 | 50 284
252340
2653 45
2915
3834
3966
4098
4230
.077799.084362
7912
8024
8137
8250
CORR.
FOR
SEC.
.078928.085691
9041
9154
9268
9382
.079495.086360
-
.078362.085025 30 66
5158 40 89
5291 45 100
5424 50 ||
5558
5825
5958
6092
6226
CORR.
FOR
SEC.
+
" Corr.
10
57
15 85
20 113
4495
Corr.
Corr.
4627 10 22 10 57
15 86
4760 15 33
4892 20 44
20 114
30 170
40 226
45 254
50 283
" Corr.
10 57
15 85
43 20 114
Ja
30 171
40 228
45 257
50 285
TANGENT 1
404026
4365
4703
5042 3
5380
.405719
6058
0-234
5
6
6397 7
6736 8
7075 9
407414 10
7753 11
8092 12
8432 13
8771 14
.4091115
9450 16
9790 17
.410130 18
0470 19
.410810 20
115021
1490 22
1830 23
2170 24
.41251 25
2851 26
3192 27
3532 28
3873 29
.414214 30
4554 31
4895 32
5236 33
5577 34
.415919 35
6260 36
6601 37
6943 38
7284 39
.417626 40
7967 41
8309 42
865 43
8993 44
.419335 45
967746
.420019 47
0361 48
0704 49
.421046 50
1388 51
1731 52
207453
2416 54
422759 55
3102 56
3445 57
3788 58
4132 59
.42447560
484
0.390731
0999
1267
I
234 56789
5.392070
11
12
13
234
14
10.393407
6678σ)
16
17
18
19
22222 222
8 666 gа 55555 £55±5 @wwww wwww XNXG EN~~~
15.394744 30 134
501 40 178
5278 45 200
5546 50 223
5813
21
22
20.396080
23
24
26
27
28
29
32
25.397415
33
34
36
37
38
39
40
41
30.398749
9016
9282
9549
9816
42
43
44
SINE
35.400082
0349
0616
0882
1149
46
1534
1802
47
48
49
2337
2605
2872
3140
51
52
53
54
3674 " Corr.
3942 10 45
4209 15 67
4477 20 89
56
6347
6614
6881
7148
57
7682
7949
8216
8482
50.404078
4344
4610
4876
5142
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
55.405408
5673
5939
6205
6471
60.406737
.401415
1681
" Corr.
1948 10
10 44
2214 15 67
2480 20 89
COSINE
.920505
0391
0277
0164
0050
.919936
9822
9707
9593
9479
·
.919364
9250
" Corr.
9135 10 19
9021 15 29
8906 20 38
.918791 30
867640 77
8561 45 86
8446 50 96
8331
918216
8101
7986
7870
7755
45.402747 30 133.915312 30 59
3013 40 178
3279 45 200
3545 50 222
3811
5194 40 78
5077 45 88
4960 50 98
4842
917639
7523
7408
7292
7176
.917060
6944
6828
6712
6596
.916479
6363
6246
6130
6013
CORR.
FOR
SEC.
+
.914725
4607
4490
4372
4254
.915896
5780
"Corr.
5663 10
20
5546 15 29
5429 20 39
914136
4018
3900
3782
3664
913546
23°

VERSINE
.079495.086360
9609 6495
9723
6629
6763
6898
9836
9950
.080064.087033
0178
0293
0407
0521
7168 " Corr.
7302 10 23
7438 15 34
7573 20 45
.080636.087708 30 68
7844 40 90
7979 45 102
EXSEC
0750
0865
0979
1094 8251
57.081209||.088387
1324
1439
1554
1669
.081784.089068
1899 9205
2014
9342
9479
2130
2245
9616
3056
3172
3288
3404
.082361.089753
2477
3870
3987
8523
8659
8795
8932
9890 "Corr.
2592.090028 10 23
2708 0166 15 34
28241 0303 20 46
.082940.090441 30 69
0579 40 92
0717 45 104
0855 50 115
0994
083521.091132
3637
3754
45 258
8115 50 113 || 50 287
5393
5510
5628
5746
1271
1410
.084104.091827
4220
4337
4454
4571
1548
1688
CORR.
FOR
SEC.
+
1966
2105
2245
2384
" Corr.
.084688.092524
4806 2664
4923 2804 10 23
5040 2944 15 35
5158 3085 20 47
.085275.093225 30 70
3366 40 94
3506 45 105
3647 50 17
3788
.085864.093929
5982
6100
6218
6336
086454.094636
4070
4212
4353
4495
CORR.
FOR
SEC.
+
" Corr.
10 57
15 86
20 115
30 172
40 230
" Corr.
10 58
15 86
20 115
30 173
40 231
45 259
50 288
"Corr.
10 58
15 87
20 116
|| 30 174
40 232
45 261
50 290
TANGENT
1
424475 0
4818
5162 2
5505 3
5849 4
·
.426192 5
6536 6
6880 7
7224 8
7568 9
•
427912 10
8256 ||
429634 15
9978 16
.430323 17
8600 12
8945 13
9289 14
.431358 20
1703 21
0668 18
1013 19
·
·
2048 22
2393 23
2739 24
.434812 30
5158 31
5504 32
5850 33
6197 34
433084 25
3430 26
3775 27
412128
4466 29
.436543 35
6889 36
7236 37
.43827640
862241
8969 42
931643
9663 44
7582 38
7929 39
440010 45
0358 46
070547
1053 48
1400 49
441748 50
2095 51
2443 52
2791 53
3139 54
4434871 65
3835 56
4183 57
4532 58
4880 59
445229 60
485
0.406737
7002
7268
7534
7799
0123#
4
56789 0-23=
5.408065
8330
8596
8862
9127
10.409392
11
12
9658 Corr.
992310 44
13.410188 15 66
0454 20 88
15.410719 30 133
0984 40 177
1249 45 199
1514 50 221
1780
14
16
17
18
19
20.412044
2310
2574
2840
3104
22222 22222 67887 1888 FIRE BONO 85
21
23
24
25.413369
3634
3899
4164
4428
26
27
28
29
30.414693
4958
5223
31
32
33 5487
34
5752
35.416016
6281
6545
6810
7074
36
SINE
37
38
39
41
42
43
44
40.417338
7603
" Corr.
7867|10 44
813115 66
8396 20 88
46
45.418660
47
48
49
85887
50.419980
51.420244
52 0508
53 0772
54
1036
56
57
55.421300
1563
1827
2091
2355
60.422618
58
59
VERSINES, EXSECANTS, AND TANGENTS
24°

11
CORR.
FOR
SEC.
+
30 132
8924 40 176
9188 45 198
9452 50 220
9716
COSINE
.913546
3427
3309
3190
3072
.912953
2834
2715
2596
2478
.911164
1044
0924
0804
0684
.912358
2239 Corr.
212010 20
200115 30
1882 20 40
.911762 30 60
1642 40 80
1523 45 90
1403 50 100
1284
1.910564
0443
0323
0202
0082
.909961
9841
9720
9599
9478
.909357
9236
9115
8994
8872
CORR.
FOR
SEC.
IS
+
.908751
8630 " Corr.
850810 20
8387 15 30
8265 20 41
.907533
7411
7289
7166
7044
.908143 30 61
802140 81
1.906922
6799
6676
6554
6431
.906308
7900 45 91
7778 50 101
7655
VERSINE
.086454.094636
6573 4778
6691
4920
6810
5062
6928
5204
7761
7880
.087047.095347
7166
"Corr.
7285
7404
5489
5632 10 24
5775 15 36
7522 5917 20 48
.087642.096060 30 72
6204 40 95
6347 45 107
6490 50 119
6634
7999
8118
EXSEC
.088238.096777
8358
8477
8597
8716
6921
7065
.088836.097498
8956
9076
9196
9316
0159
0280
0401
0522
7209
7353
.089436.098221
9557
9677
9798
9918
7642
7787
7931
8076
2589
2711
2834
2956
.090039.09894830
8366 "Corr.
8510
8657 15
8802 20
.090643.099678
0764 9824
0885 9971
1006. 100118
1128 0264
.091249.100411
1370
1492
1613
1735
CORR.
FOR
SEC.
+
0558
0706
0853
1000
30 176
140 234
909440
9240 45 109 45 263
9386 50 121 50 293
9532
.091857.101148
1979
2100
2222
2345
.092467.101888 30 74
2036 40 99
2185 45 11
2334 50 123
2482
.093078.102631
3201
3324
ន58ស
1296
" Corr.
1444110 25
159215 37
1740 20 49
2780
2930
3446 3079
3569 3228
.093692. 103378
CORR.
FOR
SEC.
+
"Corr.
10 58
15 87
20 116
" Corr.
10 59
15 88
120 117
TANGENT
.445229
5577
5926
6275
"Corr.
10 59
15 88
20 118
30 177
40 235
45 265
50 294
↑
0123➡
30 175.44871910
40 233
45 262
50 291
6624 4
5
.446973
7322 6
7671 7
8020 8
8369 9
906811
9418 12
9768 13
.45011714
.450467 15
0817 16
1167| 17
1517 18
1868 19
.452218 20
2568|21
291922
3269 23
3620 24
.453971 25
4322 26
4673 27
5024 28
5375 29
.455726 30
6078 31
6429 32
678133
713234
.457484 35
7836 36
8188 37
8540 38
8892 39
.459244 40
9596 41
9949 42
.46030143
065444
.46100645
1359 46
1712 47
2065 48
241849
.462771 50
3124 51
3478 52
3831 53
418454
.464538 55
4892 56
5246 57
5600 58
5954 59
.466308 60
486
1
0.422618
2882
3146
3409
3672
1234
5.423936
66789 0-~M➡ BO789
11
10.425253
12
13
14
16
17
18
19
15.426569 30 132
6832 40 175
7095 45 197
7358 50 219
7621
22222 2020≈ 87087 ❤ DE NON CONDO
23
20.427884
26
27
32
33
34
SINE
37
25.429198
9461
9723
9986
29.430248
30.430511
42
43
44
4199
4463
4726
4990
46
47
48
49
35.431823
2086
2348
2610
2873
51
52
5516 " Corr.
577910 44
6042 15 66
6306 20 88
40.433135
53
8147
8410
8672
8935
56
57
0774
1036
1299
1561
45.434445 30 131
4707 40 175
50.435755
6017
6278
6540
6802
3397
365910
" Corr.
44
3921 15 65
4183 20 87
55.437063
7325
7587
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
4969 45 196
5231 50 218
5493
58 7848
59
8110
60.438371
COSINE
.906308
6185
6062
5939
5815
.905692
5569
5445
5322
5198
.903834
3709
3585
3460
3335
905075
4951
" Corr.
4827 10 21
470315 31
4579 20 41
.904455 30 62
433140 83
4207 45 93
4082 50 103
3958
.903210
3086
2961
2836
2710
.902585
2460
2335
2209
2084
.901958
1832
1707
1581
1455
.901329
1203
1077 10
095115
0825 20
CORR.
FOR
SEC.
.900698 30
.900065
.899939
9812
9685
9558
+
.899431
9304
9176
9049
8922
.898794
" Corr.
21
32
42
0572 40 84
0445 45 95
0319 50 105
0192
25°

VERSINE
EXSEC
.093692. 103378
3815 3528
3938
3678
4061
3828
4185
3978
.094308.104128
4431
4555
4678
5049
5173
4278" Corr.
4429 10 25
4580 15 38
4802 4730 20 50
.094925.104881 30 76
5032 40 101
5184 45 113
5297 5335 50 126
5421 5486
.095545. 105638
5669 5790
5793 5942
5918 6094
6042 6246
.096166.106398
6291
6415
6540
6665
6551
6703
6856
7009
.096790.107162
6914
7039
7164
7290
21
7315 Corr.
7468 10 26
7621 15 38
7775 20 51
|.097415.107928|30
7540
7665
7791
7916
.098042.108699
8168
8293
8419
8545
8082 40 103
8236 45 115
839050 128
8544
8853
9008
9163
9318
.098671.109473
8797
8923
9049
CORR.
FOR
SEC.
+
9628
9783
9938
9175.110094
63.099302. 110250
9428 0406
" Corr.
9555 0562 10 26
0718 15 39
0874 20 52
.099935. 111030 30 78
.100061 1187 40 104
0188 1344 45 117
0315 1500 50 130
0442 1657
9681
9808
. 100569. 111814
0696
0824
0951
1078
1972
2129
2286
2444
.101206 112602
CORR.
FOR
SEC.
+
" Corr.
10 59
15 89
20 118
77 || 30 179
40 238
45 268
50 298
" Corr.
10 60
15 89
20 119
" Corr.
10
60
15 90
20 120
TANGENT
.466308
30 180
40 239
45 269
50 299
OI23
9499
0
6662
7016 2
7370 3
7725 4
468080
8434 6
8789
9144
30 178.469854| 10
40 236 470209 II
45 266
50 296
0564 12
0920 13
1275 14
5
5678 a
9
.471631 15
1986 16
2342 17
2698 18
3054 19
.47341020
3766 21
412222
4478 23
4835 24
475191 25
5548 26
5905 27
6262 28
6618 29
.476976 30
733331
7690 32
8047 33
8405 34
.478762 35
9120 36
9477 37
9835 38
.48019339
480551 40
090941
126842
1626 43
1984 44
482343 45
270146
306047
3419 48
377849
.484137 50
4496 51
4855 52
5214 53
5574 54
.48593355
6293 56
6653 57
701358
7373 | 59
.487733 60
487
0.438371
8633
8894
9155
9417
0123♬
4
5.439678
9939
7.440200
0462
0723
6789σ
||
234 5678
10.440984
1245 "Corr.
1506 10 43
1767 15 65
2028 20 87
15.442289 30 130
2550 40 174
2810 45 196
3071 50 217
3332
12
13
14
16
17
18
19
20
22222
234
23
24
15555 £55±5 wwwww www.BBBB
26
27
28
29
25.444896
5156
5417
5677
5938
31
32
33
34
30.446198
6458
6718
6979
7239
36
44
35.447499
7759
8019
8279
8539
46
40.448799
9509 8588Z DONOR 8
47
SINE
48
49
•
51
443593
3853
4114
52
4375
4635
53
54
50.451397
56
57
58
59
9059
" Corr.
9319 10 43
9579 15 65
9839 20 87
1656
1916
2175
2435
55.452694
VERSINES, EXSECANTS, AND TANGENTS
26°

CORR.
FOR
SEC.
+
2954
3213
3472
3731
60.453990
45.450098 30 130.892979 30
0358 40 173
061845 195
0878 50 216
1137
COSINE
896794
8666
8539
8411
8283
.898156
8028
7900
7772
7643
.897515
7387 " Corr.
7258 10 21
713015 32
7001 20 43
896228
6099
5970
.896873 30 64
6744 40 86
6615 45 97
6486 50 107
6358
5841
5712
.895582
5453
5323
5194
5064
.894934
4804
4675
4545
4415
.894284
4154
4024
3894
3763
.893633
CORR.
FOR
SEC.
" Corr.
22
3502
337110
324115 33
3110 20 44
+
.892323
2192
2061
1929
1798
2848 40 87
2717 45 98
2586 50 109
2455
891666
1534
1402
1270
1138
.891006
VERSINE
101206
1334
1461
1589
1717
2613
2742
2870
2999
.101844.113393
1972
2100
2228
2357
•
.102485.114187 30 80
4347 40 106
4506 45 119
4666 50 133
4826
5196
5325
5455
5585
EXSEC
.103127.114985
3256
3385
3514
3642
103772.115787
3901
4030
4159
4288
| 12602
2760
2918
3076
3234
6106
6237
.104418.116592
4547
4677
4806
4936
3552" Corr.
371010 27
3869 15 40
4028 20 53
6753 " Corr.
691510 27
7077 15 41
7238 20 54
105066.17400 30 81
7562 40 108
7725 45 122
788750 135
8050
5145
5306
5466
5626
.105716.118212
5846
5976
7808
7939
8071
8202
5948
6108
6269
6431
8730
8862
. 108994
.106367.119028
6498
6629
6759
6890
65.107021 ||.119847
►
8375
8538
8701
8865
9192
9355
9519
9683
7152.120012 "Corr.
7283 0176 10 27
7414 0340 15 41
0505 20 55
7545
.107677.12067030
CORR.
FOR
SEC.
108334.121496
8466
8598
82
083540 110
1000 45 124
1165 50 137
1331
1662
1828
1994
2160
122326
CORR.
FOR
SEC.
+
" Corr.
10 60
15 90
20 120
30 181
40 241
45 271
50 301
" Corr.
10 61
15 91
20 121
30 182
40 242
45 272
50 303
" Corr.
TANGENT
10 61
15 91
20 122
487733
8093
8453
8813
9174
I
01234 56789
.489534
9895 6
.490256
7
0617
0978 9
491339 10
1700 1
206112
2422113
2784 14
.493145 15
3507 16
3869 17
4231 18
4593 19
.494955 20
531721
5679 22
6042 23
6404 24
.496767 25
7130 26
7492 27
7855 28
8218 29
.504042 45
440646
4771 47
513648
5502 49
30 183.505867 50
40 244 6232 51
45 274 6598 52
50 304 6963 53
7329 54
498582 30
8945 31
9308 32
9672 33
.500035 34
.50039935
0763 36
| 127 37
149138
1855 39
.50221940
2583 41
2948 42
331243
3677 44
.507695 55
8061 56
8427 57
8793 58
9159 59
.509525 60
488
1
0.453990
4250
4509
4768
5027
01234
5.455286
5545
5804
6063
6322
000 Noo
6
8
9
||
12
23
10.456580
6839 " Corr.
7098 10 43
7357 15 65
7615 20 86
15.457874 30 129
8132 40 172
8391 45 194
8650 50 215
18
19 8908
13
14
16
17
20.459166
9425
9683
9942
24.460200
25.460458
21
22
23
2 2073 2m~m~ ❤❤♪❤8 2029 29700 8508Z DONDO O
26
28
29
32
33
34
30.461749
2007
2265
2522
2780
36
35.463038
3296
3554
3812
4069
37
38
39
41
40.464327
42
43
44
46
SINE
47
48
49
45.46561430 129
5872 40 172
612945 193
6387 50 214
6644
51
52
0716
0974
1232
1491
50.466901
7158
7416
7673
7930
53
54
56
57
55.468187
8444
8701
8958
9215
58
59
4584 " Corr.
4842 10 43
5100 15 64
5357 20 86
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
60.469472
COSINE
.891006
0874
0742
0610
0478
.890345
0213
0080
.889948
9815
888350
8217
8083
7949
7815
.889682
9549
" Corr.
9416 10 22
928315 33
9150 20 44
|.889017 30 67
8884 40 89
8751 45 100
861750 111
8484
.887682
7548
7413
7279
7145
887011
6876
6742
6608
6473
.886338
6204
6069
5934
5799
CORR.
FOR
SEC.
.884310
4174
4038
3902
3766
+
885664
5529
" Corr.
5394 10 23
5258 15 34
5123 20 45
.28498830 68
4852 40 90
4717 45 102
458150 13
4445
.883630
3493
3357
3221
3084
.882948
27°

VERSINE
.108994
9126
9258
9390
9522
EXSEC
.122326
2493
2659
2826
2993
9920
.110052
.109655. 123160
9787 3327
" Corr.
3494 10 28
3662 15 42
0185 3829 20 56
.110318.123997 30 84
0451 4165 40 112
0584 4333 45 126
4501 50 140
0717
0850 4669
.110983.124838
1116
1249
1383
1516
5006
5175
5344
5513
.111650. 125682
1783
5851
1917
6021
2051
6190
2185 6360
.112318.126530
2452
2587
2721
2855
3124
3258
3392
3527 8066
.113662.128237
3796 8409
3931
8581
8752
4066
4201
8924
6700
"Corr.
" Corr.
6870 10 28 10 62
704115 43 15 92
721120 57 20 123
.112989.127382 30 85
30 185
7553 40 114 40 246
7724 45 128 ||45 277
7895 50 142 50 308
.114336.129096
4471
4606
4742
4877
5283
5419
5555
9269
9441
9614
9786
.115012. 129959
5148.130132 "Corr.
0306 10
0479 15 43
0652 20 58
.115690.130826 30 87
5826 1000 40 116
1174 45 130
5962
6098
6234
CORR.
FOR
SEC.
+
.116370.131696
6507
6643
6779
6916
.117052.132570
1871
2045
CORR.
FOR
SEC.
+
2220
2395
30 186
40 248
45 279
1348 50 145 150 310
1522
5
.511359
1726
2093 7
6
" Corr.
10 61
15 92
2460 8
20 122
2828 9
30 184.513195 10
40 245
3562 11
45 276
3930 12
50 306
4298 13
466614
" Corr.
10 62
15 93
20 124
TANGENT
.509525
9892
.510258 2
0625
0992 4
0123‡
.515034 15
5402 16
5770 17
6138 18
6507 19
.516876 20
7244 21
7613 22
7982 23
8351 24
.518720 25
9089 26
9458 27
9828 28
.520197 29
.520567 30
0937 31
1307 32
1677 33
2047 34
.522417 35
2787 36
3158 37
3528 38
3899 39
.524270 40
4641 41
501242
5383 43
5754 44
526126 45
6497 46
6868 47
724048
7612 49
.527984 50
8356 51
8728 52
9100 53
9473 54
.529845 55
.530218 56
0591 57
096358
1336 59
.531709 60
489
0.469472
9728
9985
3.470242
0499
I
234 56789
5.470755
1012
1268
1525
1782
10.472038
11
12
234
13
14
16
17
18
19
22222 22
20.474600
4856
5112
5368
5624
21
23
24
26
15.473320 30 128.880891 30
3576 40 171
383245 192
4088 50 213
4344
25.475880
6136
27 6392
6647
6903
28
29
SINE
30.477159
7414
31
32 7670
33
7926
34 8181
36
37
38
39
35.478436
8692
8947
9203
9458
41
43
44
46
47
48
49
55
11
40.479713
9968 Corr.
42.480224 10 42
0479 15
15 64
0734 20 85
45.480989 30 127
1244 40 170
1499 45 191
1754 50 212
2009
51
52
11
"
2294 Corr.
2551 10 43
2807 15
15 64
3063 20 85
50.482263
2518
2773
3028
3282
53
54
58
59
55.483537
56 3792
4046
57
VERSINES, EXSECANTS, AND TANGENTS
28°

CORR.
FOR
SEC.
+
4301
4555
60.484810
COSINE
.882948
2811
2674
2538
2401
.882264
2127
1990
1853
11
1716
.881578
1441 Corr.
1304 10 23
116615 34
1028 20 46
880201
0063
.879925
9787
9649
.879510
9372
9233
9095
8956
0753 40 92
0615 45 103
0477 50 115
0339
878817
8678
8539
8400
8261
CORR.
FOR
SEC.
878122
7983
7844
+
.876026
5886
5746
5605
5464
7704
[1
7565
.877425
7286 Corr.
7146 TO 23
7006 15 35
6867 20 47
.876727 30
.875324
5183
5042
4902
4761
.874620
70
93
6587|40
6447 45 105
6307 50 117
6166
VERSINE
117052.132570
7189 2745
7326
2921
3096
7462
7599
3272
.117736.133448
7873
8010
8147
8284
88
4506 40 118
4683 45 133
4860 50 147
5037
69.119109 ||.135215
.118422.13432930
8559
8696
8834
8972
•
EXSEC
9937
. 120075
9247
9385
9523
966 I
.119799.136104
0213
0351
" Corr.
" Corr.
10 62
To
3624
3800 10 29
3976 15 44 15 94
4153 20 59 20 125
5392
5570
5748
5926
. 120490.136996
0628
0767
0905
1044
.123273.
3413
3553
3693
3834
6282
6460
6639
6818
IT
7176 Corr.
7355 10 30
7534 15 45
7714 20 60
121183.137893 30 90
1322
1461
1600
1739
4114
4254
4395
4536
.121878.138794
2017
2156
2296
2435
122575.139698
2714
2854.140061
2994
3133
8974
9155
9336
9517
30 188
807340 120 40 251
8253 45 135 145 283
8433 50 150 50 314
8613
9879
0242
0424
CORR.
FOR
SEC.
+
140606
0788
" Corr.
0971 10 30
115315 46
1336 20 61
.123974.14151830
91
1701 40 122
1884 45 137
206750 152
2251
.124676.142434
4817
4958
5098
5239
.125380.143354
CORR.
FOR
SEC.
+
2618
2802
2986
3170
30 187
40 249
45 281
50 312
"Corr.
10 63
15 94
20 126
11
Corr.
10 63
15 95
20 126
TANGENT
.531709
1
0
2083 1
2456
2829
3203 4
.533576
3950
4324
4698
5072 9
.535446 10
582111
6195 12
6570 13
694514
232
5678σ
.537319 15
7694 16
8069 17
8444 18
8820 19
.53919520
9571 21
9946 22
.540322 23
0698 24
.541074 25
1450 26
1826 27
2203 28
2579 29
.542956 30
333231
3709 32
4086 33
446334
.544840 35
5218 36
5595 37
5973 38
6350 39
.546728 40
7106 41
7484 42
7862 43
824044
.548619 45
8997 46
9376 47
9755 48
.550134 49
30 190.550512 50
40 253 0892 51
45 284
1271 52
50 316 1650 53
2030 54
552409 55
2789 56
3169 57
3549 58
3929 59
.55430960
490
→
0.484810
01234 BONEK
6
8
5.486081
9
11
12
234
13
10.487352
14
16
17
18
19
27222 222~~ ~~~~~ ❤
7606 " Corr.
7860 10 42
8114 15 63
8367 20 85
15.48862130 127
8875 40 169
9129 45 190
9382 50 211
9636
20.489890
21.490143
23
24
25
27
28
29
34
36
38
39
SINE
41
42
43
44
5064
5318
30.492424
2677
2930
3183
3436
46
47
48
49
5573
5827
35.493689
3942
4195
4448
4700
3.NOJ DONO
6335
6590
51
6844
7098
52
53
54
0397
0650
0904
17
40.494953
5206 Corr.
5459 10 42
571115 63
5964 20 84
45.496216 30 126
6469 40 168
6722 45 189
6974 50 210
7226
56
491 157
1410
1664
1917
2170
50.497479
7731
7983
8236
8488
55.498740
8992
57 9244
58
59
9496
9748
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
60.500000
COSINE
.874620
4479
4338
4196
4055
.873914
3772
3631
3489
3348
.873206
3064 " Corr.
2922 10 24
278015 36
2638 20 47
.872496 30 71
2354 40 95
2212 45 107
2069 50 18
1927
.871784
1642
1499
1357
1214
.871071
0928
0785
0642
0499
.870356
0212
0069
.869926
9782
869639
9495
9351
9207
9064
CORR.
FOR
SEC.
.868920
8776
"Corr.
8632 10 24
8487 15 36
8343 20 48
|.868199 30 72
8054 40 96
7910 45 108
7766 50 120
7621
867476
7331
7187
7042
6897
.866752
6607
6461
6316
6171
866025
29°

VERSINE
.125380.143354
5521
5662
5804
5945
6936
7078
7220
7362
EXSEC
.126086. 144278
6228
6369
6511
4463
" Corr.
4648 10 31
483415 46
6652 5020 20 62
.126794.145206 30 93
5392 40 124
5578 45 140
5764 50 155
5950
3538
3723
3908
4093
.127504.146137
7646
7788
7931
8073
8501
8643
8786
.128216. 147073
8358
.129644
9788
9931
.130074
0218
6324
6511
6698
6885
7260
7448
7636
7824
.128929.148012
9072
9215
9358
9501
8200 Corr.
8389 10 32
8578 15 47
8766 20 63
148956 30 95
95
9145 40 126
9334 45 143
9524 50 158
9713
.130361.149903
0505. 150093
0649
0283
0473
0793
0936 0664
.131080. 150854
1224
1368
1513
1657
1045
1236
1427
1618
CORR.
FOR
SEC.
+
31
.131801.151810
1946
2090
2234
2379
.132524.152769 30 96
2669 2962 40 128
3154 145 144
3347 50 160
2813
2958
3103
3540
.133248.153733
3393
3539
3684
3829
.133975.154700
11
2002 Corr.
2193 TO
32
2385 15 48
2577 20 64
3926
4120
4313
4507
CORR.
FOR
SEC.
+
11
Corr.
10 64
15 95
20 127
30 191
40 254
45 286
50 318
Corr.
10 64
15 96
20 128
30 192
40 256
45 288
50 320
11
Corr.
10 64
15 97
20 129
30 193
40 258
45 290
50 322
TANGENT
554309
4689
5070
5450
5831
·
556212
•
01234
7736
6593 6
6974 7
7355
5878 σ
.558118 10
84991
8881 12
9263 13
964514
9
.560027 15
04091 16
0791 17
174 18
1556 19
561939 20
232221
2705 22
3088 23
3471 24
.563854 25
423826
4621 27
5005 28
5389 29
565773 30
615731
6541 32
6925 33
7310 34
.567694 35
807936
8464 37
8849 38
9234 39
.56961940
.570004 41
0390 42
077643
1161 44
571547 45
1933 46
2319 47
2705 48
3092 49
573478 50
386551
425252
4638 53
5026 54
.57541355
5800 56
6187 57
6575 58
6962 59
577350 60
491
0.500000
0-23➡
4
66789 0-234 56
5.501259
1511
1762
2014
2266
11
10.502517
12
13
14
16
17
18
19
~~~~~
234
15.503774 30 126
4025 40 167
4276 45 188
452850 209
4779
20.505030
21
22
5281
5532
5783
6034
25.506285
6536
6786
7037
7288
23
24
~~~~2 67~8 ❤❤❤ 2 99 85087 DONOR O
26
27
32
33
34
30.507538
7789
8040
8290
8541
37
38
SINE
35.508791
9041
9292
9542
9792
41
0252
0504
0756
1007
42
43
44
40.510043
46
47
48
49
2768
" Corr.
3020 10 42
327115 63
3523 20 84
53
54
56
50.512542
2792
3042
3292
3541
57
58
55.513791
4040
4290
4539
4789
VERSINES, EXSECANTS, AND TANGENTS
30°

CORR.
FOR
SEC.
+
0293
0543 10
" Corr.
42
0793 15 62
1043 20 83
11220205
60.515038
COSINE
866025
5880
5734
5589
5443
.865297
5151
5006
4860
4713
.863102
2955
2808
2661
2514
45.511293 30 125.859406 30 74
1543 40 167
1793 45 187
204350 208
2293
9258 40 99
9109 45 112
8960 50 124
8811
.864567
4421 Corr.
427510 24
412815 37
3982 20 49
.863836 30 73
3689 40 98
3542 45 110
3396 50 122
3249
.862366
2219
2072
1924
1777
.861629
1482
1334
1186
1038
.860890
0742
0594
0446
0298
CORR.
FOR
SEC.
+
11
"
.860149
0001 " Corr.
.859852 10 25
.858662
8513
8364
8214
8065
9704 15 37
9555 20 50
.857916
7766
7616
7467
7317
.857167
VERSINE
.133975.154700
4120
4266
4411
4557
EXSEC
.134703.155672
4849
4994
5140
5287
5579
5725
5872
6018
4894
5089
5283
5478
.135433.15664830
.136164.157628
6311
6458
6604
6751
8518
8666
8814
8962
"
5867 Corr.
6062 10
6257 15 49 15 97
6452 20 65
20 130
.136898.158612
7045 8809
7192
9006
9204
7339
7486
9402
98
6844 40 130
7039 45 147
7235 50 163
7432
7824
8021
8218
8415
.137634.159600
7781
7928
0742
0891
1040
1189
.139110.161588
9258
9406
9554
9702
CORR.
FOR
SEC.
1788
1988
2188
2389
.139851.162589
2790
2990
9999
.140148
0296 3191
0445 3392
.140594.163594
+
11
9798 Corr.
Corr.
9996 10 33 10 65
8076. 16019515 50 15 98
8223 0393 20 66 20 131
.142084.165616
2234
2384
2533
2683
Corr.
3310 65
.138371.160592 30 100 || 30 196
0791 40 133
0990 45 149
1189 50 166
1389
40 261
45 294
50 327
"
3795 Corr.
3997 10 34
4199 15 51
4401 20 67
5819
6022
6226
6430
.142833 166633
=
.141338.164603 30 101
1487 4805 40 135
1636 500845 152
1786
5210 50 169
1935
5413
rt
CORR.
FOR
SEC.
+
30 195
40 259
45 292
50 324
=
11
" Corr.
10 66
15 99
20 132
30 197
40 263
45 296
50 329
TANGENT
.577350
7738
8126
8514
1
0-23=
8903 4
.579291
9620 6
580068
7
0457
0846
5678σ
9
.581235 10
1624 11
2014 12
2403 13
2793 14
.583183 15
3573 16
3963 17
4353 18
4743 19
.585134 20
5524 21
5915 22
6306 23
6696 24
.587088 25
7479 26
7870 27
8262 28
8653 29
.589045 30
9437 31
9829 32
.59022 33
061334
.591006 35
1398 36
1791 37
2184 38
2577 39
.592970 40
3363 41
3756 42
4150 43
4544 44
.594938 45
5331 46
5726 47
6120 48
6514 49
596908 50
7303 51
7698 52
8093 53
8488 54
598883 55
9278 56
9674 57
.600069 58
0465 59
.60086 60
492
1
0.515038
5287
5537
5786
6035
0-232 BOZOK O-2M4 WO789
5.516284
6
7
8
9
11
10.517529
12
13
14
16
17
19
15.518773 30 124
9022 40 166
9270 45 186
22222 222°~ 6.087 HOMO ERRE DONOR OF CON
23
24
18 9519 50 207
9768
20.520016
26
27
28
29
25.521258
1506
1754
2002
2250
31
32
33
34
SINE
30.522499
2747
2994
36
37
38
39
6533
6782
7031
7280
41
42
35.523738
3986
4234
4481
4729
43
44
II
7778 Corr.
8027 10 41
8276 15 62
8525 20 83
46
47
48
49
40.524977
0265
0513
0761
1010
51
52
53
54
56
3242
3490
57
58
59
45.526214
50.527450
7697
7944
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
11
5224 Corr.
5472 10 41
571915 62
5966 20 82
30 124
6461 40 165
6708 45 186
6956 50 206
7203
55.528685
8932
9179
9426
9673
60.529919
8191
8438
COSINE
.857167
7017
6868
6718
6567
.856417
6267
6117
.854156
4005
3854
3702
3551
5966
11
5816
855666
5515 Corr.
5364 10 25
521415 38
5063 20 50
854912 30 76
4761 40 101
4610 45 113
4459 50 126
4308
.853399
3248
3096
2944
2792
.852640
2488
2336
2184
2032
CORR.
FOR
SEC.
+
.851879
1727
1574
1422
11
1269
.851117
0964 Corr.
0811 10 26
0658 15 38
0505 20 51
848818
8664
8510
8356
8202
848048
31°

VERSINE
142833. 166633
2983
3132
3282
3433
EXSEC
.143583.167655
3733
3883
4034
4184
5541
5692
6837
7042
7246
7450
7860 Corr.
8065 10
34
827015 51
8476 20 69
.144334.168681|30 103
8887 40 137
9093 45 154
4786 9299 50 171
4937 9505
4485
4636
.1450881.169711
5239
5390.170124
6146
6298
6449
9918
0331
0538
.145844.170746
5995 0953
1161
1368
1576
.850352 30
0199 40 102
0046 45 115
.849893 50 128.150107
9739
.849586
9432
9279
9125
8972
7664
7816
7968 3665
.148121.173875
8273
8426
8578
8731
4085
4295
4506
4716
.148883.174927
9036
9189
9342
9495
146601.171784
6752
1993
"Corr.
2201 10 35
6904
7056 2410 15 52 15 100
261920 70 120 133
7208
.608807 20
9205 21
9604 22
.610003 23
0403 24
.610802 25
1201 26
160127
2001128
2401 29
.147360.172828 30 10530 200.612801 30
7512 3037 40 139 ||40 267
3246 45 157 45 300
3201 31
3601 32
3456 50 174
50 333
5138
5349
5560
5772
77.149648.175983
9801
9954
CORR.
FOR
SEC.
+
=
11
6195 Corr.
6407 TO 35
6619 15 53
6831 20 71
.151 182.178109
1336
8322
8536
1490
1644
8750
1798
8964
.151952.179178
CORR.
FOR
SEC.
+
0261
.150414.177044 30 106
0568 7257 40 142
0721 7469 45 160
7682 50 177
0875
1028
7896
IT
Corr.
10 66
15 99
20 132
30 199
40 265
45 298
50 331
TANGENT 1
.600861 0
1257
1653 2
2049 3
2445 4
" Corr.
10 67
5
.602842
3239 6
3635 7
4032 8
4429 9
.604827 10
522411
5622 12
6019 13
641714
.606815 15
7213 16
7611 17
8010 18
8408 19
400233
4402 34
.614803 35
5204 36
5605 37
6006 38
6408 39
.61680940
721141
761342
801443
8417 44
" Corr.
.618819 45
9221 46
10 67 9624 47
15 101.620026 48
20 134 0429 49
30 202.620832 50
40 269 1235 51
45 302 1638 52
50 336 2042 53
2445 54
.622849 55
3253 56
3657 57
406158
4465 59
.624869 60
493
1
01234 BO76G
0.529919
1.530166
8
5.531152
9
0-23➡
10.532384
11
12
13
14
#1
2630 Corr.
2876 10 41
3122 15 61
3368 20 82
15.533614 30 123
3860 40 164
4106 45 185
4352 50 205
4598
16
5678DO
17
18
19
22222
234
21
20.534844
22
23
24
222
5GOI
26
25.536072
28
29
6318
27 6563
6809
7054
31
ოოო ოოო
32
30.537300
7545
7790
8035
8281
33
34
=====
36
37
38
39
35.538526
41
43
SINE
44
0412
0659
0906
46
47
48
49
1399
1645
1891
2138
40.539751
9996 " Corr.
42.540240 TO
41
0485 15 61
0730 20 82
45.540974 30 122
121940 163
1464 45 183
1708150 204
1953
80 20000
52
87 BO788 8
5090
5336
5581
5827
53
50.542197
2442
2686
54
56
57
58
59
8771
9016
9261
9506
2930
3174
55.543419
VERSINES, EXSECANTS, AND TANGENTS
32°

3663
3907
4151
4395
CORR.
FOR
SEC.
+
60.544639
COSINE
.848048
7894
7740
7585
7431
.847276
7122
6967
6813
11
6658
846503
6348 Corr.
6193 10 26
6038 15 39
5883 20 52
.845728 30 78
5573 40 103
5417 45 116
526250 129
5106
.844951
4795
4640
4484
4328
.844172
4016
3860
3704
3548
.843391
3235
3079
2922
2766
CORR.
FOR
SEC.
.842609
2452
2296
2139
11
1982
.841825
1668
Corr.
1511 10 26
135415 39
1196 20 52
.841039 30 79
0882 40 105
0724 45 118
0567 50 131
0409
.840251
0094
839936
9778
9620
.839462
9304
9146
8987
8829
.838671
VERSINE
.151952.179178
2106
2260
9393 " Corr.
9607 TO 36
2415
9822 15 54
2569. 180037 20 72
2878
3033
3187
3342
.152724.18025230 108
0468 40 144
0683 45 162
0899 50 180
1115
.153497.181331
3652
3807
3962
4117
EXSEC
.154272.182414
4427
4583
4738
4894
5205
5360
5516
5672
FT
2631 Corr.
2848 TO 36
3065 15 55
3283 20 73
.155049.183501|30 109
3719 40 146
3937 45 164
4155 50 182
4374
1547
1763
1980
2197
. 155828.184593
5984
6140
6296
6452
7548
7704
.156609.185689
6765
6921
7078
7234
7861
8018
4812
5031
5250
5469
.157391.186790 30 111
7011 40 147
723245 166
7453 50 184
7674
9906
.160064
0222
0380
CORR.
FOR
SEC.
+
11
5909 Corr.
6129 10 37
6349 15 55
6569 20 74
.158175.187895
8332
8489
8646
8804
.159749.190120 30 112
0344 40 149
0567 45 168
0791 50 186
1015
.160538.191239
0696
0854
1013
1171
1464
1688
1913
2138
.161329.192363
CORR.
FOR
SEC.
+
" Corr.
то 68
15 101
20 135
30 203
40 270
45 304
50 338
8117
8339
8561
It
13
8783
.158961. 189006
9228 Corr. Corr.
9451 10 37 10 69
967415 56 15 103
9897 20 75
9118
9276
9433
9591
20 137
" Corr.
10 68
15 102
20 136
30 204
40 272
45 306
50 340
11
Corr.
10 68
15 103
20 137
30 206
40 275
TANGENT
45 309
50 344
.624869
5274
5679
6083
6488
.626894
7299
7704
8110
8516
01234 56789
.62892210
9327 11
9734 12
63014013
0546 14
.630953 15
1360 16
1767 17
2174 18
2581 19
.632988 20
3396 21
3804 22
42123
461924
.635027 25
5436 26
5844 27
6253 28
666 29
30 205
40 273
45 308
.63911735
9527 36
9937 37
50 342.640347 38
0757 39
.641167 40
1578 41
1989 42
2399 43
2810 44
.637070 30
7479 31
7888 32
8298 33
8707 34
.643222 45
3633 46
4044 47
4456 48
486849
.645280 50
5692 51
6104 52
6516 53
6929 54
.647342 55
7755 56
8168 57
858 58
8994 59
649408 60
494
0.544639
01234 BOT∞ a
6
7
5.545858
8
9
||
1234
12
10.547076
13
14
56789
16
17
18
19
~~~~~ ~~
0123=
15.548293 30 122
8536 40 162
8780 45 182
9023 50 203
9266
21
22
20.549509
24
9752
9995
23.550238
0481
DONOR Oma ko♪❤.
26
27
28
29
25.550724
31
32
33
34
36
SINE
30.551937
2180
2422
2664
2907
37
38
4883
5127
41
5371
5614
42
43
44
35.553149
3392
3634
3876
4118
6102
6346
6589
6833
46
47
48
49
40.554360
g g⌘ཡཿཀྱཿg
51
| 52
7320" Corr.
7563 10 41
7807 15 61
8050 20 81
53
54
56
0966
1209
1452
1694
57
58
59
45.555570 30 121
581240 161
6054 45 181
6296 50 202
6537
50.556779
7021
7262
7504
7745
55.557986
8228
8469
8710
8952
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
" Corr.
4602
4844 10 40
5086 15 60
5328 20 81
60.559193
COSINE
.838671
8512
8354
8195
8036
.837878
7719
7560
7401
7242
.837083
6924
Corr.
6764 10 27
6605 15 40
6446 20 53
.835488
5328
5168
.836286 30 80
6127 40 106
5967 45 120
5807 50 133
5648
5008
4848
.834688
4528
4367
4207
4046
.833886
3725
3565
3404
3243
.833082
2921
2760
2599
2438
CORR.
FOR
SEC.
.832277
2116
1954 10
179315
1631 20
+
.831470 30
.830661
0499
0337
0174
0012
.829850
9688
9525
9363
9200
.829038
" Corr.
27
40
54
1308 40 108
146 45 121
0984 50 135
0823
33°

VERSINE
.161329.192363
1488
1646
1805
1964
2281
2440
2599
2758
"
2589 Corr.
2814 10 38
3040 15 57
3266 20 76
.162122.193492 30 113 30 207
3718 40 151 40 276
3945 45 170 45 311
417150 189 ||50 346
4398
EXSEC
.162917.194625
3076
3236
3395
3554
4852
5080
.163714.195763
3873
4033
4193
5991 " Corr.
6219 10 38
6448 15 57
4352 6677 20 77
.164512.196906 30 15
7135 40 153
7364 45 172
4992 7593 50 191
5152 7823
4672
4832
.165312.198053
5472 8283
5633
8513
5793
8744
5954
8974
5307
5535
.166114.199205
6275
6435
6596
.167723.201523
7884
8046
8207
8369
8692
8854
9016
9177
81.168530 ||.202690
9501
9663
9826
9988
1756
1989
To
9436 "Corr. Corr.
966710 39 10 70
9898 15 58 15 105
6757.200130 20 77 20 140
.166918.200362 30 116 30 210
7079 0594 40 155 ||40 280
0826 45 174 45 315
105850 194 50 350
1291
7240
7401
7562
2223
2456
2924
" Corr.
3158 10 39
339215 59
3626 20 78
.169339.203861 30 118
4096 40 157
433145 176
4566 50 196
4801
CORR.
FOR
SEC.
+
170150.205037
0312
0475
0637
0800
.170962.206218
5273
5509
5745
5981
CORR.
FOR
SEC.
+
11
Corr.
10 69
15 104
20 138
"Corr.
10 69
15 104
20 139
30 209
40 278
45 313
150 348
11
" Corr.
10 70
15 106
20 141
30 211
40 281
45 316
50 352
TANGENT
.649408
9821
.650235
0649
1063
0
1
234 567Bσ
.651477
1892
2306
2721
8
3136 9
.65355110
3966 1
4382 12
4797 13
521314
.655629 15
6045 16
6461 17
6877 18
7294 19
.657710 20
8127 21
8544 22
8961 23
9378 24
.659796 25
.660214 26
063 27
1049 28
1467 29
661886 30
2304 31
2722 32
314133
3560 34
.663979 35
4398 36
4818 37
5237 38
5657 39
.666077 40
6497 41
691742
7337 43
775844
.66817945
8600 46
902047
9442 48
9863 49
.670284 50
0706 51
1128 52
1550 53
1972 54
.672394 55
2817 56
3240 57
3662 58
4085 59
.674508 60
495
↑
0.559193
9434
9675
9916
4.560157
5.560398
0639
0880
1121
1361
I
234
5678 M
9
10.561602
11
-232 DONOσ
12
1843 "Corr.
20831040
2324 15 60
2564 20 80
15.562805 30 120
3045 40 159
3286 45 180
18 3526 50 200
16
17
19
3766
20.564007
4247
4487
4727
4967
25.565207
5447
5687
5927
6166
13
14
22222 22222 6.2
21
1234
23
24
26
27
28
29
567
30.566406
6646
32 6886
33
7125
34 7365
31
35.567604
7844
8083
55±5 @www.
36
37
38
39
41
40.568801
42
43
4444
55
SINE
51
52
47
48
49 0952
50.571191
1430
1669
1907
2146
53
8322
8562
54
gg⌘38
56
57
58
59
9040 Corr.
9280 10 39
9519 15 60
9758 20 80
55.572384
2623
2861
3100
3338
60.573576
VERSINES, exsecants, AND TANGents
34°

31
CORR.
FOR
SEC.
+
COSINE
.829038
8875
8712
8549
8386
828223
8060
7897
7734
7571
"
827407
7244 Corr.
708110¯¯¯¯¯27
6917 15 41
6753 20 55
.825770
5606
5442
45.569997 30 119.821647 30 83
46.570236 40 159
1481 40 110
1315 45 124
0475 45 179
0714 50 199
149 150 138
.826590 30 82
6426 40 109
6262 45 123
6098 50 136
5934
5278
5114
.824949
4785
4620
4456
4291
1.824126
396 I
3796
3632
3467
1.823302
3136
2971
2806
2640
1.822475
CORR.
FOR
SEC.
+
2310 " Corr.
2144 10 28
1978 15 41
181320 55
0983
.820817
0651
0485
0318
0152
.819985
9819
9652
9486
9319
1.819152
VERSINE
1125
1288
1451
1614
.170962 ||.206218
6455
" Corr.
669210 40
692915 60
7166 20 79
.171777.207404 30 119
7642 40 159
7879 45 179
8118 50 198
8356
1940
2103
2266
2429
.172593.208594
2756
2919
3083
3247
EXSEC
.173410.209790
3574.210030
4722
4886
**
Corr.
3738 0270 10 40
3902 0510 15 60
4066 0750 |20 80
174230.210990 30 121
4394 123140 161
1472 45 181
1713 50 201
4558
1954
8833
9072
9311
9550
.175051.212196
5215
5380
5544
2922
5709 3164
6864
7029
7194
7360
.175874.213406
6039
6204
6368
6533
.178353
8519
8685
8851
9017
2438
2680
3649 Corr.
389210 41
413515 61
4378 20 81
.176698.214622 30 122
4866 40 163
5109 45 183
5354 50 203
5598
.177525.215842
7690
7856
8022
8187
9349
9515
9682
9848
6087
6332
6577
6822
CORR.
FOR
SEC.
+
"
" Corr.
217068
7314
7559 10 41
780615 62
8052 20 81
.179183.218298 30 124
8545 40 165
8792 45 185
9039 50 206
9286
180015219534
0181
9782
0348.220030
0514 0278
0681
0526
180848.220775
CORR.
FOR
SEC.
+
" Corr.
10 71
15 106
20 141
"
10 71
15 107
20 142
30 212.676627
40 283
45 318
150 354
Corr.
30 213
40 285
45 320
50 356
TANGENT
.674508
4932
5355
5779
6203
11
Corr.
10 72
15 108
20 144
0-23⇒
5678σ
7051
7475
7900
8324 9
.678749 10
9174 11
9599 12
.680025 13
0450 14
.680876 15
130216
1728 17
2154 18
2580 19
683007 20
3433 21
3860 22
4287 23
4714 24
.685142 25
5569 26
5997 27
6425 28
6853 29
687281 30
7709 31
813832
8567 33
8996 34
" Corr.
10 72
15 107
20 143
30 215.689425 | 35
40 286
9854 36
45 322.690283 37
150 358 0713 38
| 14239
.691572 40
2003 41
2433 42
2863 43
3294 44
693725 45
4156 46
4587 47
501848
5450 49
30 216.695881 | 50
40 288 6313 51
45 324 6745 52
7177 53
7610 54
50 360
698042 55
8475 56
8908 57
934158
9774 59
700208 60
496
0.573576
3815
4053
4291
4529
0123♬
4
5678 σ
5.574767
5005
5243
5481
5719
9
10.575957
34
13
" 11
11
6195 Corr.
12 6432 10 40
6670 15 59
6908 20 79
15.577145 30 119
7383 40 158
762045 178
785850 198
8095
14
16
17
18
19
20.578332
8570
8807
9044
9281
22222
U2 2
23
24
25.579518
9755
9992
28.580229
0466
30.580703
0940
1176
1413
1650
26
27
29
170m2 komm8 2 49700 85087
55
31
32
33
34
35.581886
36
37
38
39
441
42
43
44
SINE
40.583069
46
48
51
45.584250 30 118
4486 40 157
4722 45 177
4958 50 197
5194
52
50.585429
53
54
2123
2360
2596
2832
56
57
58
59
3305 Corr.
354110 39
3777 15 59
4014 20 79
55.586608
5665
5901
6137
6372
6844
7079
7314
7550
60.587785
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
COSINE
.819152
8985
8818
8651
8484
.818317
8150
7982
7815
7648
.815801
5633
5465
5296
5128
.817480
7312
" Corr.
7145 TO 28
6977 15 42
6809 20 56
.816642 30 84
6474 40 12
6306 45 126
6138 50 140
5970
.814959
4791
4622
4453
4284
.814116
3947
3778
3608
3439
.813270
3101
2931
2762
2592
CORR.
FOR
SEC.
.812423
2253
" Corr.
2084 TO 28
191415 42
1744/20 57
.811574 30
+
.810723
0553
0383
0212
0042
1404 40 113
.809871
9700
9530
1234 45 127
1064 50 142
0894
9359
9188
809017
35°

VERSINE
EXSEC
.180848.220775
11
1015
1182
1349
1023 Corr.
1272 10 42
1522 15 63.
1516 1771 20 83
.181683.222020 30 125
2270 40 167
2520 45 188
2770 50 208
2352 .3021
1850
2018
2185
.182520.223271
2688
2855
3023
3191
.183358.224527
3526
3694
3862
4030
3522
3773
4024
4276
.185041.227055
5209
5378
5547
5716
7309
7563
7818
8072
.185884.228327
6053
6222
6392
6561
187577.230886
7747
7916
8086
8256
85.188426||.232174
8596
8766
8936
9106
9447
9617
9788
9958
11431
1400
1658
1916
CORR.
FOR
SEC.
+
#
I
8582 Corr. Corr.
8837 10 43 10 73
9092 15 64 15 110
9348 20 85 20 147
.186730.229604 30 128 30 220
6899 9860 40 171 40 293
7069.2301 16 45 192
7238 0372 50 214
7408 0629
45 330
50 367
2432
" Corr.
2690 10
43
2949 15 65
3207 20 87
CORR.
FOR
SEC.
+
" Corr.
10 73
.706730 15
7166 16
7603 17
8040 18
8476 19
4779
" Corr.
503110 42
5284 15 63 15 109
5536 20 8420 146
184199.225789 30 127 30 219.708913 20
4367 6042 40 169 40 292
9350 21
4535
6295 45 190 45 328
9788 22
50 365 .710225 23
066324
4704 6548 50 211
4872
6802
.190129.234764
0300
0470
0641
0812
.190983.236068
" Corr.
10 72
5025
5285
5546
5807
15 109
20 145
30.217
40 290
45 326
50 362
n
Corr.
то 74
15 111
20 148
.189277.23346630 130 30 222
3726 40 173
398545 195
4245 50 216
4504
40 295
45 332
50 369
TANGENT
.700208
0641
1075 2
1509
1943
.702377
0123A BONBO
4
5
2812 6
3246 7
3681 8
4116 9
704552 10
4987 11
5422 12
5858 13
629414
.711101 25
1539 26
1977 27
2416 28
28541 29
.71329330
3732 31
4171 32
4611 33
5050 34
.715490 35
5930 36
6370 37
681038
7250 39
.717691|40
8132 41
8573 42
901443
9455 44
.719897 45
.720339 46
0781 47
1223 48
166549
.722108 50
2550 51
2993 52
3436 53
387954
.724323 55
4766 56
5210 57
5654 58
6098 59
.726542 60
497
0.587785
01234
3
5678σ
5.588961
9196
9431
9666
9901
6
9
||
10.590136
234
12
13
14
567
16
17
18
19
22222
~~2 22~~2 .~~~~ ❤❤♪ FIRE 99700 85087 68788
21
20.592482
2716
2950
3185
3419
25.593653
3887
4121
4355
4589
23
24
15.591310 30 117
1544 40 156
1779 45 176
2013 50 195
2248
26
27
28
31
32
30.594823
5057
5290
5524
5758
33
34
36
37
38
35.595991
6225
6458
6692
6925
SINE
42
43
44
8021
8256
8491
8726
40.597159
46
47
48
49
45.598325
51
54
56
0371 Corr.
0606 10 39
0840 15 59
1075 20 78
50.599489
9722
9955
53.600188
0420
57
58
59
55.600653
0885
1118
1350
1583
60.601815
VERSINES, EXSECANTS, AND TANGENTS
36°

CORR.
FOR
SEC.
"
11
+
7392
"Corr.
7625 10 39
785815 58
8092 20 78
8558 40 155
8791 45 175
902450 194
9256
COSINE
.809017
8846
8675
8504
8332
.808161
7990
7818
7647
7475
"
.807304
7132 Corr.
6960 10 29
6788 15 43
6617 20 57
.806445 30
805584
5411
5239
5066
4894
.804721
4548
4376
4203
4030
6273 40 115
6100 45 129
5928 50 143
5756
.803857
3684
3511
3338
3164
.802991
2818
2644
2470
2297
CORR.
FOR
SEC.
"
.802123
1950 Corr.
1776 10 29
160215 44
1428 20 58
30 117.801254 30 87
1080 40 116
090645 131
0731 50 145
0557
.800383
0208
0034
799859
9685
+
.799510
9335
9160
8986
8810
.798636
VERSINE
190983.236068
1154
1325
1496
1668
2010
2182
2353
2525
EXSEC
11
6329 Corr.
6591 10 44
685315 66
7115 20 86
.191839.237377 30 131
7639 40 175
7902 45 197
8165 50 219
8428
.192696.238691
2868
3040
3212
3383
86.193555||.240011
3727
3900
4072
4244
4589
4761
4934
5106
8955
9218
9482
9746
.194416.241336 30 133
1602 40 177
1868 45 200
2134 50 222
2400
It
0275 Corr.
0540 10 44
0805 15 67
1070 20 89
.195279.242666
5452
5624
5797
5970
7182
7356
7530
7703
2933
3200
3468
3735
.196143.244003
6316
6489
6662
6836
4270
" Corr
4538 10 45
4807 15 67
5075 20 90
.197009.245344 30 135
561340 180
5882 45 202
6152 50 225
6421
CORR.
FOR
SEC.
+
. 197877.246691
8050
8224
8398
8572
6961
7232
7502
7773
.198746.248044
8920 8315 Corr.
9094 858710 45
8858 15 68
9130 20 91
.199617.249402 30 136
9269
9443
9792
9675 40 182
9966 9947 45 205
.200141.250220 50 227
0315
0493
.200490.250766
0665
0840
1014
1190
.201364.252136
1040
1313
1587
1861
11
CORR.
FOR
SEC.
+
I
Corr
74
10
15 11
20 149
30 223
40 297
45 334
50 372.730104
" Corr.
75
10
15 112
20 150
30 224
40 299
45 337
50 374
10 75
15 113
20 151
TANGENT
.726542
6987
7432
7877
8322
30 226
40 301
45 339
50 376
.728767
9212
9658
"Corr.
10 76
15 114
20 152
0
1231
4
5678σ)
8
0550 9
730996 10
1443 11
1889 12
2336 13
2783 14
.733230 15
3678 16
4125 17
4573 18
5021 19
73996 30
"Corr..740411 31
0862 32
1312 33
1763 34
.735469 20
5917 21
6366 22
6815 23
7264 24
.737713 25
8162 26
8612 27
9061 28
951 29
.742214 35
2666 36
311737
3569 38
4020 39
744472 40
4925 41
5377 42
583043
628244
.746735 45
718946
7642 47
8096 48
8549 49
30 227 749003 50
140 303 9458 51
45 341 991252
50 379.750366 53
0821 54
751276 55
173156
2187 57
2642 58
3098 59
753554 60
498
1
0.601815
2047
2280
-232 567BG 0-23-
4
8
5.602976
11
12
10.604136
13
14
16
17
18
~~~~~
19
∞ α
15.605294 30 116
5526 40 154
5757 45 174
5988 50 193
6220
21
22
20.606451
23
24
~°~~* ommMZ HOMMA D
26
27
28
29
25.607607
7838
8069
8300
8531
32
34
37
SINE
38
30.608761
8992
9223
2512
2744
41
42
43
44
3208
3440
3672
3904
35.609915
36.610145
46
47
48
49
LOLOLOLO LO
51
" Corr.
39
4367
459910
483115 58
5062 20 77
40.611067:
52
6682
6914
7145
7376
53
54
56
57
58
59
9454
9684
45.612217 30 115
2447 40 153
2677 45 173
2907 50 192
3137
50.613367
0376
0606
0836
11
1297 Corr.
1527 10 38
1757 15 57
1987 20 77
3596
3826
4056
4285
55.614515
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
4744
4974
5203
5432
60.615662
COSINE
.798636
8460
8285
8110
7935
.797759
7584
7408
7233
7057
.796882
6706
6530 10
635415
6178 20
796002 30
.795121
4944
4768
4591
4415
.794238
4061
3884
3707
3530
.793353
3176
2999
2822
2644
5826 40 117
5650 45 132
547450 147
5297
.792467
2290
2112
19341
1757
CORR.
FOR
SEC.
.789798
9620
9441
+
" Corr.
9263
9084
.791579
1401
"Corr.
1224 10 30
1046 15 45
0868 20 59
.788905
8727
8548
8369
8190
788011
BENE
.790690 30 89
0512 40 119
0333 45 134
0155 50 148
.789977
37°

VERSINE
.201364.252136
1540
1715
1890
2065
11
2410 Corr.
2685 10 46
2960 15 69
3235 20 92
.202241.253511 30 138
3786 40 184
40621 45 207
433950 230
4615
2416
2592
2767
2943
.203118.254892
3294
3470
EXSEC
5168
5446
3646
5723
3822 6000
203998.256278
4174
4350
4526
6556 Corr.
6835 10 47
7113 15 70
4703 7392 20 93
.204879.257670 30 140
7950 40 186
8229 45 210
8509 50 233
8788
5056
5232
5409
5585
.205762.259069
5939 9349
6116
6293
•
6470.260191
.206647.260472
6824
7001
7178
7356
7710
7888
8066
8243
9629
99.0
.208421.263298
8599
8776
8954
9132
tt
0754 Corr.
1036 10
1318 15
1600 20
0559
0737
0916
.209310.264719
9488
9667
9845
.210023
CORR.
FOR
SEC.
+
3581
3865
4150
44341
.207533.261882 30 14230 232
2165 40 189 40 309
2448 45 21245 348
273 50 236 50 386
3014
.211095.267579
1273 7866
1452
8154
8442
8730
IT
5004 Corr.
5289 10
5574 15
5860 20
1631
1810
.211989.269018
11
CORR.
FOR
SEC.
+
Corr.
10 76
32535
72
15 114
20 153
" Corr.
471077
71 15 116
9420 155
30 229
40 305
45 343
50 381
It
Corr.
10 77
15 115
20 154
30 230
40 307
45 345
50 384
"Corr.
48 10 78
210202.266146 30 14330 233
0380 6432 40 191
6719 45 215
7005 50 239
7292
15 117
95 20 156
40 311
45 350
50 389
TANGENT 1
.753554
4010
4467
4923
5380 4
5
.755837
6294 6
6751 7
7209 8
7667 9
.758125 10
8583 11
9041 12
9500 13
995914
.760418 15
0877 16
1336 17
1796 18
2256 19
.762716 20
317621
3636 22
4097 23
4558 24
0-232 DOTOM O-23-
.765019 25
5480 26
594 27
•
6403 28
6865 29
767327 30
7789 31
8252 32
8714 33
9177 34
.769640 35
.770104 36
0567 37
1031 38
1495 39
.771959 40
2423 41
2888 42
335343
3818 44
.774283 45
4748 46
521447
5680 48
6146 49
.776612 50
7078 51
7545 52
8012 53
8479 54
.778946 55
941456
9881 57
.780349 58
0817 59
.78128660
499
0.615662
0-232 57 0-231 DO
6
8
5.616807
9
14
16
17
18
10.617951
22222 222
21
234
23
24
15.619094 30 114
9322 40 152
9551 45 171
9779 50 190
19.620007
20.620236
26
27
28
29
0207 600MM DIE OOO OOG
31
32
33
34
25.621376
1604
1831
2059
2287
36
37
38
39
SINE
30.622515
2742
2970
3197
3425
41
42
43
44
5891
6120
6349
6578
46
47
48
49
35.623652
3880
4107
4334
4561
7036
7265
7494
7722
51
40.624788
52
53
54
8180 " Corr.
8408 10 38
8637 15 57
8866 20 76
0464
0692
0920
1148
45.625924
56
57
58
59
50.627057
7284
7510
7737
7963
VERSINES, EXSECANTS, AND TANGENTS
38°

CORR.
FOR
SEC.
+
5016 " Corr.
5243 10 38
5470 15 57
5697 20 76
55.628189
8416
8642
8868
9094
60.629320
6150 40 151
6377 45 170
6604 50 189
6830
COSINE
.788011
7832
7652
7473
7294
.787114
6935
6756
6576
6396
.786216
6037
" Corr.
5857 TO 30
5677 15 45
5497 20 60
.784416
4235
4055
3874
3694
.785317 30 90
5137 40 120
4957 45 135
4776 50 150
4596
.783513
3332
3151
2970
2789
.782608
2427
2246
2065
1883
CORR.
FOR
SEC.
.781702
1520
1339
1157
11
0976
.780794
0612 Corr.
0430 TO 30
0248 15 46
0066 20 61
30 113.779884 30 91
9702 40 121
+.211989.269018
2168
2348
2527
.778973
8791
8608
8426
8243
9520 45 137
9338 50 152
9156
.778060
7878
7695
7512
7329
.777146
VERSINE
EXSEC
11
9307 Corr.
9596 10 48
9884 15 73
2706.270174 20 97
.212886.270463 30 145
3065 0753 40 193
1043 45 218
1333 50 242
1624
3244
3424
3604
.213784.271914
3963
4143
4323
4503
2205
2496
2788
3079
.214683.273371
4863
5043
5224
5404
5765
5945
6126
6306
11
3663 Corr.
3956 10 49
4248 15 73
4541 20 98
.215584.274834 30 147
5128 40 196
5421 45 220
5715 50 245
6009
.216487.276303
6668
6849
7030
7211
6598
6893
7188
7483
.217392.277779
7573
7754
7935
8117
8074
8370 10
Corr.
50
8667 15 74
8963 20 99
1046
1344
1643
1942
CORR.
FOR
SEC.
+
.220116.282241
0298
0480
0662
0844
=
.218298.279260 30 149
8480
8661
9557 40 198
9854 45 223
8843.280152 50 248
9024 0450
.219206.280748
9388
9570
9752
9934
11
.221940.285247
2122
2305
2488
2671
.222854.286760
5549
5851
6154
6457
"
2541 Corr.
2840 10 50
3140 15 75
3441 20 100
.221027.283741 30 151
1209
1392
4042 40 201
4343 45 226
4644 50 251
1574
1757
4944
CORR.
FOR
SEC.
+
Corr.
10
78
15 117
20 157
30 235
40 313
45 352
50 392
11
Corr.
10 79
15 118
20 158
30 237
40 316
45 355
50 394
" Corr.
то 79
15 119
20 159
30 238
40 318
45 357
50 397
Corr.
10 80
15 120
20 160
30 240
40 320
45 360
50 400
TANGENT
781286
1754
•
2692
1
0
2223 2
.783630
4100
D-232 507B α)
3161 4
6
4570
5040 8
5510 9
.78598110
645211
6922 12
739413
7865 14
788336 15
8808 16
9280 17
9752 18
.790225 19
.790698 20
1170 21
1643 22
2117 23
2590 24
793064 25
3538 26
401227
4486 28
496 29
.795436 30
591131
6386 32
6862 33
7337 34
.79781335
8290 36
8766 37
9242 38
971939
.800196 40
067441
1151 42
1629 43
210744
.802585 45
3063 46
3542 47
4021 48
4500 49
.804979 50
5458 51
5938 52
6418 53
6898 54
.807379 55
7859 56
8340 57
882158
9302 59
.809784 60
500
1
0.629320
9546
9772
9998
4.630224
5.630450
0676
0902
1127
1353
0-23+ DO7OO 0-234 FOTOσ
6
8
9
10.631578
16
If
1804 Corr.
2029 10 38
12
13 2255 15 56
2480 20 75
14
17
18
19
15.632705 30 113
2931 40 150
3156 45 169
3381 50 188
3606
22
20.633831
21
COA DONO 07~07 ❤❤♪❤
22
23
24
26
25.634955
27
28
29
31
30.636078
33
6303
32 6527
34
36
37
38
SINE
35.637200
441
42
43
44
46
47
49
39 8096
4056
4281
4506
4730
OKOM KANOO O
51
5180
5405
5629
5854
40.638320
11
8544 Corr.
8768 10 37
899215 56
9215 20 75
45.639439 30 112
9663 40 149
9886 45 168
48.6401 10 50 186
0333
52
53
54
50.640557
6751
6976
56
57
58
59
7424
7648
7872
0780
1003
55.641673
1226
1450
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
1896
2119
2342
2565
60.642788
COSINE
777146
6963
6780
6596
6413
773472
3287
3103
2918
2734
.772549
2364
2179
1994
1810
.776230
6046
5863
5679
n
5496
.775312
5128 Corr.
4944 10 31
4761|15 46
4577 20 61
|.774393 30
92
4209 40 123
402445 138
3840 50 153
3656
771625
1440
1254
1069
0884
|.767911
7725
7538
—
7352
7165
CORR.
FOR
SEC.
770699
0513
0328
#1
0142
.769957
.769771
9585 Corr.
9400 TO 31
9214 15 46
9028 20 62
768842 30 93
8656 40 124
8470 45 139
8284 50 155
8097
.766978
6792
6605
6418
6231
1.766044
+
39°

VERSINE
.222854
3037
3220
3404
3587
3954
4137
4321
4504
.223770.288278 30 152
8583 40 203
8888 45 229
9192 50 254
9498
6713
6897
7082
7266
.224688 .289803
4872 .290109
5056
0415
5239
0721
" Corr.
51
6160
5423 1028
.225607 .291335
5791 1642
5976 1949 10
225615 77
2564 20 103
.226528.292872 30 154
3181 40 205
3489 45 231
3798 50 256
4107
6344
.227451.294416
7636
7821
8006
8190
8560
8746
8931
9116
EXSEC
11
.286760
7063 Corr.
7366 10 51
7670 15 76
7974 20 102
.228375.295967 30 155
6278 40 207
6589 45 233
6900 50 259
7212
.229301.297524
9487
9672
7836
8149
9858 8461
.230043 8774
1344
1530
1716
1903
CORR.
FOR
SEC.
+
4726
" Corr.
503610 52
534615 78
5656 20 104
2648
2835
.232089.302234
2275
2462
3208
3395
3582
3769
.233956
11
2550
2867 TO
"Corr.
53
3183 15 79
3500 20 106
.233022.303818 30 159
4135 40 211
4453 45 238
4771 50 264
5089
305407
CORR.
FOR
SEC.
+
TANGENT
11
809784
Corr..810266
10 80
0748
15 121
20 161
30 242
40 322
45 362
150 403
" Corr.
10 81
15 121
20 162
"Corr.
10 81
15 122
20 163
30 244
40 326
45 367
50 407
1230
1712
" Corr.
10 82
15 124
120 165
.812195
2678
3161
30 247
40 330
45 371
150 412
0-23 DO700 0-232
24
.230229.299088
0415
0600
.829234 40
9401 Corr. Corr.
9725 41
9715 10 52 10 82.830216 42
0786 .300029 15 78 15 123
0972 0343 20 105 120 164
.231158.300658 30 157 30 246
40 328
45 369
50 410
0708 43
199 44
0972 40 209
1288 45 235
1603 50 262
1918
5
6
3644 8
4128
30 243
819462 20
40 324
9949 21
45 364.820435|22
50 405 0922 23
1409 24
.821896 25
2384 26
2872 27
3360 28
3848 29
9
.814612 10
5096 11
5580 12
6065 13
654914
.817034 15
7520 16
8005 17
8490 18
8976 19
824336 30
482531
5314 32
5803 33
6292 34
.826782 35
7272 36
7762 37
8252 38
8743 39
.831691 45
2183 46
2676 47
3169 48
3662 49
.834155 50
4648 51
5142 52
563653
6130 54
836624 55
7119 56
7614 57
8109 58
8604 59
.839100 60
501
01234 567
8
9
5.643901
01234 56
10.645013
||
12
13
14
16
11
5236 Corr.
5458 10 37
5680 15 55
5902 20 74
15.646124 30 111
6346 40 148
6568 45 166
6790 50 185
7012
17
18
19
22222 2070 8~~♡~
21
20.647233
7455
7677
23
24
26
28
29
25.648341
8563
8784
9006
9227
31
32
34
30.649448
9669
9890
33.650111
0332
36
37
38
39
SINE
642788
3010
3233
3456
3678
5555
35.650553
0774
0995
1216
1437
42
4124
4346
4568
4791
43
44
46
47
48
49
40.651657
1878 Corr.
2098 10 37
2319 15 55
2539 20 73
45.652760 30 110
2980 40 147
3200 45 165
3421 50 184
3641
JAKZ HONO8 8
7898
8120
51
52
53
54
50.653861
56
57
58
59
4081
4301
4521
4741
55.654961
VERSINES, EXSECANTS, AND TANGENTS
40°

5180
5400
5620
5840
CORR.
FOR
SEC.
+
60.656059
COSINE
.766044
5857
5670
5483
5296
.765109
4921
4734
4546
4359
762292
2104
1915
1727
1538
.764171
3984 " Corr.
3796 10 31
47
3608 15
3420 20 63
.763232 30 94
3044 40 125
2856 45 141
2668 50 157
2480
.761350
1161
0972
0784
0595
.760406
0217
0028
.759839
9650
CORR.
FOR
SEC.
+
.759461
9271
9082
8893
11
8703
.758514
8384 Corr.
8134 TO 32
7945 15 47
7755 20 63
.756615
6425
6234
6044
5854
.757565 30 95
7375 40 127
718545 142
6995 50 158
6805
.755663
5472
5282
5091
4900
.754710
VERSINE
EXSEC
л
.233956 ||.305407
5726 Corr. Corr.
6045 10 5310 83
6364 15 80 15 124
6684 20 107 20 166
4143
4330
4517
4704
5079
5266
5454
5641
.235829.308607
6016
6204
6392
6580
.234891.307004 30 160 30 249
7324 40 21440 332
7644 45 240 45 373
7965 50 267 ||50 414
8286
236768|| .310217|
6956
7144
7332
7520
7896
8085
8273
8462
.238650.313457
8839
9028
9216
9405
9783
9972
.240161
0350
8928
9250
9572
9894
2055
2245
.237708.311833 30 162 30 250
2158 40 216 40 333
2482 45 24345 375
2807 50 270 ||50 417
3132
.240539.316724
0729
0918
1107
1297
2625
2815
3005
3195
11
0540 Corr. Corr.
0863 10 54
54 10 83
1186 15 81 15 125
1510 20 108 ||20 167
.241486.318368
1676
1866
CORR.
FOR
SEC.
+
.239594.315087 30 163 30 262
5414 40 218 40 335
5741 45 245 45 377
6068 50 272 50 419
6396
4528
4718
-
7052
7381
7710
8039
4909
5100
11
K
πI
11
3782 Corr. Corr.
4108 10 54 TO
TO 84
4434 15 82 15 126
4760 20 109 ||20 168
.242435.320019 30 165
0350 40 220
0681 45 248
1013 50 275
1344
CORR.
FOR
SEC.
+
π
11
8698 Corr.
9027 TO 55
9358 15 83
9688 20 110
245290.325013
.244337.32334130 167
3675 40 222
400945 250
4344 50 278
4678
.243385.321676
11
" Corr.
3575 2009 Corr.
3766 2342 10 5610 85
2674 15 83
3008 20 ||
3956
4146
n
Corr.
TO
15 127
20 169
30 253
40 337
45 380
50 422
TANGENT
0
I
.839100
9596
.840092
0588 3
2
1084 4
15 127
20 170
1
5
.841581
2078 6
2576 7
3073 8
3571 9
.844069 10
4567 11
5066 12
5564 13
6063 14
.846562 15
7062 16
7562 17
8062 18
8562 19
849062 20
9563 21
.850064 22
0565 23
1067 24
·
.851568 25
2070 26
2573 27
3075 28
3578 29
859124 40
9630 41
84.860136| 42
0642 43
1148 44
.854081 30
4584 31
5087 32
5591 33
6095 34
.856599 35
7104 38
7608 37
8113 38
8618 39
.861655 45
2162 46
2669 47
3177 48
3685 49
.864193 50
470151
5209 52
5718 53
6227 54
30 255.866736 55
40 339 7246 56
45 382 7756 57
50 424
8266 58
2776 59
.869287 60
502
0.656059
OI234 BONBO
6
7
5.657156
8
9
||
12
234
10.658252
13
14
16
17
19
8471 " Corr.
8690 10 36
8908 15 55
9127 20 73
15.659346 30 109
9564 40 146
9783 45 164
18.660002 50 182
0220
PURE 700 80000 888
21
22
23
24
20.660439
0657
0875
1094
1312
26
27
28
29
25.661530
1748
1966
2184
2402
31
32
33
34
30.662620
2838
3056
3273
3491
36
37
38
39
5555 EDI
41
42
43
44
SINE
35.663709
3926
4144
4361
4578
46
47
48
49
6278
6498
6717
6937
40.664796
5013
" Corr.
523010 36
5448 15 54
5665 20 72
45.665882 30 108
6099 40 145
631645 163
6532 50 181
6749
51
52
53
7375
7594
7814
8033
GOVOG EN
50.666966
54
56
57
58
59
55.668049
7183
7399
7616
7833
8266
8482
8698
8914
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
60.669131
COSINE
.754710
4519
4328
4137
3946
750880
0688
0496
0303
0111
.753755
3563
3372
3181
11
2989
.752798
2606 Corr.
2415 10 32
2223 15 48
2032 20 64
.751840 30 96
1648 40 128
145645 144
1264 50 160
1072
.749919
9726
9534
9341
9148
.748956
8763
8570
8377
8184
.747991
7798
7605
7412
7218
.745088
4894
4700
4506
4312
CORR.
FOR
SEC.
#1
747025
6832 Corr.
6638 IC 32
644515 48
6251 20 65
.74605730
97
5864 40 129
5670 45 145
5476 50 161
5282
.744117
3923
3728
+
3534
3339
.743145
41°

VERSINE
245290.325013
5481
5672
5863
6054
6437
6628
6819
7011
"I
11
5348 Corr. Corr.
5684 10 56 TO 85
601915 84 15 128
6355 20 112 20 171
.246245.326692 30 168
30 256
7028 40 225 40 342
7365 45 25345 384
7702 50 281 50 427
8040
.247202.328378
7394
7585
EXSEC
9312
9504
9697
9889
.250081.333478
0274
0466
0659
0852
1237
1430
8716
9054
1623
1816
.251044.335192 30 172
5536 40 229
5880 45 258
6225 50 286
6569
7777
9392
7968 9731
.248160.330071
"
8352 0410 Corr.
" Corr.
8544 0750 10 5710 86
8736 1090 15 85 15 129
8928 1430 20 113 20 172
.249120.331771 30 170 30 258.879553 20
2112 40 227 40 343 .880069 21
2453 45 255 45 387
2794 50 284 50 429
3136
0585 22
1102 23
1619 24
.252009.336914
2202
2395
2588
2782
CORR.
FOR
SEC.
+
#1
3820 Corr. "Corr.
4163 10 5710 86
4506 15 86 15 130
4849 20 115 20 173
30 259
40 346
45 389
50 432
.255883
6077
6272
6466
6661
.256855
7259
7605
7951
8297
.252975.338643
3168
3362
3555
8990 "Corr.
9337 10 58
9684 15 87
3749.340032 20 116
.253943.340380 30 174
4136
4330
0728 40 232
1076 45 260
1425 50 289
1774
4524
4718
.254912.342123
5106
5300
5494
5688
CORR.
FOR
SEC.
+
2473 Corr.
2823 10 58
3173 15 88
3523 20 117
343874 30 175
4225 40 234
4577 45 263
4928 50 292
5280
345633
30 261
40 348
45 391
50 435
TANGENT 1
.869287 0
9798
.870309
0820
1332
.871844
2356 6
5
2868 7
3381 8
3894 9
TO
10
" Corr.
87
15 131
20 175
0-23 ♬
30 263
40 350
45 394
50 438
44
.874407 10
4920 1
5434 12
5948 13
646214
.876976 15
7491 16
8006 17
8522 18
9037 19
.882136 25
2653 26
3171 27
3689 28
4207 29
.884725 30
8882 38
9403 39
1J
.889924 40
Corr..890446| 41
10 87
15 130
20 174
0968 42
1489 43
201244
.892534 45
3057 46
3580 47
410348
4627 49
524431
5763 32
6282 33
6802 34
.887322 35
7842 36
8362 37
.895151 50
5675 51
6199 52
6724 53
7249 54
.897774 55
8299 56
8825 57
935 58
9878 59
.90040460
503
0.669131
01234 5O7∞a
6
8
5.670211
9
OI23A BONBO
||
10.671290
12
13
14
16
17
11
1505 Corr.
1721 10 36
1936 15 54
2152 20 72
15.672367 30 108
2582 40 144
2797 45 161
3012 50 179
3228
18
19
22222 222≈≈ 67~87 68788 2=22E CON UNAZ KONDA O
21
23
20.673443
3658
3873
4088
4302
32
33
34
SINE
25.674517
4732
27 4947
26
5161
5376
36
30.675590
5805
6019
6233
6448
9347
9563
41
9779
9995
35.676662
6876
7090
7304
7518
42
43
44
0427
0642
0858
1074
40.677732
46
47
48
49
56
57
45.678801 30 107
901440 142
9228 45 160
944150 178
9655
59
50.679868
51.680081
55.680934
1147
1360
1573
1786
n
7946 Corr.
8160 10
36
8373 15 53
8587 20 71
VERSINES, EXSECANTS, AND TANGENTS
42°

0295
0508
0721
CORR.
FOR
SEC.
+
60.681998
COSINE
.743145
2950
2755
2561
2366
.742171
1976
1781
I 586
1390
.739239
9044
8848
8652
8455
.738259
8063
7867
7670
7474
.737277
7081
6884
6688
6491
.741195
"Corr.
1000
0805 10 33
060915 49
0414 20 65
.740218 30 98
0022 40 130
.739827 45 147.260173
9631 50 163
9435
CORR.
FOR
SEC.
.736294
6097
5900
5703
11
5506
.735309
5112 Corr.
4915 10 33
4717 15 49
4520 20 66
.734322 30
.733334
3137
2939
2741
2543
4125 40 132
3928 45 148
3730 50 165
3532
.732345
2147
1949
1750
1552
.731354
VERSINE
.256855.345633
7050
7245
7439
7634
11
11
8024
8219
5985 Corr. Corr.
6338 10 59 TO 88
669115 89 15 132
7045 20 118 20 176
.257829.347399 30 177
7753 40 236
8107 45 266
8414 8462 50 295
8610 8817
.258805.349172
9000 9528 Corr.
9195 9884 10 59
9391.35024015 89
9586 0596 20 119
.259782.350953 30 179
9978 131040 238
1668 45 268
0369 2025 50 298
0565 2383
1]
"
1348
1545
EXSEC
.260761.352742
1152
0956 3100 " Corr.
3459 10 60
3818 15 90
417820 120
.261741.354538 30 180
4898 40 240
5258 45 270
561950 300
5980
1937
2133
2330
2526
.262723.356342
4297
4494
5875
6072
6270
6468
CORR.
FOR
SEC.
+
99.265678 ||.36180030 183
2166 40 244
11
.266666.363634
6863
7061
7259
7457
=
7853
8051
8250
8448
.268646.367328
4002
Corr.
4370 10 62
4739 15 92
5108 20 123
CORR.
FOR
SEC.
+
30 264
40 352
45 396
50 440
.264691.359972
11
4888.360337 Corr. Corr.
5085 070210 61 10 90
то
5283 1068 15 92 15 135
5480 143320 122 20 180
.267655.365477 30 185
5846 40 246
6216 45 277
6586 50 308
6957
11
Corr.
TO 88
15 133
20 177
Corr.
10 89
15 133
20 178
30 267
40 356
45 400
50 445
-
11
11
3116
10
2919 6703 Corr
7065 10 61
7428 15 91
7790 20 121
3312
15 134
3509
20 179
.263706.358153 30 182 30 268.91901035
9547 36
3903
8516 40 24240 358
4100
8880 45 272 45 403.920084 37
9244 50 30350 447
062138
9608
1159 39
.92169740
223541
2773 42
3312 43
385144
2532 45 275 45 405
289950 305 ||50 450
3267
Corr.
89
TANGENT
.900404
0931
1458
1985
2513
30 266.908336 15
40 354
8867 16
45 398
9398 17
50 443
9930 18
.910462 19
.903041
1
"Corr.
10 90
15 136
20 181
01234
5678✪
3569
4098
4627 8
5156 9
.90568510
6215 11
6745 12
7275 13
780514
.910994 20
1526 21
2059 22
2592 23
312624
.913659 25
4193 26
4727 27
5262 28
5796 29
.916331 30
6866 31
7402 32
7938 33
8474 34
30 270.924390 45
40 360 493046
5470 47
601048
6551 49
.927091 50
7632 51
8174 52
8715 53
9257 54
30 271.929800 55
40 362 .930342 56
45 407
0885 57
50 452
1428 58
1971 59
.93251560
504
0.681998
2211
2424
2636
I
234 5E7BG 0-234 FOTO α
6
9
Il
12
10.684123
13
14
16
17
18
19
22222 22
0123➡
21
15.685183 30 106
539540 141
56C7 45 159
5818 50 176
6030
22
20.686242
23
24
26
27
28
29
25.687299
+ E55±5 wwwww wwwwg
6666 1650 55555 55
31
32
33
34
36
30.688355
8566
8776
8987
9198
37
39
SINE
41
42
2849
35.689409
9620
9830
38.690041
0251
43
44
.683061
3274
3486
3698
3911
52
40.690462
53
54
4335 Corr.
4547 10 35
4759 15 53
497 20 71
6453
6665
6876
7088
56
57
7510
7721
58
7932
8144
50.692563
2773
2982
3192
3402
-
45.69151330 105
1723 40 140
1933 45 158
2143 50 175
2353
55.693611
3821
4030
4240
4449
60.694658
TI
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
Corr.
11
0672
0882 TO 35
1093 15 53
1303 20 70
COSINE
.731354
1155
0957
0758
0560
.730361
0162
.729964
9765
9566
.729367
11
9168 Corr.
8969 10 33
8770 15 50
8570 20 66
.728371 30 100
8172 40 133
7972 45 150
7773 50 166
7573
.727374
7174
6974
6774
6575
.726375
6175
5975
5775
5575
725374
5174
4974
4773
4573
724372
4172
3971
3770
3570
CORR.
FOR
SEC.
723369
3168
" Corr.
2967 10 34
2766 15 50
2565 20 67
.722364 30 101
2163 40 134
1962 45 151
1760 50 168
1559
.721357
I 156
0954
0753
0551
.720349
0148
.719946
9744
9542
.719340
43°

VERSINE
.268646.367328
8845
9043
9242
9440
1828
2028
2227
2427
EXSEC
"
11
Corr.
10 91
7698 Corr.
8070 10 62
8442 15 93 15 136
8814 20 124 ||20 182
.269639.369186 30 186 30 273
9838
40 364
.270036
9559 40 248
9932 45 280
0235.370305 50 311
45 409
50 454
0434
0678
.270633.371052
0832
1031
1230
1430
.271629.372927 30 188 30 274
40 366
45 411
50 457
.272626.374809
2826
3026
3825
4025
4225
4425
1427 " Corr
" Corr.
1801 10 63 10 91
2176 15 94 15 137
255 20 125 ||20 183
5187 " Corr.
5564 1063
5943 15 95
3425 6321 20 126
3226
CORR.
FOR
SEC.
+
3303 40 250
3679 45 282
4055 50 313
4432
.273625.376700 30 190
7079 40 253
7458 45 284
7838 50 316
8218
5828
6029
6230
6430
274626.378598
4226
5026
5227
8240
8441
CORR.
FOR
SEC.
+
.276631.382420
6832
7033
7234
7435
.278643.386275
8844
9046
9247
9449
" Corr.
TO 92
15 138
20 184
30 276
40 368
45 414
50 459
TANGENT
.932515
3059
3603
40 372
45 418
50 465
20 187
6663
TI
Corr. Corr.
7050 10 65 10 9.3
7438 15 97 15 140
7827 20 130
.279651.388215 30 194
9852 8604 40 259
280054
0256
8994 45 292 45 421
938350 32450 467
0458
9773
.280660.390164
4148
4693
.935238
5783
0
•
.948965 30
8979 " Corr
" Corr.
9518 31
9360 10 64 10 92 950071 32
9742 15 96 15 139 0624 33
5427.3801231 20 127 20 185 1178 34
.275628.380505 30 191 30 277.951733 35
0888 40 255 40 370 2287 36
127045 287 45 416
1653 50 31950 462
2037
2842 37
3397 38
3953 39
.954508 40
1234
5
6
6329 7
6875 8
7422 9
.937968 10
8515 11
9062 12
9610 13
.940158 14
.940706 15
1254 16
1803 17
2352 18
2902 19
!!
2804 Corr. "Corr.
3189 10 64 10 93
3573 15 9615 139
3958 20 129 20 186
.277636.384344 30 19330 279.95729245
7837
7849 46
840747
8966 48
9524 49
8038
4729 40 257
5115 45 289
5502 50 321
5888
.943451| 20
4001 21
4552 22
5102 23
5653 24
.946204 25
6756 26
7307 27
7860 28
8412 29
506441
5621 42
6177 43
6734 44
.960083 50
0642 51
1202 52
1761 53
2322 54
30 280.962882 55
40 374 3443 56
4004 57
4565 58
5127 59
965689 60
505
1
0.694658
4868
5077
5286
5495
1
234
5.695704
5913
6122
6330
6539
6
7
8
9
I~~~~ ~
11
12
13
14
10.696748
6956 Corr.
7165 1055
35
7374 15 52
7582 20 69
15.697790 30 104
16 799940 139
17 8207 45 156
8415 50 174
8623
18
19
20.698832
21
9040
22 9248
9456
9663
25.699871
26.700079
23
24
27
28
29
≈≈ 8.~m~ 68M08 200
31
30.700909
||17
1324
1531
1739
32
33
34
35.701946
36
2153
37 2360
38 2567
39
2774
46
47
SINE
48
49
ggg8 ཀྱ8g
40.702981
41
42
43 3601 15
444
3188 "Corr.
3395 10 34
52
3808 20 69
45.704015 30 103
4221 40 138
4428 45 155
4634 50 172
4841
50.705047
5253
5459
51
52
53
54
0287
0494
0702
56
57
58
55.706078
6284
6489
6695
6901
60.707107
59
5666
5872
VERSINES, EXSECANTS, AND TANGENTS
44°

ST
CORR.
FOR
SEC.
+
COSINE
719340
9138
8936
8733
8531
.718329
8126
7924
7721
7519
.717316
" Corr.
7113
691110 34
6708 15 51
6505 20 68
|.716302 30 101
6099 40 135
5896 45 152
.715286
5083
4880
5693 50 169
5490
4676
4473
.714269
4066
3862
3658
3454
.713250
3046
2843
2638
2434
.712230
2026
1822
1617
1413
CORR.
FOR
SEC.
+
.70916!
8956
8750
8545
8340
" Corr.
.711209
1004
0800 10 34
059515 51
0390 20 68
.710185 30 102
.709981 40 137
9776 45 154
9571 50 171
9366
.708134
7929
7724
7518
7312
707107
VERSINE
"
280660 .390164
0862 0554 Corr.
0945 10 65
1337 15 98
1728 20 131
1064
1267
1469
1874
2076
2279
2481
.281671.392120 30 196
2513 40 262
2905 45 294
3298 50 327
3692
.282684.394086
2887
3089
3292
3495
3901
4104
4307
4510
EXSEC
.283698.396059 30 198
6455 40 264
685145 297
7248 50 329
7644
.284714.398042
4917
5/20
5324
5527
5934
6138
6342
6546
8439 " Corr.
8837 10 67
9235 15 100
9634 20 133
.285731.400032 30 200
0432 40 266
0831 45 299
123150 333
1632
11
4480 Corr.
4874 10 66
5269 15 99
5664 20 132
.286750.402032
6954
7157
7362
7566
.289815
.290019
0224
0429
0634
CORR.
FOR
SEC.
+
.287770
7974
8178
8383
8587 5653
.290839
1044
1250
1455
1660
.291866
2071
2276
2482
2688
.292893
.288791.406057
8996
9200
9405
9610
#1
2433 Corr.
2834 10 67
3236 15 101
3638 20 134
404040 30 201
4443 40 268
4846 45 302
5249 50 336
6462
" Corr.
6866 10 68
7272 15 102
7677 20 135
.408083 30 203
8489 40 271
8896 45 305
9303 50 338
9710
410118
0526
" Corr.
0934 10 68
1343 15 102
1752 20 137
412161 30 205
2571 40 273
2981 45 307
3392 50 341
3802
414214
CORR.
FOR
SEC.
+
"
Corr.
10 94
15 141
20 188
30 282
40 376
45 423
50 470
11
Corr.
10 94
15 142
20 189
130 284
40 378
45 425
50 473
" Corr.
10 95
15 143
20 190
11
Corr.
10 96
15 143
20 191
TANGENT 1
965689
6251
6814
7377
7940
11
Corr.
10 96
15 144
20 192
968504
" Corr.
10 97
15 145
20 193
0-234 56789
9067 6
9632
.970196
0761
30 285.979842 25
40 380 .980413 26
45 428 098327
50 475
1554 28
2126 29
.971326 10
1892 11
2458 12
302413
3590 14
.974157 15
4724 16
5291 17
5859 18
6427 19
.976996 20
756421
8133 22
8703 23
9272 24
30 287.985560 | 35
40 382 613436
45 430 6708 37
50 478
7282 38
7857 39
.982697 30
3269 31
3842 32
4414 33
4987 34
.988432 40
9007 41
9582 42
.990158 43
0735 44
30 288.99131145
40 385
45 433
50 481
188846
2465 47
3043 48
3621 49
.994199 | 50
4778 51
5357 52
5936 53
6515 54
30 290.997095 55
40 387
45 435
50 484
7676 56
8256 57
883758
9418 59
1.00000 60
506
0.707107
0-23➡
56780 0-2MI 5O789
5.708134
9
||
10.709161
12
13
14
11
9366 Corr.
9571 10 34
9776 15 51
9981 20 68
15.710185 30 102
0390 40 137
0595 45 154
0800 50 171
1004
16
17
18
19
~~~~~ 22~°2 2~~~~
21
20.711209
234
22
23
24
26
27
28
29
25.712230
2434
2638
2843
3046
31
32
33
34
30.713250
3454
3658
3862
4066
36
37
8 66000 gggg 55555 £5555 www.
38
35.714269
4473
4676
4880
5083
39
41
42
43
SINE
40.715286
46
7312
7518
7724
7929
47
48
49
8340
8545
8750
8956
51
52
45.716302 30 101
6505 40 135
6708 45 152
6911 50 169
7113
53
54
1413
1617
1822
2026
50.717316
75191
7721
7924
8126
56
57
58
59
55.718329
8531
8733
8936
9138
60.719340
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
"
5490 Corr.
5693 10 34
5896 15 51
6099 20 68
COSINE
.707107
6901
6695
6489
6284
.706078
5872
5666
5459
5253
.702981
2774
2567
2360
2153
.705047
4841
" Corr.
4634 10 34
4428 15 52
422 20 69
704015 30 103
3808 40 138
360145 155
3395 50 172
3188
701946
1739
1531
1324
1117
.700909
0702
0494
0287
0079
.696748
6539
6330
CORR.
FOR
SEC.
.699871
9663
9456
9248
9040
.698832
" Corr.
8623
8415 1035
8207 15 52
7999 20 69
697790 30 104
7582 40 139
7374 45 156
7165 50 174
6956
6122
5913
+
.695704
5495
5286
5077
4868
.694658
45°

VERSINE
.292893.414214
3099
3305
3511
3716
EXSEC
"
4625 Corr.
5037 10 69
.293922.416275 30 207
6688 40 276
7102 45 310
7516 50 344
4128
4334
4541
4747 7931
5449 15 103
5862 20 138
8261
8469
8676
8883
.294953.418345
5366
5572
5159 8760 " Corr.
9176 TO 69
9592 15 104
5779.420008 20 139
.295985.420425 30 208
6192 0842 40 278
6399 1259 45 313
1677 50 347
2095
6605
6812
.297019.422513
7226
7433
7640
7847
CORR.
FOR
SEC.
+
.298054.424611 30 210
5032 40 280
5453 45 316
5874 50 351
6296
1:
2932 Corr.
3351 10 70
3771 15 105
4191 20 140
.299091.426718
9298
9506
9713
9921
3461
3670
3878
4087
#!
7141 Corr.
7564 10 71
7987 15 106
8410 20 141
.300129.428834 30 212
0337 9259 40 283
0544 9684 45 318
0752.430109 50 354
0960 0534
301168.430960
1377
1585
1793
2001
1386 Corr.
#!
5669 Corr.
6100 10 72
6530 15 108
6962 20 144
.304296.437393 30 216
7825 40 288
4505
4714
4923
5132
.305342.439556
CORR.
FOR
SEC.
+
8257 45 324
8690 50 360
9123
" Corr
то то
15 15
20 20
=
#1
1813 10 71 TO 10
2240 15 107
2667 20 143
.302210.433095 30 214
2418
3523 40 285
3952 45 321
45 45
2626
2835
4380 50 357 || 50 50
3044 4810
.303252.435239
30
40
45
50
"
29
39
44
49
20
242 8998
898
Corr.
15 15
20
30
40
30
40
TANGENT "
I
1.00000 0
.00058
.00116 2
.00175 3
.00233 4
5
1.00291
.00350 6
.00408 7
.00467 8
.00525 9
1.00583 10
.00642 11
.00701 12
.00759 13
.00818 14
1.00876 15
.00935 16
.00994 17
.01053 18
.01112 19
1.01170 20
.01229 21
.01288 22
.01347 23
.01406 24
1.01465 25
.01524 26
.01583 27
.01642 28
.01702 29
1.01761 30
.01820 31
.01879 32
.01939 33
.01998 34
1.02057 35
.02117 36
.02176 37
.02236 38
.02295 39
1.02355 40
.02414 41
.02474 42
.02533 43
.02593 44
1.02653 45
.02713 46
.02772 47
.02832 48
.02892 49
1.02952 50
.03012 51
.03072 52
.03132 53
.0319254
1.03252 55
.03312 56
.03372 57
.03433 58
.0349359
1.0355360
507
0.719340
0-23A BONBO
6
4.720148
5.720349
0551
0753
0954
1156
8
9
10.721357
||
K
12
I23#
13
1559 Corr.
1760 1034
1962 15 50
2163 20 67
15.722364 30 101
2565 40 134
2766 45 151
2967 50 168
3168
14
16:
5678σ
17
18
19
0-23A
20.723369
~~~~~ ~~~~~ 8~~~7
21
22
23
24
26
27
28
29
25.724372
4573
4773
4974
5174
32
33
34
30.725374
5575
5775
5975
6175
36
37
38
SINE
लवन ने ले ले ले
1234
39
9542
9744
9946
35.726375
6575
6774
41
42
43
44
3570
3770
40.727374
3971
4172
46
47
48
49
56
57
58
59
6974
7174
50.729367
51 9566
52 9765
53 9964
54.730162
55.730361
45.728371 30 100
8570 40 133
8770 45 50
8969 50 166
9168
VERSINES, EXSECANTS, AND TANGENTS
46°

CORR.
FOR
SEC.
+
"
ST
11
7573 Corr.
7773 10 33
797215 50
8172 20 66
0560
0758
0957
1155
60.731354
COSINE
.694658
4449
4240
4030
3821
.69361!
3402
3192
2982
2773
11
"
692563
2353 Corr.
2143 10 35
193315 53
1723 20 70
691513 30 105
1303 40 140
1093 45 158
088250 175
0672
.690462
0251
0041
.689830
9620
.689409
9198
8987
8776
8566
1.688355
8144
7932
7721
7510
.687299
7088
6876
6665
6453
CORR.
FOR
SEC.
Tr
1.686242
6030 Corr.
5818 10 35
5607 15 53
5395 20 71
.685183 30 106
497140 141
4759 45 159
4547 50 176
4335
.684123
3911
3698
3486
3274
.683061
2849
2636
2424
2211
1.681998
CORR.
FOR
VERSINE
SEC.
+
" Corr.
.305342 ||.439556
5551 9990
5760.4404251073
0859 15 109
1294 20 145
5970
6179
.306389.441730 30 218
2165 40 290
260145 327
3038 50 363
3475
6598
6808
7018
7227
.307437.443912
7647
7857
8067
8277
EXSEC
#1
4350 Corr.
4788 10 73
522615 110
5665 20 147
446104 30 220
6544 40 293
8907 6984 45 330
9118 7424 50 366
9328 7865
.309538.448306
.308487
8697
11
8748 Corr.
9190 10 74
9632 15 111
0380.450075 20 148
9749
9959
.310170
.310591.450518 30 222
0802 0962 40 296
1013
1224
1406 45 333
1850 50 370
2295
1434
.311645.452740
1856
2068
2279
2490
2912
3124
3335
3547
.312701.454971 30 224
5419 40 298
5867 45 336
6315 50 373
3185 "Corr.
3631 10 75
4077 15 112
4524 20 149
5453
5665
.315877.461726
6089
6302
6514
6726
6764
.313758 .457213
3970
4182
1!
7662 Corr. Corr.
811210 75 10 10
8562 15 113
9013 20 150
.314817 ||.459464 30 226
5029
4393
4605
15 15
20 21
30
31
9916 40 301 140 41
5241.460368 45 339 45 46
0820 50 376
1273
50 52
It
#!
2179 Corr.
2633 10 76
308815 114
3542 20 152
.316939.463997 30 228
445340 304
4909 45 342
5365 50 380
5822
7151
7364
7576
7789
.318002.466279
CORR.
FOR
SEC.
+
"Corr.
ΤΟ TO
15 15
20 20
30
30
40
41
45 46
50 51
8558 Col
TANGENT
1.03553
.03613
.03674
.03734
.03794
1.03855
.03915
1
0123
6
7
.03976
.04036 8
.04097 9
1.04158 10
.04218 11
.04279 12
.04340 13
.0440114
1.0446 15
.04522 16
.04583 17
.04644 18
.04705 19
1.04766 20
.04827 21
.04888 22
.04949 23
.050 10 24
1.05072 25
.05133 26
.05194 27
.05255 28
.05317 29
1.05378 30
.05439 31
.05501 32
.0556233
.05624 34
1.05685 35
.05747❘ 36
.05209 37
.05870 38
.05932 39
1.05994 40
.0605641
.0611742
.06179 43
.06241 44
1.0630345
.06365 | 46
.06427 47
.06489 48
.0655 49
1.06613 50
.06676 51
.06738 52
.06800 53
.06862 54
1.06925 55
.06987 56
.07049 57
.07112 58
.07174 59
1.07237 60
508
0.731354
01234 567BM OILMI BONEG
8
5.732345
9
12
13
10.733334
14
16
17
18
19
15.734322 30
22222
21
•234 5ØN
22
20.735309
23
24
8 88000 20000 55555 55±5 wwwww wwwww N N N N N
26
27
28
25.736294
36
37
38
39
41
30.737277
7474
7670
7867
8063
42
43
SINE
35.738259
8455
8652
8848
9044
46
1552
1750
1949
2147
47
48
49
2543
2741
2939
3137
51
52
40.739239
9435
"Corr.
96311033
9827 15 49
44.740022 20 65
53
54
11
3532 Corr.
3730 10 33
3928 15 49
4125 20 66
56
5506
5703
5900
6097
45.740218 30 98
041440 130
0609 45 147
0805 50 163
1000
57
4520 40 132
471745 148
491550 165
5112
50.741195
1390
1586
1781
1976
58
59
6491
6688
6884
708!
55.74217|
2366
256 I
2755
2950
CORR.
FOR
SEC.
+
TABLE XX.-NATURAL SINES, COSINES,
60.743145
COSINE
.681998
1786
1573
1360
1147
99.67880130 107
8587 40 142
.680934
0721
0508
0295
11
0081
.679868
9655 Corr.
9441 10 36
9228 15 53
9014 20 71
.677732
7518
7304
7090
6876
8373 45 160
8160 50 178
7946
.676662
6448
6233
6019
5805
|.675590
5376
5161
4947
4732
.674517
4302
4088
3873
3658
.673443
CORR.
FOR
SEC.
+
"Corr
3228
301210 36
2797 15 54
2582 20 72
.672367 30 108
2152 40 144
.671290
1074
0858
0642
0427
193645 161
172150 179
1505
.670211
.669995
9779
9563
9347
.669131
47°

VERSINE
.318002.466279
8214
8427
8640
8853
9279
9492
9705
.319066.468571 30 230
9031 40 306
9491 45 345
995 50 383
0345
0559
0772
0986
9919.470412
320132.470874
EXSEC
1413
1627
1840
2054
11
6737 Corr.
7195 10 77
7653 15 115
8112 20 153
3552
3767
3981
4195
.322268.475510
2482
2696
2910
3124
11
1335 Corr.
1798 10 79
2260 15 116
2723 20 155
5698
5912
6127
6342
CORR.
FOR
SEC.
+
323338.477843 30 234
831 40 312
40
.321199.473186 30 232 30
3650 40 309
4114 45 348 45
4579 50 386
5044
IT
5975 Corr.
6442 10 78
6908 15 117
7376 20 156
.324410.480187
4624
4839
5053
5268
7848
8064
8279
8495
8780 45 351
9248 50 390
9718
.325483.482542 30 236
301440 315
3487 45 354
3960 50 393
4433
15
0657 Corr.
1128 10 79
1599 15 118
2070 20 157
326557.484907
6772
6988
7203
7418
.327633.487283 30 238
CORR.
FOR
SEC.
+
.328710.489670
8926.490149
" Cor
" Corr.
off.
140 8990
#t
Corr.
9142 0628 10 80
9358
1108 15 120
1588 20 160
9573
.329789.492068 30 240
330005
0221
2549 40 320
3030 45 361
3512 50 401
0437
0653
3994
.330869.494476
15
20
16
aa
21
7760 40 318 40
8237 45 357 45
8714 50 397 50
9192
445
#t
JT
5382 Corr
5856 TO 79 10
6332 15 119 15 16
6807 20 159
20 21
227
321.08179 15
.08243 16
.08306 17
.08369 18
.08432 19
1.08496 20
.08559 21
.08622 22
.08686 23
.08749 24
42
50 53
47
Corr.
π
30 32
32 5 45 53
43
TANGENT
1.07237 0
.07299
.07362
.07425
2
3
.07487 4
48
5
1.07550
.07613 6
.07676 7
.07738
.07801 9
8
1.07864 10
.07927 ||
.07990 12
.08053 13
.08116 14
1.08813 25
.08876 26
.08940 27
.09003 28
.09067 29
1.09131 30
.09195 31
.09258 32
.09322 33
.09386 34
1.09450 35
.09514 36
.09578 37
.09642 38
.09706 39
1.09770 40
.09834| 41
.0989942
.09963 43
.10027 44
1.1009145
.1015646
.1022047
.10285 48
. 10349 49
1.10414 50
.10478 51
.10543 52
.10608 53
.10672 54
1.10737 55
. 10802 56
.10867 57
.1093 58
.10996 59
1.11061 60
509
0.743145
3339
3534
3728
3923
I
234
5.744117
5678 ✪
6
8
9
|
10.745088
234
12
13
14
DONDO ON22 20702 87~♡~ *❤~88 22
16
17
18
19
15.746057 30
6251 40 129
6445 45 145
6638 50 161
6832
21
20.747025
7218
7412
26
23 7605
24
7798
28
29
25.747991
8184
8377
8570
8763
31
32
30.748956
9148
9341
33
34
37
38
39
35.749919
36.75011|
441
42
SINE
43
4444
46
47
48
49
4312
4506
4700
4894
40.750880
9599 85887 BONGO 8
51
52
5282 " Corr.
5476 1032
5670 15 48
5864 20 65
53
54
56
50.752798
57
9534
9726
64
45.751840 30 96
2032 40 128
2223 45 144
2415 50 160
2606
58
59
0303
0496
0688
55.753755
VERSINES, EXSECANTS, AND TANGENTS
48°

2989
3181
3372
3563
CORR.
FOR
SEC.
+
1072" Corr.
12641 10
32
1456 15
1648 20
3946
4137
4328
4519
60.754710
COSINE
.669131
8914
8698
8482
8266
1.668049
7833
7616
7399
7183
97.665882 30 108
5665 40 145
5448 45 163
5230 50 181
5013
.666966
" Corr
6749
6532 10 36
6316 15 54
6099 20 72
.664796
4578
4361
4144
3926
.663709
3491
3273
3056
2838
.662620
2402
2184
1966
1748
.661530
1312
1094
0875
0657
.660439
0220
.658252
8033
7814
CORR.
FOR
SEC.
7594
7375
.657156
6937
6717
6498
6278
.656059
ม
VERSINE
330869.494476
1086
1302
1518
1734
EXSEC
.331951.496896 30 243
2167
2384
7381 40 323
7867 45 364
2601 8353 50 404
2817 8840
333034.499327
3251
It
9814 Corr.
3468.500302 10 82
3684 0790 15 122
3901 1279 20 163
8688
8906
9125
9343
.33956.514145
9780
9998
48.659783 15 55.340217
9564 20 73
Corr.
0002 10 36
.659346 30 109
9127 40 146
8908 45 164
8690 50 182
8471
"
4960 Corr.
5443 10 81
5927 15 121
641 20 162
.335204.504221
5422
5639
5856
6074
6509
6727
6944
7162
.334118.501768 30 245
4335 2258 40 326
2748 45 367
3239 50 408 | 50
3730
4552
45
4770
4987
.336291.506685 30 247
7179 40 329
7674 45 371
8169 50 412
8664
.337380.509160
7598
CORR.
FOR
SEC.
+
4713 Corr.
5205 10 82
5698 15 124
6192 20 165
1310
1529
9657 " Corr.
7816.510154 TO
10 83
8034 0651 15 125
1149 20 166
.338470.511647 30 249
8252
2146 40 332
2645 45 374
3145 50 416
3645
2406
2625
n
.341748.519176
1967
" Corr.
11
9682 Corr.
2186.520188 TO
85
0694 15 127
1201 20 169
3063
3283
3502
3722
.343941.524253
CORR.
FOR
SEC.
+
4646
5148 10
10 84
5650 15 126
15
0436 6152 20 168 ||20
.340654.516655 30 252
0873 7158 40 336
7662 45 378
8166 50 419
8671
1092
.342844.521709 30 254
2217 40 339
2725 45 381
3234 50 423
3743
11
TO
15
20
Corr.
"I
16
22
30
40
"Corr.
IT
17
22
33
44
49
55
엉​태영 ​심히
​30 33
40 45
45 50
50 56
TANGENT
1.11061 0
11126 1
.11191 2
·
?
.11256 3
.11321 4
5
1.11387
.11452 6
.11517 7
.11582 8
.11648 9
1.11713 10
.11778 ||
.11844 12
.11909 13
.11975 14
1.1204115
.12106 16
.12172 17
. 12238 18
.12303 19
1.12369 20
.12435 21
.1250122
. 12567 23
.12633 24
1.12699 25
12765 26
❤
. 12831 27
.12897 28
.12963 29
1.13029 30
.13096 31
.13162 32
. 13228 33
. 13295 34
1.13361 35
.13428-36
. 13494 37
.13561 38
.13627 39
1.1369440
. 13761 41
. 13828 42
.13894 43
.1396144
1.14028 45
.14095 46
.1416247
.14229 48
.14296 49
1.14363 50
.14430 51
.14498 52
.14565 53
.14632 54
1.1469955
.14767 56
14834 57
.14902 58
.14969 59
1. 15037 60
•
510
0.754710
4900
5091
-UMI BOZOK DIET CONDOM2 20702 87~~7 ❤❤♪❤.
234
2
3
4
6
5.755663
5854
6044
6234
6425
8
9
12
13
14
16
10.756615
6805 Corr.
6995 TO 32
718515 47
7375 20 63
17
18
19
15.757565 30
7755 40 127
7945 45 142
8134 50 158
8324
20.758514
21
8703
22 8893
23
24
26
25.75946 |
29
9650
9839
28.760028
0217
31
32
30.760406
33
34
36
37
38
35.761350
39
SINE
5282
5472
41
42
43
44
46
47
48
gགླ73་ྒུ གྲུཤྩསྒྱུ རྐྱོཆོ
40.762292
49
51
9082
9271
52
2480 " Corr.
2668 10 31
2856 15 47
3044 20 63
45.763232 30 94
3420 40 125
3608 45 141
3796 50 157
3984
53
54
50.764171
4359
4546
4734
4921
56
0595
0784
0972
1161
57
58
59
1538
1727
1915
2104
55.765109
5296
5483
5670
5857
60.766044
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
COSINE
.656059
5840
5620
5400
5180
.654961
4741
4521
4301
4081
.653861
" Corr.
3641
3421 10 37
3200 15 55
2980 20 73
95.652760 30 110
2539 40 147
2319 45 165
2098 50 184
1878
.651657
1437
1216
0995
0774
.650553
0332
0111
.649890
9669
.649448
9227
9006
8784
8563
.648341
8120
7898
7677
7455
CORR.
FOR
SEC.
645013
4791
4568
4346
4124
+
.647233
" Corr
7012
6790 10 37
656815 55
6346 20 74
.646124 30 111
5902 40 148
5680 45 166
5458 50 185
5236
.643901
3678
3456
3233
3010
642788
49°

VERSINE
.343941.524253
4160
4380
4600
4820
5259
5479
.345039.526809 30 256
7322 40 342
7835 45 384
8349 50 427
5919 8863
5699
6800
7020
EXSEC
.346139.529377
4763
" Corr.
5274 10 85
9668
9889
.350110
0331
578515 128
6297 20 171
6359 9892 Corr.
6579.530408 1086
.348343.534549
8563
8784
9005
9226
.347240 ||.531957 30 259
7461 2475 40 345
2992 45 388
351 50 431
4030
7681
7902
8122
1216
1437
11
Corr.
10 π
092415 129 15 17
1440 20 172 20 23
1880
2102
2323
2545
CORR.
FOR
SEC.
+
350552.539769
.349447.537153 30 261
7675 40 348
8198 45 392
8721 50 435
9245
11
6322
6544
5069 Corr.
5589 TO 87
6110 15 131
663 20 174
0773.540294 " Corr.
0994 0819 TO 88
"
11
.351659.542397 30 264
2924 40 351
3452 45 395
3980 50 439
4509
354987.550356
5209
5432
5654
5876
134415 132
1871 20 176
6767
6990
.357212.555724
.352767.545038
11
tt
2988
3210
5567 Corr. Corr.
6097 10 89 10 12
3432 662815 133 15 17
3654 7159 20 177 20 23
.353876.547691 30 266
8223 40 355
4320 8755 45 399
4542
9288 50 443
4764 9822
4098
#1
0890 Corr.
1425 10 89
196115 134
2497 20 179
.356099.553034 30 269
3571 40 358
4108 45 403
4646 50 448
5185
CORR.
FOR
SEC.
+
30
34
40 46
45
50
30
40
45
50
51
57
35
46
52
58
TANGENT
1.15037
.15104
.15172
.15240
.15308
1.15375
.15443
.15511
.15579
.15647
01234 56789
4
1.1571510
.15783
.15851 12
.15919 13
.15987 14
1.1605615
.16124 16
. 16192 17
.16261 18
.16329 19
•
1.16398 20
.16466 21
.16535 22
. 16603 23
.16672 24
1.1674125
. 16809 26
.1687827
.16947 28
.17016 29
1.17085 30
.1715431
.17223 32
.17292 33
.17361 34
1.17430 35
.17500 36
.17569 37
.17638 38
.17708 39
1.17777 40
.17846 41
.17916 42
.17986 43
.18055 44
1.18125 45
.18194 46
.18264 47
.18334 48
.18404 49
1.18474 50
.18544 51
.18614 52
.1868453
.18754 54
1.18824 55
.18894 56
.1896457
.1903558
.19105 59
1.19175 60
511
0.766044
6231
6418
6605
6792
0-234
5.766978
66789 0-2M7 DO780
||
10.767911
12
13
141
16
17
18
19
15.768842 30
~~~~~ 222
V37
21
23
24
20.769771
9957
22.770142
0328
0513
26
27
28
29
25 .770699
ommm2 6088 FIRE 29500 850MJ HONO
32
33
34
36
30.771625
1810
1994
2179
2364
37
38
39
SINE
41
35.772549
2734
2918
3103
3287
42
43
44
7165
7352
7538
7725
40.773472
46
47
48
49
8097 Corr.
8284 1031
8470 15
8656 20 62
51
52
53
54
0884
1069
1254
1440
45.774393
56
57
58
9028 40 124
921445 139
9400 50 155
9585
50.775312
5496
5679
5863
6046
11
VERSINES, EXSECANTS, AND TANGENTS
50°

CORR.
FOR
SEC.
+
55.776230
6413
6596
6780
6963
60.777146
11
3656 Corr.
3840 TO 31
4024 15 46
4209 20 61
30
4577 40 123
4761 45 138
4944 50 153
5128
COSINE
.642788
2565
2342
2119
1896
.641673
1450
1226
1003
0780
.640557
46.63988615
0333 " Corr.
0110 TO 37
93.639439 30 112
921540 149
8992 45 168
8768 50 186
8544
56
9663 20 75
.638320
8096
7872
7648
7424
.637200
6976
6751
6527
6303
CORR.
FOR
SEC.
.636078
5854
5629
5405
5180
.634955
4730
4506
4281
4056
+.357212.555724
7435
7658
" Corr.
.633831
3606
3381 10 38
3156 15 56
293 20 75
92.632705 30 113
2480 40 150
2255 45 169
2029 50 188
1804
.631578
1353
1127
0902
0676
.630450
0224
.629998
9772
9546
.629320
VERSINE
11
6263 Corr.
6804 10 90
734415 136
7881
8104 7885 20 181
8550
8774
9220
.358327.558427 30 271
8969 40 361
9512 45 407
8997.560055 50 452
0598
EXSEC
.359443.56|142
9667
9890
.360114
0337
.36056.563871 30 27430
0785 4418 40 365
4966 45 410
5514 50 456
6063
1008
1232
1456
.361680.566612
1904
2128
2352
2576
3473
3697
.362800.569366 30 276
3024
9919 40 368
3249.570472 45 414
1025 50 460
1579
CORR.
FOR
SEC.
+
.363922.572134
5270
5494
"
1687 Corr.
2232 TO 91
2778 15 137
15
3324 20 182 20
5719
5944
"
7162 Corr.
7712 10 92
8263 15 138
8814 20 184
"
4146 2689 Corr.
4371
3244 TO 93
4595
3800 15 139
4820
4357 20 186
365045.574914 30 279
5472 40 372
6030 45 418
6589 50 465
7148
7520
7745
7971
8196
.366169.577708
11
6394 8268 Corr.
6619 8829 10 94
6844 9390 15 141
7069 9952 20 188
.367295.580515 30 281
1078 40 375
1641 45 422
2205 50 469
2770
.368422.583335
11
8647 3900 Corr.
8873 4467 10 95
9098
5033 15 142
9324 5601 20 189
.369550.586168 30 284
6737 40 379
7306 45 426
7875 50 474
8445
9776
.370002
0228
0454
370680.589016
CORR.
FOR
SEC.
+
11
Corr.
10 12
18
24
40
45
50
=
36
47
53
59
Corr.
TO 12
15 18
20 24
30 36
40 48
45 55
50 61
TANGENT
1.19175
.19246
.19316
.19387
.19457
1
01234
1.19528
6
.19599
.19669 7
.19740
.19811
5878 O
I
1.19882 10
.19953 1
.20024 12
.20095 13
.20166 14
1.20237 15
.20308 16
.20379 17
.2045 18
.20522 19
1.20593 20
.20665 21
.20736 22
.20808 23
.20879 24
1.20951 25
.21023 26
.21094 27
.21166 28
.21238 29
1.21310 30
.21382 31
.21454 32
.21526 33
.21598 34
1.21670 35
.21742 38
.21814 37
.21886 38
.21959 39
1.2203140
.22104 41
.22176 42
.22249 43
.2232141
1.22394 45
.22467 46
.2253947
.2261248
.22685 49
1.22758 50
.2283 51
.2290452
.22977 53
.23050 54
1.23123 55
.23196 56
.23270 57
.23343 58
.23416 59
1.23490 60
512
0.777146
7329
7512
7695
7878
0-234 BOZBO
5.778060
6
8
||
12
13
14
10.778973
234
17
18
19
22222
21
15.779884 30 91
16.780066 40 121
0248 45 137
043050 152
0612
20.780794
0976
1157
1339
1520
23
22 20700 Am❤❤7 ❤❤MMA DERE DE
24
26
28
29
25.781702
1883
2065
2246
2427
31
32
30.782608
2789
2970
3151
3332
33
34
36
37
38
39
35.783513
3694
3874
4055
4235
SINE
41
42
43
44
40.784416
8243
8426
8608
8791
46
47
48
49
g g⌘7 ཀྱྰཀྨng
51
JI
9156 Corr.
9338 10 30
9520 15 46
9702 20 61
45.785317 30
52
53
54
50.786216
6396
6576
6756
6935
56
57
58
59
CORR.
FOR
SEC.
+
55.787114
7294
7473
7652
7832
TABLE XX.-NATURAL SINES, COSINES,
5497 40 120
5677 45 135
5857 50 150
6037
60.788011
COSINE
.629320
9094
8868
8642
8416
.628189
7963
7737
7510
7284
.624788
456||
4334
4107
3880
11
.627057
6830 Corr.
6604 10 38
6377 15 57
615020 76
.625924 30 113
5697 40 151
5470 45 170
5243 50 189
5016
.623652
3425
3197
2970
2742
.622515
2287
2059
1831
1604
.621376
1148
0920
0692
0464
4596
" Corr.
" Corr.
4776 1030.619779 10
4957 15 45 955115 57
5137 20 60 9322 20 76
.620236
0007
C
CORR.
FOR
SEC.
.617951
7722
7494
7265
7036
+
.616807
6578
6349
6120
5891
.615662
51°

VERSINE
.370680.589016
0906
1358
1584
9587 " Corr.
1132.590158 10 96
0731 15 143
1303 20 191
2037
2263
2490
2716
.371811.591877 30 287
2450 40 382
3025 45 430
3600 50 478
4175
EXSEC
.375212.600542
5439
5666
5893
6120
.372943.594751
st
3170
3396
3623
19
20
25
5328 Corr.
" Corr.
5905 TO 97 10 12
648215 145 15
3850 7061 20 193
.374076.597639 30 290
4303 8219 40 386
8799 45 434
9379 50 483
9960
4530
4757
4984
6575
6803
7030
7258
8169
8396
.376348.603458 30 292
4043 40 390
4628 45 439
5214 50 487
5801
.377485.606388
7713
7941
9308
9536
CORR.
FOR
SEC.
+
1124
1706 TO
2290 15 146
2873 20 195
1592
1820
" Corr.
97
.378624.609332 30 295
8852
9923 40 394
9080.610514 45 443
1106 50 492
1698
.379764.612291
6976
" Corr.
7564 10 98
8153 15 148
8742 20 197
9993
38.380221
0449
0678
90.619094 30 114.380906.615264 30 298
8866 40 152
5860 40 397
1134
1363
8637 45 171
8408 50 190
6457 45 447
705450 497
7652
8180
.382049.618251
#1
2278 8850 Corr.
2506 9450 TO 100
2735.620050 15 151
2964 065 20 201
3422
3651
3880
4109 3665
.384338.624269
.383193.621253 30 301
1855 40 401
2458 45 452
30650 502
#1
Corr.
2884 Corr.
3478 10 99 10 13
4073 15 149 15 19
4668 20 199 ||20 25
CORR.
FOR
SEC.
+
30
40 O
45
50
=
#1
37
50
56
62
30 38
40 51
45 57
50
63
TANGENT
T
1.23490 0
.23563
.23637 2
.23710 3
.23784 4
1.23858 5
.2393
.24005 7
6
.24079 8
.24153 9
1.24227 10
.2430||||
.24375 12
.24449 13
.24523 14
1.24597 15
.24672 16
.24746 17
.24820 18
.24895 19
1.24969 20
.25044 21
.25118 22
.25193 23
.25268 24
1.25343 25
.25417 26
.25492 27
.25567 28
.25642 29
1.2571730
.25792 31
.25867 32
.2594333
.2601834
1.26093 35
.26169 36
.26244 37
.26320 38
.26395 39
1.26471 40
.26546 41
.26622 42
.2669843
.26774 44
1.26849 45
.2692546
.27001 47
.27077 48
.27153 49
1.27230 50
.27306 51
.2738252
.27458 53
.27535 54
1.27611 55
.27688 56
.27764 57
.27841 58
.2791759
1.27994 60
513
1
0.788011
8190
8369
8548
8727
I
234
5.788905
9084
9263
9441
9620
5678 —
6
8
9
I
234
10.789798
9977 " Corr.
12.790155 10
30
033315 45
0512 20 59
15.790690 30 89
086840 119
1046 45 134
1224 50 148
1401
13
14
16
17
18
19
20.791579
1757
1934
2112
2290
22222 20722 87887 H8M88 222
21
23
24
25.792467
2644
2822
2999
3176
26
28
29
30.793353
3530
3707
3884
4061
31
32
33
34
35.794238
4415
4591
4768
4944
36
37
38
39
SINE
40.795121
41
42
43
44
46
47
48
49
45.796002 30 88
6178 40 117
6354 45 132
6530 50 147
6706
51
52
53
54
50.796882
7057
7233
7408
7584
56
57
58
ggགྨཐ8
55.797759
7935
8110
8285
8460
59
VERSINES, EXSECANTS, AND TANGENTS
52°

#1
5297 Corr.
5474 10 29
565015 44
5826 20 59
CORR.
FOR
SEC.
+
60.798636
COSINE
.615662
5432
5203
4974
4744
.614515
4285
4056
3826
3596
.613367
"
3137 Corr.
2907 10 38
2677 15 57
2447 20 77
.611067
0836
0606
0376
0145
.612217 30 115
1987 40 153
1757 45 173
1527 50 192
1297
.609915
9684
9454
9223
8992
.608761
8531
8300
8069
7838
.607607
7376
7145
6914
6682
CORR.
FOR
SEC.
+
.604136
3904
3672
3440
3208
.606451
6220
" Corr.
5988 10 39
58
575715
5526 20 77
605294 30 116
5062 40 154
4831 45 174
4599 50 193
4367
602976
2744
2512
2280
2047
.601815
VERSINE
.384338.624269
4568
4797
5026
5256
5715
5944
6174
6404
7323
7553
EXSEC
.385485.627300 30 304
7908 40 405
8517 45 456
9126 50 507
9736
386633.630346
6863
7093
4874 " Corr.
5480 10 101
0546
0777
1008
6086 15 152
6693 20 203
.388933.636483
9164
9394
9624
9855
2624
2855
3086
3318
CORR.
FOR
SEC.
+
0957 "Corr.
1569 10 102
218115 153 15
2794 20 205
20
.387783.633407 30 307 30
8013 4021 40 409 40 52
4636 45 460 45
525 50 512 50 65
5866
8243
8473
8703
390085.639574 30 310
0316.640194 40 413
0814 45 465
1435 50 517
2057
7100 " Corr.
7717 10 103
.391239.642680
1469
1700
1931
2162
4243
4474
8336 15 155
8954 20 207
3303 " Corr.
39261 10 104
392393.64580130 313
6427 40 417
705445 470
768 50 522
8309
455115 157
5175 20 209
.393549.648938
3780
"
9567 Corr.
4012.650197 10 105
0827 15 158
1458 20 211
394706.652090 30 316
4938
5169
5401
5633
.395864.655258
CORR.
FOR
SEC.
+
6096 5893" Corr.
6328 6529 10 106
6560
7166 15 160
7803 20 213
6792
" Corr.
10
397024.65844130 319
7256
9080 40 426
7488
9719 45 479
7720.660359 50 532
7953 0999
.398185.661640
10
GEET BO
" Corr.
២០០៥ ៩០។
13
15
ជូនដ
20
30
2722 40 421 40 53
3355 45 474 45 60
3988 50 527 50 66
4623
20
TANGENT
1
1.27994
.28071
.28148
.28225
.28302 4
0123A
1.28379 5
.28456 6
.28533 7
.28610 8
.28687 9
1.2876410
.2884211
.28919 12
.28997 13
.29074 14
1.29152 15
.29229 16
.29307 17
.29385 18
.29463 19
1.29541 20
.29618 21
.29696 22
.29775 23
.29853 24
1.29931 25
30009 26
.30087 27
.30166 28
.30244 29
1.30323 30
30401131
.30480 32
.30558 33
.30637 34
1.30716 35
.30795 36
.30873 37
.30952 38
.31031 39
1.31110 40
.31190 41
.31269 42
.3134843
26 .31427 44
40 1.31507 45
.31586 46
.31666 47
.3174548
.31825 49
1.31904 50
.31984 51
.32064 52
.32144 53
.32224 54
1.32304 55
32384 56
.32464 57
.3254458
.32624 59
1.3270460
514
0.798636
01234 5
6
5.799510
9685
9859
8.800034
0208
789
9
O-NET DOM 222 D*7** 87~♡~ HMMMM DI
10.800383
12
13
14
16
17
21
15.801254|30 87
1428 40 116
1602 45 131
1776 50 145
1950
23
24
20.802123
2297
2470
2644
2818
26
27
25.802991
3164
3338
3511
3684
28
29
31
32
30.803857
4030
4203
4376
4548
33
34
SINE
36
38
39
8810
8986
9160
9335
35.804721
4894
5066
5239
5411
41
42
43
44
49
OGODZ KONO. 8
0557 " Corr.
0731 10 29
0906 15 44
1080 20 58
40.805584
5756
" Corr.
5928 10 29
610015 43
6273 20 57
51
46
45.806445 30 86
661740 115
47 6788 45 129
48
6960 50 143
7132
52
53
54
50.807304
56
57
58
59
7475
7647
7818
7990
55.808161
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
8332
8504
8675
8846
60.809017
COSINE
.601815
1583
1350
1118
0885
.600653
0420
0188
.599955
9722
.597159
6925
6692
6458
6225
.595991
5758
5524
5290
5057
.594823
4589
4355
4121
3887
.593653
3419
3185
2950
2716
.590136
589901
CORR.
FOR
SEC.
.592482
2248 " Corr.
2013 10 39
1779 15 59
1544 20 78
|.591310|30 117
107540 156
0840 45 176
0606 150 195
0371
9666
9431
9196
+
.588961
8726
8491
8256
8021
587785
9580
.599489
9812
.400045
0278
.400511.668086
9256 " Corr. 0744 8734" Corr.
9024 10 39 0976 938310 108
879115 58
1209.670033 15 163
8558 20 78 1442 0683 20 217
1.598325 30 117.401675.671334 30 326
8092 40 155
1908 1985 40 434
2142 2637 45 489
2375 3290 50 543
2608 3943
7858 45 175
7625 50 194
7392
53°

VERSINE
.398185.661640
8417
8650
8882
9115
EXSEC
.399347.66485530 322
5500 40 430
6146 45 484
6792 50 537
7439
"
2282 Corr.
2924 TO 107
3567 15 161
421 20 215
.402841.674597
3075
3308
3542
3775
6581
6815
7050
7284
CORR.
FOR
SEC.
+
5252 " Corr.
590710 110
404009.677877 30 329
8535 40 438
4242
4476
4710
4943 .680512
6563 15 164
7220 20 219
.405177.681173
5411
5645
5879
6113
9193 45 493
9852 50 548
.406347.684486 30 332
5150 40 443
5816 45 498
648150 554
7148
1274
1509
1744
1834 " Corr.
2496 10 ||
315915 166
382220 222
.407518.687815
7752
7987
8221
.408690 .691161 30 336
8925 1833 40 447
9160 2504 45 503
3177 50 559
3850
9394
9629
.409864.694524
410099
0334
5199
" Corr.
5874 10 113
655015 170
0569
0804 7227 20 226
411039.697904 30 339
8582 40 452
9261 45 509
9941 50 565
8483
" Corr.
" Corr.
9152 10 112 10 14
982115 168 15 21
8456.69049120 224 20 28
1979.700621
.412215.701302
CORR.
FOR
SEC.
+
" Corr.
10
14
20
15
20
27
30 41
40
45
50
54
61
68
30 42
40 55
45 62
50 69
TANGENT 1
1.32704
.32785
.32865
.32946
.33026
0123♬
4
1.33107 5
.33188
.33268 7
.33349
.33430
9
5678
1.3351110
.33592|11
.33673 12
.33754 13
.3383514
1.3391615
.33998 16
.34079 17
.34160 18
.34242 19
1.34323 20
.34405 21
.34487 22
.34568 23
.34650 24
1.34732 25
.3481426
.34896 27
.34978 28
.35060 29
1.35142 30
.3522431
.35307 32
.35389 33
.3547234
1.35554 35
.35637 36
.35719 37
.35802 38
.35885 39
1.35968 40
.3605141
.36134 42
.36217 43
.3630044
1.3638345
.36466 46
36549 47
.36633 48
.36716 49
50
1.36800
.3688351
36967 52
.3705053
.37134 54
1.37218 55
.37302 56
.37386 57
.37470 58
.37554 59
1.37638 60
515
0.809017
01234 56789
5.809871
6.810042
7 0212
0383
0553
10.810723
"
1234
12
0894 Corr.
1064 10 28
1234 15 42
1404 20 57
15.811574 30 85
1744 40 113
1914 45 127
2084 50 142
2253
13
14
16
17
18
19
22222
20.812423
2592
2762
2931
3101
25.813270
3439
3608
3778
3947
21
23
24
12 207°° 27~37 ❤❤♪88 22
26
28
29
31
30.814116
4284
4453
33
34
36
35.814959
5128
37 5296
الا
38 5465
39
5633
42
40.81580I
43
44
SINE
46
47
9188
9359
9530
9700
སྤྱ⌘གྲུ⌘སྒྱུ ༄3གྲུ རྐྱོཆོ;
48
49
52
45.816642 30 84
6809 40 112
6977 45 126
7145 150 140
7312
53
54
4622
4791
50.817480
7648
7815
7982
8150
56
57
58
55.818317
8484
8651
8818
8985
60.819152
VERSINES, EXSECANTS, AND TANGENTS
54°

CORR.
FOR
SEC.
+
5970 " Corr.
6138 10 28
630615 42
6474 20 56
COSINE
587785
7550
7314
7079
6844
586608
6372
6137
5901
5665
.585429
5194
"Corr.
4958 10 39
4722 15 59
4486 20 79
584250 30 118
401440 157
.583069
2832
2596
2360
2123
3777 45 177
3541 50 197
3305
.581886
1650
1413
1176
0940
580703
0466
0229
579992
9755
.579518
9281
9044
8807
8570
CORR.
FOR
SEC.
.578332
8095
" Corr.
7858 10 40
7620 15 59
7383 20 79
.577145 30 119
6908 40 158
6670 45 178
6432 50 198
6195
575957
5719
5481
+
5243
5005
574767
4529
4291
4053
3815
1.573576
VERSINE
412215.701302
2450
2686
2921
3156
3628
3863
4099
4335
.413392.704716 30 342
5401 40 457
6087 45 514
6773 50 571
7460
7640
7877
EXSEC
.414571.708148
4806 8836 " Corr.
5042 9525 10 15
5278.710215 15 173
5514 0906 20 231
.415750.711597 30 346
5986 2289 40 461
2982 45 519
3675 50 577
4369
6223
6459
6695
.416931.715064
7168
7404
1983 " Corr.
2665 10 14
334815 171
4032 20 228
0719
0956
1193
1430
418114.718548 30 350
9248 40 466
9947 45 524
8824.720648 50 583
1349
8350
8587
9060
.419297.722051
9534
9771
.420008
0245
CORR.
FOR
SEC.
+
2380
2617
5759 " Corr.
645610 116
715215 175
7850 20 233
420482.725571 3C 353
6277 4C 471
6984 45 530
7692 50 588
8400
3092
3330
3568
3805
.421668.729110
1905
2753 "Corr.
3457 10 118
416115 177
4866 20 235
9820 "
ff Corr.
2142.730530 10 119
1241 15 178
1954 2C 238
5709
5947
6185
.426424
.422855.732666 30 357
3380 40 476
4094 45 535
4809 50 595
5525
.424043 .736241
4281
4519
4757
4995
6958 " Corr.
7676 10 120
8395 15 180
9114 20 240
.425233739835 30 360
5471.740556 40 480
1277 45 541
2000 50 601
2723
743447
CORR.
FOR
SEC.
+
" Corr.
14
IC
15
21
20 28
30 43
40 57
45 64
50 71
" Corr.
10 15
15 22
120 29
44
30
40
45 66
58
50 73
TANGENT
1.37638
37722
.37807
.37891
.37976
1.38C6C
0-234
4
5878α
.38145 6
.38229
.38314 8
.38399
1.38484 10
.38568 11
.3865312
.38738 13
.3882414
1.38909 15
.3899416
.39C79 17
.39165 18
.39250 19
9
1.39336 20
.39421 21
.39507 22
.39593 23
.39679 24
་
1.39764 25
.39850 26
.39936 27
.40022 28
.40109 29
1.4019530
.4028131
.40367 32
.40454 33
40540 34
1.40627 35
.40714 36
.40800 37
.40887 38
.40974 39
1.41061 40
.411484
.41235 42
.41322 43
.4140944
1.41497 45
.41584 46
.41672 47
.41759 48
.41847 49
•
1.41934 50
.42022 151
,42110 52
.42198 53
.4228654
1.42374 55
.42462 56
.42550 57
.42638 58
42726 59
1.42815 60
516
0.819152
0-23 =†
4
6678 σ
9
5.819985
6.820152
0318
0485
0651
01234 5
10.820817
11
12
13
0983 " Corr.
1149 TO 28
131515 41
14 1481 20 55
16
17
-- OUNCE DO700 87087 98588 DIRE
15.821647 30
18
21
22
20.822475
2640
2806
2971
3136
23
24
26
27
25.823302
3467
3632
3796
3961
28
31
30.824126
4291
4456
4620
4785
SINE
44
35.824949
5114
5278
5442
5606
9319
9486
9652
9819
40.825770
46
47
48
8 66466 200g 55:
49
51
52
45.826590 30 82
6753 40 109
6917 45 123
7081 50 136
7244
53
54
50.827407
7571
7734
1813 40 110
1978 45 124
2144 50 138
2310
56
57
58
59
55.828223
8386
8549
CORR.
FOR
SEC.
+
5934
" Corr.
6098 10 27
626215 41
6426 20 55
7897
8060
TABLE XX.-NATURAL SINES, COSINES,
8712
8875
60.829038
COSINE
.573576
3338
3100
2861
2623
.572384
2146
1907
1669
1430
571191
83.569997 30 119
9758 40 159
9519 45 179
9280 50 199
9040
.568801
8562
8322
8083
7844
0952 " Corr.
0714 10 39
0475 15
60
0236 20 80
.567604
7365
7125
6886
6646
[]
566406
6166
5927
5687
5447
|.565207
4967
4727
4487
4247
564007
CORR.
FOR
SEC.
+
3766 " Corr.
3526 10 40
3286 15 60
3045 20 80
.562805|30 120
2564 40 159
2324 45 180
2083 50 200
1843
561602
1361
1121
0880
0639
560398
0157
559916
9675
9434
.559193
55°

VERSINE
426424.743447
6662
6900
EXSEC
7854
8093
8331
8570
7139
7377 6350 20 243
427616.747078 30 364
7806 40 486
8535 45 546
9265 50 607
9996
4172 " Corr.
4897 10 121
5623 15 182
.428809.750727
9048
9286
9525
9764
1460 " Corr.
2192 TO 123
292615 184
3661 20 245
.430003.754396 30 368
C242 5132 40 491
0481 5869 45 552
6606 50 613
7345
0720
0960
2635
2875
3114
3354
.431199.758084
1438
1678
2156
.432396.761791 30 372
2534 40 496
3279 45 558
4024 50 620
4770
CORR.
FOR
SEC.
+
8824 "Corr.
9564 10 124
1917.760306 15 186
1048 20 248
.433594.765517
3834
4073
4313
4553
5273
5513
5753
6265" Corr.
7013 10 125
7762 15 188
8512 20 251
.434793.769263 30 376
5033.770015 40 501
0767 45 564
1520 50 626
2274
.435993.773029
6234
6474
6714
6955
3784 " Corr.
4541 10 127
529815 190
6056 20 253
.437195.776815 30 380
7436
7676
7574 40 506
8334 45 570
9096 50 633
9857
7917
8157
.438398.780620
8639
8879
9120
9361
1384
" Corr.
2148 10 128
2913 15 192
3679 20 256
439602.784446 30 384
521340 512
5982 45 576
675150 640
7521
9843
440084
0325
0566
.440807.788292
CORR.
FOR
SEC.
+
" Corr.
10 15
15 22
20
30
30
40
45
50
45
60
67
75
"Corr.
10 15
23
15
120
31
30 46
40 61
45 69
50 77
TANGENT
1.42815
.42903
.42992
.43080
.43169
·
1.43258
•
43347 6
43436 7
.43525 8
.43614 9
1.43703 10
•
.43792 11
.43881 12
►
OI234 SOT@α 0-234
43970 13
44060 14
1.4414915
44239 16
.4432917
.44418 18
.44508 19
5
•
1
1.44598 20
44688 21
.44778 22
44868 23
.44958 24
1.45048 25
.45139 26
.4522927
45320 28
.45410 29
1.45501 30
45592 31
.45682 32
.45773 33
.4586434
1.45955 35
.46046 36
46137 37
.46229 38
.46320 39
1.46411 40
.46503 41
.46595 42
.46686 43
.4677844
1.4687045
.46962 46
47054 47
.47146 48
.47238 49
1.47330 50
.47422 51
.47514 52
.4760753
.47699 54
•
1.47792 55
47885 56
.47977 57
48070 58
.48163 59
1.48256 60
•
517
01234
0.829038
56789 0-234 5
5.829850
6.830012
0174
0337
0499
||
10.830661
12
13
14
16
17
18
19
22222 202~~ 67087 68788 25221 99509 OZAMI HONOR
15.831470|30
23
24
20.832277
2438
2599
2760
2921
26
28
25.833082
3243
3404
3565
3725
31
32
33
34
36
30.833886
4046
4207
4367
4528
37
SINE
35.834688
4848
5008
38 5168
5328
41
42
43
44
9200
9363
9525
9688
46
40.835488
47
49
0823 " Corr.
0984 10 27
114615 40
1308 20
52
45.836286
53
54
1631 40 108
1793 45 121
1954 150 135
2116
56
58
59
50.837083
7242
7401
VERSINES, EXSECANTS, AND TANGENTS
56°

CORR.
FOR
SEC.
+
11
5648 Corr.
5807 10 27
5967 15 40
6127 20 53
55.837878
8036
8195
8354
8512
30 80
6446 40 106
6605 45 120
6764 50 133
6924
7560
7719
WNO GENE
60.838671
.559193
8952
8710
8469
8228
COSINE
.557986
7745
7504
7262
7021
.556779
6537 " Corr.
6296 10 40
605415 60
5812 20 81
.554360
4118
3876
3634
3392
.555570 30 121
5328 40 161
5086 45 181
4844 50 202
4602
.553149
2907
2664
2422
2180
1.551937
1694
1452
1209
0966
.550724
0481
0238
549995
9752
·
·
549509
CORR.
FOR
SEC.
9266
"Corr.
9023 10
41
8780 15 61
8536 20 81
1.547076
6833
6589
+
.548293 30 122
8050 40 162
7807 45 182
7563 50 203
7320
6346
6102
545858
5614
5371
5127
4883
.544639
VERSINE
2255
2496
2738
2979
1772
440807.788292
1048 9063 " Corr.
1290 983610 129
1531.790609 15 194
1383 20 259
442014.792158 30 388
2934 40 517
3710 45 582
4488 50 646
5266
EXSEC
3704
3946
.443221 ||.796045
3463 6825
" Corr.
7605 TO 31
8387 15 196
4188 9169 20 261
.444430 ||.799952 30 392
4672.800736 40 523
4914 152145 588
5156 2307 50 653
5398 3094
6366
6608
.445640 ||.803881
5882 4669 " Corr.
5458 TO 132.
6124
624815 198
7039 20 264
CORR.
FOR
SEC.
+
.446851.807830 30 396
8623 40 528
9416 45 594
7578.810210|50 660
1005
7C93
7336
7820
.448063.811801
8306
8548
8791
9034
9519
9762
.450005
0248
2598 "Corr.
3395 10 133
419415 200
4993 20 267
449276.815793 30 400
6594 40 534
739645 601
8198 50 667
9002
450491.819806
11
0734.820612 Corr.
0977 1418 10 135
1220 222515 202
1464 3033 20 270
.451707.823842 30 405
1950 4651 40 540
5462 45 607
6273 50 675
7085
2193
2437
2680
452924.827898" Corr.
3167 8712 10 136
3411 952715 205
3654.830343 20 273
3898
1160
454142.831977 30 409
2796 40 546
3615 45 614
4386
4629
4873
5117
.455361.836078
4435 50 682
5256
CORR.
FOR
SEC.
+
#1
Corr.
то 16
15 24
20
31
30
47
63
40
45 71
50 79
" Corr.
10 16
15 24
20
32
30
40
45
50
50
48
66
73
81
TANGENT
1.48256
.48349
.48442
.48536
.48629
1
01234
1.48722 5
.48816 6
.48909 7
.49003 8
.49097 9
1.49190 10
.49284||||
49378 12
.49472 13
49566 14
1.49661 15
.49755 16
49849 17
.49944 18
.50038 19
1.50133 20
.50228 21
.50322 22
.50417 23
.50512 24
1.50607 25
.50702 26
.50797 27
.50893 28
.50988 29
1.51084|30
.5117931
.51275 32
.51370 33
.51466 34
1.51562 35
.51658 36
.51754 37
.5185038
.51946 39
1.52043 40
.5213941
.52235 42
.5233243
.5242944
1.52525 45
.52622 46
.52719 47
.5281648
.52913 49
1.53010 50
.53107 51
.53205 52
.53302 53
.53400 54
1.53497 55
.53595 56
.53693 57
.5379158
.53888 59
1.53986 60
518
0.838671
OI232 BON∞O
6
5.839462
7
8
01234
9.840094
10.840251
II
12
13
14
" Corr.
0409
0567 10 26
072415 39
0882 20 52
15.841039 30 79
1196 40 105
1354 45 118
151150 131
1668
6678σ
16
17
18
19
OUR 00 87087 KOMM8 222 29700 65000 66
26
27
20.841825
1982
21
22
2139
23 2296
24
2452
28
25.842609
2766
2922
3079
3235
29
31
32
30.843391
3548
3704
3860
4016
33
34
36
37
35.844172
4328
4484
4640
4795
38
39
441
42
43
444
SINE
40.844951
46
47
48
49
8829
8987
9146
9304
0-23
9620
9778
9936
51
52
45.845728 30 78
5883 40 103
6038 45 116
6193 50 129
6348
50.846503
6658
6813
6967
7122
53
54
687888
56
57
58
55.847276
7431
7585
7740
7894
59
5106 " Corr.
5262 10 26
541715 39
5573 20 52
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
60.848048
COSINE
.544639
4395
4151
3907
3663
1.543419
3174
2930
2686
2442
.539751
9506
9261
9016
8771
TE
.542197
1953 Corr.
1708 10 41
146415 61
1219 20 82
540974 30 122
073040 163
0485 45 183
0240 50 204
.539996
.538526
8281
8035
7790
7545
.537300
7054
6809
6563
6318
536072
5827
5581
5336
5090
CORR.
FOR
SEC.
.534844
4598
" Corr.
4352 10 41
4106 15 61
3860 20 82
..532384
2138
1891
+
533614 30 123
3368 40 164
3122 45 185
2876 50 205
2630
1645
1399
.531152
0906
0659
0412
0166
1.529919
57°

VERSINE
.455361.836078
5605
5849
6093
6337
6826
7070
7314
7558
6901 Corr.
7725 10 138
855015 207
9375 20 276
456581.840202 30 414
1029 40 552
1857 45 621
2687 50 689
3517
.457803.844348
8047
8292
8536
8781
9760
.46C0C4
EXSEC
5180 "Corr.
6012 10 139
684615 209
768 20 279
459026 .848516 30 418
9270 9352 40 558
9515.850190 45 627
1028 50 697
1867
.460249.852707
0494
C739
0984
1229
1719
1965
2210
.461474.856922 30 423
7767 40 564
4173
4419
4664
491C
CORR.
FOR
SEC.
+
3548 " Corr.
4390 10 141
2455.860310
.462700 .861159
2946
3191
3437
3682
5648
5894
6140
5233 15 211
6077 20 282
8614 45 634
9461 50 705
.463928.865420 30 427
6275 40 570
7131 45 641
7988 50 712
8845
.468848
9094
9341
9588
9834
.470081
11
2009 Corr.
2860 10 142
371315 214
4566 20 285
.465156.869704
11
5402.870564 Corr.
1424 110 144
2286 15 216
3148 20 288
466386 .874012 30 432
6632 4876 40 576
6878 5742 45 648
7124 6608 50 720
7370 7476
.467616.878344
7862
8355
8601
9213 "Corr.
8109.880083 10145
C954 15 218
1827 20 291
882700 30 437
3574 40 583
4449 45 655
5325 50 728
6202
887080
10
15
20
CORR.
FOR
SEC.
+
" Corr.
17
25
30
40
45
150
50
66
75
83
TANGENT
"Corr.
10 17
15 26
120 34
30 51
40 68
45 77
50 85
1.53986
.54085
.54183
.54281
.54379
1.54478
.54576
.54675
.54774
.54873
01234
4
6678 σ)
1.54972 10
.5507111
.55170 12
.55269 13
33.55368 14
5
9
234
1.5546715
.55567 16
.55666 17
.55766 18
.55866 19
567
1.55966 20
.56065 21
.56165 22
.56265 23
.56366 24
1.56466 25
.56566 26
.56667 27
.56767 28
.56868 29
1.56969 30
.57C69 31
.57170 32
.5727 33
.57372 34
1.57474 35
.57575 36
.57676 37
.57778 38
.5787939
1.57981 40
.5808341
.58184 42
.58286 43
.58388 44
1.5849045
.58593 46
58695 47
.58797 48
.5890049
1.59002 50
.5910551
.59208 52
.5931153
.5941454
1.5951755
.59620 56
.59723 57
.59826 58
.5993059
1.6003360
519
2222
0-231 DO7OK DIEME DOOR OM2 20702 87~m7 no♪❤. I 9º 85887 KANAA
0.848048
8202
8356
8510
8664
4
5.848818
8972
9125
9279
9432
6
8
9
10.849586
12
14
9739 " Corr.
9893 10 26
13.850046 15 38
0199 20 51
16
17
15.850352 30 77
0505 40 102
0658 45 15
081150 128
0964
18
19
20.851117
21
22
23
24
26
25.851879
28
29
31
32
30.852640
34
36
33 3096
3248
37
38
35.853399
39
41
42
43
44
SINE
46
47
40.854156
48
49
51
52
1269
1422
1574
1727
53
54
2032
2184
2336
2488
45.854912 30 76
5063 40 101
5214 45 113
5364 50 126
5515
56
2792
2944
50.855666
57
58
59
3551
3702
3854
4005
55.856417
4610 15 38
4761 20 50
VERSINES, EXSECAnts, and TANGENTS
58°

5816
5966
6117
6267
CORR.
FOR
SEC.
+
6567
6718
6868
7017
60.857167
COSINE
529919
9673
9426
9179
8932
.528685
8438
8191
7944
7697
.527450
7203 " Corr.
6956 10 41
6708 15 62
6461 20 82
.524977
4729
4481
4234
3986
|.526214 30 124
5966 40 165
5719 45 186
5472 50 206
5224
.523738
3490
3242
2994
2747
.522499
2250
2002
1754
1506
.521258
1010
0761
0513
0265
.520016
CORR.
FOR
SEC.
+
9519 10 41
9270 15 62
9022 20 83
.518773 30 124
8525 40 166
.517529
7280
7031
6782
6533
8276 45 186
8027 50 207
7778
.516284
6035
5786
5537
5287
.515038
VERSINE
4308 "Corr..519768 "Corr..480232
4459 TO 25
470C81
C327
0574
0821
887080
7959 " Corr.
883910 147
9720 15 221
1068.890602 20 295
1562
1809
2056
2303
.471315.891484 30 442
2368 40 589
3253 45 663
4139 50 737
5026
.472550.895914
2797
3044
3292
3539
EXSEC
.473786.900368 30 447
4034 1262 40 596
2156 45 670
4528 3052 50 745
4776 3949
4281
.475023.904847
5271
5519
5766
6014
6758
7006
7253
8246
8494
.476262.909351 30 452
6510.910255 40 603
1160 45 678
2066 50 753
2973
6803 " Corr.
7692 10 149
8583 15 223
9475 20 298
.477501.913881
7750
7998
CORR.
FOR
SEC.
+
0481
0730
0978
11
5746 Corr.
6646 10 151
754615 226
8448 20 301
1475
1724
2222
8990
.478742.918436 30 457
9350 40 609
9239.920266 45 686
9487 1182 50 762
9735 2099
.479984.923017
4790 " Corr.
5700 10 152
6615 229
7523 20 305
.482471.932258
2720
2969
3218
3467
3937 " Corr.
" Corr.
4857 10 154 10 18
5778 15 231 15 27
670120 308 120 36
.481227.927624 30 462 130
CORR.
FOR
SEC.
+
54
8549 40 616 40 72
9475 45 693 45 81
1973.93040150 770 50 90
1329
3188 " Corr.
411810 156
5050 15 234
5984 20 312
" Corr.
10 18
15 26
20 35
.483716.936918 30 468
3965 785340 623
4214
8789 45 701
9726 50 779
4463
4713.940665
.484962.941604
30 53
40 70
45
79
50
88
TANGENT -1
1.60033 0
.60137
.60241
60345
.60449
OI23➡
6787 UI
5
1.60553
.60657
.60761
.60865
.60970 9
1.61074 10
.6117911
.61283 12
.61388 13
.61493 14
1.6159815
.61703 16
.61808 17
.61914 18
.62019 19
1.62125 20
.62230 21
.62336 22
.62442 23
.62548 24
1.62654 25
.62760 26
.62866 27
.62972 28
.63079 29
1.6318530
.63292 31
.63398 32
.63505 33
.6361234
1.63719 35
.63826 36
.63934 37
64041 38
.64148 39
1.64256 40
.6436341
.64471 42
.64579 43
.64687 44
1.64795 45
.64903 46
.65011 47
.65120 48
.65228 49
1.65337 50
.65445 51
.65554 52
.6566353
.6577254
1.65881 55
.65990 56
.66099 57
.66209 58
.66318 59
1.66428 60
520
0.857167
7317
7467
7616
I
234 5E7BQ 0-234
6
5.857916
8065
8214
8364
8513
8
9
11
10.858662
12
14
16
17
18
13 9109 15 37
9258 20 50
OUR DON~~ Am~m~ HOMMA 2=997
21
15.859406 30 74
9555 40 99
9704 45 112
9852 50 124
19.8600CI
20.860149
23
24
29
33
34
SINE
25.860890
IC38
1186
1334
1482
36
7766
30.861629
1777
1924
2072
2219
39
35.862366
2514
2661
2808
2955
42
44
LOLOLOL
8811 "Corr.
8960 10 25
40.863102
46
47
48
49
C298
C446
0594
0742
51
52
53
54
45.863836 30 73
3982 40 98
4128 45 110
4275 50 122
4421
50.864567
4713
4860
5006
5151
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
3249 " Corr.
3396 10 24
354215 37
3689 20 49
55.865297
56 5443
5589
5734
5880
57
58
59
60 1.866025
COSINE
.515038
4789
4539
4290
4040
.513791
3541
3292
3042
2792
510043
.509792
.512542
42
2293 " Corr.
2043 10
179315 62
1543 20 83
.511293 30 125
1043 40 167
0793 45 187
0543 50 208
C293
9542
9292
9041
.5C8791
8541
8290
8040
7789
507538
7288
7037
6786
6536
.506285
6034
5783
5532
5281
5C5030
CORR.
FOR
SEC.
4779 " Corr.
4528 TC 42
427615 63
4025 20 84
.503774 30 126
3523 4C 167
3271 45 188
3020 50 209
2768
.502517
2266
2014
1762
1511
+
.501259
1007
C756
0504
0252
5000CO
59°

VERSINE
.484962
5211
5461
5710
5960
.486209
6459
6708
6958
.490208
0458
0708
0959
EXSEC
7208.950108
.487458.951058
7707
7957
8207
8457
.491209
1459
1710
1960
2211
.941604
2544 "Corr.
348610 158
4429 15 236
5372 20 315
3966
4217
4468
4719
946317 30 473
7263 40 630
8210 45 709
9158 50 788
.488707.955825 30 478
8957
9207
9457
9707
.489957.960621
2009 " Corr.
2962 10 159
3915 15 239
4870 20 319
CORR.
FOR
SEC.
+
30
6782 40 638 40
774045 718
8699 50 797
9659
45
50
8993
9244
.492462.970294
2712
2963
3214
3464
1583 " Corr.
254610 161
35115 242
4477 20 323
965444 30 484
6411 40 645
738045 726
8351 50 807
9322
.493715.975174 30 490
6153 40 653
7133 45 734
811550 816
9097
1268 " Corr.
2243 10 163
3218 15 245
4195 20 326
.494970.980081
5221
5472
5724
5975
.496226.985017 30 495
6477
6729
6980
7232
.497483.989982
7734.990979
30
6008 40 660 40
7000 45 743 45
7993 50 826 50
8987
7986
" Corr.
1976 10 167
297515 251
8489 3975 20 334
8238
.498741.994976 30 501
5979 40 668
" Corr
10
15
20
"Corr.
1066 Corr.
205210 165 10
19
3039 15 248 15 29
4028 20 330
20
38
6982 45 752
7987 50 835
8993
9496
9748
500000.00000
CORR.
FOR
SEC.
+
19
28
37
56
74
83
93
57
76
86
96
TANGENT 1
1.66428 0
.66538
..66647
.66757
.66867
OI232 BONBO
4
1.66978
.67088
.67198
.67309 8
.67419 9
6
1.67530 10
.67641 |
.67752 12
.67863 13
.67974 14
1.68085 15
.68196 16
.68308 17
.6841918
.68531 19
1.68643 20
.68754 21
.68866 22
.68979 23
.69091 24
1.69203 25
.6931626
.69428 27
.69541 28
.69653 29
1.69766 30
.69879 31
.6999232
.70106 33
.7021934
1.70332 35
.7C446 36
.7056037
.70673 38
.70787 39
1.7090140
.7101541
.71129 42
.71244 43
.7135844
1.71473 45
.71588 46
.71702 47
.71817 48
.71932 49
1.72047 50
.72163 51
.72278 52
.72393 53
.72509 54
1.7262555
.72741 56
.72857 57
.72973 58
.73089 59
1.73205 60
521
0-234 BONBO
0.866025
6
8
5.866752
9
||
234 5ON
10.867476
12
13
14
16
17
7621" Corr.
7766 10 24
7910 15 36
8054 20 48
15.868199 30
18
19
22222
21
8 666 Fogg 55555 £55±5 wwwxx wwwww ~~~~~ 28
23
20.868920
9064
9207
9351
9495
24
26
27
29
25.869639
9782
9926
28.870069
0212
31
32
33
34
30.870356
0499
0642
0785
0928
36
37
38
39
35.871071
1214
1357
1499
1642
41
42
SINE
43
44
6171
6316
40.871784
46
6461
6607
47
48
49
6897
7042
7187
7331
52
53
54
45.872496 30
56
57
834340 96
8487 45 108
8632 50 120
8776
58
59
50.8732C6
3348
3489
3631
3772
VERSINES, EXSECANTS, AND TANGENTS
CORR.
FOR
SEC.
+
55.873914
4055
4196
4338
4479
60.874620
1927 " Corr.
2069 10 24
221215 36
2354 20 47
2638 40 95
2780 45 107
2922 50 118
3064
COSINE
500000
9748
9496
9244
8992
.498740
8488
8236
7983
7731
.497479
72.496216 30 126
5964 40 168
5711 45 189
5459 50 210
5206
7226 " Corr.
6974 10 42
6722 15 63
6469 20 84
.494953
4700
4448
4195
3942
.493689
3436
3183
2930
2677
.492424
2170
1917
1664
1410
.491157
0904
C650
0397
0143
489890
9636
CORR.
FOR
SEC.
+
" Corr.
9382 10 42
9129 15 63
8875 20 85
71.48862130 127
8367 40 169
8114 45 190
7860 50 211
7606
.487352
7098
6844
6590
6335
.486081
5827
5573
5318
5064
.484810
60°

VERSINE
EXSEC
5000001.OCCOC
C252 .CC101
0504.00202
C756 .00303
1008|| .00404
.501260 1.00505
1512.00607
" Corr.
1764 .00708 TO T7
2017 .00810 15 25
.00912 20 34
34
.502521.0101430
2269
2774
.0111640
.0111640
3026 .01218 45
3278 .01320 50
3531 .01422
.5037841.01525
4036 .01628
4289
.01730
.01833
4541
4794 .01936
5050471.02039
5300
5552
5805
6058
.02143
.02246
.02349
.02453
•
CORR.
FOR
SEC.
+
5063111.02557
6564 .02661
11 Corr.
6817.02765 10 17
7070.0286915 26
7323 .02973 20 35
.507576 1.03077 30 52
7830 .03182 40 70
8083.0328645 78
8336 .03391 50 87
8590 03496
.508843 1.03601
9C96 .03706
9350 .038||
9603.03916
9857 .04022
.5101101.04128
0364 .04233
0618
0871
.04339
.04445
1125 .C4551
.513919 1.05727
4173.05835
4427.05942
4682.06050
4936.06158
.515190.06267
.511379 1.04658
1633 .04764 " Corr.
1886 .04870 10 18
2140 .04977 15 27
2394 .05084 20 36
.512648 1.05191 30 53
2902.05298 40 71
3156 .05405 45 80
3410.05512 50 89
3665.05619
CORR.
FOR
SEC.
+
51 30 59
6840 78
76 45
85 50
88
98
"Corr.
10 20
15 29
20
39
" Corr.
10 20
15 30
20 40
30 60
40 80
45 90
50 100
"Corr.
TO
20
15 31
20 41
30 61
40 82
45 92
||50 102
TANGENT
1.732C5
.73321
.73438
.73555
.73671
T
0-23#
·
4
5
1.73788
.73905 6
.74022 7
.74140 8
.74257 9
1.74375 10
.74492 1
74610 12
.74728 13
.7484614
1.74964 15
.75082 16
.7520017
.75319 18
.75437 19
1.75556 20
.75675 21
.7579422
.75913 23
.76032 24
1.76151 25
.76271 26
.76390 27
.76510 28
.76630 29
1.76749 30
.76869 31
.76990 32
.7711033
.7723034
1.77351 35
.77471 36
.77592 37
.77713 38
.77834 39
1.77955 40
.7807741
.78198 42
.7831943
.78441 44
1.78563 45
.78685 46
.78807 47
.78929 48
.79051 49
1.79174 50
.79296 51
.79419 52
.7954253
.79665 54
1.79788 55
.79956
.80034 57
.80158 58
.8028 59
1.80405 60
522
0.87462C
0-234 BOT∞∞
6
7
5.875324
8
9
0-231 BONOO ON 22222
14
16
10.876026
17
18
19
21
22
23
6166 " Corr.
6307 10 23
6447 15
35
6587 20 47
15.876727 30 70
6867 40 93
7006 45 105
7146 50 17
7286
♡=
24
20.877425
7565
7704
7844
7983
597
26
27
28
29
31
32
33
34
25.878122
8261
8400
8539
8678
23.
MA KAMA NE BODO 858
SINE
30.878817
8956
9095
9233
9372
36
37
4761
4902
42
43
44
5042
5183
117
48
49
35.879510
9649
9787
9925
39.880063
40.880201
5464
5605
51
5746
5886
52
0339
" Corr
C477 10 23
061515 34
0753 20 46
45.880891 30 69
IC28 40 92
116645 103
1304 50 115
1441
50.881578
1716
1853
56
57
g g878
58
59
53 1990
54
2127
55.882264
2401
2538
2674
2811
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
60.882948
COSINE
.484810
4555
4301
4C46
3792
.483537
3282
3028
2773
2518
482263
2009
" Corr.
1754 10 42
1499 15 64
1244 20 85
.479713
9458
9203
8947
8692
.480989 30 127
0734 40 170
0479 45 191
0224 50 212
.479968
.478436
8181
7926
7670
7414
.477159
6903
6647
6392
6136
CORR.
FOR
SEC.
+
.472038
1782
1525
1268
1012
.475880
5624
5368
5112
11
4856
.474600
4344 Corr.
4088 TO 43
383215 64
3576 20 85
.470755
0499
0242
.469985
9728
.469472
61°

VERSINE
EXSEC
.515190 1.06267
5445.06375
5699.06483
5954.06592
6208.06701
.516463.06809
6718.06918 " Corr.
6972 .07027 10 18
7227.07137 15 27
7482.07246 20 37
37
.5177371.07356 30
7991.07465 40
.07575 45
.0768550
8246
8501
8756 .07795
.5190111.07905
9266.08015
9521.08126
9776.08236
.520032 .08347
.520287 1.08458
0542 .08569
C797 .C8680
1053.08791
1308.08903
.5215641.09014
.5228411.09574 30 56
3097 .09686 40 75
3353 .09799 45 84
3608.099|||50 94
386410024
4632
.5241201.10137.
4376.10250
.10363
4888.10477
5144 .10590
525400.10704
5656 .10817
5912.10931
6168.11045
6424 .11159
.473320 30 128.526680 ||1.11274
3063 40 171
2807 45 192
2551 50 213
2294
CORR.
FOR
SEC.
+
1819.09126 " Corr.
2074 .0923810 19
2330.0935015 28
2586 .09462 20 37
•
" Corr.
10 21
15 31
20 42
55 30 63
7340 83
82 ||45 94
9150 104
.5292451.12425
9501 .12540
9758.12657
.530015.12773
0272
.530528.13005
527962 1.11847 30 58
8218 .11963 40 77
8475.12078 45 86
8732 .12193 50 96
8988 .12309
.12889
CORR.
FOR
SEC.
+
6937 .11388
" Corr.
" Corr.
7193 1150310 19 10 22
11617 15 29 15 33
.11732 20 38 20 44
7449
7706
" Corr.
10 21
15 32
20 43
30 64
40 85
45 96
50 106
30 65
40 87
45 98
50 109
TANGENT
1.80405
8-231
.80529
.80653 2
.80777
.80901
4
1.81025
7
.81150
.81274
.81399 8
.81524 9
1.81649 10
.81774 11
.81899 12
.8202513
.82150 14
5678σ
1.82276 15
.82402 16
.82528 17
.82654 18
.82780 19
1.82906 20
.83033 21
.83159 22
.83286 23
.83413 24
1.83540 25
.83667 26
.83794 27
.83922 28
.84049 29
1.84177 30
.84305 31
.8443332
.8456133
.84689 34
1.84818 35
.84946 36
.85075 37
.85204 38
.85333 39
1.85462 40
.85591 41
.85720 42
.85850 43
.8597944
1.86109 45
.86239 46
.86369 47
.86499 48
.8663049
1.86760 | 50
.86891 51
.8702152
.87152 53
.87283 54
1.87415 55
.87546 56
.87677 57
.87809 58
.87941 59
1.88073 60
523
0.882948
3084
3221
3357
3493
1
·~34
2
6787 UI
5.883630
3766
3902
4038
4174
9
10.884310
11
12
234
13
14
16
17
18
19
~~~~~ ~~~~~ 8~~~~ ❤COMMO
15.88498830 68
5123 40 90
5258 45 102
5394 50 113
5529
20.885664
5799
5934
6069
6204
21
22
23
24
26
25.886338
6473
27 6608
28
29
36
SINE
30.887011
7145
32 7279
33 7413
34
7548
37
35.887682
7815
7949
41
42
43
44
38 8083
39
8217
5 £555
40.888350
Č
4445 Corr.
4581 10 23
4717 15
34
4852 20 45
3 གླ་ུགྨ་ྒུ གྱགཤྩ ངྒོ=ཪྐོ#
6742
6876
46
47
45.889017 30 67
9150 40 89
9283 45 100
9416 50 ||
57
49 9549
VERSINES, EXSECANTS, AND TANGENTS
62°

8484 " Corr.
8617 10 22
8751 15 33
8884 20 44
50.889682
9815
52 9948
53.890080
54 0213
CORR.
FOR
SEC.
+
55.890345
56 0478
0610
58 0742
59 0874
60.891006
COSINE
469472
9215
8958
8701
8444
.468187
7930
7673
7416
7158
"Corr.
.466901
6644
6387 10 43
6129 15 64
5872 20 86
464327
4069
3812
3554
3296
.465614 30 129
5357 40 172
5100 45 193
4842 50 214
4584
463038
2780
2522
2265
2007
1.461749
1491
1232
0974
0716
460458
0200
459942
9683
9425
CORR.
FOR
SEC.
+
456580
6322
6063
5804
5545
.459166
8908 Corr.
8650 10 43
839115 65
8132 20 86
457874 30 129
7615 40 172
7357 45 194
7098 50 215
6839
455286
5027
4768
4509
4250
.453990
VERSINE
EXSEC
.530528 ||| .13005
0785 .13122
1042.13239
1299 .13356
1556 .13473
.5318131.13590
2070
.13707 " Corr.
2327 .13825 10 20
2584 .13942 15 30
2842 .14060 20 39
.533099|||.14178 30 59
14296 40 79
.14414 45 89
3871 .14533 50 98
4128 .14651
.534386.14770
4643 .14889
4900 .15008
.15127
3356
3613
5158
5416 .15246
.535673.15366
5931
.15485
6188
.15605
6446
.15725
6704 .15845
536962.15965
CORR.
FOR
SEC.
+
11
7478
7220 .16085 Corr.
.16206 10 20
.16326 15 30
.16447 20 40
7735
7993
.538251
8509
1.16568 30 61
.16689 40 81
8768.16810 45 91
9026 .16932 50 101
9284 .17053
539542.17175
9800 .17297
.540058 .17419
0317 .17541
0575 .17663
.540834 1.17786
1092 .17909
1350 .18031
1609 .18154
1868 .18277
.542126.18401
2385 .18524
2643
2902
3161
"Corr.
.18648 10 21
.18772 15 31
.18895 20 41
.543420 1.19019 30 62
3678 .1914440 83
3937 .19268 45 93
4196 .19393 50 104
4455 .19517
1.544714 1.19642
4973
5232
.19767
.19892
5491 .20018
5750.20143
|.546010 ||1.20269
CORR.
FOR
SEC.
+
" Corr.
10 22
15 33
20
44
30
67
40 89
45 100
50 11|
" Corr.
10 23
" Corr.
10 23
15 35
120 47
TANGENT
30 70
40 93
45 105
50 116
1
01234 5O789
1.88073
.88205
.88337
.88469
.88602 Ц
1.88734
.88867 6
.89000
.89133
.89266
1.8940010
.89533 11
.89667 12
.89801 13
.89935 14
1.9006915
.90203 16
.90337 17
.90472 18
.90607 19
1.90741 20
.90876 21
.91012 22
.91147 23
.91282 24
15 34
20 45
30 68 1.92098|30
40 91.92235 31
45 102 .9237132
50 114 .92508 33
.92645 34
1.9141825
.91554 26
.91690 27
.91826 28
.91962 29
1.92782 35
.92920 36
.93057 37
.93195 38
.93332 39
1.93470 40
.9360841
.93746 42
.93885 43
.94023 44
1.9416245
.94301 46
.94440 47
.94579 48
.94718 49
1.94858 50
.94997 51
.95137 52
95277 53
.95417 54
1.95557 55
.95698 56
.95838 57
.95979 58
.96120 59
1.9626160
524
0-234 SON∞σ
0.891006
6
7
5.891666
8
9
π
NMI DONDO OUNCE KON** 67~m7 ❤❤MMM DIE 99
12
10.892323
13
14
16
2455 " Corr.
2586 10 22
2717 15 33
2848 20 44
15.892979|30 65
3110 40 87
324145 98
3371 50 109
3502
17
18
19
21
22
23
24
20.893633
26
27
28
29
25.894284
31
32
33
34
36
30.894934
37
38
39
41
42
SINE
43
35.895582
44
1138
1270
1402
1534
46
47
48
49
1798
1929
2061
2192
51
52
53
54
40.896228
g gགླ⌘
56
3763
3894
4024
4154
57
58
59
4415
4545
4675
4804
5064
5194
5323
5453
45.896873 30 64
7001 40 86
7130 45 97
725850 107
7387
50.897515
7643
7772
7900
8028
5712
5841
5970
6099
55.898156
8283
8411
8539
8666
" Corr.
6358
6486 10 21
661515 32
6744 20 43
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
60.898794
COSINE
.452694
2435
2175
1916
1656
•
453990
3731
3472
3213
2954
•
448799
8539
8279
8019
7759
447499
7239
6979
6718
6458
446198
5938
5677
5417
5156
.444896
4635
4375
4114
3853
5473061.20900
.451397
7565 .21026 " Corr.
7825 .21153 10 21
8084 .21280 15 32
8344 .2140720 42
5486031.21535| 30 64
1137 " Corr. 8863 .21662 40 85
0878 10 43 9122 .21790 45 96
C618|15 65 9382.2191850 106
0358 20 87 9642 .22045
.450098 30 130.549902||1.22174
449839 40 173 550161 .22302
957945 195 042.22430
9319 50 216 0681 .22559
9059
0941 .22688
551201 1.22817
1461 .22946
1721 .23075
1981 .23205
2241 .23334
CORR.
FOR
SEC.
.443593
3332 Corr.
307110 43
281015 65
2550 20 87
440984
0723
0462
0200
439939
+
rt
.439678
9417
9155
8894
8633
.438371
63°

VERSINE
546010 1.20269
6269 .20395
6528 .20521
6787 .20647
7046 .20773
EXSEC
.552501 1.23464
2761
3021
.23594 " Corr.
.23724 10 22
.23855 15 33
3542 .23985) 20 44
3282
4062
4323
4583
4844
553802 1.24116 30 65
.24247 40 87
.24378 45 98
.24509 50 109
.24640
CORR.
FOR
SEC.
+
.5551041.24772
5365.24903
5625 .25035
5886 .25167
6147 .25300
.5564071.25432
6668 .25565
6929 .25697
7190 .25830
7450 .25963
442289 30 130.5577||||1.26097
2028 40 174 7972 .26230
" Corr.
176745 196
8233 .26364 10 22
1506 50 217 8494 .2649815 34
1245
8755 .26632 20
45
.559016 1.26766 30
9277
9538
9800
560061
67
.26900 40 90
.27035 45 101
.27169 50 112
.27304
560322 1.27439
0583 .27574
0845 .27710
1106 .27845
1367 .27981
561629 1.28117
CORR.
FOR
SEC.
+
15
20
6
" Corr.
24
1.96969 5
.971
.97253 7
.97395 8
10
36
48 .97538 9
30
40 95
45 107
50 119
711.97680 10
.97823 |
30
73
40 97
45 110
50 122
TANGENT 1
"Corr.
10 25
15 37
20 50
1.96261 0
.96402 1
.96544 2
.96685 3
.96827 4
30
40 100
1.99841 25
"Corr. .99986 26
10 242.0013127
15 37 .00277 28
20 49 .004231 29
2.00569 30
.00715 31
.00862 32
.01008 33
.0115534
45 112
50 125
.97966 12
.98110 13
.98253 14
1.9839615
.98540 16
.98684 17
.98828 18
.98972 19
1.99116 20
.9926121
.99406 22
.99550 23
.99695 24
2.01302 35
.0144936
.01596 37
.C1743 38
.01891 39
2.0203940
.0218741
.02335 42
.0248343
.02631 44
752.03526 50
.03675 51
.0382552
.03975 53
.04125 54
2.02780 45
.0292946
.03078 47
.03227 48
.03376 49
2.04276 55
.04426 56
.04577 57
.04728 58
.04879 59
2.05030 60
525
0.898794
01234 567BQ 0-234 5
8
9
5.899431
12
13
14
16
17
18
19
10.900065
22222 202≈≈ 87087 2888 2=22E CON UNOZ PONAI
N
23
24
15.900698 30 63
0825 40 84
0951 45 95
1077 50 105
1203
26
20.901329
1455
1581
1707
1832
SINE
32
8922
9049
25.901958
2084
2209
2335
2460
37
38
9176
9304
30.902585
2710
2836
2961
3086
41
42
9558
9685
9812
9939
43
4444
35.903210
3335
3460
3585
3709
46
.0192 "Corr.
031910 21
044515 32
0572 20 42
40.903834
3958 Corr.
408210 21
4207 15 31
4331 20 41
56
45.904455
57
58
59
30 62
4579 40 83
4703 45
47
93
48 4827 50 103
49
4951
50.905075
51
5198
52
5322
53 5445
54
5569
55.905692
5815
5939
VERSINES, EXSECANTS, AND TANGENTS
64°

CORR.
FOR
SEC.
+
TI
6062
6185
60.906308
COSINE
.438371
8110
7848
7587
7325
.437063
6802
6540
6278
6017
.435755
.433135
2873
2610
2348
2086
5493 " Corr.
5231 110 44
4969 15 65
4707 20 87
434445 30 131
4183 40 175
392145 196
3659 50 218
3397
.431823
1561
1299
1036
0774
.430511
0248
429986
9723
9461
429198
8935
8672
8410
8147
CORR.
FOR
SEC.
+
.427884
7621
" Corr.
7358 10 44
7095 15 66
6832 20 88
.425253
4990
4726
4463
4199
426569 30 132
6306 40 175
604245 197
5779 50 219
5516
423936
3672
3409
3146
2882
|.422618
CORR.
FOR
SEC.
+
.5616291.28117
1890.28253" Corr.
.2839010 23
.2852615 34
.28663 20 46
.562937|||.28800 30
69
.28937 40 92
.29074 45 103
.2921150 114
.29349
VERSINE
2152
2413
2675
3198
3460
3722
3983
EXSEC
.5642451.29487
4507.29625
4769.29763
5031.29901
5293.30040
.565555 1.30179
5817 .30318
6079.30457 IC 23
6341.30596 15 35
6603.30735 20 47
7127
7390
7652
7914
.5668651.30875 30 70
.31015 40 93
.31155 45 105
.31295 50 117
.31436
.5681771.31576
8439
8701
8964
.31999
9226 .32140
.31717
.31858
.572116.33708
2379 .33852
2642
2905
.33996
.34140
3168 .34284
.569489.32282
9752 .32424 "Corr.
.570014 .32566 10 24
0277.3270815 36
0539 .32850 20 48
.5708021.32993 30 72
1065 .33135 40 95
1328 .33278 45 107
1590.33422 50 119
1853 .33565
#
.574747 1.35154 30 73
5010 .353CC 40 97
5274 .35446 45 110
5537 .35592 50 122
5801 .35738
576064 1.35885
6328 .36031
6591 .36178
6854 .36325
7118 .36473
.5773821.36620
CORR.
FOR
SEC.
+
"
10
15
20
#1
" Corr
2.05030
.05182
.05333
.05485
51.C5637
Corr.
25
38
30 76
40 102
45 114
50 127
Corr.
10 26
15 39
20 52
30 78
40 104
45 117
150 130
It
Corr.
10 26
15 40
20 53
5734311.34429
3694
3958
Corr.
4221
.34573 Corr.
.34718 10 24 10 27
.34863|15 37 15 40
4484 .35009 20 49 120 54
30 81
140 108
45 121
50 134
30 79
40 106
45 119
50 132
TANGENT
1
0
I
•234 56789
2.05790
.05942 6
.06094
.06247
.06400 9
2.06553 10
.06706 II
.06860 12
.07014 13
.07167 14
2.07321 15
.07476 16
.07630 17
.07785 18
.07939 19
2.08094|20
.08250 21
.08405 22
.0856C 23
.08716 24
2.08872 25
.09028 26
.0918427
.09341 28
.09498 29
2.09654 30
.09811|31
.09969 32
.10126 33
.10284 34
2.10442 35
.10600 36
.10758 37
.10916 38
.11075 39
2.1123340
.11392 41
.11552 42
.11711 43
.11871 44
2.1203045
.12190 46
.12350 47
.12511 48
.12671 49
2.12832 50
.12993 51
.13154 52
.1331653
.1347754
2.13639 55
.138C1 56
.13963 57
.14125 58
.14288 59
2.14451 60
526
01234 BO766
0.906308
8
—
5.906922
9
›INMA BONBO
10.907533
16
19
12
13
|| 7655" Corr.
7778 10 20
7900 15 30
14 8021 20 41
15.908143 30 61
8265 40 81
8387 45 91
18 8508 50 101
17
8630
27222 227°* ..❤❤7 HOMMA DE 99599 8UNOZ PONO8 8
24
20.908751
26
28
25.909357
9478
9599
9720
984!
32
33
34
SINE
30.909961
31.910082
41
6431
6554
6676
6799
42
43
44
7044
7166
7289
7411
35.910564
0684
0804
0924
1044
47
48
49
" Corr.
40.911164
1284
140310
152315 30
1642 20 40
20
8872
8994
9115
9236
45.911762
56
57
0202
0323
0443
58
59
50.912358
2478
2596
2715
2834
55.912953
3072
3190
3309
3427
30 60
1882 40 80
2001 45 90
2120 50 100
2239
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
60.913546
COSINE
422618
2355
2091
1827
1563
.421300
1036
0772
0508
0244
.419980
.417338
7074
6810
9716" Corr.
9452 10 44
918815 66
8924 20 88
418660 30 132
8396 40 176
8131 45 198
7867 50 220
7603
6545
6281
416016
5752
5487
5223
4958
.414693
4428
4164
3899
3634
.413369
3104
2840
2574
2310
.412044
CORR.
FOR
SEC.
+
1780 Corr.
1514 10 44
1249 15 66
0984 20 88
8596
8330
408065
7799
7534
7268
7002
406737
65°

VERSINE
EXSEC
.5773821.36620
7645.36768 " Corr.
7909 .3691610 25
8173 .3706415 37
8437 .37212 20 50
.578700 1.37361 30 75
8964 .37509 40 99
9228.37658 45 112
9492 .37808 50 124
9756 .37957
.5800201.38106
0284.38256
0548 .38406
0812 .38556
1076 .38707
5813401.38857
11
1604 .39008 Corr.
1869.39159 10 25
2133.3931115 38
2397.39462 20 51
.582662.39614 30 76
2926.39766 40 102
3190.39918 45 114
3455.40070 50 127
3719.40222
.5839841.40375
4248 .40528
4513.40681
4777 .40835
5042
.40988
CORR.
FOR
SEC.
+
.5853071.41142
5572
5836
6101
.41296 "Corr.
.4145010 26
.41605 15 39
6366 .41760 20 52
.5866311.41914 30 78
6896 42070 40 104
7160.42225 45 117
7426 .42380 50 130
7690 .42536
.5879561.42692
8220.42848
8486.43005
8751.43|62
9016 .43318
.410719 30 133.589281|1.43476
0454 140 177
9546 .43633 " Corr.
0188 45 199 9812.43790 10 26
.409923 50 221.590077.43948 | 15 40
9658
.409392
9127
0342 .44106 20 53
.590608.44264 30 79
0873 44423 40 106
1138 .44582 45 119
44741 50 132
8862
1404
1670 .44900
•
591935.45059
2201 .45219
2466.45378
2732 .45539
2998 .45699
.593263.45859
CORR.
FOR
SEC.
+
Corr.
10 27 .14777
15 41
20 55
30 82
40 110
45 123
50 137
" Corr.
10 28
15 42
20 56
30 84
40 112
45 126
50 140
#1
Corr.
10 28
15 43
20 57
30 85
40 114
45 128
50 142
" Corr.
TO 29
15 44
20 58
130 87
40 116
145 131
50 145
TANGENT
2.14451
.14614
.14940
.15104
2.15268
.15432
.15596
.15760
1
0-23I DO70. 0-234 5
4
5
6
8
.15925 9
2.16090 10
.16255 11
.16420 12
.16585 13
.16751 14
2.16917 15
.17083 16
.17249 17
.17416 18
.17582 19
2.17749 20
.17916 21
.18084 22
.18251 23
.18419 24
2.18587 25
.18755 26
.18923 27
.19092 28
.19261|29
2.1943030
.19599 31
.19769 32
.19938 33
.20108 34
2.20278 35
.20449 36
.20619 37
.20790 38
.2096 39
2.21132 40
.21304 41
.21475 42
.21647 43
.21819 44
2.21992 45
.22164 46
.22337 47
.22510 48
.22683 49
2.22857 50
.23030 51
.23204 52
.23378 53
.23553 54
2.23727 55
.23902 56
.24077 57
.24252 58
.24428 59
2.24604 60
527
01234 5676K
0.913546
8
5.914136
9
DINBE DO 22
10.914725
||
12
13
14
2222 222
16
17
18
19
4842 "Corr.
4960 10 20
5077 15 29
5194 20 39
15.915312 30
5429 40 78
5546 45 88
5663 50 98
5780
21
20.915896
22
23
24
26
27
25.916479
6596
6712
28 6828
29
6944
41
42
30.917060
7176
31
32
7292
33 7408
34
7523
43
44
SINE
35.917639
36
7755
7870
37
38
7986
39 8101
40.918216
46
47
3664
3782
3900
4018
g ggགུ གgགྲུགg དྩོཆོ་ོ;
48
4254
4372
4490
4607
49
51
52
6013
6130
6246
6363
45.918791 30 57
8906 40 77
9021 45 86
9135 50 96
9250
53
54
50.919364
9479
9593
9707
9822
57
58
59
55.919936
56.92C050
VERSINES, EXSECANTS, AND TANGENTS
8331 " Corr.
844610
19
856115 29
8676 20 38
CORR.
FOR
SEC.
+
0164
0277
C391
60.920505
COSINE
.406737
6471
6205
5939.
5673
.405408
5142
4876
4610
4344
404078
3811
" Corr.
3545 10 44
327915 67
3013 20 89
401415
1149
0882
0616
0349
1.400082
.399816
9549
9282
9016
59.402747 30 133.597253 ||1.48295
2480 40 178
2214 45 200
1948 50 222
1681
.398749
8482
8216
7949
7682
.397415
7148
6881
6614
6347
CORR.
FOR
SEC.
396080
5813 " Corr.
5546 10 45
5278 15 67
50120 89
394744 30 134
4477 140 178
4209 45 200
3942 50 223
3674
.393407
3140
2872
2605
2337
.392070
1802
1534
1267
0999
390731
66°

VERSINE
EXSEC
.593263 ||1.45859
3529 .46020 11 Corr.
3795.4618110 27
4061 .4634215 41
4327 .46504 20 54
.5945921.46665 30 81
.46827 40 108
5124 .46989 45 122
5390 .47152 50 135
5656 .47314
4858
.5959221.47477
6189 47640
6455
6721
.47804
.47967
6987 .48131
•
7520.48459 " Corr.
7786 .48624 10 27
8052 .4878915 41
8319
.48954 20 55
.598585.4911930
8851
9118
9384
82
.49284 40 110
.49450 45 124
.49616 50 137
9651.49782
CORR.
FOR
SEC.
+
1.5999181.49948
.600184
0451
0718
0984
.50115 " Corr.
.50282 10 28
.5044915 42
.50617 20 56
.6012511.50784 30 84
1518.50952 40 112
.51120 45 126
2051 .51289 50 140
2318 .51457
1784
.6025851.51626
2852 .51795
3119 .51965
3386 .52134
3653 .52304
.6052561.53329 30 85
5523.53500 40 114
5791.53672 45 128
6058 .53845 50 142
6326 .54017
606593 1.54190
6860.54363 "Corr.
7128.54536 |10 29
7395.54709|15 43
7663 .54883 20 58
CORR.
FOR
SEC.
+
6C9269.55930
" Corr.
10 30
15 44
20
59
30 89
40 118
45 133
50 148
"Corr.
IC 30
15 45
20
60
603920|1.52474
" Corr.
4187.52645
" Corr.
4454.52815 10 28 10 31
4722.52986||5 43 15 46
4989.53157 20 57 20 62
30
90
40 120
45 135
50 150
" Corr.
10 31
15 46
20 61
30 92
40 122
45 137
150 153
30 93
40 124
45 139
50 155
" Corr.
10 32
15 47
20 63
1.6079301.55057 30 87
30 95
40 126
8198.55231|40 116
8466 .55405 45 130 45 142
8733.55580 50 145 150 158
9001 .55755
TANGENT
2.24604
·
0123♬
.24780
.24956
.25132
.25309 4
5678 ❤
2.25486
.25663
.25840
.26018 8
.26196
9
2.26374 10
.26552 11
.26730 12
.26909 13
.27088 14
2.27267 15
.27447 16
.27626 17
.27806 18
.27987 19
2.28167 20
.28348 21
.28528 22
.2871C 23
.28891 24
2.29073 25
.29254 26
.2943727
.29619 28
.29801 29
2.29984|30
.30167 31
.303532
.3053433
.30718 34
2.30902 35
.31086 36
.3127137
.31456 38
.31641 39
2.31826 40
.3201241
.3219742
.32383 43
.3257044
2.32756 45
.3294346
.33130 47
.33317 48
.3350549
2.33693 50
.3388151
.3406952
.34258 53
.34447 54
2.34636 55
.34825 56
.3501557
.35205 58
.35395 59
2.35585 60
528
0.920505
0618
0732
0846
0959
0123➡
4
5.921072
1185
1299
1412
1525
6789 σ
8
9
10.921638
22
O-NIH BOMB 0702 87887 HMMMM DEIRE
||
12
13
14
1750 " Corr.
186310 19
1976 15 28
2088 20 37
15.922201 | 30
56
2313 40 75
2426 45 84
2538 50 94
2650
16
17
18
19
20.922762
2874
2986
3098
3210
21
22
23
24
25.923322
26
28
29
31
32
30.923880
33
34
36
37
38
39
35.924435
41
SINE
42
43
44
40.924989
46
47
g gg8gཁྱུགཡུག Čཆོī་
48
3434
3545
49
3657
3768
51
52
53
54
3991
4102
4213
4324
56
57
58
59
45.925540 30 55
5651 40 73
5761 45 83
5871 50 92
5980
4546
4657
4768
4878
50.926090
6200
6310
6419
6529
5099 " Corr.
521010 18
5320 15 28
5430 20 37
55.926638
6747
6857
6966
7075
TABLE XX.—NATURAL SINES, cosines,
CORR.
FOR
SEC.
+
60.927184
COSINE
.390731
0463
0196
.389928
9660
.389392
9124
8856
8588
8320
.388052
7784 " Corr.
7516 10 45
7247 15 67
6979 20
89
.385369
5101
4832
4564
4295
384027
3758
3490
3221
2952
.38671130 134
6443 40 179
6174 45 201
5906 50 224
5638
382683
2415
2146
1877
1608
.381339
1070
0801
0532
0263
.379994
CORR.
FOR
SEC.
+
.377302
7033
6763
6494
6224
8
9725 " Corr.
9456 10 45
9187 15 67
8918 20 90
.375955
5685
5416
5146
4876
.374607
.378649 30 135
8379 40 180
8110 45 202
7841 50 224
7571
67°

VERSINE
0876
1144
6092691.55930
9537.56106 " Corr.
9804 .5628210 30
.610072 .5645815 цц
0340 .56634 20 59
.6106081.56811 30 89
.56988|40 118
.57165 45 133
.57342 50 148
1680 .57520
.6119481.57698
2216 .57876
2484.58054
2753 .58233
3021.58412
.6132891.58591
3557
3826
.5877|| " Corr.
.58950 TO 30
4094 .5913015 45
4362 .5931 20 60
.614631 ||1.59491|30 90
4899 .59672 40 120
5168 .59853 45 135
5436.60035 50 150
5705.60217
1412
EXSEC
.615973 1.60399
CORR.
FOR
SEC.
+
6242 .60581 " Corr.
6510 .60763|10
31
46
61
6779 .6094615
7046 .61129 20 61
.617317 1.61313 30 92
7585 .61496 40 122
7854.61680 45 138
8123 .61864 50 153
8392|| .62049
.6186611.62234
8930 .62419
9199 .62604
9468 .62790
9737 .62976
|.620006 ||1.63162
0275 .63348
" Corr.
0544 .6353510 31
0813 .63722 15 47
1082.63909|20 62
2159
2429
|.621351||1.64097 30 94
1621|| .64285 40 125
1890 .64473 45 140
.64662 50 156
.64851
.622698 1.65040
2967 .65229
" Corr.
3237.65419 10 32
3506 .6560915
3776 .65799 20
.6240451.65989 30
48
64
95
4315 .66180 40 127
4584.6637145 143
4854 .66563 50 159
5124 .66755
.625393 1.66947
CORR.
FOR
TANGENT
SEC.
+
"Corr.
2.35585
.35776
10 32 .35967
.36158
15 48
20 64
.36349
30 96
40 129
45 145
50 161
" Corr.
10 33
15 49
20
20 65
30 98
40 130
45 147
50 163
" Corr.
10 33
15 50
20
20 65
30 99
40 133
45 149
50 166
" Corr.
10 34
15 51
20 67
30 101
40 135
145 152
50 168
" Corr.
10 34
15 51
20 69
30 103
40 137
45 154
50 171
2.36541
.36733
.36925
.37118
0123# 50780
6
7
.37311 9
2.3750410
.37697 11
.37891|12
.38084 13
.38279 14
2.38473 15
.38668 16
.38862 17
.39058 18
.39253 19
2.39449 20
.39645 21
.39841 22
.40038 23
.40235 24
2.40432 25
.40629 26
.40827 27
.41025 28
.4122329
2.41421 30
.4162031
.41819 32
.42019 33
.4221834
2.42418 35
.42618 36
.42819 37
.4301938
.43220 39
2.4342240
.4362341
.43825 42
.44027 43
.44230 44
2.44433 45
.4463646
.44839 47
.4504348
.45246 49
2.45451 50
.45655 51
.45860 52
.4606553
.4627054
2.46476 55
.4668256
.46888 57
.4709558
.4730259
2.47509 60
529
01234
0.927184
5678σ
9
5.927728
7836
7945
8053
8161
10.928270
||
234
12
13
14
16
17
18
19
333
27222 222≈≈ 87~~7 6888 E CONDO UNA CON
15.928810 30 54
8917 40 72
9025 45 81
9133 50 90
9240
20.929348
9455
9562
23 9669
24
9776
26
28
25.929884
9990
27.930097
0204
0311
31
33
34
30.930418
0524
32 0631
SINE
41
42
43
7293
7402
7510
7619
35.930950
1056
1162
1268
1374
44
40.931480
46
47
48
49
8378" Corr.
8486 10 18
859415 27
8702 20 36
51
52
0737
0843
56
45.932008 30 53
2113 40 70
2219 45 79
2324 50 88
2429
50.932534
2639
2744
2849
2954
57
58
59
VERSINES, EXSECANTS, AND TANGENTS
68°

CORR.
FOR
SEC.
+
55.933058
3163
3267
3372
3476
60.933580
1586
" Corr.
1691|10 18
1797 15 26
1902 20 35
COSINE
.374607
4337
4067
3797
3528
.373258
2988
2718
2448
2178
.371908
1638 " Corr.
136810 45
109815 68
0828 20 90
8936
8665
8395
8125
.367854
7584
7313
7042
6772
.366501
6231
5960
5689
5418
.365148
4877
4606
4335
4064
.370557 30 135
0287 40 180
0017 45 203
.369747 50 225.630253
9476
.369206
.363793
CORR.
FOR
SEC.
3522
"Corr.
3251 10 45
2980 15 68
2709 20 90
.362438 30 136
2167 40 181
1896 45 203
1625 50 226
1353
.361082
0811
0540
0268
359997
359725
9454
9182
8911
8640
.358368
VERSINE
6253931.66947
5663
5933
6203
6472
EXSEC
.628092 1.68884
8362.69079 " Corr.
8632 .6927510 33
8902 .69471 15 49
9172 .69667 20 66
.6294431.69864 30 98
9713.70061 40 131
.70258 45 148
.70455 50 164
0524 .70653
9983
2416
2687
2958
3228
CORR.
FOR
SEC.
+
7012
.6267421.67911 30 9730 104
.68105 40 129 40 139
7282 .68299 45 145
7552 .68494 50 162
7822 .68689
45 157
50 174
.67139 " Corr.
.6733210 32
.67525 15 48 15
.67718 20 65 20
.630794 1.70851
1064
1335
1605
1875
.71050 " Corr.
" Corr.
.71249 10 33 10 36
.71448 15 50 15 54
.71647 20 67 | 20 72
.6321461.71847 30 100 30 108
.72047 40 133 40 143
.72247 45 150 45 161
.72448 50 167 50 179
.72649
ย
.6348521.73862 30 102
CORR.
FOR
SEC.
+
2.47509
" Corr.
10
.47716
.47924
35
52 .48132
70
.48340
.633499||1.72850
3769
.73052
Corr.
" Corr.
.7325410 3410 36
.73456 15 51 15 55
4582 .73659 20 68
4040
4311
20 73
30 109
5123 .74065 40 136 40 146
5394 .7426945 152 45 164
5665 .74473 50 169 50 182
5936 .74677
6389181.76945
9189
9460
.637562.75909 30 103
7833 .76116 40 138
8104 .76323 45 155
8375 .76530 50 172
8647 .76737
.6362071.74881
6749
6478 .75086 " Corr. Corr.
.75292 10 34 10 37
.7549715 52 15 55
.75703 20 69
7020
7291
120 74
=
2.49597 10
.49807 |
.50018 12
.50229 13
20 71 .50440 14
30 1062.50652 15
40 141 .50864 16
11 Corr.
10 35
15 53
45 159
50 177
.51076 17
.51289 18
.51502 19
2.51715 20
.51929 21
.52142 22
.52357 23
.5257124
11
30 111
40 148
45 166
50 185
TANGENT
"
.77154 Corr.
.77362 10 35
" Corr.
10 37
9732 .77571 15 52 15 56
.640003 .77780 20 70 20 75
.6402751.77990 30 10530 112
0546 .78200 40 140 40 150
0818.78410 45 157 45 169
1089 .78621 50 175 50 187
1360
.78832
.641632.79043
2.48549
.48758
.48967
.49177
.49386
01234 5O789
6
2.52786 25
.53001 26
.53217 27
.53432 28
.53648 29
2.5386530
.54082 31
.54299 32
.54516 33
.54734 34
2.54952 35
.55170 36
.55389 37
.55608 38
.55827 39
2.56046 40
.5626641
.56487 42
.56707 43
.56928 44
2.57150 45
.57371 46
.57593 47
.57815 48
.58038 49
2.5826 50
.58484 51
.58708 52
.58932 53
.59156 54
2.5938155
.59606 56
.5983 57
.60057 58
.60283 59
2.60509 60
530
0.933580
01234
5678 O
9
5.934101
||
234
12
13
10.934619
141
DONOR 20
16
18
22222 2000* 0.087 COMO DE BO
23
24
15.935135 30 52
5238 40 69
5341 45 77
5444 50 86
5547
26
27
20.935650
5752
5855
5957
6060
28
29
31
25.936162
6264
6366
6468
6570
32
33
34
SINE
36
3685
3789
30.936672
6774
6876
6977
7079
37
38
39
3893
3997
41
42
43
44
4204
4308
46
47
48
49
4412
4515
35.937181
7282
7383
7485
7586
40.937687
51
52
53
54
g gཀླ78
4722 Corr.
4826 10 17
4929 15 26
5032 20 34
56
57
58
45.938191
50.938694
8794
8894
8994
9094
7788" Corr.
7889 10 17
7990 15 25
8091 20 34
55.939194
9294
9394
9494
9593
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
30 50
8292 40 67
8392 45 76
8493 50 84
8593
60.939693
COSINE
.358368
8096
7825
7553
7281
.357010
6738
6466
6194
5923
.355651
5379 " Corr.
5107 10 45
483515 68
4563 20 91
.352931
2658
2386
2114
1842
.354291 30 136
4019 40 181
3747 45 204
3475 50 227
3203
.351569
1297
1025
0752
0480
.350207
.349935
9662
9390
9117
.348845
8572
8299
8027
7754
.347481
CORR.
FOR
SEC.
+
7208 " Corr.
6936 10 45
6663 15 68
6390 20 91
.344752
4479
4206
3933
3660
.346117 30 136
5844 40 182
5571 45 205
5298 50 227
5025
.343386
3113
2840
2567
2294
.342020
69°

VERSINE
EXSEC
CORR.
FOR
SEC.
+
.641632 1.79043
1904.79254 " Corr.
2175.79466|10 36
2447.79679 15 53
2719.79891 20 71
.642990 ||| .80104 30 107
3262.80318 40 142
3534.80531 45 160
3806.80746 50 178
4077.80960
.645709 ||| .82254 30 108
5981.8247140 145
6253.82688 45 163
6525 .82906 50 181
6797.83124
.644349 1.81175
4621.81390 " Corr.
" Corr.
4893.81605 10 36 10 39
5165.81821|15 54 15 58
5437.82037 20 72 20 77
.650065.85767 " Corr.
0338.85990 10 37
0610.86213 15 56
0883.86437 20 75
.651 155 1.86661 30 112
1428.86885 40 149
1701.87109 45 168
1973 .87334 50 187
2246 .87560
|.655248 ||1.90063
CORR.
FOR
SEC.
+
.647069 1.83342
7342 .83561
" Corr.
"Corr.
7614.83780|10 37 10 39
7886 .8399915 55 15 59
8158.84219 20 73 20 78
.648431 1.84439|30 110
8703.84659 40 147
8975.84880 45 165
9248 .85102 50 184
9520 .85323
30 118
40 157
45 177
50 196
.649793 1.85545
11
Corr.
10 38
15 57
20 76
5521.90293 " Corr.
5794.90524 10 39
6067 .90754 15 58
6340.90986 20 77
30 114
40 152
45 171
50 190
.656614 1.91217 30 116
6887 .91449 40 156
7160.91681 45 174
7433.91914 50 193
7706 .92147
.6579801.92380
30 116
40 156
45 174
50 193
.652519 1.87785
2792 .88011 " Corr. " Corr.
3064.88238|10 38 10 40
3337.88465|15 57 15 61
3610.88692 20 76 20
81
30 121
4156.89148 40 152 40 162
4429.89376 45 171
4702 .89605 50 190
4975.89834
.653883.88920|30 114
45 182
50 202
" Corr.
10 40
15 60
20 80
" Corr.
10 41
15 62
20 82
TANGENT
30 123
40 165
45 185
50 206
2.60309
.60736
.60963
.61190
.61418
2.61646
.61874
OI234 DON»« 0-234
0
6
.62103
.62332 8
.62561 9
2.62791 10
.6302111
.63252 12
.6348313
.6371414
2.6394515
.64177 16
.64410 17
.64642 18
.64875 19
2.65109 20
.6534221
.65576 22
.6581 23
.66046 24
30 120
2.68653 35
40 160 .68892 36
45 179.69131 37
150 199 .6937 38
.69612 39
2.69853 40
.70094 41
.70335 42
.70577 43
.70819 44
2.6628|| 25
.66516 26
.66752 27
.66989 28
.67225 29
2.67462 30
.67700 31
.67937 32
.68175 33
.68414 34
2.71062 45
.71305 46
.71548 47
.71792 48
.72036 49
2.72281 50
.72526 51
.72771 52
.73017 53
.73263 54
2.73509 55
.73756 56
.74004 57
.7425 58
.74499 59
12.74748 60
531
0.939693
9792
9891
9991
234
-237 CO700 CODE COM 2 DO700 8.~m~ nom❤❤ 9 29509 85885 D8588 8
ოოოო
4.940090
5.940189
0288
0387
0486
0585
6
7
8
9
10.940684
12
13
14
21
20.941666
1764
1862
1960
2058
22
23
24
16
17
15.941176 30 49
1274 40 66
1372 45 74
18 1470 50 82
1569
19
26
25.942155
27
28
29
31
30.942642
2739
32 2836
34
33 2932
3029
36
37
35.943126
39
41
42
38 3416
3512
43
SINE
44
40.943608
46
47
48
49
0782 " Corr.
C881 TC 16
097915 25
1078 20 33
2252
2350
2447
2544
45.944089 130 48
4185 40 64
428145 72
4376 50 80
4472
51
53
54
50.944568
3223
3319
4663
52 4758
56
57
59
3705
"Corr.
380110 16
3897 15 24
3993 20 32
55.945044
5139
5234
5329
5424
VERSINES, EXSECANTS, AND TANGENTS
70°

CORR.
FOR
SEC.
4854
4949
60.945519
COSINE
.342020
1747
1473
1200
0926
.340653
C380
0106
339832
9559
339285
9012 " Corr.
8738 IC 46
846415 68
8190 20 91
.337917 3C 137
7643 40 183
7369 45 205
7095 50 228
6821
.336548
6274
6000
5726
5452
1.335178
4903
4629
4355
4081
.333807
3533
3258
2984
271C
.332436
2161
1887
1612
1338
CORR.
FOR
SEC.
331063
0789 " Corr.
0514 10 46
0240 15 69
329965 20 92
.329691 30 137
9416 40 183
9141 45 206
8867 50 229
8592
328317
8042
7768
7493
7218
.326943
6668
6393
6118
.5843
.325568
VERSINE
EXSEC
9074
657980.92380
8253.92614 " Corr.
8527.92849 10 39
88CC.9308315 59
.93318 20 78 | 20
.6593471.93554 30 118
9620.93790 40 157
9894 .94026 45 177
660168 94263 50 196
0441 .94500
.66C7151.94737
0988.94975 " Corr.
1262.9521310 40
1536.95452 15 60
1810.95691 20 80
.6620831.95931 30 120
2357.96171 40 160
2631 .96411 45 180
2905.96652 50 200
3179.96893
.663452.97135
CORR.
FOR
SEC.
+
3726 .97377 " Corr.
4000 .97619 10 41
.97862 15 61
4548.98106 20 81
4274
.6648221.98349 30 122
5097.98594 40 163
5371 .98838 45 183
5645 .99083 50 203
5919.99329
.6661931.99574
6467.99821 Corr.
6742 2.00067|10 41
7016 .00315 15 62
7290 .CC562 20 83
.667564 2.00810 30 124
7839 .CIC59 40 166
8113 .C1308 45 186
8388 .01557 50 207
8662 .01807
11
.671683 2.04584
.668937 2.02057
9211 .02308 " Corr.
9486.02559 10 42
9760 .02810 15 63
.670035 .03062 20 84
.670309 2.03315 30 126
0584 .03568 40 169
0859 .0382145 190
1133 .04075|50 211
1408 .04329
"
1958 .04839 Corr.
2232 .0509410 43
2507 .05350 15 64
2782 .05607 20 86
1.673057 2.05864 30 129
3332 .06121 40 172
3607 .0637945 193
3882 .06637 50 214
4157 .06896
.674432 2.07155
CORR.
FOR
SEC.
+
"Corr.
10 42
15 63
84
30 125
30 125
40 167
45 188
50 209
" Corr.
10 42
15 64
||20 85
130 127
30 127
40 170
45 191
50 212
" Corr.
10 43
15 65
20 86
30 130
40 173
45 194
50 216
" Corr.
10 44
15 66
20 88
130 132
40 176
45 198
50 220
" Corr.
10 45
15 67
20 89
30 134
40 179
45 201
50 223
" Corr.
10 45
15 68
20 91
30 136
40 182
45 204
50 227
TANGENT
2.74748
.74997
.75246
.75496
.75746
2.75996
0-234 DONO O-232
5
7
.76247
.76498
.76750 8
.77CC2 9
6
2.77254 10
.77507 11
.7776 12
.7801413
78269 14
2.78523 15
.78778|16
.79033 17
.79289 18
.79545 19
2.79802 20
.8005921
.80316 22
.80574 23
.80833 24
2.8109125
.81350 26
.81610 27
.81870 28
.82130 29
2.82391 30
.8265331
.82914 32
.83176 33
.83439 34
2.83702 35
.83965 36
.84229 37
.84494 38
.84758 39
2.8502340
.85289 41
.85555 42
.8582243
.86089 44
2.86356 45
.86624 46
.86892 47
.8716148
.87430 49
2.87700 50
.87970 51
.88240 52
.88511 53
.8878354
2.89055 55
.89327 56
.89600 57
.89873 58
.9C147 59
2.90421 60
532
0.945519
5613
5708
5802
5897
OI234 KONWO
5.94599!
6
7
8
9
01234 5
10.946462
11
12
13
14
6556 "Corr.
6649 10 16
674315 23
6837 20 31
15.946930 30 47
7024 40 62
7117 45 70
7210 50 78
7304
16
17
18
19
22222 2070 200
23
20.947397
7490
7583
7676
7768
26
29
25.947861
7954
8046
8139
8231
31
32
33
34
30.948324
8416
8508
8600
8692
36
37
38
39
SINE
35.948784
8876
8968
9C60
9151
41
42
43
44
6085
6180
40.949243
6274
6368
3889 23ཀྲུགg
51
52
49.950063
50.950154
53
54
45.949699 30 46
46
9790 40 61
47 988 45 68
48
9972 50 76
56
57
58
59
55.950606
0696
0786
0877
0967
60.951056
9334 " Corr.
9426 10 15
9517 15 23
9608 20 30
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
0244
0335
0425
0516
COSINE
.325568
5293
5018
4743
4468
.324193
3917
3642
3367
3092
320062
.319786
.322816
2541
" Corr.
2266 10 46
1990 15 69
1715 20 92
.321440 30 138
1164 40 184
0888 45 207
0613 50 229
0337
9511
9235
8959
.318684
8408
8132
7856
7580
.317305
7029
6753
6477
6201
.315925
5649
5373
5097
4821
CORR.
FOR
SEC.
+
.314545
4269
" Corr.
399210 46
371615 69
3440 20 92
.311782
1506
1229
0953
0676
.313164 30 138
2888 40 184
26145 207
2335 50 230
2059
.310400
C123
309847
957C
9294
.309017
71°

VERSINE
EXSEC
.674432 2.07155
4707 .C7415 " Corr.
4982 .07675 10 44
5257 .0793615 66
5532 .08197 20 87
.675807 2.08459130 131
6083 .08721 40 175
6358.08983 45 197
6633 .09246 50 218
6908.0951C
.677184 2.09774
CORR.
FOR
SEC.
+
7459 .10038 " Corr.
7734 .1030310 44
8010 .1056815 67
82850834 20 89
.678560 2.11101 30 133
8836 .11367 40 178
9112 .11635 45 200
9387 1190350 222
9663 .12171
.679938 2.12440
.680214 .12709
" Corr.
0489 .12979 10 45
0765 .1324915 68
1041 .13520 20 91
681316 2.13791130 136
1592 .14063 40 181
1868 .14335 45 204
2144 .14608 50 226
21120 .!488!
.682695 2.15155
11
CORR.
FOR
SEC.
+
" Corr.
10 46
15 69
20 92
.686836 2.19322 30 141
7112 .19604 40 188
7389 .19886 45 211
7665 .20169 50 235
7941
.20453
30 139
40 185
45 208
50 231
.688218 2.20737
8494 .21021
" Corr.
.2130610 48
.2159215 72
.21878 20 96
8771
9047
9324
.689600 2.22165 30 144
9877 .22452 40 192
.690153 .22740 45 215
0430 .23028 50 239
0706 .23317
.690983 2.23607
" Corr.
10 47
15 70
20 94
30 141
40 188
45 211
50 235
" Corr.
10 48
15 72
20 96
2971 .15429 Corr.
3247 .157041046
"Corr.
10 49
3523 .1597915 69 15 73
3799 .16255 20 9220 97
30 143
40 191
45 215
50 239
.684075 2.16531 30 138 30 146
4351 .16808 40 185 40 194
4627 .17085 45 208
4903 .17363 50 231
5179 .17641
45 219
50 243
.685455 2.17920
5731 .18199
" Corr.
" Corr.
6008 .1847910 47 10 49
.1875915 70 15 74
6560 .19040 20 94 20 99
6284
30 148
4C 198
45 223
50 247
" Corr.
10 50
15 76
20 101
30 151
40 201
45 227
50 252
TANGENT
2.90421
.90696
.90971
.91246 3
.91523 4
2.91799
.92076
.92354
.92632
.92910 9
2.93189 10
.93468 11
.93748 12
.94028 13
.94309 14
BENBO
• QM=
2.94590 15
.94872 16
.95155 17
.95437 18
.95720 19
2.96004 20
.96288 21
.96573 22
.96858 23
.97144 24
2.97430 25
.97717 26
.98004 27
.98292 28
.98580 29
2.98868 30
.99158 31
.99447 32
.99738 33
3.00028 34
3.00319 35
.00611 36
.00903 37
.0119638
.01489 39
3.0178340
.02077 41
.02372 42
.02667 43
.0296344
3.0326045
.03556 46
.03854 47
.C4152 48
.04450 49
3.04749 50
.05049 51
.05349 52
.05649 53
.05950 54
3.06252 55
.06554 56
.06857 57
.07160 58
.07464 59
3.07768 60
533
0.951056
1146
1236
1326
1415
234 567BG 0-23-
1
5.951505
8
9
10.95195!
11
12
13
2040 " Corr.
2129 10 15
2218 15 22
2307 20 30
15.952396 30 44
2484 40 59
2573 45 67
2662 50 74
2750
14
16
17
18
19
22222
234
21
20.952838
22
222
$23
24
26
25.953279
27
28
29
31
+32
33
34
8 66L6G Fgggg 55555 £55±5 8www.
30.953717
3804
3892
3979
4066
36
37
38
35.954153
4240
4327
4414
4501
42
43
44
SINE
40.954588
46
47
48
49
1594
1684
1773
1862
51
555
52
45.955020 30 43
5106 40 58
5192 45 65
5278 50 72
5364
53
54
2926
3015
3103
3191
50.955450
5536
5622
5707
5793
3366
3454
3542
3629
56
57
58
59
55.955878
5964
6C49
6134
6220
60.956305
VERSINES, EXSECANTS, AND TANGENTS
72°

CORR.
FOR
SEC.
+
4674 " Corr.
476 IC 14
4847 15 22
4934 20 29
COSINE
.309017
8740
8464
8187
7910
.307633
7357
7080
6803
6526
.306249
.303479
3202
2924
2647
2370
5972 " Corr.
5695 10 46
5418 15 69
5141 20 92
.302093
1815
1538
1261
0983
.300706
0428
0151
.299873
9596
.299318
9041
8763
8486
8208
CORR.
FOR
SEC.
+
2920
3197
3474
.693751 2.26531
4028 .26827
" Corr.
4305 .27123 10 50
4582 .27420 15 75
.27717 20 99
.304864 30 138.695136 ||2.2801530 149
.28313 40 199
.28612 45 224
.28912 50 249
.29212
4859
4587 40 185
4310 45 208
4033 50 231
3756
.297930
7653
Corr.
7375 10 46
7097 15 69
6819 20 93
.295152
4874
4596
4318
4040
11
.293762
3484
3206
2928
2650
.292372
.296542 30 139
6264 40 185
5986 45 208
5708 50 231
5430
VERSINE
.690983 2.23607
1260
1536
.23897
"Corr.
.24187 10 49
1813 .24478 15 73
2090 24770 20 98
EXSEC
5413
5690
•
5967
6244
.692367 2.25062 30 146 30 154
2643 .25355 40 195 40 205
.25648 45 220 45 231
.25942 50 24450 256
.26237
CORR.
FOR
SEC.
+
.696521 2.29512
6798 .29814 "Corr.
7076 .30115 10 51
7353 .3041815 76
7630 .30721 20 101
.697907 2.31024|30 152
8185 .31328 40 203
8462 .31633 45 228
8739 .31939 50 253
9017 .32244
.702070 2.35649
2347
2625
2903
3181
.704848 2.38808
" Corr.
10
A A A
51
15 77
20 103
"
5126
5404
.39128 Corr.
.39448 10 54
5682 .39768 15 81
5960.40089 20 107
.699294 2.32551
9572 .32858 " Corr.
9849 .33166 10 5210
.700127 .3347415 78
0404 .33783 20 103
.7C0682 2.34092 30 155
0959 .34403 40 207
1237 .34713 45 233
1514 .35025 50 258
1792 .35336
706238 2.40411 30 161
6516 .40734 40 215
6794 .41057 45 242
7072 41381 50 269
7350 .41705
.707628 2.42030
CORR.
FOR
SEC.
+
10 52
15 78
20 104
30 157
40 209
45 235
50 261
Corr.
" Corr.
10 53
15 80
20 106
30 159
40 213
45 239
50 266
.35962 " Corr.
Corr.
15 83
20 110
.36276 10 53 10 55
.36590 15 79
.36905 20 105
703458 2.37221 30 158
3736 .37537 40 211
4C14 .37854 45 237
4292 .38171|50 263
4570 .38489
" Corr.
54
15 81
20 108
30 162
30 162
40 217
45 244
50 271
-
11
30 166
40 221
45 248
50 276
"Corr.
10 56
15 84
20 112
30 169
40 225
45 253
50 281
TANGENT
1
3.07768
.08073
.08379
.C8685
.C8991 4
01234
3.09298
.09606
.09914 7
.10223
.10532
6898σ
❤
5
3.10842 10
.153 11
.11464 12
.11775 13
.1208714
3.12400 15
.12713 16
.13027 17
.13341 18
.13656 19
3.13972 20
.14288 21
.14605 22
.14922 23
.1524C) 24
3.15558 25
.15877 26
.16197 27
.16517 28
.16838 29
3.17159 30
.17481 31
.17804 32
.18127 33
.18451 34
3.18775 35
.191001 36
.19426 37
. 19752 38
.20079 39
3.20406 40
.2073441
.2106342
.21392 43
.21722 44
3.22053 45
.2238446
.22715 47
.23048 48
.23381 49
3.23714 50
.24049 51
.2438352
.2471953
.25055) 54
3.25392 55
.25729 56
.26067 57
.26406 58
.26745 59
3.2708560
534
0.956305
6390
6475
6560
6644
I
234 5E7BQ 0-2MI DON∞α
5.956729
6
8
9
"I
12
10.957151
13
14
16
17
18
19
15.957571 30
21
22
23
24
22 20222 87~M7 ❤❤♪❤
26
2.7
20.957990
SINE
36
37
38
39
6814
6898
25.958406
8489
8572
8654
8737
6982
7067
30.958820
8902
8985
9067
9150
41
42
43
44
35.959232
9314
9396
9478
9560
46
47
48
49
gགླག༄གྲུཐརྐྱོ
7235 " Corr.
732010
14
740415
7488 20
21
28
40.959642
51
52
53
54
8073
8156
8239
8323
7655 40 56
7739 45 63
7822 50 70
7906
58
59
9724 " Corr.
9805 TO 14
9887 15 20
9968 20 27
45.960050130
50.960456
0537
0618
0698
0779
CORR.
FOR
SEC.
+
TABLE XX.-NATURAL SINES, COSINES,
55.960860
56 0940
57
1021
1101
1182
41
013140 54
0212 45 61
0294 50 68
0375
60.961262
COSINE
.292372
2094
1815
1537
1259
289589
9310
" Corr.
903210 46
875315 70
8475 20 93
42.288196|30 139
7918 40 186
7639 45 209
7360 50 232
7082
.290980
0702
0424
0146
289867
.286803
6525
6246
5967
5688
.285410
5131
4852
4573
4294
.284015
3736
3458
3178
2900
.282620
2342
2062
1783
1504
CORR.
FOR
SEC.
+
.281225
0946
" Corr.
066710 47
0388 15 70
0108 20 93
.278432
8153
7874
7594
7315
.279829 30 140
9550 40 186
9270 45 209
8991 50 233
8712
.277035
6756
6476
6196
5917
.275637
73°

VERSINE
.707628 2.42030
7906
8185
8463
8741
EXSEC
CORR.
FOR
SEC.
+
.709020 2.43666 30 164
9298 .43995 40 219
9576 .44324 45 247
9854 44655 50 274
.710133 .44986
.42356 " Corr.
.4268310 55
.4301015 82
.43337 20 110
.7104112.45317
0690 .45650 " Corr.
0968 .45983 10 56
1247 .46316 15 84
1525.46651 20 112
.711804 2.46986 30 168
2082 .47321 40 224
.47658 45 252
.47995 50 280
.48333
2361
2640
2918
.713197 2.48671
3475 .49010 "Corr.
3754 .4935010 57
4033 4969115 86
4312.50032 20 114
.714590 2.50374 30 171
4869 .50716 40 228
5148 .51060 45 257
5427 .51404 50 285
5706 .51748
.715985 2.52094
6264.52440 " Corr.
6542 .52787 10 58
6822 .5313415 87
7100.53482 20 117
1009
1288
.717380 2.53831 30 175
7658 .54181 40 233
7938 .54531 45 262
8217 .54883 50 291
8496 .55234
718775 2.55587
9054.55940 Corr.
9333.56294 10 59
9612.56649 15 89
9892.57005 20 119
.720171 2.57361 30 178
0450 .57718 40 238
0730.58076 45 268
.58434 50 297
.58794
.721568 2.59154
3804
4083
1847.59514 " Corr.
2126.59876 10 61
2406.60238 15 91
2685 .60601 20 122
.722965 2.60965 30 182
3244 .61330 40 243
.61695 45 273
.6206150 304
.62428
3524
.724363 2.62796
CORR.
FOR
SEC.
+
" Corr.
10 57
15 86
20 115
30 172
40 229
45 258
50 287
" Corr.
10
15
20 117
58
88
30 175
40 234
45 263
50 292
" Corr.
10 59
15 89
20 119
30 179
40 236
45 268
50 298
" Corr.
10 61
15 91
20 122
30 182
40 243
45 273
50 304
" Corr.
10 62
15 93
20 124
30 186
140 248
45 279
50 310
" Corr.
10 63
15 95
20 126
30 190
40 253
45 285
50 316
TANGENT
3.27085
.27426
.27767
.28109
.28452
3.28795
0-231 DO7OK O-232 D
4
5
.29139 6
.29483
.29829 8
.30174
9
3.3052110
.30868 11
.31216 12
.31565 13
.3191414
3.32264 15
.3261416
.32965 17
.33317 18
.3367019
3.34023 20
34377 21
.34732 22
.35087 23
.35443 24
3.35800 25
.36158 26
.36516 27
.36875 28
.37234 29
3.37594 30
.3795531
.38317 32
.38679 33
.39042 34
3.39406 35
.39771 36
.40136 37
.4050238
.40869 39
3.41236 40
.4160441
.41973 42
.42343 43
.4271344
3.43084 45
.43456 46
.43829 47
.4420248
.44576 49
3.44951 50
.45327 51
.45703 52
.46080 53
.4645854
3.46837 55
.47216 56
.47596 57
.47977 58
.48359 59
3.48741 60
535
OU234 BO7BK OI2M4 Borwa 0-234
0.961262
6
8
5.961662
9
11
12
13
14
10.962059
16
17
19
~~~~~ ~~~~~ ~~~
15.962455 30
2534 40
2613 45
21
22
23
24
53
59
18 2692 50 66
2770
20.962849
2928
3006
3084
3163
567
26
27
28
29
SINE
25.963241
3319
3397
3475
3553
31
32
33
34
1342
1422
ოოო ოო
1502
1582
1741
1821
36
1900
1980
30.963630
3708
3786
3863
3941
OGNDA HONO. 8
35.964018
4095
4173
37
38
4250
39 4327
2139 " Corr.
2218 10 13
229715 20
2376 20 26
54
56
57
58
59
40.964404
11
41
42
4481 Corr.
4557 TO 13
4634 15 19
44 471 20 25
43
50.965169
5245
5321
45.964787 30 38
46
4864 40 51
47
4940 45 57
48 5016 50 64
49
5093
51
52
53 5397
5473
55.965548
VERSINES, EXSECANTS, AND TANGENTS
74°

CORR.
FOR
SEC.
+
5624
5700
5775
5850
60.965926
COSINE
.275637
5358
5078
4798
4519
12.72840
2560 " Corr.
2280 TO 47
2000 15 70
172C 20 93
40.271440 30 140
1160 40 187
0880 45 210
0600 50 233
0320
.274239
3959
3679
340C
3120
.270040
269760
•
9480
9200
8920
.268640
8359
8079
7799
7519
1.267238
6958
6678
6397
6117
1.265837
5556
5276
4995
4715
.261628
1347
1066
0785
0504
CORR.
FOR
SEC.
264434
4154 " Corr.
3873 10 47
359215 70
3312 20 94
.263031 30 140
2751 40 187
2470 45 210
2189 50 234
1908
.260224
259943
+
9662
9381
9100
.258819
VERSINE
EXSEC
.724363 2.62796
4642 .63164 " Corr.
4922.63533 10 62
52C2 .63903 15 93
5481.64274 20 124
CORR.
FOR
SEC.
+
.725761 2.64645 30 186
6041 .65018 40 248
6321.65391 45 279
66CC .65765 50 310
6880.66140
11
.727160 2.66515
744C
7720
8000
.66892 Corr.
.67269 10 63
.67647 15 95
8280.68025 20 127
.728560 2.684C5 30 190
8840 .68785 40 254
9120 .69167 45 285
9400 .69549 50 317
9680 .69931
.729960 2.70315
730240 .70700 Corr.
0520
.71085 10 65
0800 .7147115 97
1080 .71858 20 130
.731360 2.72246 30 194
1641 .72635 40 259
1921 .73024 45 292
2201 .73414 50 324
2481 .73806
.732762 2.74198
3603
3883
3C42 .74591 Corr.
3322.74984 10 66
.75379 15 99
.75775 20 132
.734163 2.76171 30 199
4444 .76568 40 265
4724 .76966 45 298
5005 .77365 50 331
5285 .77765
735566 2.78166
5846 .78568
" Corr.
6127 .78970 10 68
6408 .79374 15 102
6688 .79778 20 135
.736969 2.80183 30 203
7249.80589 4C 271
7530.80996 45 3C5
7811 .81404 50 338
8092 .81813
.738372 2.82223
8653.82633 " Corr.
8934 .83045 TO 69
9215 .8345715 104
9496.83871 20 138
.739776 2.84285 30 208
.74C057 .84700 40 277
0338 .85116 45 311
0619
0900 .85951
.85533 50 346
741181 2.86370
CORR.
FOR
SEC.
+
" Corr.
10 65
15 97
20 129
30 194
40 258
45 290
50 323
" Corr.
IC 66
15 99
20 132
30 198
40 263
45 296
50 329
"Corr.
10 67
15 ICI
20 134
30 202
4C 269
45 303
50 336
"Corr.
IG 69
15 103
20 137
30 206
40 275
45 309
150 343
" Corr.
10 70
15 105
20 140
30 210
40 281
45 316
50 351
" Corr.
10 72
15 107
20 143
30 215
40 287
45 322
50 358
TANGENT
3.48741
2
.49125
49509
.49894 3
.50279 Ц
3.50666
.51053 6
.51441 7
.51829 8
.52219 9
5
0-23=
3.5260910
.53COL
.53393 12
.53785 13
.5417914
3.54573 15
.54968 16
.55364 17
.5576118
.56159 19
3.56557 20
.5695721
57357 22
.57758 23
.58160 24
3.58562 25
.58966 26
.59370 27
.59775 28
.6018129
3.60588 30
.60996 31
.6140532
.61814 33
.62224 34
3.62636 35
.63048 38
.63461 37
.6387438
.64289 39
3.64705 40
.6512141
.6553842
.65957 43
.66376 44
3.66796 45
.67217 46
.67638 47
.68061 48
.68485 49
3.68909 50
.6933551
.69761 52
.70188 53
.7061654
3.71046 55
.71476 56
.71907 57
•
.72338 58
.72771 59
3.73205 60
536
0.965926
01234 5O7∞O
6
8
5.966301
9
OURED COM O DO7°) 8.~❤Z MOMMA DIE DOOD ODORO BONO88
11
12
14
10.966675
16
22
23
24
26
27
31
SINE
20.967415
7489
7562
7636
7709
32
33
34
6001
6076
37
49
15.967046 30
7120 40
719445 56
7268 50 62
7342
25.967782
7856
7929
8002
8075
36
6151
6226
38
30.968148
8220
8293
8366
8438
6376
6451
42
43
44
6526
6600
35.968511
8583
37 8656
46
49
6749 "Corr.
6823 10
6898 15
6972 20 25
53
54
40.968872
8728
8800
50.969588
9659
9730
9801
9872
TABLE XX. -NATURAL SINES, COSINES,
75°

CORR.
FOR
SEC.
+
45.969231 30 36
9302 40 48
9374 45 54
9445 50 60
9517
55.969943
56.970014
8944 " Corr.
9016 10 12
9088 15 18
9159 20 24
0084
0155
0225
12
60.970296
OGEN MON
L
COSINE
.258819
8538
8257
7976
7695
.257414
7133
6852
6570
6289
.253195
2914
2632
2351
2069
.256008
5727 "Corr.
5446 10 47
51645 70
4883 20 94
.254602 30 141
4321 40 188
4039 45 211
3758 50 234
3477
251788
1506
1225
0943
0662
.250380
0098
.249817
9535
9253
.248972
8690
8408
8126
7844
CORR.
FOR
SEC.
.244743
4461
4179
3897
3615
+
.247563
7281
" Corr.
6999 10 47
671715 70
6435 20 94
246153 30 141
5871 40 188
5589 45 211
5307 50 235
5025
.243333
3051
2768
2486
2204
.241922
VERSINE
EXSEC
CORR.
FOR
SEC.
+
741181 2.86370
1462.86790 " Corr.
1743 .87211 10 71
2024.87633 15 106
2305 .88056 20 142
.742586 ||2.88479 30 212
2867 .88904 40 283
3148.89330 45 319
3430.89756 50 354
3711.90184.
=
CORR.
FOR
SEC.
+
" Corr.
10
15 110
20 146
.743992 2.90612
4273 .91042 " Corr.
4554.91473 10 72
4836 .91904 15 109
5117.92337 20 145
.745398 2.92770 30 217
5679
5961
30 225
.93204 40 290 40 299
.93640 45 32645 337
6242.94076 50 362
6523.94514.
50 374
3.73205
.73640
73 .74075
.74512
.74950 4
||30 220
40 293
45 330
||50 366
11
.09063 Corr.
5821.09535 10 79
6103.1000915 119
6385 .10484 20 159
.756667 3.10960 30 239
6949 .11437 40 318
7232 .11915 45 358
.12394 50 398
7796.12875
7514
.7580783.13357
Corr.
746805 2.94952
7086.95392 31
Corr.
7368.95832 10 74
77
7649.96274| 15 ||| 15 115
7931.96716 20 148 20 153
10
" Corr.
10 75
15 112
20 150
=
.748212 2.9716030 22230 230
8494 .97604 40 296 40 306
8775.98050 45 333
9057.98497 50 370
9338.98944|
45 344
50 383
" Corr.
10 82
15 123
20 164
TANGENT
30 246
40 328
45 369
50 410
01234
5
3.75388
.75828 6
.76268 7
.76709 8
.77152 9
3.77595 10
.78040 11
.78485 12
.78931 13
.79378 14
.749620 2.99393
#1
9902.99843 Corr.
.750183 3.00293 10 76
0465.00745 15 114
" Corr.
10 78
15 117
45 352
50 391
3.91364 40
11
0747 .01198 20 152 20 157
.751028 3.0165230 22730 235
1310.02107 40 303 40 313
1592 .0256345 341
1874 .03020 50 379
2156 .03479
752437 3.03938
2719.04398 Corr. " Corr. .91839 41
3001.04860 TO 78 10 80.9231642
3283.0532215 116 15 120 .92793 43
3565 .05786 20 155
.9327 44
.753847 3.06251 30 233
3.93751 45
4129 .06717 40 310
.9423246
4411 .07184 45 349
94713 47
4693 .07652 50 388
.95196 48
4975 .08121
.95680 49
.755257 3.08591
5539
20 160
30 240
40 320
45 360
50 400
3.79827 15
.80276 16
.80726 17
.81177 18
.81630 19
3.82083 20
.82537 21
.82992 22
.83449 23
.83906 24
3.84364 25
.84824 26
.85284 27
.85745 28
.86208 29
3.86671 30
.87136 31
.87601 32
.88068 33
.88536 34
3.89004 35
.89474 36
89945 37
.90417 38
.90890 39
3.96165 50
.9665151
.97139 52
.97627 53
.9811754
3.98607 55
.99099 56
.99592 57
4.00086 58
.00582 59
4.01078 60
537
0.970296
1234 5O7OQ 0-234 5O
6
8
5.970647
9
12
13
14
10.970995
16
17
18
19
22222 222~~ ❤❤❤❤❤ mmmmm d=IRE BOND 5885
21
15.971342 30 35
1411 40 46
1480 45 52
1549 50 58
1618
23
234
20.971687
1755
1824
1893
1961
24
5678∞∞
26
27
28
29
31
25.972029
(32
33
34
36
37
30.972370
2438
2506
38
39
SINE
42
0366
0436
0506
0577
35.972708
2776
2843
2910
2978
43
44
23.
LOLOLO5
0716
0786
0856
0926
47
48
49
40.973045
52
53
54
1065 " Corr.
113410 12
120415 17
1273 20 23
2098
2166
2234
2302
56
2573
2641
50.973712
3778
3844
45.973379 30 33
3446 40 44
3512 45 50
3579 50 56
3645
VERSINES, EXSECANTS, AND TANGENTS
76°

55.974042
CORR.
FOR
SEC.
+
3112 " Corr.
3179 10
324615
3312 20 22
3910
3976
4108
57 4173
58 4239
59 4305
60 .974370
•
COSINE
241922
1640
1357
1075
0793
240510
0228
.239946
9663
9381
.239098
8816" Corr.
8534 10 47
825115 71
7968 20 94
.236273
5990
5708
5425
5142
.237686 30 141
7403 40 188
7121 45 212
6838 50 235
6556
.234859
4577
4294
4011
3728
.233445
3162
2880
2597
2314
.232031
1748
1465
1182
0899
CORR.
FOR
SEC.
+
.227784
7501
7218
6935
6651
.226368
6085
5801
5518
5234
.224951
VERSINE
EXSEC
9207
758078 3.13357
8360 .13839
" Corr.
8643 .14323 10 81
8925.1480915 122
.15295 20 163
.759490 3.15782 30 244
.16271 40 326
.760054 .16761 45 367
0337 .17252 50 407
0619 .17744
CORR.
FOR
SEC.
+
11
.760902 3.18238
1184 .18733 Corr.
1466 .19228 10 83
1749 .19725 15 125
2032 .20224 20 167
.762314 3.20723 30 250
2597 .21224 40 334
2879 .21726 45 376
3162 .22229 50 417
3444 .22734
.7637273.23239
4010 .23746 " Corr.
4292 .2425510 85
4575 .24764 15 128
4858 .25275 20 171
5423
5706
5989
.765141 3.25787 30 257
.26300 40 342
.26814 45 385
.27330 50 428
6272.27847
.766555 3.28366
6838 .28885 " Corr.
7120.29406 TO 88
7403 .29929 15 132
7686.30452 20 175
.767969 3.30977 30 263
8252 .31503 40 351
8535 .3203145 395
8818 .32560 50 438
9101.33090
CORR.
FOR
SEC.
+
" Corr.
10 84
15 126
20 168
.773632 3.41759 30 277
3915.4231240 369
4199.42867 45 415
4482 .43424 50 461
4766.43982
.775049 3.44541
30 252
40 336
45 378
50 420
" Corr.
10 86
15 129
20 172
30 258
40 344
45 387
50 430
" Corr.
10 88
15 132
20 176
.769384 3.33622
" Corr.
.230616
0333
" Corr.
0050 10 47
17.229767 15 71
9667 .34154 " Corr.
9950 .3468910 90
110
92
|.770233 .3522415 135 15 139
9484 20 94 0516 .3576120 180 20 185
.229200 30 142.770800 3.36299 30 270
891740 189 1083 .36839 40 360
8634 45 212
1366 .37380 45 405
8351 50 236
1649 .37923 50 450
8068
1932 .38466
.772216 3.39012
30 264
40 352
45 396
50 440
" Corr.
10 90
15 135
||20 180
30 270
40 361
45 406
50 451
30 277
40 370
45 416
50 462
2499.39558 "Corr." Corr.
2782 .40106 10 92 110 95
3065 .40656 15 138 15 142
3349 .41206 20 185 120 189
130 284
40 379
45 426
150 474
TANGENT
4.01078 0
.01576
.02074
.02574
.03076
4.03578
.04081
01234 5O789
6
.04586
.05092 8
.05599
4.06107 10
.06616 |
.07127 12
.07639 13
.0815214
4.08666 15
.09182 16
.09699 17
.1021618
.10736 19
4.1125620
.11778 21
.12301 22
.12825 23
.13350 24
4.13877 25
.14405 26
.14934 27
.15465 28
.15997 29
4.16530 30
.17064 31
.17600132
..18137 33
.18675 34
4.19215 35
.19756 38
.20298 37
.20842 38
.21387 39
4.21933 40
.22481 41
.23030 42
.23580 43
.24132 44
4.24685 45
.25239 46
.25795 47
.26352 48
.26911 49
4.27471 50
.28032 51
.28595 52
.29159 53
.2972454
4.30291 55
.30860 56
.31430 57
.32001 58
.32573 59
4.33148 60
538
0.974370
4436
4501
4566
4631
0-237 DONOM O-NMI NON * **70* 8.~♡~ -87.8
-20
222
5.974696
4761
4826
4891
4956
6
7
8
9
10.975020
12
13
14
16
17
18
19
15.975342 30
5406 40 43
5471 45 48
5534 50 53
5598
21
20.975662
5726
5790
5853
5917
22
23
24
26
25.975980
27
28
29
31
32
33
30.976296
34
36
37
38
39
35.976610
41
42
43
44
SINE
46
47
48
8739 23ཤྩg čརྴ
49
40.976922
51
52
5085
"Corr.
5149 10 ||
5214 15 16
5278 20 21
53
54
6044
6107
6170
6233
56
45.977231 30 31
7293 40 41
7354 45 46
7416 50 51
7477
57
6359
6422
6484
6547
50.977539
7600
7661
58
59
6672
6735
6797
6859
55.977844
7905
7966
8026
8087
60.978148
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
6984
"Corr.
7046 10
10
7108 15 15
7169 20 21
7722
7783
COSINE
.224951
4668
4384
4101
3817
.222116
1832 Corr.
1548 10 47
126515 71
098 20 95
32.22069730 142
0414 40 189
0130 45 213
.219846 50 236
9562
.223534
3250
2967
2683
2399
.219279
8995
8711
8427
8143
.217859
7575
7292
7008
6724
.216440
6156
5872
5588
5304
.215019
4735
4451
4167
3883
CORR.
FOR
SEC.
.210756
0472
0187
.209903
9619
+
.213599
33.15 Corr.
3030 10 47
2746 15 71
2462 20 95
.212178 30 142
1893 40 189
1609 45 213
1325 50 237
1040
.209334
9050
8765
8481
8196
.207912
77°

VERSINE
EXSEC
.7750493.44541
5332 .45102
5616
.45664
.46228
5899
6183 46793
776466 3.47360
6750 .47928
.48498
7033
7317
.49069
7601 .49642
.50781
.777884 3.50216
8168
8452
8735 .51947
9019 .52527
.51368
.54863
.55451
.779303 3.53109
9586 .53692
9870 .54277
.780154
0438
.7807213.56041
1005 .56632
1289.57224
1573 .57819
1857 .58414
782141 3.59012
2425 .59611
2708 .60211
2992 .60813
3276 .61417|
.783560 3.62023
3844 .62630
4128 .63238
4412 .63849
4696
.64461
784981 3.65074
5265 .65690
5549.66307
5833.66925
6117 .67545
.786401 3.68167
6685 .68791
6970 .69417
7254 || . 70044
7538.70673
.787822 3.71303
8107 .71935
8391.72569
8675
.73205
8960.73843
.789244 3.74482
9528
.75123
9813.75766
.76411
0381.77057
790097
790666 3.77705
0950
.78355
1235.79007
1519.7966|
1804.80316
.792088 3.80973
DIFF. 10"
93.5
93.7
94.0
94.2
94.5
94.7
95.0
95.2
95.5
95.7
95.8
96.2
96.5
96.7
97.0
97.2
97.5
97.7
98.0
98.3
98.5
98.7
99.2
99.2
99.7
99.8
100.0
100.3
100.7
101.0
101.2
101.3
101.8
102.0
102.2
102.7
102.8
103.0
103.3
103.7
104.0
104.3
104.5
104.8
105.0
105.3
105.7
106.0
106.3
106.5
106.8
107.2
107.5
107.7
108.0
108.3
108.7
109.0
109.2
109.3
95.8
96.2
96.5
96.7
96.8
97.2
97.3
97.7
98.0
98.0
98.5
98.7
98.8
99.2
99.3
99.7
100.0
100.2
100.5
100.7
101.0
101.2
101.5
101.7
102.0
102.3
102.5
102.8
103.2
103.3
103.7
103.8
104.2
104.5
104.7
105.0
105.3
105.5
105.8
106.2
106.3
106.7
107.0
107.3
107.5
107.8
108.2
108.3
108.8
109.0
109.2
109.7
109.8
110.2
110.5
110.8
111.0
111.5
11.7
112.0
TANGENT
4.33148 0
.33723 1
34300 2
:34879 3
.35459
·
4.36040 5
.36623 6
.37207 7
.37793 8
.38381 9
4.38969 10
.39560 11
.40152 12
•
4.41936 15
42534 16
43134 17
·
40745 13
41340 14
43735 18
.44338 19
·
4.44942 20
.45548 21
.46155 22
46764 23
.47374 24
·
4.47986 25
.48600 26
.49215 27
.49832 28
.5045! 29
4.51071 30
.5169331
.52316 32
.52941 33
.53568 34
4.54196 35
.54826 36
.55458 37
.5609 38
.56726 39
4.57363 40
.5800141
.58641 42
.59283 43
.59927 44
4.6057245
.6121946
.61868 47
.62518 48
.63171 49
4.63825 50
.64480 51
.65138 52
.6579753
.66458 54
4.6712155
.67786 56
.68452 57
.69121 58
.69791 59
4.70463 60
539
0.978148
8208
8268
8329
8389
1234
5.978449
6678σ
9
01234
10.978748
11
12
13
14
56789 0-234
16
17
18
15.97904630 30
9105 40 40
9164 45 44
9223 50 49
9282
19
22222 22~~~ ~~~~Õ HAMMA DE22 8700
20.979341
21
23
9399
9458
9517
9575
25.979634
9692
9750
9809
9867
24
5678
26
27
28
29
31
33
34
30.979925
9983
32.98C040
C098
0156
36
37
SINE
35.980214
0271
C329
0386
0443
38
39
41
42
43
44
8509
8569
8629
8689
55555
40.980500
0558
C615/10
0672 15
C728 20
46
47
8808 "Corr.
8867 10 10
8927 15 15
8986 20 20
48
49
45.980785 30
08421 40
0899 45
0955 50
1012
51
52
53
54
50.981068
1124
1180
1237
1293
56
57
58
59
CORR.
FOR
SEC.
+
55.981349
1404
1460
1516
1572
VERSINES, EXSECANTS, AND TANGENTS
78°

60.981627
" Corr.
9
14
Corr
19
2333
28
38
43
47
COSINE
.207912
7627
7343
7058
6773
.206489
6204
592C
5635
535C
.205066
4781 " Corr.
4496 | 10 47
4211 15 71
3926 20 95
202218
1933
1648
1363
1078
.203642 30 142
3357 40 190
3072 45 214
2787 50 237
2502
.200793
0508
0223
.199938
9653
.199368
9083
8798
8513
8228
.197942
7657
7372
7C87
6802
.196517
CORR.
FOR
SEC.
+
6231 " Corr.
5946 TC 48
5661 15 71
5376 20 95
.193664
3378
3093
2807
2522
.192236
1951
1666
1380
IC94
.190809
VERSINE
792088 3.80973
2373.81633
2657.82294
2942 .82956
3227.83621
793513.84288
3796 .84956
4080.85627
4365 .86299
4650.86973
EXSEC
794934 3.87649
5219.88327
5504.89007
5789 .89689
6074 .90373
.796358 3.91058
6643 .91746
6928
.92436
7213
.93128
7498.93821
.797782 3.94517
8067 .95215
2352.95914
8637.96616
8922.97320
.799207 3.98025
9492.98733
9777 .99443
.800062 4.00155
0347 .00869
.800632 4.01585
0917 .023C3
1202 .03024
1487 .03746
1772 .04471
8020584.05197
2343 .05926
2628 .06657
2913 .07390
3198 .08125
.195090 30 143.804910 4.12583
4805 40 190
4520 45 214
4234 50 238
3949
.8C3483 4.08863
3769 .09602
4054 .10344
4339
4624 .11835
.11088
5195 .13334
5480 .14087
5766
6051
6622
6907
7193
.14842
.15599
.806336 4.16359
.17121
.17886
.18652
7478.19421
.8077644.20193
8049
.20966
8334
.21742
8620
.22520
8906 .23301
.809191 4.24084
DIFF. 10"
110.0 112.3
110.2
112.7
110.3
112.8
110.8
111.2
111.3
111.8
112.0
112.3
112.7
113.0
113.3
113.7
114.C
114.2
114.7
115.0
115.3
115.5
116.0
118.0
118.3
118.7
119.0
119.3
119.7
120.2
120.3
120.8
121.C
121.5
121.8
116.3
116.5
117.0 119.5
122.2
122.5
123.0
117.3 119.7
117.5
120.0
123.2
123.7
124.0
124.5
124.7
125.2
125.5
125.8
126.2
126.7
127.0
127.5
113.3
113.5
127.7
128.2
128.7
113.8
114.2
114.5
114.8
115.2
128.8
129.3
129.7
130.2
130.5
115.3
115.8
116.2
116.3
116.8
117.0
117.3
117.8
118.0
118.3
118.8
119.0
120.5
120.8
121.0
121.5
121.8
122.2
122.5
122.8
123.2
123.5
124.0
124.2
124.7
125.0
125.3
125.8
126.0
126.5
126.8
127.2
127.5
128.0
128.3
128.7
129.2
129.3
129.8
130.3
130.5
131.0
131.5
131.7
132.2
132.7
132.8
肇
​TANGENT
4.70463
.71137
.71813 2
.72490
.73170 4
01234
4.73851 5
.74534 6
75219 7
.75906 8
.76595 9
4.77226 10
.77978 11
.78673 12
.7937C 13
.80068 14
4.80769 15
.81471 16
.82175 17
.82882 18
.83590 19
4.84300 20
.85013 21
.85727 22
.86444 23
.87162 24
4.87882 25
.88605 26
.89330 27
.90056 28
.90785 29
4.91516 30
.92249 31
.92984 32
.93721 33
.94460 34
4.95201 35
.95945 36
.96690 37
.97438 38
.98188 39
4.98940 40
.99695 41
5.00451 42
.01210 43
.019744
5.0273445
.03499 46
.04267 47
.05037 48
.05809 49
5.06584 50
.07360 51
.08139 52
.08921 53
.0970454
5.10490 55
.11279 56
.12069 57
.12862 58
.13658 59
5.14455 60
540
01.981627
1234 5O7
6
8
9
5.981904
0-232 5O7ɑa
||
12
13
124
10.982178
16
17
18
19
21
22
NA DANO0 870M2 CON DE 99700 850HZ HONOR 8
23
24
26
5.982450 30 27
2505 40 36
2559 45
2613 50 45
2667
20.982721
2774
2828
2882
2935
27
28
29
31
25.982989
3042
3096
3149
3202
32
33
34
36
30.983255
3308
3361
3414
3466
37
38
39
SINE
41
1683
1738
42
1793
1848
35.983519
3572
3624
3676
3729
43
44
1959
2014
46
2069
2123
40.983781
47
48
49
51
2233
228710
234215
2396 20
45.984041
58
50.984298
4350
4401
4452
4503
CORR.
FOR
SEC.
+
55.984554
4605
4656
4707
4757
TABLE XX.-NATURAL SINES, COSINES,
" Corr.
" Corr.
3833
3885 TO
9
3937 15 13
3989 20
17
60.984808
FEWN TÕEL
30 26
4092 40 35
414445 39
4196 50
4247
43
14
COSINE
.190809
0523
0238
.189952
9667
.189381
9095
8810
8524
8238
.185095
4809
4523
4237
3951
.187953
7667 " Corr.
738110 48
709615 71
6810 20 95
.186524 30 143
6238 40 190
5952 45 214
5667 50 238
5381
.183665
3380
3094
2808
2522
.182236
1950
1664
1377
1091
180805
0519
0233
CORR.
FOR
SEC.
.176512
6226
5940
5653
5367
+
.179947
9661
.179375
9088 " Corr.
8802 TO 48
8516 15 72
8230 20 95
.177944 30 143
7657 40 191
737145 215
7085 50 239
6798
.175080
4794
4508
4221
3935
.173648
79°

VERSINE
EXSEC
.8091914.24084
9477 .24870
9762
.810048
.25658
.26448
0333 .27241
.810619 4.28036
0905 .28833
1190 .29634
1476.30436
1762 .31241
.812047 4.32049
2333 .32859
2619.33671
2904 .34486
3190 .35304
.813476 4.36124
3762 .36947
4048 .37772
4333 .38600
4619.39430
.8149054.40263
5191 .41099
5477.41937
5763
.42778
6049 .43622
.8163354.44468
6620 .45317
6906 .46169
7192 .47023
7478 .47881
.817764 4.48740
·
8050 49603
8336.50468
8623 .51337
8909.52208
.819195 4.53081
9481 .53958
9767.54837
.820053
.55720
0339 .56605
.820625 4.57493
0912.58383
1198.59277
1484.60174
1770 .61073
.62881
|.822056 4.61976
2343
2629
2915 .64701
3202.65616
.63790
.823488 4.66533
3774 .67454
4060 .68377
4347.69304
4633 .70234
.824920 4.71166
5206 .72102
5492 .73041
5779 .73983
6065 74929
.826352 4.75877
DIFF. 10"
131.0 133.5
131.3 133.7
131.7 134.2
132.2 134.7
132.5 134.8
132.8
133.5
133.7
134.2
134.7
135.0
135.3
135.8
136.3
136.7
139.3
139.7
140.2
140.7
141.0
141.5
142.0
142.3
143.0
143.2
137.2
139.7
140.0
140.3
137.5
138.0
138.3 140.8
138.8 141.3
146.2
146.5
147.2
147.5
148.0
148.3
149.0
149.5
149.8
150.5
150.8
151.5
151.8
152.5
152.8
135.5
135.7
136.3
153.5
153.8
154.5
155.0
155.3
136.5
137.0
143.8
146.2
144.2
146.7
144.8
147.2
145.2 147.7
145.5 148.0
156.0
156.5
157.0
157.7
158.0
137.5
137.8
138.3
138.7
139.2
141.7
142.2
142.7
143.0
143.5
144.0
144.3
144.8
145.3
145.8
148.5
149.0
149.5
150.0
150.5
150.8
151.3
152.0
152.3
152.8
153.3
154.0
154.3
154.8
155.3
155.8
156.3
156.8
157.5
157.8
158.5
158.8
159.5
160.0
160.5
TANGENT
5.14455
.15256 1
.16058 2
.16863
.1767! L
བ-ས་པ་
5.18480 F
.19293 €
.20107
.20925 8
.21744
9
5.22566 10
.2339111
.24218 12
.25048 13
.25880 14
BENB σ
5.2671515
.27553 16
.28393 17
.29235 18
.30080 19
5.30928 20
.31778 21
.32631 22
.33487 23
.34345 24
5.35206 25
.36070 26
.36936 27
.37805 28
.38677 29
5.39552 30
.4042931
.4130932
.42192 33
.43078 34
5.43966 35
.44857 36
.4575137
.46648 38
.47548 39
5.48451 40
.4935641
.5026442
.51176 43
.52090 44
5.53007 45
.53927 46
.5485147
.55777 48
.5670649
5.57638 50
.5857351
.5951 52
.60452 53
.61397 54
•
5.6234455
.63295 56
64248 57
.65205 58
.6616559
5.67128 60
541
0.984808
4858
4909
4959
5009
1
234 56789
5.985059
01234 DO789 O
10.985309
11
12
13
14
5358
" Corr.
5408 10 8
5457 15 12
5507 20 16
15.985556 30 25
5605 40 33
5654 45 37
5704 50 41
5752
16
17
18
~~~~~ ~
19
20.985801
21
22
5850
5899
5948
5996
25.986044
6093
6141
*** ***** 8~~♡~
23
24
26
27
28
29
31
32
33
34
5 £5555 @www.
30.986286
6334
6382
6429
6477
36
37
38
39
35.986525
6572
6620
41
42
43
44
SINE
40.986762
46
47
go to 5:
5109
5159
48
49
5209
5259
52
53
54
45.986996 30
7043 40
7090 45
7136 50
7183
g g⌘ཐg¢
6189
6238
56
50.987229
7275
7322
7368
7414
57
58
59
6667
6714
55.987460
7506
7551
7597
7643
6809
6856 10
6903 15
6950 20
CORR.
FOR
SEC.
+
VERSINES, EXSECANTS, AND TANGENTS
80°

60.987688
" Corr.
8
12
16
COSINE
31
35
.173648
3362
3075
2789
2502
.172216
1929
1642
1356
1069
.167916
7629
7342
7C56
6769
.166482
6195
5908
5621
5334
.17C783
0496 " Corr.
0210 10 48
.169923 15 72
9636 20 96
.169350 30 143
9C63 40 191
8776 45 215
8489 50 239
8203
.165048
4761
4474
4187
3900
.163613
3326
3039
2752
2465
-
.162178
1891 " Corr.
16C4 TO 48
1317 15 72
1030 20 96
23.160743 30 144
0456 40 191
016845 215
39.15988150 239
9594
.159307
9020
8732
8445
8158
CORR.
FOR
SEC.
.157871
7584
7296
7009
6722
.156434
+
VERSINE
EXSEC
.826352 4.75877
6638.76829
6925 .77784
7211 .78742
7498.797C3
8931
.827784 4.80667
8C71 .81635
8358 .82606
8644.83581
.84558
.829217 4.85539
9504.86524
9790 .875||
.830077.88502|
0364.89497
.8306504.90495
0937.91496
1224.92501
1511.93509
1797.94521
.8320844.95536
2371.96555
2658 .97577
2944.98603
3231.99633
.833518 5.CC666
3805 .01702
4092
.02743
4379.03787
.04834
4666
.8392575.22113
9544 .23226
9832.24343
.840119 .25464
0406 .26590
.84C693 5.27719
0980 .28853
1268 .2999|
1555 .31133
1842 .32279
DIFF. 10"
.842129 5.33429
2416 .34584
27C4 .35743
2991 .36906
3278 .38073
.843566 5.39245
158.7 161.0
159.2
161.7
159.7
162.2
160.2
162.7
160.7
163.2
161.3
161.8
162.5
162.8
163.5
.8349525.05886
5239.06941
5526 .08000
5813.09062
6100 .10128
.836387 5.11199
6674 .12273
6961.13350
7248 .14432
7535 .15517
182.2
.837822 5.16607
8109 .17700
8396.18797
8683 .19898
8970 .21004 184.3
182.8
183.7
184.8
164.2
164.5
167.7
165.2
165.8 168.2
166.3 168.8
172.7
173.5
174.0
174.5
175.3
166.8 169.3
167.5 169.8
168.0 170.5
168.7 171.2
169.3
171.5
169.8
170.3
171.0
171.7 174.2
172.2
174.5
175.8
176.5
177.0
177.7
178.5
179.0
179.5
180.3
180.8
181.7
185.5
186.2
163.7
164.3
186.8
187.7
188.2
164.8
165.3
166.0
189.0
189.7
190.3
191.0
191.7
166.5
167.0
192.5
193.2
193.8
194.5
195.3
172.3
172.8
173.3
175.3
175.8
176.3
177.0
177.7
178.3
179.0
179.5
180.2
180.7
181.5
182.0
182.8
183.3
184.0
184.7
185.3
185.8
186.7
187.3
188.0
188.7
189.3
190.0
190.7
191.3
192.0
192.8
193.5
194.2
194.8
195.7
196.3
197.0
197.7
TANGENT
5.67128
.68094
.69064
.70037
.71013
5.71992
01234
5
.72974 6
.73960 7
.74949 8
.75941 9
5.76937 10
.77936 IE
.78938 12
.7994413
.80953 14
5.81966 15
.82982 16
.84001 17
.85024 18
.86051 19
5.8708020
.88114 21
.89151 22
.90191 23
.91236 24
5.92283 25
.93335 26
.9439027
.95448 28
.96510 29
5.9757630
98646 31
.99720 32
6.00797 33
.01878 34
6.02962 35
.0405136
.0514337
.06240 38
.07340 39
6.08444 40
.0955241
.1066442
.1177943
.1289944
6.1402345
.1515146
.16283 47
.17419 48
.18559 49
6.1970350
.20851 51
.22003 52
.23160 53
.2432154
6.2548655
.26655 56
.27829 57
.29007 58
.30189 59
6.31375 60
542
-
0.987688
7734
7779
7824
7870
OI23A BON89
6
7
1234
10.988139
||
12
13
14
16
17
18
19
22222
21
15.988362 30
8406 40 29
8450 45 33
8494 50 37
8538
20.988582
22
23
24
~~~~~ AMM❤~ ❤❤♪❤. DIE OO9 85087 ONOO O
26
27
25.98880C
8843
8886
893C
8973
28
29
234
32
33
30.989016
9059
9102
9144
9187
34
36
37
38
39
35.989230
9272
9315
9357
9399
SINE
41
42
43
44
.987915
7960
8005
8050
8094
46
47
40.989442
48
49
51
52
8184 "
8228 10 Corr
8273 15 ||
8317 20
15
8626
8669
8713
8756
53
45.989651 30 21
9693 40 28
9735 45 31
9776 50 35
9818
50.989859
9900
9942
9983
56
57
58
59
54.990024
55.990065
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
9484
" Corr.
9526 10
7
9568 15 10
9610 20 14
0106
0146
0187
0228
60.990268
COSINE
.156434
6147
5860
5572
5285
11
.153561
3273 Corr.
2986 10 48
2698 15 72
2411 20 96
22.152123 30 144
1836 40 192
1548 45 216
1261 50 240
0973
.154998
4710
4423
4136
3848
.150686
0398
0111
.149823
9535
.149248
8960
8672
8385
8097
.147809
7522
7234
6946
6658
.146371
6083
5795
5508
5220
.144932
CORR.
FOR
SEC.
4644 " Corr.
4356 10 48
4068 15 72
3780 20 96
.143493 30 144
3205 40 192
2917 45 216
2629 50 240
2341
.142053
1765
1477
1189
0901
.140613
0325
0037
+
.139749
9461
.139173
81°

VERSINE
843566 5.39245
3853 .40422
4140 .41602
4428 .42787
4715.43977
845002 5.45171
5290.46369
5577
.47572
5864 .48779
6152 .49991
EXSEC
.846439 5.51208
6727
7014
7302
.52429
.53655
54886
7589 .56121
.849314 5.63633
9602
.64902
.66176
9889
.850177
.850752 5.70027
.71321
.72620
.73924
.75233
1040
1328
207.5
.847877 5.57361
8164 .58606
8452 .59855 208.2
8739
9027
.61110 209.2
.62369
209.8
210.7
1615
1903
211.5
212.3
213.0
.67454
0465 .68738 214.0
214.8
.8521915.76547
2478
.77866
2766
.79191
3054
.80521
3342 .81856
.853629 5.83196
4492 .87250
4780 .88612
855068 5.89979
5356.91352
5644
5932
6220
.92731
.94115
.955C5
.856507 5.96900
6795
7083
.98301
.99708
7371 6.CI 120
7659
.02538
857947 6.03962
DIFF. 10"
9963
.860251
0539
196.2 198.5
196.7 199.2
197.5
200.0
198.3
200.7
199.0
201.5
3917 .84542
224.3
4205.85893 225.2
226.2
227.0
227.8
859387 6.11171
9675.12630
199.7
200.5
201.2
202.0
202.8
203.5
204.3
205.2
205.8
206.7
.14096
.15568
.17046
.860827 6.18530
215.7
216.9
217.3
218.2
219.0
219.8
220.8
221.7
222.5
223.3
8235 .05392 238.3
8523 .06828 239.3
8811 .08269 240.2
9099 .09717 241.3
242.3
228.8
229.8
230.7
231.7
232.5
233.5
234.5
235.3
236.3
237.3
243.2
244.3
245.3
246.3
247.3
2C2.2
202.8
203.7
204.5
205.2
206.0
206.8
207.5
208.4
209.0
210.0
210.7
211.5
212.3
213.0
2.4.0
214.7
215.7
216.3
217.3
218.0
218.8
219.8
220.7
221.5
222.3
223.2
224.0
225.0
225.8
226.8
227.5
228.7
229.3
230.3
231.3
232.2
233.2
234.0
235.0
236.0
236.8
237.8
238.8
239.7
240.8
241.7
242.7
243.8
244.7
245.7
246.7
247.8
248.7
249.8
TANGENT 1
6.31375 0
.32566 I
.33761 2
.34961 3
.36165 4
6.37374 5
.38587 6
39804 7
.41026 8
.42253 9
6.43484 10
.44720 1
.4596 12
.47206 13
.48456 14
6.49710 15
.50970 16
.52234 17
.53503 18
.54777 19
6.56055 20
.57339 21
.58627 22
.59921 23
.61219 24
6.62523 25
.63831 26
.65144 27
.66463 28
.67787 29
6.6911630
.70450 31
.71789 32
.73133 33
.74483 34
6.75838 35
.77199 36
.78564 37
.7993638
.8131239
6.82694 40
.8408241
.85475 42
.86874 43
.88278 44
6.89688 45
.9110446
.92525 47
.93952 48
.9538549
6.96823 50
.98268 51
.99718 52
7.01174 53
.02637 54
7.04105 55
.05579 56
.07059 57
.08546 58
.10038 59
7.1153760
543
0.990268
0308
C349
0389
0429
0123→
4
66700 01ZMI DODOO 20
5.990469
0510
0549
0589
0629
8
9
10.990669
12
13
14
16
17
15.990866 30
18
19
COME DONE0 87037 ❤❤ I DO 87 KONAX 8
20.991061
21
22
23
~~~~
24
26
25.991254
27
28
29
SINE
31
32
33
34
36
30.991445
1483
1521
1558
1596
37
38
39
" Corr.
0708
0748 10 7
0787 15 10
0827 20 13
20
0905 40 26
0944 45 29
0983 50 33
1022
35.991634)
1671
1709
1746
1783
441
42
43
44
1100
1138
1177
1216
49
1292
1331
1369
1407
40.991820
1857
1894 10
1931 15
1968 20
54
45.992005| 30
2042 40
207845
CORR.
FOR
SEC.
+
50.992187
2224
226C
VERSINES, EXSECANTS, AND TANGENTS
82°

2296
2332
55.992368
2404
2439
2475
2511
60.992546
" Corr.
6
28
2115 50 31
2151
DON 250.
12
COSINE
.139173
8885
8597
8309
8021
137733
7444
7156
6868
6580
.133410
3121
2833
2545
2256
.136292
" Corr.
6004
48
5716 10
5427 15 72
5139 20 96
.13485130 144
4563 40 192
427445 216
3986 50 240
3698
.131968
1680
1391
1103
0815
.130526
0238
. 129949
9661
9372
127642
" Corr.
7353
7065 10 48
6776 15 72
6488 20 96
18.12619930
129084
8796
8507
8219
7930
CORR.
FOR
SEC.
.124756
4467
4179
+
144
5910 40 192
5622 45 216
5333 50 241
5045
3890
3602
.123313
3024
2736
2447:
2158
.121869
VERSINE
.860827 6.18530
1115 .20020
1403 .21517]
1691 .23019,
1979 .24529
EXSEC
862267 6.26044
2556 .27566
2844 .29095|
3132 .30630
3420 .32171
.863708 6.33719
3996 .35274
4284 .36835
4573 .38403
4861 .39978
865149 6.41560
5437 .43148
5726 .44743
6014 .46346
6302 .47955
.866590 6.49571
6879 .51194
7167 .52825
7455 .54462
7744 .56107
.868032 6.57759
8320 .59418
8609 .61085
8897 .62759
9185 .64441
.869474 6.66130|
9762 .67826
.69530
0339 .71242
0628 .72962
.870051
.8709166.74689
1204 .76424
1493
.78167
1781 .79918
2070 .81677
.872358 6.83443
2647 .85218
2935 .87001
3224 .88792|
3512 .90592
.873801 6.92400
4090 .94216
4378.96040
4667.97873
.99714
.875244 7.01564
5533 .03423
5821 .05291
4955
6110 .07167
6398 .09052
.876687 7.10946
DIFF. 10"
6976
7264
7553 .16681
7842 .18612
.878131 7.20551
248.3
249.5
250.3
251.7
252.5
253.7
254.8
255.8
256.8
258.0
259.2
260.2
261.3
262.5
263.7
264.7
265.8
267.2
268.2
269.3
270.5
271.8
272.8
274.2
275.3
276.5
277.8
279.0
280.3
281.5
282.7
284.0
285.3
286.7
287.8
289.2
290.5
291.8
293.2
294.3
295.8
297.2
298.5
300.0
301.3
302.7
304.0
305.5
306.8
308.3
309.8
311.3
312.7
314.2
315.7
.12849 317.2
.14760 318.5
320.2
321.8
323.2
250.8
251.8
253.0
253.8
255.2
256.0
257.2
258.3
259.3
260.5
261.5
262.7
263.7
265.0
266.0
267.2
268.3
269.5
270.7
271.8
273.0
274.2
275.3
276.5
277.8
279.0
280.2
281.5
282.8
283.8
285.3
286.3
287.8
289.0
290.3
291.2
293.0
294.2
295.7
296.8
298.3
299.5
301.0
302.3
303.7
205.2
306.5
308.0
309.3
310.7
312.3
313.7
315.0
316.7
318.0
319.7
321.0
322.7
324.2
325.7
TANGENT
7.11537
.13042
.14553 2
.16071
0123➡
.17594 4
7.19125 5
.20661 6
.22204 7
.23754 8
.25310 9
7.26873 10
28442 11
.30018 12
.31600 13
.3319014
7.34786 15
36389 16
.37999 17
.39616 18
.41240 19
7.42871 20
.44509 21
.46154 22
.47806 23
.49465 24
7.51132 25
.52806 26
.54487 27
.56176 28
.57872 29
7.5957530
.61287 31
.63005 32
.64732 33
.66466 34
7.68208 35
.69957 36
.71715 37
.73480 38
.75254 39
7.77035 40
.78825 41
80622 42
.82428 43
.84242 44
7.86064 45
.87895 46
.89734 47
.91582 48
.93438 49
7.95302 50
.97176 5!
.99058 52
8.0094853
.02848 54
8.04756 55
.06674 56
.08600 57
.10536 58
.12481 59
8.14435 60
544
0.992546
01231 5O7BO
6
5.992722
8
9
||
12
13
14
233
10.992896
BOO CO2 PON** 8~~M7 HOMMA DIRE DE 85885
16
17
21
15.993068 30
22
23
24
20.993238
3272
3306
3339
3373
26
27
28
29
25.993406
3440
3473
3506
3539
31
32
33
34
SINE
30.993572
3605
3638
3670
3703
36
37
2582
2617
2652
2687
38
39
2757
2792
2827
2862
35.993736
3768
3800
3833
3865
241
42
43
44
46
47
48
49
1
2931 Corr.
2966. 10
3000 15
3034 20 11
40.993897
53
54
56
57
58
59
3103 40 23
3137 45 26
3171 50 29
3204
45.994056|30
50.994214
4245
4276
CORR.
FOR
SEC.
+
TABLE XX.-NATURAL SINES, COSINES,
4307
4338
55.994369
4400
4430
4461
4491
" Corr.
3929
396110 5
3993 15 8
4025 20 II
6
60.994522
9
16
4088140 21
4120 45
24
4151 50 26
4182
COSINE
.121869
1581
1292
1003
0714
.120426
0137
.119848
9559
9270
.118982
" Corr.
8693
8404110 48
811515 72
7826 20 96
.116093
5804
5515
5226
4937
.114648
4359
4070
3781
3492
.113203
2914
2625
2336
2047
.111758
1469
1180
0891
C602
CORR.
FOR
SEC.
17.117537 30 144.882463 7.50793
7248 40 193
6960 45 217
6671 50 241
6382
944515 72
9156 20 96
.10886730 145
8578 40 193
8288 45 217
7999 50 241
7710
.107421
7132
6842
6553
6264
.105975
5686
5396
5107
4818
.104528
83°

VERSINE
EXSEC
.878131 7.20551
8419 .22500 324.8
8708 .24457
8997 .26425
326.2
328.0
9286 .28402
.879574 7.30388
.32384
9863
.880152
.3439C
0441 .36405
0730 .38431
.881018 7.40466
1307
.42511
1596
.44566
1885 .46632
2174 .48707
2752.52889
3040 .54996
3329 .57113
3618.59241
.8839077.61379
4196 .63528
4485 .65688
4774 .67859
5063.70041
.885352 7.72234
5641 .74438
5930
.76653
6219
.78880
6508 .8!!!8
.886797 7.83367
7086 .85628
7375 .87901
.90186
7664
7953.92482
.8882427.94791
8531 .97111
8820.99444
9109 8.01788
9398 .04146
DIFF. 10"
.891133 8.18553
1422 .20999
1712
.23459
2001
.25931
2290 .28417
.8925798.30917
2868 .33430
3158
.35957
3447 .38497
3736.41052
894025 8.43620
4314.46203
4604 .48800
4893 .51411
5182 .54037
895472 8.56677
329.5
331.0
332.7
334.3
335.8
337.7
339.2
340.8
342.5
344.3
345.8
347.7
349.3
351.2
352.8
354.7
256.3
358.2
360.C
361.8
363.7
365.5
367.3
369.2
371.2
373.0
374.8
.110313
.8896878.06515
399.2
0023 " Corr.
397.0
9977.08897
.109734 10 48.890266 .11292
.13699
0555
0844 .16120
376.8
378.8
380.8
386.7
388.8
401.2
403.5
405.5
4C7.7
410.0
412.0
414.3
416.7
327.2
328.7
330.3
332.0
333.5
418.8
421.2
423.3
425.8
428.0
335.0
336.8
430.5
432.8
435.2
437.7
440.0
338.3
340.0
341.7
343.2
345.0
379.2
381.3
383.2
382.7
385.2
384.8 387.2
346.7
348.3
350.2
389.2
391.2
390.7 393.3
395.2
393.0
394.8
397.5
351.8
353.5
355.2
357.2
358.8
360.5
362.5
364.2
366.2
367.8
369.8
371.7
373.5
375.5
371.3
399.3
401.7
403.7
405.8
407.8
410.2
412.3
414.7
416.7
419.0
421.3
423.5
425.8
428.3
430.5
432.8
435.2
437.7
440.0
442.5
TANGENT
0
8.14435
.16398 1
.18370 2
.20352 3
.22344 4
8.24345 5
.26355 6
.28376 7
.30406 8
.32446 9
8.34496 10
.36555 11
.38625 12
40705 13
.42795 14
8.44896 15
.4700716
.4912817
51259 18
.53402 19
8.55555 20
.5771821
.59893 22
.62078 23
.64275 24
8.66482 25
.68701 26
7C931 27
.73172 28
.75425 29
8.77689 30
.79964 31
.82252 32
.84551 33
.86862 34
8.89185 35
.9152036
.93867 37
.96227 38
.98598 39
9.00983 40
.0337941
.05789 42
.0821 43
.10646 44
9.13093 45
.15554 46
.18028 47
.20516 48
.2301649
9.25530 50
.28058 51
.30599 52
.33154 53
.35724 54
9.38307 55
.40904 56
.43515 57
.46141 58
.4878 59
9.51436 60
545
1
0.994522
1234 56700 0-234 5
8
9
5.994673
11
12
13
10.994822
14
16
17
6 666 6ººº 55555 £55±5 88*** wwwww ~*~*~ IN TOD
18
21
22
23
24
26
15.994968 30
4998 40
5027 45 22
5056 50 24
5084
20.995113
5142
5170
5199
5227
27
28
29
25.995256
5284
5312
5340
5368
31
32
33
34
30.995396
5424
5452
5480
5507
36
37
38
39
SINE
35.995534
5562
5589
5616
5644
41
4552
4582
42
4613
4643
43
44
4703
4733
4762
4792
40.99567!
46
47
48
49
4851
4881 10
4910 15
4939 20
52
54
59
50.995937
5963
5989
6015
6041
45.995805 30 13
5832 40 18
5858 45 20
5884 50 22
5911
ga
55.996067
6093
6118
6144
6169
VERSINES, EXSECANTS, AND TANGENTS
84°

CORR.
FOR
SEC.
+
60.996195
4
5698 "
Corr.
5725 10
5752 15 7
5778 20 9
Corr.
5
7
10
15
COSINE
.IC4528
4239
3950
3660
3371
.101635
1346 " Corr.
1056 10 48
0767 15 72
C478 20 96
.ICC188 30 145
20.099899 40 193
9609 45 217
9320 50 241
9030
.103082
2792
2503
2214
1924
.098741
8451
8162
7872
7583
.097293
7004
6714
6425
6135
.095846
5556
5267
4977
4688
.094398
4108
3819
3529
3240
.090053
.089764
CORR.
FOR
SEC.
092950
2660
" Corr.
2371 10 48
2081 15 72
1791 20 97
.091502 30 145
1212 40 193
0922 45 217
0633 50 241
0343
9474
9184
8894
+
.088605
8315
8025
7735
7446
.087156
VERSINE
EXSEC
.8954728.56677
5761 .59332
6050 .62002
634C .64687
6629.67387
896918 8.70103
7208 .72833
7497 .75579
7786 .78341
8076 .81119
.8983658.83912
8654 .86722
8944.89547
9233 .92389
9522.95248
.899812 8.98123
900101 9.01015
0391 .03923
0680 .06849
0970 .09792
.9012599.12752
1549 .15730
1838
.18725
2128 .21739
2417.24770
.902707 9.27819
2996 .30887
3286 .33973
3575 .37077
3865 .40201
.9041549.43343
4444 .46505
4733
.49685
5023 .52886
5312.56106
905602 9.59346
5892 .62605
6181.65885
6471 .69186
6760 .72507
907050 9.75849
7340 .79212
7629 .82596
7919 .86001
8209 .89428
908498 9.92877
8788.96348
9078.99841
9367 10.0336
9657
.0689
90994710.1045
.910236 .1404
0526 .1765
C816 .2128
1106 .2493
.911395 10.2861
1685 .3231
1975 .3604
2265 .3979
2554 .4357
.912844 10.4737
DIFF. 10"
442.5 445.0
445.0 447.5
447.5
449.8
450.0
452.5
452.7 455.0
455.0
457.7
460.3
463.1
465.5
468.3
470.8
473.7
476.5
479.2
482.0
484.7
487.7
490.5
493.3
496.3
499.2
5C2.3
505.2
508.2
511.3
514.2
517.3
520.7
523.7
527.0
530.0
533.5
536.7
540.0
543.2
546.5
550.2
553.5
557.0
560.5
564.C
567.5
571.2
574.8
578.5
582.2
58.8
59.3
59.8
60.2
60.5
60.8
61.3
61.7
62.2
62.5
63.C
63.3
457.5
460.2
462.7
465.3
468.0
470.7
473.5
476.0
478.8
481.7
484.3
487.2
1
[
49.3
49.5
50.0
50.2
50.3
50.8
51.0
51.3
51.8
51.8
52.3
52.7
53.0
53.2
53.7
53.8
54.3
54.5
54.8
55.3
55.5
56.0
56.3
56.7
57.0
57.3
57.7
58.2
58.3
59.0
59.2
59.5
60.0
60.3
60.8
61.2
61.5
62.0
62.3
62.8
63.2
63.7
TANGENT
9.51436
.54106
.56791
.59490
.62205
9.64935
.67680
.70441
.73217
.76009
01234
5678 σ
9
9.7881710
.81641 11
.8448212
.8733813
.9021114
9.93101 15
.96007 16
.98930 17
10.0187 18
.0483 19
10.0780 20
. 1080 21
.1381 22
. 1683 23
.1988 24
10.2294 25
.2602 26
.291327
.3224 28
.3538 29
10.3854 30
.4172 31
.4491132
.4813 33
.5136 34
10.5462 35
.5789 36
.6118 37
.6450 38
.6783 39
10.7119 40
.7457|41
.7797 42
.8139 43
.8483 44
10.882945
.917846
.952847
.9882 48
11.0237 49
11.0594 50
.095451
.1316 52
.1681 53
.2048 54
11.2417 55
.2789 56
.3163 57
.3540 58
.3919 59
11.4301 60
546
0-234 BON∞O
0.996195
6
7
5.996320
8
9
O-NMI DOM ONE 0700 om♡♡Z KOMMA DI
16
10.996444
17
19
21
TI
6468 Corr.
12
6493 10 4
13 6517 15 6
14
8
6541 20
15.996566 30
6590 40 16
6614 45 18
22
20.996685
23
18 6637 50 20
6661
24
26
27
28
25.996802
29
31
33
34
30.996917
36
6940
32 6963
37
38
SINE
39
*** **
6220
6245
6270
6295
35.997030
41
6345
6370
6395
6420
42
43
44
46
47
6708
6732
6756
6779
40.997141
ggg238g
6825
6848
51
52
53
54
6872
6894
56
57
58
59
6985
7008
7053
7075
7097
7119
45.997250 30
50.997357
7378
7399
7420
7441
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
11
7272 40 14
7293 45
48 7314 50 18
49
7336
7163
7185 10
55.997462
7482
7503
7523
7544
60.997564
7207 15
7229 20
" Corr.
E57
4
-46∞
COSINE
.087156
6866
6576
16
6286
5997
1.085707
5417
5127
4837
4547
.081359
1069
0779
0489
0199
12.082808 30 145.917192 11.0761
2518 40 193
2228 45 217
1938 50 242
1649
.079909
9619
9329
084258
" Corr.
3968
3678 10 48
3388 15 72
3098 20 97
9039
8749
078459
8169
7879
7589
7299
.077009
6719
6429
6139
5849
85°

CORR.
FOR
SEC.
072658
2368
2078
1788
1497
+
.071207
0917
0627
0337
0047
.069756
1.075559
5269
" Corr.
10 48
4979 10
468915 73
4399 20 97
.074108 30 145
3818 40 193
3528 45 218
3238 50 242
2948
VERSINE
EXSEC
.912844 10.4737
3134 .5120
3424
.5505
3714
.5893
4003 .6284
914293 10.6677
4583 .7073
4873
.7471
5163
.7873
5453 .8277
.9093
.915742 10.8684
6032
6322
6612 .9921
690211.0340
.9506
7482 .1185
7772
.1612
8062
.2043
8351 .2476
.9186411.2913
8931 .3352
9221 3795
9511 .4241
9801 .4690
92009111.5142
0381 .5598
.6057
.6520
.6986
0671
0961
1251
921541
1.7455
1831
.7928
2121
.8404
2411
.8884
2701 .9368
922991 11.9855
328112.0346
92444112.2347
4731 .2857
5021 .3371
5311 .3889
56CI
.4411
.925892 12.4937
6182 .5468
6472
6762
7052 .7084
DIFF.
10"
927342 12.7631
7632 .8183
7922
.8739
8212 .9300
8503 .9865
63.9
64.3
928793 13.0435
9083 .1010
9373 .1589
9663
.2173
9953 .2762
.930244 13.3356
64.8
65.3
65.7
66.1
66.5
67.0
67.4
67.9
68.3
68.8
69.3
69.8
70.3
70.8
71.3
71.8
72.3
72.8
82.0
82.6
3571 .0840
3861 .1339
4151 .1841 83.9
83.3
84.5
73.3
73.9
74.5
75.0
75.5
76.1
76.6
77.2
77.8
78.3
78.9
79.5
80.2
80.8
81.4
88.5
89.2
.6002
.6541 89.9
90.6
91.3
85.1
85.8
86.4
87.1
87.8
92.1
92.8
93.6
94.4
95.2
95.9
96.6
97.5
98.3
99.2
TANGENT
11.4301 0
.4685 I
.5072
2
.5461 3
.5853 4
11.6248
.6645 6
.7045 7
.7448 8
.7853 9
•
11.8262 10
.8673 11
.9087 12
.9504 13
.992314
12.0346 15
.0772 16
.1201 17
.1632 18
.2067 19
12.2505 20
.2946 21
.3390 22
.3838 23
4288 24
12.4742 25
.5199 26
.5660 27
.6124 28
.6591 29
12.7062 30
.7536 31
.801432
.8496 33
.8981 34
12.9469 35
.9962 36
13.045837
.0958 38
.1461 39
13.1969 40
.2480 41
.299642
.3515 43
.403944
13.4566 45
.5098 46
.5634 47
.6174 48
.6719 49
13.7267 50
.7821 51
.8378 52
.8940 53
.9507 54
14.0079 55
.0655 56
.1235 57
.182158
.241 59
14.3007 60
547
0.997564
01234
5678 σ
9
5.997664
11
12
234
10.997763
13
14
16
17
18
22 2~~~2 。z~m7 ❤❤♪❤
34
24
20.997953
21
7972
22 7990
23
8008
8027
15.997859 30
7878 40 13
7897 45 14
7916 50 16
19 7934
26
27
25.998045
8063
8081
32
33
34
28 8099
29
8117
37
30.998135
8152
8170
8188
8205
SINE
35.998222
8240
8257
8274
8291
41
42
43
44
7584
7604
7624
7644
46
47
40.998308
8325
8 66986 gggg 55:
7684
7704
7724
7743
48
49
"
8342 10 Corr.
3
835815 4
8375 20 5
45.998392 30 8
8408 40 ||
8424 45 12
844150
8457
14
51
52
53
54
7782 " Corr.
7802 10
782115
7840 20 6
50.998473
8489
8505
8521
8537
56
57
58
59
55.998552
8568
8584
8599
8614
60.998630
VERSINES, EXSECANTS, AND TANGENTS
86°

CORR.
FOR
SEC.
+
COSINE
.069756
9466
9176
8886
8596
.068306
8015
7725
7435
11
7145
066854
6564 Corr.
6274 10
48
598415 73
5693 20 97
9.06540330 145
5113 40 194
4823 45 218
4532 50 242
4242
.063952
3661
3371
3081
2790
.062500
2210
1920
1629
1339
.061048
0758
0468
0178
.059887
.059597
9306
9016
8726
8435
.055241
4950
4660
4369
4079
CORR.
FOR
SEC.
.058145
7854 " Corr.
7564 10 48
7274 15 73
6983 20 97
056693 30 145
6402 40 194
6112 45 218
5822 50 242
5531
053788
3498
3207
+
2917
2626
.052336
VERSINE
EXSEC
.930244 ||13.3356
0534 .3955
0824
.4559
1114
.5168
1404 .5782
.931694 13.6401
1985 .7026
2275 .7656
2565 .8291
2855 .8932
.933146 13.9579
343614.0231
3726 .0889
4016 .1553
4307 .2222
.934597 14.2898
4887 .3579
5177
.4267
5468
.4961
5758 .5661
.936048 14.6368
6339 .7081
6629
.7801
6919
.8527
7210 .9260
.937500 14.9999
7790 15.0746
8080 .1500
8371 .2261
8661 .3029
.938952 15.3804
9242 .4587
9532
.5377
9822 .6175
.940113 .6981
.940403 15.7794
0694 .8616
0984 .9446
1274 16.0283
1565 .1130
.941855 16.1984
2146 .2848
2436
.3720
2726
.4600
3017 .5490
.94330716.6389
3598 .7298
3888
.8215
4178
.9142
4469 |17.0079
.944759 17.1026
5050 .1983
5340 .2950
5631
.3927
5921 .4915
.946212 17.5914
.6923
6502
6793
.7944
7083 .8975
737418.0019
947664 18.1073
DIFF.
10"
99.9
100.8
101.7
102.5
103.4
104.3
105.2
106.0
106.9
107.9
108.9
109.8
110.8
111.8
112.8
113.8
114.8
115.8
116.9
117.9
118.9
120.1
121.2
122.3
123.4
124.6
125.8
126.9
128.2
129.4
130.6
131.9
133.1
134.4
135.7
137.1
138.4
139.8
141.2
142.6
144.1
145.5
146.9
148.4
150.0
151.5
153.0
154.7
156.4
158.0
159.6
161.3
163.0
164.8
166.6
168.4
170.2
172.0
174.C
175.9
TANGENT
14.3007
.3607
4212
.4823
.5438
14.6059
.6685
.7317
.7954
.8596
1
0
1
234 56789
14.9244 10
.9898 1
15.0557 12
.1222 13
1893 14
15.2571 15
.3254 16
.3943 17
.4638 18
.5340 19
15.604820
.6762 21
.7483 22
.82 23
.8945 24
15.9687 25
16.043526
.1190 27
.1952 28
.2722 29
16.3499 30
.4283 31
.5075 32
.5874 33
.668134
16.7496 35
.8319 36
.9150 37
9990 38
17.0837 39
17.1693 40
.255841
.3432 42
431443
.520544
17.6106 45
.7015 46
.7934 47
.8863 48
.980249
18.0750 50
.1708 51
.2677 52
.3655 53
.4645 54
18.5645 55
.6656 56
.7678 57
.8711 58
.9755 59
19.081 60
548
0.99863C
01234 5O789
6
5.998705
11
1234
12
13
10.998778
14
16
17
18
19
2222 KON** 2.087 HOMMA DIE DOOD OKOON HONOR 8
21
23
24
26
15.998848 30
8862 40
27
20.998917
8931
8944
8957
8971
28
29
31
32
25.998984
8997
9010
9023
9036
36
37
38
39
30.999048
9061
9073
33 9086
34
9098
41
42
35.99911|
9123
9135
9147
9159
43
44
SINE
46
8645
8660
47
8675
8690
40.999171
48
49
8719
8734
51
8749
8763
52
53
45.999229
56
57
8792
8806 10
882015
8834 20
58
59
8876 45 10
889C50
12
8904
50.999285
9296
9307
9318
9328
TABLE XX.-NATURAL SINES, COSINES,
CORR.
FOR
SEC.
+
"Corr.
55.999339
9350
9360
9370
9381
9183
9194 10
92C6 15
9218 20
30
9240 40
9252 45
60.999391
YUMI MOON
" Corr.
INMI 6000
9263 50 10
9274
8
COSINE
9
.052336
2046
1755
1464
1174
.050884
0593
0302
0012
.049721
.046525
6235
5944
5654
5363
.045072
4782
4491
4201
3910
.043619
3329
3038
2748
2457
6.039260
.049431
9140 JI Corr.
885C 10 48
855915 73
8269 20 97
.047978 30 145
7688 40 194
7397 45 218
7106 50 242
6816
.042166
1876
1585
1294
1004
.C40713
2
0422
" Corr.
C132 10 48
3 .03984115 73
9550 20 97
87°

CORR.
FOR
SEC.
.037806
7516
7225
6934
6644
30 145
8969 40 194
8678 45 218
8388 50 242
8097
.036353
6062
5772
5481
5190
.034900
+
VERSINE
EXSEC
947664 18.1073
7954 .2140
8245
.3218
8536
.4309
8826 .5412
949116 18.6528
9407
.7656
9698
.8798
9988 .9952
.950279 19.1121
95056919.2303
0860 .3499
1150 .4709
1441 .5934
1731 .7174
.95202219.8428
2312 .9698
2603 20.0984
2894 .2285
3184
.3603
.953475 20.4937
3765 .6288
4056 .7656
4346 .9041
4637 21.0444
5218
5509
.954928 21.1865
.33C5
.4764
5799
.6241
6090 .7739
956381 21.9256
6671 22.0794
6962 .2352
7252 .3932
7543 .5533
.957834 22.7156
8124 .8802
8415 23.0471
8706 .2164
8996 .3880
.959287 23.5621
9578 .7387
9868 .9179
:960160 24.0997
0450 .2841
960740 24.4713
1031 .6613
1322 .8542
1612 25.0499
1903 .2487
.962194 25.4505
2484 .6555
2775 .8636
3066 26.0750
3356 .2898
.963647 26.5080
3938 .7298
4228 .9551
4519 27.1842
4810 .4170
.965100 27.6537
DIFF.
10"
177.9
179.9
181.9
184.0
186.1
188.2
190.4
192.6
194.9
197.2
199.4
201.8
204.3
206.8
209.4
211.9
214.4
216.9
219.7
222.4
225.3
228.2
231.0
233.9
237.0
240.I
243.2
246.3
249.7
253.0
256.4
259.8
263.4
267.0
270.6
274.5
278.3
282.3
286.2
290.3
294.3
298.7
303.2
307.5
312.1
316.8
321.6
326.3
331.4
336.5
341.7
347.0
352.5
358.1
363.8
369.7
375.7
381.9
388.2
394.7
TANGENT
19.0811
T
19.6273
.1879
.2959 2
.4051
.5156 4
0123-
5
.7403 6
.8546 7
.9702 8
20.0872 9
20.2056 10
3253 11
.4465 12
.56913
.6932 14
20.818815
.9460 16
21.0747 17
.2049 18
.3369 19
21.4704 20
.6056 21
.7426 22
.8813 23
22.0217 24
22.1640 25
.308126
.4541 27
,6020 28
.7519 29
22.903830
23.057731
.2137 32
.3718 33
.5321 34
23.6945 35
.8593 36
24.0263 37
.1957 38
.3675 39
24.5418 40
.718541
.8978 42
25.0798 43
.2644 44
25.451745
.6418 46
.8348 47
26.0307 48
.2296 49
26.4316 50
.636751
.8450 52
27.055653
.2715 54
27.4899 55
.7117 56
.9372 57
28.1664 58
.3994 59
28.6363 60
549
0.999391
01234 56700 0-234
8
9
5.999440
||
12
13
14
10.999488
5678σ
16
17
19
~~~~~ ~~~~~ ~m~m. ❤❤
21
15.999534 30
9542 40
9551 45
234
22
23
24
26
27
28
29
18 9560 50 7
9568
20.999577
9585
9594
9602
9610
33
34
25.999618
9626
9634
9642
9650
SINE
41
43
44
9401
9411
30.999657
9665
9672
9680
9687
46
47
9421
9431
g878 238གཤྩ 55
35.999694
9702
9709
48
9450
9460
49
9469
9479
40.999729
52
53
54
9497
9507 10
9516 15
9525 20
56
57
45.999762❘ 30
58
59
9716
9722
50 .999793
51
VERSINES, EXSECANTS, AND TANGents
88°

CORR.
FOR
SEC.
+
55.999821
9827
9832
9837
9843
976840
9774 45
978150
9787
9799
9804
9810
9816
9736 " Corr.
9743 10
9749 15
9756|20
Corr.
I
2
60.999848
23
1677
I
22 3I55
COSINE
.034900
4609
4318
4C27
3737
.033446
3155
2864
2574
2283
.029085
8794
8503
8212
7922
C27631
7340
7049
6758
6468
.031992
1702 Corr.
141110
48
112015 73
0829 20 97
.C30538 30 145
C248 40 194
.029957 45 218
966650 242
9376
.026177
5886
5595
5305
5014
.024723
4432
4141
3851
3560
S
.020361
0C70
.019779
9488
9197
.018907
8616
8325
8034
7743
.017452
-
CORR.
FOR
SEC.
.023269
2978 Corr.
2687 10 48
2396 15 73
2106 20 97
.021815 30 145
1524 40 194
1233 45 218
0942 50 242
0652
+
-
VERSINE
EXSEC
965100 27.6537
5391 27.8944
5682 28.1392
5973 28.3881
6263 28.6414
.966554 28.8990
6845 29.1612
7136 29.4280
7426 29.6996
7717 29.9761
968C08 30.2576
8298 30.5442
8589 30.8362
8880 31.1337
9171 31.4367
.969462 31.7455
9752 32.0603
970043 32.3812
033432.7083
0624 33.0420
.970915 33.3823
1206 33.7295
1497 34.0838
1788 34.4454
2078 34.8145
.972369 35.1914
2660 35.5763
295135.9695
3242 36.3713
3532 36.7818
.973823 37.2016
411437.63C7
4405 38.0696
4695 38.5185
4986 38.9780
975277 39.4482
5568 39.9296
585940.4227
6149 40.9277
6440 41.4452
.976731 41.9757
7022 42.5196
731343.0775
7604 43.6498
7894 44.2372
.97818544.8403
8476 45.4596
8767 46.0960
9058 46.75CC
9348 47.4224
979639 48.1141
993048.8258
9802249.5584
0512 50.3129
0803 51.0903
.981C93 51.8916
1384 52.7179
1675 53.5705
1966 54.4505
2257 55.3595
.982548 56.2987
DIFF.
10"
401
408
415
422
429
478
487
496
505
515
525
535
545
556
567
445
453
461
29.8823
437
30.1446 6
30.4116 7
30.6833 8
469
30.9599 9
579
591
603
615
628
641
655
670
685
700
715
731
748
766
784
803
822
842
863
884
907
930
954
979
ICC5
1032
1061
1090
1121
1153
1186
1221
1258
1296
1336
TANGENT
1378
1421
1467
1515
1566
1
28.6363 0
28.8771 I
29.1220 2
29.3711 3
29.6245 4
31.2416 10
31.5284 ||
31.8205 12
32.1181 13
32.4213 14
32.7303 15
33.0452 18
33.3662 17
33.6935 18
34.0273 19
34.3678 20
34.7151 21
35.0695 22
35.4313 23
35.8006 24
36.1776 25
36.5627|26
36.9560 27
37.3579 28
37.7686 29
38.1885 30
38.617731
39.0568 32
39.5059 33
39.9655 34
40.4358 35
40.917436
41.410637
41.9158 38
42.4335 39
42.9641 40
43.5081 41
44.0661 42
44.6386 43
45.2261 444
45.8294 45
46.448946
47.0853 47
47.7395 48
48.4121 49
49.1039 50
49.8157 51
50.5485 52
51.303253
52.0807 54
52.882 55
53.7086 56
54.561357
55.4415 58
56.3506 59
57.2900 60
550
:
0.999848
9853
9858
I
·234 5O7BC)
6
8
5.999872
9
01234
11
10.999894
12
6678α
~~~~~
9898
By
9902 Inspec-
13 9906 tion
14
9910
16
17
18
15.999914
19
21
22
23
24
~34 DONO mõ
20.999932
26
27
28
29
ოოო
3=
25.999948
32
33
34
ოოოოო
667
36
30.999962
37
38
39
441
42
43
44
TABLE XX.—NATURAL TRIGONOMETRIC FUNCTIONS
SINE
35.999974
46
47
48
49
9862
9867
400 85887 HONOR 8
9877
9881
9886
9890
51
52
53
54
40.999983
56
9918
9922
9925
9929
57
58
9936
9939
45.999990
9992
9993
9994
9995
59
9942
9945
60
9951
9954
9957
9959
50.999996
9964
9967
9969
9971
9976
9978
9980
9981
9985
9986
9988
9989
55.999999
9997
9997
9998
9998
9999
One
One
One
CORR.
FOR
SEC.
+
One
COSINE
.017452
7162
6871
6580
6289
.015998
57C7
5416
5126
4835
.CI 1635
1344
1054
0763
0472
.010181
.009890
.C14544
4253 " Corr.
3962 10 48
367115 73
3380 20 97
.013090 30 145
279940 194
250845 218
2217 50 242
1926
9599
9308
9017
.008726
8436
8145
7854
7563
.007272
6981
6690
6400
6109
.005818
CORR.
FOR
SEC.
5527 " Corr.
52360 48
494515 73
4654 20 97
3782 45 218
3491 50 242
3200
002909
2618
2327
2036
1745
.001454
1164
0873
0582
0291
89°

Zero
VERSINE
EXSEC
.982548 56.2987
2838 57.2698
3129 58.2743
3420 59.3141
37160.3910
.984002 61.5072
4293 62.6646
4584 63.8657
487465.1130
5165 66.4093
.985456 67.7574
5747 69.1605
6038 70.6221
632972.1458
6620 73.7359
.98691075.3966
7201 77.1327
7492 78.9497
778380.8532
8074 82.8495
988365 84.9456
8656 87.1492
8946 89.4689
9237 91.9139
9528 94.4947
.004363 30 145.995637 ||228.184
4072 40 194
5928 244.554
6218 263.443
6509285.479
6800 311.523
.98981997.2230
.990110100.112
0401103.176
0692 106.431
0983109.897
.991274 113.593
1564 17.544
1855121.778
2146126.325
2437 131.222
.992728136.511
3019142.241
3310148.468
3600155.262
3891162.703
.994182170.888
4473179.935
4764189.987
5055 201.221
5346 213.860
.997091342.775
7382 380.972
7673 428.719
7964 490.107
8255571.958
998546 686.550
8836 858.437
91271144.92
9418 1717.87
9709 3436.75
One
Infin.
DIFF.
10"
1619
1674
1733
1795
1860
1929
2002
2079
2161
2247
2339
2436
2540
2650
2768
2894
3028
3173
3327
3494
3673
3866
4075
4302
4547
482
511
543
578
616
659
706
758
816
881
955
1.038
1.132
1.240
1.364
1.508
1.675
1.872
2.107
2.387
2.728
3.148
3.673
4.341
5.209
6.366
7.958
10.231
13.642
19.099
26.648
47.75
95.49
286.48
TANGENT
1
57.290C
58.2612 1
59.2659 2
60.3058
61.3829 4
01234 DO78Q
62.4992 5
63.6567 6
64.8580
66.1055
7
67.4019 9
68.750110
70.1533 |
71.6151 12
73.1390 13
74.7292 14
76.3900 15
78.126316
79.9434 17
81.8470 18
83.8435 19
85.9398 20
88.1436 21
90.4633 22
92.9085 23
95.4895 24
98.2179 25
101.107 26
104.171 27
107.426 28
110.892 29
114.589 30
118.540 31
122.774 32
127.321|33
132.219 34
137.507 35
143.237 36
149.465 37
156.259 38
163.700 39
171.88540
180.93241
190.98442
202.219 43
214.858 44
229.182 45
245.552 46
264.44147
286.478 48
312.52149
343.774 50
381.971 51
429.718 52
491.106 53
572.957 54
687.549 55
859.436 56
145.92 57
1718.8758
3437.75 59
Infin. 60
551
No. 100 Log 000
N.
100
101
0
000000
4321
102 8600
103 C12837
104 7033
105 021189
106
107 9384
108 | 033424
109
5306
Diff.
434
433
432
431
430
429
428
427
426
425
424
423
422
421
420
419
418
417
416
415
414
413
412
411
410
409
408
407
406
405
404
403
402
401
400
399
398
397
396
395
394
393
392
391
390
389
388
TABLE XXI.—LOGARITHMS OF NUMBERS
1
AA AAAAAA
1
4
5
6
3891 432
8174❘ 428
3259
0434 0868 1301 1734 2166 2598 3029 3461
4751 5181 5609 6038 6466 6894 7321 7748
9026 9451 9876 *0300 *0724 *1147 | *1570 *1993 *2415| 424
3680 4100 4521 4940 5360 5779 6197 6616 420
8700 9116 9532 9947 *0361 | *0775 | 416
2428 2841 3252 3664 4075 4486 4896 412
6533 6942 7350 7757 8164 8571 8978 408
0195 | *0600 | *1004 | *1408 | *1812 *2216 | *2619 | *3021 404
4628 5029 5430 5830
8620 9017 9414 9811 *0207 *0602 *0998 397
7451 7868 8284
1603 2016
5715 6125
9789
4227
6230 6629 7028 400
3826
7426 7825
8223
43
43
43
43
43
43
43
43
43
43
42
42
42
22 222
42
42
42
42
42
42
42
41
4:
41
41
41
41
41
41
41
41
40
40
40
40
40
40
40
40
40
*****888** **
40
39
39
39
39
39
39
39
2
00 00 00 00 00
87
87
86
86
86
86
86
85
84
84
83
8888889
80
85
85
85
85
127
84 127
84
126
84
126
80
79
79
79
79
00000000 00000
126
125
125
83
125
83 125
83
121
83
121
82
124
82
123
82
123
79
82
123
82
122
81
122
81
122
81
122
81
121
81
121
80
121
80
120
80 120
78
2
78
78
3
78
78
130
130
130
129
129
PROPORTIONAL PARTS
129
128
128
128
128
127
3
120
119
119
119
119
118
118
118
117
117
4
174
173
173
172
172
172
171
171
170
170
170
169
169
168
168
168
167
167
166
166
166
165
165
164
164
164
163
163
162
162
162
161
161
160
160
160
159
159
158
158
158
157
157
156
156
117
116 155
เค
5
217
217
216
216
215
215
214
214
213
213
212
212
211
211
210
210
209
209
208
208
207
207
206
206
205
205
204
204
203
203
202
202
201
201
200
200
199
199
198
198
197
197
196
196
195
156 195
194
6
260
260
259
259
258
257
257
256
256
255
254
254
253
253
252
251
251
250
250
249
248
248
247
247
246
245
245
244
244
243
2.12
242
241
241
240
239
239
238
238
237
236
236
235
235
234
233
233
7
304
303
302
302
301
300
300
299
298
298
297
296
295
295
294
293
293
292
291
291
290
289
288
288
287
286
286
285
234
284
283
282
281
281
280
279
279
278
277
277
276
275
274
274
273
No. 109 Log 040

7
| 8 | 9
89
347
346
346
345
344
343
342
342
341
340
339
338
338
337
336
335
334
33-4
333
332
331
330
330
329
328
327
326
326
325
324
323
322
322
321
320
319
318
318
317
316
315
314
314
313
312
272 311
272 310
391
390
389
388
387
386
385
384
383
383
382
381
380
379
378
377
376
375
374
374
373
372
371
370
369
368
367
366
365
365
364
363
362
361
360
359
358
357
356
356
355
354
353
352
351
Diff.
350
349
Diff.
434
433
432
431
430
429
428
427
426
425
424
423
422
421
420
419
418
417
416
415
414
413
412
411
410
409
408
407
406
405
404
403
402
401
400
399
398
397
396
395
394
393
392
391
390
389
388
552
No. 110 Log 041
N.
0
110 041393
111
No. 124 Log 096
1
2
3
4
5
Diff.
112
113
6 7 8 9 1
1787 2182 2576 2969 3362 3755 4148 4540 4932 393
5323 5714 6105 6495 6885 7275 7664 8053 8442 8830 390
9218 9606 9993 *0380 *0766 *1153 *1538 *1924
053078 3463 3846 4230 4613 4996 5378 5760
7286 7666 8046 8426 8805
9185 9563
1075 1452 1829 2206 2582 2958 3333
4832 5206 5580
5953
6326 6699 7071
8557 8928 9298 9668 0038 | *0407 | *0776 *1145 *1514 370
2250 2617 2985 3352 3718 4085 4451 4816 5182 366
5912 6276 6640 7004 7368 7731 8094 8457
*2309 *2694 386
6142 6524 383
9942 | *0320 379
3709 4083 376
7443 7815 373
114 6905
115 060698
116 4458
117 8186
118 071882
5547
119
8819 363
123
124
120 079181
121 082785
122
9543 9904 *0266 *0626 *0987 *1347 *1707 *2067 *2426 360
3144 3503 3861 4219 4576 4934 5291 5647 6004❘ 357
6360 6716 7071 7426 7781 8136 8490 8845 9198 9552❘ 355
9905 *0258 | *0611 | *0963 *1315 | *1667 | *2018 | *2370 *2721 *3071
|
| 352
093422 3772 4122 4471 4820 5169 5518 5866 6215
6562 349
Diff.
387
386
385
384
383
382
381
380
379
378
377
376
375
374
373
372
371
370
369
368
367
366
365
364
363
362
361
360
359
358
357
356
355
354
353
352
351
350
349
348
347
TABLE XXI.-LOGARITHMS OF NUMBERS

1
80000 00000000 mm
39
39
39
38
38
38
38
38
38
38
38
38
38
37
37
37
37
37
37
37
37
37
37
36
36
8888888
36
36
36
36
36
36
36
36
35
35
35
35
es escriure ciuitatis
35
35
35
35
2
ZZ ZZZZZZZZ
77
77
77
77
77
76
76
76
76
74
HH~~~~~~~~ ~~ZZZZZORA 888
74
73
76
75
75
75 113
75
112
75
74
74
74
73
73
73
73
72
72
72
71
71
71
109
108
72 108
71
3
70
PROPORTIONAL PARTS
116
116
116
115
115
115
114
114
72 108
107
70
114
113
113
113
69
112
112
111
111
111
110
110
110
110
109
109
71 106
70 106
70 105
105
70
107
107
107
106
4
155
154
154
154
153
153
152
152
152
151
151
150
150
150
149
149
148
148
148
147
147
146
146
146
145
145
144
144
144
143
143
142
142
142
141
141
140
140
5
194
193
193
192
192
191
191
190
190
189
189
188
188
187
187
186
186
185
185
184
184
183
183
182
182
181
181
180
180
179
179
178
178
177
177
176
176
175
175
174
105 140
139
104
104
139 174
6
232
232
231
230
230
229
229
228
227
227
226
226
225
224
224
223
223
222
221
221
220
220
219
218
218
217
217
216
215
215
214
214
213
212
212
211
211
210
7
271
270
270
269
268
267
267
266
265
265
264
263
263
262
261
260
260
259
258
258
257
256
256
255
254
253
253
252
251
251
250
249
249
248
247
246
246
245
209 244
209
244
208
243
8
310
309
308
307
306
306
305
304
303
302
302
301
300
299
298
298
297
296
295
294
294
293
292
291
290
290
289
288
287
286
286
285
284
283
282
282
281
280
9
348
347
347
346
345
344
343
342
332
331
341
379
340 378
339 377
338
376
338
375
337
374
336
373
335
372
334
371
333
370
330
329
329
328
327
326
325
324
323
322
321
320
320
Diff.
387
386
385
384
383
382
381
380
279
314
278
313
278 312
369
368
367
366
365
364
363
362
361
360
359
358
357
356
355
319
354
318 353
317
352
316
351
315
350
349
348
347
553
No. 125 Log 097
N.
125096910
126 100371
127 3804
128 7210
129 110590
130 | 113943
131 7271
132120574
133 3852
134 7105
135130334
136
137
Diff.
347
346
345
344
343
342
341
340
339
338
337
336
335
334
333
332
331
330
329
328
327
326
325
324
323
322
321
320
319
318
317
316
315
314
313
312
311
310
0 1
4277
4611
6276
7603
7934
9586
6608 6940 333
9915 *0245| 330
3198 3525 328
6456 6781 325
0903
1231 1560 1888
2871
4504 4830 5156
6131
8399
2260
4178
7429 7753 8076
9045 9368 9690 *0012 323
0655 0977 1298 1619
2580 2900 3219 321
3539 3858 4177 4496 4814 5133
5769 6086 6403 318
6721 7037 7354 7671 7987 8303 8618 8934 9249 9564 316
138 9879 *0194 *0508 *0822 *1136 *1450*1763 *2076 *2389 *2702 314
139 | 143015
3327 3639 3951 4263 4574 4885 5196 5507 5818 311
5451
309
308
307
1
♡♡
10 10
35
35
35
34
34
34
HHHH HH
34
34
34
34
34
34
34
33
333333333~~~~~~~ ~~~~~~
32
32
32
32
32
32
32
32
32
32
31
31
31
31
31
31
0 00 00
TABLE XXI.—LOGARITHMS OF NUMBERS
31
31
1234 4 5 6 7 8
7257 7604 7951 8298
0715 1059 1403
1747
4146 4487
5169
7549
7888
8565
0926
1263
1934
2
69
69
69
283 88*3****** *PPRRAGA88 88822222** **⠀⠀⠀⠀⠀⠀
69
68
68
68
68
68
67
67
67
67
67
66
66
66
66
66
65
65
65
65
65
64
64
64
64
64
63
63
63
63
63
62
62
62
62
62
61
3
104
104
104
103
103
103
102
102
102
101
101
101
101
100
100
100
99
99
99
98
98
98
98
97
97
97
96
96
GGO 8809
95
95
95
95
94
94
94
93
183 322
4828
8227
1599
PROPORTIONAL PARTS
93
4944
8265
93
92
92
4
139
138
138
138
137
137
136
136
136
135
135
134
134
134
133
133
132
132
96 128
127
127
132
131
131
130
130
130
129
129
128
128
126
126
126
125
125
124
124
124
123
123
5278
8595
er
5
174
173
173
172
172
171
171
170
170
169
169
168
168
167
167
166
166
165
165
164
164
163
163
162
162
161
161
160
8644
2091
5510
8903
2270
160
159
159
158
158
157
157
156
156
155
5611
8926
2216
5481
8722
1939
6
8990 9335 9681 *0026 346
2434 2777 3119 3462 343
5851 6191 6531 6871 341
9241 9579 9916 *0253 338
2605 2940 3275 3609❘ 335
208
208
207
206
206
205
205
201
203
203
202
202
201
200
200
199
199
198
197
197
196
196
195
194
194
193
193
192
5943
9256
188
188
187
187
186
2544
5806
7
243
242
242
241
240
239
239
238
237
237
236
235
235
234
233
232
232
231
No. 139 Log 145

230
230
229
228
228
227
226
225
225
221
191
223
191
223
190
222
190 221
189
221
220
219
218
218
217
155
185 216
154
185
216
154 184
215
8
278
277
276
275
274
274
273
272
271
270
270
269
268
267
266
266
265
264
255
254
254
253
252
251
250
9 Diff.
250
249
248
9
247
246
246
312
311
311
310
309
308
263
296
262
295
262
294
261
293
260
293
259
292
258
291
258 290
257
289
256
288
307
306
305
304
303
302
302
301
300
299
298
297
287
286
285
284
284
283
282
281
280
279
278
277
276
Diff.
347
346
345
344
343
342
341
340
339
338
337
336
335
334
333
332
331
330
329
328
327
326
325
324
323
322
321
320
319
318
317
316
315
314
313
312
311
310
309
308
307
554
No. 140 Log 146
N.
140
141
142
143
144
145
146
147
148
149
Diff.
306
305
304
303
302
301
300
299
298
297
296
295
294
293
292
291
290
289
288
287
286
285
284
283
282
281
280
0
279
278
277
276
275
274
273
272
271
150
176091
6381
6670
151
8977
9264
9552
152
2129
2415
153
4975 5259
7803
154
8084
0612 0892
181844
4691
7521
155 190332
156 3125
157 5900
6453
6176
158 8657 8932 9206
201397 1670 1943
3403
3681
159
146128
9219
152288
152288
TABLE XXI.-LOGARITHMS OF NUMBERS
5336
8362
161368
4353
7317
170262
3186
1
2200808 2....22222 2222
31
31
30
30
30
30
30
30
30
30
30
30
29
29
29
29
29
29
29
29
29
29
28
28
28
28
28
28
2222~~~~
28
28
8177∞ ∞ (
28
28
27
27
27
1
27
6438
9527
2594
5640 5943
8664
1667
4650
7613
2
7777888 80........ CONDAD999 999 NO NO NO NO NO
61
61
6748 7058 7367 7676 7985
8603
8294
8911 309
*1982 307
9835 *0142 *0449 *0756*1063*1370*1676 *1982
2900 3205
3510 3815 4120 4424 4728 5032 305
6246 6549 6852 7154
8061 303
7457 7759
9266
9868 0168 *0469 *0769 *1068 301
3161 3460 3758 4055 299
9567
2266
2564
2863
4947 5244
5541
5838 6134
6430
6726
7022 297
7908
8497
8203
0555 0848 1141
3478 3769 4060
8792 9086 9380
1726 2019 2311
4641 4932 5222
9968❘ 295
2895| 293
1434
4351
5512 5802 291
61
60
60
58
58
58
57
57
57
57
57
56
56
56
56
56
2
55
55
55
55
54
54
8965
1967
3
92
92
91
91
91
90
90
88
88
88
87
15000 50000 88
3
87
86
90
120
89
119
89
119
89 118
89
118
86
86
86
85
85
85
84
84
118
117
117
116
87 116
83
6959
7248
9839 *0126
2700 2985
5542
8366
1171
3959
6729
9481
2216
4
122
122
122
121
121
120
120
PROPORTIONAL PARTS
116
115
115
114
114
4 5 6
114
113
113
112
112
84 112
111
111
110
110
7536 7825 8113
*0413 *0699 *0986
3270 3555 3839
6391
6674
9209
9490
2010
4514 4792
5
153
153
152
152
151
151
150
150
149
149
148
148
147
147
146
146
145
145
144
144
143
143
142
5825 6108
8647
8928
1451
1730
2289
2567
2846 279
5623 278
4237
5069
5346
7005 7281
7556 7832
8107 8382 276
9755 *0029 | *0303 | *0577 | *0850*1124
2488 2761 3033
274
3848 272
3305 3577
142
141
141
140
140
139
83
83
83
82 110
82
109
137
82
109
136
81
108 136
139
138
138
137
6
184
183
182
182
181
181
180
179
179
178
178
177
176
176
175
175
174
173
173
172
172
171
170
170
169
169
168
167
167
166
166
165
164
164
163
163
7
214
214
213
212
211
211
210
209
209
208
207
207
206
205
204
204
203
202
202
201
200
200
199
No. 159 Log 203

789
198
197
197
196
195
195
194
193
193
192
191
190
190
9674
2603
8401
*1272
8
4123
6956
9771
245
244
243
242
242
241
240
239
238
238
237
236
235
234
234
233
232
231
230
230
229
228
227
226
226
225
224
223
222
222
221
220
219
218
218
217
8689 289
*1558 287
*1558
4407 | 285
7239 283
*0051 | 281
0051
9
275
275
274
273
272
271
270
269
268
267
266
266
265
264
Diff
263
262
261
260
259
258
257
257
256
255
254
253
252
251
250
249
248
248
247
246
245
244
Diff.
306
305
304
303
302
301
300
299
298
297
296
295
294
293
292
291
290
289
288
287
286
285
284
283
282
281
280
279
278
277
276
275
274
273
272
271
555
No. 160 Log 204
N.
·|·|
160 204120
161
6826
162 9515
163 | 212188
164 4844
165217484
166 220108
167
2716
168
5309
169 7887
170230449
2996
5528
8046
Diff.
171
172
173
174 | 240549
175 | 243038
176
5513
177 7973
178 250420
179
272
271
270
269
268
267
266
265
264
263
262
261
260
259
258
257
256
255
254
253
252
251
250
TABLE XXI.—LOGARITHMS OF NUMBERS
249
248
247
246
245
244
243
242
241
240
1
~~~ ~~~~722222 22290
27
27
0960 1215 1470
3501 3757
4011
6033
6285
6537
8297
8548
8799
9049
0799
1048 1297
1546
3286
3534
6499
3782 4030
6252
8464 8709
0664 0908 1151
2853 3096 3338 3580
5759 6005
8219
8954
1395
3822
27
26
26
26
26
26
26
26
26
26
25
25
25
112125
1 |
25
25
25
25
25
24
24
24
24
24
0704
3250
5781
2
HHH HUM
54
54
54
54
54
9247 269
*1921 | 267
4579 266
4391 4663 4934 5204 5475
5746 6016 6286 6556 271
7096 7365 7634 7904 8173 8441 8710 8979
9783 *0051*0319 *0586 | *0853 | *1121 *1388 *1654
2454 2720 2986
3252 3518 3783 4049 4314
5109 5373 5638 5902
6166
6430 6694 6957
7747 8010 8273
9060 9323 9585
0370 0631
1675 1936 2196
4792
7372 7630 258
9938 *0193 256
7221 264
9846 262
8536
8798
0892
1153
1414
2456| 261
2976
3236 3496
3755
4015
5051 259
5568
5826
6084
6342
4274 4533
6600 6858 7115
9170
8144
8400
8657 8913
9426
9682
53
53
53
~~~~ ~~HHHHHOOD DO******oo oooo
52
52
52
52
52
51
51
51
51
51
50
50
50
50
50
49
2
49
49
49
49
48
48
48
3
FXR ZALIZZZZZZ ZIIII***** 00 00 00 00
82
81
80
80
80
79
79
78
77
106
105
79
105
78
101
78 101
7-t
PROPORTIONAL PARTS
75
3
104
103
77
103
77 102
77
102
102
101
101
74
76
76
76
75
100
75 100
74
74
74
73
73
73
33322
4
72
109
108
108
72
108
107
107
106
106
100
99
99
98
98
98
97
4
97
96
5880
er
5
136
136
135
135
134
134
133
133
132
132
131
131
130
125
124
5
124
123
123
122
1724
4264
6789
9299
1795
4277
6745
9198
1638
4064
122
121
121
120
6
163
163
162
130
129
129
128
128
127
152
127
152
126
151
126
151
125 150
161
161
160
160
159
158
158
157
157
156
6
1979
4517
7011
9550
2044
4525
6991
9443
149
149
148
148
147
146
146
145
145
144
1881
4306
7
4772
7237
9687
2234
4770
2488 2742 255
5023 5276 253
7544 7795 252
7292
9800 *0050 | *0300] 250
2293 2541 2790 249
5019
5266| 248
246
9932 *0176| 245
7482
7728
2368 2610 243
4790 5031 242
2125
4548
190
190
189
188
188
187
186
186
155
181
155
181
154
180
151 179
153
179
178
177
176
176
175
185
184
183
183
182
789
No. 179 Log 255

174
174
173
172
172
171
170
169
169
168
8
218
217
216
215
214
214
213
212
211
210
210
209
208
207
206
206
205
201
203
202
202
201
200
199
198
198
197
196
195
194
194
193
192
9
245
244
243
242
241
210
239
239
238
237
236
235
234
233
232
Diff.
231
230
230
229
228
227
226
225
224
223
222
221
221
220
219
218
217
216
Diff.
272
271
270
269
268
267
266
265
264
263
262
261
260
259
258
257
256
255
254
253
252
251
250
249
248
247
246
245
244
243
242
241
240
556
No. 180 Log 255
N.
180 255273
181 7679
182 260071
183
2451
184 4818
185 267172
186 9513
187271842
188 4158
6462
189
190 278754
191 281033
192 3301
193
194
5557
7802
290035
195
196
197
0
2256
4466
198
6665
199 8853
Diff.
239
238
237
236
235
234
233
232
231
230
TABLE XXI.—LOGARITHMS OF NUMBERS
200 301030
201
202
203
204
229
228
227
226
225
224
223
222
221
220
219
218
217
216
215
214
213
212
1
24
24
24
24
24
23
***** **~*~~~~~~ 22222227
23
23
23
23
23
23
23
23
23
22
22
22
22
22
1
21
21
21
9439
8982 9211
1261 1488
1715
3979
3527 3753
5782 6007
8026 8249 8473
6232
0702
0257 0480
2478 2699
2920
4687 4907
6884 7104
9071
9289
1247
3412
5566
2
48
48
47
47
47
47
47
46
46
AAA AA
46
46
46
45
45
45
45
45
44
6958
9355
1739 1976
4109 4346
6467 6702
5514 5755 5996 6237 6477 6718
7918 8158 8398 8637 8877 9116
0310 0548 0787 1025 1263 1501
2688 2925 3162 3399 3636 3873
5054 5290 5525 5761 5996 6232
7406
7641 7875 8110 8344 8578 8812 9046
9746
9980 *0213 *0446 *0679 *0912 *1144 *1377
2074 2306 2538 2770 3001 3233 3464 3696
4389 4620 4850 5081 5311 5542 5772 6002
6692 6921 7151
7380 7609 7838 8067 8296
** ********
44
44
44
44
43
2114
3196
4275
2764 2980 217
4921 5136 216
7068 7282 215
7924 8137 8351
9417 213
9204
8778 8991
212
9630 9843 *0056 *0268 *0481 *0693 *0906 *1118 *1330 *1542
2331 2547
4491 4706
6854
5351
5996 6211 6425 6639
7496
7710
8564
43
43
2
43
43
42
3
1464
1681
3628 3844
5781
PPPARGAO 8822228828 80IIIZZZZZ
3
72
71
71
71
71
70
70
70
69
69
69
68
5127
7323
9507
68
68
67
64
4
96
95
95
94
94
9.4
93
93
222 2228228*** *******
92
92
PROPORTIONAL PARTS
92
91
90
90
89
89
4
88
87
87
86
85
85
9667
1942
4205
6456
8696
0925
3141
5347
7542
9725
1898
4059
5
120
119
119
118
118
117
117
116
116
115
115
114
114
113
113
112
112
10
111
111
110
5
110
109
109
108
108
107
107
106
4431
6681
8920
1147
3363
5567
7761
9943
9895 *0123 *0351 *0578
2169 2396
6
143
143
142
142
141
140
140
139
139
138
137
137
6
136
136
135
4656
6905
9143
1369
3584
9366
1591
3804
5787 6007
7979
8198
0161 *0378
|
134
134
133
133
132
131
131
130
130
129
128
128
127
7
7
167
167
166
2622 2849
4882 5107
7130
7354
9589
1813
4025
6226
8416
*0595
165
165
No. 204 Log 311

164
163
162
162
161
160
160
159
158
158
157
156
155
155
154
153
153
152
151
151
150
149
148
7439 241
7198
9594 9833 239
8
8
191
190
190
189
188
187
186
186
185
184
183
182
182
181
180
179
178
178
177
176
175
174
174
173
172
171
170
170
9
2214 238
4582 237
6937 235
9279❘ 234
| 233
*1609
3927| 232
6232 230
8525 229
*0806 228
3075 227
5332 226
7578| 225
9812 223
2034 222
4246 221
Diff.
6446 220
8635 219
*0813 218
|
9
215
214
213
212
212
211
210
209
208
207
206
205
204
203
203
202
201
200
199
198
197
196
195
194
194
193
192
191
Diff.
239
238
237
236
235
234
233
232
231
230
229
228
227
226
225
224
223
222
221
220
219
218
217
216
215
214
213
212
557
No. 205 Log 311
N.
205
311754
206
3867
207
5970
208
8063
209 320146
210 322219
211
212
213
214
215
216
4454
217
6460
218
8456
219 340444
230
231
232
233
234
Diff.
212
211
210
220 342423
221
222
209
208
207
206
205
204
203
202
201
200
TABLE XXI.-LOGARITHMS OF NUMBERS
199
198
197
196
195
194
193
192
191
190
• 1 2 3 4 5 6 7 8
0
189
188
4282
6336
8380
330414
332438
1
2817 3014 3212
4785 4981 5178
2620
4392 4589
6353
8305 8500 8694
6549 6744 6939 7135
223
8889
9083
224 350248 0442 0636 0829
1023
225
352183
2375 2568 2761
2954
226
4108
4493 4685
6408 6599
4876
6790
227
6026
4301
6217
7935 8125 8316
9076
9835 *0025 *0215 *0404 *0593 *0783 *0972
| | | | | |
228
8506 8696 8886
229
222 2222222222 2222------ ga
21
21
21
21
20
20
20
20
20
20
20
20
20
20
19
19
19
19
19
19
19
2426
2633 2839
4488 4694 4899
6541 6745
8583 8787
0617
6950
8991
0819 1022
2640 2842 3044 3246
4655 4856 5057
6660 6860 7060
8656 8855 9054
0642 0841 1039
4078 4289
6180 6390
8272 8481
0354 0562
1966 2177 2389 2600 2812 3023 3234 3445 3656 211
4710 4920 5130 5340 5551 5760 210
6809 7018 7227 7436 7646 7854 209
9106 9314 9522 9730
0977 1184 1391 1598 1805
8898
9938 208
2012 207
42
42
42
42
42
41
41
41
41
41
40
40
** **888***** **
40
1917
361728
2105 2294 2482 2671 2859 3048 3236 3424 188
3612 3800 3988 4176 4363 4551 4739 4926 5113 5301 188
5488 5675 5862 6049 6236 6423 6610
7169
6796 6983
187
7542 7729 7915 8101 8287 8473 8659 8845 9030 186
9216 9401 9587 9772 9958 *0143 *0328 *0513 *0698 *0883
|
7356
185
40
39
39
39
39
39
38
38
38
2 3 4 5 6
64
38
38
aa aaaaa☺☺☺uo 8822288888 ***
63
63
63
62
62
62
62
61
61
61
60
60
4499
6599
8689
0769
60
59
59
59
59
58
58
58
57
57
57
PROPORTIONAL PARTS
56
*** *88*
85
84
81
84
83
83
82
82
82
81
81
*88 800**OFFER OP
80
80
79
79
78
78
78
3046
5105
7155
9194
3252
5310
7359
9398
1225 1427
3447
5458
5257
7260 7459
9253
1237
77
77
76
76
76
75
106
106
105
105
104
101
103
103
102
102
101
101
9451
1435
1 ****** **
100
3409
5374
100
99
98
98
97
7330
9278
1630
3649
5658
7659
9650
1632
4077 206
3458 3665 3871
5516 5721 5926
7563
7767 7972
6131 205
8176 204
*0211 203
2236 | 202
9601
9805 *0008
1832
2034
4051
4253 202
6059
6260❘ 201
7858 8058 8257❘ 200
9849 | *0047 | *0246
199
198
1830
2028 2225
3802
5766
7525
7720
1216
3999 4196 197
5962 6157 196
7915 8110 195
9472 9666
194
9860*0054
1410 1603 1796 1989 193
3339 3532 3724 3916 193
5068 5260 5452 5643 5834 192
6981 7172 7363
7554 7744 191
9266 9456 9646 190
*1161 *1350*1539
|
3147
189
3606
5570
127
127
126
125
125
124
124
123
122
122
121
121
120
119
119
118
118
117
116
116
115
115
114
3850
5859
113
113
7
No. 234 Log 370

148
148
147
146
146
145
144
144
143
142
141
141
140
139
139
138
137
137
136
135
134
134
133
132
132
8
170
169
168
9
167
166
166
165
164
163
162
162
161
160
159
158
158
157
156
155
154
154
153
152
151
150
9
Diff.
191
190
189
187
186
185
185
181
183
182
181
180
179
178
177
176
176
175
174
173
172
171
188 209
208
207
206
205
204
203
202
201
200
170
169
Diff.
212
211
210
199
198
197
196
195
194
193
192
191
190
189
188
558
No. 235 Log 371
N.
235
236
237
238
239
240 | 380211
241
242
243
5606
244 7390
245 389166
246
390935
247
Diff.
2697
248 4452
249 6199
187
186
185
184
183
182
181
180
371068
2912
1253 1437
3096 3280
4932 5115
€577 6759 6942
8580 8761
4748
8398
179
178
177
176
175
174
173
172
171
170
0123456789
169
168
167
166
165
164
2017
3815
250
397940
251 9674
252 401401
253 3121
254 4834
255 406540
256 8240
257 9933
258 411620 1788 1956
259 3300
1
야​야야 ​야야​야야​야야​야야
​19
19
19
18
18
18
18
18
18
18
TABLE XXI.—LOGARITHMS OF NUMBERS
18
18
18
17
17
17
17
17
==
17
17
17
17
0392
2197
3995
5785
7568
9343
260 414973 5140 5307 5474
261 6641 6807 6973 7139
8301 8467 8633 8798
5641 5808
5974 6141 6308 6474 167
7306 7472 7638 7804 7970 8135 166
8964 9129 9295 9460 9625 9791 165
263 9956 *0121 | *0286 *0451 *0616 *0781 *0945 *1110 *1275 *1439 165
264 421604 1768 1933 2097 2261 2426 2590 2754 2918 3082 164
262
17
16
9520
1112 1288
2873 3048
4627 4802
6374 6548
6710
6881
8410 8579
*0102 *0271
2
co co co co co co co w w w wo
7051
8749
*0440
2124
3467 3635 3803
37
37
37
8114 8287 8461 8634 8808 8981
9847 *0020 0192 *0365 *0538 *0711
1573 1745 1917 2089 2261 2433
3292 3464
3807
3635
3978
5005 5176
5346
5517
5688
7221
7391
8918
9087
*0609 *0777
2293 2461
3970 4137
37
37
36
36
36
36
0573
2377
4174
36
5964
7746
35
35
35
35
35
34
34
** **≈≈≈≈
34
34
34
33
33
33
33
3
698555 HOBHHNNNHH HORADI
54
54
54
53
53
53
53
52
52
52
51
51
51
50
50
0754
2557
4353
50
50
49
1622 1806 1991 2175 2360 2544 2728 184
3464 3647 3831 4015 4198 4382 4565 184
5298 5481
5664 5846 6029 6212 6394 183
7124 7306
7488
7670 7852 8034
8943
8216❘ 182
9124
9306
9487 9668
9849
0030 181
0934
2737
4533
6142
6321
7923
8101
9698
9875 *0051 | *0228
1464
1641
3224 3400 3575 3751
4977 5152 5326 5501
6722 6896 7071 7245
4
***
75
74
74
74
73
322
73
PROPORTIONAL PARTS
72
72
****** *888JJJZIN
72
71
71
70
70
70
69
69
68
68
68
67
67
66
66
66
HIMNIECO 8888
94
93
93
92
5 | 6
92
91
91
90
1115
2917
588 6**8
86
4712
4891
5070
6499 6677 6856
8279 8456
86
85
85
84
84
83
83
82
89 106
88
106
88
105
87 104
87 104
7212 178
8634
8989 178
*0405 *0582 *0759 177
8811
1817 1993 2169 2345 2521 176
4277 176
3926 4101
5850
5676
6025 175
7766
7766 174
7419
7592
1296 1476
3097 3277
112
112
111
110
2605
2777
4492
4149
4320
5858
7561
6029 6199
7731 7901
9257 9426 9595
*0946 *1114 *1283
2629 2796 2964
4305 4472 4639
110
109
90
107
125
89 107 125
124
123
123
122
103
103
102
127
109
127
108 126
101
101
100
100
99
98
7
9154 9328
9501 173
*0883 *1056 *1228 173
131
130
130
129
128
No. 264 Log 423

121
120
120
119
118
118
1656
3456
5249
7034
117
116
116
115
8
150
149
148
147
146
146
145
144
143
142
142
141
140
139
138
138
137
136
1837 181
3636 180
5428❘ 179
135
134
134
133
132
131
2949 172
4663 171
6370 171
8070 170
9764
9764 169
*1451 169
3132 168
4806 167
9
168
167
167
166
165
Diff.
164
163
162
161
160
159
158
158
157
156
155
154
153
152
151
150
149
149
148
Diff.
187
186
185
184
183
182
181
180
179
178
177
176
175
174
173
172
171
170
169
168
167
166
165
164
559
No. 265 Log 423
N.
266
265 423246 3410 3574 3737
4882 5045 5208 5371
6511 6674 6836 6999
8135 8297 8459 8621
269 9752 9914 *0075 *0236
267
268
271
272
273
274
Diff.
01
164
163
162
161
160
159
158
157
156
155
154
153
152
151
150
TABLE XXI.-LOGARITHMS OF NUMBERS
149
148
147
146
145
144
143
1
7468 7623 7778
9015 9170 9324
280 447158 7313
281 8706 8861
282 450249 0403
283 1786
1940
3318
0557 0711
284
2093 2247
3471 3624 3777
285454845 4997 5150 5302
286 6366 6518 6670
287 7882 8033 8184
288
9392 9543 9694
289460898
6821
8336
9845
1048 1198 1348
00000
2997
290 462398 2548 2697 2847
291 3893 4042 4191 4340 4490
292 5383 5532 5680 5829 5977
293 6868 7016 7164 7312 7460
294 8347 8495 8643 8790 8938
295469822 9969 0116 0263 *0410
|
296 471292 1438 1585 1732 1878
297 2756 2903 3049 3195 3341
4216 4362 4508 4653 4799
5671 5816 5962
298
299
16
270431364
2007
3610
1525 1685 1846
2167 2328 2488 2649 2809 161
2969 3130 3290 3450
3770 3930 4090 4249 4409 160
4569 4729 4888 5048 5207 5367 5526 5685 5844 6004 159
6163 6322 6481 6640 6799
6957 7116 7275 7433 7592 159
7751 7909 8067 8226 8384 8542 8701 8859 9017 9175 158
275439333 9491 9648 9806 9964 *0122 | *0279 | *0437 | *0594 *0752 158
276 440909 1066 1224 1381 1538 1695 1852 2009 2166
277 2480 2637 2793 2950
3576 3732
4045 4201 4357
4981 5137 5293
5604 5760
2323 157
3106
3263
3419
3889 157
278
4669
4513
5915 6071
279
6226
6537 6692
6848
16
16
16
16
16
16
16
16
16
15
15
---
15
10110
15
15
--
10 10 10 10 10 HT
15
15
{ port pod
1
15
15
15
14
14
2
32222
33
33
32
32
31
31
31
31
31
30
30
30
2
8822222
30
30
29
29
29
29
3
49
49
49
48
48
3
48
47
47
47
47
46
46
46
45
45
45
44
44
44
44
43
43
4
66
65
aaooooo 8822888:32 22338
PROPORTIONAL PARTS
65
64
64
64
63
63
62
62
4
3901
5534
62
61
61
60
60
60
59
59
58
3487 3633
2318
3779
4944 5090 5235
6107 6252 6397 6542 6687 6832
58
58
57
4065 4228
5697
5860
7324 7486
8944 9106
7161
8783
*0398 | *0559 | *0720 *0881
5
***** 82O**NNOCP P**2222
7933
8242
8397 8552 155
9478
9787
9941 *0095
154
1479 1633 154
1326
0865 1018
2400
2553
2706
2859
3012
3165 153
3930
4082
4235
4387
4540
4692❘ 153
5454
5606
6973
7125
8487
5758 5910 6062
7276 7428 7579
8789 8940 9091
*0146 *0296 *0447
1948
8638
6214 152
7731 152
9242 151
*0597 *0748 151
2098 2248 150
9995
1499
1649
1799
81
81
80
4825
6382
3146
3296 3445
4639 4788 4936
6126 6274 6423
7608 7756 7904 8052
9085 9233 9380
*0557 *0704 *0851 *0998 *1145
2025 2171
80
79
79
78
78
77
77
76
76
75
8088
9633
1172
75
74
74
73
73
72
72
5 | 6 | 7 | 8 | 9
82
98
115
131
148
82
98
114
130
147
113
130
146
129
145
113
112
128
144
6
*ONNO BO
97
97
96
95
789
4392 4555 4718 164
6023 6186 6349 163
7648 7811 7973 162
9268 9429 9591 162
*1042 *1203 161
95
94
94
93
92
.92
91
No. 299 Log 476

**EEO 8888888
91
90
89
89
111
111
110
109
109
108
107
106
106
105
3594 3744 150
5085 5234 149
6571 6719 149
8200 148
9527 9675 148
147
146
2464 2610
3925 4071 146
5526 146
6976❘ 145
5381
5449 156
7003❘ 155
127
126
126
125
124
123
122
122
121
120
104 119
104
118
103
118
117
102
102
116
87
86
101
86 100
115
114
Diff.
143
142
141
140
140
139
138
137
136
135
134
133
132
131
131
130
129
Diff.
164
163
162
161
160
159
158
157
156
155
154
153
152
151
150
149
148
147
146
145
144
143
560
No. 300 Log 477
N.
307
308
309
314
315
316
310 | 491362
311
312
313
317
318
319
320
321
322
338
339
Diff.
330 518514
331 9828
332521138
142
141
140
0
300 477121
7266 7411 7555 7700 7844
8422
301
9863
8566
480007
302
1299
303
1443
7989 8133 8278
8711 8855 8999 9143 9287 9431 9575 9719
0151 0294 0438 0582 0725 0869 1012
1156
1586 1729 1872 2016 2159 2302 2445 2588 2731
304 2874
3016
3159 3302 3445 3587 3730 3872
305 | 484300 4442 4585 4727 4869 5011 5153 5295
306
6005 6147
5721 5863
6289 6430 6572
6997
6714
7138 7280 7421
7704 7845 7986 8127 8269 8410
8692
8551
8833 8974 9114 9255 9396 9537 9677 9818 141
9958 *0099 *0239 *0380*0520 *0661 *0801 *0941 *1081 *1222 140
| | |
4015
4157
5437
5579
6855
7563
139
138
137
136
135
134
133
TABLE XXI.-LOGARITHMS OF NUMBERS
132
131
130
2760
4155
5544
6930
498311
9687
501059
2427
3791
129
128
127
8646 8777 8909 9040 9171
9959 *0090 *0221 *0353 *0484
1269 1400 1530 1661
1792
2575 2705 2835
2966
3096
3876 4006 4136
4266
4396
5693
336
333 2444
334 3746
335 525045 5174 5304 5434 5563
6339 6469 6598 6727 6856
6985
7630 7759 7888 8016 8145 8274
8917 9045 9174 9302 9430
9559
530200 0328 0456 0584 0712 0840
337
1
HAT
14
505150
6370 136
7721 135
5286 5421
6505 6640 6776
7856
9203
324 | 510545
323
9740
5557 5693 5828 5964 6099 6234
6911 7046 7181 7316 7451 7586
7991 8126 8260 8395 8530 8664 8799 8934 9068
9337 9471 9606
9874 *0009 *0143 *0277 *0411
0679 0813 0947 1081 1215 1349 1482
1750 134
2017 2151 2284 2418 2551 2684 2818
3084 133
3218 3351 3484 3617 3750 3883 4016 4149
4548 4681 4813 4946 5079
5211 5344 5476
5874 6006 6139 6271 6403 6535 6668 6800
329 7196 7328 7460 7592 7724 7855 7987 8119
1616
2951
325 511883
326
4282 4415 133
327
328
5609 5741 133
6932 7064 132
8382 132
8251
14
14
1
14
14
14
14
14
13
13
13
13
13
333
2
*** ~~22222222
1502 1642 1782 1922 2062
2900 3040 3179 3319
3458
4294 4433 4572 4711 4850
5822 5960 6099 6238
7344 7483 7621
8724 8862 8999 9137
5683
7068
7206
8448
8586
9824
9962
1196
1333 1470
2564
2700
2837
3927
4063
4199
28
28
28
∞ ∞ N N N N NO22 222
28
28
27
27
27
27
2
27
26
26
26
26
26
25
3
3223
43
42
42
3
42
41
41
41
41
40
40
40
39
39
8888888888
PROPORTIONAL PARTS
39
38
38
4
2201
3597
4989
6376
7759 7897 8035
9275 9412
0099 *0236 *0374 *0511 *0648 *0785
1607
2154
1744
3109
2973
1880 2017
3246 3382
4607 4743
3518
4335
4471
4878
57
56
56
56
55
55
54
4
54
54
53
53
❤❤NN NH
52
52
52
51
51
10
5
*** RA*2288888 J22
71
71
70
70
69
69
68
68
56789
67
67
66
66
65
65
64
64
6
500 8888
85
85
84
83
83
82
$2
81
80
80
79
ZEN ZII****
9303 9434 9566 9697 131
*0615 *0745 *0876 *1007
1922
2183
131
2314 131
3616 130
3226
3486
4785
4915 130
6210 129
4526
4656
5822
5951
7114
7243
8402 8531
9687
9815
1096
0968
79
78
77
77
76
7
2341 2481 2621 140
3737 3876 4015 139
5267 5406 139
6653 6791 139
5128
6515
8173 138
138
2053
3356
No. 339 Log
*** NOKKIIIII 888
99
98
114
99 113
112
97
97
96
94
8
90
9550
*0922
2291
107
93
106
92
106
92
105
91 104
90 103
111
125
110
124
110
123
95 109 122
95
108
122
102
89 102
137
137
3655❘ 136
5014 136
6081
7372 7501 129
8660 8788 129
9943 *0072 128
1223 1351 128
Diff
9
145
144
144
143
143
142
142
141
128
127
126
531

121
120
119
118
117
116
115
114
135
134
Diff.
142
141
140
139
138
137
136
135
134
133
132
131
130
129
128
127
561
No. 340 Log 531
N.
347
348
349
340 531479
341
342
343
344
345
346
356
357
358
359
360
361
362
350 | 544068
351
352
363
364
365
366
0
Diff.
128
127
126
125
124
123
122
121
120
537819
9076
540329
119
118
117
116
1607 1734 1862
2754 2882 3009
4026 4153 4280
3136
4407
5294
5421 5547 5674
6685
6558
6811 6937
7945
8071 8197
9202 9327
0455
1579 1704
0580
1829
2825
2950
3074
3481
367
4666
368 5848
369 7026
TABLE XXI.-LOGARITHMS OF NUMBERS
4316
4440 4564
5678 5802
5555
6666 6789
7036
5307
6543
6913
353 7775 7898 8021 8144
354 9003 9126 9249 9371
355550228 0351 0473 0595 0717
8267
9494
1938
1450 1572 1694 1816
2668
3155
3883
4368
5094
5578
1
T
0000
13
556303
6423 6544 6664
7507 7627 7748 7868
13
13
13
22222 2222
1
12
12
12
12
12
12
12
4192
5431
8709 8829 8948 9068
9188
99070026 *0146 *0265 *0385
561101 1221 1340 1459
562293
2412 2531
3600 3718
2650
3837
1578
2769
3955
4784 4903 5021 5139
5966 6084 6202 6320
7262 7379 7497
7144
12
12
2790 2911
4004 4126
5215 5336
2
2222~~~~~
10 10 LO LO LLLL
26
25
25
25
25
25
2
24
24
24
2222
4433
24
24
23
23
3
370 568202
371 9374
|
372
570543
1709
8788 8905 9023 9140 9257 117
9959 *0076 *0193 *0309 *0426 117
1126 1243 1359 1476 1592 117
2291
3336 3452
4494 4610
373
8319 8436 8554 8671
9491 9608 9725 9842
0660 0776 0893 1010
1825 1942 2058 2174
2988 3104 3220
4147 4263 4379
5303 5419 5534 5650 5765
6687 6802 6917
2407 2523
374
2872
3684
3568
4726
375
574031
4841
376
5188
5880
5996
377
7147
6341 6457 6572
7032
378 7492 7607 7722
8066 8181
379 8639 8754 8868 8983 9097 9212 9326
7836 7951
8295
9441
co co co co co co co wwwwww
38
38
38
38
37
37
37
36
3
36
36
9452
0705
1953
3199
35
35
3033
4247
5457
35
4
55000* 00 00
51
51
49
49
48
4
PROPORTIONAL PARTS
48
48
47
47
46
6785
7988
1990 2117 2245 2372
3391 3518
3264
3645
4534
4661
4787
4914
5800
5927
6053 6180
7063
7189
7315 7441
8322
8448
8574
9578
9703
9829
0830
0955
1080
2078 2203
2327
3323
3447
3571
LO
5
64
64
63
63
*“ug 82288232%
62
5
62
61
61
60
4688
5925
7159
8389
60
59
9616
0840
2060
3276
4489
5699
59
58
6905
7026
7146
8108 8228
8349
9308 9428
*0504 | *0624
1698 1817
9548
*0743
1936
2887 3006 3125
4074
5257 5376 5494
6437 6555 6673
7614 7732 7849
4192 4311
6
งงงงงงงง
77
76
76
75
74
74
73
73
72
FHOO
6
71
71
4812
6049
7282
8512
9739
0962
70
70
2181
3398 3519
4610
4731
5820
5940
7
8635
9861
1084
2303
2500
2627 128
3899❘ 127
3772
5041
5167❘ 127
6306
6432 126
7567 7693 126
8699 8825 8951 126
9954 *0079 | *0204 125
1205
1330 1454 125
2452
2576
2701 125
3944 124
3696
3820
4936 5060
6172 6296
7405
7
********* 8888
90
89
87
86
85
85
84
No. 379 Log 579

83
83
82
81
8
8
5183 124
6419 124
7529 7652 123
8758 8881
123
123
9984 | *0106
1206 1328 122
2425 2547 122
3640
3762 121
4852
4973 121
6061
6182 121
7267
8469
9667
9787❘ 120
*0863 *0982 119
2055
2174 119
3244
3362 119
4429
4548 119
5730 118
6909 118
8084 118
5612
6791
7967
102
102
88 101
88
100
99
112
98
111
98 110
97
109
96
108
9 Diff.
7387 120
8589 120
8410
9555
2639 2755 116
3800
3915 116
4957
5072 116
6111 6226 115
7262 7377 115
8525 115
9669 114
95
94
94
93
9
115
114
113
113
Diff.
128
127
126
125
124
123
122
121
120
107
119
106
118
105 117
104 116
562
No. 380 Log 579
N.
387
388
389
396
397
0
400 | 602060
3144
4226
401
402
403
5305
404
6381
405
607455
406
407
8526
9594
408 610660
409 1723
Diff.
115
114
113
112
111
110
109
108
107
106
105
104
103
TABLE XXI.—LOGARITHMS OF NUMBERS
1176
2214
1
|
380 579784
381580925
382 2063
383
9898 *0012 *0126
1039 1153 1267
2177 2291 2404
*0241 *0355 *0469 *0583 *0697 *0811
| 114
1381
1495 1608 1722 1836 1950 114
2631 2745 2858 2972 3085 114
3879 3992 4105
5235
2518
3199
3312
3426
3539
3652
3765
4218 113
384
4331
4444
4557
4670
4783
4896
5009 5122
5348 113
385
585461
5574
5686 5799
5912
6024
6137
6362
6475 113
386
6587
6700
7037
7149
7262
7486
7599 112
7711
6812 6925
7823 7935 8047
8944 9056
8160
8272
8384
8496 8608
8832
9503
9167 9279 9391
9950 *0061 | *0173 | *0284 | *0396 | *0507 | *0619
8720 112
9615 9726 9838 112
*0730*0842 *0953| 112
12
11
11
11
11
11
11
11
11
1
391
2843 2954 3064
2510
3618
392
3729
3950
4061
393
4724
4834
4945
5055
5165
394
5827
5937
6047
6157
6267
6377
390 591065
1176 1287 1399 1510 1621 1732
2177 2288 2399
2621 2732
3286 3397 3508
3840
4393 4503 4614
5496 5606 5717
395596597 6707 6817 6927
7695 7805 7914 8024
8791 8900 9009 9119 9228
9883 9992 *0101 *0210] *0319
399 600973 1082 1191 1299 1408
7037 7146
7256 7366
7476
8572
9665
8134 8243 8353 8462
9337 9446 9556
*0428 *0537 *0646 *0755
1517 1625 1734 1843
398
11
11
10
10
2
23
23
23
22
22
22
~~~~~~~
2211ARA
2
22
22
3313
411
4370
3842
4897
412
413
5950 6055 6160
414
7734
410612784 2890 2996 3102
3419 3525
3630
3736❘ 106
3947 4053 4159
4475 4581 4686 4792 106
5003 510S 5213
5424 5529 5634 5740
5845 105
6265 6370 6476 6581 6686
6790 6895 105
7000 7105 7210 7315 7420 7525 7629
7839 7943 105
8153 8257 8362 8466 8571 8676
8884 8989 105
9198 9302 9406 9511 9615 9719
0240 0344 0448 0552 0656 0760
1280 1384 1488 1592 1695
1799
2318 2421 2525 2628 2732 2835
415
8780
618048
416 9093
417 | 620136
9928 *0032
104
104
0968 1072
9824
0864
1903
2939
418
2007 2110 104
30-12 3146 104
419
21
21
21
21
21
3
3
32H80 2~~~~~~
35
34
34
34
33
33
2711
2819 2928
3036 108
2169 2277 2386
3253 3361 3469
3794
3902
4010
4118 108
4334 4.142 4550
4658
4766
4874
4982
5089
5197 108
5413 5521
5736
5844
5951
6059
6166
6274 108
6489 6596
5628
6704 6811
7777 7884
6919
7026
7133
7241 |
7348 107
7991
8098 8205
8312
8419 107
9381 9488 107
7562 7669
8633 8740 8847 8954 9061 9167 9274
9701 9808 9914 *0021 *0128 *0234 *0341 *0447 *0554
| | |
0767
0873 0979
1829 1936 2042
1086 1192 1298 1405
107
1511 1617 106
2572 2678❘ 106
2148 2254
2360 2466
33
32
32
32
32
31
31
4
AAABA
4
46
46
PROPORTIONAL PARTS
45
45
44
44
44
43
43
42
42
42
41
2494
3577
3207
4264
5319
5
BANGOR BIJ8888
58
57
57
5
56
56
55
55
54
54
53
2603
3686
53
52
52
6
8802233 822008
69
68
68
6
67
67
66
65
65
64
64
63
62
62
7
TOCOON PORNNI
81
80
79
78
78
77
76
76
75
7
74
74
No. 419 Log 623

6250
7374
73
72
1843 1955 2066 111
3175 111
4282 111
5386 110
6487 110
7586 110
110
109
109
1951 109
4171
5276
8
8 | 5
200000 788888
92
91
90
89
88
86
86
104
103
90 102
101
100
99
85
9 Diff.
8681
9774
*0864
84
83
9
98
97
96
95
95
94
93
Diff.
115
114
113
112
111
110
109
108
107
106
105
104
103
563
No. 420 Log 623
N.
420 623249
421
422
·
423
6340
424 7366
425 628389
426 9410
427 | 630428
428
429
448
449
TABLE XXI.-LOGARITHMS OF NUMBERS
4282
5312
Diff.
430 633468
3569
431
4477 4578
5484 5584
432
433
6488
434 7490
435 638489
436
9486
437 | 64048]
438
439
104
103
102
101
100
99
98
97
96
95
7468
8491
9512
0530
1444 1545
2457 2559
1474
2465
450 653213
451
4177
452 5138
453
6098
454
7056
455
658011
456
8965
457
9916
458 660865
459 1813
440 643453
3551
441
4439
4537
442
5422
5521
443
6404 6502
444 '7383 7481
8458
445 648360
446 9335❘ 9432
447650308 0405
1278
2246
12 ··
3353 3456
4385 4488
5415 5518
6443 6546
7571
8593
9613
0631
1647
2660
1
10
10
10
10
10
10
10
10
10
10
9586
0581
1573
2563
2852
460 662758
2947 3041
461 3701 3795 3889 3983
462 4642
463
464
5685
5785
6688
6789
7790
6588
7590 7690
8589 8689
9686 9785
8789
8888
9885
0779 0879
0680
1672 1771 1871
2662
2761
2860
3670
4679
2
22222 222AA
3650
4636
21
21
20
20
20
20
20
5619
5717
6600
6698
7579
7676
8555
8653
8750
9530 9627 9724
0502 0599 0696
1375 1472 1569 1666
2343 2440
2536 2633
19
19
19
3
3771
4779
22222 22222
3230
3135
4078
4172
4736 4830 4924 5018 5112
5581 5675 5769 5862 5956 6050
6518
31
31
31
4
30
30
3559 3663 3766 3869
4591 4695 4798 4901
5621 5724 5827 5929
6648 6751 6853
7775 7878
7673
8695
8797 8900
9715
9817
0733
0835
1748 1849
2761 2862
3749 3847
4734
4832
30
29
29
29
29
3872
4880
5886
6889
7890
4
5815
6796
7774
42
41
41
40
40
40
39
39
38
38
LO
5
NNHHO 89****
51
51
50
PROPORTIONAL PARTS
50
49
3973
5004
6032
6956 7058
7161
8185
7980 8082
9002 9104 9206
9919 *0021 *0123 *0224
0936 1038 1139 1241
1951 2052 2153 2255
2963
3064
3165
3266
49
48
48
3973
4074
4981 5081
5986
6989
7990 8090
8190
8988 9088
9188
9984 *0084 *0183
0978 1077
1970
2069
3058
2959
8848
9821
0793
1762
2730
3946
4044
4931 5029
5913
6894
7872 7969
3695
4658
5619
3309 3405
4273 4369
5235
4946
5906
3502 3598
3791 3888 3984
4465 4562
4754 4850
5331 5427 5523
5715 5810
6002
6290 6386 6482 6577 6673 6769 6864 6960 96
7247 7343 7438 7534 7629 7725
8107 8202 8298 8393 8488 8584
9060 9155 9250 9346 9441
6194
7152
7820 7916 96
8679 8774 8870 95
9631 9726
95
9821
*0581 *0676 *0771
| |
1529 1623 1718
95
9536
| |
*0011 *0106 *0201 *0296 *0391 *0486
0960 1055 1150 1245 1339 1434
1907 2002 2096 2191 2286 2380
2475 2569
2663
6
46666 82233
6087
7089
61
6011
6992
61
60
59
59
3324
4266
5206
6143
z「。
7
8
58
58
57
4175
5182
6187
7189
6237 6331 6424 94
6612 6705 6799 6892 6986 7079 7173 7266 7360 94
No. 464 Log 667

4076
5107
6135
1177
1276
2168 2267
3156 3255
4276
5283
6287
7290
8290
9287
*0283
22688 JIZZZ
69
69
67
67
8067
8945 9043
9919 *0016
0890 0987
1084 1181
1859 1956 2053 2150
2826 2923 3019 3116 97
9 Diff.
4143 4242
4340 98
5127 5226 5324 98
6110 6208
7089
4179❘ 103
5210 103
6238 103
7263❘ 103
8287 102
9308 | 102
*0326❘ 102
1342 102
***** ****
2356❘ 101
3367❘ 101
6306 98
7187 7285 98
8165 8262 98
9140 9237
0113 *0210
97
97
97
3418 3512
3607
4360 4454 4548
5299 5393 5487
5 | 6 | 7 | 8 | 91
Diff.
52
62
73
83
52
62
72
71
71
82
82
81
70
80
79
78
4376 101
5383 101
6388 100
7390 100
8389 100
9387❘ 100
*0382
99
1375 99
99
2366
3354 99
78
77
76
8888
4080 96
96
5042
96
94
93
92
91
90
9999
HONE 8888
97
89
87
86
86
AA LLLLL
95
95
94
94
94
104
103
102
101
100
99
98
97
96
95
564
No. 465 Log
N.
465
466
123 3
7826
8759
667453 7546 7640 7733
8386 8479
467 9317 9410
468 670246 0339
469 1173 1265
8572 8665
9503 9596
9689
0431 0524 0617
1358 1451 1543
480
481
470
672098
471
3021
472
3942
473
4861
474
475
676694
476
7607
5778 5870 5962 6053
6785 6876 6968
7698 7789 7881
8609 8700 8791
9519 9610
0426 0517
477
8518
478
9700
9428
479 680336
0607
Diff.
482
483
484
485
486
6636 6726
487
7529 7618
488 8420 8509
489 9309 9398
***HA 888
94
93
TABLE XXI.-LOGARITHMS OF NUMBERS
Log 667
92
91
90
0
491
1081
492
490690196 0285 0373 0462
1170 1258 1347
2053 2142 2230
2935 3023 3111
3991
1965
493
2847
494
3727
495 694605
3815 3903
4693
5482 5569 5657
4781
4868
496
497
6356 6444 6531
498 7229 7317 7404
499 8101 8188 8275
89
87
86
85
681241 1332 1422
2145 2235 2326
3047 3137 3227
3947 4037 4127
4935
5025
5831
4845
685742
1
2190 2283 2375 2467
3113 3205 3297 3390
4034 4126 4218
4953 5045 5137
66666
9
∞∞∞∞∞ ∞∞0777
aaaa O)
9
9
9
9
9
2775
3635
4494
5179 5265 5350
6035 6120 6206
6718 6803 6888 6974 7059
19
234
19
1513
2416
3317
4217
5114
5921
6010
6815 6904
7707 7796
8598
9486
8687
9575
18
18
9751 87
0617 87
500698970
501 9838
502 700704
503 1568
504 2431
505 703291
506 4151 4236 4322 4408
507 5008 5094
508 5864 5949
509
9057 9144 9231 9317 9404 9491 9578 9664
9924 0011 *0098 *0184 *0271 0358 *0444 *0531
0790 0877 0963
1136 1222 1309 1395 1482 86
1654 1741 1827
1999 2086 2172 2258 2344 86
2517 2603 2689
2861 2947 3033 3119 3205 86
3721❘ 3807 3893 3979 4065 86
4579 4665
4751 4837 4922 86
5436 5522 5607 5693 5778 86
6291 6376 6462 6547 6632 85
7144) 7229 7315 7400 7485 85
3377 3463 3549
18
18
18
17
17
17
OOOOOONN NOaaa
28
28
28
27
27
27
26
26
26
26
4
38
37
37
36
36
36
35
PROPORTIONAL PARTS
35
34
34
4310
5228
5320
6145
6236
7059
7151
7972 8063
8973
8882
9791
9882
0698
0789
1050
1913
5
7920
8852
5
9782
0710
1636
47
47
46
46
45
2560
3482
4402
45
44
44
43
43
2759 88
0550 0639 0728 0816 0905 0993 89
1435 1524 1612 1700 1789 1877 88
2318 2406 2494 2583 2671
3199 3287 3375 3463 3551 3639 88
4078 4166 4254 4342 4430 4517 88
4956 5044 5131 5219 5307 5394 88
5832 5919 6007 6094 6182 6269 87
6618 6706 6793 6880 6968 7055 7142 87
7491 7578 7665 7752 7839 7926 8014 87
8362 8449 8535 8622 8709 8796 8883 87
5744
2055
90
2957
90
3407
3857
90
4307
4396
1603 1693 1784 1874 1964
2506 2596 2686 2777 2867
3497 3587 3677 3767
4486 4576 4666 4756 90
5383 5473
5563 5652 90
6279 6368 6458 6547 89
7351 7440 89
8331 89
5204
5294
6100
6189
7083 7172
7261
6994
7886
8064
8153
8242
7975
8776 8865
9664 9753
8953
89
9841
6
8013
8945
9875
0802 0895 0988 1080 93
1728 1821 1913 2005 93
68051 MONNA
55
55
6
2652 2744
3574 3666
4494
4586
5412 5503
6419
2836 2929 92
3758 3850 92
4677 4769 92
5595 5687 92
6511
6328
7242
6602 92
7333 7424 7516 91
8154 8245 8336 8427 91
9064 9155 9246 9337 91
99730063 *0154
0879 0970 1060
0245
91
1151 91
54
53
53
52
52
51
8106 8199 8293 93
9038 9131 9224 93
9967 *0060 *0153
7
***** **PRO
7
66
No. 509 Log 707

65
64
89
9012 9131 9220
9930 *0019 *0107 89
64
63
62
62
61
60
60
| 8 | 9
777
75
74
74
73
72
Diff.
71
70
70
69
68
**** 20ONE
85
**** 22222
84
83
82
81
93
80
79
78
77
77
8888
Diff.
***H* **0000000
94
93
92
91
90
89
88
87
86
85
565
No. 510 Log 707
N. 0
510
511
512
513
710117
514
0963
515 711807
516
517
518
519
552
553
554
Diff.
******* 2
86
85
520716003
6087 6170
521 6838 6921 7004
7671 7754 7837 7920
8502 8585 8668 8751
522
523
9331 9414 9497
9580
524
525 |720159
526 0986
0407
0242
1068
1811 1893
2634
528
2716
529 3456 3538
527
0325
1151 1233
1975 2058
2798 2881
3620 3702
84
TABLE XXI.-LOGARITHMS OF NUMBERS
707570
8421
9270
83
82
2650
3491
4330
5167
81
80
79
4522
4604
531
5176 5258 5340
5422
532
533
534
530 724276 4358 4440
5095
5912 5993 6075 6156 6238
6727 6809 6890 6972 7053
7541
7623 7704 7785 7866
535728354 8435 8516 8597 8678
536 9165 9246 9327 9408 9489
537 9974 *0055 | *0136 | *0217 *0298
|
538 730782 0863 0944 1024 1105
539 1589 1669 1750 1830 1911
540 732394
541
3197
2474 2555 2635 2715
3278 3358 3438 3518
4079 4160 4240 4320
542
3999
543
4800
4880
5599 5679
4960 5040 5120
5759
6556
5838 5918
6635 6715
7352 7431 7511
544
545 736397 6476
546 7193 7272
547 7987
8781 8860 8939 9018 9097
9572 9651 9731 9810 9889
8067 8146 8225 8305
548
549
1
∞∞∞∞ ∞ ∞ ∞
9
9
8
1
8
8
8
8
7655 7740 7826
8506 8591 8676
9355 9440 9524
0202
0287 0371
1048
1132 1217
1892
1976 2060
2734
2902
3575
3742
4414
4581 4665
5251
5418
5502
2
ZENE900 o
17
17
17
2
17
16
2818
3659
4497
5335
16
16
16
0836
551
550 | 740363 0442 0521 0600 0678 0757
0915 0994 1073 79
1152 1230 1309 1388 1467 1546 1624 1703 1782 1860 79
1939 2018 2096 2175 2254 2332 2411 2489 2568 2647 79
2725 2804 2882 2961 3039 3118 3196 3275 3353 3431 78
3510 3588 3667 3745 3823 3902 3980 4058 4136 4215 78
3
3
22******
26
26
25
25
25
24
24
24
4
4
34
34
34
33
7911
8761
9609
0456
1301
6337
6254
7088 7171
8003
8834
9663
0490
1316
2140
2963 3045
3784
3866
1**≈≈≈≈ 32
33
32
32
2144
2986 3070
3826 3910
4749
5586
PROPORTIONAL PARTS
5
**22778
43
43
42
56 7 8 9 1 Diff.
8166 8251 8336 85
9015 9100 9185 85
9863 9948 *0033 85
0710 0794 0879 85
1639 1723 84
2481
2566 84
3323 3407 84
4162 4246 84
5000 5084 84
5836 5920 84
7996 8081
8846 8931
9694 9779
0540 0625
1385
1470
2229
2313
3154
3994
4833
5669
42
41
41
6320 6401
7134 7216
7948 8029
8759 8841
9570
9651 9732 9813 9893
*0378 *0459 | *0540 *0621 *0702
| | | |
40
40
4685
5503
6421
6504
6588 6671
7254
7338
7421 7504
8253
6754 83
7587 83
8419 83
9248 83
9994 *0077 83
8336
9165
8086 8169
8917 9000 9083
9745 9828 9911
0573 0655 0738
1398 1481
2222 2305
0903 83
1728 82
2552 82
3374
3127
82
4194 82
3948
2796
3598
4400
5200
5998
6795
7590
8384
9177
9256
9968 *0047
4767
4849
4931
5585 5667 5748
6
1186 1266 1347 1428 1508
1991 2072 2152 2233 2313
1554
2397
3238
2588*9* to the
4078
4916
5753
51
50
50
49
49
48
0821
1563 1646
2387 2469
3209 3291
4030
4112
47
80
2876 2956
3679 3759
4480 4560
5279 5359
3037 3117
3839 3919 80
4640 4720 80
5439 5519 80
6237 6317 80
7034 7113
6078 6157
6874
6954
7670
7749
7829 7908 79
8543 8622 8701 79
8463
7
No. 554 Log 744

7460 81
6483 6564
7297 7379
8110 8191 8273 81
8922 9003 9084
81
81
81
81
81
888MANY 1
60
60
59
58
9335
9414 9493 79
*0126 | *0205 | *0284 79
57
57
56
55
8
5013 82
5830 82
6646 82
2 p*8:208
69
68
67
66
66
65
64
63
9
**** 8888
2 ZZPISES
77
77
2222
76
75
74
73
72
71
888888*2** ****000
80
Diff.
B*****8 2
86
85
84
83
82
81
80
79
566
No. 555
555 Log 744
N.
555 744293
556
557
558
6634
559 7412
0
564 1279
565 752048
566 2816
567 3583
568
569
TABLE XXI.-LOGARITHMS OF NUMBERS
570755875
6636
7396
596
597
598
599
Diff.
4371 4449
5075 5153 5231
5855 5933
6712
7489
78
77
76
NIZALEZ
6103
571
5951 6027
6712 6788
7472 7548
6864
572
573
8155 8230 8306
7624
8382
574 8912 8988 9063 9139
575759668 9743 9819 9894
576 760422 0498 0573 0649
577 1176 1251 1326 1402
1928 2003
2153
578
579
2078
2679 2754 2829
2904
75
580763428 3503 3578
581
582
4176 4251 4326
4923 4998 5072
5669 5743 5818
584 6413 6487 6562
583
585767156 7230 7304 7379
8860
586 7898 7972 8046 8120
587 8638 8712 8786
9377 $151 9525
589770115 0189 0263
588
9599
0336
74
73
72
8266
8343
9040
9118
560 748188
8421 8498 8576
561 8963
9195 9272 9350
562 9736 9814
9891 9968 *0045 *0123
563750508 0586 0663 0740 0817 0894
1356 1433 1510 1587 1664
2125 2202 2279 2356
2893
3660
2433
2970
3047
3123
3736
3813
3889
4578 4654
4348 4425
5112 5189
4501
5265
5341
5417
1
1
8
∞∞∞∞777
8
2
590 | 770852
1073
1146
0926 0999
1587 1661 1734
591
1808
1881
592
2542
2615
593
2322 2395 2468
3055 3128 3201 3274 3348
594 3786 3860 3933 4006 4079
595774517 4590 4663 4736 4809
5246 5319 5392 5465 5538
5974 6047 6120 6193 6265
6701 6774 6846 6919 6992
7427 7499 7572 7644 7717
8
8
4528
5309
6089
6011
6790 6868
7567 7645
2
3
16
15
15
15
15
15
14
3
3653
4400
5147
23
23
23
23
22
22
22
5892
6636
***
4
2222222
4
31
31
30
30
30
29
29
4606 4684
5387 5465 5543
6167 6245 6323
6945 7023 7101
7722
7800
7878
3727
4475
5221
5966
6710
7453
8194
8934
9673
0410
5
39
6180 6256
6332
6408
6940
7016
7092
7168
7700
7775
7851
7927
8609
8685
8458 8533
9214 9290 9366 9441
9970 *0045 0121 *0196
0724 0799 0875 0950
1477 1552 1627 1702
2228 2303 2378 2453
3053 3128
2978
3203
PROPORTIONAL PARTS
wo wo wo wo wo wo co
5
39
38
38
37
37
36
4730
5494
8653
8731 8808 8885 77
9427 9504 9582 9659 77
*0200 | *0277 | *0354 | *0431
0971 1048 1125 1202 77
1895 1972 77
1741
1818
2509
2586
2663
3200
3277
3353 3430
4119 4195
3966 4042
4807
4883 4960
5722
5570
5646
7527
8268
9008 9082
9746 9820
0484 0557
7064
7789
6
*******
6
47
46
5621
4762 4840 4919 4997 78
5699 5777 78
6479 6556 78
7256 7334 78
8033 8110 78
6401
7179
7955
46
45
3802
3877
4550 4624
3952
4699
5296 5370 5445 5520
6041 6115 6190
6264
6785 6859 6933
7007
7749
7082 74
7823 74
7675
8416
8490 8564 74
9156
9230 9303 74
9894
9968 | *0042 74
0705
0631
0778
74
44
44
43
7601
8342
7
1220 1293
1367
1440
1514
1955
2028
2102
2175
74
2248 73
2908 2981
73
2688
2762
2835
4298 4371 4444
5100
3421 3494 3567 3640 3713 73
4152 4225
4882 4955
5610 5683
6338 6411 6483
7137 7209 7282
7862 7934 8006
5028
5756
5173
5902
5829
73
73
73
73
73
72
6556
6629
No. 599 Log 778
89 Diff.

7
BAKKNES
55
54
53
53
52
51
50
6484
7244
8
6560 76
8003
76
76
76
7320 76
8079
8761 8836
9517 9592
*0272 *0347
|
75
1025 1101 75
1778 1853 75
2529 2604 75
3278 3353 75
62
62
61
4027 4101
4774 4848
***8-88
2740 77
3506 77
4272 77
60
5036 76
59
58
58
5799 76
75
75
5594 75
6338 74
7354
8079
0000000000 EEEEEEEE88 29
9
2008785
77
69
68
68
67
66
65
ZZZZZZZZZZ
Diff.
ZIPRZEZ
78
77
76
75
74
73
72
567
No. 600 Log 778
N.
600
601
602
603780317
1037
604
605 | 781755
606
607
608 3904
609 4617
0
620 792392
3092
621
622
Diff.
88 2222
TABLE XXI.-LOGARITHMS OF NUMBERS
778151
8874
9596
73
72
2462
3162
3790 3860
623
4488
4558
5254
624
5185
625
795880
5949
626
6574
6644
627
7268
628 7960
629 8651
71
70
2473
3189
69
68
5401 5472 5543
5615
6893
6964
7035
7673
610 785330
611 6041 6112 6183 6254 6325
612 6751 6822
613 7460 7531 7602
614 8168 8239 8310
615|788875 8946 9016
9581 9651 9722
617790285
0988
7744
8381 8451
9087
9157
616
9792
9863
0426
0567
0356
1059
618
1129
619
1691
1761
1831 1901
1
1
8224 8296 8368
8947 9019 9091
9669
9741
9813
0389
0461
0533
1109
1181
1253
1827 1899 1971
2544
2616 2688
3260
3332
3403
3975 4046
4118
4689 4760 4831
7227
งง
630 799341 9409 9478 9547
631 800029 0098 0167 0236
632 0717 0786 0854 0923
633 1404 1472 1541 1609
634 2089 2158 2226 2295 2363
635 | 802774 2842 2910 2979 3047
3457 3525 3594 3662 3730
4139 4208 4276 4344
4821 4889 4957 5025
5501 5569 5637
636
637
638
639
5705
2
7337 7406 7475
8029 8098 8167
8720
8789 8858
640 806180 6248 6316 6384 6451
641 6858 6926 6994 7061 7129
642 7535 7603 7670 7738 7806
643 8211 8279 8346 8414 8481
644 8886 8953 9021 9088 9156
645809560 9627 9694 9762 9829
646810233 0300 0367 0434 0501
647 0904 0971 1039 1106 1173
648 1575 1642 1709 1776 1843
649
2
คง
15
14
14
LLL LL
14
14
14
3
3
0496
1199
2532 2602
2672
3231 3301
3371
3930 4000 4070
4627 4697
5324 5393
6019 6088 6158
4767
5463
6713 6782 6852
7545
8236
8927
2222 22
21
21
21
20
***
4
4
22** *7
29
29
0605
1324
2042
2759
3475
4189
4902
2245 2312 2379 2445 2512 2579 2646 2713
28
28
28
8441 8513 8585
9163 9236 9308 9380
9957 *0029 *0101
| |
9885
27
5686
5757
5828
5899
5970 71
6680 71
6396
6467
6538
6609
7106
7177
7248
7319
8027
7815 7885
8522 8593
7390 71
7956
8098 71
8663 8734 8804
9228 9299 9369 9440 9510 71
9933 *0004 *0074 | *0144 *0215 70
0637 0707
1269 1340
0778 0848 0918 70
1480 1550 1620
2181
70
1971
2041
2252
2322
70
PROPORTIONAL PARTS
5
coco co co co como
5
9616 9685 9754
0305 0373 0442
37
36
36
35
2742
3441
4139
4836
5532
6227
6921
7614
8305
8996
35
34
8658 8730 8802 72
9452 9524 72
*0173 *0245 72
0677 0749 0821 0893 0965 72
1396 1468 1540 1612 1684 72
2258
2329 2401 72
2974 3046 3117 72
3689
3761 3832 71
4475 4546 71
5187 5259
2114
2186
2831
2902
3546
3618
4261
4332
4403
4974
5045
5116
6
6519
7197
6
1410
2111
9823
0511
0992 1061 1129 1198
1678 1747
1815
2500
1884
2568
2637 2705
2432
3116
3184
3252 3321 3389
3798 3867
4412 4480 4548
68
3935 4003 4071 68
4616 4685
5093 5161 5229 5297
5773
5365
4753
5433
6112
5841
5908
5976
6044
2812
2882
3511 3581
4209
4279
4906 4976
5672
6366
7060
44
43
43
42
!44
5602
6297
41
6990
7683
8374
9065
41
6587
7264
7
7752
8443
9134
7
6655
6723
7332 7400
6790
7467
7873 7941 8008 8076 8143
8549 8616
8684 8751 8818
9223 9290 9358 9425 9492
9896 9964 *0031 | *0098 | *0165
0569 0636 0703 0770 0837
1240 1307 1374 1441
1910 1977 2044 2111
500 000 000
51
No. 649 Log 812

49
48
48
8
2952
3022
70
3721
3651
4349
70
4418 70
5045 5115
70
5741 5811
7821
8513
9203
6436 6505 69
7129 7198
69
69
69
69
9892
0580
1266
9 Diff
9961
0648
1335
1952 2021
58
58
8
3 | 5
9
en er eenen en
7890
8582
9272
57
56
55
54
28 ****
?????22%** ***⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀8822222 2222222222 22;
67
67
1508 67
2178 67
2780 2847 67
66
65
64
63
70
62
61
69
69
69
69
68
68
69
68
68
68
68
67
67
Diff.
** SENS
73
72
71
70
69
68
568
No. 650 Log 812
N.
650
812913
651
3581
652
4248
653 4913
654 5578
660819544
661820201
670826075
4314 4381
4980 5046
5644 5711
6308 6374
6970
7036 7102
7565 7631 7698 7764
658 8226 8292 8358 8424
659 8885 8951 9017 9083
655816241
656 6904
657
680832509
681 3147
682
662 0858
663
1514
664 2168
665 | 822822
3474 3539 3605
4126 4191
668 4776 4841
669 5426 5491
666
667
Diff.
TABLE XXI.-LOGARITHMS OF NUMBERS
67
66
65
*22*62
64
63
12
1
7776 ∞ ∞
6
6
9741
0399
9610 9676
0267 0333
0924 0989
1579 1645
2233 2299
1055
1710
2364
2887 2952 3018
6269
6917
671
6140 6204
6723 6787 6852
7369 7434 7499
8015 8080 8144
672
7563
673
674
8209
8853
9497
8660 8724 8789
675 829304 9368 9432
676 9947 *0011 *0075 *0139 *0204
677830589 0653 0717 0781 0845
678 1230 1294 1358 1422
679 1870 1934 1998
1486
2126
2062
2980 3047
4514
4048
4647 4714
5312 5378
5179
5246
3114
3648 3714 3781
4447
5113
5777 5843
6440 6506
7169
7830 7896 7962 8028
8490 8556 8622 8688
9149 9215 9281 9346
5910
5976
6042
6573
6639
6705
7235
7301
7367
3
3670
4256 4321
4906 4971
5556 5621
2
13
13
13
13
13
12
3
222-**
4
20
20
20
19
19
19
3181
3848
4
27
26
26
26
25
25
1775
2430
3083
3735
4386
5036
5686
9807
9873
0464
9939
0595
1120 1186 1251
0530
1841 1906
2495 2560
3148 3213
3800 3865
4451 4516
5101 5166
5751 5815
5
2573 2637
3211 3275
2700
3338
3975
2764 2828
3402 3466
2956
2892
3530
3020 3083 64
3593 3657 3721
64
3784 3848 3912
683
4484 4548
4039 4103 4166 4230 4294 4357 64
4611 4675
5120 5183 5247 5310
4739
4421
5056
685 835691
64
4802
5437
684
4866 4929 4993
5500 5564 5627
5373
5944
6071
6134 6197 6261 63
686
687
5754 5817 5881
6007
6514 6577 6641
7146 7210 7273
7778 7841 7904 7967 8030
689 8219 8282 8345 8408
8534
6324 6387 6451
6957 7020 7083
7588 7652 7715
6704
7336
688
8471
3247 3314
3914 3981
4581
PROPORTIONAL PARTS
233222
6
0909
1550
2189
9164 9227
63
693
694
690 838849 8912 8975 9038 9101
9289 9352 9415 63
691 9478 9541 9604 9667 9729 9792 9855 9918 9981 *0043
692 840106 0169 0232 0294 0357 0420 0482 0545 0608 0671 63
0733 0796 0859 0921
0984 1046 1109 1172 1234 1297
1359 1422 1485 1547 1610 1672 1735 1797 1860 1922
2172 2235 2297 2360 2422 2484 2547
2796 2859 2921 2983 3046 3108 3170
3420 3482 3544 3606
4042 4104 4166 4229
4664 4726 4788 4850
695
841985 2047
2609 2672
3233 3295
2110
2734
63
63
62
62
62
696
697
3357
698 3855 3918 3980
699 4477 4539 4601
3669
4291
4912
3731 3793
4353 4415 62
4974 5036
62
34
› | S
31
6
7
40
40
39
38
38
37
6334 6399 6464 6528
6981 7046 7111 7175
7628 7692 7757 7821
8273 8338 8402 8467
8918 8982 9046 9111
9561 9625 9690 9754 9818 9882 64
6593 6658 65
7240 7305 65
7886 7951 65
8531 8595 64
9175 9239 64
*0268 *0332 *0396
*0460 *0525 64
1037 1102 1166
64
No. 699 Log 845

7
3381 3448 3514 67
4114 4181 67
4780 4847 67
5445 5511 66
6109 6175 66
6771 6838 66
7433 7499 66
8094 8160 66
8754 8820 66
9412 9478 66
*0004 | *0070] *0136 66
0661
0727
3279
3930
4581
1317 1382 1448 66
1972 2037 2103 65
2626
2691 2756 65
3344 3409 65
3996 4061 65
4646 4711 65
5296
5361 65
5945 6010 65
5231
5880
0973
1614 1678 1742 1806
2253
2317
2381 2445
47
46
46
45
44
43
8
8
XANHAA
54
53
52
0792
51
9 Diff.
50
50
6767
6830 6894 63
7399 7462 7525 63
8093 8156 63
8597 8660 8723 8786 63
9
5578888888 888
646558
60
59
66
59
PPARA AA
57
FFF JEFF❤❤❤❤❤* ***
64
64
63
88885
Diff.
232***
67
66
65
64
63
62
569
No. 700 Log 845
N.
To│··
700845098
701
702
703
704
705
706 8805
707 9419
708 850033
709 0646
710 851258
711
1870
2480
712
713
3090
714
3698
715 854306
4913
716
717
718
719
727
728
729
720 | 857332
7935
721
722
723
724 9739
725 | 860338
726
5160 5222 5284
5718 5780 5842 5901
6337 6399 6461 6523
6955 7017 7079
7573
848189
7634 7696
8251 8312
8866 8928
9481 9542
0095 0156
0707 0769
730 863323
731
Dia.
** 222
62
61
60
TABLE XXI.-LOGARITHMS OF NUMBERS
1320 1381
1931 1992
2541 2602
3150 3211
3759
3820
4367
4428
4974
5034
5519 5580
5640
6124 6185
6729
59
58
1
1 2 3 4 5 6 7 8 9
696
6245
6789 6850
6
6
732
3917
4511
5104
5696
733
734
735
866287
6465 6524
736
6878
6937
7055 7114
737
7467 7526
7585 7644 7703
738 8056 8115 8174 8233 8292
739 8644
8703
7393
7453
7995
8056
8657 8718
8537 8597
9138
9318
9918
9198 9258
9799 9859
0398 0458 0518
0937 0996 1056 1116 1176
1534 1594 1654 1714 1773
2131 2191 2251 2310 2370
2728 2787 2847 2906
2966
2
222 22
6405
6996
3382 3442 3501 3561
3977 4036 4096 4155
4570 4630 4689 4748
5163 5222 5282 5341
5755 5814 5874
5933
6346
3
0 00 00 00 m
19
6585
7819
8435
7141 7202
7758
8374
8989 9051
9604| 9665
0217 0279
0830
0891
18
7513
8116
18
5701 5761
6306
6366
6910
6970
18
17
1442
1503
1564 1625
2053
2114
2175
2236
2785
2663 2724
3272 3333
3881 3941
3394
4002
4063
4488 4549 4610
4670
5277
5095 5156
5216
5822
5882
6427
6487
7031 7091
5400
5992
6583
7173
7762
8350
8762 8821 8879 8938
5346 5408 5470 5532 5594 5656 62
5966 6028 6090 6151 6213 6275 62
6646 6708 6770 6832 6894
7264
7326
7388
8004
7449 7511 62
8066 8128 62
7881
7943
8497
8559
8743 62
9297 9358 61
9911 9972 61
0524 0585 61
1136 1197 61
4
*** **
7574
8176
8778
9379
9978
0578
25
24
24
7821
8409
8997
9584
|
59
9290 9349 9408 9466 9525
9642 9701 9760 59
9877 9935 9994 *0053 *0111*0170 *0228 *0287 | *0345
| |
0462 0521 0579 0638 0696 0755
1047 1106 1164 1223 1281 1339
1631
0813 0872 0930
740 869232
741 9818
742 870404
743 0989
1573
745 872156
58
1398 1456 1515 58
1981
744
1690 1748 1806
1865
1923
2040 2098 58
2215 2273 2331 2389 2448
746
2739 2797 2855 2913 2972 3030
747
3321 3379 3437 3495 3553 3611
3902 3960 4018 4076 4134 4192
749 4482 4540 4598 4656 4714 4772
748
PROPORTIONAL PARTS
24
23
5
9112
9726
0340
0952
228 88
31
31
30
1833
2430
3025
30
29
7634
7694
7875
60
8236
8297
8477 60
8838
60
8898 8958 9018 9078 60
9439 9499| 9559| 9619| 9679
*0038 *0098 *0158 *0218 *0278
|
0637 0697
60
0757 0817 0877 60
1355 1415 1475 60
1236 1295
1952 2012
2072 60
1893
2489
3085
2549 2608
3144
3204
6
coco co was caused
9174
9788
8620 8682
9235
9849
0401 0462
1075
1014
6642
7232
36
37
37
35
59
3620 3680 3739
4214 4274 4333
4808
4926
3799 3858
4392 4452
4985
59
4867
5045 59
5459 5519 5578 5637 59
6051
6110 6169 6228 59
35
1686
2297
1747
1809 61
2419 61
2358
2846 2907 2968 3029 61
3455 3516 3577 3637 61
4124 4185 4245 61
4731 4792 4852 61
5337 5398 5459 61
5943 6003
6064 61
6548
6608
6668 60
7152
7212
7272
7755
7815
8357 8417
7
No. 749 Log 875

8468
9056
2564
3146
2506
2622 2681 58
3088
3669
3727
4250 4308
3204 3262 58
3785 3844 58
4366 4424 58
4830 4888 4945 5003 58
*****
6701 6760 6819 59
43
43
42
7291 7350 7409 59
7880
41
41
Jand Jarak
8
Diff
999999998
88888
50
62
7939 7998 59
8527 8586 59
9114 9173 59
49
48
2668 60
3263 60
88 8888888888 **DAAAAAAA AAAAAI
47
46
60
9
56
55
54
53
52
Diff.
STO AX
62
61
60
59
58
570
No. 750 Log 875
N.
750
875061
751
5640
752
6218
753
6795
754 7371
755 877947
756
757
758
759
767
768
0
787
788
789
TABLE XXI.—LOGARITHMS OF NUMBERS
780 892095
781
Diff.
8522
9096
9669
880242
7760700
57
56
55
54
782
783
2150 2206 2262
2651 2707 2762 2818
3207 3262 3318 3373
3762 3817 3873 3928
784 4316
4427 4482
785 894870
4980 5036
5588
6140
4371
4925
5423 5478 5533
786
5975 6030 6085
6526 6581 6636
7077 7132 7187
1
1
CO CO CO LO
6829
770 886491 6547 6604
771 7054 7111
7617 7674
7392
772
7730
7955
7898
8460
773
8179
8516
774
8236 8292
8797 8853
9358
8741
775 889302
8909
9414
9974
7786 7842
8348 8404
8965 9021 9077
9470 9526 9582 9638
*0030 | *0086 | *0141 | *0197
0533 0589 0645 0700 0756
1259 1314
1816 1872
776
9918
9862
890421
777
778
0980
1537
0477
1035 1091 1147
1593 1649 1705
779
6
6
6
5119 5177
5698 5756
6276 6333
6853 6910 6968
7429 7487 7544
8004 8062 8119
8579 8637 8694
9153 9211
9268
9841
9726 9784
0299 0356
0413
760
880814
761
1385
0871
1442
1955 2012
1042 1099 1156 1213 1271
1613 1670 1727 1784 1841
2183 2240 2297
2354 2411
2923
762
763
2525
2695
2752
2809
2866
3264
3321
3377
1328 57
1898 57
2468 57
2980 3037 57
3548 3605 57
4059 4115 4172 57
4625
5192
5700 5757
3434
3491
0928 0985
1499 1556
2069 2126
2581
2638
764 3093 3150 3207
765 883661 3718 3775
4229 4285 4342
4795 4852 4909
5361 5418 5474
769 5926 5983 6039
3945
4002
766
4739 57
3832 3888
4399 4455
4965 5022
5531
4512 4569
5078 5135
5644
4682
5248
5587
5813
5305 57
5870 57
6434 56
6096
6152
6209
6265
6321
6378
2
111 pol
2
11
Josh Janda Jatok,
792
793
790 897627 7682 7737 7792 7847 7902 7957
791 8176 8231 8286 8341 8396 8451 8506
8725 8780 8835 8890 8944 8999 9054
9273 9328 9383 9437 9492 9547 9602
794 9821 9875 9930 9985 | *0039 | *0094 | *0149
795 900367 0422 0476 0531 0586 0640 0695
796 0913 0968 1022 1077 1131 1186 1240
797 1458 1513 1567 1622 1676 1731 1785
798 2003 2057 2112 2166 2221 2275 2329
799 2547 2601 2655 2710 2764 2818 2873
11
3
11
11
3
17
17
17
16
4
6716 6773
6660
7167 7223 7280 7336
8752
9325
9898
0471
5466 5524 5582 58
5235 5293 5351 5409
5813 5871 5929
6391 6449 6507
5987 6045
6102
6680
6160 58
6737 58
6564
6622
7026
7083
7141
7199
7256
7602
7659
7717
7774
7314 58
7889 58
8407 8464
7832
8177
8234 8292
8981 9039
8349
8809 8866 8924
9383 9440
9956 *0013
9497 9555 9612 57
*0070 *0127 *0185
|
0528 0585 0642 0699
57
0756 57
4
2222
23
1203
1760
LO
5
PROPORTIONAL PARTS
5
2822
2317 2373
2873 2929
3429
3484
3984
4538
5091
5644
6195
3540 3595
4039 4094
4593 4648
5146 5201 5257
5699 5754
5809
6251 6306 6361
6802 6857 6912
7352 7407 7462
6692 6747
7242 7297
29
27
6
6
****
34
34
33
32
7
6885
7449
7
8011
8573
2429 2484 2540
2985 3040 3096
3651
6942 6998 56
8067
7505 7561 56
8123 56
8629 8685 56
9190 9246 56
9694 9750 9806
*0253 | *0309 | *0365
9134
56
56
0812 0868 0924
56
1370 1426 1482 56
1928
1983
2039
No. 799 Log 903

988888888
8
40
39
4150 4205
4704 4759
39
38
8
2595
3151
3706
4261
4814
5312 5367
5864 5920
6416 6471
6967 7022
7517
7572
46
45
44
43
9
9 Diff.
5889
8012
8067
$122
55
8561 8615 8670 55
9109 9164 9218 55
9656 9711 9766 55
0203 | *0258 | *0312 55
0749 0804 0859 55
1295 1349 1404 55
1840 1894 1948 54
2384 2438 2492 54
2927 2981 3036 54
51
50
ཡུག་གཡུགཐག ཐགགགགཐགག
50
57
49
eer eeeeeggggg goods
56
56
56
56
55
55
55
55
55
55
55
Diff.
57
56
55
54
571
No. 800 Log 903
N.
800
801
802
903090
3633
4174
803
4716
804
5256
805 905796
806
6335
807
6874
808
7411
809
7949
811
812
813
814
815
816
817
818
819
0
827
828
829
TABLE XXI.—LOGARITHMS OF NUMBERS
Diff.
55
54
53
52
3920
4449
820913814 3867
821 4343 4396
822 4872 4925 4977
5400 5453 5505
5927 5980 6033
823
824
825 916454
6507 6559
826
6980 7033 7085
7506
7558 7611
8083 8135
8607
8030
8555
8592 8646
9128 9181
810908485 8539
9021 9074
9556 9610 9663 9716
910091
0197 0251
0304
0144
0624 0678
911158 1211
1690 1743
2222 2275
0731 0784 0838
1264 1317 1371
1797 1850 1903
2328 2381 2435
2753 2806 2859 2913
3284 3337 3390
2966
3496
3443
1
1
6
5
CO LO LO LO
2
6927 6981
7465 7519
8002 8056
5
3144 3199 3253 3307
3687 3741 3795
3849
4229
4283
4337
4391
4770 4824
4878
4932
5310
5364
5418
5472
5850
5904
5958
6389
6443 6497
2
11
11
11
10
GO
7663
8188
8659 8712
3
3
700046
830919078
831 9601
832 920123
833
0645
834
1166
1374
835
921686
1738
1790
1842
1894
836
2310 2362 2414
837
2829
2881 2933
2206 2258
2725 2777
838 3244 3296 3348 3399 3451 3503
839 3762 3814 3865 3917
3969 4021
17
16
16
4
3361
3904
4445
4986
5526
6012 6066
6551
6604
7089
7143
7035
7573 7626
8110
7680
8163
8217
16
3973
4502
5030
5558
6085 6138
6612
6664
7138
7190
9287
9340
9130 9183 9235
9653 9706 9758 9810
9862
0176 0228 0280 0332 0384
0697
0749 0801 0853
0906
1426
1218
1270
1322
1946
8699
9235
9770
4
2222
21
21
4555
5083
5611
7716
8240
8764
5
8753
9289
5
82285
27
27
PROPORTIONAL PARTS
26
6
9823
0358
0891
1424
1956 2009
2488
2541
3019
3549
4499
5040
5580
6119
6
3416 3470
3958 4012
4553
6658
7196
7734
8270
52
5725 5776 52
6240
6188
840 924279 4331 4383 4434 4486 4538 4589 4641 4693 4744 52
841 4796 4848 4899 4951 5003 5054 5106 5157 5209 5261
842 5312 5364 5415 5467 5518 5570 5621 5673
843 5828 5879 5931 5982 6034 6085 6137
844 6342 6394 6445 6497 6548 6600 6651 6702 6754
845 926857 6908 6959 7011 7062 7114
846 7370 7422 7473 7524 7576 7627
847 7883 7935 7986 8037 8088 8140
848 8396 8447 8498
849
6291 51
6805 51
7216|
7268
7319 51
7165
7678
8191
7730
7781
7832 51
8293 8345 51
8549 8601 8652 8703
8959 9010 9061 9112 9163 9215
8242
8754 8805 8857 51
9266 9317 9368 51
8908
2222
8807
9342
9877
32
31
3072
3602
33
32
4026 4079
4132
53
4608
5136
53
4184
4660 4713
5189 5241
5664 5718 5769
6243 6296
4237 4290
4766 4819
5294
5347 53
5822 5875 53
6349 6401 53
6927 53
6191
6717 6770 6822 6875
7243 7295 7348
7768
7453 53
7400
7820 7873 7925
8293
7978 52
8345 8397 8450 8502
8921 8973 9026
52
8816
8869
52
7
5634
6173
6712
7250
7787
8324
3524 3578 54
4066 4120 54
4607 4661 54
5094 5148 5202 54
5688 5742 54
6227 6281 54
6766 6820 54
7304 7358 54
7841 7895
54
8378
8860
9396
9930
7
8914
8967
54
9449 9503 54
9984 *0037 53
0411 0464 0518 0571 53
0944 0998
1477
No. 849 Log 929

89 D
52
9392
9914
52
0436
0489 0541 0593
52
0958 1010 1062 1114 52
1582 1634
1478 1530
1998
2050
2518 2570
2985 3037 3089
2466
1051 1104 53
1530
1584
1637 53
2063
2116
2169 53
2594
2647
2700 53
3125
3178 3231
3655 3708 3761
8888888
9444 9496 9549
9967 *0019 | *0071
39
38
37
36
Diff.
8431 54
2622 2674 52
3140 3192 52
3555 3607 3658 3710 52
4072 4124 4176 4228 52
8
XJS XXXxx:
44
43
42
42
❤x xxxxxXKONN N~~~~~~~~N NNI
53
9
53
2102 2154 52
50
49
48
47
52
Diff.
55
54
53
52
572
No. 850 Log 929
N.
858
859
861
862
863
864
865
866
850
929419
851
852
853
854
9470 9521 9572
9930 9981 *0032 *0083
930440 0491 0542 0592
0949 1000 1051 1102
1458 1509 1560 1610
855 931966 2017 2068
2474 2524 2575
2626
2981 3031 3082 3133
3487 3538 3589 3639
3993 4044 4094 4145
2118
856
857
0
890
891
Diff.
51
50
TABLE XXI.-LOGARITHMS OF NUMBERS
5003
5507
6011
6514
937016
49
48
1
4650
4700
4751
4852
860 934498 4549 4599
5054 5104
5154
5205
5255
5356
5558
5608
5658
5759 5809 5860
6061
6111
5709
6162 6212 6262 6313 6363
6665 6715 6765 6815 6865
7167 7217 7267 7317 7367
6564 6614
7066 7117
7518 7568 7618
7668
7718
8169
8219
867 8019 8069 8119
868 8520 8570 8620
869 9020 9070 9120 9170 9220
8670
8720
LOLO
1
5
5
LOLO
944483 4532 4581 4631
4680
880
881
4729
4976 5025 5074 5124 5173 5222
882
5469 5518 5567 5616
883
884
885
5665
5961 6010 6059 6108 6157
6452 6501 6551 6600 6649
946943 6992 7041 7090 7140
886 7434 7483 7532 7581 7630
7924 7973 8022 8070 8119
8413 8462 8511 8560
889 8902 8951 8999 9048
887
888
5
2
5
2
10
10
892
949390 9439 9488 9536 9585
9878 9926 9975 *0024 *0073 *0121
950365 0414 0462 0511 0560 0608
893 0851 0900 0949 0997 1046 1095
894 1338 1386 1435 1483
895951823 1872 1920 1969
2308 2356 2405 2453
2792 2841 2889 2938
1532
2017
1580
2066
896
2502 2550
2986 3034
3083
897
898
3276 3325 3373 3421 3470 3518 3566
899 3760 3808 3856 3905 3953 4001 4049
10
10
3
3
55 55
15
15
-
15
14
870 939519 9569 9619 9669 9719
9769
9819
9869 9918
9968 50
871 940018
873
1462
874
0068 0118 0168 0218 0267 0317 0367 0417 0467 50
872 0516
0566 0616 0666 0716 0765 0815 0865 0915 0964
1014 1064 1114 1163 1213 1263 1313 1362 1412
1511 1561 1611 1660 1710 1760 1809 1859 1909 1958 50
942008 2058 2107 2157 2207 2256 2306 2355 2405 2455 50
2504 2554 2603 2653| 2702 2752 2801 2851 2901 2950 50
3000 3049 3099 3148 3198 3247 3297 3346 3396 3445 49
878 3495 3544 3593 3643 3692 3742 3791 3841 3890 3939 49
879 3989 4038 4088 4137
4186 4236 4285 4335 4384 4433 49
875
876
877
4
4
22 22
20
20
20
19
2677
3183 3234 3285 3335
3740
3690
3791 3841
4296 4347
4195
4246
5 | 6
9623 9674 9725 9776 9827 9879 51
*0134 *0185 *0236 *0287 *0338 *0389 51
0643 0694 0745 0796 0847 0898 51
1153 1204 1254 1305 1356 1407
1661 1712 1763 1814 1865 1915
2220 2271 2322 2372 2423
51
51
2169
51
2727 2778 2829 2879
51
2930
3437
51
3386
3892
3943 51
4397
4448 51
8609
9097
PROPORTIONAL PARTS
5
224 225
8770
9270
26
25
24
7769 7819 7869 7919
8269
4801
5306
38 22
31
7
7 | 8 | 9
6 7
30
29
29
No. 899 Log 954

8320 8370 8420
8820 8870 8920
9320 9369 9419
1629 1677
2114 2163
2599
36
35
4902
5406
5910
6413
34
34
6305
4779 4828 4877 4927 49
5272 5321
5370 5419 49
5715 5764 5813 5862 5912 49
6207 6256
6698 6747 6796
7189 7238 7287 7336 7385 49
7679 7728 7777 7826 7875 49
8168 8217 8266 8315 8364 49
8657 8706 8755 8804 8853
9146 9195 9244 9292 9341 49
6354 6403 49
6845 6894 49
49
8
9634 9683 9731 9780 9829 49
*0170 *0219 *0267 *0316 49
0657 0706
1143 1192
0754 0803 49
1240 1289 49
1726 1775 49
2211 2260 48
2696 2744 48
2647
3131
3180 3228 48
3615 3663 3711 48
4098 4146 4194 48
6463 50
6916 6966 50
7418 7468 50
Diff.
4953 50
41
40
5457 50
39
38
5960 50
AAAAAAAADA A....
7969 50
8470 50
8970 50
9469 50
9
46
45
44
43
50
50
G9 2005
Diff.
51
50
49
48
573
No. 900 Log 954
N.
0
923
924
TABLE XXI.-LOGARITHMS OF NUMBERS
900
954243 4291
4725 4773
901
4821
902
5207
5255
5303
903
5688
5736
5784
904
6216
6265
906
7224
6168
905 956649
7128
907 7607
908 8086
909
6697 6745
7176
7655 7703
8134 8181
8659
936
937
938
939
920 963788 3835
921
922
8564 8612
Diff.
910 959041 9089 9137 9185
911 9518 9566 9614 9661
912 9995 *0042 *0090 *0138
| |
913 960471 0518 0566 0613
914 0946
915961421
0994 1041
1469 1516
916 1895 1943 1990
917 2369 2417 2464
918 2843 2890 2937
919 3316 3363 3410
49
48
47
46
930 968483
931
8950
932
9416
933 9882
934 970347
935970812
1
4260 4307 4354
4731 4778 4825
5202 5249 5296
5672
1
2
3882 3929
4401
4872
5343
925966142
5719 5766 5813
6189 6236 6283
6658 6705 6752
7127 7173 7220
6799
7267
926 6611
927 7080
928 7548 7595 7642 7688 7735
929 8016 8062 8109 8156 8203
LO LO LO LO
5
4339 4387
4869
5
2
10
10
9
3
F
3
5351 5399
5832 5880
6313
6361
6793
6840
7272
7320
7751
7799
8229
8277
8707
8755
15
14
14
14
4
8716
9183
*0114
0579
8530 8576 8623 8670
8996 9043 9090 9136
9463 9509 9556 9602 9649
9928 9975 *0021 *0068
0393 0440 0486 0533
0858 0904 0951
1276 1322 1369 1415
1740 1786 1832 1879
2203 2249 2295 2342
2666 2712 2758 2804
0997
1044
1461
1508
1925
1971
2388
2851
0661
1089 1136
1563
1611
2038
2085
2511
2559
2985 3032
3457 3504
4435 4484
4918
4966
4
20
19
19
18
5
5860
6329
5447
5928
6409
6888
5
7368
7847 7894
8325
8373
8803
8850
25
24
24
23
9232
9280
9328 9375
9423
9471
9709
9757
9804 9852 9900
9947
*0185 *0233 *0280 *0328 *0376 *0423
| | |
0756 0804 0851
1231 1279 1326
1753 1801
0709
1184
1658
1706
2132
2180
2227
2275
2606
2653
2701 2748
3079
3126 3174 3221
3693
3552
3599
3646
6845
7314
7782
8249
PROPORTIONAL PARTS
2434
2897
6
6
7
4532
4580
5014
5062
4628 4677 48
5110 5158 48
5592 5640 48
5495
5543
5976
6024 6072 6120 48
6457 6505 6553
48
6601
6936 6984 7032 7080 48
7416 7464
7512
7990
7559 48
8038 48
8468
8946
940 | 973128
3497 3543
46
941
942
46
3174 3220 3266 3313 3359 3405 3451
3590 3636 3682 3728 3774 3820 3866 3913 3959 4005 46
4051
4143
4097
4189 4235 4281 4327 4374 4420 4466
943 4512 4558 4604 4650 4696 4742 4788 4834 4880 4926 46
944 4972 5018 5064 5110 5156 5202 5248 5294
5478
945975432
5524 5570 5616 5662 5707 5753
946 5891 5937 5983 6029 6075 6121 6167 6212
6350 6396 6442 6488 6533 6579 6625 6671
6808 6854 6900 6946 6992 7037 7083 7129
7266 7312 7358 7403 7449 7495 7541 7586
5340
46
5386
5799 5845 46
6258 6304 46
6717 6763 46
7175 7220 46
7632
46
7678
947
948
949
2288
7942
8421
8898
29
29
No. 949 Log 977

7
3977 4024 4071 4118 4165 4212 47
4448 4495 4542 4590 4637 4684 47
4919
5013 5061 5108 5155 47
5390
5484 5531 5578 5625 47
5954 6001 6048 6095
6423 6470 6517
4966
5437
5907
6376
6564
7033
6892 6939 6986
7361
7408 7454 7501 47
7829
7922
7969 47
8296
8390
8436 47
7875
8343
8
34
34
33
32
9 Diff.
8516
8
8994
39
38
38
37
000000000
8763
8810
8856 8903 4.7
9229
9276 9323 9369 47
9695 9742 9789 9835 47
*0161 0207 *0254 *0300 47
0626 0672 0719 0765 46
1090 1137 1183 1229 46
1554 1601 1647 1693 46
2018 2064 2110 2157 46
2481| 2527
2943 2989
2573 2619 46
3035
3082 46
20 00 00 00 00 00 00 00 00
48
9
48
48
48
0899 48
1374 48
1848 47
2322 47
44
43
42
41
48
2795 47
3268 47
3741 47
47
47
47
HAAHAHA HAHA
Diff.
49
48
47
46
574
No. 950 Log 977
N.
950
951
952
953
9093
954
9548
955 980003
956
957
958
959
0
972
973
TABLE XXI.-LOGARITHMS OF NUMBERS
960 982271
961
977724
Diff.
7769 7815
8181 8226
8637
8683
9138
9594 9639
4443
0458
0912
1366
1819
46
45
970 986772 6817
971
2723
962
3175
963
3626
964 4077
965 984527
966
4617
5067
4122 4167
4572
4977 5022
967 5426 5471
5516
968 5875 5920 5965
969 6324 6369 6413
1
980 991226
981 1669
2111
1270 1315
1713 1758
2156 2200
982
983
2554 2598 2642
2995
985 | 993436
984
986
987
988
989
1
0049
0503
0957
1411
1456
1864 1909
LO LO THE
5
2
2316 2362 2407 2452
2769 2814 2859 2904
3220 3265 3310
3356
3671 3716 3762
3807
1359
1802
2244
2686
3039 3083 3127
3480 3524 3568
3921 3965 4009
4361 4405 4449
4757
4801 4845 4889
5196 5240 5284
3877
4317
5328
2
0094
0549
1003
6906 6951
6861
7219 7264 7309
6996 7040
7443 7488
7353
7398
7666
7711 7756
7800
7845
7890
7934
8113
8157 8202
8247
8291
974
8604 8648
8336 8381
8559
8693 8737 8782 8826
975 989005 9049 9094 9138 9183 9227 9272
976 9450 9494 9539 9583 9628 9672| 9717
977 9895) 9939 9983 *0028 *0072] *0117 0161
| | |
978 990339 0383 0428 0472 0516
979 0783 0827 0871 0916 0960
----
9
9
7861 7906
8272 8317 8363
8728 8774 8819
9184 9230
9275
9685
9730 9776
0231
0140
0185
0594
0640
0685
1139
1048
1093
1547
2000
9
9
3
3
1488
1501
1954
14
14
4212
4662
5112
5561
6010
6458
13
13
4
4
18
18
18
17
||||
4257
4707
5157
5606
5651
6055 6100
6144
6503 6548 6593
PROPORTIONAL PARTS
2497 2543
2588
3040
2949 2994
3401
3446
3491
3852 3897 3942
4302
4392
4842
5292
5741
6189
6637
5
7952
7998 8043 8089 8135 46
8409 8454 8500 8546 8591 46
8865 8911
9321 9366
8956
9412
9002
9457
9912
9047 46
9503 46
9958 46
9821
9867
0276 0322
0730 0776
1184 1229
1592 1637 1683
2045 2090 2135
0367 0412 45
0821 0867 45
1275 1320 45
1728 1773 45
2181
2226 45
4347
4797
2222
4752
5202 5247
5696
23
23
2730
3172
3613
4053
4493
4933 4977 5021 5065
5372 5416
5460
5504
6
7085
7532
7979
990 | 995635
5898
5942
5986 6030
44
991
5679 5723
6074 6117 6161
6512 6555| 6599
6949 6993 7037 7080 7124 7168 7212 7255 7299 7343 44
7386 7430 7474 7517
5767 5811 5854
6205 6249 6293 6337 6380 6424 6468 44
6643 6687 6731 6774 6818 6862 6906 44
992
993
994
995
996
7561 7605 7648 7692 7736 7779 44
7998 8041 8085 8129 8172 8216 44
8434 8477 8521 8564 8608 8652 44
8869 8913 8956 9000 9043 9087 44
9131 9174 9218 9261 9305 9348 9392 9435 9479 9522 44
999 9565 9609 9652 9696 9739 9783 9826 9870 9913 9957
997823 7867 7910 7954
8259 8303 8347 8390
8695 8739 8782 8826
997
998
43
822808
27
No. 999 Log 999

26
26
7
8425
8871
9316
9761 9806 9850
*0206 | *0250 | *0294
0561 0605 0650 0694 0738
1004 1049 1093 1137 1182
22283
3987
4437
31
30
7130
7577
8024
8470
3536
2633 2678 45
3085 3130 45
3581 45
4032 45
4482 45
4887 4932 45
5337 5382
45
5786
6234
6682
8
9
5830
6279
1403 1448
1846 1890
2288
2333
1492 1536 1580 1625 44
1935 1979 2023 2067 44
2377 2421 2465 2509 44
2863 2907 2951 44
3392 44
3833 44
2774
2819
3216 3260 3304 3348
3657 3701 3745 3789
4097 4141 4185 4229 4273
4537 4581 4625 4669
4713
co co co co co co
6727
37
7175
7622
36
35
34
8068
8514
5108 5152
5547 5591
Diff.
AAAA
9
AAA AAAA
41
41
40
39
S
10 10 10 10 10 10 1
8916 8960 45
9361 9405 45
44
44
44
44
45
45
45
45
45
45
45
44
44
44
** **:
44
Dir.
46
45
44
43
575
TABLE XXI-A.—7-PLACE LOGARITHMS OF NUMBERS FROM
1 to 100

NO.
-2MIS Cramo -
3
5
10
14
15
LOG.
NO.
LOG.
NO.
0.0000000 26 1.4149733 51
0.3010300 27 1.4313638 52
0.4771213 28 1.4471580 53
0.6020600 29 1.4623980
0.6989700 30 1.4771213
0.7781513
0.8450980
0.9030900
0.9542425
1.0000000
12 1.0791812
13
1.0413927
~~~~~
12345
~~~~M MMMMM M
wwwww
36
37
1.1139434 38
1.1461280 39
1.1760913 40
31 1.4913617 56
32 1.5051500 57
33 1.5185139 58
34 1.5314789 59
35 1.5440680 60
5 FEES
ין
200 GE
54
ii in inno
CONSTANT
55
1.5563025 61
1.5682017 62
1.5797836
16
1.2041200 41 і 1.6127839 66
1.2304489
17
18 1.2552725
1.6232493 67
1.6334685 68
1.6434527 69
19 1.2787536
20 1.3010300 45 1.6532125 70
Ratio of circum. to diameter
Ratio of dia. to circumference
I radian. Deg. in arc = radius
Minutes in arc equal to radius
Length of 1° arc, unit radius
।。
Length of arc, unit radius
Base of hyperbolic logarithms
63
1.5910646 64
1.6020600 65
MIS
~~
71
2345
72
73
74
SYMBOL
E-IF
CONSTANTS
ח
LOG.
1.7075702
1.7160033
1.7242759
1.7323938 79
1.7403627 80
π
πT
180°
1.7481880 ει
1.7558749
1.7634280
1.7708520
1.7781513
21 1.3222193
1.8512583
1.8573325
22
23
46 1.6627578
1.3424227 47 1.6720979
1.3617278 48 1.6812412
1.3802112 49 1.6901961
25 1.3979400 50 1.6989700 75 1.8750613 100 2.0000000
96
97
1.8633229 98
1.8692317
1.9822712
1.9867717
1.9912261
1.9956352
24
99
LOG.
1.8808136
1.8864907
78 1.8920946
76
77
1.8976271
1.9030900
180°
π
10800°
W
NO.
π
1 0800°
2345
82
1.7853298 86
1.7923917 87
1.7993405 88
1.8061800 89
1.8129134 90
NUMBER
83
92
1.8195439 91
1.8260748
1.8325089
1.8388491
1.8450980
TABLE XXI-B.—7-PLACE LOGARITHMS OF USEFUL
84
85 1.9294189
MIS
93
94
1.9084850
1.9138139
1.9190781
1.9242793
95
1.9344985
1.9395193
1.9444827
1.9493900
1.9542425
1.9590414
1.9637878
1.9684829
1.9731279
1.9777236

LOG.
3.1415927
0.4971499
0.3183099 9.5028501
57.295780
1.7581226
3437.7468
3.5362739
0.01745329 8.2418774
0.00029089 6.4637261
2.7182818 0.4342945
576
TABLE XXII.—LOGARITHMIC SIN, COS, TAN, AND COT
Explanation of Use When Angles to Seconds Are Involved
Angles Less Than 3°.-Owing to the rapid variation in sine,
tangent, and cotangent of small angles, the usual straight-
line method of interpolation for seconds may not be sufficiently
precise. A closer method is based upon the fact that natural
sines and tangents of small angles are closely proportional to
the angles themselves.
In this table, ƒ refers to a trigonometric function, and s
refers to the number of seconds in an angle. Although log f
varies rapidly, (log ƒ—log s) varies so slowly that it may be
determined from values at 1' intervals.
EXAMPLE.-Find log tan 0° 38′ 42″.
tan 38'42″ 38′ 42″
tan 38' 38'
2322″
2280"
or log tan 38′ 42″=log 2322+(log tan 2280”—log 2280)
log f=log s +(log f-logs)
log-s=3.365862
or
log ƒ-log s=4.685593 (−10)
log tan 38′ 42″=8.051455 (−10)
In the reverse process of finding an angle to seconds when
its logarithmic sine, tangent, or cotangent is given, first
determine by inspection the value of (log f-log s) or of
(log f+log s). Then subtract the given log ƒ in such a way as
to leave +logs, and find the required angle.
Angles Greater Than 3°.-In the bulk of the table the use of
the Diff. per 1" is replaced by a quicker method in the form
of sets of corrections for multiples of 10" and 15". The num-
ber of these sets varies so as to limit the maximum error for a
50" correction to one in the sixth place of the logarithm any-
where within the indicated range.
Thus, log sin 36° 18′ 50″ is found quickly to be 9.772331
+144-9.772475. Because the sets of corrections are exact
at the middle of the indicated range, slightly greater precision
would be obtained in this example by adding a correction of
143, since the 50″ correction at 36° 30′ is 142. This gives log
sin 36° 18′ 50″-9.772474. Many computers subtract the
10" correction of 29 from log sin 36° 19′.
Obviously, corrections for any number of seconds could be
obtained speedily, if the field work justifies, by combining the
corrections and shifting the decimal point.
577
0°
0-23 HA COM¤¬
4
======*=** ********** 8*88*88688 *=*
20
21
22
23
24
25
26
27
28
29
30
31
33
34
TABLE XXII.—LOGARITHMIC SIN, COS, TAN, AND COT
35
36
37
39
40
42
43
44
45
46
**** C**KKAHEKAO
47
48
49
50
31
52
53
54
55
56
57
58
90°
59
60
//
S
0
60
120
180
240
300
360
420
480
540
600
660
720
780
840
900
960
1020
1080
1140
1200
1260
1320
1380
1440
1500
1560
1620
1680
1740
1800
1860
1920
1980
2040
2100
2160
2220
2280
2340
2400
2460
2520
2580
2640
2700
2760
2820
2880
2940
3000
3060
3120
3180
3240
3300
3360
34 20
3480
3540
3600
$
"/
Sin.
log f
-Inf.
6.463726
6.764756
6.940847
7.065786
7.162696
.241877
.308824
.366316
.417968
7.463 726
.505118
.542906
.577668
.609853
7.639816
.667845
.694173
.718997
.742478
7.764754
.785943
.806146
.825451
.843934
7.861662
.878695
.895085
.910879
.926119
7.940842
.955082
.968870
.982233
7.995198
8.007787
.020021
.031919
.043501
.054781
8.065776
.076500
.086965
.097183
.107167
8.116926
.126471
.135810
.144953
.153907
8.162681
.171280
.179713
.187985
.196102
8.204070
.211895
.219581
.227134
.234557
8.241855
log f
Cos.
log f
575
575
575
575
575
575
575
575
574
574
574
574
574
574
574
573
573
573
573
573
572
572
572
572
571
571
571
570
570
570
569
569
569
568
568
567
567
566
566
566
-log s
4.685
565
565
564
564
563
562
562
561
561
560
560
559
558
558
557
556
556
555
554
554
553
4.685
log f
575
575
575
575
575
575
575
575
576
576
576
576
577
577
577
578
578
578
579
579
580
580
581
581
582
583
583
584
584
585
586
587
587
588
589
590
591
592
593
593
594
595
596
598
599
600
601
602
603
604
605
607
608
609
611
612
613
615
616
618
619
-log s
Tan.
log f
- Inf.
6.463726
6.764756
6.940847
7.065 786
7.162696
.241878
.308825
.366817
.417970
7.463727
.505120
.542909
.577672
.609857
7.639820
.667849
.694 179
.719003
.742484
7.764761
.785951
.806155
.825460
.843944
7.861674
.878708
.895099
.910891
.926134
7.940858
.955 100
.968889
.982253
7.995219
8.007809
.020044
.031945
.043527
.054809
8.065806
.076531
.086997
.097217
.107203
8.116963
.126510
.135851
.144996
.153952
8.162727
.171328
.179763
.188036
.196156
8.204 126
.211953
.219641
.227195
.234621
8.24 1921
log f
Cot.
Col.
log f
+ Inf.
3.536274
3.235244
3.059153
2.934214
2.837304
.758122
.691175
.633183
.582030
2.536273
.494880
.457091
.422328
.390143
2.360180
.332151
.305821
.280997
.257516
2.235239
.214049
.193845
.174540
.15 605 6
2.138326
.121292
.104901
.089106
.073866
2.059142
.044900
.031111
.017747
2.004781
1.992191
.979956
.968055
.956473
.945 191
1.934194
.923469
.913003
.902783
.892797
1.883037
.873490
.864149
.855004
.84 6048
1.837273
.828672
.820237
.811964
.803844
1.795874
.788047
.780359
.772805
.765379
1.75 8079
log f
Tan.
log f
+log s
5.314
425
425
425
425
425
425
425
425
4 24
424
424
424
423
423
423
422
422
422
421
421
420
420
419
419
418
417
417
416
416
415
414
413
413
412
411
4 10
409
408
407
407
406
405
404
402
401
400
399
398
397
396
395
393
392
391
389
398
387
385
384
382
381
5.314
log f
+log s
Cos.
ten
ten
Len
ten
ten
ten
9.999999
.999999
.999999
.999999
9.999998
.999998
.999997
.999997
.999996
9.999996
.999995
.999995
.999994
.999993
9.999993
.999992
.999991
.999990
.999989
9.999989
.999988
.999987
.999986
.999985
9.999983
.999982
.999981
.999980
.999979
9.999977
.999976
.999975
.999973
.999972
9.999971
.999969
.999968
.999966
.999964
9.999963
.999961
.999959
.999958
.999956
9.999954
.999952
.999950
.999948
.999946
9.999944
.999942
.999940
.999938
.999936
9.999934
Sin.
179°
60
59
58
57
56
55
54
53
52
33 88959****= =8*58*388= 88:
51
50
49
48
47
46
45
44
43
42
41
40
39
37
36
34
33
32
31
30
29
28
27
26
25
24
**** ***======= CROTON+~~HO
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
2
89°
578
1°
91°
--23 EN LO CO 1 ∞ ∞
0
1
4
5
6
7
8
9
RICCIELE** 22******** A*88*88583 7********* A****BAKE
10
11
12
14
15
16
17
18
19
20
21
23
24
25
26
27
28
29
30
31
32
33
34
36
37
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
"1
$
3600
3660
3720
3780
3840
3900
3960
4020
4080
4140
4200
4260
4320
4380
4440
4500
4560
4620
4680
4740
4800
4860
4920
4980
5040
5100
5160
5220
5280
5340
5400
5460
5520
5580
5640
5700
5760
5820
5880
5940
6000
6060
6120
6180
6240
6300
63 60
6420
6480
6540
6600
6660
6720
6780
6840
6900
6960
7020
7080
7140
7200
S
"I
TABLE XXII.-LOGARITHMIC SINES,
Sin.
log f
8.24 1855
.249033
.256094
.263042
.269881
8.276614
.283243
.289773
.296207
.302546
8.308794
.314954
.321027
.327016
.332924
8.338753
.344504
.350181
.355 783
.361315
8.366777
.372171
.377499
.382762
.387962
8.393101
.398179
.403199
.408161
.413068
8.417919
.422717
.427462
.432156
.436800
8.441394
.445941
.450440
.454893
.459301
8.463665
.467985
.472263
.476498
.480693
8.484848
.488963
.493040
.497078
.501080
8.505045
.508974
.5 12867
.5 16726
.520551
8.524343
.528102
.531828
.535523
.539186
8.542819
log f
Cos.
log f
- log s
4.685
553
552
551
551
550
549
548
547
547
546
545
544
543
542
541
540
539
539
538
537
536
535
534
533
532
531
530
529
527
526
525
524
523
522
521
520
518
517
516
5 15
514
512
511
510
509
507
506
505
503
502
501
499
498
497
495
494
492
491
490
488
487
log f
619
620
622
623
625
627
628
630
632
633
635
637
638
640
642
644
646
648
649
65 1
4.685
653
655
657
659
661
663
665
668
670
672
674
676
679
681
683
685
688
690
693
695
697
700
702
705
707
710
713
715
718
720
723
726
729
731
734
737
740
743
745
748
751
-log s
Tan.
log f
8.241921
.249102
.256165
.263115
.269956
8.276691
.283323
.289856
.296292
.302634
8.308884
.315046
.321122
.327114
.333025
8.338856
.344610
.350289
.355895
.361430
8.366895
.372292
.377622
.382889
.388092
8.393234
.398315
.403338
.408304
.413213
8.418068
.422869
.427618
.432315
.436962
8.441560
.446110
.450613
.455070
.459481
8.463849
.468172
.472454
.476693
.480892
8.485050
.489170
.493250
.497293
.501298
8.505267
.509200
.513098
.516961
.520790
8.524586
.528349
.532080
.535779
.539447
8.543084
log f
Cot.
Cot.
log f
1.758079
.75 0898
.743835
.736885
.730044
1.723309
.716677
.710144
.703708
.697366
1.691116
.684954
.678878
.672886
.666975
1.661144
.655390
.649711
.644105
.638570
1.633105
.627708
.622378
.617111
.611908
1.606766
.601685
.596662
.591696
.586787
1.581932
.577131
.572382
.567685
.563038
1.558440
.553890
.549387
.544930
.540519
1.536151
.531828
.527546
.523307
.519108
1.5 14950
.510830
.506750
.502707
.498702
1.494733
.490800
.486902
.483039
.479210
1.475414
.471651
.467920
.464221
.460553
1.456916
log f
Tan.
log f
+log s
5.314
381
380
378
377
375
373
372
370
368
367
365
363
362
360
358
356
354
352
351
349
347
345
343
341
339
337
334
332
330
328
326
324
321
319
317
315
312
310
307
305
303
300
298
295
293
290
287
285
282
280
277
274
271
269
266
263
260
257
255
252
249
5.314
log f
+log s
Cos.
9.999934
.999932
.999929
.999927
.999925
9.999922
.999920
.999918
.999915
.999913
9.999910
.999907
.999905
.999902
.999899
9.999897
.999894
.999891
.999888
.999885
9.999882
.999879
.999876
.999873
.999870
9.999867
.999864
.999861
.999858
.999854
9.999851.
.999848
.999844
.999841
.999838
9.999834
.999831
.999827
.999824
.999820
9.999816
.999813
.999809
.999805
.999801
9.999797
.999794
.999790
.999786
.999782
9.999778
.999774
.999769
.999765
.999761
9.999757
.999 753.
.999748
.999744
.999740
9.999735
Sin.
178°

S***ANK A
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
ZEN NNNNNNNNN. ********** **
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
99876 NEON - O
10
5
3
2
1
88°
579
2°
O-~3 HAS TO EX9
1
2
4
92°
5
6
8
10
11
12
13
14
15
16
17
18
19
********** ***8*88588 ***
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
AENCE
50
51
52
53
54
55
56
57
38
59
60
"1
$
7200
7260
7320
7380
7440
7500
7560
7620
7680
7740
7800
7860
7920
7980
8040
8100
8160
8220
8280
8340
8400
8460
8520
8580
8640
8700
8760
8820
8880
8940
9000
9060
9120
9180
9240
9300
9360
9420
9480
9540
9600
9660
9720
9780
9840
9900
9960
10020
10080
10140
10200
10260
10320
10380
10440
10500
10560
10620
10680
10740
10800
S
"/
COSINES, TANGENTS, AND COTANGENTS
Sin.
log f
8.542819
.546422
.549995
.553539
.557054
8.560540
.563999
.567431
.570836
.574214
8.577566
.580892
.584193
.587469
.590721
8.593948
.597152
.600332
.603489
.606623
8.609734
.612823
.615891
.618937
.621962
8.624965
.627948
.630911
.633854
.636776
8.639680
.642563
.645428
.648274
.651102
8.653911
.656702
.659475
.662230
.664968
8.667689
.670393
.673080
.675751
.678405
8.681043
.683 665
.686272
.688863
.691438
8.693998
.696543
.699073
.701589
.704090
8.706577
.709049
.711507
.713952
.716383
8.718800
log f
Cos.
log f
487
485
484
482
481
479
478
476
475
473
-log s
4.685
471
470
468
467
465
463
462
460
458
457
455
453
451
450
448
446
444
443
441
439
437
435
433
431
430
428
426
424
422
420
418
416
414
412
410
408
406
404
402
400
398
396
394
392
389
387
385
383
381
379
376
4.685
log f
751
754
757
760
763
766
769
773
776
779
782
785
788
792
795
798
802
805
808
812
815
818
822
825
829
833
836
840
843
847
851
854
858
862
866
869
873
877
881
885
889
893
897
900
905
909
913
917
921
925
929
933
937
942
946
950
955
959
963
968
972
-log s
Tan.
log f
8.543084
.546691
.550268
.553817
.557336
8.560828
.564291
.567727
.571137
.574520
8.577877
.581208
.584514
.587795
.591051
8.594283
.597492
.600677
.603839
.606978
8.610094
.613189
.616262
.619313
.622343
8.625352
.628340
.631308
.634256
.637184
8.640093
.642982
.645853
.648704
.65 1537
8.654352
.657149
.659928
.662689
.665433
8.668160
.670870
.673563
.676239
.678900
8.681544
.684172
.686784
.689381
.691963
8.694529
.697081
.699617
.702139
.704646
8.707140
.709618
.712083
.714534
.716972
8.719396
log f
Cot.
Col.
log f
1.456916
.453309
.449732
.446183
.442664
1.439172
.435709
.432273
.428863
.425480
1.422123
.418792
.415486
.412205
.408949
1.405717
.402508
.399323
.396161
.393022
1.389906
.386811
.383738
.380687
.377657
1.374648
.371660
.368692
.365744
.362816
1.359907
.357018
.354 147
.351296
.348463
1.345648
.34 2851
.340072
.337311
.334567
1.331840
.329130
.326437
.323761
.321100
1.318456
.315828
.3 13216
.310619
.308037
1.305471
.302919
.300383
.297861
.295354
1.292860
.290382
.287917
.285466
.283028
1.280604
log f
Tan.
Jog f
+log s
5.314
249
246
243
240
237
234
231
227
224
221
218
215
212
208
205
202
198
195
192
188
185
182
178
175
171
167
164
160
157
153
149
146
142
138
134
131
127
123
119
115
111
107
103
100
095
091
087
083
079
075
071
067
063
058
054
050
045
041
037
032
028
5.314
log f
+log s
Cos.
9.999735
.999731
.999726
.999722
.999717
9.999713
.999708
.999704
.999699
.999694
9.999689
.999685
.999680
.999675
.999670
9.999665
.999660
.999655
.999650
.999645
9.999640
.999635
.999629
.999624
.999619
9.999614
.999608
.999603
.999597
.999592
9.999586
.999581
.9995 75
.999570
.999564
9.999558
.999553
.999547
.999541
.999535
9.999529
.999524
.999518
.9995 12
.999506
9.999500
.999493
.999487
.999481
.999475
9.999469
.999463
.999456
.999450
.999443
9.999437
.999431
.999424
.9994 18
.999411
9.999404
Sin.
177°

**********
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
* =88F8***~~ 22*~****~~ 822094CCC. --~20
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
21
19
18
17
16
15
14
13
12
11
10
9
8
7
6
4
1
/
87°
580
3°
01234567∞om
8
9
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
******** SENDIK
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
93°
Sin.
8.718800
.721204
.723595
.725972
.728337
8.730688
.733027
.735354
.737667
.739969
8.742259
.744536
.746802
.749055
.751297
8.753528
.755747
.757955
.760151
.762337
8.764511
.766675
768828
.770970
.773101
8.775223
.777333
.779434
.781524
.783605
8.785675
.787736
.789787
.791828
.793859
8.795881
.797894
.799897
.801892
.803876
8.805852
.807819
.809777
.811726
.813667
8.815599
.817522
.819436
.821343
.823240
8.825130
.827011
.828884
.830749
.832607
8.834456
.836297
.838130
.839956
.841774
8.843585
Cos.
TABLE XXII.-LOGARITHMIC SINES,
D. 1".
40.07
39.85
39.62
39.42
39.18
38.98
38.78
38.55
38.37
38.17
37.95
37.77
37.55
37.37
37.18
36.98
36.80
36.60
36.43
36.23
36.07
35.88
35.70
35.52
35.37
35.17
35.02
34.83
34.68
34.50
34.35
34.18
34.02
33.85
33.70
33.55
33.38
33.25
33.07
32.93
32.78
32.63
32.48
32.35
32.20
32.05
31.90
31.78
31.62
31.50
31.35
31.22
31.08
30.97
30.82
30.68
30.55
30.43
30.30
30.18
D. 1".
Cos.
9.999404
.999398
,999391
.999384
.999378
9.999371
.999364
.999357
.999350
.999343
9.999336
.999329 " Corr.
.999322 10
.999315 15
.999308
20
9.999301 30
.999294
40
.999287 45
.999279 50
.999272
9.999265
.999257
.999250
.999242
.999235
9.999227
.999220
.999212
.999205
.999197
9.999189
.999181
.999174
.999166
.999158
9.999150
.999142
.999134
.999126
.999118
9.999110
.999102
.999094 10
.999086
.999077 20
9.999069
30
.999061 40
.999053
45
.999044 50
.999036
9.999027
.999019
.999010
.999002
.998993
9.998984
.998976
.998967
Corr.
for
Sec.
.998958
.998950
9.998941
Sin.
-
1*18857
1
224 LOS CO
2
2
4
5
5
6
Corr.
1
2
3
4
to co t
6
6
7
Tan. D. 1".
8.719396
.721806
.724204
.726588
.728959
8.731317
.733663
.735996
.738317
.740626
8.742922
.745207
.747479
.749740
.751989
8.754227
.756453
.758668
.760872
.763065
8.765246
.767417
.769578
.771727
.773866
8.775995
.778114
.780222
.782320
.784408
8.786486
.788554
.790613
.792662
.794701
8.796731
.798752
.800763
.802765
.804758
8.806742
.808717
.810683
.812641
.814589
8.816529
.818461
.820384
.822298
.824205
8.826103
.827992
.829874
.831748
.833613
8.835471
.837321
.839163
.840998
842825
8.844644
Cot.
40.17
39.97
39.73
39.52
39.30
39.10
38.88
38.68
38.48
38.27
38.08
37.87
37.68
37.48
37.30
37.10
36.92
36.73
36.55
36.35
36.18
36.02
35.82
35.65
35.48
35.32
35.13
34.97
34.80
34.63
34.47
34.32
34.15
33.98
33.83
33.68
33.52
33.37
33.22
33.07
32.92
32.77
32.63
32.47
32.33
32.20
32.05
31.90
31.78
31.63
31.48
31.37
31.23
31.08
30.97
30.83
30.70
30.58
30.45
30.32
D. 1".
Cot.
1.280604
278194
60
59
275793 58
273412 57
,271041 56
1.268683
55
54
.266337
.264004 53
.261683 52
.259374
51
176°

1.257078
50
254793
49
.252521
48
.250260
47
.248011 46
1.245773
45
44
.243547
.241332 43
.239128 42
.236935
41
1.234754
40
.232583 39
.230422 38
.228273 37
226134 36
1.224005
35
34
.221886
.219778 33
.217680 32
215592
31
1.213514
30
.211446 29
.209387 28
.207338 27
.205299 26
1.203269
25
201248
.199237
24
23
.197235 22
.195242
21
~*******2;
1.193258
20
19
.191283
.189317 18
.187359 17
1.173897
.172008
.170126
.168252
.166387
1.164529
.162679
.160837
.159002
.157175
1.155356
.185411 16
1.183471 15
.181539 14
.179616 13
.177702 12
.175795 11
Tan.
10
9
8
7
SOLO SEM NÍO
6
1
0
86°
581
4°
01234 19 6 1 00 –
5
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
94°
50
51
52
53
54
55
56
57
58
59
60
39993
COSINES, TANGENTS, AND COTANGENTS
Sin. D. 1".
8.843585
.845387
.847183
.848971
.850751
8.852525
.854291
.856049
.857801
.859546
8.861283
.863014
.864738
.866455
.868165
8.869868
.871565
.873255
.874938
.876615
8.878285
.879949
.881607
.883258
.884903
8.886542
.888174
.889801
.891421
.893035
8.894643
.896246
.897842
.899432
.901017
8.902596
.904169
.905736
.907297
.908853
8.910404
.911949
.913488
.915022
.916550
8.918073
919591
.921103
.922610
.924112
8.925609
.927100
.928587
.930068
.931544
8.933015
.934481
.935942
.937398
.938850
8.940296
Cos.
30.03
29.93
29.80
29.67
29.57
29.43
29.30
29.20
29.08
28.95
28.85
28.73
28.62
28.50
28.38
28.28
28.17
28.05
27.95
27.83
27.73
27.63
27.52
27.42
27.32
27.20
27.12
27.00
26.90
26.80
26.72
26.60
26.50
26.42
26.32
26.22
26.12
26.02
25.93
25.85
25.75
25.65
25.57
25.47
25.38
25.30
25.20
25.12
25.03
24.95
24.85
24.78
24.68
24.60
24.52
24.43
24.35
24.27
24.20
24.10
D. 1".
Cos.
9.998941
.998932
.998923
.998914
.998905
9.998896
.998887
998878
.998869
.998860
9.998804
.998795
.998785
.998776
.998766
9.998851
.998841
.998832 10
.998823 15
.998813
9.998757
.998747
.998738
.998728
.998718
9.998708
.998699
.998689
.998679
.998669
9.998659
.998649
.998639
.998629
.998619
9.998609
.998599
.998589
.998578
.998568
.998495
.998485
998474
.998464
9.998453
.998412
.998431
.998421
.998410
9.998399
Corr.
for
Sec.
.998388
.998377
.998366
.998355
9.998344
Sin.
#
20
30
9.998558
.998548
.998537
.998527 15
.998516 20
9.998506 30
40
5555
45
50
#1
10
40
45
50
Corr.
IN 2 3 1 CO 2 ∞
2
2
5
6
7
8
Corr.
1233 in t− ∞ σ
3
3
5
7
8
9
Tan.
8.844644
.846455
.848260
.850057
.851846
8.853628
.855403
.857171
.858932
,860686
8.862433
.864173
.865906
.867632
.869351
8.871064
.872770
.874469
876162
.877819
8.879529
.881202
882869
.884530
.886185
S.887833
.889476
.891112
.892742
.894366
8.895984
.897596
.899203
.900803
.902398
8.903987
.905570
.907147
.908719
,910285
8.911846
.913401
.914951
.916495
.918034
8.919568
.921096
.922619
.924136
.9256-19
8.927156
.928658
930155
931647
.933134
8.934616
.936093
.937565
939032
.940494
8.941952
Cot.
►
D. 1".
30.18
30.08
29.95
29.82
29.70
29.58
29.47
29.35
29.23
29.12
29.00
28.88
28.77
28.65
28.55
28.43
28.32
28.22
28.12
28.00
27.88
27.78
27.68
27.58
27.47
27.38
27.27
27.17
27.07
26.97
26.87
26.78
26.67
26.58
26.48
26.38
26.28
26.20
26.10
26.02
25.92
25.83
25.73
25.65
25.57
25.47
25.38
25.28
25.22
25.12
25.03
24.95
24.87
24.78
24.70
24.62
24.53
24.45
24.37
24.30
D. 1".
Cot.
1.155356
60
.153545 59
.151740 58
.149943 57
.148154 56
1.146372 55
.144597
54
.142829 53
.141068 52
.139314
51
175°

1.137567
50
.135827 49
.134094 48
.132368 47
.130649 46
1.128936 45
.127230 44
.125531 43
.123838
42
.122151 41
1.120471
40
.118798 39
.117131 38
.115170 37
.113815
36
35
1.112167
.110524
34
.108888 33
.107258 32
.105634
31
1.096013
.094430
.092853
.091281
.089715
5555555
1.104016
30
.102404
29
.100797 28
.099197 27
.097602
26
25
24
23
1.072844
.071342
.069845
.06$353
.066866
1.065384
.063907
.062435
1.088154
21
20
.086599 19
.085019 18
.083505 17
.081966
1.080432
16
15
.078904 14
.077381 13
.075864 12
.074351 11
.060968
.059506
1.058048
Tan.
********22
10
9
876543210
85°
582
5°
01234 LO CO 1 ∞ σ
5
6
7
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
95°
Sin.
8.940296
.941738
.943174
.944606
.946034
8.947456
.948874
.950287
.951696
.953100
8.954499
.955894
.957284
.958670
.960052
8.961429
.962801
.964170
.965534
.966893
8.968249
.969600
.970947
.972289
.973628
8.974962
.976293
.977619
.978941
.980259
8.981573
.982883
.984189
.985491
.986789
8.988083
.989374
.990660
.991943
.993222
8.994497
.995768
.997036
.998299
.999560
9.000816
.002069
.003318
.004563
.005805
9.007044
.008278
.009510
.010737
.011962
9.013182
.014400
.015613
.016824
.018031
9.019235
Cos.
TABLE XXII.-LOGARITHMIC SINES,
D. 1".
24.03
23.93
23.87
23.80
23.70
23.63
23.55
23.48
23.40
23.32
23.25
23.17
23.10
23.03
22.95
22.87
22.82
22.73
22.65
22.60
22.52
22.45
22.37
22.32
22.23
22.18
22.10
22.03
21.97
21.90
21.83
21.77
21.70
21.63
21.57
21.52
21.43
21.38
21.32
21.25
21.18
21.13
21.05
21.02
20.93
20.88
20.82
20.75
20.70
20.65
20.57
20.53
20.45
20.42
20.33
20.30
20.22
20.18
20.12
20.07
D. 1".
Cos.
9.998344
.998333
.998322
.998311
.998300
9.998289
.998277
.998266
,998255
.998243
9.998232
.998220
.998209 10
Corr.
2
.998197 15 3
9.998116
.998104
.998092
.998080
.998068
9.998056
.998044
.998032
.998020
.998008
9.998174 30
.998163 40
.998151 45
.998139 50
.998128
9.997996
.997984
.997972
.997959
.997947
9.997935
,997922
.997910
.997897
.997885
Corr.
for
Sec.
9.997872
.997860
I
.960473
.998186 20 4 .961866
8.963255
.964639
.966019
.967394
.968766
.997822
9.997809
.997797
.997784
.997771
.997758
428948
9.997745
.997732
.997719
.997706
.997693
9.997680
.997847 10
.997667
.997654
.997641
.997628
9.997614
Sin.
#
.997835 15
428048
30
45
50
16890
Corr.
2
3
4
6
Tan.
8
10
11
8.941952
.943404
.944852
.946295
.947734
8.949168
.950597
.952021
.953441
.954856
8.956267
.957674
.959075
8.970133
.971496
.972855
.974209
.975560
8.976906
.978248
.979586
.980921
.982251
8.983577
.984899
.986217
,987532
.988842
8.990149
.991451
.992750
.994045
.995337
8.996624
.997908
.999188
9.000465
.001738
9.003007
.004272
.005534
.006792
.008047
9.009298
.010546
.011790
.013031
.014268
9.015502
.016732
.017959
.019183
.020403
9.021620
Cot.
D. 1".
24.20
24.13
24.05
23.98
23.90
23.82
23.73
23.67
23.58
23.52
23.45
23.35
23.30
23.22
23.15
23.07
23.00
22.92
22.87
22.78
22.72
22.65
22.57
22.52
22.43
22.37
22.30
22.25
22.17
22.10
22.03
21.97
21.92
21.83
21.78
21.70
21,65
21.58
21.53
21.45
21.40
21.33
21.28
21.22
21.15
21.08
21.03
20.97
20.92
20.85
20.80
20.73
20.68
20.62
20.57
20.50
20.45
20.40
20.33
20.28
D. 1".
Cot.
174°

8885
1.058048
.056596 59
.055148
.053705 57
.052266 56
1.050832
55
.049403 54
.047979 53
.046559 52
.045144
51
50
1.043733
.042326 49
.040925 48
.039527 47
.038134 46
1.036745 45
60
.035361 44
.033981 43
,032606 42
.031234
41
40
1.029867
.028504 39
.027145 38
.025791 37
.024440
1.023094
36
35
34
.021752
.020414 33
.019079 32
.017749
31
1.016423
30
.015101 29
.013783 28
.012468
27
.011158 26
1.009851
25
.008549 24
.007250 23
.005955
.004663 21
1.003376
20
.002092 19
.000812 18
0.999535 17
.998262 16
15
0.996993
.995728 14
.994466 13
.993208 12
.991953 11
********22
0.990702 10
.989454
9
.988210
8
7
.986969
6
.985732
0.984498
.983268
.982041
.980817
.979597
0.978380
Tan.
OLEST
5
1
0
84°
583
6°
0123410 CZ∞ σ
6
7
8
9
10
11
12
13
14
15
16
17
18
19
222222
20
21
23
24
25
26
27
28
29
96°
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Sin.
COSINES, TANGENTS, AND COTANGENTS
9.019235
.020435
.021632
.022825
.024016
9.025203
.026386
.027567
.028744
.029918
9.031089
.032257
.033421
.034582
.035741
9.036896
.038048
.039197
.040342
.041485
9.042625
.043762
.044895
.046026
.047154
9.048279
.049400
.050519
.051635
.052749
9.053859
.054966
.056071
.057172
.058271
9.059367
.060460
.061551
.062639
.063724
9.064806
.065885
.066962
.068036
.069107
9.070176
.071242
.072306
.073366
.074424
9.075480
.076533
.077583
.078631
.079676
9.080719
.081759
.082797
.083832
.084864
9.085894
Cos.
D. 1".
20.00
19.95
19.88
19.85
19.78
19.72
19.68
19.62
19.57
19.52
19.47
19.40
19.35
19.32
19.25
19.20
19.15
19.08
19.05
19.00
18.95
18.88
18.85
18.80
18.75
18.68
18.65
18.60
18.57
18.50
18.45
18.42
18.35
18.32
18.27
18.22
18.18
18.13
18.08
18.03
17.98
17.95
17.90
17.85
17.82
17.77
17.73
17.67
17.63
17.60
17.55
17.50
17.47
17.42
17.38
17.33
17.30
17.25
17.20
17.17
D. 1".
Cos.
9.997614
.997601
.997588
.997574
.997561
9.997547
.997534
.997520
.997507
.997493
9.997341
.997327
.997313
9.997480
tt
.997466
.997452 10
·.997439 15
.997425 20
9.997411 30
.997397 40 9
.997383 45 10
.997369 50 12
.997355
.997299
.997285
9.997271
.997257
.997242
.997228
.997214
9.997199
.997185
.997170
.997156
.997141
9.997127
.997112
.997098
.997083
.997068
9.996904
.996889
.996874
.996858
.996843
9.996828
.996812
.996797
.996782
.996766
9.996751
Corr.
for
Sec.
Sin.
9.997053
W
.997039
.997024 10
.997009 15
.996994 20
9.996979 30
.996964 40
7
10
.996949 45 11
.996934 50 12
.996919
Corr.
5ŏoJawNl
Corr.
2
4
Tan.
9.021620
.022834
.024044
.025251
.026455
9.027655
.028852
.030046
.031237
.032425
9.033609
.034791
.035969
.037144
.038316
9.032185
.040651
.041813
.042973
.044130
9.045284
.046434
.047582
.048727
.049869
9.051008
.052141
.053277
.054407
.055535
9.056659
.057781
.058900
.060016
.061130
9.062210
.063348
.064453
065556
.066655
9.067752
.068816
.069938
.071027
.072113
9.073197
.074278
.075356
.076132
.077505
9.078576
.0796-44
.080710
.081773
.082833
9.083891
.0849.17
.086000
.087050
.088098
9.089144
Cot.
D. 1".
20.23
20.17
20.12
20.07
20.00
19.95
19.90
19.85
19.80
19.73
19.70
19.63
19.58
19.53
19.48
19.43
19.37
19.33
19.28
19.23
19.17
19.13
19.08
19.03
18.98
18.93
18.88
18.83
18.80
18.73
18.70
18.65
18.60
18.57
18.50
18.47
18.42
18.38
18.32
18.28
18.23
18.20
18.15
18.10
18.07
18.02
17.97
17.93
17.88
17.85
17.80
17.77
17.72
17.67
17.63
17.60
17.55
17.50
17.47
17.43
D. 1".
Cot.
0.978380
60
.977166 59
.975956 58
,974749 57
.973545 56
0.972345
55
.971148 54
.969954 53
.968763
52
.967575
51
0.966391
.965209
.964031
48
.962856 47
.961684
173°

46
45
0.960515
.959349 44
.958187 43
.957027 42
.955870 41
50
49
0.954716
40
39
.953566
.952418 38
.951273
37
.950131
36
0.948992
35
.947856
34
.946723 33
.945593 32
.944465 31
0.943341
.942219
.941100
.939984 27
.938870 26
0.937760
25
.936652 24
.935547 23
0.921424
.920356
.919290
.918227
.917167
0.916109
.915053
.914000
.912950
.911902
0.910856
288888998
.934444 22
.933345 21
Tan.
0.932248
20
.931154 19
.930062 18
.928973 17
.927887 16
0.926803 15
.925722 14
.924644 13
.923568 12
.922495
11
30
10
9
8
6
DLO « BO-O
3
2
1
0
/
83°
584
7°
0123HELLO KO DO ∞∞
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
97°
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
Sin.
9.085894
.086922
.087947
.088970
.089990
9.091008
.092024
.093037
.094047
.095056
9.096062
.097065
.098066
.099065
.100062
9.101056
.102048
.103037
.104025
.105010
9.105992
.106973
.107951
.108927
.109901
9.110873
.111842
.112809
.113774
.114737
9.115698
.116656
.117613
.118567
.119519
9.120469
.121417
.122362
.123306
.124248
9.125187
.126125
.127060
.127993
.128925
9.129854
.130781
.131706
.132630
.133551
9.134470
.135387
.136303
53
.137216
54
.138128
55
9.139037
56
.139944
57 .140850
58
.141754
59
.142655
60
9.143555
Cos.
TABLE XXII.-LOGARITHMIC SINES,
D. 1".
17.13
17.08
17.05
17.00
16.97
16.93
16.88
16.83
16.82
16.77
16.72
16.68
16.65
16.62
16.57
16.53
16.48
16.47
16.42
16.37
16.35
16.30
16.27
16.23
16.20
16.15
.16.12
16.08
16.05
16.02
15.97
15.95
15.90
15.87
15.83
15.80
15.75
15.73
15.70
15.65
15.63
15.58
15.55
15.53
15.48
15.45
15.42
15.40
15.35
15.32
15.28
15.27
15,22
15.20
15.15
15.12
15.10
15.07
15.02
15.00
D. 1".
Cos.
9.996751
.996735
.996720
.996704
.996688
9.996673
.996657
.996641
.996625
.996610
9.996594
.996578 ม Corr.
.996562 10 3
.996546 15 4
5
.996530 20
9.996514 30
.996498 40 11
8
.996482 45 12
.996465 50 13
.996449
9.996433
.996417
.996400
.996384
.996368
9.996351
.996335
.996318
.996302
.996285
9.996269
..996252
.996235
.996219
.996202
9.996185
.996168
.996151
.996134
.996117
9.996100
.996083
Corr.
.996066
10
3
.996049 15 4
.996032 20
9.996015 30
6
9
11
.995998 40
.995980 45 13
.995963 50 14
.995946
9.995928
.995911
.995894
.995876
.995859
9.995841
.995823
.995806
.995788
.995771
9.995753
Corr.
for
Sec.
Sin.
H
Tan.
9.089144
.090187
.091228
.092266
.093302
9.094336
.095367
.096395
.097422
.098446
9.099468
.100487
.101504
.102519
.103532
9.104542
.105550
.106556
.107559
.108560
9.109559
.110556
.111551
.112543
.113533
9.114521
.115507
.116491
.117472
.118452
9.119429
.120404
.121377
.122348
.123317
9.124284
.125249
.126211
.127172
.128130
9.129087
.130041
.130994
.131944
.132893
9.133839
.134784
.135726
.136667
.137605
9.138542
.139476
.140409
.141340
.142269
9.143196
.144121
.145044
.145966
.146885
9.147803
Cot.
D. 1".
17.38
17.35
17.30
17.27
17.23
17.18
17.13
17.12
17.07
17.03
16.98
16.95
16.92
16.88
16.83
16.80
16.77
16.72
16.68
16.65
16.62
16.58
16.53
16.50
16.47
16.43
16.40
16.35
16.33
16.28
16.25
16.22
16.18
16.15
16.12
16.08
16.03
16.02
15.97
15.95
15.90
15.88
15.83
15.82
15.77
15.75
15.70
15.68
15.63
15.62
15.57
15.55
15.52
15.48
15.45
15.42
15.38
15.37
15.32
15.30
D. 1".
Cot.
172°

0.910856
.909813
60
59
.908772 58
.907734 57
.906698
0.905664
56
55
.904633 54
.903605
53
.902578 52
.901554 51
0.900532 50
.899513 49
.898496 48
.897481 47
.896468
0.895458
46
45
.894450 44
.893444 43
.892441 42
.891440
41
0.890441
40
39
.889444
.888449 38
.887457 37
.886467 36
0.885479 35
.884493 34
.883509 33
.882528 32
.881548 31
0.880571
30
.879596 29
.878623
28
.877652
.876683 26
0.875716 25
.874751 24
.873789 23
.872828
.871870 21
0.870913
20
.869959 19
.869006 18
.868056 17
.867107 16
0.866161 15
.865216 14
.864274 13
.863333 12
.862395 11
0.861458
.860524
.859591
.858660
.857731
0.856804
.855879
.854956
.854034
.853115
0.852197
Tan.
22*7****≈≈ 22*!
10
9
8
7
6
1043210
5
82°
585
8°
01234UCZBO
7
8
9
10
11
12
13
14
15
16
17
18
19
EO ES ES ED ES MO
20
21
22
23
24
25
26
27
28
29
* 30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
7000000
48
49
50
51
52
53
54
55
56
57
58
888888
59
60
98°
COSINES, TANGENTS, AND COTANGENTS
Sin. D. 1"
9.143555
.144453
.145349
.146243
.147136
9.148026
.148915
.149802
.150686
.151569
9.152451
.153330
.154208
.155083
.155957
9.156830
.157700
.158569
.159435
.160301
9.161164
.162025
.162885
.163743
.164600
9.165454
.166307
.167159
.168008
.168856
9.169702
.170547
.171389
.172230
.173070
9.173908
.174744
.175578
.176411
.177242
9.178072
.178900
.179726
.180551
.181374
9.182196
.183016
.183834
.184651
.185466
9.186280
.187092
.187903
.188712
.189519
9.190325
.191130
.191933
.192734
.193534
9.194332
14.97
14.93
14.90
14.88
14.83
14.82
14.78
14.73
14.72
14.70
14.65
14.63
14.58
14.57
14.55
14.50
14.48
14.43
14.43
14,38
14.35
14.33
14.30
14.28
14.23
14.22
14.20
14.15
14.13
14.10
14.08
14.03
14.02
14.00
13.97
13.93
13.90
13.88
13.85
13.83
13.80
13.77
13.75
13.72
13.70
13.67
13.63
13.62
13.58
13.57
13.53
13.52
13.48
13.45
13.43
13.42
13.38
13.35
13.33
13.30
Cos. D. 1".
Cos.
9.995753
.995735
.995717
.995699
.995681
9.995664
.995646
.995628
.995610
.995591
9.995390
.995372
.995353
.995334
.995316
9.995573
.995555
Corr.
.995537 10
3
.995519 15
5
.995501 20
6
9.995482 30
9
.995464 40
12
.995446 45
14
.995427 50 15
.995409
9.995297
.995278
.995260
,995241
.995222
9.995203
.995184
,995165
.995146
.995127
9.995108
.995089
.995070
.995051
.995032
9.994818
,994798
.994779
.994759
.994739
9.994720
.994700
.994680
Corr.
for
Sec.
#1
9.995013
.994993
.994974 10
.994955 15
.994660
.994640
9.994620
Sin.
Corr.
3
5
.994935 20
6
9.994916 30 10
.994896 40 13
.994877 45 15
16
.994857
50
.994838
M
Tan. D. 1".
9.147803
.148718
.149632
.150544
.151454
9.152363
.153269
.154174
.155077
.155978
9.156877
.157775
.158671
.159565
.160457
9.161347
.162236
.163123
.164008
.164892
9.165774
.166654
.167532
.168409
.169284
9.170157
.171029
.171899
.172767
.173634
9.174499
.175362
.176224
.177084
.177942
9.178799
.179655
.180508
.181360
.182211
9.183059
.183907
.184752
.185597
.186439
9.187280
.188120
.188958
.189794
.190629
9.191462
.192294
.193124
.193953
.194780
9.195606
.196430
.197253
.198074
.198894
9.199713
Cot.
15.25
15.23
15.20
15.17
15.15
15.10
15.08
15.05
15.02
14.98
14.97
14.93
14.90
14.87
14.83
14.82
14.78
14.75
14.73
14.70
14.67
14.63
14.62
14.58
14.55
14.53
14.50
14.47
14.45
14.42
14.38
14.37
14.33
14.30
14.28
14.27
14.22
14.20
14.18
14.13
14.13
14.08
14.08
14.03
14.02
14.00
13.97
13.93
13.92
13.88
13.87
13.83
13.82
13.78
13.77
13.73
13.72
13.68
13.67
13.65
D. 1".
Cot.
0.852197 60
.851282 59
.850368 58
.849456 57
.848546 56
0.847637
55
846731 54.
.845826 53
.844923 52
.844022
51
171°

0.843123
50
.842225
49
.841329 48
.840435 47
.839543 46
0.838653
45
,837764 44
.836877 43
.835992 42
.835108
41
0.834226
40
.833346 39
.832468 38
.831591 37
..830716 36
0.829843 35
.828971 34
.828101 33
.827233 32
.826366 31
0.825501
30
.824638 29
.823776 28
.822916 27
.822058 26
0.821201 25
.820345 24
.819492 23
.818640 22
.817789 21
0.816941
20
.816093 19
.815248 18
.814403 17
.813561 16
0.812720
15
14
.811880
.811042 13
.810206 12
.809371 11
0.808538
.807706
.806876
.806047
.805220
0.804394
.803570
.802747
,801926
.801106
0.800287
Tan.
10
9
8
78143210
6
2.
81°
586
9°
01234 LOCO DI ∞ σ
5
6
8
9
10
11
12
13
14
15
16
17
18
19
272****7
20
21
23
24
25
26
28
29
30
31
32
33
34
35
36
37
38
39
27234
99°
40
41
44
45
46
47
48
49
Sin.
9.194332
.195129
.195925
.196719
.197511
9.198302
.199091
.199879
.200666
.201451
9.202234
.203017
.203797
.204577
.205354
9.206131
.206906
.207679
.208452
.209222
9.209992
.210760
.211526
212291
.213055
9.213818
.214579
.215338
.216097
.216854
9.217609
.218363
.219116
.219868
.220618
9.221367
.222115
.222861
.223606
.224349
9.225092
,225833
.226573
.227311
.228048
9.228784
.229518
.230252
.230984
.231715
50
9.232444
51
.233172
52
.233899
53
.234625
54
.235349
55
9.236073
56
.236795
57
,237515
58
.238235
59
.238953
60 9.239670
Cos.
TABLE XXII.-LOGARITHMIC SINES,
D. 1".
13.28
13.27
13.23
13.20
13.18
13.15
13.13
13.12
13.08
13.05
13.05
13.00
13.00
12.95
12.95
12.92
12.88
12.88
12.83
12.83
12.80
12.77
12.75
12.73
12.72
12.68
12.65
12.65
12.62
12.58
12.57
12.55
12.53
12.50
12.48
12.47
12.43
12.42
12.38
12.38
12.35
12.33
12.30
12.28
12.27
12.23
12.23
12.20
12.18
12.15
12.13
12.12
12.10
12.07
12 07
12.03
12.00
12.00
11.97
11.95
D. 1".
Cos.
9.994620
.994600
.994580
.994560
.994540
9.994519
.994499
.994479
.994459
.994438
9.994418
.994398
.994254
.994233
Corr.
.994377 10
3
.994357 15
5
.994336 20 7
9.994316 30 10
.994295 40
.994274 45
14
15
17
50
9.994212
.994191
.994171
.994150
.994129
9.994108
.994087
.994066
.994045
.994024
9.994003
.993982
.993960
.993939
.993918
9.993897
.993875
.993854
.993832
.993811
"
9.993572
.993550
.993528
.993506
.993484
9.993462
.993440
Corr.
for
Sec.
9.993789
.993768 11 Corr.
.993746 10
4
.993725 15
5
7
.993703 20
9.993681 30 11
.993660 40 14
.993638 45 16
.993616 50 18
.993594
.993418
.993396
.993374
9.993351
Sin.
Tan.
9.199713
.200529
.201345
.202159
.202971
9.203782
.204592
.205400
.206207
.207013
9.207817
.208619
.209420
.210220
.211018
9.211815
.212611
.213405
.214198
.214989
9.215780
.216568
.217356
.218142
.218926
9.219710
.220492
.221272
.222052
.222830
9.223607
.224382
.225156
.225929
.226700
9.227471
.228239
.229007
.229773
.230539
9.231302
.232065
.232826
233586
.234345
9.235103
.235859
.236614
.237368
.238120
9.238872
.239622
.240371
.241118
.241865
9.242610
243354
.244097
.244839
.245579
9.246319
Cot.
D. 1".
13.60
13.60
13.57
13.53
13.52
13.50
13.47
13.45
13.43
13.40
13.37
13.35
13.33
13.30
13.28
13.27
13.23
13.22
13.18
13.18
13.13
13.13
13.10
13.07
13.07
13.03
13.00
13.00
12.97
12.95
12.92
12.90
12.88
12.85
12.85
12.80
12.80
12.77
12.77
12.72
12.72
12.68
12.67
12.65
12.63
12.60
12.58
12.57
12.53
12.53
12.50
12.48
12.45
12.45
12.42
12.40
12.38
12.37
12.33
12.33
D. 1".
Cot.
0.800287
60
.799471 59
.798655 58
.797841 57
.797029
56
0.796218
55
54
.795408
.794600 53
..793793 52
.792987 51
0.792183 50
.791381 49
.790580 48
.789780 47
.788982 46
0.788185 45
.787389 44
.786595 43
.785802 42
.785011
41
0.784220
40
.783432 39
.782644 38
.781858 37
170°

.781074
0.780290
36
35
.779508 34
.778728 33
777948
32
.777170 31
0.776393
30
.775618 29
.774844 28
.774071 27
0.772529
.773300 26
25
.771761 24
.770993 23
.770227 22
.769461 21
0.768698
20
.767935 19
.767174 18
.766414 17
.765655 16
0.764897 15
.764141 14
.763386 13
.762632 12
.761880 11
0.761128
,760378
.759629
.758882
.758135
0.757390
.756646
.755903
.755161
.754421
0.753681
Tan.
10
9
· CO LO TE CONHO
2
80°
587
10°
0123 HIS CO Z-89
6
7
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
100°
COSINES, TANGENTS, AND COTANGENTS
Sin.
9.239670
.240386
.241101
241814
.242526
9.243237
,243947
.244656
.245363
.246069
9.246775
.247478
.248181
.248883
.249583
9.250282
.250980
.251677
.252373
.253067
9.253761
.254453
.255144
.255834
.256523
9.257211
.257898
.258583
.259268
.259951
9.260633
.261314
.26 1994
.262673
.263351
9.264027
.264703
.265377
.266051
.266723
9.267395
.268065
.268734
.269402
.270069
9.270735
.271400
.272064
.272726
.273388
9.274049
.274708
.275367
.276025
.276681
9.277337
.277991
.278645
.279297
.279948
9.280599
Cos.
D. 1".
11.93
11.92
11.88
11.87
11.85
11.83
11.82
11.78
11.77
11.77
11.72
11.72
11.70
11.67
11.65
11.63
11.62
11.60
11.57
11.57
11.53
11.52
11.50
11.48
11.47
11.45
11.42
11.42
11.38
11.37
11.35
11.33
11.32
11.30
11.27
11.27
11.23
11.23
11.20
11.20
11.17
11.15
11.13
11.12
11.10
11.08
11.07
11.03
11.03
11.02
10.98
10.98
10.97
10.93
10.93
10.90
10.90
10.87
10.85
10.85
D. 1".
Cos.
9.993351
.993329
.993307
.993284
.993262
9.993240
.993217
.993195
.993172
.993149
9.992898
.992875
.992852
.992829
.992806
9.992783
.992759
.992736
9.993127
.993104
Corr.
.993081 10
4
.993059 15 6
.993036 20
8
9.993013 30 11
.992990
1.40
15
17
.992967
45
.992944 50 19
.992921
.992713
.992690
9.992666
.992643
.992619
.992596
.992572
9.992549
.992525
,992501
.992478
.992454
9.992190
.992166
.992142
.992118
.992093
9.992069
"1
9.992430
.992406
.992382 10
,992359 15
.992335 20
8
9.992311 30 12
.992044
.992020
.991996
.991971
9.991947
Corr.
for
Sec.
Sin.
11
Corr.
4
∞
.992287 40 16
.992263 45
.992239 50
.992214
6
600080
18
20
Tan.
9.246319
.247057
.247794
.248530
.249264
9.249998
.250730
.251461
.252191
.252920
9.2536-18
254374
.255100
.255824
.256547
9.257269
.257990
.258710
.259429
.260146
9.260863
.261578
.262292
.263005
.263717
9.264428
.265138
.265847
.266555
.267261
9.267967
.268671
.269375
.270077
.270779
9.271479
.272178
.272876
.273573
.274269
9.274964
.275658
.276351
.277043
.277734
9.278424
.279113
.279801
.280488
.281174
9.281858
.282542
.283225
.283907
.284588
9.285268
.285947
.286624
.287301
.287977
9.288652
Cot.
D. 1".
12.30
12.28
12.27
12.23
12.23
12.20
12.18
12.17
12.15
12.13
12.10
12.10
12.07
12.05
12.03
12.02
12.00
11.98
11.95
11.95
11.92
11.90
11.88
11.87
11.85
11.83
11.82
11.80
11.77
11.77
11.73
11.73
11.70
11.70
11.67
11.65
11.63
11.62
11.60
11.58
11.57
11.55
11.53
11.52
11.50
11.48
11.47
11.45
11.43
11.40
11.40
11.38
11.37
11.35
11.33
11.32
11.28
11.28
11.27
11.25
D. 1".
Cot.
0.753681
.752943
60
59
.752206 58
.751470 57
.750736 56
0.750002 55
.749270 54
.748539 53
.747809 52
.747080 51
169°

0.746352
50
49
.745626
.744900 48
.744176 47
.743453
0.742731
46
45
.742010 44
.741290 43
.740571 42
.739854
41
0.739137
40
.738422 39
.737708 38
.736995 37
36
.736283
0.735572
35
.734862
34
.734153 33
.733445
32
.732739
31
0.732033
30
29
.731329
.730625 28
.729923
27
26
.729221
0.728521
228
25
.727822 24
.727124 23
.726427 22
.725731 21
0.718142
.717458
.716775
.716093
715412
0.714732
.714053
.713376
.712699
.712023
0.711348
0.725036
20
19
.723649 18
.724342
.722957
17
.722266 16
0.721576 15
.720887 14
.720199 13
.719512 12
.718826 11
Tan.
A
098765432IO
10
1
79°
588
11°
01234BU7∞" OER
6
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
101°
Sin.
9.280599
.281248
.281897
.282544
.283190
9.283836
.284480
.285124
.285766
.286108
9.287048
.287688
.288326
.288964
.289600
9.290236
.290870
.291504
.292137
.292768
9.293399
.294029
.294658
.295286
.295913
9.296539
.297164
297788
.298412
.299034
9.299655
.300276
.300895
.301514
.302132
9.302748
.303364
.303979
.304593
..305207
9.305819
.306430
.307041
.307650
.308259
9.308867
.309474
.310080
.310685
.311289
9.311893
.312495
.313097
.313698
:314297
9.314897
.315495
.316092
.316689
.317284
9.317879
Cos.
TABLE XXII.-LOGARITHMIC SINES,
D. 1".
10.82
10.82
10.78
10.77
10.77
10.73
10.73
10.70
10.70
10.67
10.67
10.63
10.63
10.60
10.60
10.57
10.57
10.55
10.52
10.52
10.50
10.48
10.47
10.45
10.43
10.42
10.40
10.40
10.37
10.35
10.35
10.32
10.32
10.30
10.27
10.27
10.25
10.23
10.23
10.20
10.18
10.18
10.15
10.15
10.13
10.12
10.10
10.08
10.07
10.07
10.03
10.03
10.02
9.98
10.00
9.97
9.95
9.95
9.92
9.92
D. 1".
Cos.
9.991947
.991922
.991897
.991873
.991848
9.991823
.991799
.991774
.991749
.991724
9.991699
.991674
.991649 10
.991624 15
.991599 20
9.991574
30 13
.991549
17
.991524 45 19
.991498
21
.991473
9.991448
.991422
.991397
.991372
.991346
9.991321
.991295
.991270
.991244
.991218
9.991193
.991167
.991141
.991115
.991090
9.991064
.991038
.991012
.990986
.990960
9.990803
11
88948
9.990671
.990645
.990618
.990591
.990565
40
9.990934
.990908
.990882
.990855 15
.990829 20
9.990538
.990511
.990485
.990458
.990431
9.990404
Sin.
50
Corr.
for
Sec.
་
Corr.
10 4
7
9
30 13
.990777 40 18
.990750 45
.990724 50
.990697
Corr.
468
22223
20
Tan.
9.288652
.289326
.289999
.290671
.291342
9.292013
.292682
.293350
.294017
,294684
9.295349
.296013
.296677
.297339
.298001
9.298662
.299322
,299980
.300638
.301295
9.301951
.302607
.303261
.303914
.304567
9.305218
.305869
.306519
.307168
.307816
9.308463
.309109
.309754
.310399
.311042
9.311685
.312327
.312968
.313608
.314247
9.314885
.315523
.316159
.316795
.317430
9.318064
.318697
.319330
.319961
.320592
9.321222
.321851
.322479
.323106
.323733
9.324358
.324983
.325607
.326231
.326853
9.327475
Cot.
D. 1".
11.23
11.22
11.20
11.18
11.18
11.15
11.13
11.12
11.12
11.08
11.07
11.07
11.03
11.03
11.02
11.00
10.97
10.97
10.95
10.93
10.93
10.90
10.88
10.88
10.85
10.85
10.83
10.82
10.80
10.78
10.77
10.75
10.75
10.72
10.72
10.70
10.68
10.67
10.65
10.63
10.63
10.60
10.60
10.58
10.57
10.55
10.55
10.52
10.52
10.50
10.48
10.47
10.45
10.45
10.42
10.42
10.40
10.40
10.37
10.37
D. 1".
Cot.
0.711348
60
.710674 59
.710001
.709329
.708658
0.707987
55
.707318 54
.706650 53
.705983 52
.705316 51
168°

1889156
0.704651 50
.703987 49
.703323 48
.702661 47
.701999 46
0.701338 45
.700678 44
.700020 43
.699362 42
.698705 41
57
0.698049
40
.697393 39
.696739 38
.696086
99999
37
.695433 36
0.694782 35
.694131 34
.693481
33
.692832 32
.692184 31
0.678778
.678149
.677521
.676894
.676267
0.675642
0.691537
30
.690891 29
.690246 28
.689601
27
.688958 26
0.688315 25
.687673 24
.687032
.686392
23
.685753 21
.675017
.674393
.673769
.673147
0:672525
Tan.
0.685115 20
.684477 19
.683841 18
.683205 17
.682570 16
0.681936
.681303
.680670
15
14
13
.680039 12
.679408 11
********22 2-**
10
9
7
5
4
3210
78°
589
12°
01234 LE LO II 00 C
5
7
8
9
10
11
12
13
14
15
16
17
18
19
27222*27*2 ******BAR GERAI4-*** ADI
23
24
25
26
28
29
20 9.329599
21
.330176
.330753
.331329
.331903
9.332478
.333051
.333624
.334195
.334767
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
098998
102°
COSINES, TANGENTS, AND COTANGENTS
Sin.
9.317879
.318473
.319066
.319658
.320249
9.320840
.321430
.322019
.322607
.323194
9.323780
.324366
.324950
.325534
.326117
9.326700
.327281
.327862
.328442
.329021
9.335337
.335906
,336475
.337043
.337610
9.338176
.338742
,339307
.339871
.340434
9.340996
341558
.342119
.342679
.343239
9.343797
.344355
.344912
.345469
.346024
9.346579
.347134
.347687
.348240
.348792
9.349343
.349893
.350443
.350992
.351540
9.352088
Cos.
D. 1".
9.90
9.88
9.87
9.85
9.85
9.83
9.82
9.80
9.78
9.77
9.77
9.73
9.73
9.72
9.72
9.68
9.68
9.67
9.65
9.63
9.62
9.62
9.60
9.57
9.58
9.55
9.55
9.52
9.53
9.50
9.48
9.48
9.47
9.45
9.43
9.43
9.42
9.40
9.38
9.37
9.37
9.35
9.33
9.33
9.30
9.30
9.28
9.28
9.25
9.25
9.25
9.22
9.22
9.20
9.18
9.17
9.17
9.15
9.13
9.13
D. 1".
Cos.
9.990404
.990378
.990351
,990324
.990297
9.990270
.990243
.990215
.990188
.990161
9.990134
.990107
.990079 10
.990052 15
.990025 20
9.989997
30
9.989860
.989832
.989804
.989777
.989749
9.989721
.989693
.989665
.989637
.989610
9.989582
.989553
.989525
.989497
.989469
9.989441
.989413
.989385
.989356
.989328
Corr.
for
Sec.
.989970
.989942 45 21
.989915 50 23
.989887
9.989014
.988985
.988956
.988927
.988898
9.988869
.988840
.988811
.988782
.988753
9.988724
Sin.
11
Corr.
5
7
9
14
40 18
88948
9.989300
.989271 " Corr.
.989243 10 5
.989214
15
7
.989186 20
9.989157
.989128 40
.989100 45
.989071 50
.989042
10
30 14
19
21
28948
24.
Tan.
9.327475
.328095
.328715
.329334
.329953
9.330570
.331187
.331803
.332418
.333033
9.333646
.334259
.334871
.335482
.336093
9.336702
.337311
.337919
.338527
.339133
9.339739
.340344
.340948
.341552
.342155
9.342757
.343358
.343958
.344558
.345157
9.345755
.346353
.346949
.347545
.348141
9.348735
.349329
.349922
.350514
.351106
9.351697
.352287
.352876
.353465
.354053
9.354640
.355227
.355813
.356398
.356982
9.357566
.358149
.358731
.359313
.359893
9.360474
.361053
.361632
.362210
.362787
9.363364
Cot.
D. 1".
10.33
10.33
10.32
10.32
10.28
10.28
10.27
10.25
10.25
10.22
10.22
10.20
10.18
10.18
10.15
10.15
10.13
10.13
10.10
10.10
10.08
10.07
10.07
10.05
10.03
10.02
10.00
10.00
9.98
9.97
9.97
9.93
9.93
9.93
9.90
9.90
9.88
9.87
9.87
9.85
9:83
9.82
9.82
9.80
9.78
9.78
9.77
9.75
9.73
9.73
9.72
9.70
9.70
9.67
9.68
9.65
9.65
9.63
9.62
9.62
D. 1".
Cot.
0.672525
.671905
.671285 58
.670666 57
.670047 56
0.669430 55
.668813 54
.668197 53
.667582 52
.666967 51
167°

0.666354
50
.665741 49
.665129 48
.664518
47
.663907
46
0.663298
45
.662689
44
.662081 43
.661473 42
.660867 41
0.642434
.641851
.641269
.640687
5899
0.660261
40
39
.659656
.659052
38
.658448 37
.657845 36
0.657243 35
.656642 34
.656042 33
.655442 32
.654843 31
0.654245 30
.653647 29
.653051 28
.652455 27
.651859 26
0.651265 25
.650671 24
.650078 23
.649486 22
.648894 21
.640107
0.639526
60
0.648303 20
.647713 19
.647124 18
.646535 17
.645947 16
0.645360 15
.644773 14
.644187 13
.643602 12
.643018 11
.638947
.638368
.637790
.637213
0.636636
Tan.
10
9
7
6
LO *
2
/
77°
590
13°
0123 ALONG
8
9
10
11
12
13
14
15
16
17
*22222222222
19
20
21
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
103°
Sin.
9.352088
.352635
.353181
.353726
.354271
9.354815
.355358
.355901
.356443
.356984
9.357524
.358064
.358603
.359141
.359678
9.360215
.360752
.361287
.361822
.362356
9.362889
.363422
.363954
,364485
.365016
9.365546
.366075
.366604
.367131
.367659
9.368185
.368711
.369236
.369761
.370285
9.370808
.371330
.371852
.372373
.372894
9.373414
.373933
.374452
.374970
.375487
9.376003
.376519
.377035
.377549
.378063
9.378577
.379089
.379601
.380113
.380624
9.381134
.381643
.382152
.382661
.383168
9.383675
Cos.
TABLE XXII.-LOGARITHMIC SINÈS,
D. 1".
9.12
9.10
9.08
9.08
9.07
9.05.
9.05
9.03
9.02
9.00
9.00
8.98
8.97
8.95
8.95
8.95
8.92
8.92
8.90
8.88
8.88
8.87
8.85
8.85
8.83
8.82
8.82
8.78
8.80
8.77
8.77
8.75
8.75
8.73
8.72
8.70
8.70
8.68
8.68
8.67
8.65
8.65
8.63
8.62
8.60
8.60
8.60
8.57
8.57
8.57
8.53
8.53
8.53
8.52
8.50
8.48
8.48
8.48
8.45
8.45
D. 1".
Cos.
9.988724
.988695
.988666
.988636
.988607
9.988578
.988548
.988519
.988489
.988460
9.988430
.988401
.988371 10
.988342 15
.988312 20
9.988282 30
.988252 40
.988223 45
.988193 50 25
.988163
9.988133
.988103
.988073
.988043
.988013
9.987983
.987953
.987922
.987892
.987862
9.987832
.987801
.987771
.987740
.987710
9.987679
.987649
.987618
.987588
.987557
9.987217
.987186
.987155
.987124
.987092
9.987061
.987030
.986998
.986967
.986936
9.986904
-
Sin.
10
Corr.
for
Sec.
$1
9.987526
.987496
Corr.
.987465 10
5
.987434 15
8
.987403 20 10
9.987372 30 15
.987341
21
.987310
.987279
.987248
40
5555
45
Corr.
5
7
10
15
50
24823
20
23
26
Tan.
9.363364
.363940
.364515
.365090
.365664
9.366237
.366810
.367382
.367953
.368524
9.369094
.369663
.370232
.370799
.371367
9.371933
.372499
.373064
.373629
.374193
9.374756
.375319
.375881
.376442
.377003
9.377563
.378122
.378681
.379239
.379797
9.380354
.380910
.381466
.382020
.382575
9.383129
.383682
.384234
384786
.385337
9.385888
.386438
.386987
.387536
.388084
9.388631
.389178
.389724
.390270
.390815
9.391360
.391903
.392447
.392989
.393531
9.394073
.394614
.395154
.395694
.396233
9.396771
Cot.
D. 1".
9.60
9.58
9.58
9.57
9.55
9.55
9.53
9.52
9.52
9.50
9.48
9.48
9.45
9.47
9.43
9.43
9.42
9.42
9.40
9.38
9.38
9.37
9.35
9.35
9.33
9.32
9.32
9.30
9.30
9.28
9.27
9.27
9.23
9.25
9.23
9.22
9.20
9.20
9.18
9.18
9.17
9.15
9.15
9.13
9.12
9.12
9.10
9.10
9.08
9.08
9.05
9.07
9.03
9.03
9.03
9.02
9.00
9.00
8.98
8.97
D. 1".
Cot.
60
0.636636
.636060 59
.635485 58
.634910 57
.634336
56
55
0.633763
.633190 54
.632618 53
.632047 52
.631476 51
0.630906 50
.630337 49
.629768 48
.629201 47
.628633
46
45
0.628067
.627501
44
.626936 43
.626371 42
.625807 41
0.625244
40
.624681 39
.624119 38
.623558 37
.622997 36
0.622437 35
.621878 34
.621319 33
.620761 32
.620203 31
166°

30
0.619646
.619090
29
.618534 28
.617980 27
.617425 26
0.616871 25
.616318 24
.615766 23
.615214 22
.614663 21
0.608640
.608097
.607553
.607011
.606469
0.605927
.605386
.604846
.604306
.603767
0.603229
0.614112
20
.613562 19
.613013 18
.612464 17
.611916 16
0.611369 15
.610822 14
13
.610276
.609730 12
.609185 11
Tan.
********Z
10
9
8
7
·CK 43210
5
76°
591
14°
0123 ELO CN89
7
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
104°
COSINES, TANGENTS, AND COTANGENTS
Sin.
9.383675
.384182
.384687
.385192
.385697
9.386201
.386704
.387207
,387709
.388210
9.388711
.389211
.389711
.390210
.390708
9.391206
.391703
.392199
.392695
.393191
9.393685
.394179
.394673
.395166
.395658
9.396150
.396641
.397132
.397621
.398111
9.398600
.399088
.399575
.400062
400549
9.401035
.401520
.402005
.402489
.402972
9.403455
.403938
.404420
.404901
.405382
9.405862
.406341
.406820
.407299
.407777
9.408254
.408731
.409207
.409682
.410157
9.410632
.411106
.411579
.412052
.412524
9.412996
Cos.
D. 1".
8.45
8.42
8.42
8.42
8.40
8.38
8.38
8.37
8.35
8.35
8.33
8.33
8.32
8.30
8.30
8.28
8.27
8.27
8.27
8.23
8.23
8.23
8.22
8.20
8.20
8.18
8.18
8.15
8.17
8.15
8.13
8.12
8.12
8.12
8.10
8.08
8.08
8.07
8.05
8.05
8.05
8.03
8.02
8.02
8.00
7.98
7.98
7.98
7.97
7.95
7.95
7.93
7.92
7.92
7.92
7.90
7.88
7.88
7.87
7.87
D. 1".
Cos.
9.986904
.986873
.986841
.986809
.986778
9.986746
.986714
.986683
.986651
.986619
9.986587
9.986266
.986234
.986202
.986169
་་
Corr.
.986555
.986523 10
5
8
.986491 15
.986459
20 11
9.986427 30
16
.986395 40 21
.986363 45 24
.986331 50 27
.986299
.986137
9.986104
.986072
.986039
.986007
.985974
9.985942
.985909
.985876
.985843
.985811
9.985778
.985745
.985712
.985679
.985646
9.985280
.985247
.985213
.985180
.985146
9.985113
9.985613
.985580
Corr.
.985547
10
6
.985514 15
8
.985480 20 11
9,985447 30 17
.985414 40
.985381 45 25
.985347 50
28
.985314
.985079
.985045
.985011
.984978
9.984944
Sin.
Corr.
for
Sec.
"J
222
Tan.
9.396771
.397309
.397846
.398383
.398919
9.399455
.399990
.400524
.401058
.401591
9.402124
.402656
.403187
.403718
.404249
9.404778
.405308
.405836
.406304
.406892
9.407419
.407945
.408471
.408996
.409521
9.410045
.410569
.411092
.411615
.412137
9.412658
.413179
.413699
.414219
.414738
9.415257
.415775
.416293
.416810
.417326
9.417842
.418358
.418873
.419387
.419901
9.420415
.420927
.421440
.421952
.422463
9.422974
.423484
.423993
.424503
.425011
9.425519
.426027
.426534
.427041
.427547
9.428052
Cot.
D. 1".
8.97
8.95
8.95
8.93
8.93
8.92
8.90
8.90
8.88
8.88
8.87
8.85
8.85
8.85
8.82
8.83
8.80
8.80
8.80
8.78
8.77
8.77
8.75
8.75
8.73
8.73
8.72
8.72
8.70
8.68
8.68
8.67
8.67
8.65
8.65
8.63
8.63
8.62
8.60
8.60
8.60
8.58
8.57
8.57
8.57
8.53
8.55
8.53
8.52
8.52
8.50
8.48
8.50
8.47
8.47
8.47
8.45
8.45
8.43
8.42
D. 1″.
Cot.
0.603229
.602691
60
59
.602154 58
.601617 57
.601081 56
55
.600010 54
0.600545
.599476 53
.598942 52
.598409 51
165°

0.597876
50
.597344 49
.596813 48
.596282 47
46
.595751
0.595222
45
.594692
44
.594164
43
.593636 42
.593108
41
0.592581 40
.592055
39
.591529 38
.591004 37
.590479
0.589955
36
35
34
.589431
.588908 33
.588385 32
.587863 31
0.587342 30
.586821 29
.586301 28
.585781 27
26
25
.584225 24
.583707
23
.583190 22
.582674 21
.585262
0.584743
0.582158
.581642 19
.581127 18
.580613 17
.580099 16
0.579585 15
0.577026
.576516
.576007
.575497
.574989
0.574481
.573973
.573466
.572959
.572453
0.571948
222
Tan.
20
.579073 14
.578560 13
.578048 12
.577537 11
10
9876543210
75°
592
15°
012341 618
5
9
10
11
12
13
14
15
16
17
18
19
2222
20
21
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
105°
Sin. D. 1".
9.412996
.413467
.413938
.414408
.414878
9.415347
.415815
.416283
.416751
.417217
9.417684
.418150
.418615
.419079
.419544
9.420007
.420470
.420933
.421395
.421857
9.422318
.422778
.423238
.423697
.424156
9.424615
.425073
.425530
.425987
.426443
9.426899
.427354
.427809
.428263
.428717
9.429170
.429623
.430075
.430527
.430978
9.431429
.431879
.432329
.432778
.433226
9.433675
.434122
.434569
.435016
.435462
9.435908
.436353
.436798
.437242
.437686
9.438129
.438572
.439014
TABLE XXII.-LOGARITHMIC SINES,
.439456
.439897
9.440338
Cos.
7.85
7.85
7.83
7.83
7.82
7.80
7.80
7.80
7.77
7.78
7.77
7.75
7.73
7.75
7.72
7.72
7.72
7.70
7.70
7.68
7.67
7.67
7.65
7.65
7.65
7.63
7.62
7.62
7.60
7.60
7.58
7.58
7.57
7.57
7.55
7.55
7.53
7.53
7.52
7.52
7.50
7.50
7.48
7.47
7.48
7.45
7.45
7.45
7.43
7.43
7.42
7.42
7.40
7.40
7.38
7.38
7.37
7.37
7.35
7.35
D. 1".
Ccs.
9.984944
.984910
.984876
.984842
.984808
9.984774
.984740
.984706
.984672
.984638
9.984603
Corr.
.984569
.984535
10
.984500
6
15 9
.984466 20 11
30
9.984432
17
.984397 40 23
.984363
45 26
.984328 50 29
.984294
9.984259
.984224
.984190
.984155
.984120
9.984085
.984050
.984015
.983981
.983946
9.983911
.983875
.983840
.983805
.983770
9.983735
.983700
.983664
.983629
.983594
Corr.
for
Sec.
9.983202
.983166
.983130
9.983558
.983523
.983487 10
.983452 15
.983416 20 12
9.983381 30 18
.983345 40
.983309 45
.983273 50 30
.983238
.983094
.983058
9.983022
.982986
.982950
.982914
.982878
9.982842
Sin.
=
IT
Corr.
6
9
12220
24
27
Tan. D. 1".
9.428052
.428558
.429062
.429566
.430070
9.430573
.431075
.431577
.432079
.432580
9.433080
.433580
.434080
.434579
.435078
9.435576
.436073
.436570
.437067
.437563
9.438059
.438554
.439048
.439543
.440036
9.440529
441022
.441514
.442006
.442497
9.442988
.443479
.443968
.444458
.441947
9.445435
.445923
.446411
.446898
.447384
9.447870
.448356
.448841
.449326
.449810
9.450294
.450777
.451260
.451743
.452225
9.452706
.453187
.453668
.454148
.454628
9.455107
.455586
.456064
.456542
.457019
9.457496
Cot.
8.43
8.40
8.40
8.40
8.38
8.37
8.37
8.37
8.35
8.33
8.33
8.33
8.32
8.32
8.30
8.28
8.28
8.28
8.27
8.27
8.25
8.23
8.25
8.22
8.22
8.22
8.20
8.20
8.18
8.18
8.18
8.15
8.17
8.15
8.13
8.13
8.13
8.12
8.10
8.10
8.10
8.08
8.08
8.07
8.07
8.05
8.05
8.05
8.03
8.02
8.02
8.02
8.00
8.00
7.98
7.98
7.97
7.97
7.95
7.95
D. 1".
Cot.
0.571948
60
59
.571442
.570938 58
.570434 57
.569930
56
0.569427
55
.568925
54
.568423
53
.567921
52
.567420 51
0.566920
.566420
.565920
.565421
.564922
0.564424
.563927
.563430
164

.562933
.562437
50
49
48
47
46
45
44
43
42
41
0.561941
40
39
.561446
.560952 38
.560457 37
.559964 36
35
0.559471
.558978 34
.558486
33
.557994
32
.557503 31
0.557012
30
29
.556521
.556032 28
.555542 27
0.554565
.555053 26
25
24
.554077
.553589 23
.553102 22
.552610 21
0.552130 20
.551644 19
.551159 18
.550674 17
.550190 16
0.549706
.549223
15
14
.548740 13
.548257 12
.547775 11
0.547294 10
.546813
.546332
.545852
.515372
0.544893
.544414
.543936
.543458
.542981
0.542504
Tan.
9
8
7
6
› LOTS & HO
4
3
2
1
0
74
593
16°
012341 CZ ∞ —
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
23
22242722
25
26
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
55
56
57
58
59
60
106°
COSINES, TANGENTS, AND COTANGENTS
Sin.
9.440338
.440778
.441218
.441658
.442096
9.442535
.442973
.443410
.443847
.414284
9.444720
445155
.445590
446025
.446459
9.446893
447326
4-47759
448191
.448623
9.449054
449485
449915
.450345
.450775
9.451204
.451632
.452060
452488
.452915
9.453342
.453768
.454194
.454619
.455044
9.455469
.455893
.456316
.456739
.457162
9.457584
.458006
.458427
.458848
.459268
9.459688
.460108
.460527
.460946
.461364
9.461782
.462199
.462616
.463032
.463448
9.463864
.464279
.464694
.465108
.465522
9.465935
Cos.
D. 1".
7.33
7.33
7.33
7.30
7.32
7.30
7.28
7.28
7.28
7.27
7.25
7.25
7.25
7.23
7.23
7.22
7.22
7.20
7.20
7.18
7.18
7.17
7.17
7.17
7.15
7.13
7.13
7.13
7.12
7.12
7.10
7.10
7.08
7.08
7.08
7.07
7.05
7.05
7.05
7.03
7.03
7.02
7.02
7.00
7.00
7.00
6.98
6.98
6.97
6.97
6.95
6.95
6.93
6.93
6.93
6.92
6.92
6.90
6.90
6.88
D. 1".
Cos.
9.982842
.982805
.982769
.982733
.982696
9.982660
.982624
.982587
.982551
.982514
9.982109
.982072
.982035
.981998
.981961
9.981924
.981886
.981849
.981812
.981774
9.982477
.982441
Corr.
.982404 10
6
.982367 15
9
.982331 20 12
9.982294
.982257
.982220
982183
.982146
9.981737
.981700
.981662
,981625
.981587
9.981549
.981512
.981474
.981436
.981399
Corr.
for
Sec.
9.980981
.980942
.980904
.980866
.980827
9.980789
.980750
.980712
.980673
.980635
9.980596
Sin.
11
89985
30 18
40
45
50
=
2225
28
9.981361
.981323
.981285
10
.981247 15
13
.981209 20
9.981171 30 19
.981133 40 25
.981095 45 29
.981057 50
.981019
31
Corr.
6
10
2322
Tan.
9.457496
.457973
.458449
.458925
.459400
9.459875
460349
.460823
.461297
.461770
9.462242
.462715
.463186
.463658
.464128
9.464599
.465069
.465539
.466008
.466477
9.466945
.467413
.467880
.468347
.468814
9.459280
.469746
.470211
.470676
.471141
9.471605
.472069
.472532
.472995
.473457
9.473919
.474381
474842
.475303
.475763
9.476223
.476683
.477142
.477601
.478059
9.478517
.478975
.479432
.479889
.480345
9.480801
.481257
.481712
.482167
.482621
9.483075
.483529
.483982
.484435
.484887
9.485339
Cot.
D. 1".
7.95
7.93
7.93
7.92
7.92
7.90
7.90
7.90
7.88
7.87
7.88
7.85
7.87
7.83
7.85
7.83
7.83
7.82
7.82
7.80
7.80
7.78
7.78
7.78
7.77
7.77
7.75
7.75
7.75
7.73
7.73
7.72
7.72
7.70
7.70
7.70
7.68
7.68
7.67
7.67
7.67
7.65
7.65
7.63
7.63
7.63
7.62
7.62
7.60
7.60
7.60
7.58
7.58
7.57
7.57
7.57
7.55
7.55
7.53
7.53
D. 1".
Cot.
60
0.542504
.542027 59
.541551 58
541075
57
.540600
56
0.540125
55
.539651
54
.539177
53
.538703 52
.538230
51
0.537758
50
.537285
49
.536814 48
.536342 47
.535872 46
0.535-101
45
.534931 44
.534461
43
42
.533523 41
.533992
163°

0.533055
40
.532587 39
.532120 38
.531653 37
.531186 36
0.530720
35
34
.530254
.529789 33
.529324 32
.528859 31
0.528395
30
.527931
29
.527468 28
.527005 27
.526543 26
0.526081
25
.525619 24
.525158
23
.524697
.524237 21
0.519199
.518743
.518288
.517833
.517379
0.523777
20
.523317
19
.522858 18
17
.522399
.521941 16
0.521483 15
.521025 14
.520568 13
.520111 12
.519655 11
0.516925
.516171
.516018
.515565
.515113
0.514661
Tan.
********22
10
9
8
7
6543210
73°
594
17°
01234167∞‹
8
9
10
11
12
13
14
15
16
17
18
19
222******≈ 8-03-0688
20
21
23
24
25
26
27
28
29
31
32
33
34
35
37
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
99999
107°
Sin.
TABLE XXII.-LOGARITHMIC SINES,
9.465935
.466348
.466761
.467173
.467585
9.467996
.468407
.468817
.469227
.469637
9.470046
.470455
.470863
.471271
.471679
9.472086
.472492
.472898
.473304
.473710
9.474115
.474519
.474923
.475327
.475730
9.476133
.476536
.476938
.477340
.477741
9.478142
.478542
.478942
.479342
.479741
9.480140
.480539
.480937
.481334
.481731
9.482128
.482525
.482921
.483316
.483712
9.484107
.484501
.484895
.485289
.485682
9.486075
.486467
486860
.487251
.487643
9.488034
.488424
.488814
.489204
.489593
9.489982
Cos.
D. 1".
6.88
6.88
6.87
6.87
6.85
6.85
6.83
6.83
6.83
6.82
6.82
6.80
6.80
6.80
6.78
6.77
6.77
6.77
6.77
6.75
6.73
6 73
6.73
6.72
6.72
6.72
6.70
6.70
6.68
6.68
6.67
6.67
6.67
6.65
6.65
6.65
6.63
6.62
6.62
6.62
6.62
6.60
6.58
6.60
6.58
6.57
6.57
6.57
6.55
6.55
6.53
6.55
6.52
6.53
6.52
6.50
6.50
6.50
6.48
6.48
D. 1".
Cos.
9.980596
.980558
.980519
.980480
.980442
9.980403
.980364
.980325
.980286
.980247
9.980208
.980169
Corr.
.980130
10
7
.980091 15 10
,980052 20
9.980012
13
30 20
.979973 40 26
.979934 45 29
.979895 50 33
.979855
9.979816
.979776
.979737
.979697
.979658
9.979618
.979579
.979539
.979499
979459
9.979420
.979380
.979340
.979300
.979260
9.979220
.979180
.979140
.979100
.979059
Corr.
for
Sec.
9.978615
.978574
.978533
.978493
.978452
9.978411
.978370
.978329
.978288
.978247
9.978206
Sin.
11
9.979019
.978979
Corr,
7
.978939 10
.978898 15 10
.978858 20 13
9.978817 30 20
.978777 40 27
.978737 45 30
.97S696 50 34
.978655
222223
11
Tan. D. 1".
9.485339
.485791
.486242
.486693
.487143
9.487593
.488043
.488492
.488941
.489390
9.489838
.490286
.490733
.491180
.491627
9.492073
.492519
.492965
.493410
.493854
9.494299
.494743
.495186
495630
.496073
9.496515
.496957
.497399
.497811
.498282
9.498722
.499163
.499603
.500012
500481
9.500920
.501359
.501797
.502235
.502672
9.503109
.503546
.503982
.504418
.504854
9.505289
.505724
.506159
.506593
.507027
9.507460
.507893
.508326
.508759
.509191
9.509622
.510054
.510485
.510916
.511346
9.511776
Cot.
7.53
7.52
7.52
7.50
7.50
7.50
7.48
7.48
7.48
7.47
7.47
7.45
7.45
7.45
7.43
7.43
7.43
7.42
7.40
7.42
7.40
7.38
7.40
7.38
7.37
7.37
7.37
7.37
7.35
7.33
7.35
7.33
7.32
7.32
7.32
7.32
7.30
7.30
7.28
7.28
7.28
7.27
7.27
7.27
7.25
7.25
7.25
7.23
7.23
7.22
7.22
7.22
7.22
7.20
7.18
7.20
7.18
7.18
7.17
7.17
D. 1".
Cot.
.512857
0.512407
0.514661
.514209
.513758 58
.513307 57
56
55
.511957 54
.511508 53
.511059 52
.510610 51
162°

60
59
0.510162 50
.509714 49
.509267 48
.508820 47
.508373 46
0.507927 45
.507481 44
.507035 43
,506590 42
.506146 41
9999
0.505701
40
39
.505257
.504814 38
.504370 37
.503927
0.503485
36
35
.503043 34
.502601 33
.502159 32
.501718 31
0.501278 30
.500837 29
.500397 28
.499958 27
.499519
26
0.499080 25
.498641 24
******** 22
.498203 23
.497765 22
.497328 21
0.496891 20
0.492540
.492107
.491674
.491241
.490809
0.490378
.489946
.489515
.489084
.488654
0.488224
.496454 19
Tan.
.496018 18
.495582 17
.495146
0.494711
16
15
.494276 14
.493841 13
.493407 12
.492973 11
10
9
8
76543210
72°
595
18°
0123D FLO ES t∞a
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
*******
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
108°
COSINES, TANGENTS, AND COTANGENTS
Sin. D. 1".
9.489982
490371
.490759
.491147
.491535
9.491922
.492308
.492695
.493081
.493466
9.493851
.494236
.494621
.495005
495388
9.495772
.496154
.496537
.496919
.497301
9.497682
.498064
.498444
.498825
.499204
9.499584
.499963
.500342
.500721
.501099
9.501476
.501854
.502231
.502607
.502984
9.503360
.503735
.504110
.504485
.501860
9.505234
.505603
.505981
.506354
.506727
9.507099
.507471
.507843
.508214
.508585
9.508956
.509326
.509696
.510065
.510434
9.510803
.511172
.511540
511907
.512275
9.512642
Cos.
6.48
6.47
6.47
6.47
6.45
6.43
6.45
6.43
6.42
6.42
6.42
6.42
6.40
6.38
6.40
6.37
6.38
6.37
6.37
6.35
6.37
6.33
6.35
6.32
6.33
6.32
6.32
6.32
6.30
6.28
6.30
6.28
6.27
6.28
6.27
6.25
6.25
6.25
6.25
6.23
6.23
6.22
6.22
6.22
6.20
6.20
6.20
6.18
6.18
6.18
6.17
6.17
6.15.
6.15
6.15
6.15
6.13
6.12
6.13
6.12
D. 1".
Cos.
9.978206
.978165
.978124
.978083
.978042
9.978001
.977959
.977918
.977877
.977835
9.977377
.977335
.977293
9.977794
.977752
.977711 10
Corr
7
.977669 15 10
.977628 20 14
9.977586 30 21
.977544 40 28
.977503 45 31
.977461 50 35
.977419
.977251
.977209
9.977167
.977125
.977083
.977041
.976999
9.976957
.976914
.976872
.976830
.976787
9.976745
.976702
.976660
.976617
.976571
9.976103
.976060
.976017
Corr.
for
Sec.
.975974
.975930
9.975887
.975844
.975800
.975757
.975714
9.975670
Sin.
"
9.976532
.976489
Corr.
7
.976446 10
.976404 15 11
.976361 20 14
9.976318 30 .21
.976275 40 29
.976232 45 32
.976189 50 36
.976146
Tan.
9.511776
.512206
.512635
.513064
.513493
9.513921
.514349
.514777
.515204
.515631
9.516057
.516481
.516910
.517335
.517761
9.518186
.518610
.519034
.519458
.519882
9.520305
.520728
.521151
.521573
.521995
9.522417
.522838
.523259
.5236.80
.524100
9.524520
.524940
.525359
.525778
.526197
9.526615
.527033
.527451
.527868
.528285
9.528702
.529119
.529535
.529951
.530366
9.530781
.531196
.531611
.532025
.532139
9.532853
.533266
.533679
.534092
.534504
9.534916
.535328
.535739
.536150
.536561
9.536972
Cot.
D. 1".
7.17
7.15
7.15
7.15
7.13
7.13
7.13
7.12
7.12
7.10
7.12
7.10
7.08
7.10
7.08
7.07
7.07
7.07
7.07
7.05
7.05
7.05
7.03
7.03
7.03
7.02
7.02
7.02
7.00
7.00
7.00
6.98
6.98
6.98
6.97
6.97
6.97
6.95
6.95
6.95
6.95
6.93
6.93
6.92
6.92
6.92
6.92
6.90
6.90
6.90
6.88
6.88
6.88
6.87
6.87
6.87
6.85
6.85
6.85
6.85
D. 1".
Cot.
0.488224
60
.487794 59
.487365 58
.486936 57
.486507 56
0.486079
55
.485651 54
.485223 53
.484796 52
.484369
51
-161°

0.483943
50
.483516
49
.483090 48
.482665
47
46
.482239
0.481814
45
.481390 44
.480966 43
.480542 42
.480118 41
0.479695
40
.479272 39
.478849 38
.478427 37
.478005 36
0.477583 35
.477162 34
.476741 33
.476320 32
.475900 31
0.475480
30
29
.475060
.474641 28
.474222 27
.473803 26
0.473385
25
.472967
24
.472549
.472132
.471715 21
0.471298
20
.470881 19
0.467147
.466734
.466321
.465908
.465496
0.165084
.464672
.464261
.463850
.463439
0.463028
***22 22:
.470465 18
.470049 17
.469634 16
0.469219 15
Tan.
23
.46SS04 14
.468389 13
.467975 12
.467561 11
10
9
8
CICLO A3210
7
6
71°
596
19°
0123E LA CD 2– 00 σ
5
6
8
9
10
11
12
13
14
15
16
17
18
19
228
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
109°
Sin.
9.512642
.513009
.513375
.513741
.514107
9.514472
.514837
.515202
.515566
.515930
9.516294
.516657
.517020
.517382
.517745
9.518107
.518468
.518829
.519190
.519551
9.519911
.520271
.520631
,520990
.521349
9.521707
.522066
.522424
.522781
.523138
9.523495
.523852
.524208
.524564
.524920
9.525275
.525630
.525984
.526339
.526693
9.527046
.527400
.527753
.528105
.528458
9.528810
.529161
.529513
.529864
.530215
9.530565
.530915
.531265
.531614
.531963
9.532312
.532661
.533009
.533357
.533704
9.534052
Cos.
TABLE XXII.-LOGARITHMIC SINES,
D. 1".
6.12
6.10
6.10
6.10
6.08
6.08
6.08
6.07
6.07
6.07
6.05
6.05
6.03
6.05
6.03
6.02
6.02
6.02
6.02
6.00
6.00
6.00
5.98
5.98
5.97
5.98
5.97
5.95
5.95
5.95
5.95
5.93
5.93
5.93
5.92
5.92
5.90
5.92
5.90
5.88
5.90
5.88
5.87
5.88
5.87
5.85
5.87
5.85
5.85
5.83
5.83
5.83
5.82
5.82
5.82
5.82
5.80
5.80
5.78
5,80
D. 1".
Cos.
9.975670
.975627
.975583
.975539
.975496
9.975452
.975408
.975365
.975321
.975277
9,975233
.975189
9.974792
.974748
.974703
,974659
.974614
9.974570
.974525
.974481
.974436
.974391
Corr.
.975145 10
7
.975101 15 11.
.975057 20
9.975013 30
15
.974969 40 29
.974925 45 33
.974880 50 37
.974836
9.974347
.974302
.974257
.974212
.974167
9.974122
.974077
.974032
.973987
.973942
H
9.973444
.973398
.973352
.973307
.973261
9.973215
.973169
.973124
.973078
.973032
9.972986
Sin.
Corr.
for
Sec.
9.973897
.973852
.97380710
.973761 15
.973716 20
9.973671 30
Corr.
8
11
15
23
.973625 40 30
.97358045 34
.973535 50 38
.973489
"
F42285
Tan. D. 1".
9.536972
.537382
.537792
.538202
.538611
9.539020
.539429
.539837
.540245
.540653
9.541061
.541468
.541875
.542281
.542688
9.543094
.543499
'.543905
.544310
.544715
9.545119
.545524
.545928
.546331
.546735
9.547138
.547540
.547943
.548345
.548747
9.549149
.549550
.549951
.550352
.550752
9.551153
.551552
.551952
.552351
.552750
9.553149
.553548
.553946
.554344
.554741
9.555139
.555536
.555933
.556329
.556725
9.557121
.557517
.557913
.558308
.558703
9.559097
.559491
.559885
.560279
.560673
9.561066
Cot.
6.83
6.83
6.83
6.82
6.82
6.82
6.80
6.80
6.80
6.80
6.78
6.78
6.77
6.78
6.77
6.75
6.77
6.75
6.75
6.73
6.75
6.73
6.72
6.73
6.72
6.70
6.72
6.70
6.70
6.70
6.68
6.68
6.68
6.67
6.68
6.65
6.67
6.65
6.65
6.65
6.65
6.63
6.63
6.62
6.63
6.62
6.62
6.60
6.60
6.60
6.60
6.60
6.58
6.58
6.57
6.57
6.57
6.57
6.57
6.55
D. 1".
Cot.
0.463028
.462618
.462208
160°

8885
.461798 57
.461389 56
0.460980 55
.460571 54
.460163 53
.459755 52
.459347 51
60
0.458939
50
.458532 49
.458125 48
.457719 47
.457312
46
0.456906
45
.456501
44
.456095 43
.455690 42
.455285 41
0.442879
.442483
.442087
.441692
.441297
0.440903
59
0.454881
40
39
.454476
.454072 38
.453669 37
.453265 36
0.452862
35
.452460 34
.452057 33
.451655 32
.451253 31
.440509
.440115
.439721
.439327
0.438934
0.450851
30
.450450 29
.450049 28
.449648 27
.449248 26
0.448847 25
.448448 24
.448048 23
.447649 22
.447250 21
Tan.
0.446851 20
.446452 19
.446054 18
.445656 17
.445259 16
0.444861 15
.444464 14
.444067 13
.443671 12
.443275 11
10
9
8
7
6
LOTES ON TO
2
0
70°
597
20°
0123T LO CO 1 0σ
6
7
8
9
10
11
12
13
14
15
16
17
18
*1 222 223
19
21
24
25
26
27
28
29
20 9.540931
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
COSINES, TANGENTS, AND COTANGENTS
Sin.
110°
Corr.
for
Sec.
9.534052
.534399
* Corr.
.534745 10 58
.535092 15 86
.535438 20 115
9.535783 30 173
.536129 40 230
.536474 45
259
.536818 50 288
.537163
9.537507
11
.537851 Corr.
.538194 10 57
.538538 15 86
.538880 20 114
9.539223 30 171
.539565 40 228
.539907 45 257
.540249 50 285
.540590
.541272
Corr.
.541613 10 57
,541953 15 85
.542293 20 113
9.542632 30 170
.542971 40 226
.543310 45 255
.543649 50 283
.543987
"
9.544325
.544663
Corr.
.545000 10 56
.545338 15 84
.545674 20 112
9.546011 30 168
.546347 40 224
.546683 45 252
.547019
50
280
.547354
Corr.
9.547689
.548024
,548359 10 56
.548693 15 83
.549027 20 111
9.549360 30 167
.549693 40 222
.550026 45 250
.550359 50 278
.550692
9.551024
.551356 1 Corr.
.551687 10 55
.552018 15 83
.552349 20 110
9.552680 30 165
.553010 40 220
57
.553341 45 248
58
.553670
50
275
59
.554000
60 9.554329
Cos.
Г
Cos.
9.972986
.972940
.972894
.972848
.972802
9.972755
.972709
.972663
.972617
.972570
9.972058
.972011
.971964
.971917
.971870
9.971823
.971776
.971729
.971682
.971635
1
9.972524
.972478 Corr.
.972431 10 8
.972385 15 12
.972338 20 16
9.972291 30 23
.972245 40 31
.972198 45 35
.972151 50 39
.972105
9.971588
.971540
.971493
.971446
.971398
9.971351
.971303
.971256
.971208
.971161
9.970635
.970586
.970538
9.971113
.971066
Corr.
.971018 10 8
.970970 15 12
.970922 20 16
9.970874 30 24
.970827 40 32
.970779 45 36
.970731 50 40
.970683
.970490
.970442
9.970394
.970345
.970297
.970249
.970200
9.970152
Sin.
Corr.
for
Sec.
n
Tan.
R
9.561066
.561459 Corr.
.561851 10 65
.562244 15 98
.562636 20 131
9.563028 30 196
.563419 40 261
.563811 45 294
.564202 50 326
.564593
9.564983
.565373 Corr.
.565763 10 65
.566153 15 97
.566542 20 130
9.566932 30 194
.567320 40 259
.567709 45 292
.568098 50 324
.568486
9.568873
.569261 Corr.
.569648 10 64
.570035 15 97
570422 20
129
9.570809 30 193
.571195 40 258
17
.571581 45 290
.571967 50 322
.572352
W
Corr.
for
Sec.
11
9.572738
.573123
Corr.
.573507 10 64
.573892 15 96
320
n
.574276 20 128
9.574660 30 192
.575044 40 256
.575427 45 288
50
.575810
.576193
9.576576
.576959
Corr.
.577341 10 64
.577723 15 95
.578104 20 127
9.578486 30 191
.578867 40 254
.579248 45 286
.579629 50 318
.580009
9.580389
.580769 Corr.
.581149 10 63
.581528 15 95
.581907 20 126
9.582286 30 189
.582665 40 252
.583044 45 284
50 316
.583422
.583800
9.584177
Cot.
"
Cot.
56
55
0.438934
60
.438541 59
.438149 58
.437756 57
.437364
0.436972
.436581 54
.430189 53
.435798 52
.435407 51
0.435017 50
.434627 49
.434237 48
.433847 47
.433458 46
0.433068
45
44
.432680
.432291
43
.431902
42
.431514
41
40
0.431127
39
.430739
.430352 38
.429965
37
.429578 36
0.429191
.428805
35
34
.428419 33
.428033
.427648
159°

0.419611
.419231
.418851
.418472
.418093
0.417714
.417335
.416956
.416578
.416200
0.415823
Tan.
32
0.427262
.426877
.426493
.426108
.425724
0.425340
25
21
20
.424956 24
.424573 23
.424190
.423807
0.423424
.423041 19
.422659 18
.422277 17
.421896 16
0.421514 15
.421133 14
.420752 13
.420371 12
.419991
11
27 82*7****22
31
30
29
26
10
98765 TE 2 HO
0
69°
598
21°
0123456789
222****7*2
10 9.557606
11
12
13
14
15
16
17
18
19
20
21
23
24
25
26
28
29
30
31
32
33
34
35
36
37
588 GILRINGE**
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Sin.
111°
9.554329
.554658
Corr.
.554987 10 55
.555315 15 82
.555643 20 109
9.555971 30 164
.556299 40 218
.556626 45 246
.556953 50 273
.557280
TABLE XXII.-LOGARITHMIC SINES,
.557932 A
Corr.
.558258 10 54
.558583 15 81
.558909 20 108
9.559234 30 162
,559558 40 217
.559883 45 244
.560207 50 271
.560531
9.560855
Corr.
.561178 n
.561501 10 54
.561824 15 80
.562146 20 107
9.562468
30 161
.562790
40 215
.563112 45 242
.563433 50 268
.563755
9.570435
.570751
Corr.
for
Sec.
ה
9.564075
.564396
Corr.
.564716 10 53
.565036 15 80
.565356 20 106
9.565676 30 160
.565995 40 213
.566314 45 240
.566632 50 266
.566951
.57 1066
.571380
.571695
9.57 2009
.572323
.572636
.572950
.573263
9.573575
Cos.
9.567269
Corr.
.567587
.567904 10 53
.568222 15 79
.568539 20 106
9.568856 30 158
.569172 40 211
.569488 45 237
.569804 50 264
.570120
11
T
" Corr.
10 52
15 78
20 105
30 157
40
209
45 235
50 262
Cos.
9.970152
.970103
.970055
.970006
.969957
9.969909
.969860
.969811
.969762
.969714
Corr.
8
9.969665
.969616
.969567 10
.969518 15 12
.969469 20 16
9.969420 30 25
.969370 40 33
.969321 45 37
.969272 50 41
.969223
9.969173
.969124
.969075
.969025
.968976
9.968926
.968877
.968827
.968777
.968728
9.968678
.968628
.968578
.968528
.968479
9.968429
.968379
.968329
.968278
.968228
Corr.
8
10
9.968178
.968128
.968078
.968027 15 13
.967977 20 17
9.967927 30
.967876 40 34
.967826 45 38
.967775
.967725
25
9.967674
.967624
.967573
.967522
.967471
9.967421
.967370
.967319
.967268
.967217
9.967166
Sin.
Corr.
for
Sec.
n
55555
50 42
Tan.
9.584177
.584555 Corr.
.584932 10 63
.585309 15 94
.585686 20 125
9.586062 30 188
.586439 40 251
.586815 45 282
.587190 50 314
.587566
9.587941
.588316 Corr.
.588691 10 62
.589066 15 94
.589440 20 125
9.589814 30 187
.590188 40 249
.590562 45 280
.590935 50 312
.591308
9.591681
n
H
P
.592054 Corr.
.592426 10 62
.592799 15 93
.593171 20 124
9.593542 30 186
.593914 40 248
.594285 45 279
.594656 50 310
.595027
"
Corr.
for
Sec.
9.595398
.595768 Corr.
.596138 10 62
.596508 15 92
.596878 20 123
9.597247 30 185
.597616 40 246
.597985 45 277
.598354 50 308
.598722
9.599091
.599459 Corr.
.599827 10 61
.600194 15 92
.600562 20 122
9.600929 30 184
.601296 40 245
.601663 45 275
.602029 50 306
.602395
9.602761
.603127
Corr.
.603493 10 61
.603858 15 91
.604223 20 122
9.604588
30
182
.604953 40 243
.605317 45 274
.605682 50 304
.606046
9.606410
Cot.
W
1
Cot.
0.415823 60
.415445 59
.415068 58
.414691 57
.414314
56
0.413938 55
.413561
54
.413185 53
.412810 52
.412434 51
0.412059 50
.411684 49
.411309 48
.410934 47
.410560 46
0.410186 45
.409812 44
.409438 43
.409065 42
.408692 41
0.408319 40
.407946 39
.407574 38
.407201 37
.406829 36
0.406458 35
.406086 34
.405715 33
.405344 32
.404973
31
28
0.404602 30
.404232 29
.403862
.403492
.403122
0.402753
27
26
25
.402384 24
.402015 23
.401646
.401278 21
0.400909 20
.400541 19
.400173 18
.399806 17
.399438 16
0.399071 15
.398704 14
.398337 13
.397971 12
.397605 11
0.397239
.396873
.396507
158°

.395777
0.395412
.395047
.394683
10
9
8
.396142 7
.394318
.393954
0.393590
Tan.
********22
CLO 4~10
6
68°
599
22°
0123 NË LA CO Z-89
6
7
10
11
12
13
14
15
16
17
18
19
272*******
23
24
25
26
20 9.579777
21
28
29
30
31
32
33
34
35
36
37
38
382=234
39
40
41
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
112°
COSINES, Tangents, AND COTANGENTS
Sin.
9.573575
.573888
Corr.
.574200 10 52
.574512 15 78
.574824 20 104
9.575136 30 156
.575447 40 208
.575758 45 234
.576069 50 259
.576379
"
9.576689
.576999
Corr.
.577309 10 51
.577618 15 77
.577927 20 103
9.578236
30
154
.578545 40 206
.578853 45 232
.579162 50 257
.579470
"
Corr.
for
Sec.
.580085 11 Corr.
.580392 10 51
.580699 15 77
.581005 20 102
9.581312 30 153
.581618 40 204
.581924 45 230
.582229 50 255
,582535
9.582840
.583145 1 Corr.
.5834-49 10 51
.583754 15 76
.584058 20 101
9.584361 30 152
.584665 40 202
.584968 45 228
.585272 50 253
.585574
.591580
9.591878
Cos.
9.585877
.586179 Corr.
.586482 10 50
.586783 15 75
.587085 20 100
9.587386 30
151
.587688
40 201
.587989 45 226
.588289 50 251
.588590
1
9.588890
.589190
Corr.
.589489 10 30
.589789 15 75
.5900$8 20 100
9.590387 30 149
.590686 40 199
.590984 45 224
.591282 50 249
17
Cos.
9.967166
.967115
.967064
.967013
.966961
9.966910
.966859
.966808
.966756
.966705
9.966136
.966085
.966033
10
Corr.
9.966653
.966602
.966550 10 9
.966499 15 13
.966447 20 17
9.966395 30 26
.966344 40 34
.966292 45 39
.966240 50 43
.966188
.965981
.965929
9.965876
.965824
.965772
.965720
.965668
9.965615
.965563
.965511
.965458
.965406
9.965353
.965301
.965248
.965195
.965143
9.964560
.964507
.964454
Corr.
9
9.965090
.965037
.964984 10
.964931 15 13
.964879 20 18
9.964826 30 26
.964773 40 35
.964720 45 40
.964666 50 44
.964613
.964400
.964347
9.964294
.964240
.964187
.964133
964080
9.964026
Sin.
Corr.
for
Sec.
"
Tan.
9.606410
.606773
Corr.
.607137 10 60
.607500 15 91
.607863 20 121
9.608225 30 181
.608588 40 242
.608950 45 272
.609312 50 302
.609674
9.610036
.610397 Corr.
.610759 10 60
.611120 15 90
.611480 20 120
9.611841 30 180
.612201 40 240
.612561 45 270
.612921 50 300
.613281
9.613641
.614000 Corr.
.614359 10 60
.614718 15 90
.615077 20 119
9.615435 30 179
.615793 40 239
.616151 45 269
.616509 50 299
.616867
買
​M
11
Cot.
9.617224
.617582 Corr.
.617939 10 59
.618295 15 89
.618652 20 119
9.619008 30 178
.619364 40 238
.619720 45 267
.620076 50 297
.620432
9.620787
.621142
Corr.
.621497 10 59
.621852 15 89
.622207 20 118
177
236
9.622561
30
.622915 40
.623269 45 266
295
.623623 50
.623976
11
Corr.
for
Sec.
1
55555
9.624330
.624683 Corr.
.625036 10 59
.625388 15 88
.625741 20 117
n
9.626093 30 176
.626445 40 235
.626797 45 264
.627149 50 293
.627501
9.627852
Cot.
0.393590 60
.393227 59
.392863 58
.392500 57
.392137 56
0.391775 55
.391412 54
.391050 53
.390688 52
.390326 51
0.389964 50
.389603 49
.389241 48
.388880 47
.388520 46
0.388159 45
.387799 44
.387439 43
.387079 42
.386719 41
0.386359 40
.386000 39
.385641 38
.385282 37
.384923 36
0.384565
.384207
35
34
.383849 33
.383491 32
.383133 31
0.382776 30
.382418 29
.382061 28
.381705 27
.381348 26
0.380992 25
.380636 24
.380280 23
.379924 22
.379568 21
0.379213 20
.378858 19
.378503 18
.378148 17
.377793 16
0.377439 15
.377085 14
.376731 13
.376377 12
.376024 11
0.375670
.375317
.374964
.374612
.374259
157°

0.373907
.373555
.373203
.372851
.372499
0.372148
Tan.
896
10
7
6
4321O
67°
600
23°
CH234LOLO
0
1
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
8838
28
29
30
31
32
33
34
35
36
37
BBGELRE
38
39
41
42
43
44
45
46
47
48
49
50
51
52
40 9.603594
53
54
55
56
57
58
59
60
Sin.
113°
TABLE XXII.-LOGARITHMIC SINES,
9.591878
.592176
n Corr.
,592473 10 49
.592770 15 74
.593067 20 99
9.593363 30 148
.593659 40 198
.593955 45 222
.594251 50 247
.594547
9.594842
Corr.
.595137
.595432 10 49
.595727 15 74
.596021 20 98
9.596315 30 147
.596609 40 196
.596903 45 221
.597196 50 245
.597490
Corr.
for
Sec.
n
9.597783
Corr.
.598075
.598368 10 49
.598660 15 73
n
.598952 20 97
9.599244 30 146
.599536 40 194
.599827 45 219
.600118 50
243
.600409
9.600700
.600990
Corr.
.601280 10 48
.601570 15 72
.601860 20 96
9.602150 30 145
.602439 40 193
.602728 45 217
50
.603017
241
.603305
=
.603882
Corr.
.604170 10 48
.604457 15 72
.604745 20 96
9.605032 30 144
.605319 40 191
.605606 45 215
.605892 50 239
.606179
1
9.606465
.606751
Corr.
.607036 10 47
.607322 15 71
.607607 20 95
9.607892 30 142
.608177 40 190
.608461 45 214
.608745 50 237
.609029
9.609313
Cos.
Cos.
9.964026
.963972
.963919
.963865
.963811
9.963757
..963704
.963650
.963596
.963542
и
9.963488
.963434
.963379 10
.963325 15
Corr.
9
14
.963271 20 18
9.963217 30 27
.963163 40 36
.963108 45 41
.963054 50 45
.962999
9.962945
.962890
.962836
.962781
.962727
9.962672
.962617
.962562
.962508
.962453
9.962398
.962343
.962288
.962233
.962178
9.962123
.962067
.962012
.961957
.961902
9.961846
.961791
.961735 10
Corr.
9
.961680 15 14
.961624 20 19
9.961569 30 28
.961513 40 37
.961458 45 42
.961402 50 46
,961346
9.961290
.961235
.961179
.961123
.961067
9.961011
.960955
.960899
.960843
.960786
9.960730
Corr.
for
Sec.
Sin.
Я
Tan.
9.627852
.628203
Corr.
.628554 10 58
.628905 15 88
.629255 20 117
9.629606
30 175
.629956 40 234
.630306 45 263
.630656 50 292
.631005
9.631355
.631704
Corr.
.632053 10 58
.632402 15 87
.632750 20 116
9.633099 30 174
.633447 40 232
.633795 45 261
.634143 50 290
.634490
9.634838
.635185 Corr.
.635532 10 58
.635879 15 87
.636226 20 115
9.636572
30 173
.636919 40 231
.637265 45 260
.637611 50 289
.637956
.
Corr.
for
Sec.
n
E
9.638302
50 287
.638647 Corr.
.638992 10 57
.639337 15 86
.639682 20 115
9.640027 30 172
.640371 40 230
.640716 45 258
.641060
.641404
9.641747
.642091
Corr.
.642434 10 57
.6412777 15 86
.643120 20 114
9.643463
30 171
.643806 40 228
.644148 45 257
.644490 50 286
.644832
ה
1
9.645174
.645516 Corr.
.645857 10 57
.646199 15 85
.646540 20 114
9.646881 30 170
.647222 40 227
.647562 45 256
.647903 50 284
.648243
9.648583
Cot.
n
•
Cot.
156°

0.372148
.371797
.371446 58
.371095 57
.370745 56
0.370394 55
.370044 54
.369694 53
.369344 52
.368995 51
60
59
Tan.
******
41
0.368645 50
.368296 49
.367947 48
.367598 47
.367250 46
0.366901 45
.366553
44
.366205 43
.365857 42
.365510
0.365162 40
.364815
39
.364468 38
.364121 37
.363774 36
0.363428 35
.363081 34
.362735 33
.362389 32
.362044
31
30
0.361698
.361353 29
.361008 28
.360663 27
.360318 26
0.359973
25
.359629 24
.359284 23
.358940
.358596
1222
0.358253 20
.357909 19
.357566 18
.357223 17
.356880 16
0.356537 15
.356194 14
.355852 13
.355510 12
.355168 11
0.354826
.354484
.354143
.353801
.353460
0.353119
.352778
.352438
.352097
.351757
0.351417
10
9
876543210
66°
601
24°
0123AL CZ Ca
7
HANN
8
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
114°
COSINES, TANGENTS, AND COTANGENTS
Sin.
9.609313
.609597
Corr.
.609880
10 47
.610164 15 71
.610447 20 94
9.610729
30 141
.611012 40 188
.611294 45 212
50 236
.611576
.611858
Corr.
for
Sec.
.616060
9.61633S
.616616
9.612140
.612421
.612702
.612983
Corr.
10 47
15 70
.613264 20 93
30
9.613545
140
.613825 40 187
.614105
.614385
45 210
50 234
.614665
9.614944
.615223 Corr.
.615502 10 46
.615781 15
70
93
20
30
139
40
186
45
209
50 232
.616894
.617172
.617450
H
A
11
9.617727
.618004
Corr.
.618281 10 46
.618558 15 69
.618834 20 92
9.619110 30 138
.619386 40 184
.619662 45 207
.619938 50 230
.620213
11
9.620488
.620763
Corr.
.621038
10 46
.621313
15
69
.621587 20 91
9.621861 30 137
.622135 40 183
.622409
45 206
.622682 50 228
.622956
R
9.623229
.623502
Corr.
.623774 10 45
.624047 15 68
M
.624319 20 91
9.624591 30 136
.624863 40 181
.625135 45 204
.625406 50 227
.625677
9.625948
Cos.
Cos.
9.960730
.960674
.960618
.960561
.960505
9.960448
.960392
.960335
.960279
.960222
9.959310
.959253
.959195
.959138
.959080
Corr.
9.960165
.960109
.960052 10 9
.95999515 14
.959938 20 19
9.959882 30 28
.959825 40 38
.959768
.959711
.959654
45
43
47
9.959596
.959539
.959482
.959425
.959368
9.959023
.958965
.958908
.958850
.958792
9.958734
.958677
.958619
.958561
.958503
9.957863
.957804
.957746
.957687
.957628
9.957570
.957511
.957452
.957393
.957335
9.957276
Sin.
Corr.
for
Sec.
1
55595
9.958445
.958387 Corr.
.958329 10 10
,958271 15 15
.958213 20 19
9.958154 30 29
.958096 40 39
45
44
.958038
.957979 50 49
.957921
50
H
Tan.
9.648583
.648923 Corr.
.649263 10 56
.649602 15 85
.649942 20 113
9.650281 30 170
.650620 40 226
45 254
.650959
50
.651297
283
.651636
9.651974
.652312 Corr.
.652650 10 56
.652988 15 84
.653326 20 112
30
9.653663
169
40
.654000
.654337
.654674
225
253
45
50
281
.655011
"
9.655348
.655684
Corr.
.656020 10 56
15
.656356
84
112
.656692
20
9.657028
.657364
.657699
.658034
.658369
Corr.
for
Sec.
Cot.
30
40
45
50
11
W
168
224
252
280
.667682 45 248
.668013 50 275
.668343
9.668673
Cot.
9.658704
.
.659039
Corr.
.659373 10 56
.659708 15 83
.660042 20 111
9.660376 30 167
.660710 40 223
.661043 45 250
.661377 50 278
.661710
9.662043
.662376
Corr.
.662709 10 55
.663042 15 83
.663375 20 111
9.663707 30 166
.664039 40 222
45
.664371
249
.664703 50 277
.665035
9.665366
0.334634
.665698 Corr. .334302
.666029 10 55
.666360 15 83
.666691 20 110
9.667021 30 165
.667352 40 220
0.351417 60
.351077 59
.350737 58
.350398 57
.350058 56
0.349719 55
.349380 54
.349041 53
.348703 52
.348364 51
0.348026 50
.317688 49
347350 48
.347012 47
46
.346674
0.346337
45
.346000
44
.345663
43
42
41
..345326
.344989
40
0.344652
39
.344316
38
.343980
.343644 37
.343308
0.342972
36
35
34
.342636
.342301
33
.341966 32
.341631
31
155°

0.341296
.340961
.340627
.340292
.339958
0.339624
26
25
.339290 24
.338957 23
.338623
.338290
.333971
.333640
.333309
0.332979
.332648
.332318
.331987
,331657
0.331327
30
29
28
2227******
Tan.
21
0.337957 20
.337624 19
.337291 18
.336958 17
.336625 16
0.336293 15
.335961 14
.335629 13
.335297 12
.334965 11
10
9
8
6
5
L&&&1 -
4
3
2
0
65°
602
25°
0123HLO 6 7 ∞-
4
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
115°
Sin.
TABLE XXII.-LOGARITHMIC SINES,
9.625948
.626219
Corr.
.626490 10 45
.626760 15 67
.627030 | 20 90
9.627300 30 135
.627570 40 180
.627840 45 202
.628109 50 225
.628378
9.628647
.628916 1 Corr.
.629185 10 45
.629453 15 67
.629721 20 89
9.629989 30 134
.630257 40 179
.630524 45 201
.630792 50 223
.631059
9.633984
9.631326
.631593 Corr.
.631859 10 44
.632125 15 66
.632392 20 89
9.632658 30 133
.632923 40 177
.633189 45 199
.633454 50 221
.633719
Corr.
for
Sec.
A
E
.634249
.634514 10
Corr.
44
.634778 15 66
.635012 20 88
9.635306 30 132
.635570 40 176
.635834 45 198
.636097 50 220
.636360
11
9.636623
.636886 " Corr.
.637148 10 44
.637411 15 65
.637673 20 87
9.637935 30 131
.638197 40 175
.638458 45 196
.638720 50
218
.638981
9.639242
.639503 M Corr.
.639764 10 43
.640024 15 65
.640284 20 87
9.640544 30 130
.640804 40 173
.641064 45 195
.641324 50 217
.641583
9.641842
Cos.
Cos.
9.957276
.957217
.957158
.957099
.957040
9.956981
.956921
.956862
.956803
.956744
"
9.956684
.956625 Corr.
.956566 10 10
.956506 15 15
.956447 20 20
9.956387 30 30
.956327 40 40
.956268 45 45
.956208 50 50
.956148
9.956089
.956029
.955969
.955909
.955849
9.955789
.955729
.955669
.955609
.955518
9.955488
.955428
.955368
.955307
.955247
9.955186
.955126
.955065
.955005
.954944
Corr.
for
Sec.
9.954274
.954213
.954152
.954090
.954029
9.953968
W
9.954883
.954823
.954762 | To
.954701
Corr.
10
15 15
.954640 20 20
9.954579 30 30
.954518 40 41
.954457 45 46
.954396 50 51
.954335
.953906
.953845
.953783
.953722
9.953660
Sin.
!
Tan.
9.668673
.669002 Corr.
.669332 10 55
.669661 15 82
.669991 20 110
9.670320 30 164
.670649 40 219
.670977 45 246
.671306 50 274
.671635
H
.677194
.677520
.677846
.678171
9.671963
.672291
.672619
.672947
.673274
9.673602
.673929 Corr.
.674257 10 54
.674584 15 82
.674911 20 109
9.675237
.675564
.675890
.676217
.676543
9.676869
"
GABO NE
Corr.
for
Sec.
30
163
40 218
45 245
50 272
9.678496
.678821 Corr.
.679146 10 54
.679471 15 81
.679795 20 108
9.680120 30
162
.680144 40 216
.680768 45 243
.681092 50 270
.681416
Π
9.681740
.682063
.682387
.682710
.683033
9.683356
.683679 Corr.
.684001 10 54
.684324 15 80
.684646 20
9.684968
.685290
.685612
.685934
.686255
107
9.686577
.686898
.687219
.687540
.687861
9.688182
Cot,
1
30 161
40 214
45 241
50 268
Cot.
154°

0.331327 60
.330998 59
.330668 58
.330339 57
.330009 56
0.329680 55
.329351 54
.329023 53
.328694 52
.328365 51
0.328037 50
.327709 49
.327381 48
.327053 47
.326726 46
0.326398 45
.326071 44
.325743 43
.325416 42
.325089 41
0.324763 40
.324436 39
.324110 38
.323783 37
.323457 36
0.323131 35
.322806 34
.322480 33
.322154 32
.321829 31
0.321504 30
.321179 29
.320854 28
.320529 27
.320205 26
0.319880 25
.319556 24
.319232 23
.318908 22
.318584 21
0.318260 20
.317937 19
.317613 18
.317290 17
.316967 16
0.316644 15
.316321 14
.315999 13
.315676 12
.315354
0.315032
.314710
.314388
.314066
.313745
0.313423
.313102
11
10
9
8
.312781
.312460
.312139
0.311818
Tan.
6
4432-0
1
64°
603
26°
01234 LE LO DI COσ
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
**2*70**
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
116°
COSINES, TANGENTS, AND COTANGENTS
Sin.
9.641842
Corr.
for
Sec.
10
.642101 . Corr.
.642360
43
.642618 15 65
.642877 20 86
9.643135
30 129
.643393 40 172
.643650 45 194
.643908 50 215
.644165
9.644423
.644680 " Corr.
.644936 10 43
.645193 15 64
.645450 20 85
9.645706 30 128
.645962 40 171
.646218 45 192
.616474 50 213
.646729
9.646984
.647240
Corr.
.647494 10 42
.647749 15 64
.618001 20 85
9.648258 30 127
.648512 40 170
.648766 45 191
.649020 50 212
.649274
W
9.649527
.649781
Corr.
.650034 10 42
.650287 15 63
.650539 20 84
9.650792 30 126
.651044 40 168
.651297 45 189
.651519 50 210
.651800
9.654558
.654808
.655058
.
9.652052
.652304 1
Corr.
.652555 10 42
.652806 '15 63
.653057 20 84
9.653308 30 125
40
.653558
167
.653808 45
188
.651059 50 209
.654309
1
Corr.
10 41
.655307 15 62
.655556 20 83
9.655805 30 124
.656054 40 166
.656302 45 187
.656551 50 207
.656799
9.657017
Cos.
Cos.
9.953660
.953599
.953537
.953475
.953413
9.953352
.953290
.953228
.953166
.953104
9.952419
.952356
.952294
.952231
.952168
9.952106
9.953042
.952980
Corr..
.952918 10 10
.952855 15 16
.952793 20 21
9,952731 30 31
40
.952669
42
.952606
45 47
.952544 50 52
.952481
.952043
.951980
.951917
.951854
9.951791
.951728
.951665
.951602
.951539
9.951476
.951412
.951349
.951286
.951222
9.950522
.950458
.950394
.950330
.950266
9.950202
.950138
.950074
.950010
.949945
9.949881
Sin.
Corr.
for
Sec.
9.951159
.951096
Corr.
.951032 10 11
.950968 15 16
.950905 20 21
9.950841 30 32
.950778 40 42
.950714 45
48
.950650 50 53
.950586
H
Tan.
9.688182
.688502 Corr.
.688823 10 53
.689143 15 80
.689463 20 106
9.689783 30 160
.690103 40 213
.690423 45 240
.690742 50
266
.691062
.696470
.696787
9.691381
.691700
.692019
.692338
.692656
9.692975
.693293 Corr.
.693612 10 53
.693930 15 79
.694248 20 106
9.694566
.694883
.695201 45
.695518 50 264
.695836
30 159
40 212
238
9.696153
.697103
.697420
Corr.
for
Sec.
R
#
9.697736
.698053 " Corr.
.698369 10 53
.698685 15 79
.699001 20 105
9.699316 30 158
.699632 40 210
.699947 45 236
.700263 50 263
.700578
A
9.700893
.701208
.701523
.701837
.702152
9.702466
.702781 Corr.
.703095 10 52
.703409 15 78
.703722 20 104
9.70-1036
.704350
.704663 45
.704976 50
.705290
30 157
40 209
235
261
9.705603
.705916
.706228
.706541
706854
9.707166
Cot.
Cot.
153°

0.311818 60
.311498 59
.311177 58
.310857 57
.310537 56
0.310217 55
.309897 54
.309577 53
.309258 52
.308938 51
0.308619 50
.308300 49
.307981 48
.307662 47
.307344 46
0.307025 45
.306707 44
.306388 43
.306070 42
.305752 41
0.305434
40
39
.305117
.304799
.304482
38
37
.304164 36
0.303847
35
.303530 34
.303213 33
.302897 32
.302580 31
Tan.
30
0.302264
.301947
29
.301631 28
.301315 27
.300999
0.300684
********22
26
25
.300368 24
.300053 23
.299737
.299422 21
0.299107
20
19
.298792
.298477 18
.298163
17
.297848 16
0.297534 15
.297219 14
.296905 13
.296591 12
.296278 11
0.295964 10
.295650 9
8
.295337
.295024
.294710
0.294397
.294084
7
.293772
.293459
.293146
0.292834
654321O
0
63°
604
27°
01234 6789
5
10
11
12
13
14
15
16
17
18
19
222222
23
24
25
26
27
28
29
30
31
32
33
20 9.661970
21
34
35
36
37
38
39
2223
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
Sin.
117°
TABLE XXII.-LOGARITHMIC SINES,
9.657047
.657295
Corr.
.657542 10 41
.657790 15 62
.658037 20 82
9.658284 30 123
.658531 40 164
.658778 45 185
.659025 50 205
.659271
9.659517
.659763
.660009
.660255
.660501
9.660746
.660991
.661236 10
.662214
.662459
.662703
.662946
9.663190
"
.663433
.663677
.663920
.664163
Corr.
for
Sec.
R
.661481 15
.661726 20 81
.671134
.671372
9.671609
Cos.
Corr.
41
61
៨៩៨
122
30
40
163
45
183
50 203
9.664406
.664648 . Corr.
.664891 10 40
.665133 '15 60
.665375 20 80
9.665617 30 121
.665859 40 161
.666100 45 181
.666342 50 201
.666583
9.666824
.667065
.667305
.667546
.667786
9.668027
.66$267 Corr.
.668506 10 40
.668746 15 60
.668986 20 80
"
9.669225
30 119
.669464
.669703
40 159
45 179
.669942 50 199
.670181
9.670419
.670658
.670896
Cos.
9.949881
.949816
.949752
.949688
.949623
9.949558
.949494
.949429
.949364
.949300
9.949235
.949170
Corr.
.949105 10 11
.949040 15 16
.948975 20 22
9.948910 30 33
.948845 40 43
45 49
,948780
.948715 50 54
.948650
9.948584
.948519
.948454
.948388
.948323
9.948257
.948192
.948126
.948060
.947995
9.947929
.947863
.947797
.947731
.947665
9.947600
.947533
.947467
.947401
.947335
9.946604
.946538
.946471
.946404
.946337
9.946270
.946203
.946136
W
9.947269
.947203
Corr.
.947136 10 11
.947070 15 17
.947004 20 22
9.946937 30 33
40
.946871
44
.946804 45 50
.946738
50 55
.946671
.916069
.946002
9.945935
Sin.
Corr.
for
Sec.
曾
​Tan.
9.707166
.707478 "Corr.
.707790 10 52
.708102 15 78
.708414 20 104
9.708726 30 156
.709037 40 207
.709349 45 233
.70966050 259
.709971
9.710282
.710593
.710904
.711215
.711525
15
77
9.711836
.712146
Corr.
.712456 10 51
712766
.713076 20 103
9.713386
30 155
.713696
40 206
.714005
45 232
.714314 50 258
.714624
9.714933
.715242
.715551
.715860
.716168
Corr.
for
Sec.
9.716477
716785
Corr.
.717093 10 51
.717401 15 77
.717709 20
102
9.718017
30 154
.718325 40 205
.718633 45 231
.718940 50 256
.719248
9.719555
.719862
.720169
.720476
.720783
9.721089
.721396
Corr.
.721702
10 51
.722009 15 76
.722315 20 102
9.722621
.722927
.723232
.723538
.723844
9.724149
.724454
.724760
.725065
.725370
9.725674
Cot.
n
30 153
40 204
45 229
50 255
Cot.
152°

60
53
0.292834
.292522 59
.292210 58
.291898 57
.291586 56
0.291274 55
.290963 54
.290651
.290340 52
.290029 51
0.289718 50
289407 49
.289096 48
.288785 47
.288475 46
0.288164 45
.287854 44
.287544 43
42
.287234
.286924 41
40
0.286614
39
.286304
.285995
.285686
38
37
.285376 36
0.285067 35
,284758 34
.284449 33
.284140 32
.283832 31
0.283523 30
.283215 29
.282907 28
.282599 27
.282291 26
0.281983
25
.281675 24
.281367 23
.281060 22
.280752 21
0.280445 20
.280138 19
.279831 18
.279524 17
.279217 16
0.278911 15
.278604 14
.278298 13
.277991 12
.277685 11
0.277379
.277073
.276768
.276462
.276156
0.275851
.275546
,275240
.274935
,274630
0.274326
Tan.
10
9
8
7
654321O
0
62°
605
28°
100 g 100 yol yand
0123456780-6
9
10
11
12
13
14
15
16
17
18
19
FAA 2~~~2227
20
21
23
24
25
26
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
118°
COSINES, TANGENTS, AND COTANGENTS
Sin.
.
9.671609
.671847
Corr.
.672084 10 39
.672321 15 59
.672558 20 79
9.672795 30 118
.673032 40 158
.673268 45 177
.673505 50 197
.673741
9.673977
.674213
.674448
.674684
.674919
9.675155
.675390 #
Corr.
.675624 10 39
.675859
.676094
9.676328
15
58
78
.676562
.676796
.677030
.677264
9.677498
.677731
.677964
.678197
.678430
9.680982
.681213
.681443
.681674
.681905
9.682135
Corr.
for
Sec.
CAAU NEd
20
Cos.
9.678663
.678895
17 Corr.
.679128 10 39
.679360 15 58
.679592 20 77
9.679824 30 116
.680056 40 154
.680288 45 174
.680519 50 193
.680750
2888888
30 117
40
156
45 175
50 195
"
.682365 Corr.
.682595 10 38
.682825 15 57
.683055 20 76
9.683284 30 115
.683514 40 153
.683743 45 172
.683972 50 191
.684201
9.684430
.684658
.684887
.685115
.685343
9.685571
Cos.
9.945935
.945868
.945800
.945733
.945666
9.945598
.945531
.945464
9.944582
.944514
944446
.944377
.944309
9.944241
.944172
.944104
.944036
.943967
.945396
.945328
9,945261
.945193 Corr.
.945125 10 11
.945058 15 17
.944990 20 23
9.944922 30 34
40
.944854
45
.944786 45 51
.944718 50 57
.944650
9.943899
.943830
.943761
.943693
.943624
9.943555
.943486
.943417
.943348
.943279
9.942517
.942448
.942378
.942308
.942239
9.942169
.942099
.942029
.941959
.941889
9.941819
Corr.
for
Sec.
Sin.
"
Tan.
9.725674
.725979
Corr.
.726284 10 51
.726588 15 76
.726892 20 101
9.727197 30 152
.727501 40 203
.727805 45 228
.728100 50 253
.728412
.733558
.733860
.734162
.734463
9.728716
.729020
.729323
.729626
.729929
9.730233
.730535 Corr.
.730838 10 50
.731141 15 76
.731444 20 101
9.731746
.732048
.732351
.732653
.732955
9.733257
Corr.
for
Sec.
9.737771
.738071
.738371
·
9.943210
.943141 R Corr.
.943072 10 12
.943003 15 17
.942934 20 23
9.942864 30 35
.942795 40 46
Corr.
.942726 45 52 .739870 10 50
.942656 50 58
.942587
.740169 15 75
.740468 20 100
9.740767
30 149
40 199
.741066
.741365 45 224
.741664 50 249
.741962
.738671
.738971
9.739271
*.739570
9.742261
.742559
.742858
.743156
.743454
9.743752
Cot.
Ħ
9.734764
.735066
" Corr.
.735367 10 50
.735668 15 75
.735969 20 100
9.736269 30
150
.736570 40 200
.736870 45 225
.737171 50 250
.737471
GRAW NE
30 151
40 201
45 227
50 252
C
GAAO NHI
Cot.
151°

0.274326 60
.274021 59
.273716 58
.273412 57
.273108 56
0.272803 55
.272499 54
.272195 53
.271891 52
.271588 51
0.271284
39
50
.270980 49
.270677
48
.270374 47
.270071 46
0.269767 45
.269465 44
.269162 43
.268859 42
.268556 41
0.268254 40
.267952
.267649 38
.267347 37
.267045 36
0.266743 35
.266442 34
.266140 33
.265838 32
.265537 31
0.265236 30
.264934 29
.264633 28
.264332 27
.264031 26
0.263731 25
.263430 24
.263130 23
.262829 22
.262529 21
0.262229 20
.261929 19
.261629 18
.261329 17
.261029 16
0.260729 15
.260430 14
.260130 13
.259831 12
.259532 11
0.259233
.258934
.258635
.258336
.258038
0.257739
.257441
.257142
.256844
.256546
0.256248
Tan.
10
9
876543210
61°
606
29°
01234L ON ∞a o
gye
8
9
10
11
12
9.687843
.688069
.688295
.688521
.688747
9.688972
.689198 R
Corr.
.689423 10 37
.689648 15 56
.689873
20 9.690098
20 75
21
13
14
15
16
17
18
19
272 ******* *-***
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
**2 www
40
41
42
43
44
45
46
47
48
49
588888888
Sin.
59
60
119°
TABLE XXII.-LOGARITHMIC SINES,
9.685571
.685799 ก Corr.
.686027 10 38
.686254 15 57
.686482 20 76
9.686709 30 113
.686936
40 151
.687163 45 170
.687389 50 189
.687616
.690323
.690548
.690772
.690996
9.691220
.691444
.691668
.691892
.692115
Corr.
for
Sec.
CABO NI
.698751
9.698970
Cos.
30 112
40 150
45 168
50 187
9.692339
.692562
Corr.
.692785 10 37
.693008 15 56
.693231 20
74
9.693453 30 111
.693676 40 148
.693898 45 167
.694120 50 185
.694342
50
51
9.696775 30 110
.696995 40 147
.697215 45 165
52
53
.697435 50 183
54
.697654
55 9.697874
.698094
56
57
.698313
.698532
1
9.694564
.694786
.695007
.695229
.695450
9.695671
.695892 Corr.
.696113 10 37
.696334 15 55
.69655-1 20 73
ה
Cos.
9.941819
.941749
,941679
.941609
.941539
9.941469
.941398
.941328
.941258
.941187
9.941117
.941046
Corr.
.940975 10 12
.940905 15 18
.940834 20 24
9.940763 30 35
.940693 40 47
.940622 45 53
.940551 50 59
.940480
9.940409
.940338
.940267
.940196
.940125
9.940054
.939982
.939911
.939840
.939768
9.939697
.939625
.939554
.939482
.939410
9.939339
.939267
.939195
.939123
.939052
9.938258
.938185
.938113
.938040
.937967
9.937895
.937822
.937749
Corr.
for
Sec.
9.938980
.938908
Corr.
.938836 10 12
.938763 15 18
.938691 20 24
9.938619 30 36
40
.938547
48
.938475
.938402
.938330
.937676
.937604
9.937531
Sin.
1
45
50
54
60
Tan.
9.743752
.744050 Corr.
.744348 10 49
.744645 15 74
.744943 20 99
9.745240 30 149
.745538 40 198
.745835 45 223
.746132 50 248
.746429
9.746726
.747023
.747319
.747616
.747913
9.748209
.748505
.748801
.749097
.749393
9.749689
.749985
.750281
.750576
.750872
9.751167
.751462
.751757
.752052
.752347
9.755585
.755878
.756172
#
Corr.
for
Sec.
Cot.
Corr.
49
74
20 98
UAAC NEW
9.752642
.752937 11 Corr.
.753231 10 49
.753526 15 74
.753820 20 98
9.754115 30 147
.754409 40 196
.754703 45 220
.754997 50 245
.755291
148
197
45 222
50 246
.756465
.756759
9.757052
.757345
Corr.
.757638 10 49
.757931 15 73
.758224 20 97
9.758517 30 146
.758810 40 195
.759102 45 219
.759395 50 244
.759687
9.759979
.760272
.760564
.760856
.761148
9.761439
ย
Cot.
0.256248 60
.255950 59
.255652 58
.255355 57
.255057 56
0.254760 55
.254462 54
.254165 53
.253868 52
.253571 51
150°

0.253274
50
.252977 49
.252681 48
.252384 47
.252087 46
0.251791 45
,251495 44
.251199 43
.250903 42
.250607 41
0.250311
.250015
.249719 38
.249424 37
.249128 36
0.248833 35
.248538 34
.248243 33
.247948 32
.247653 31
0.247358 30
.247063 29
.246769 28
.246474 27
.246180 26
0.245885 25
.245591 24
.245297 23
.245003
.214709 21
0.244415
20
.244122 19
.243828 18
.243535 17
.243241 16
0.242948 15
.242655 14
.242362 13
.242069 12
.241776
11
0.241483
.241190
.240898
.240605
.240313
0.240021
.239728
.239436
239144
.238852
0.238561
Tan.
40
39
2222
10
9
8
7
6
DL43210
5
60°
607
30°
SANG là con 9
0
1
2
6
8
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
10 9.701151
11
30
31
32
33
34
35
36
37
38
39
50
51
52
53
54
55
56
57
58
59
60
COSINES, TANGENTS, AND COTANGENTS
Sin.
120°
9.698970
.699189 "
Corr.
.699407 10 36
.699626 15 54
.699844 20 73
9.700062 30 109
.700280 40 145
.700498 45 163
.700716 50 181
.700933
.701368
.701585
.701802
.702019
9.702236
40 9.707606
41
42
43
44
45
46
47
48
49
.702452
.702669 10
.702885 15 54
.703101 20 72
.704610
.704825
.705040
.705254
9.703317 30 108
.703533 40 144
.703749 45 162
.703964 50 180
.704179
9.704395
Corr.
for
Sec.
"
9.705469
.705683
Corr.
.705898 10 36
.706112 15 53
706326 20
71
9.706539 30 107
.706753 40 142
.706967 45 160
.707180 50 178
.707393
9.709730
709941
.710153
.710364
.710575
9.710786
.710997
.711208
711419
.711629
9.711839
Cos.
11
.707819
.708032
.708245
.708458
9.708670
.708882
.709094
.709306
.709518 20
11
10
Corr.
35
15 53
70
1948 8948
Cos.
.936357
Corr.
.936284
36
30 106
40 141
45 158
50 176
9.937531
.937458
.937385
.937312
.937238
9.937165
.937092
.937019
.936946
.936872
9.936799
.936725 Corr.
.936652
12
10
.936578 15 18
.936505 20 25
9.936431
30 37
40 49
45 55
.936210 50 61
.936136
9.936062
.935988
.935914
.935840
.935766
9.935692
.935618
.935543
.935469
.935395
1
9.933445
.933369
.933293
.933217
.933141
9.933066
Sin.
Corr.
for
Sec.
9.935320
.935246
.935171
.935097
.935022
9.934948
.934873
.934798
.934723
.934649
9.934574
.934499
Corr.
.934424 10 13
.934349 15 19
.934274 20 25
9.934199 30 38
.934123 40 50
.934048
56
45
.933973 .50 63
.933898
9.933822
.933747
.933671
.933596
.933520
"
Tan.
9.761439
.761731
.762023
.762314
.762606
9.762897
.763188
Corr.
763479 10 48
.763770 15 73
.761061 20 97
9.764352 30 145
.764643 40 194
.764933 45 218
.765224 50 242
.765514
9.765805
.766095
.766385
766675
.766965
9.767255
.767545
.767834
.768124
.768414
9.768703
.768992
769281
.769571
.769860
9.770148
.770437
.770726
.771015
.771303
9.771592
.771880
772168
.772457
.772745
I
n
Corr.
for
Sec.
1040 2048
48
72
20 96
15
Corr,
30 144
n
45 217
50 241
9.773033
.773321
.773608
.773896
.774184
9.774471
.774759
Corr.
.775046 10 48
.775333 15 72
.775621 20 96
9.775908
.776195
.776482
.776768
.777055
9.777342
.777628
.777915
.778201
.778488
9.778774
Cot.
CAAW STU
Cot.
30 144
40 191
45 215
50 239
0.238561
.238269
.237977
.237686
.237394
149°

0.229852
.229563
193
.229274
.228985
60
59
58
57
56
55
0.237103
.236812 54
.236521 53
.236230 52
.235939
51
0.235648
50
.235357
49
48
.235067
.234776 47
.234486 46
0.234195
45
44
.233905
.233615
43
42
.233325
.233035 41
0.232745 40
.232455 39
.232166 38
.231876 37
.231586 36
0.231297 35
.231008 34
.230719 33
.230429 32
.230140
31
30
29
28
27
.228697 26
0.228408
.228120
25
24
.227832 23
.227543 22
.227255 21
*********~
Tan.
0.226967
20
.226679 19
.226392 18
17
16
.226104
.225816
0.225529 15
.225241 14
.224954 13
,224667 12
.224379 11
0.224092 10
9
.223805
.223518 8
,223232
.222945
0.222658
.222372
7
6
5
.222085
.221799
.221512
0.221226
43210
59°
608
31°
01234B67∞ of
8
9
11
12
13
14
10 9.713935
15
16
17
18
** 222******* AMABI38588 GEN
19
20
21
23
24
25
26
27
28
29
30
31
32
33
34
36
39
40
41
42
43
44
45
46
47
48
49
KIBONDO
54
Sin.
55
56
57
58
59
60
121°
TABLE XXII.-LOGARITHMIC SINES,
9.711839
.712050
Corr.
.712260 10 35
.712469 15 52
.712679 20 70
9.712889
.713098
30 105
40 140
.713308 45 157
.713517 50 174
.713726
.714144
714352
.714561
.714769
.716224
.716432
.716639
.716846
9.717053
9.714978
.715186 Corr.
.715394 10 35
.715602 15 52
.715809 20 69
9.716017
.717259
.717466
.717673
.717879
50 9.722181
51
.722385
52
53
.722588
.722791
.722994
9.723197
9.720140
.720345
.720549
.720754
.720958
9.721162
11
Corr.
for
Sec.
ה
9.718085
.718291
Corr.
.718497 10 34
.718703 15 51
718909 20 68
9.719114 30 103
.719320 40 137
719525 45 154
719730 50 171
.719935
.723400
.723603
.723805
.724007
9.724210
Cos.
30 104
40 138
45 155
50 173
.721366
Corr.
.721570 10 34
.721774
51
.721978
11
曾
​AAU NE
15
20 68
30 102
40 135
45 152
50
169
Cos.
9.933066
.932990
.932914
.932838
.932762
9.932685
.932609
.932533
.932457
.932380
9.932304
.932228 Corr.
.932151 10 13
.932075
15 19
.931998 20 26
9.931921 30
38
.931845 40 51
.931768 45
58
.931691 50 64
.931614
9.931537
.931460
.931383
.931306
.931229
9.931152
.931075
.930998
.930921
.930843
9.930766
,930688
.930611
.930533
.930456
9.930378
Corr.
for
Sec.
.930300
.930223
"1
.930145
.930067
9.929989
.929911 Corr.
.929833 10 13
.929755 15 20
.929677 20 26
9.929599
.929521
30 39
40 52
.929442
45
59
65
.929364
.929286
9.929207
.929129
.929050
.928972
.928893
9.928815
.928736
.928657
.928578
.928499
9.928420
Sin.
n
535999999
50
Tan.
9.778774
.779060
779346
.779632
.779918
9.780203
.780489
Corr.
.780775 10 48
71
95
15
.781060
.781346
9.781631
.781916
.782201
.782486
.782771
9.783056
.783341
783626
.783910
.784195
9.784479
.784764
.785048
9.787319
.7876Q3
787886
.788170
.788453
9.788736
.789019
.789302
.789585
.789868
n
9.792974
.793256
793538
.793819
.794101
.785332
.785616
9.785900
.786184
Corr.
.786468 10 47
.786752
15 71
9.794383
.794664
794946
Corr.
for
Sec.
CABO 87
.795227
.795508
9.795789
Cot.
20
30 143
40 190
45 214
50 238
.787036 20 95
#
48 8848
9.790151
.790434
.790716
.790999
.791281
9.791563
.791846
.792128
.792410 15 70
.792692 20 94
Corr.
47
10
30
142
40 189
45 213
50 236
ה
30 141
40 188
45 211
50 235
Cot.
0.221226 60
.220940 59
.220654 58
.220368 57
.220082 56
0.219797 55
.219511 54
.219225 53
.218940 52
.218654 51
0.218369 50
.218084 49
.217799 48
47
.217514
.217229 46
0.216944 45
.216659 44
.216374 43
.216090 42
.215805 41
0.215521
40
.215236 39
.214952 38
.214668 37
.214384 36
0.214100 35
.213816 34
.213532 33
.213248 32
.212964 31
0.212681
30
.212397 29
.212114 28
.211830 27
.211547 26
0.211264
25
.210981 24
.210698 23
.210415
.210132
148°

0.205617
.205336
.205054
21
0.209849
20
.209566 19
.209284 18
.209001 17
.208719 16
0.208437 15
.208154 14
.207872 13
.207590 12
.207308 11
0.207026
.206744
.206462
.206181
205899
.204773
.204492
0.204211
Tan.
*******22
10
9
8
•COLO HMOTO
6
5
3
0
58°
609
32°
01234 LO
5
7
∞
10
11
12
13
14
9 .726024
15
16
17
18
222*******
23
24
25
.726827
.727027
9.727228
.727428
.727628 10 33
.727828
15 50
19 .728027 20 66
21
20 9.728227
.728427
.728626
.728825
.729024
9.729223
26
27
28
29
30
31
32
33
34
35
36
37
8888888
38
39
40
41
42
43
44
45
46
47
48
49
COSINES, TANGENTS, AND COTANGENTS
Sin.
50-
51
52
53
54
55
56
57
58
59
60
122°
9.724210
.724412
.724614
Corr.
10 34
.724816 15 50
.725017 20 67
9.725219 30 101
.725420 40 134
.725622 45 151
.725823 50 168
9.726225
.726426
.726626
.729422
.729621
.729820
.730018
9.732193
.732390
.732587
.732784
.732980
9.733177
9.734157
.734353
.734549
R
.734744
.734939
9.735135
.735330
.735525
.735719
735914
9.736109
Corr.
for
Sec.
獎
​Cos.
9.730217
.730415 Corr.
.730613 10 33
.730811 15 49
731009 20 66
9.731206 30 99
.731404 40 132
.731602 45 148
.731799 50 164
.731996
1242 8048
30 100
45
50
Corr.
П
11
.733373 Corr.
.733569 10 33
.733765 15 49
.733961 20 65
133
149
166
30 98
40 130
45
50
147
163
Cos.
9.926831
.926751
.926671
.926591
.926511
9.926431
.926351
.926270
.926190
.926110
9.928420
.928342
.928263
.928183
.928104
9.928025
.927946
.927867
.927787
.927708
9.927629
.927549 Corr.
.927470 10 13
.927390 15 20
.927310 20 27
9.927231 30 40
.927151 40 53
.927071 45 60
.926991 50 66
.926911
9.926029
.925949.
.925868
.925788
.925707
9.925626
.925545
.925465
.925384
.925303
9.924409
.924328
.924246
.
.924164
.924083
9.924001
.923919
.923837
.923755
.923673
9.923591
Sin.
9.925222
.925141 Corr.
.925060 10 14
.924979 15 20
.924897 20 27
9.924816 30 41
.924735 40 54
.924654 45 61
.924572 50 68
.924491
Corr.
for
Sec.
W
Tan.
9.795789
.796070
.796351
.796632
.796913
9.797194
.797474 Corr.
.797755 10 47
.798036 15 70
.798316 20 93
9.798596
.798877
.799157
.799437
.799717
9.799997
.800277
.800557
.800836
.801116
9.804187
.804466
.804745
.805023
.805302
9.805580
.805859
.806137
9.801396
.801675
.801955
.802234
.802513
9.802792
.803072
Corr.
.803351 10 46
.803630
.803909 20 93
.806415
.806693
"
9.806971
.807249
Corr.
for
Sec.
30 140
40 187
45 210
50 234
1
19323
15 70
30 139
40 186
45 209
50 232
.807527
.807805
.808083
9.808361
Corr.
.808638
.808916 10 46
.809193
.809471
9.809748
15 69
20 92
.810025
.810302
.810580
.810857
9.811134
.811410
.811687
.811964
.812241
9.812517
Cot.
R
2 8948
30 139
40 185
45 208
50 231
Cot.
147°

0.204211
60
.203930 59
.203649 58
.203368 57
.203087 56
0.202806 55
.202526 54
.202245 53
.201964 52
.201684 51
0.201404
50
.201123
49
,200843
48
.200563 47
46
.200283
0.200003
.199723
45
44
.199443
43
.199164 42
.198884 41
ཐ832
Tan.
0.198604
.198325
.198045
.197766
36
.197487
0.197208
35
.196928 34
.196649 33
.196370 32
.196091 31
0.195813
.195534
.195255
.194977
.194698 26
0.194420
.194141
25
24
.193863 23
.193585 22
0.193029
16
15
.193307 21
20
.192751 19
.192473 18
.192195 17
.191917
0.191639
.191362 14
.191084 13
.190807 12
.190529 11
0.190252 10
.189975 9
.189698 8
.189420
.189143
0.188866
.188590
7
.188313
.188036
.187759
0.187483
40
39
38
37
2207837
29
6543210
57°
610
33°
0123
4
7
•∞- DEN
-
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22232322≈≈ 8-****UNADDING**
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58888888
59
60
123°
Sin.
TABLE XXII.—LOGARITHMIC SINES,
9.736109
.736303
Corr.
.736498 10 32
.736692 15 48
.736886 20 65
9.737080 30 97
.737274 40 129
.737467 45 145
.737661 50 161
.737855
Corr.
for
Sec.
9.743792
.743982
.744171
.744361
.744550
9.744739
.744928
9.738048
.738241
.738434
.738627
.738820
9.739013
.739206
Corr.
.739398 10 32
.739590 15 48
.739783 20 64
9.739975
.740167
.740359
.740550
.740742
9.740934
.741125
.741316
.741508
.741699
.
9.745683
.745871
.746060
.746248
.746436
9.746624
.746812
.746999
.747187
.747374
9.747562
Cos.
"
9.741889
,742080 M Corr.
.742271 10 32
.742462 15 48
.742652 20 63
9.742842 30 95
.743033 40 127
.743223 45 142
.743413 50 158
.743602
30 96
40 128
45 144
50 160
"
745117 10
.745306
.745494
AAW NEO
20
Corr.
31
58
30 94
40 125
45 141
Cos.
9.923591
.923509
.923427
.923345
.923263
9.923181
.923098
.923016
.922933
.922851
9,922768
.922686 Corr.
.922603 10 14
.922520 15 21
.922438 20 28
9.922355 30 41
.922272 40 55
.922189 45 62
.922106 50 69
.922023
9.921940
.921857
.921774
.921691
.921607
9.921524
.921441
.921357
.921274
.921190
9.921107
.921023
.920939
.920856
.920772
9.920688
,920604
.920520
.920436
.920352
9.919424
.919339
.919254
.919169
.919085
9.920268
.920184 Corr.
.920099 10 14
.920015 15 21
.919931 20 28
9.919846 30 42
.919762 40 56
.919677 45 63
.919593 50 70
.919508
9.919000
.918915
.918830
.918745
.918659
9.918574
#
Sin.
Corr.
for
Sec.
M
Tan.
9.812517
.812794
.813070
.813347
.813623
9.813899
.814176
.814452
.814728
.815004
9.815280
.815555 Corr.
.815831 10 46
.816107 15 69
.816382 20 92
9.816658 30 138
.816933 40 184
.817209 45 207
.817484 50 230
.817759
9.818035
.818310
.818585
.818860
.819135
9.819410
.819684
.819959
.820234
.820508
9.820783
.821057
.821332
.821606
.821880
9.822154
.822429
822703
.822977
.823251
Corr.
for
Sec.
9.826259
.826532
.826805
.827078
.827351
9.827624
.827897
.828170
.828442
.828715
9.828987
Cot.
W
9.823524
.823798 Corr.
.824072 10 46
.824345 15 68
.82461920 91
9.824893 30 137
.825166 40 182
.825439 45 205
.825713 50 228
.825986
Cot.
146

0.187483
60
.187206
59
.186930 58
.186653 57
.186377
56
0.186101 55
.185824 54
.185548 53
.185272 52
.184996 51
0.184720 50
.184445 49
.184169 48
.183893 47
.183618 46
0.183342 45
.183067 44
.182791 43
.182516 42
.182241
41
40
0.181965
.181690
39
.181415
38
.181140
37
.180865
36
0.180590 35
.180316 34
180041 33
.179766 32
.179492 31
0.179217 30
.178943 29
.178668
28
.178394
27
.178120
26
0.177846
25
.177571 24
.177297 23
.177023 22
.176749 21
0.176476 20
.176202 19
.175928 18
.175655 17
.175381 16
0.175107 15
.174834 14
.174561 13
.174287 12
.174014
0.173741
.173468
.173195
11
10
9
8
.172922
7
.172649
6
0.172376
.172103
.171830
.171558
.171285
0.171013
Tan.
4~NI O
3
56°
611
34°
01234 20 40 2 0 0
7
8
9
10
11
12
13
14
15
16
17
18
19
20
2222
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
124°
COSINES, TANGENTS, AND COTANGENTS
Sin.
9.747562
.747749
.747936
.748123
.748310
9.7484197
.748683
Corr.
.748870
31
.749056 15 47
.749243 20 62
9.749429
.749615
.749801
.749987
.750172
9.750358
.750513
.750729
.750914
.751099
9.751284
.751469
.751654
.751839
.752023
9.752208
.752392
.754229
.754412
.754595
.751778
9.754960
.755143
.755326
.755508
.755690
9.755872
Corr.
for
Sec.
.756418
.756600
.752576 10
.752760 15
.752944 20
9.753128
.753312
.753495
.753679
.753862
9.754016
9.756782
.756963
.757144
.757326
.757507
9.757688
54
55
56
.757869
57
.758050
58
.758230
59
.758411
60 9.758591
Cos.
1
1940 000
30 93
40 124
45 140
155
50
n
.756054
.756236 10
GAAU NEO
5558
30
40
50
11
Corr.
30
45
20 61
1992 090
15
30
40
Corr.
45
50
91
121
136
151
Cos.
9.918574
.918489
.918404
.918318
.918233
9.918147
.918062
.917976
.917891
..917805
9.916859
.916773
.916687
9.917719
.917634
.917548
Corr.
10 14
15 22
.917462
.917376 20 29
9.917200
30 43
.917204
40
57
.917118 45 64
.917032
50
72
.916916
.916600
.916514
9.916427
.916341
.916254
.916167
.916081
9.915994
.915907
.915820
.915733
.915646
9.915559
.915472
.915385
.915297
.915210
9.914246
.914158
.914070
11
.913982
.913894
9.913806
.913718
.913630
.913541
.913453
9.913365
Sin.
Corr.
for
Sec.
9.915123
.915035
Corr.
.914948 10 15
.914860 15 22
.914773 20 29
9.914685 30 44
,914598 40 58
.914510 45 66
.914422 50 73
.914334
11
NOC*N
Tan.
9.828987
.829260
,829532
.829805
.830077
9.830349
.830621
.830893
.831165
.831437
9.831709
.831981
Corr.
.832253 10 45
.832525 15 68
.832796 20 91
9.833068 30
136
.833339 40 181
.833611 45 204
,833882 50 226
.834154
9.834425
.834696
.834967
.835238
.835509
9.835780
.836051
.836322
.836593
.836864
9.837134
.837405
.837675
.837946
.838216
9.838487
.838757
.839027
.839297
.839568
9.842535
.842805
.843074
.843343
9.839838
.840108
Corr.
.840378 10 45
.840648 15 67
.840917 20 90
9.841187 30 135
.841457 40 180
.841727 45 202
.841996 50 225
.842266
.843612
9.843882
.844151
.844420
.844689
.844958
9.845227
#1
Cot.
Corr.
for
Sec.
A
Cot.
145°

0.171013
60
.170740 59
.170468
58
.170195
57
.169923
56
0.169651
55
.169379 54
.169107 53
.168835 52
.168563 51
0.168291 50
.168019 49
.167747 48
.167475 47
.167204
0.166932
46
45
44
43
.166661
.166389
.166118 42
.165846 41
0.165575
40
.165304 39
.165033 38
.164762 37
.164491
36
35
0.164220
.163949 34
.163678 33
.163407 32
.163136 31
0.162866
30
.162595 29
.162325 28
.162054
27
26
.161784
0.161513
25
.161243 24
.160973 23
.160703 22
.160432 21
0.160162 20
.159892 19
.159622 18
.159352 17
.159083 16
0.158813 15
.158543 14
.158273 13
.158004 12
.157734 11
0.157465 10
.157195
9
.156926
.156657
.15638S
0.156118
.155849
.155580
.155311
.155042
0.154773
Tan.
8765432IO
1
0
55°
612
35°
01234567∞ CF23,
8
9
11
10 9.760390
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
888 GENE-
36
37
38
39
40
41
42
43
Sin.
46
47
48
49
125°
9.758591
.758772
.758952
.759132
.759312
9.759492
.759672
Corr.
.759852 10 30
.760031 15
.760211 '20
.760569
.760748
.760927
.761106
9.761285
.761464
.761642
.761821
.761999
9.765720
.765896
.766072
766247
44 .766423
45 9.766598
9.762177
.762356
.762534
.762712
.762889
9.763067
.763600
.763777
9.763954
.764131
.764308
.764485
.764662
9.764838
.765015
.765191
.765367
.765544
TABLE XXII.-LOGARITHMIC SINES,
.763245
Corr.
.763422 10 30
15 44
20 59
50
51
52
53
54
55 9.768348
56
57
58
59
60
9.767475
.767649
.767824
.767999
.768173
n
.768522
.768697
.768871
.769045
9.769219
Cos.
CANO NM
W
Corr.
for
Sec.
42 8948
30
40
45
50
1
.766774
.766919
.767124
.767300
20
120
134
149
GAAC NEW
89
118
133
148
Corr.
10 29
44
20 58
30 88
40
117
131
50 146
Cos.
9.913365
.913276
.913187
.913099
.913010
9.912922
.912833
.912744
.912655
.912566
9.912477
.912388
Corr.
.912299 10 15
.912210 15 22
.912121 20 30
9.912031 30 45
.911942 40 60
.911853 45 67
.911763 50 74
.911674
9.911584
.911495
.911405
.911315
.911226
9.911136
.911046
.910956
.910866
.910776
9.910686
.910596
.910506
.910415
.910325
9.910235
.910144
.910054
.909963
.909873
•
Corr.
for
Sec.
n
1
9.909782
.909691 Corr.
.909601 10 15
.909510 15 23
.909419 20 30
9.909328 30 45
.909237 40 61
.909146 45 68
.909055 50 76
.908964
9.908873
.908781
.908690
.908599
.908507
9.908416
.908324
.908233
.908141
.908049
9.907958
Sin.
Tan.
9.845227
.845496
.845764
.846033
.846302
9.846570
.846839
.847108
.847376
.847644
9.847913
.848181
Corr.
.848449 10 45
.848717 15 67
.848986 20 89
9.849254 30 134
.849522 40 179
.849790 45 201
.850057 50 223
.850325
9.850593
.850861
.851129
.851396
.851664
9.851931
.852199
.852466
.852733
.853001
9.853268
.853535
.853802
.854069
.854336
9.854603
.854870
.855137
.855404
.855671
Corr.
for
Sec.
9.855938
.856204 ក Corr.
.856471 10 44
.856737 15 67
.857004 20 89
9.857270 30 133
.857537 40 178
.857803 45 200
.858069 50 222
.858336
9.858602
.858868
.859134
.859400
.859666
9.859932
.860198
.860464
.860730
.860995
9.861261
Cot.
Cot.
0.154773
60
,154504 59
.154236 58
.153967 57
.153698 56
144°

0.153430 55
.153161 54
.152892 53
.152624 52
.152356 51
0.152087 50
.151819 49
.151551 48
.151283 47
.151014 46
0.150746 45
.150478 44
.150210 43
.149943 42
.149675 41
0.149407 40
.149139 39
.148871 38
.148604 37
.148336
36
0.148069
35
.147801
34
.147534 33
32
31
.147267
.146999
0.146732
.146465
.146198
30
29
28
********27
.145931
.145664 26
0.145397
25
.145130 24
.144863 23
.144596 22
.144329 21
0.144062
20
.143796 19
.143529 18
.143263 17
.142996 16
0.142730 15
.142463 14
.142197 13
.141931 12
.141664
0.141398
.141132
11
.140866
.140600
.140334
0.140068
.139802
.139536
.139270
.139005
0.138739
Tan.
10
9
8
7
6
U5432-O
54*
613
36°
01234 L ∞ Z ∞ ~
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
126°
COSINES, TANGENTS, AND COTANGENTS
Sin.
9.769219
.769393
.769566
.769740
.769913
9.770087
.770260 Corr.
.770433 10 29
.770606 15 43
.770779 20 58
9.770952
.771125
.771298
.771470
.771643
9.771815
.771987
.772159
.772331
.772503
9.772675
.772847
.773018
.773190
.773361
9.774388
.774558
.774729
.774899
,775070
9.775240
.775410
775580
.775750
.775920
9.773533
.773704
Corr.
.773875 10 28
.774046 15 43
.774217 20 57
9.776090
.776259
.776429
.776598
.776768
9.776937
.777106
.777275
.777444
.777613
9.777781
.777950
.778119
11
.778792
.778960
.779128
.779295
9.779463
Cos.
Corr.
for
Sec.
30 86
40 115
45 130
50 144
M
3888-88
11
30 85
40 114
45 128
50 142
10520
Corr.
28
42
56
30 84
40 112
45 126
.778287 50 140
.778455
9.778624
Cos.
9.907958
.907866
.907774
.907682
.907590
9.907498
.907406
.907314
.907222
.907129
9.906111
.906018
.905925
.905832
.905739
9.907037
.906945 Corr.
.906852 10 15
15
.906760
23
.906667 20 31
9.906575 30 46
.906482 40 62
.906389 45 69
.906296 50 77
.906201
9.905645
.905552
.905459
.905366
.905272
9.905179
.905085
.904992
.904898
.904804
9.904711
.904617
.904523
.904429
.904335
9.903298
.903203
.903108
.903014
.902919
9.902824
.902729
.902634
.902539
F
9.904241
.904147
Corr.
.901053 10 16
.903959
15 24
.903864 20 31
9.903770 30 47
.903676 40 63
.903581 45 71
.903487 50 79
.903392
.902444
9.902349
Sin.
Corr.
for
Sec.
"
Tan.
9.861261
.861527
.861792
.862058
.862323
9.862589
,862854
.863119
.863385
.863650
9.866564
.866829
0.137411
0.138739 60
.138473 59
.138208 58
.137942 57
.137677 56
55
.137146 54
.136881 53
.136615 52
.136350 51
9.863915
0.136085 50
.864180 Corr.
.135820 49
.864445 10 44 .135555 48
.864710 15
.135290
66
47
.864975 20 88 .135025 46
9.865240 30 132 0.134760 45
.865505 40 177 .134495 44
.865770 45 199 .134230 43
.866035 50 221 .133965 42
.866300
.133700
41
.867094
.867358
.867623
9.867887
.868152
.868416
.868680
.868945
9.869209
.869473
.869737
.870001
.870265
9.870529
.870793
.871057
.871321
.871585
9.874484
.874747
.875010
1
9.871849
.872112
Corr.
.872376 10 44
,872610 15 66
.872903 20 88
9.873167 30 132
.873430 40 176
.873694 45
198
.873957 50 220
.874220
.875273
.875537
9.875800
.876063
.876326
.876589
.876852
9.877114
Cot.
Corr.
for
Sec.
n
Cot.
40
0.133436
.133171
39
.132906
38
.132642 37
.132377
36
0.132113
.131848
35
34
.131584 33
.131320 32
.131055
0.130791
31
30
.130527 29
.130263 28
.129999 27
.129735 26
0.129471
25
143°

.129207 24
.128943
.128679
.128415 21
.123937
.123674
0.128151
.127888
127624 18
.127360 17
.127097 16
0.126833 15
.126570 14
.126306 13
.126043 12
.125780
0.125516
.125253
.124990
11
.124727
.124463
0.124200
.123411
.123148
0.122886
***22 20
Tan.
23
19
09876O LA SED ON TO
10
3
0
53°
614
37°
0
OH~~ LO SE89
2
5
6
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
127°
Sin.
9.781134
.781301
.781468
781634
.781800
9.781966
9.779463
.779631
.779798
.779966
.780133
9.780300
.780467 Corr.
.780634 10 28
.780801 15 42
.780968 20 56
782132
.782298
.782464
.782630
9.782796
.782961
.783127
.783292
.783458
9.784447
.784612
.784776
784941
.785105
9.785269
785433
.785597
.785761
.785925
9.783623
.783788
Corr.
.783953 10 27
784118
.784282
9.786089
.786252
786416
.786579
786742
9.786906
.787069
9.787720
.787883
.788045
TABLE XXII.-LOGARITHMIC SINES,
.788208
.788370
9.788532
788694
.788856
.789018
.789180
9.789342
Ħ
Cos.
Corr.
for
Sec.
30 83
40 111
45 125
139
50
W
05205
15
.787232
10
Corr.
27
.787395
41
15
,787557 20 54
FI
15535
41
30
82
40 110
45 123
50 137
30 81
40 108
45 122
50 135
Cos.
9.902349
.902253
.902158
.902063
.901967
9.901872
.901776
.901681
.901585
.901490
9.901394
.901298 Corr.
.901202
16
10
.901106 15 24
.901010 20 32
9.900914 30 48
.900818 40 64
.900722 45 72
.900626 50 80
.900529
9.900433
.900337
.900240
.900144
.900047
9.899951
.899854
.899757
.899660
.899564
9.899467
.899370
.899273
.899176
.899078
9.898981
.898884
.898787
.898689
:898592
9.897516
.897418
.897320
.897222
.897123
9.897025
.896926
.896828
Corr.
for
Sec.
17
9.898494
.898397 " Corr.
.898299 10 16
.898202 15 24
.898104 20 33
9.898006 30 49
.897908 40 65
.897810 45 73
.897712 50 82
.897614
.896729
.896631
9.896532
Sin.
Tan.
9.877114
.877377
.877640
.877903
.878165
9.878428
.878691
.878953
.879216
.879478
9.879741
.880003 Corr.
.880265 10 44
.880528 15 66
.880790 20 87
9.881052 30 131
.881314 40 175
45
.881577
197
.881839 50 218
.882101
9.882363
.882625
.882887
.883148
.883410
9.883672
883934
.884196
.884457
.884719
9.884980
.885242
.885504
.885765
.886026
9.886288
.886549
.886811
.887072
.887333
9.890204
.890165
.890725
9.887594
.887855
Corr.
.888116 10 43
.888378 15 65
.888639 20 87
9.888900 30 130
.889161 40 174
889421 45 196
.889682 50 217
S89943
,890986
.891247
M
9.891507
.891768
.892028
.892289
.892549
9.892810
Cot.
Corr.
for
Sec.
π
Cot.
142°

0.122886
60
122623 59
.122360 58
.122097 57
.121835 56
55
0.121572
.121309 54
.121047 53
.120784
52
.120522
51
0.120259
50
.119997
49
.119735 48
.119472 47
.119210 46
0.118948 45
.118686 44
.118423 43
.118161 42
.117899 41
0.117637 40
.117375 39
.117113 38
.116852 37
.116590
36
0.116328 35
.116066 34
.115804 33
.115543 32
.115281
31
0.115020
30
.114758 29
.114496 28
.114235
27
.113974
.108232
.107972
.107711
.107451
0.107190
Tan.
26
25
24
23
0.113712
.113451
21
.113189
.112928
.112667
0.112406 20
.112145 19
.111884 18
.111622 17
.111361 16
0.111100 15
.110839 14
.110579 13
.110318 12
.110057 11
0.109796
.109535
.109275
.109014
.108753
0.108493
2222
10
9
8
7
COLD 43 21O
0
52°
615
38°
0123TELO SO Z Ca
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
22222
20
21
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
TOOOO DAAAAPZADƏ w
40
41
42
43
44
45
46
47
43
43
50
51
52
53
54
55
56
57
58
59
60
88888
128°
COSINES, TANGENTS, AND COTANGENTS
Sin.
9.789342
.789504
.789665
.789827
.789988
9.790149
.790310 Corr.
.790471 10 27
.790632 15 40
.790793 20 54
9.790954
.791115
.791275
.791436
.791596
9.791757
.791917
.792077
.792237
.792397
.793832
.793991
9.794150
.794308
.794467
.794626
.794781
9.794942
.795101
.795259
.795417
.795575
9.792557
.792716
.792876
793035
.793195
9.793354
11
.793514
Corr.
.793673 10 26
15 40
20 53
9.795733
.795891
.796049
.796206
.796364
9.796521
.796679
.796836
.796993
.797150
9.797307
.797464
.797621
.797777
.797934
9.798091
Corr.
for
Sec.
"
.798247
.798403
.798560
.798716
9.798872
Cos.
30 80
40 107
45 121
50 134
142 8048
30 79
106
119
132
45
50
11 Corr.
26
39
52
10
1033
102 2008
15
78
30
40 105
118
45
50
131
Cos.
9.896532
.896433
.896335
.896236
.896137
9.896038
.895939
.895840
.895741
.895641
.89494145
.894846
.894746
.894646
9.895542
.895443 Corr.
.895343 10 17
.895244 15 25
895145 20 33
9.895045 30 50
40 66
45 75
50 83
9.894546
.894446
.894346
.894246
.894146
9.894046
.893946
:893846
.893745
.893645
9.893544
.893444
.893343
.893243
.893142
9.893041
.892940
.892839
.892739
..892638
9.891523
.891421
.891319
.891217
.891115
9.892536
.892435
Corr.
.892334 10 17
.892233 15 25
.892132 20 34
9.892030 30 51
.891929 40 68
891827 45 76
50
.891726
.891624
84
9.891013
.890911
.890809
.890707
.$90605
9.890503
Sin.
Corr.
for
Sec.
H
Tan.
9.892810
.893070
.893331
.893591
.893851
9.894111
.894372
.894632
.894892
.895152
9.895412
.895672 Corr.
.895932 10 43
.896192 15 65
.896-452 20 87
30
9.896712
130
40 173
.896971
45 195
.897231
50 216
.897491
.897751
9.898010
.898270
.898530
.898789
.899049
9.899308
.899568
.899827
.900087
.900346
9.900605
.900864
.901124
.901383
.901642
9.901901
.902160
.902420
.902679
.902938
9.905785
.906043
.906302
.906560
.906819
1
9.903197
.903456
Corr.
.903714 10 43
.903973 15 65
20
.904232
86
30 129
9.904491
40 173
.904750
194
45
216
50
.905008
.905267
.905526
9.907077
.907336
.907594
.907853
.908111
9.908369
Cot.
Corr.
for
Sec.
11
Cot.
0.107190 60
.106930 59
.106669 58
.106409 57
.106149 56
55
0.105889
.105628 54
.105368 53
.105108 52
.104848 51
0.104588 50
.104328 49
.104068 48
.103808 47
.103548 46
45
0.103288
.103029 44
.102769 43
.102509 42
.102249 41
0.101990 40
.101730 39
.101470 38
.101211
37
.100951
36
0.100692
35
.100432
.100173
0.094215
.093957
.093698
.093440
.093181
0.092923
.092664
.092406
.092147
141°

.091889
0.091631
.099913 32
.099654
31
30
0.099395
.099136 29
.098876 28
.098617
27
.098358 26
0.098099
25
24
.097840
.097580
23
20
19
22
.097321
.097062 21
0.096803
.096544
18
.096286
17
.096027
.095768 16
15
0.095509
.095250 14
.094992
13
12
.094733
.094474
11
Tan.
6330
34
33
10
9
P
0781843210
5
51°
616
39°
0123456789
10
11
12
13
14
15
16
17
18
19
222******
20
21
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
BGE-RIKON**
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
129°
Sin.
9.798872
.799028
.799184
.799339
.799495
9.799651
.799806
Corr.
.799962 10 26
.800117 15
.800272 20
9.801201
.801356
.801511
.801665
.801819
9.801973
.802128
.802282
.802436
.802589
9.802743
.802897
•
30
40
78
103
9.800427
.800582
.800737 45 116
.800892 50 129
.801047
.803050
.803204
.803357
9.803511
.803664
,803817
.803970
,804123
9.804276
.804428
.804581
.804734
.804886
TABLE XXII.—LOGARITHMIC SINEs,
9.806557
.806709
.806860
.807011
.807163
9.807314
.807465
.807615
.807766
59
.807917
60 9.808067
Cos.
Corr.
for
Sec.
Q
CABU NE
Corr.
10 26
15 38
20 51
22 894
9.805039
.805191
.805343
.805495
.805647
9.805799
.805951 n Corr.
.806103
.806254
25
38
.806106 20 50
22
30 77
40 102
45 115
128
50
10
15
8*38 85
30
76
40 101
45 114
126
50
Cos.
9.890503
.890400
.890298
.890195
.890093
9.889990
.889888
.889785
.889682
.$89579
9.889477
" Corr.
.889374
.889271 10 17
.889168 15 26
889064 20 34
9.888961 30 52
.888858 40 69
.888755 45 77
.888651 50 86
.888548
9.888414
.888341
.888237
.888134
.888030
9.887926
.887822
.887718
.887614
.887510
9.887406
.887302
.887198
.887093
.886989
9.886885
.886780
.886676
.886571
.886466
9.886362
.886257 Corr.
.886152 10 18
.886047 15 26
.885942 20 35
9.885837 30 53
.885732 40 70
45 79
.885627
.885522 50 88
.SS5416
9.885311
.885205
.885100
.884994
.884889
9.884783
.884677
.884572
.884466
.884360
9.884254
Corr.
for
Sec.
Sin.
"
Tan.
9.908369
.908628
.908886
,909144
.909402
9.909660
.909918
.910177
.910435
.910693
9.910951
.911209
Corr.
.911467 10 43
.911725 15 64
.911982 20 86
9.912240 30 129
.912498 40 172
.912756 45 193
.913014 50 215
.913271
9.913529
.913787
.914044
.914302
.914560
9.914817
.915075
.915332
.915590
.915817
9.916104
.916362
.916619
.916S77
.917134
9.917391
.917648
.917906
.918163
.918420
9.921247
.921503
.921760
.922017
.922274
W
9.918677
.918934
Corr.
.919191 10 43
.919448 15 64
.919705 20 86
9.919962 30 128
.920219 40 171
45 193
.920476
.920733 50 214
.920990
9.922530
.922787
.923044
.923300
.923557
9.923814
Cot.
Corr.
for
Sec.
A
Cot.
0.091631
60
.091372 59
.091114 58
.090856 57
.090598 56
0.090340 55
.090082 54
.089823 53
.089565 52
.089307 51
50
0.089049
49
.088791
.088533 48
.088275 47
.088018 46
0.087760 45
.087502 44
.087244 43
.086986
42
.086729 41
0.086471
40
.086213 39
.085956 38
.085698 37
.085440
0.085183
.084925 34
36
35
.084668 33
.084410 32
.081153
31
0.083896
.083638
.083381
.083123
.0$2866
0.082609
.082352
.082094
.081837
,081580
140°

0.078753
.078497
.078240
.077983
.077726
0.077470
.077213
.076956
.076700
.076443
0.076186
Tan.
30
29
28
27
26
25
24
222222
21
20
0.081323
.081066
.080809
19
18
17
16
.080552
080295
0.080038 15
.079781 14
.079524 13
.079267 12
.079010 11
23
10
9
8
5
4
3210
50°
617
40°
01234 LO ∞ 1− ∞✪
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
130°
COSINES, TANGENTS, AND COTANGENTS
Sin.
9.808067
.808218
.808368
.808519
.808669
9.808819
.808969
.809119
.809269
.809419
9.809569
.809718 " Corr.
.809868 10 25
.810017 15 37
.810167 20 50
9.810316 30 75
.810465 40 99
.810614 45 112
.810763 50 124
.810912
9.811061
.811210
.811358
,811507
.811655
9.811801
.811952
.812100
.812248
.812396
9.812544
.812692
.812840
.812988
.$13135
9 813283
.813430
.813578
.813725
.813872
50 9.815485
51
.815632
52
.815778
53
.815924
54
.816069
55
56
57
58
59
60
9.814019
.814166
Corr.
.S14313 10 24
.814460 15 37
.814607 20
49
9.814753 30 73
.814900 40 98
.815046 45 110
.815193 50 122
.815339
9.816215
.816361
.816507
Corr.
for
Sec.
.816652
.816798
9.816943
Cos.
"
Cos.
9.884254
.884148
.884012
.$83936
.883829
9.883723
.883617
.883510
.883404
.883297
9.882121
.882014
.881907
.881799
.881692
9.881584
9.883191
.883084 Corr..
.882977 10 18
.882871 15 27
882764 20 36
9.882657 30 53
.882550 40 71
.882443 45 80
.8S2336 50 89
.882229
.881477
.881369
.SS1261
.881153
9.881016
.880938
.880830
.880722
.880613
9.880505
.880397
.880289
.880180
.SS0072
9.878875
.878766
.S78656
.878547
.878438
9.878328
.878219
.878109
.S77999
.877890
9.877780
9.879963
.879855
Corr.
.879746 10 18
.879637 15
27
.879529 20 36
9.879420 30 54
.879311 40 73
.879202 45 82
.879093
50 91
.878984
Sin.
Corr.
for
Sec.
"
Tan.
9.923814
.924070
.924327
.924583
.924840
9.925096
.925352
.925609
.925865
.926122
9.926378
.926634
Corr.
.926890 10 43
.927147 15 64
.927403 20 85
9.927659
30 128
.927915 40 171
.928171 45 192
.928427 50 213
.928684
9.928940
.929196
.929452
.929708
.929964
9.930220
.930175
.930731
.930987
.931243
9.931499
.931755
.932010
.932266
.932522
9.932778
.933033
.933289
.933545
.933800
9.934056
.934311
Corr.
.934567 10 43
.934822 15 64
.935078 20 85
9.935333 30 128
.935589 40 170
.935844 45 192
.936100 50 213
.936355
9.936611
.936866
.937121
.937377
.937632
9.937887
.938142
.938398
Corr.
for
Sec.
.938653
.938908
9.939163
Cot.
A
Cot.
0.076186
60
.075930 59
.075673 58
.075417 57
.075160 56
0.074904 55
.074648 54
.074391 53
.074135 52
.073878 51
0.073622
.073366
50
49
.073110 48
.072853 47
.072597 46
0.072341 45
.072085 44
.071829 43
.071573 42
.071316 41
0.071060
40
.070804 39
.070548
38
37
.070292
.070036
36
0.069780
35
.069525 34
.069269 33
.069013 32
.068757
31
139°

0.068501
.068245
.067990
.067734
.067478
0.067222
25
.066967 24
.066711 23
.066455
.066200 21
0.065944
20
.065689 19
.065433 18
.065178 17
.064922 16
0.064667 15
.064411 14
.064156 13
.063900 12
.063645 11
.061858
.061602
82*7****27 22:
.061347
.061092
0.060837
Tan.
30
29
0.063389 10
.063134 9
.062879
8
.062623
7
.062368
0.062113
26
6
5
4
3210
0
49°
618
41°
01234 LO CON CO σ
6
7
8
9
DI234
10
11
12
13
14
15
16
17
18
19
**********
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
20 9.819832
21
.819976
22
.820120
23
.820263
24
.820406
9.820550
.820693
.820836
.820979
.821122
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
88888
Sin.
131°
9.816943
.817088
.817233
.817379
.817524
9.817668
.817813
.817958
.818103
.818247
9.818392
.818536 Corr.
.818681 10 24
.818825 15 36
.818969 20 48
9.819113 30 72
.819257 40 96
.819401 45 108
.819545 50 120
.819689
9.821265
.821407
.821550
.821693
.821835
9.821977
.822120
.822262
.822404
.822546
TABLE XXII.-LOGARITHMIC SINES,
9.824104
.824245
.824386
.824527
.824668
9.824808
.824949
.825090
.825230
.825371
9.825511
Cos.
A
9.822688
.822830
Corr.
.822972 10 24
.823114 15 35
.823255 20 47
9.823397 30 71
.823539 40 94
.823680 45 106
.823821 50 118
.823963
Corr.
for
Sec.
"
Cos.
9.877780
.877670
.877560
.877450
.877340
9.877230
.877120
.877010
.876899
.876789
9.876678
.876568 Corr.
.876457 10 18
.876347 15 28
.876236 20 37
9.876125 30 55
.876014 40 74
.875904 45
83
.875793 50 92
.875682
9.875571
.875459
.875348
.875237
.875126
9.875014
.874903
.874791
.874680
.874568
9.874456
.874344
.874232
.874121
.874009
9.873896
.873784
.873672
.873560
.873448
Corr.
for
Sec.
"
9.872208
.872095
.871981
.871868
.871755
9.871641
.871528
.871414
.871301
.871187
9.871073
Sin.
9.873335
.873223 17
Corr.
.873110 10 19
.872998 15 28
.872885 20 38
9.872772 30 56
.872659 40 75
.872547 45 85
.872434 50 94
.872321
Tan.
9.939163
.939418
.939673
.939928
.940183
9.940439
.940694
.940949
.941204
.941459
9.941713
.941968 Corr.
.942223 10 42
.942478 15 64
.942733 20 85
9.942988 30 127
.943243 40
170
.943498 45 191
.943752 50 212
.944007
9.944262
.944517
.944771
.945026
.945281
9.945535
.945790
.946045
.946299
.946554
9.946808
.947063
.947318
.947572
.947827
9.948081
.948335
.948590
.948844
.949099
11
9.951896
.952150
.952405
.952659
.952913
9.953167
.953421
.953675
.953929
.954183
9.954437
Cot.
Corr.
for
Sec.
9.949353
.949608 Corr.
.949862 10 42
950116 15 64
.950371 20 85
9.950625 30 127
.950879 40 170
.951133 45 191
.951388 50 212
.951642
n
Cot.
0.060837
60
.060582 59
.060327 58
.060072 57
.059817
56
55
0.059561
.059306 54
.059051 53
.058796 52
.058541 51
0.058287 50
.058032 49
.057777 48
.057522 47
.057267 46
0.057012 45
.056757 44
.056502 43
.056248 42
.055993 41
0.055738 40
.055483 39
.055229 38
.054974 37
.054719
36
35
0.054465
.054210 34
.053955 33
.053701
32
.053446 31
0.053192
30
29
.052937
.052682 28
.052428 27
26
.052173
0.051919
25
.051665 24
.051410 23
.051156 22
.050901 21
0.050647 20
.050392 19
.050138 18
.049884 17
.049629 16
0.049375 15
.049121 14
.048867 13
.048612 12
.048358 11
138°

0.048104 10
.047850
.047595
9
8
.047341
7
.047087
0.046833
.046579
.046325
.046071
.045817
0.045563
Tan.
6
• LO CONIO
4
3
2
0
48°
619
42°
0123 TE LO LO DI ∞ m
4
5
6
7
8
10
11
12
13
14
15
16
17
18
19
222*******♬
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
20 9.828301
.828439
21
.828578
.828716
.828855
9.828993
38
39
40
41
42
43
44
45
46
47
48
49
COSINES, TANGENTS, AND COTANGENTS
Sin.
132°
9.825511
.825651
.825791
825931
.826071
9.826211
.826351
.826491
.826631
.826770
9.826910
.827049 Corr.
.827189 10 23
.827328 15 35
.827467 20 46
9.827606 30 70
.827745 40 93
.827884 45 104
.828023 50 116
.828162
.829131
.829269
.829407
.829545
9.829683
.829821
.829959
.830097
.830234
9.830372
.830509
.830646
.830784
.830921
50 9.832425
51
.832561
52
.832697
53
.832833
54
.832969
55 9.833105
56
.833241
57
.833377
58
59
60
n
9.831058
.831195 Corr.
.831332 10 23
.831469 15 34
.831606 20 46
9.831742 30 68
.831879 40 91
.832015 45 103
.832152 50 114
.832288
.833512
.833648
9.833783
Cos.
Corr.
for
Sec.
11
Cog.
9.871073
.870960
.870846
.870732
.870618
9.870504
.870390
.870276
.870161
9.868785
.868670
.868555
.868440
.868324
9.868209
.868093
.867978
.867862
.867747
.870047
9.869933
.869818 Corr.
.869704 10 19
.869589 15 29
.869474 20 38
9.869360 30 57
.869245 40 76
.869130 45 86
96
.869015 50
.868900
9.867631
.867515
.867399
.867283
.867167
9.867051
.866935
.866819
.866703
.866586
9.865302
.865185
.865068
.864950
S64833
9.861716
.864598
.864481
.864363
.864245
9.864127
Sin.
n
Corr.
for
Sec.
9.866470
866353 Corr.
.866237 10 19
.866120 15 29
.866004 20 39
9.865887
30 58
.865770 40 78
.865653 45
.865536 50 97
.865419
N
88855
Tan.
9.954437
.954691
.954946
.955200
.955454
9.955708
.955961
.956215
.956469
.956723
9.956977
.957231 Corr.
.957485 10 42
.957739 15 63
.957993 20 85
9.958247 30 127
.958500 40 169
.958754 45 190
.959008 50 211
.959262
9.959516
.959769
.960023
.960277
,960530
9.960784
.961038
.961292
.961545
.961799
9.962052
.962306
.962560
.962813
.963067
9.963320
.963574
.963828
.964081
.95-1335
9.967123
.967376
.967629
.967883
.968136
9.964588
.964842
" Corr.
.965095 10 42
.965349 15 63
.965602 20 84
9.965855 30 127
.966109 40 169
.966362 45 190
.966616 50 211
.966869
9.968389
.968643
Corr.
for
Sec.
n
.968896
.969149
.969403
9.969656
Cot.
Cot.
137°

0.045563 60
.045309 59
.045054 58
.044800 57
.044546 56
0.044292
55
.044039 54
.013785 53
.043531 52
.043277 51
0.043023 50
.042769
49
.042515 48
.042261
.042007
0.041753
47
46
45
.041500 44
.041246 43
.040992
42
.040738 41
0.040484
40
39
.040231
.039977 38
37
.039723
.039470 36
0.039216
35
.038962
34
33
.038708
.038455 32
.038201 31
CAGEUKIANG AJ:
Tan.
www.**
30
0.037948
.037694
29
.037440 28
.037187
.036933
0.036680
26
25
.036426 24
.036172 23
.035919
22
.035665 21
0.035412
20
.035158 19
.034905
18
17
16
.034651
.034398
0.034145
15
.033891 14
.033638 13
.033384 12
.033131
0.032877
.032624
11
.032371
.032117
.031864
0.031611
.031357
.031104
.030851
.030597
0.030344
AR******27 22
10
· 9 6 1
8
·CO LO SE ES
0
47°
620
43°
01234 10 6 1 00¬
7
8
9
10
11
12
13
14
15
16
17
18
19
2228
20
21
22
23
24
25
I****** A
26
27
28
29
30
31
32
33
34
35
36
5888
37
38
39
40
41
42
43
44
45
46
47
48
49
58
59
60
Sin.
133°
9.833783
.833919
.834054
.834189
.834325
9.834460
.834595
.834730
.834865
.834999
9.836477
.836611
.836745
9.835134
.835269 Corr.
.835403 10 22
.835538 15
34
.835672 20 45
9.835807 30 67
40 90
45 101
50 112
.835941
.836075
.836209
.836343
.836878
.837012
9.837146
.837279
.837412
.837546
.837679
9.837812
.837945
.838078
.838211
.838344
9.838477
.838610
.838742
.838875
.839007
50
51
52
53
54
55 9.841116
56
.841247
57
.841378
.841509
.841640
9.841771
9.840459
.840591
.840722
.840854
.840985
TABLE XXII.—LOGARITHMIC SINES,
9.839140
#T
.839272 Corr.
.839404 10 22
.839536 15 33
.839668 20 44
9.839800
.839932
40
30 66
88
99
45
.840064
.840196
50 110
.840328
Cos.
Corr.
for
Sec.
"
Cos.
9.864127
.864010
.863892
.863774
.863656
9.863538
.863419
.863301
.863183
.863064
Corr.
20
9.862946
.862827
.862709 10
.862590 15 30
.862471 20 40
9.862353 30 59
.862234 40 79
.862115 45 89
50
.861996
99
.861877
9.861758
.861638
.861519
.861400
.861280
9.861161
.861041
.860922
.860802
.860682
9.860562
.860442
.860322
.860202
.860082
9.859962
"
9.858151
.858029
.857908
.857786
.857665
9.857543
.857422
.857300
.857178
.857056
9.856934
Sin.
Corr.
for
Sec.
.859842
.859721
.859601
.859480
9.859360
.859239 Corr.
.859119 10 20
.858998 15 30
.858877 20 40
9.858756 30 60
.858635 40 81
45
.858514
91
50
.858393
101
.858272
"I
Tan.
9.969656
.969909
.970162
.970416
,970669
9.970922
.971175
.971429
.971682
.971935
9.972188
.972441
Corr.
.972695 10 42
.972948 15 63
.973201 20 84
9.973454 30 127
.973707 40 169
45 190
.973960
50 211
.974213
.974466
9.974720
.974973
.975226
.975479
.975732
9.975985
.976238
.976491
.976744
.976997
9.977250
.977503
.977756
.978009
.978262
9.978515
.978768
.979021
.979274
.979527
B
9.982309
.982562
.982814
.983067
.983320
9.983573
.983826
.984079
.984332
.984584
9.984837
Cot.
Corг.
for
Sec.
9.979780
.980033
Corr.
.980286 10 42
.980538 15 63
.980791 20 84
9.981044 30 126
169
.981297 40
45 190
.981550
50 211
.981803
.982056
ย
Cot.
0.030344
60
.030091 59
.029838
.029584
58
57
.029331 56
0.029078 55
54
.028825
.028571 53
.028318
52
.028065
51
0.027812
.027559
.027305
.027052
.026799
0.026546
.026293
.026040
.025787
.023534
0.025280
.025027
.024774
.024521
.024268
0.024015
.023762
.023509
.023256
.023003
136°

0.017691
.017438
.017186
.016933
.016680
0.016427
.016174
.015921
.015668
.015416
0.015163
Tan.
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
0.022750
.022497
.022244
.021991
.021738
0.021485
24
.021232
.020979
.020726
23
22
20
.020473 21
0.020220
.019967 19
.019714 18
.019462 17
.019209 16
0.018956 15
.018703 14
.018450 13
.018197 12
.017944 11
********Z
25
10
9
8
7854~NIO
6
2
1
46°
621
44°
01234567"" OFN
8
9
10
11
12
13
14
15
16
17
18
** 272******* SAN
19
20
21
23
24
25
26
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
57
58
59
60
8888888
134°
COSINES, TANGENTS, AND COTANGENTS
Sin.
9.841771
.841902
.842033
.842163
.812294
9.842424
.842555
.842685
.842815
.842946
10
Corr.
22
32
9.843076
.843206
.843336
.843466 15
.843595 20 43
9.843725 30 65
.843855 40 86
.843984 45 97
.844114 50 108
.844243
9.844372
.844502
.844631
.844760
.844889
9.845018
.845147
.845276
.845405
.845533
9.845662
.815790
.845919
.846017
.846175
9.846304
.846432
.846560
.846688
.846816
55 9.818852
56
.848979
.849106
.849232
.849359
9.849485
Cos.
9.848218
.848345
.848472
.848599
.818726
Ħ
9.846944
Corr.
.847071
.847199
10 21
.847327 15 32
.847454 20 42
9.847582 30 64
.847709 40 85
.847836 45 96
.847964 50 106
.848091
Corr.
for
Sec.
P
Cos.
9.856934
.856812
.856690
.856568
.856446
9.856323
.856201
.856078
.855956
.855833
9.854480
.854356
.854233
"
9.855711
.855588 Corr.
.855465 10 21
.855342 15 31
.855219 20 41
9.855096 30 62
.854973 40 82
.854850 45 92
.854727 50 103
.854603
.854109
.853986
9.853862
.853738
.853614
.853490
.853366
9.853242
.853118
.852994
.852869
.852745
9.852620
.852496
.852371
.852247
.852122
9.850745
.850619
.850493
.850368
.850242
9.851997
.851872 Corr.
.851747 10 21
.851622 15 31
.851497 20 42
9.851372 30 63
.851246 40 83
.851121 45 94
.850996 50 104
.850870
9.850116
.849990
.849864
.849738
.849611
9.849485
Sin.
Corr.
for
Sec.
"
Tan.
9.984837
.985090
.985343
.985596
.985848
9.986101
.986354
.986607
.986860
.987112
9.987365
.987618 Corr.
.987871 10 42
.988123 15 63
.988376 20 84
9.988629 30 126
.988882 40 169
.989134 45 190
.989387 50 211
.989640
9.989893
.990145
.990398
.990651
.990903
9.991156
.991409
.991662
.991914
.992167
9.992420
.992672
.992925
.993178
.993431
9.997473
.997726
.997979
.998231
.998484
W
9.993683
.993936
.994189
.994441
.994694
9.994947
.995199 Corr.
.995452 10 42
.995705 15 63
.995957 20 84
9.996210 30 126
.996463 40 168
.996715 45 190
50 211
.996968
.997221
9.998737
.998989
.999242
.999495
.999747
0.000000
Cot.
Corr.
for
Sec.
=
Cot.
0.015163 60
.014910 59
.014657 58
.014404 57
.014152 56
0.013899
55
.013646
54
50
.013393 53
.013140 52
.012888 51
0.012635
.012382 49
.012129 48
.011877 47
.011624 46
0.011371 45
.011118 44
.010866 43
.010613
42
.010360
41
135°

0.007580
.007328
0.010107
.009855
.009602 38
.009349 37
.009097 36
0.008844
35
.008591 34
.008338 33
.008086 32
.007833
31
.007075
.006822
.006569
0.006317
INH 80**
40
39
222782
30
29
26
25
,006064 24
.005811 23
.005559 22
.005306 21
0.005053 20
.004801 19
.004548 18
.004295 17
.001043 16
0.003790 15
.003537 14
.003285 13
.003032 12
.002779 11
0.002527 10
.002274 9
.002021
8
.001769
.001516
0.001263
.001011
.000758
.000505
.000253
0.000000
Tan.
76543210
45°
622
Minutes
024
8
10
12
14
16
18
20
22
24
26
28
♡♡♡♡♡♡♡♡♡♡♡♡ ****~ ~*
30
32
34
36
38
40
42
44
46
48
50
52
C =
.75
C= 1.00
TABLE XXIII.-STADIA REDUCTIONS

0°
Hor. Vert. Hor. Vert. Hor. Vert.
Dist. Dist. Dist. Dist. Dist.
Dist.
1.74
1.80
100.00 .00 99.97
100.00 .06 99.97
100.00 .12 99.97 1.86
100.00 .17 99.96 1.92
100.00 .23 99.96 1.98
100.00 .29 99.96 2.04
100.00 .35 99.96
100.00 .41 99.95
100.00
100.00
100.00
100.00
100.00
99.99
99.99
99.99
1°
.52 99.95
.58 99.95
2.09
2.15
.47 99.95 2.2I
99.99
.93 99.93
99.99 .99 99.93
99.99 1.05 99.92
99.99 1.11 99.92
99.99 1.16 99.92
2.97
99.99 1.22 99.91
99.98 1.28 99.91 3.02
99.98 1.34 99.90 3.08
99.98 1.40 99.90 3.14
99.98 1.45 99.90 3.20
99.98 1.51 99.89 3.26
54 99.98 1.57 99.89 | 3.31
99.97 1.63 99.89 3.37
99.97 1.69 99.88 3.43
99.971.74 99.88 3.49
56
58
60
.64 99.94
.70 99.94
.76 99.94 2.50
.81 99.93
2.56
.87 99.93
2.62
.75 ,ΟΙ .75
1.00 .ΟΙ 1.00
C
= 1.25 1.25 .02 1.25
2.27
2.33
2.38
2.44
2.67
2.73
2.79
2.85
2.91
2.91
.02
.03
.03
99.88
3.49
99.87
99.87 3.55
99.87
3.60
99.87
3.66
99.86 3.72
99.86 3.78
2°
99.83
99.83
99.82
99.82
3°
99.82
4.24
99.81 4.30
99.81 4.30
99.81
4.36
Hor. Vert.
Dist. Dist.
99.85 3.84
99.85 3.84
99.69 5.57
99.85 3.89 99.68 5.63
99.84 3.95 99.68 5.69
99.84 4.01 99.67 5.75
99.83 4.07
99.83 4.07 99.66 5.80
99.80 4.42
99.80 4.42
99.80 4.47
.75
1.00
1.25
99.73 5.23
99.72 5.28
99.71
99.71 5.34
99.71
99.71
5.40
99.70 5.46
99.69 5.52
4.13 99.66 5.86
4.13
4.18
4.18 99.65 5.92
99.64 || 5.98
99.63 | 6.04
99.63|| 6.09
99.62
6.15
99.61 | 6.21
99.79 4.53 99.61 6.27
99.79
4.59
4.59
99.78 4.65
99.78 4.65 99.59 6.38
99.60 | 6.32
99.78 4.71 99.58 6.44
99.77 4.76|99.58 | 6.50
99.77 4.82 99.57 6.56
99.76 4.88 99.56 6.61
99.76 4.94 99.55 6.67
99.75 4.99 99.55 6.73
99.74 5.05 99.54 6.79
99.74 5.II 99.53 6.84
99.73 5.17 99.52 6.90
99.73 5.23 99.51 6.96
.03 .75 .05
.04
1.00 .06
.05
1.25
.08
623
Minutes
10
N N N N
0246∞ 0
12
14
16
18
20
22
99.51 6.96 99.24
99.5I 7.02 99.23
99.50 7.07 99.22
99.49 7.13 99.21
8 99.48 7.19 99.20
99.47 7.25 99.19
24
26
28
30
www.y
GAAAA Aw
32
34
36
38
40
42
44
46
48
LO LO LO 101
52
54
56
TABLE XXIII.—STADIA REDUCTIONS

4°
58
Hor. Vert.
Dist. Dist.
60
Hor.
Dist.
9.60 98.71 11.30 98.28 13.00
9.65 98.69 11.36 98.27 13.05
9.71 98.68 11.42 98.25 13.11
9.77 98.67 11.47 98.24 13.17
9.83 98.65 11.53 98.22 13.22
99.33 8.17 99.01 9.88 98.64 11.59 98.20 13.28
99.32 8.22 99.00 9.94 98.63 11.64 98.19 13.33
99.31
8.28 98.99 10.00 98.61 11.70 98.17 13.39
99.30 8.34 98.98 10.05 98.60 11.76 98.16 13.45
50 99.29 8.40 98.97 10.11 98.58 11.81 98.14 13.50
99.46 7.30 99.18
99.46 7.36 99.17
99.45 7.42 99.16
99.44
7.4899.15
99.43 7.53 99.14
99.42 7.59 99.13
99.41 7.65 99.11
99.40 7.71 99.10
99.39 7.76 99.09
99.38 7.8299.08
5°
99.38 7.88 99.07
99.37 7.9499.06
99.36 7.99 99.05
99.35 8.05 99.04
99.34 8.11 99.03
6°
go
99.28 8.45 98.96 10.17 98.57 11.87 98.13 13.56
99.27 8.51 98.94 10.22 98.56 11.93 98.11 13.61
99.26 8.57 98.93 10.28 98.54 11.98 98.10 13.67
99.25 8.63 98.92 10.34 98.53 12.04 98.08 13.73
99.24 8.68 98.91 10.40 98.51 12.10 98.06 13.78
.75 .75 .06 .75 .07 .75 .08 .74
C = = 1.00 1.00 .08 1.00
C = 1.25 1.25 .10 1.24
с
.10 .99
.II
.99
.12
1.24
.14
I.24
Vert. Hor. Vert. Hor. Vert.
Dist. Dist. Dist. Dist. Dist.
8.68 98.91 10.40 98.51 12.10
8.74 98.90 10.45 98.50 12.15
8.80 98.88 10.51 98.49 12.21
8.85 98.87 10.57 98.47 12.27
8.91 98.86 10.62 98.46 12.32
8.97 98.85 10.68 98.44 12.38
9.03 98.83 10.74 98.43 12.43
9.08 98.82 10.79 98.41 12.49
9.14 98.81 10.85 98.40 12.55
9.20 98.80 10.91 98.39 12.60
9.25 98.78 10.96 98.37 12.66
9.31 98.77 11.02 98.36 12.72
9.37 98.76 11.08 98.34 | 12.77
9.43 98.74 11.13 98.33 12.83
9.48 98.73 11.19 98.31 12.88
9.54 98.72 11.25 98.30 12.94
.10
.13
.16
624
Minutes
12
14
16
18
20
222
02 46∞
30
A w w w w w
32
17.10
17.16
96.36 18.73
96.34 | 18.78
98.06 13.78 97.55 15.45 96.98
98.05 13.84 97.53 15.51 96.96
98.03 13.89 97.52 15.56 96.94
98.01 13.95 97.50 15.62 96.92 17.26 96.29 18.89
8 98.00 14.01 97.48 15.67 96.90 17.32 96.27 18.95
97.98 14.06 97.46 15.73 96.88 17.37 96.25 19.00
17.21 96.32 | 18.84
|
10
|
34
36
28
97.97 14.12 97.44 15.78 96.86 17.43 96.23 | 19.05
97.95 14.17 97.43 15.84 96.84 17.48 | 96.21 | 19.11
97.93 14.23 97.41 15.89 96.82 17.54 96.18 19.16
97.92 14.28 97.39 15.95 96.80 17.59 96.16 19.21
97.90 14.34 97.37 16.00 96.78 17.65 96.14 19.27
22 97.88 14.40 97.35 16.06 96.76 17.70 96.12 19.32
24 97.87 14.45 97.33 16.11 96.74 17.76 96.09 19.38
26 97.85 14.51 97.31 16.17 96.72 17.81 96.07 19.43
97.83 14.56 97.29 16.22 96.70 17.86 96.05 19.48
97.82 14.62 97.28 16.28 96.68 17.92 96.03 19.54
97.80 14.67 97.26 16.33 96.66 17.97 96.00 19.59
97.78 14.73 97.24 16.39 96.64 18.03 95.98 19.64
|
|
97.76 14.79 97.22 16.44 96.62 18.08 95.96 19.70
97.75 14.84 97.20 16.50 96.60 18.14 95.93 19.75
97.73 14.90 97.18 16.55 96.57 18.19 95.91 | 19.80
97.7114.9597.16| 16.61 | 96.55 18.24 95.89 19.86
97.69 15.01 97.14 16.66 96.53 18.30 95.86| 19.91
97.68 15.06 97.12 16.72 96.51 18.35 95.84 19.96
97.66 15.12 97.10 16.77 96.49 18.41 95.82 20.02
97.64 15.17 97.08 16.83 96.47|18.46 95.79 20.07
| |
97.62 15.23 97.06 16.8896.45 18.51 95.77 20.12
97.61 15.28 97.04 16.94 96.42 | 18.57 95.75 20.18
97.59 15.34 97.02 | 16.99 | 96.40 18.62 95.72 20.23
|
97.57 15.40 97.00 17.05 96.38 18.68 95.70 20.28
97.55 15.45 96.98 | 17.10|96.36|18.73|95.68 20.34
|
38
40
42
44
46
48
45
50
555
52
Our cry o
54
TABLE XXIII.—STADIA REDUCTIONS

8°
56
9°
10°
Hor. Vert. Hor. Vert. Hor. Vert.
Dist. Dist. Dist. Dist. Dist. Dist.
II°
Hor. Vert.
Dist. Dist.
C = .75
.74 .II .74 .12 .74 .14 .73 .15
C = 1,00 .99 .15 .99 .17 .98
1.23
.18
.98
.20
C = 1.25 1.24 .18 1.23
.2I
.23
1.22
.25
625
Minutes
02 46∞
рассаду росту рослинни
2 46 ∞ o
12
14
16
95.68 20.34 94.94 21.92 94.15 23.47 93.30 25.00
|
95.65 20.39 94.91 21.97 94.12 23.52 93.27 25.05
95.63 20.44 94.89 22.02 94.09 23.58 93.24 25.10
95.61 20.50 94.86 22.08 94.07 23.63 93.21 | 25.15
8 95.58 20.55 94.84 22.13 94.04 23.68 93.18 25.20
10 95.56 20.60 94.81 22.18 94.01 23.73 93.16 | 25.25
|
18
20
22
24
26
Over en en en in A☆ ☆ ☆ Awwww ww.
28
30
32
34
36
38
40
42
44
46
48
50
52
54
56
58
TABLE XXIII.—STADÍA RÉDUCTIONS

12°
60
с
C
C =
1.00
.75
=
C = 1.25
13°
14°
Hor. Vert. Hor. Vert. Hor. Vert. Hor. Vert.
Dist. Dist. Dist. Dist. Dist. Dist.
Dist. Dist.
.73 .16
.98
.22
I.22
.27
|
95.53 20.66 94.79 22.23 93.98 23.78 93.13 25.30
95.51 20.71 94.76 22.28 93.95 23.83 93.10 25.35
95.49 20.76 94.73 22.34 93.93 23.88 93.07 25.40
95.46 20.81 94.71 22.39 93.90 23.93 93.04 25.45
95.44 20.87 94.68 22.44 93.87 23.99 93.01 | 25.50
95.41 20.92 94.66 22.49 93.84 24.04 92.98 25.55
95.39 20.97 94.6322.54 93.82 24.09 92.95 25.60 –
95.36 21.03 94.60 22.60 93.79 24.14 92.92 | 25.65
95.34 21.08 94.58 22.65 93.76 24.19 92.89 25.70
95.32 21.13 94.55 22.70 93.73 24.24 92.86 25.75
95.29 21.18 94.52 22.75 93.70 24.29 92.83 25.80
95.27 21.24 94.50 22.80 93.67 24.34 92.80 25.85
95.24 21.29 94.47 22.85 93.65 24.39 92.77 | 25.90.
95.22 21.34 94.44 22.91 93.62 | 24.44 | 92.74 25.95
95.19 21.39 94.42 22.96 93.59 24.49 92.71 26.00
95.17 21.45 94.39 23.01 93.56 24.55 92.68 26.05
95.14 21.50 94.36 23.06 93.53 24.60 92.65 26.10
95.12 21.55 94.34 23.11 93.50 24.65 92.62 26.15
95.09 21.60 94.31 23.16 93.47 24.70 92.59 26.20
95.07 21.66 94.28 23.22 93.45 24.75 92.56 | 26.25
95.04 21.71 94.26 23.27 93.42 24.80 92.53 26.30
95.02 21.76 94.23 23.32 93.39 24.85 92.49 26.35
94.99 21.81 94.20 23.37 93.36 24.90 | 92.46 | 26.40
94.97 21.87 94.17 23.42 93.33 24.95 92.43 26.45
94.94 21.92 | 94.15 23.47 93.30 25.00 92.40 26.50
| | |
15°
.73
.97
1.22
.18 .73
.23 .97
.29 1.21
.19 .72 .20
.25 .96
.27
•31
1.20
.33
626
Minutes
02 46∞
jong pang pH
22
24
26
28
30
♡ ♡ ♡ M4
2 46∞ O
92.40 26.50 91.45 27.96 90.45 29.39 89.40 30.78
92.37 26.55 91.42 28.01 90.42 29.44 89.36 30.83
92.34 26.59 91.39 28.06 90.38 29.48 89.33 30.87
|
92.31 26.64 91.35 28.10 90.35 29.53 89.29 30.92
8 92.28 26.69 91.32 28.15 90.31 29.58 89.26 30.97
10 92.25 26.74 91.29 28.20 90.28 29.62 89.22 31.01
32
34
36
38
20
12 92.22 26.79 91.26 28.25 90.24 29.67 89.18 31.06
14 92.19 26.84 91.22 28.30 90.21 29.72 89.15 31.10
16 92.15 26.89 91.19 28.34 90.18 29.76 89.11 31.15
18 92.12 26.94 91.16 28.39 90.14 29.81 89.08 31.19
92.09 26.99 91.12 28.44 90.11 29.86 89.04 31.24
92.06 27.04 91.09 28.49 90.07 29.90 89.00 31.28
92.03 27.09 91.06 28.54 90.04 29.95 88.97 31.33
92.00 27.13 91.02 | 28.58 | 90.00 30.00 88.93 31.38
91.97 27.18 90.99 28.63 89.97 30.04 88.89 31.42
91.93 27.23 90.96| 28.68 | 89.93 30.09 | 88.86 31.47
91.90 27.28 90.92 28.73 89.90 30.14 88.82 31.51
91.87 27.33 90.89 28.77 89.86 30.18 88.78 31.56
91.84 27.38 90.86 28.82 89.83 30.23 88.75 31.60
91.81 27.43 90.82 28.87 89.79 30.28 88.71 31.65
91.77 27.48 90.79 28.92 89.76 30.32 88.67 31.69
91.74 27.52 90.76 28.96 89.72 30.37 88.64 31.74
91.71 27.57 90.72 29.01 89.69 30.41 88.60 | 31.78
91.68 27.62 90.69 29.06 89.65 30.46 88.56 31.83
91.65 27.67 90.66 29.11 89.61 30.51 88.53 31.87
91.61 27.72 90.62 29.15 89.58 30.55 88.49 31.92
91.58 27.77 90.59 29.20 | 89.54 30.60 88.45 31.96
91.55 27.81 90.55 29.25 89.51 30.65 88.41 32.01
91.52 27.86 90.52 29.30 89.47 30.69 88.38 32.05
91.48 27.91 90.49 29.34 89.44 30.74 88.34 32.09
91.45 27.9690.45 29.39 89.40 30.78 88.30 32.14
.72 .23 .7I .24 .7I .25
40
42
44
46
48
50
TABLE XXIII.-STADIA REDUCTIONS

16°
52
54
56
58
бо
17°
18°
19°
Hor. Vert. Hor. Vert. Hor. Vert. Hor. Vert.
Dist. Dist. Dist. Dist. Dist. Dist. Dist. Dist.
C = .75
.72
.21
C = 1.00 .96 .28 .95 ⚫30 .95 .32
C = 1.25 1.20 .36 1.19 .38 1.19
.40
•94 .33
.42
1.18
627
Minutes
0246∞
12
14
16
1.18
20
J
22
24
26
28
genereren er A ☆ ☆ ☆ â wwww w
30
88.30 32.14 87.16 33.46 85.97 34.73 84.73 35.97
88.26 32.18 87.12 33.50 85.93 34.77 84.69 36.01
88.23 32.23 87.08 33.54 85.89 34.82 84.65 36.05
88.19 32.27 87.04 33.59 85.85 34.86 84.61 36.09
8 88.15 32.32 87.00 33.63 85.80 34.90 84.57 36.13
10 88.11 32.36 86.96 33.67 85.76 34.94 84.52 36.17
36
38
40
42
56
58
бо
TABLE XXIII.—STADIA REDUCTIONS

20°
21°
22°
23°
Hor. Vert. Hor. Vert. Hor. Vert. Hor. Vert.
Dist. Dist. Dist. Dist. Dist. Dist. Dist. Dist.
C = .75
C= 1.00 .94 .35 .93
C = 1.25
1.17 .44 1.16
88.08 32.41 86.92 33.72 85.72 34.98 84.48 36.21
88.04 32.45 86.88 33.76 85.68 35.02 84.44 36.25
88.00 32.49 86.84 33.80 85.64 35.07 84.40 36.29
87.96 32.54 86.80 33.84 85.60 35.11 84.35 36.33
87.93 32.58 86.77 33.89 85.56 35.15 84.31 36.37
87.89 32.63 86.73 33.93 85.52 35.19 84.27 36.41
87.85 32.67 86.69 33.97 85.48 35.23 84.23 36.45
87.81 32.72 86.65 34.01 85.44 35.27 84.18 36.49
87.77 32.76 86.61 34.06 85.40 35.31 84.14 36.53
87.74 32.80 86.57 34.10 85.36 35.36 | 84.10 36.57
87.70 32.85 86.53 34.14 85.31 35.40 84.06 36.61
87.66 32.89 86.49 34.18 85.27 35.44 84.01 36.65
87.62 32.93 86.45 34.23 85.23 35.48 83.97 36.69
87.58 32.98 86.41 34.27 85.19 35.52 83.93 36.73
87.54 33.02 86.37 34.31 85.15 35.56 83.89 36.77
87.51 33.07 86.33 34.35 85.11 35.60 83.84 36.80
87.47 33.11 86.29 34.40 85.07 35.64 83.80 36.84
87.43 33.15 86.25 34.44 85.02 35.68 83.76 36.88
87.39 33.20 86.21 34.48 84.98 35.72 83.72 36.92
87.35 33.24 86.17 34.52 84.94 35.76 83.67 36.96
87.31 33.28 86.13 34.57 84.90 35.80 83.63 37.00
87.27 33.33 86.09 34.61 84.86 35.85 83.59 37.04
87.24 33.37 86.05 34.65 84.82 35.89 83.54 37.08
87.20 33.41 86.01 34.69 84.77 35.93 83.50 37.12
87.16 33.46 85.97 34.73 84.73 35.97 83.46 37.16
.26
.70
.70 .27 .69 .29 .69
.37 .92 .38 .92
.46
1.15
.48
.30
.40
1.15 .50
628
Minutes
12
14
16
18
20
02 46∞
28
2 3
32
34
36
38
83.46 37.16 82.14 38.30 80.78 39.40 79.39 40.45
83.41 37.20 82.09 38.34 80.74 39.44 79.34 | 40.49
83.37 37.23 82.05 38.38 80.69 39.47 79.30 40.52
83.33 37.27 82.01 38.41 80.65 39.51 | 79.25 | 40.55
8 83.28 37.31 81.96 38.45 80.60
83.24 37.35 81.92 38.49 80.55
83.20 37.39 81.87 38.53 80.51
83.15 37.43 81.83 38.56 80.46
10
39.58
39.54 79.20 | 40.59
79.15 | 40.62
79.11 40.66
39.65 79.06 | 40.69
39.61
22
83.11 37.47 81.78 38.60 80.41 39.69 | 79.01 | 40.72
83.07 37.51 81.74 38.64 80.37 39.72 78.96| 40.76
83.02 37.54 81.69 38.67 80.32 39.76 78.92 40.79
82.98 37.58 81.65 38.71 80.28 39.79 78.87 40.82
82.93 37.62 81.60 38.75 80.23 39.83 78.82 40.86
26 82.89 37.66 81.56 38.78 80.18 39.86 78.77 40.89
82.85 37.70 81.51 38.82 80.14 39.90 78.73 40.92
30 82.80 37.74 81.47 38.86 80.09 39.93 78.68 40.96
24
40
42
FOOD AT000
48
50
46
82.76 37.77 81.42 38.89 80.04 39.97 78.63 40.99
82.72 37.81 81.38 38.93 80.00 40.00 78.58 41.02
82.67 37.85 81.33 38.97 79.95 40.04 78.54 41.06
82.63 37.89 81.28 39.00 79.90 40.07 78.49 41.09
82.58 37.93 81.24 39.04 79.86| 40.11 78.44 41.12
82.54 37.96 81.19 39.08 79.81 40.14 78.39 41.16
44 82.49 38.00 81.15 39.11 79.76 40.18 78.34 41.19
82.45 38.04 81.10 39.15 79.72 40.21 78.30 41.22
82.41 38.08 81.06 39.18 79.67 40.24 78.25 41.26
82.36 38.11 81.01 39.22 79.62 40.28 78.20 41.29
82.32 38.15 80.97 39.26 79.58 40.31 78.15 41.32
82.27 38.19 80.92 39.29 79.53 40.35 78.10 41.35
82.23 38.23 80.87 39.33 79.48 40.38 78.06 41.39
82.18 38.26 80.83 39.36 79.44 40.42 78.01 41.42
82.14 38.30 80.78 39.40 79.39 40.45 77.96 41.45
.68 .3I .68 .32 .67 .33 .67 .35
.4I
.90
.43
.89
.45 .89
.52 1.13
.54
I.12
.46
.56 I.II .58
52
gur er en in
54
56
58
бо
TABLE XXIII.-STADIA REDUCTIONS

C =
.75
C = 1.00
C = 1.25
24°
25°
.91
1.14
26°
27°
Vert.
Hor.
Vert. Hor. Vert. Hor. Vert. Hor.
Dist. Dist. Dist. Dist. Dist. Dist. Dist. Dist.
629
Minutes
02 46∞
8
10
12
£2
24
26
28
30
32
34
36
34
38
40
42
44
gen er en en in A☆☆
46
48
50
52
56
58
60
C=
.75
C = 1.00
C = 1.25
TABLE XXIII.-STADIA REDUCTIONS

28°
Vert.
Hor.
Dist. Dist.
77.96 41.45
77.91 41.48
77.86
41.52
77.81
77.77
77.72
41.55
41.58
41.61
77.33 41.87
77.28 41.90
77.23 41.93
76.20
77.67 41.65
77.62 41.68 76.15
77.57 41.71 76.10
77.52 41.74 76.05
77.48 41.77 76.00
77.42 41.81 75.95
77.38 41.84
Hor. Vert.
Dist.
Dist.
76.50
76.45
76.40
76 35
76.30
76.25
29°
.88 .48
I.10
.60
42.40
42.43
42.46
42.37 75.05
42.40 75.00
.36 .65
.87
1.09
42.49
42.53
42.56 | 74.75
42.68
42.71
43.04
43.07
76.94 42.12 75.45
76.89 42.15 75.40
76.84 42.19 75.35 43.10
76.79 42.22 75.30 43.13
76.74
76.69
76.64 42.31 75.15
42.34 75.10
76.59
76.55
76.50
.66
30°
Hor.
Vert.
Dist. Dist.
42.59 74.70
42.62
74.65
42.65
74.60
75.00 43.30
74.95 43.33
74.90
74.90 43.36
74.85 43.39
74.80
43.42
43.45
42.74
42.77
75.90
75.85
74.34 43.67
42.80
75.80 42.83 74.29 43.70
75.75
42.86 | 74.24
43.73
77.18
75.70
42.89
41.97
77.13 42.00 75.65
74.19 43.76
42.92 74.14 43.79
77.09 42.03 75.60 42.95
42.06 75.55
76.99❘ 42.09 75.50
77.04
74.09 43.82
42.98 74.04 43.84
43.01
73.99 43.87
.37
.49
.62
43.47
43.50
43.53
74.55
43.56
74.49 43.59
73.93 43.90
73.88 43.93
73.83 43.95
73.78 43.98
42.25 75.25 43.16 73.73
42.28 75.20 43.18
44.01
43.2I
43.24
73.68 44.04
73.63 44.07
73.58 44.09
43.27 73.52 44.12
43.30
73.47 44.15
74.44
43.62
74.39 43.65
.65 .38
.86
.5I
1.08 .63
630

American Railway Engineering Association
PLAN NO. 910-41
Properties
of Switches
Col.
Col.
Col.
Col
2 (3) 4
Frog Number
Length of
5 Switch Rail
TABLE XXIV.-TURNOUT AND CROSSOVER DATA
Ft In
Switch
Angle
Actual
Lead
Closure
Distance
Col.
5
Straight
Closure Rail
TURNOUT AND CROSSOVER DATA
FOR STRAIGHT SPLIT SWITCHES
Adopted March, 1941. Supersedes Plan No. 900, Revised March, 1934.
Col.
6
Curved
Closure Rail
Ft In
28-4
33-0
Lead Curve
Col.
7
Radius of
Center Line
Col.
8
Degree
of
Curve
Col.
9
Gage Line Offsets
Col.
10
Col.
Θ
Col. Col
(11) (12) (13)
Col. Col. Col.
(14)
(15)
16
Tangent adjacent
to Switch Rail
Tangent adjacent
to toe of Frog
0 / 11
Ft In
Ft In
Feet
0 / 11
Ft In
Ft In Inches Inches Ft In Feet
Feet
28-0
177,80 32-39-56
18-0
5 11-0 2-39-34 42-6
611-02-39-34 47-6
Ft In
25-0
32-0 | 11
11202-8 0.00 0.78
32-9
258.57 22-17-58 19-2 27-4 35-62 12 21 2-10 0.00 1.75
7 16-6 1-46-22 62-1 40-102 41-136559 15-43-16| 26-21|35-10 45-6 11 19 2-6 0.01 0.00
8 16-6 1-46-22 68-0 46-5 46-72 487.2811-46-44 27-738-8 49-9 11 20 2-8
916-6 1-46-22 72-3½ 49-5 49-74 615.12 9-19-30 28-1041-253-62 12 212-9, 0.00 0.17
ΤΟ 16-6 1-46-22] 78-9 55-10 56-0 779.39 7-21-24 29-1 12 43-556-111211 21 2-8 2.08 0.00
|
0.64 0.00
B
3
II
22-01-19-46 91-10
12
7
62-1063-0 92 7.27 6-10-56 37-853-5 69-112212-93 0.00 0.1 3
22-01-19-461 96-8 66-102 67-0 1104.63 5-11-20 38-855-5 72-1글 ​12급 ​21층 ​2-9급 ​0.00 0.50
14 22-01-19-46 107-076-576-63158 1.20 3-37-28 41-1460-279-32 12 22 2-10 0.24 0.00
15 | 30-00-58–30 126-4½ 86-11 87-01720.77 3-19-48 51-9 73-6 95-3 12 212-93 1.56 0.00
30-0 0-58-30 131-4 91-11
30-0 0-58-30 140-11299-11
20 30-0 0-58-30 151-11½ 110-11
16
18
92-0 2007.12 2-51-18] 53-0
100-0 2578.79 2-13-201 55-0
111-0 3289.29 1-44-32 57-9
76-0 99-0
80-0 105-0
85-6 113-3
12 21 2-10 0.66 0.00
121222-10 0.57 0.00
13% 222-11 2.47 0.00
631

Frog Number
557
Properties of Frogs
Col.
17
0 / 11
5 11-25-16|
6
18
20
Frog
Angle
TABLE XXIV.-TURNOUT AND CROSSOVER DATA
Col.
18
Overall
Length
Col.
19
Ft
Toe Length
Ft
Col. Col. Col.
(20
(21)
(22)
In.
3-6 1/2
3-9
4-8/2/2
5-1
꼬
​Heel Length
In.
Inches Inches Ft
9-0
5-571316-10
9-31-38 10-0
6-3
8-10-16
12-0
7 13 20-52/2
7-3 7/13 24-0큽
​7-11 7급 ​12층 ​27-7ㅎ
​9-72 8 13/631-13/
8 7-09-10 13-0
9
6-21-35 16-0
5-43-29 16-6
5-12-18 18-822
6-42
6-5 10-1
10
7층 ​12층
​34-8ㅎ
​│
7ㅎ
​13
38-2늘
​7-01 11-8
7-9글 ​12-6일
​12
4-46-19 20-4
7
13
41-8층
​14
1348-94
7
15
18
15
4-05-27 23-7 8-7글​14-11호 ​6급
​3-49-06 24-49-5 14-11/2/27 12/6 52-3/7/20
163-34-47 26-0 9-5 16-7 6% 12 55-9홉
​3-10-56 29-3 | 11-01/
11-0호 ​18-2호 ​6급 ​12동 ​62-9 급
​30-10½|| 11—0—2
6급 ​12층 ​69-10
16
2-51-51
19-10
Toe
Spread
Heel
Spread
F+ In.
13-0" Track Centers
Col.
(24)
Col.
(23
Data for Crossovers
Straight Crossover
Track
Track
In.
Ft In.
18-1 3/23
For change of 1-0
in Track Centers
6
Straight Crossover
Track
Track
Ft
In.
4-1116
21-62
5-11호
​24-11층
​6-1116
28-4급
​7-1195/55
31-103/
8-11
35-3급
​9-11
16
10-11 3/4
38-9호
​42-34
11-1144
49-21
13-11
52-8
14-1116
56-21/2/2 15-1118
63-217-1 |
13
16
70-2 19-11 금 ​20-01/1
6
FI In.
TO
13
5-02
8
6-01/2
7-06
8-03-2
9-08
10-016
11-01/1/00
12-04/
14-04
15-03
16-0
18-0
632

1/8" POINT OF SWITCH
LENGTH OF
SWITCH RAIL
SWITCH ANGLE
3
2
9
TABLE XXIV.-TURNOUT AND CROSSOVER DATA
HEEL SPREAD 6-1/4"
C.
(12)
STRAIGHT CLOSURE RAIL
TANGENT ADJACENT TO
SWITCH RAIL
(10)
15
13
NOTES
ACTUAL LEAD
IT
CURVED
CLOSURE RAIL
RAD.
RADIUS OF CENTER LINE (7)
1. TURNOUTS AND CROSSOVERS
RECOMMENDED
For main line high speed movements, No. 16 or No.
20. For main line slow speed movements, No. 12 or
No. 10. For yards and sidings, to meet general con-
ditions, No. 8.
TANGENT ADJACENT TO
TOE OF FROG
TOE SPREAD
2. The data shown are computed for turnouts out of
straight standard 4' 8½" gage track. If the wheel base
of the equipment used requires wider gage for the
switch alinement or curvature shown, the lead and
alinement of curved closure rail shall be maintained,
and inside stock and curved rails shall be moved out
the necessary amount. The gage of straight track
through switch will then be increased and the straight
(14)
(21)
P. T.
TOE
LENGTH
19
~1/2" POINT OF FROG
HEEL LENGTH
29
OVERALL LENGTH(
STRAIGHT
(18)
23
HEEL SPREAD 22
17
FROG ANGLE
29.
1/2" POINT OF FROG
closure rail shall be bent to true alinement in advance
of toe of frog.
3. FROG DESIGNS
For short spring rail type frogs, the straight closure,
and for solid manganese frogs, the straight and curved
closures, shall be lengthened to conform.
4. MODIFICATION OF ALINEMENT
The alinement specified may be used without appreci-
able wear for switch points made in accordance either
with Detail 4000 (thickness of point-14"), or with
Detail 5000 (thickness of point-0").
5. For length of closure rails, see Plan Basic No. 911.
6. For bills of timber for various turnouts and crossovers
see Plan Basic No. 912.
TRACK
CENTERS
633

American Railway Engineering
Association
LOCATION OF JOINTS FOR TURNOUTS
WITH STRAIGHT SPLIT SWITCHES
PLAN NO. 911-41 Adopted March, 1941.
Supersedes in part Plans No. 901 to 908 Incl.,
Revised March, 1934.
Frog Number
Length of
Switch Rail
5111-0
6
7
8
9
10
#
Toe Length
to ½ Pt.
T
TABLE XXIV.-TURNOUT AND CROSSOVER DATA
#
Lead
PS.to Pt
Straight Rail
Curved Rail
Rail "A" Rail "B" | Rail "C" Rail"D" Rail "E" Rail"F"
11'-0":
16-6"
16′6″ 5'-1"
19-6" 8-98
19'-6" 13-5
27-0″ | 14-0}" |
27-019-7
27-0" 22-6"
25-11 30-0"
39-0″ 123-117″.
39-0" 27-113
16-6" 6-47″
16-6" 6-5" 78-9
39-0" 27′10″
|
TE 22:0" 7:0* 91-10
12 22-0* 7-9 96-8"
14 22-0 8-7" 107-02″ 30-0″ 16-5″ 30-0″ 30-0" 19-62 7-0″
15 30-0" 9-5
16 30-0
39-0″
9-5*
126-439-0" 33-0" 14-11-133-0" 24-030-0′ 33-0" 30-01
131-433-0"|33-0" 125-11" 133-0" 19-1139-0" 33-0" 33-0"
18
| |
30-0″ 11-0½″ 140-11" 39-033-0" 27-11″ 39-0″ 33-0″ |27-1137_39-0″ 39-0′
20 30-0
1 1-0 2" || 15 1-11" 39-0" 39-0" 32-11" 39-0" 39-0" 32-11"_30-0" 30-0 30-0"
"
3-6 42-6" 28-01
32-97
3-9"
47-6″
4-8 62-1" 27-013-102″
68-0" 30-0 16-5"
72-3″ 30-0"| 19-5"
33′ 0″ 22-10
39-0 23-10
"
NOTES
1. Type "L" turnouts are for general use, except
for No. 5 and 6 turnouts.
2. Alternate Type "S" turnouts are for use with
No. 5 and 6 turnouts, and where limited space
requires location of joints ahead of switch nearer
to switch point.
3. Insulated joints shown thus
on plans
are located to conform to AAR Signal Section
Plans No. 1634-B, 1635-B and 1637-B, approved
in July, 1937, and if other rail lengths are used,
the stagger of the joints must not exceed 4'6".
Closure Rails
Length in Feet and Inches
CLOSURE RAILS AND JOINT LOCATION FOR RIGHT HAND TURNOUT TYPE "L"
30
Turnout Rails
Length in Feet and Inches
#
4. Rail lengths provide for 8" end posts at insu-
lated joints.
5. For additional turnout data see Plan Basic No.
910.
$
6. For bills of timber for various turnouts see Plan
Basic No. 912.
* Distances from stock rail joints to point of switch
are 13′11″ on curved stock rail; 12′3″ on
straight stock rail. These distances are reversed
for left hand turnouts. (See sketch on next page.)
Straight Rail
Curved Rail
Rail"G" Rail"H" Rail "J" Rail"K" Rail"M" Rail N°
15-01
15-0
19-6*
15-01
27-01
16-0'
30-01
19-6"
33-0'
16′-0″] 19-6"
39-0
16-0" 27-0"
33'-0"
33-0"
16-6″ 19-6'
16-6" 19-6"
27-019-6"
19-6" 39-0"
33-033-0"
19-6" 27-0"
19-630-0"
33-0"
39-0″
Straight
Stock Rail
Curved
Stock Rail
Plus
Stagger
30′-0″| 30′0″|_1-3;
30-0" 30-0" 1 - 17″
39-0″ 39-0″|
39-0″ 39-0″| 1-5
39-0 39-0"
39-0 39-0*|
60-0″ 60-0″ 1-10
60-0" 60′-0″| 1-10
60-0" 60-0" 1-9
60-0"| 60-0″| 2-9
60-0" 60-0"
60-0" 60-0″
60-0″| 60-0″|
"
Minus
Slogger
Lap"X"
Lap*Y*
10-107.1 9-94″
15-1087|10-3°
1-11 20-6 8-4″
23-07 11-3″
2-0211-44" 12÷6+1
1-1" 10-4" | 13-0″]
9'-5" 11-1"]
14-315-11"
14-2 20-3
21-615-7
3-81119-0117-7″ |
3-315-2″ 15-2-″
3-319-214-27
634

12-3-
Point of Switch-
13-11-
Frog Number
Length of
Switch Rail
Toe Length
·
td 201
Curved Stock Rail Type "L"
Switch Rail
Switch Rail
Straight Stock Rail Type L'
Lead
PS. to Pt
#
3-9* 47-6 32-9″
TABLE XXIV.-TURNOUT AND CROSSOVER DATA
Plus Stagger
·
•
140-11
151-117"
Rail K
Rail 'A'
Minus Stagger;
·
Rail D'
Closure Rails
Length in Feet and Inches
Rail "G"
"
Straight Rail
Curved Rail
Straight Rail
Curved Rail
5 11-0° 3-6 42-6
Rail A Rail "B" Rail"C" Rail"D" Rail E Rail "F" Rail G Rail H Rail J Rail K Rail"M" Rail"N"]
28-0″
19-6°
15-07
6 11-0
19-6
15-0*
7
16-6
27-0"
16-0
8
4-84"
16-6 5-1"
6-4
19-6" 8-9
19-613-5
25-01″ 16-0"
27-0" 19-75″
27-022-677
25-1130-0'
23-11139-0"
33-0"
9
62-1" | 27-0″ | 1340 |
68-0 27-0" 19-5"
72-3+ 30-0″ 19-5
78-9″ | 30′0″ |25-10″
91-101″ 39-0″ |23-101
16-6"
33-0"
16-0″
16-019-6"
10
16-6*
6-5"
39-0″
11
22:0"
7-0°
7-9
16-6" 27-0"
33-0"
33-0"
12 22-0
1422-0"
39-0*
96-8* 39-0 27-101
39-0" 27-11"
8-7" 107-07 30-0″ 30-0" 16-5" 16-627-0" 33-0"
9-5° | 126-41
15 30-0"
19-6" 30-0″.
33-0° 33′0″ 20-11" 30-0" 18-0" 39-0 33-0" 33-0"
|
9-5131-4" 33-0" 33-0" 25-11" 33-0" 19-11 39'0"
16
18
30-0*
39′0″ 30-0*
30-0″ |
|
11-0
39-0" 33-0" 27-11" 39-0" 33-0" 27-11" 39′-0″ 39-0"
|
| | |
30-011-0
30-0" 30-0" 30-0"
20
39-0" 32-11" 39-0" 39-0" 39-0" 32-11
Rail M
事​馨
​Rail *B*
"
Lead
Rail E
Rail*H*
Turnout Rails
Length in Feet and Inches
Rail "N
·
CLOSURE RAILS AND JOINT LOCATION FOR LEFT HAND TURNOUT TYPE L
Rail *C*
N
Rail 'F'
Rail J
|_ Lap "X"
16-6" 19-6"
19-6" 27-0"
33-0" 30-0"
33-0" 33'-0"
19-6° 27-0" 33-0″| 60-0″|| 60-0″
19-6" 30-0" 39-0"| 60-0" 60-0"
Lapy
30-0″ | 30-0″
30-0"|30-0"
39-039-0"
39-039-0"
39-0" 39-0"
39-0" 39-0″ 1-00
60-0" 60-0". 2-0
60-0" 60-0") 3-6
60-0″ 60-0" 1-11-27
60-0" 60-0" 2-11
60-0″ | 60-0*
3″
2-17
2-97
1-7
"Point of Frog
9-2 6-11
14-21-17
18-10 10-0°
4 24-10 9-11"
421 9:814-2
8-214-8"
10-912-91
12-71-7"
12-61-571
15-4814-371
2-0 17-416-3′
1-713-616-10
1-73" 17-6" 15-10"
635

American Railway Engineering
Association
LOCATION OF JOINTS FOR TURNOUTS
WITH STRAIGHT SPLIT SWITCHES
Frog Number
Length of
56
PLAN NO. 911-41 Adopted March, 1941.
Supersedes in part Plans No. 901 to 908 Incl.,
Revised March, 1934.
14
15
16
18
20
| 1'-0"
11-0"|
16-6"
16-6"
16-6"
7
8
9
10
||
12 22-0"
to ½" Pt.
普遍
​5'-1
6-45
6-5"
16-6"
22:0″ 7-0"
W
TABLE XXIV.-TURNOUT AND CROSSOVER DATA
W
Lead
PS.to "Pt.
#
11
3-6 42-6" 28'-0"
3'-9"
47-6" 32-9"
16-0"
4-83"
62-1" 27-0″ 13-10-2
19-6″
19-6″
27-0"
68-0" 27-0" | 19-5"
72-3" 30-0" 19-5"
78-933-0" 22-10"
91-1023-1039-0"
33-0"
27-0″
7-9
30-0"
39'-0"
22-0"
30-0*
30-0" 27-0"
96-8" 39-0" 12 7-10-2
8-7 107-0 30-0" 16-530-0" 22-63" 27-0" 27-0
126-439-0″ |33-0" |14-11-139-0" 18-0" 30-0"
9'-5
9-5131-4" 133-0" 33-0" 25-11" 39-0" 19-11 33-0" 30-0" 30-0"
30-0" | 1'-0"|40-11" 39-0" |27-11″ 33-0" 39-0" 33-01 27-11 33-0"| 39-0"
30-0″| |1′04″ 15 1-114″ 39-0″ 39′0″ |32-11″ | 39′0″ |39-0″ |32-11"
30-0*
KAJ
NOTES
1. Type "L" turnouts are for general use, except
for No. 5 and 6 turnouts.
2. Alternate Type "S" turnouts are for use with
No. 5 and 6 turnouts, and where limited space
requires location of joints ahead of switch nearer
to switch point.
3. Insulated joints shown thus
on plans
are located to conform to AAR Signal Section
Plans No. 1634-B, 1635-B and 1637-B, approved
in July, 1937, and if other rail lengths are used,
the stagger of the joints must not exceed 4′6″.
CLOSURE RAILS AND JOINT LOCATION FOR RIGHT HAND TURNOUT TYPE "S"
Closure Rails
Length in Feet and Inches
13-315-0"
13-519-6"
18-0123-0*
30-0" 16-7
30-0" 19-6
5/
Straight Rail
Curved Rail
Straight Rail
Curved Rail
•
Rail "A" Rail "B" | Rail"C" | Rail "D′| Rail "E′| Rail"F" | Rail"G″| Rail"H"] Rail "J' Rail"K" | Rail"M" | Rail"N"
30-0 25-11
29-11 33-0"
27-11 39-0"|
Turnout Rails
Length in Feet and Inches
#
#
4. Rail lengths provide for 38" end posts at insu-
lated joints.
5. For additional turnout data see Plan Basic No.
910.
16-0"
19-6*|
16-0″]
16-0″
16-0"| 19-6″
27-0°
6. For bills of timber for various turnouts see Plan
Basic No. 912.
33-0"
19-6″| 19-6′
33-0 27-0″.
* Distances from stock rail joints to point of switch
are 7′3″ on curved stock rail; 5′7″ on straight
stock rail. These distances are reversed for left
hand turnouts. (See sketch on next page.)
Straight
Stock Rail
Han
30-0" 30-0"
30-0"|30-0"|
39-0" 39-0"|
39-0″ 39-0″
39-0"|39-0"
39-039-0"
60-060-0"
60-0" 60-0" |
60-060-0"
60°0*160-0"
60-0" 60'-0"
19-6" 39-0"
3-5"
19-6" 33-0" 19-6″ 60-0" 160-0"| 3-4
30-0″| 30-0″ 19-6 19-6" 27-0" 39-0" 60-0" | 60-0"
3-4
Curved
Stock Rail
Plus
Stagger
1'-8"
1-72/"
1-63
1-0
3-0"
1173
[12]]
Minus
Stagger
Lap*x*
.
1119-218-17]
1'-3' 8-2
2-5″ 10-4"
Lap*y*
7-1"
9-2″
19-107 15-1"
2.
24-2211-10
✔
1-2 12-4
11-910-5-1
10-712-3"
15-013-7
1·0%" 13-4714-1131
19-10" 16-11
16-014-6
13-618-01
U
636


Frog Number
Length of
Switch Rail
Point of Switch-
7-3"
18
20
5-7″-
"
Toe Length
to ½ Pt.
3'-6"
3-9"
4-8"
#
T
Curved Stock Rail Type "S"
Switch Rail
TABLE XXIV.-TURNOUT AND CROSSOVER DATA
Switch Rail
Straight Stock Rail Type "S'
Lead
PS to Pt
Minus Stagger
42-628-0″
47-6" 132-9″
62-1" 27-0″ 13-102″
68-0" 27-0" 19-5″
72-330-0" | 19-5"
78-9" 133-0" 22-10"
91-1023-1039-0"
Rail K
Rail "A"
Rail "D"
Rail *G*
Closure Rails
Length in Feet and Inches
Straight Rail
Curved Rail
Rail"A" Rail "B" | Rail "C" | Rail "D" Rail *E'
S 11-0″]
6
1 1'-0"
16-0"
7
16-6"
13-3 15-0
13-5119-6"
16-0" 25-07″
30-0" 16-7
33′-0″ 16-6
19-6"
8 16-6″| 5-1
19-6″
9
16-6"
6-4
27:0"
10 16-6" 6-5"
33-0"
||
22-0"1 7'-0"
30-0" 25-113"
29-11 33-0"
30-019-6"
19-6"
12
22-0" 7-9½"
96-8" 39-0″ |27-10½)
14
22'-0"
22-627-0'
8-7½″ | 107-03″ 30-0″ | 16-530-0" |22′6″| 27-0″ |27-0″
15 30-0" 9:5"
17-5" 30-0"
39-0"
30-0"| 27′-0″ ·
30-0" 9'-5" 131-4" 33-0" 33-01
33-0° 25-11" 39-0" 19-1133-0 33-0" 39-0"
126-439-0" 33'-0" | 14-11" 39-0" 18-030-0"
16
30-0" 11-01" 140-112″ 39′-0″ |27-11″ 33-0"|39-0" 33-0" 27-11" 33-0" | 39-0″
30-0 11-02 151-11" 39-0″ 39-0" 32-11" 39-0"|39-032411
30-0" 30-0"
5/1
Plus Stagger
Rail M
Rail "F"
Rail "B"
Rail E
Rail *H*
Lead
CLOSURE RAILS AND JOINT LOCATION FOR LEFT HAND TURNOUT TYPE "S
C
Turnout Rails
Length in Feet and Inches
Rail "N"
#
15-0″
19-6"
16-0"
16-6"
16′-0″| 16-6″
27-0
Straight Rail
Curved Rail
Rail "G" | Rail"H" Rail "J" | Rail "K" || Rail"M" | Rail "N"
30-0*
19'-6" 1 9-6″
33-0" 19-6"
16-6″ 39-0"
16-6
"
Rail C
Rail "F*
19-6′ 16-6" 28-0"
Rail *J*
Lap"X" +
33-0" 19-6"] 60-0″ | 60-0″
28-0″ 39-0″| 60-0″ | 60-0″
Lapy"
30-0" 30-0"i
30-0" 30-0"
39-0" 39-0"|
93
39-0″ | 39-0″ 3-4
39-0" 39-0"
39-0" 39'-0" 3-22
60-0" 60'-0" 2-8
60-060-0"
60-0" 60'-0"
60-0" 60-0″ |
60'-0" 60-0"
2-76
2-71
78
F
87
GIB
5"
2'-1"
2:0
2-0
*Point of Frog
17-6119-9
9"
7-6 8-9
1-3" 8-8 10-10″ |
18-216-9"
22-013-6
12-6" 14'-0"
10-112-1
11-1113-11" |
13-415-3
19-216-7½"
21-218-7"
1 7-4″ | 16-2+
13-10% 19-8 +″
637
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
22.
23.
24.
25.
26.
27.
L
sin A =
cos A =
tan A =
cot A =
sec A =
TABLE XXV.-TRIGONOMETRIC FORMULAS
A
csc A =
H
=
a
b
с
a
b
ام
b
a
vers A 1
tan A =
с
b
с
a
General Formulas
covers A =
coexsec A =
exsec A = sec A
-
vers A =
C
B-
る
​1
In the right-angled triangle ABC, let AB = c, BC= a, CA = b.
O
sin A
COS A
cos A
cot A =
sin A
cos A
-
a
a
C
a
=
a
cos A = 2 cos² 1
A
2
K
1
sin A = 2 sin 1A cos
2
D
1 =
=
-
חח
E
C
-
b
vers B
ព
G
2
b
=
exsec B
b
=
covers B
sin 2A
1 + cos 2A
sin 2A
cos 2A
cos Asin A tan
A = V 1
=
21.
coexsec B
1 = 1 - 2 sin² 1
2
2
vers 2A
sin 2A
sin 2A
vers 2A
Let A = angle BAC =
AF = AB = AH = 1.
exsec A = sec A 1 = tan A tan
2
sin A = BC
A
cos A = AC
tan A = DF
A =
vers A = CF = BE
exsec A = BD
Area =
chord A = BF
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
1 ab
2
= 2 sin² 1A
2
vers A
cos A
tan 1 A
2
b =
C =
a = c sin A =
C =
a =
arc BF, and let radius
Then,
C =
=
a
cos B
cos² A = tan A cos A
A = cos² 1A
2
b = √ c²
c cos A =
b = c sin B =
and
If
a
b
sin A
cos A
a = c cos B = b cot B
-
coexsec A = BG
Va +
covers A = BK = LH
chord 2 A = BI = 2 BC
CF vers A
BC
sin A
c²
b2
csc A = AG

sec A = AD
cot A = HG
a²
sin² 1A
2
-
b tan A
b
sin B
a cot A
=
=
C = 90° = A + B
Then,
a tan B
vic
-
b) (c + b)
√(c − a) (c + a)
638
No.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
Given
A, B, a
A, a, b
TABLE XXV.-TRIGONOMETRIC FORMULAS
C, a, b
a, b, c
Sought
C, b, c
1 2
Area
B, C, c
Area
с
(A + B)
1 (A - B)
2
A, B
O
Area
A
Area
A
C = 180°
b =
Area =
sin B =
Area =
A =
C =
C =
H2H
sin
1
B =
2
COS
tan
C =
1
Area =
2
sin A
a
C = 180°
1
tan (A B)
2
Let s =
1
2
c =Va² b2
1 (A + B) = 90° -
2
-
B

b
(A + B)
1ab sin C
2
a
sin A
sin A =
A =
cos A =
(a + b) x
A =
1 A =
2
a
(A + B)
a
× sin (A + B) =
sin A
sin A
(A + B) +
1ab sin C
2
x b
G
× sin B
#1
Area = V S(S
2ab cos C
1c
× sin C
1ab sin C
2
a + b + C
2
(S
(A + B)
S(S
(s
COS
2 Vs(s
«
12 1
2
a
a + b
C
11 09
2
-
M
bc
(A
COS (A + B)
a2 sin B sin C
2 sin A
b
(A
b² + c² a2
2bc
}
s(s
www
Formula
b) (s
bc
a)
C
x tan
// (A - B)
-
B)
B)
-
b) (s c)
a)
a) (s
bc
-
C)
2
a
sin A
(A + B)
= (a - b) ×
b) (s
S
a) (s - b) (8 - c)
× sin C
c)
sin (A + B)
sin(A
-
B)
639
No. Squares.
HOM➡UW N∞ a o
I
2
3
8
9
II
12
13
Generen en een eug Ab±ˆˆ±**** wwwwwwwwww.BxNŏ…I☹NNU EXILAI
20
21
22
23
25
26
27
28
30
31
32
33
34
58
TABLE XXVI.—SQUARES, cubes, square roots,
CUBE ROOTS AND RECIPROCALS

∞OWN DAS
100
121
144
169
196
225
256
272 GEHAANDAA 225
676
729
784
841
900
1024
1089
1156
1225
1296
1369
1444
1521
1600
1681
1764
1849
1936
2025
2116
2209
2304
2401
2500
2601
2704
2809
2916
3025
3136
3249
3354
3481
Cubes.
18
27
64
125
216
343
512
729
1000
1331
1728
2197
2744
3375
4096
4913
5832
6859
8000
9261
10648
12167
13824
15625
17576
19683
21952
24389
27000
29791
32768
35937
39304
42875
46656
50653
54872
59319
64000
68921
74088
79507
85184
91125
97336
103823
110592
117649
125000
132651
140608
148877
157464
166375
175616
185193
195112
205379
Square
Roots.
1.0000000
.4142136
.7320508
2.0000000
2.2360680
.4494897
.6457513
.8284271
3.0000000
3.1622777
.3166248
.4641016
6055513
7416574
3.8729833
4.0000000
•
•
1231056
.2425407
.3588989
•
4.4721360
.5825757
.6904158
.7958315
.8989795
5.0000000
.0990195
1961524
.2915026
.3851648
•
5.4772256
•5677644
.6568542
.7445626
.8309519
5.9160798
6.0000000
.0827625
.1644140
.2449980
6.3245553
.4031242
.4807407
• 5574385
6332496
6.7082039
7823300
.8556546
.9282032
7.0000000
•
7.0710678
.1414284
2111026
•
2801099
.3484692
7.4161985
.4833148
5498344
.6157731
,6811457
Cube
Roots.
1.0000000
2599210
4422496
5874011
I.7099759
8171206
9129312
2.0000000
.0800837
·
•
·
•
2. 1544347
2239801
2894286
•
.3513347
.4101422
2.4662121
.5198421
5712816
6207414
.6684016
•
2.7144177
.7589243
.8020393
2.9240177
9624960
·
3.0000000
.0365889
.0723168
8438670
8844991
3.1072325
1413806
.1748021
.2075343
2396118
3.2710663
.3019272
• 3322218
3619754
.3912114
..
•
3.4199519
.4482172
.4760266
.5033981
.5303483
3.5568933
5830479
.6088261
•
.6342411
.6593057
3.6840314
7084298
.7325111
.7562858
7797631
3.8029525
.8258624
.8485011
.8708766
.8929965
Recipro-
cals.
1.000000000
500000000
.333333333
250000000
.200000000
166666667
.142857143
125000000
.IIIIIIIII
•
•
•
•
•
2500000
.058823529
5555556
2631579
.050000000
.047619048
100000000
10
090909091
11
083333333 12
.076923077
1428571
13
14
.066666667
5454545
3478261
1666667
040000000
.038461538
•
7037037
5714286
4482759
.033333333
1250000
0303030
.029411765
.028571429
7777778
No.
12345O 70 σ
1739130
1276600
0833333
0408163
I
.020000000
.019607843
9230769
8867925
8518519
.018181818
8
7857143
7543860
7241379
6949153
9
15
16
17
18
19
20
21
30
2258065 31
32
22
23
24
25
26
27
28
29
33
34
35
ვნ
7027027 37
6315789
38
5641026 39
♡~♡♡♡♡♡♡♡♡ ****:
.025000000
4390244
3809524
42
3255814 43
2727273
.022222222
44
45
46
40
41
49
50
CCCCCCCCCC
53
57
58
59
1
640
No. Squares.
85SSYKOAX8 ANNNNKO Too of
777
бо
62
63
64
65
66
67
68
69
70
72
73
74
75
76
77
78
79
80
81
82
83
84
15855 82akinana:
go
91
93
94
200
IOI
102
103
104
105
106
107
108
Iog
IIO
III
112
113
114
115
116
117
118
119
TABLE XXVI.—SQUARES, CUBES, SQUARE ROOTS,

3600
3721
3844
3969
4096
4225
4356
4489
4624
4761
4900
5041
5184
5329
5476
5625
5776
5929
6084
6241
6400
6561
6724
6889
7056
7225
7396
7569
7744
7921
8100
8281
8464
8649
8836
9025
9216
9409
9604
9801
10000
10201
10404
10609
10816
11025
11236
11449
11664
11881
12:00
12321
12544
12769
12996
13225
13456
13689
13924
14161
Cubes.
216000
226981
238328
250047
262144
274625
287496
300763
314432
328509
343000
357911
373248
389017
405224
421875
438976
456533
474552
493039
512000
531441
551368
571787
592704
614125
636056
658503
681472
704969
729000
753571
778688
804357
830584
857375
884736
912673
941192
970299
1000000
1030301
1061208
1092727
II24864
1157625
1191016
1225043
1259712
1295029
1331000
1367631
1404928
1442897
1481544
1520875
1560896
1601613
1643032
1685159
Square
Roots.
7.7459667
.8102497
.8740079
.9372539
8.0000000
8.0622577
.1240384
.1853528
.2462113
.3066239
8.3666003
.4261498
.4852814
5440037
6023253
8.6602540
•
.7177979
.7749644
.8317609
.8881944
8.9442719
9.0000000
.0553851
.1104336
.1651514
9.2195445
2736185
.3273791
•3808315
•4339811
9.4868330
• 5393920
.5916630
.6436508
.6953597
9.7467943
·
.7979590
.8488578
•
•
·
•
8994949
9498744
10.0000000
.0498756
.0995049
1488916
.1980390
10.2469508
.2956301
•3440804
•3923048
•4403065
10.4880885
.5356538
.5830052
.6301458
.6770783
10.7238053
.7703296
.8166538
.8627805
.9087121
Cube
Roots.
3.9148676
.9364972
9578915
.979057I
4.0000000
4.0207256
0412401
.0615480
.0816551
1015661
•
4.1212853
1408178
1601676
•
.1793392
1983364
4.2171633
2358236
•
.2543210
.2726586
.2908404
4.3088695
.3267487
.3444815
.3620707
.3795191
4.3968296
.4140049
.4310476
.4479602
.464745I
4.4814047
.4979414
.5143574
.5306549
.5468359
4.5629026
.5788570
5947009
.6104363
.6260650
·
4.6415888
6570095
.6723287
6875482
7026694
•
•
•
4.7176940
.7326235
.7474594
.7622032
.7768562
4.7914199
.8058955
.8202845
.834588I
8488076
4.8629442
.8769990
•
.8909732
.9048681
.9186847
Recipro-
cals.
.016666667 60
6393443 61
6129032 62
5873016 63
5625000
64
.015384615 65
5151515 66
4925373 67
4705882
4492754
.012500000
2345679
2195122
2048193
1904762
No.
.014285714
4084507 71
3888889 72
3698630 73
3513514 74
.013333333 75
3157895 76
2987013 77
2820513 78
2658228 79
8588*KU♪08 UNNSIDE
0416667
0309278
0204082
ΟΙΟΙΟΙΟ
68
010000000
.009900990
70
80
81
82
83
84
.011764706
1627907
1494253
1363636 88
1235955
89
.OIIIIIIII
go
0989011
91
0869565 92
0752688
0638298
.010526316
93
JD6585 Lääkäkohõa
94
95
97
98
99
100
ΙΟΙ
9803922 102
9708738
9615385
103
104
.009523810 105
106
9433962
9345794
107
9259259 108
9174312
109
.009090909
IIO
III
9009009
8928571 112
8849558 113
8771930 114
.008695652 115
8620690 116
8547009 117
8474576 118
8403361 119
641
No.
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
тбо
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
Squares.
14400
14641
14884
15129
15376
15625
15876
16129
16384
16641
16900
17161
17424
17689
17956
18225
18496
18769
19044
19321
19600
19S81
20164
20449
20736
21025
21316
21609
21904
22201
22500
22801
23104
23409
23716
24025
24336
24649
24964
25281
25600
25921
26244
26569
26896
27225
27556
27889
28224
28561
28900
29241
29584
29929
30276
30625
30976
31329
31684
32041
CUBE ROOTS AND RECIPROCALS

Cubes.
1728000
1771561
1815848
1860867
1906624
1953125
2000376
2048383
2007152
2146689
2197000
2248091
2299968
2352637
2406104
2460375
2515456
2571353
2628072
2685619
2744000
2803221
2863288
2924207
2985984
3048625
3112136
3176523
3241792
3307949
3375000
3442951
3511808
3581577
3652264
3723875
3796416
3869893
3944312
4019679
4096000
4173281
4251528
4330747
4410944
4492125
4574296
4657463
4741632
4826809
4913000
5000211
5088448
5177717
5268024
5359375
5451776
5545233
5639752
5735339
Square
Roots.
10.9544512
II.0000000
0453610
0905365
.1355287
II. 1803399
•
.2249722
.2694277
.3137085
.3578167
II.4017543
4455231
.4891253
5325626
.5758369
II.6189500
.6619038
7046999
•
7473401
.7898261
11.8321596
.8743421
.9163753
.9582607
12.0000000
12.0415946
0830460
.1243557
1655251
.2065556
•
12.2474487
2882057
3288280
.3693169
.4096736
12.4498996
•
.4899960
5299641
.5698051
.6095202
12.6491106
.6885775
.7279221
.7671453
.8062485
12.8452326
.8840987
.9228480
.9614814
13.0000000
13.0384048
0766968
1148770
1529464
1909060
•
•
•
13.2287566
.2664992
.3041347
3416641
.3790882
Cube
Roots.
4.9324242
9460874
.9596757
.9731898
.9866310
5.0000000
•
0132979
.0265257
.0396842
.0527743
•
5.0657970
.0787531
.0916434
1044687
.1172299
5.1299278
.1425632
1551367
1676493
1801015
•
•
5. 1924941
204S279
.2171034
.2293215
.2414828
5.2535879
2656374
.2776321
2895725
• 3014592
•
5.3132928
• 3250740
• 3368033
•3484812
.3601084
5.3716854
.3832126
3946907
4061202
.4175015
•
5.4288352
• 4401218
•4513618
4625556
•
4737037
5.4848066
•4958647
•5068784
•5178484
.5287748
•
5.5396583
•5504991
•5612978
•5720546
.5827702
5.5934447
.6040787
.6146724
.6252263
.6357408
Recipro-
cals.
120
121
008333333
8264463
8196721
8130081 123
8064516 124
122
.008000000
125
7936508 126
7874016 127
7812500 128
7751938 129
.007692308
7633588
7575758
7518797
7462687
.007407407
7352941
No.
.007142857
006896552
6849315
130
131
132
7299270 137
7246377 138
7194245 139
133
134
135
136
140
7092199 141
7042254 142
6993007 143
6944444 144
.006250000
6211180
6172840
6134969
6097561
.006060606
145
146
6802721 147
6756757 148
6711409 149
150
151
006666667
6622517
6578947 152
6535948 153
6493506 154
155
6410256 156
6369427 157
6329114 158
6289308
.006451613
159
160
161
162
163
164
165
166
6024096
5988024 167
5952381 168
5917160
169
170
171
172
.005882353
5847953
5813953
5780347
5747126 174
.005714286 175
5681818 176
173
5649718 177
5617978 178
55S6592 179
642
No. Squares. Cubes.
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
TABLE XXVI.-SQUARES, CUBES, SQUARE ROOTS,

32400
32761
33124
33489
33856
34225
34596
34969
35344
35721
36100
36481
36864
37249
37636
38025
38416
38809
39204
39601
40000
40401
40804
41209
41616
42025
42436
42849
43264
43681
44100
44521
44944
45369
45796
46225
46656
47089
47524
47961
48400
48841
49284
49729
50176
50625
51076
51529
51984
5244I
52900
53361
53824
54289
54756
55225
55696
56169
56644
57121
5832000
592974!
6028568
6128487
6229504
6331625
6434856
6539203
6644672
6751269
6859000
6967871
7077888
7189057
7301394
7414875
7529536
7645373
7762392
7880599
8000000
8120601
8242408
8365427
84S9664
8615125
8741816
8869743
8998912
9129329
9261000
9393931
9528128
9663597
9800344
9938375
10077696
10218313
10360232
10503459
10648000
10793861
10941048
11089567
11239424
11390625
11543176
11697083
11852352
12008989
12167000
12326391
12487168
12649337
12812904
12977875
13144256
13312053
13481272
13651919
Square
Roots.
13.4164079
.4536240
4907376
•
5277493
5646600
13.6014705
.6381817
.6747943
.7113092
.7477271
•
13.7840488
.8202750
.8564065
.8924440
.9283883
13.9642400
14.0000000
0356688
.0712473
.1067360
•
14.1421356
1774469
2126704
2478068
.2828569
14.3178211
•
•
• 3527001
.3874946
4222051
.4568323
•
14.4913767
.5258390
.5602198
5945195
.6287388
14.6628783
.6969385
.7309199
7648231
.7986486
14.8323970
8660687
.8996644
.9331845
.9666295
15.0000000
.0332964
.0665192
.0996689
1327460
•
15.1657509
1986842
2315462
.2643375
.2970585
15.3297097
.3622915
.3948043
.4272486
.4596248
Cube
Roots.
5.6462162
6566528
6670511
6774114
6877340
5.69S0192
7082675
.7184791
.7286543
.7387936
•
•
·
5.7488971
7589652
.7689982
7789966
7889604
5.7988900
8087857
.8186479
.8284767
.8382725
•
•
5.8480355
.8577660
.8674643
.8771307
.8867653
5.8963685
9059406
.9154817
.9249921
.9344721
•
5.9439220
.9533418
9627320
9720926
9814240
5.9907264
6.0000000
•
•
·
0092450
.0184617
.0276502
•
6.0368107
.0459435
.0550489
0641270
•
.0731779
6.0822020
.0911994
.1001702
.1091147
1180332
•
6. 1269257
.1357924
1446337
.1534495
.1622401
6.1710058
1797466
1884628
.1971544
.2058218
Recipro-
cals.
.005555556
5524862
5494505 182
5464481 183
5434783 184
.005405405 185
5376344 186
5347594 187
5319149 188
5291005 189
7
.005263158
190
5235602
191
5208333 192
5181347 193
5154639
194
005128205
No.
180
181
.005000000
4975124
4950495
4926108
4901961
.004878049
4854369
4830918
4807692
4784689
197
5102041
5076142
5050505 198
5025126
199
195
196
.004545455
200
201
202
203
204
205
206
207
208
209
210
.004761905
4739336
211
212
4716981
4694836
213
4672897 214
.004651163 215
4629630
216
4608295 217
4587156 218
4566210 219
220
4524887 221
4504505 222
4484305
4464286
223
224
225
226
.004444444
4424779
440528€
4385965
4366812 229
227
228
.004347826230
231
4329004
4310345 232
4291845 233
4273504
234
235
.004255319
4237288 236
4219409 237
4201681 238
4184100 239
643
No. Squares.
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
299
29
292
293
294
295
296
297
298
299
57600
58081
58564
59049
59536
60025
60516
61009
61504
62001
62500
63001
63504
64009
64516
65025
65536
66049
66564
67081
67600
68121
68644
69169
69696
70225
70756
71289
71824
72361
72900
7344I
73984
74529
75076
75625
76176
76729
77284
77841
78400
78961
79524
80089
80656
81225
81796
82369
82944
83521
84100
84681
85264
85849
86436
87025
87616
88209
88804
89401
CUBE ROOTS AND RECIPROCALS

Cubes.
13824000
13997521
14172488
14348907
14526784
14706125
14886936
15069223
15252992
15438249
15625000
15813251
16003008
16194277
16387064
16581375
16777216
16974593
17173512
17373979
17576000
17779581
17984728
18191447
18399744
18609625
18821096
19034163
19248832
19465109
19683000
19902511
20123648
20346417
20570824
20796875
21024576
21253933
21484952
21717639
21952000
22188041
22425768
22665187
22906304
23149125
23393656
23639903
23887872
24137569
24389000
24642171
248970S8
25153757
25412184
25672375
25934336
26198073
26463592
26730899
Square
Roots.
15.4919334
.5241747
5563492
5584573
6204994
15.6524758
6843871
7162336
7480157
.7797338
•
•
•
•
15.8113883
8429795
8745079
.9059737
• 9373775
15.9687194
16.0000000
•
•
.0312195
·0623784
.0934769
16. 1245155
1554944
1864141
2172747
2480768
16.2788206
·
•
•
•
• 3707055
• 4012195
16.4316767
4620776
• 4924225
• 5227116
•
5529454
16.5831240
.6132477
.6433170
.6733320
• 7032931
•
3095064
3401346
16.7332005
7630546
7928556
8226038
8522995
16.8819430
·
•
.9115345
.9410743
.9705627
17.0000000
17.0293864
0587221
.0880075
1172428
1464282
17.1755640
2046505
.2336879
•
2626765
.2916165
·
Cube
Roots.
6.2144650
2230843
.2316797
.2402515
.2487998
6.2573248
.2658266
.2743054
.2827613
.2911946
6.2996053
.3079935
.3163596
.3247035
3330256
6.3413257
3496042
• 3578611
.3660968
• 3743111
•
6.3825043
.3906765
.3988279
.4069585
.4150687
6.4231583
•4312276
• 4392767
.4473057
.4553148
6.4633041
.4712736
.4792236
.4871541
•4950653
6.5029572
.5108300
.5186839
.5265189
5343351
•
6.5421326
5499116
.5576722
.5654144
5731385
6.5808443
.5885323
5962023
.6038545
.6114890
•
•
6.6191060
6267054
.6342874
.6418522
.6493998
6.6569302
.6644437
.6719403
.6794200
.6868831
Recipro-
cals.
.004166667
4149378
240
241
4132231 242
4115226
243
4098361 244
.004081633
245
4065041 246
4048583 247
4032258 248
4016064 249
004000000 250
3984064 251
3968254 252
3952569
3937008
•003921569
253
254
255
3906250 256
3991051 257
3875969 258
3861004
259
.003846154
260
3831418 261
3816794 262
3S02281
263
3787879 264
003773585 265
3759398 266
3745318 267
3731343 268
3717472
269
.003703704 270
271
272
273
274
3690037
3676471
3653004
3649635
No.
.003636364
3623188
3610108
277
3597122 278
3584229
279
.003571429
3558719
3546099
280
281
282
3533569 283
3521127 284
.003508772
275
276
.003448276
285
286
287
3496503
3484321
3472222 288
3460208 289
290
3436426 291
3424658 292
3412969
293
3401361 294
.003389831 295
3378378 296
3367003 297
3355705 298
3344482 299
644
No. Squares.
300
301
302
303
304
305
306
307
308
309
310
3II
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
337
338
339
340
341
342
343
344
345
346
347
348
349
TABLE XXVI.—SQUARES, CUBES, SQUARE ROOTS,

90000
90601
357
358
359
91204
91809
92416
93025
93636
94249
94864
95481
96100
96721
97344
97969
98596
99225
99856
100489
101124
101761
102400
103041
103684
333 110889
334
111556
335
336
104329
104976
105625
106276
106929
107584
108241
108900
109561
II0224
112225
112896
113569
114244
114921
115600
116281
116964
117649
118336
119025
119716
120409
121104
121801
350
351
352
353
354
125316
355
126025
356 126736
122500
123201
123904
124609
127449
128164
128881
Cubes.
27000000
27270901
27543608
27818127
28094464
28372625
28652616
28934443
29218112
29503629
29791000
30080231
30371328
30664297
30959144
31255875
31554496
31855013
32157432
32461759
32768000
33076161
33386248
33698267
34012224
34328125
34645976
34965783
35287552
35611289
35937000
36264691
36594368
36926037
37259704
37595375
37933056
38272753
38614472
38958219
39304000
39651821
40001688
40353607
40707584
41063625
41421736
41781923
42144192
42508549
4287.5000
4324355I
43614208
43986977
44361864
44738875
45118016
45499293
45882712
46268279
Square
Roots.
17.3205081
3493516
.3781472
.4068952
.4355958
17.4642492
.4928557
•
17.6068169
.6351921
.6635217
.6918060
.720045I
17.7482393
.7763888
.8044938
.8325545
.8605711
•
17.8885438
.9164729
.9443584
9722008
18.0000000
18.0277564
·
.0554701
.0831413
•
·
•
5214155
5499288
5783958
18.1659021
.1934054
2208672
2482876
2756669
18.3030052
• 3303028
• 3575598
3847763
.4119526
•
•
•
18.4390889
4661853
.4932420
.5202592
.5472370
18.5741756
•
1107703
1383571
6010752
.6279360
.654758I
.6815417
18.7082869
.7349940
7616630
•
.7882942
8148877
•
18.8414437
8679623
.8944436
9208879
.9472953
Cube
Roots.
6.6943295
.7017593
.7091729
.7165700
.7239508
6.7313155
.7386641
7459967
.7533134
7606143
•
•
6.7678995
7751690
.7824229
7896613
•7968844
6.8040921
•
•
.8112847
.8184620
.8256242
.8327714
6.8399037
.8470213
.8541240
.8612120
.8682855
6.8753443
.8823888
8894188
.8964345
⚫9034359
6.9104232
.9173964
.9243556
9313008
•9382321
6.9451496
.9520533
•9589434
.9658198
• 9726826
•
6.9795321
.9863681
•9931906
7.0000000
.0067962
7.0135791
.0203490
.0271058
.0338497
0405806
7.0472987
.0540041
0606967
0673767
•
.0740440
7.0806988
.0873411
.0939709
1005885
.1071937
Recipro-
cals.
.003333333
300
3322259 301
3311258 302
3300330 303
3289474 304
003278689
•
305
3267974 306
.003225806
310
3215434 311
3205128 312
3194888 313
3184713 314
.003174603 315
316
3257329 307
3246753 308
3236246
309
•
No.
.003125000
3164557
3154574 317
3144654 318
3134796 319
320
3115265 321
3105590 322
3095975 323
3086420 324
003076923
3067485
3058104 327
3048780 328
3039514 329
.003030303 330
3021148 331
3012048 332
3003003 333
2994012
334
.002985075
2976190
325
326
2906977
002898551
2967359 337
2958580 338
2949853
339
.002857143
335
336
002941176
2932551
2923977
2915452 343
344
345
2890173 346
2881844 347
2873563 348
2865330 349
340
341
342
350
2849003 351
2840909
2832861
352
353
2824859 354
.002816901 355
356
2808989
2801120 357
2793296 358
2785515 359
645
No. Squares.
ვნი
361
362
363
364
365
366
367
368
36g
370
371
372
373
374
375
376
377
378
379
333
389
397
398
399
380
381
382
145924
383 146689
384 147456
129600
130321
131044
131769
132496
407
408
409
133225
133956
134689
385
148225
386 148996
387 149769
388
410
411
412
413
414
135424
136161
415
416
136900
137641
138384
417
418
139129
139876
439
140625
141376
390
152100
391 152881
153664
142129
142884
143641
392
393
154449
394
155236
395 156025
396 156816
144400
145161
150544
151321
400
401
402
403
404
405
406 164836
165649
166464
167281
157609
158404
159201
160000
160801
161604
162409
163216
164025
168100
168921
169744
170569
171396
172225
173056
173889
174724
175561
CUBE ROOTS AND RECIPROCALS

Cubes.
46656000
47045881
47437928
47832147
48228544
48627125
49027896
49430863
49836032
50243409
50653000
51064811
51478848
51895117
52313624
52734375
53157376
53582633
54010152
54439939
54872000
55306341
55742968
56181887
56623104
57066625
57512456
57960603
58411072
58863869
59319000
59776471
60236288
60698457
61162984
61629875
62099136
62570773
63044792
63521199
64000000
64481201
64964808
65450827
65939264
66430125
66923416
67419143
67917312
68417929
68921000
69426531
69934528
70444997
70957944
71473375
71991296
72511713
73034632
73560059
Square
Roots.
18.9736660
19.0000000
0262976
.0525589
.0787840
•
19. 1049732
1311265
•
.1572441
1833261
.2093727
•
•
19.2353841
.2613603
.2873015
.3132079
• 3390796
19.3649167
•3907194
4164878
•
4422221
.4679223
•
19.4935887
.5192213
•5448203
.5703858
• 5959179
19.6214169
.6468827
.6723156
.6977156
.7230829
19.7484177
.7737199
•7989899
.8242276
.8494332
19.8746069
.8997487
.9248588
9499373
•9749844
•
20.0000000
.0249844
0499377
.0748599
•
.0997512
20. 1246118
1494417
.1742410
.1990099
.2237484
•
20.2484567
.2731349
• 2977831
.3224014
• 3469899
20.3715488
• 3960781
.4205779
• 4450483
•4694895
Cube
Roots.
7.1137866
1203674
.1269360
1334925
1400370
7.1465695
.1530901
1595988
1660957
.1725809
•
·
•
7.1790544
.1855162
1919663
.1984050
2048322
7.2112479
.2176522
•
•
.2240450
.2304268
2367972
•
7.2431565
.2495045
2558415
.2621675
2684824
7.2747864
.2810794
2873617
2936339
.2998936
•
7.3061436
3123828
.3186114
3248295
·3310369
7.3372339
•3434205
3495966
3557624
.3619178
·
•
7.3680630
.3741979
3803227
3864373
•3925418
7.3986363
4047206
•
•
•
.4107950
.4168595
.4229142
7.4289589
•4349938
.4410189
.4470342
.4530399
7.4590359
.4650223
.4709991
.4769664
.4829242
Recipro-
cals.
.002777778
2770083
2762431
2754S21
2747253
.002739726
2732240
2724796
2717391
2710027
•
No.
.002702703 370
2695418 371
2688172 372
2680965 373
2673797
.002666667
374
360
361
362
363
364
365
366
367
002531646
368
369
2659574
2652520 377
2645503 378
2638522 379
.002631579 380
2624672
381
2617801
382
2610966 383
2604167 384
.002597403
385
2590674
386
2583979 387
2577320 388
2570694 389
.002564103 390
2557545 391
.002500000
375
376
2551020 392
2544529 393
2538071 394
395
396
.002439024
2433090
2427184
2421308
2525253
2518892 397
2512563 398
2506266 399
400
2493766 401
2487562 402
2481390 403
2475248
.002469136
2463054
404
405
406
2457002 407
2450980 408
2444988 409
410
411
412
413
2415459 414
415
.002409639
2403846 416
2398082
417
2392344
418
2356635 419
646
No. Squares.
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
44I
442
443
444
445
446
447
448
449
450
451
452
457
458
459
4888888
TABLE XXVI.-SQUARES, CUBES, SQUARE ROOTS,

469
176400
177241
178084
178929
179776
180625
181476
477
478
479
182329
183184
184041
184900
185761
186624
187489
188356
189225
190096
190969
191844
192721
193600
194481
195364
196249
453
454
455
207025
456 207936
208849
197136
198025
198916
199809
200704
201601
202500
203401
204304
205209
206115
460
461
462
463
464 215296
465
466
467
209764
210681
211600
212521
213444
214369
216225
217156
218089
219024
219961
470
471
472 222784
473
474 224676
475 225625
476 226576
Cubes.
220900
221841
74088000
74618461
75151448
75686967
76225024
76765625
77308776
77854483
78402752
78953589
79507000
80062991
80621568
81182737
81746504
82312875
82881856
83453453
84027672
84604519
85184000
85766121
86350888
86938307
87528384
88121125
88716536
89314623
89915392
90518849
91125000
91733851
92345408
92959677
93576664
94196375
94818816
95443993
96071912
96702579
97336000
97972181
98611128
99252847
99897344
100544625
101194696
101847563
103823000
104487111
105154048
223729 105823817
102503232
103161709
106496424
107171875
107850176
227529 108531333
228484
229441
109215352
109902239
Square
Roots.
20.4939015
.5182845
.5426386
.5669638
5912603
20.6155281
.6397674
.6639783
.6881609
.7123152
►
20.7364414
.7605395
.7846097
.8086520
.8326667
20.8566536
.8806130
9045450
.9284495
•9523268
•
20.9761770
21.0000000
.0237960
.0475652
.0713075
21.0950231
.1187121
.1423745
.1660105
.1896201
21.2132034
.2367606
.2602916
.2837967
• 3072758
21.3307290
3541565
•3775583
4009346
.4242853
·
21.4476106
4709106
.4941853
•5174348
5406592
21.5638587
.5870331
.6101828
•
•
.6333077
.6564078
21.6794834
.7025344
.7255610
7485632
.7715411
21.7944947
8174242
8403297
.8632111
8860686
Cube
Roots.
7.4888724
.4948113
5007406
5066607
•
5125715
7.5184730
5243652
5302482
•
·
•
5361221
•5419867
•
7.5478423
•5536888
5595263
5653548
• 5711743
7.5769849
.5827865
5885793
•
5943633
.6001385
·
7.6059049
.6116626
.6174116
.6231519
.6288837
7.6346067
.6403213
.6460272
.6517247
.6574138
7.6630943
.6687665
.6744303
.6800857
.6857328
7.6913717
6970023
⚫7026246
7082388
.7138448
•
7.7194426
• 7250325
• 7306141
.7361877
.7417532
7.7473109
.7528606
7584023
7639361
.7694620
7.7749801
7804904
.7859928
7914875
7969745
7.8024538
.8079254
.8133892
.8188456
.8242942
•
•
Recipro-
cals.
No.
.002380952
420
421
2375297
2369668 422
2364066
423
2358491
424
.002352941 425
2347418 426
2341920
2336449
2331002 429
427
428
430
431
.002325581
2320186
2314815
2309469 433
2304147 434
432
.002298851
2293578
2288330
2283105
2277904 439
435
436
437
438
.002272727
2267574
2262443 442
440
441
2257336 443
444
2252252
.002247191
445
2242152
446
2237136 447
2232143 448
2227171 449
.002222222
450
2217295 45I
2212389 452
2207506 453
2202643 454
.002197802 455
2192982 456
2188184 457
2183406 458
2178649 459
.002173913
460
2169197
461
2164502
462
2159827 463
464
2155172
.002150538 465
2145923
466
2141328 467
2136752 468
2132196 469
.002127660
470
2123142
471
2118644
472
2114165 473
474
2109705
.002105263 475
2100840
476
2096436 477
2092050 478
2087683 479
647
No. Squares.
482
483
484
485
86
480
230400
110592000
481 231361 111284641
111980168
112678587
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
232324
233289
234256
515
516
235225
236196
237169
238144
239121
240100
241081
242064
243049
244036
245025
246016
247009
248004
249001
250000
251001
252004
253009
254016
255025
256036
265225
266256
267289
517
518 268324
519
269361
520
521
522
523
524
274576
525 275625
526 276676
270400
271441
272484
CUBE ROOTS AND RECIPROCALS

Cubes.
527
277729
528 278784
529 279841
260100
132651000
261121 133432831
512
262144 134217728
513
263169
135005697
514 264196 135796744
280900
281961
113379904
114084125
114791256
257949
25S064
259081 131872229
115501303
116214272
116930169
287296
288369
537
538 289444
539
290521
117649000
118370771
119095488
119823157
120553784
121287375
122023936
122763473
123505992
124251499
125000000
125751501
126506008
127263527
128024064
128787625
129554216
130323843
131096512
136590875
137388096
138188413
273529 143055667
143877824
138991832
139798359
140608000
141420761
142236648
144703125
145531576
146363183
147197952
148035889
530
531
532 283024
533
284089
534 285156
535 286225 153130375
536
153990656
154854153
155720872
156590819
148877000
149721291
150568768
151419437
152273304
Square
Roots.
21.9089023
.9317122
•9544984
.9772610
22.0000000
22.0227155
.0454077
.0630765
.0907220
.1133444
22.1359436
.1585198
1810730
.2036033
2261108
22.2485955
2710575
2934968
·
• 3159136
.3383079
•
22.3606798
.3830293
.4053565
.4270615
⚫4499443
22.4722051
•4944438
5166605
.5388553
.5610283
•
22.5831796
.6053091
.6274170
.6495033
.6715681
22.6936114
.7156334
7376340
7596134
.7815715
•
•
22.8035085
.8254244
.8473193
.8691933
.8910463
22.9128785
•9346899
•9564806
9782506
23.0000000
•
•
23.0217289
.0434372
.0651252
.0867928
1084400
23.1300670
.1516738
1732605
1948270
.2163735
•
•
Cube
Roots,
7.8297353
.8351688
.8405949
.8460134
.8514244
7.8568281
.8622242
.8676130
.8729944
.8783684
7.8837352
.8890946
.8944468
8997917
9051294
•
7.9104599
.9157832
.9210994
.9264085
9317104
•
7.9370053
9422931
.9475739
.9528477
.9581144
•
7.9633743
.9686271
•9738731
.9791122
•9843444
7.9895697
9947886
8.0000000
•
.0052049
.0104032
8.0155946
.0207794
.0259574
.0311287
.0362935
8.0414515
.0466030
.0517479
.0568862
0620180
8.0671432
.0722620
.0773743
0824800
.0875794
8.0926723
.0977589
1028390
1079128
1129803
8.1180414
.1230962
1281447
•
.1331870
1382230
•
Recipro-
cals.
.002083333
2079002
2074689
2070393
2066116
.002061856
2057613
2053388
2049180
2044990
.002040816
490
2036660
491
2032520 492
2028398 493
2024291
494
.002020202
.001980198
No.
480
481
482
483
484
2016129
2012072 497
2008032 498
2004008
499
485
486
487
488
489
.002000000
500
1996008
501
1992032 502
1988072 503
1984127 504
1945525
.001941748
495
496
1976285
1972387 507
1968504 508
1964637 509
.001923077
.001960784
510
1956947
511
1953125
512
1949318 513
514
001904762
505
506
515
1937984 516
1934236 517
1930502 518
1926782 519
520
1919386 521
1915709 522
1912046 523
1908397 524
525
1901141 526
1897533 527
1893939 528
1890359 529
.001886792 530
1883239 531
1879699 532
1876173
533
1872659 534
.001869159 535
1865672 536
1862197 537
1858736 538
1855288
539
648
No. Squares.
540
54I
542
543
544
545
546
547
548
549
550
551
552
553
554
ANA BESOJOO 588 RENANI
559
565
566
555 308025
556
557
309136
558
567
569
570
571
TABLE XXVI.-SQUARES, CUBES, SQUARE ROOTS,

291600
292681
293764
294849
295936
573
297025
298116
574
299209
300304
301401
302500
303601
587
588
304704
305809
306916
560
56x
313600
175616000
314721 176558481
562 315844
563 316969
177504328
178453547
564 318096 179406144
310249
311364
312481
319225
320356
321489
332929
334084
335241
580
581
336400
337561
582 338724
583 339889
584
341056
585
586
Cubes.
324900 185193000
326041
572 327184 187149248
186169411
157464000
158340421
159220088
160103007
160989184
161878625
162771336
342225
343396
344569
163667323
164566592
165469149
345744
589 346921
166375000
167284151
168196608
328329 188132517
329476 189119224
575 330625 190109375
576
577
331776 191102976
578
579
590 348100
591 349281
592 350464
593 351649
594 352836
595
596
354025
355216
597
356409
598 357604
599
358801
169112377
170031464
322624 183250432
323761 184220009
170953875
171879616
172808693
173741112
174676879
180362125
181321496
182284263
192100033
193100552
194104539
195112000
196122941
197137368
198155287
199176704
200201625
201230056
202262003
203297472
204336469
205379000
206425071
207474688
208527857
209584584
210644875
211708736
212776173
213847192
214921799
Square
Roots.
23.2379001
2594067
.2808935
3023604
•3238076
23.3452351
3666429
3880311
.4093998
.4307490
•
23.4520788
⚫4733892
⚫4946802
.5159520
•5372046
23.5584380
5796522
.6008474
.6220236
.6431808
•
·
23.6643191
.6854386
7065392
7276210
7486842
23.7697286
•
•
• 7907545
.8117618
.8327506
.8537209
23.8746728
.8956063
.9165215
9374184
.9582971
23.9791576
24.0000000
.0208243
•0416306
.0624188
24.0831891
1039416
.1246762
1453929
1660919
24.1867732
2074369
.2280829
•
.2487113
•2693222
24.2899156
3104916
• 3310501
• 3515913
3721152
24.3926218
•
•
4131112
•4335834
•4540385
.4744765
•
•
•
Cube
Roots.
8. 1432529
1482765
1532939
1583051
1633102
8.1683092
•
•
1733020
.1782888
.1832695
.1882441
•
8.1932127
1981753
.2031319
2080825
•
2130271
8.2179657
.2228985
2278254
2327463
.2376614
•
·
•
8.2425706
.2474740
.2523715
.2572633
.2621492
8.2670294
•
.2719039
.2767726
.2816355
.2864928
8.2913444
2961903
•
.3010304
3058651
3106941
8.3155175
.3203353
.3251475
.3299542
.3347553
•
•
8.3395509
.3443410
.3491256
.3539047
3586784
8.3634466
3682095
.3729668
•
•
3777188
.3824653
•
8.3872065
.3919423
3966729
.4013981
4061180
8.4108326
•
.4155419
.4202460
.4249448
.4296383
Recipro-
cals.
543
.001851852
1848429
1845018
1841621
1838235 544
.001834862
545
1831502 546
1828154 547
1824818 548
1821494 549
.001818182 550
1814882 551
1811594 552
1808318 553
1805054
.001801802
554
.001754386
No.
1798561
1795332 557
1792115 558
1788909
559
.001785714
560
1782531 561
1779359 562
1776199 563
564
1773050
.001769912 565
1766784
566
1763668 567
1760563 568
1757469 569
.001724138
1721170
1718213
1715266
1712329
.001709402
1706485
1703578
1700680
1697793
540
54I
542
.001694915
555
556
570
1751313 571
1748252 572
1745201 573
1742160 574
.001739130 575
1736111 576
1733102 577
1730104
578
1727116 579
.001680672
1677852
580
8 0 0 0 0 0 0 0 0
58x
583
584
585
586
587
589
590
1692047 591
1689189 592
1686341 593
1683502
594
595
596
1675042
597
1672241 598
1669449 599
649
No. Squares.
600
601
602
603
604
605
606
607
608
бод
610
ότι
612
613
614
615
616
617
618
619
627
628
629
360000
361201
362404
363609
364816
366025
367236
368449
369664
370881
372100
373321
374544
375769
376996
378225
379456
380689
620
384400
621 385641
622
386884
388129
623
624 389376
625 390625
391876
626
381924
383161
$59959
630 396900
631 398161
632
399424
633 400689
634 401956
635 403225
404496
405769
636
637
638 407044
639 408321
640
409600
410881
641
642 412164
643
644
413449
414736
416025
645
646 417316
647 418609
648
419904
649 421201
650 422500
651
652
423801
425104
653 426409
654 427716
655 429025
656
657
658
430336
431649
432964
43428I
CUBE ROOTS AND RECIPROCALS

Cubes.
216000000
217081801
218167208
219256227
220348864
221445125
222545016
223648543
224755712
225866529
226981000
228099131
229220928
230346397
393129
394384 247673152
395641 248858189
231475544
232608375
233744896
234885113
236029032
237176659
238328000
239483061
240641848
241804367
242970624
244140625
245314376
246491883
250047000
251239591
252435968
253636137
254840104
256047875
257259456
258474853
259694072
260917119
262144000
263374721
264609288
265847707
267089984
268336125
269586136
270840023
272097792
273359449
274625000
275894451
277167808
278445077
279726264
281011375
282300416
283593393
284890312
286191179
Square
Roots.
24.4948974
5153013
.5356883
5560583
5764115
24.5967478
.6170673
.6373700
.6576560
.6779254
•
24.6981781
7184142
7386338
7588368
.7790234
24. 7991935
.8193473
.8394847
.8596058
.8797106
24.8997992
.9198716
9399278
.9599679
.9799920
25.0000000
.0199920
.0399681
.0599282
.0798724
25.0998008
.1197134
.1396102
.1594913
1793566
25. 1992063
.2190404
2388589
.2586619
2784493
25.2982213
.3179778
.3377189
• 3574447
• 3771551
25.3968502
.4165301
•4361947
•4558441
.4754784
25.4950976
.5147016
• 5342907
5538647
•
• 5734237
25.5929678
.6124969
.6320112
.6515107
.6709953
Cube
Roots.
8.4343267
4390098
.4436877
.4483605
.4530281
8.4576906
.4623479
4670001
•
.4716471
.4762892
8.4809261
.4855579
•4901848
.4948065
4994233
8.5040350
5086417
.5132435
.5178403
5224321
•
•
8.5270189
5316009
.5361780
5407501
.5453173
8.5498797
.5544372
5589899
5635377
•5680807
•
8.5726189
•5771523
5816809
5862047
5907238
8.5952380
•
5997476
.6042525
.6087526
.6132480
•
8.6177388
.6222248
.6267063
.6311830
.6356551
8.6401226
•6445855
.6490437
.6534974
.6579465
8.6623911
•6668310
.6712665
.6756974
.6801237
8.6845456
.6889630
.6933759
.6977843
.7021882
Recipro-
cals.
600
601
.001666667
1663894
1661130 602
1658375 603
1655629 604
.001652893 605
1650165 606
1647446 607
1644737 608
1642036 бод
.01639344 610
1636661 611
1633987 612
1631321 613
1628664
614
.001626016
No.
.001612903
1610306
1607717
1605136
1602564
.001600000
617
1623377
1620746
1618123 618
1615509
619
615
616
.001587302
1579779
1577287
620
621
622
1597444
627
1594896
1592357 628
1589825
629
623
624
625
626
630
1584786 631
1582278
632
633
634
.001574803
1572327
1569859 637
1567398 638
1564945 639
640
PASAN JOS88
.001562500
1560062
1557632
1555210
1552795
.001550388 645
1547988 646
1545595 647
1543210 648
644
1540832 649
.001538462
650
651
1536098
1533742 652
1531394 653
1529052 654
.001526718
655
1524390 656
1522070
657
658
1519757
1517451
66
TENGA
789
659
650
No.
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
9999
677
678
78
681
682
683
684
685
686
687
688
68g
697
698
699
700
701
702
703
704
705
706
TABLE XXVI.-SQUARES, CUBES, SQUARE ROOTS,

Squares. Cubes.
458329
459684
679 461041
707
708
70g
435600
436921
438244
439569
440896
680 462400
463761
465124
466489
467856
717
718
719
442225
443556
444889
446224
447561
448900
450241
451584
452929
454276
455625
456976
690
691
692
693
694
695 483025
484416
696
469225
470596
471969
473344
474721
490000
491401
492804
494209
495616
497025
498436
287496000
288804781
290117528
291434247
499849
501264
502681
710
504100
711
505521
712 506944
713 508369
714 509796
715
511225
716 512656
514089
515524
516961
292754944
294079625
295408296
296740963
298077632
299418309
300763000
302111711
303464448
304821217
306182024
476100 328509000
477481
478864
480249
481636
307546875
308915776
310288733
311665752
313046839
314432000
315821241
317214568
318611987
320013504
335702375
485809
337153536
338608873
487204 340068392
488601 341532099
321419125
322828856
324242703
325660672
327082769
329939371
331373888
332812557
334255384
343000000
344472101
345948408
347428927
348913664
350402625
351895816
353393243
354894912
356400829
357911000
359425431
360944128
362467097
363994344
365525875
367061696
368601813
370146232
371694959
Square
Roots.
25.6904652
.7099203
7293607
7487864
7681975
25.7875939
.8069758
.8263431
.8456960
.8650343
•
•
•
25.8843582
.9036677
9229628
•
•
·
25.9807621
26.0000000
•
.0192237
0384331
.0576284
26.0768096
.0959767
.1151297
.1342687
•
•
· 1533937
26.1725047
.1916017
2106848
•
·
•
26.2678511
2868789
3058929
3248932
3438797
•
•
9422435
9615100
26.3628527
.3818119
4007576
.4196896
.438608I
•
26.4575131
.4764046
.4952826
.5141472
5329983
26.5518361
5706605
5894716
.6082694
6270539
·
2297541
2488095
•
26.6458252
6645833
6833281
·
.7020598
7207784
•
26.7394839
7581763
7768557
.7955220
.8141754
Cube
Roots.
8.7065877
.7109827
.7153734
7197596
·
7241414
8.7285187
7328918
•
.7372604
.7416246
.7459846
8.7503401
.7546913
7590383
7633809
.7677192
8.7720532
7763830
7807084
·
7850296
.7893466
8.7936593
7979679
.8022721
.8065722
.8108681
8.8151598
.8194474
.8237307
•
.8280099
.8322850
8.8365559
.8408227
.8450854
.8493440
.8535985
8.8578489
.8620952
.8663375
.8705757
.8748099
8.8790400
8832661
.8874882
•
.8917063
.8959204
8.9001304
.9043366
.9085387
.9127369
.9169311
8.9211214
.9253078
•9294902
.9336687
.9378433
8.9420140
9461809
.9503438
.9545029
.9586581
Recipro-
cals.
660
661
.001515152
1512859
1510574 662
1508296 663
1506024 664
665
001503759
1501502
666
•
.001492537
1490313
1488095
1485884
1483680
.001481481
1479290
1477105
1499250 667
1497006 668
1494768
66g
.001470588
1468429
1466276
1464129
1461988
•
No.
670
671
677
1474926 678
1472754
679
.001449275
1447178
672
673
.001438849
674
675
676
.001459854
1457726
1455604 687
1453488
1451379
680
681
682
683
684
685
686
690
6gr
1445087
692
1443001 693
1440922 694
695
1436782 696
1434720 697
1432665 698
1430615
699
88898989
.001428571
700
1426534
701
1424501 702
1422475 703
1420455
704
705
.001418440
1416431
706
1414427 707
1412429 708
1410437 709
.001408451 710
1406470 711
1404494 712
1402525 713
1400560 714
.001398601 715
1396648 716
1394700 717
1392758 718
1390821 719
651
No. Squares.
720
721
722 521284
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
750
751
752
753
754
755
756
518400
519841 374805361
373248000
376367048
377933067
552049
744 553536
757
758
759
532900
534361
535824
537289
538756
745
746
747
748 5.59504
749
561001
522729
524176
379503424
525625
381078125
527076 382657176
528529
529984
531441
540225
541696
543169
544644
546121
547600
549081
550564
555025
556516
558009
CUBE ROOTS AND RECIPROCALS

Cubes.
573049
574564
576081
760 577600
761
579121
762
580644
763
764
582169
583696
765 585225
766 586756
767 588289
768 589824
769 591361
384240583
385828352
387420489
389017000
390617891
392223168
393832837
395446904
397065375
398688256
400315553
401947272
403583419
405224000
406869021
408518488
410172407
411830784
413493625
415160936
562500
421875000
564001
565504
423564751
425259008
567009 426957777
568516
428661064
416832723
418508992
420189749
570025 430368875
571536
432081216
433798093
435519512
437245479
438976000
440711081
442450728
444194947
445943744
447697125
449455096
451217663
452984832
454756609
770
592900
456533000
771
59444I
458314011
772
595984
597529
460099648
773
461889917
774
599076 463684824
775 600625 465484375
776 602176 467288576
777 603729 469097433
778 605284 470910952
779 606841 472729139
Square
Roots.
26.8328157
.8514432
8700577
.8886593
.9072481
26.9258240
.9443872
9629375
.9814751
27.0000000
•
•
27.0185122
.0370117
.0554985
.0739727
.0924344
27.1108834
1293199
.1477439
1661554
.1845544
•
27.2029410
.2213152
.2396769
.2580263
.2763634
27.2946881
3130006
•3313007
3495887
.3678644
•
27.3861279
.4043792
.4226184
•4408455
.4590604
27.4772633
.4954542
5136330
•
• 5317998
• 5499546
27.5680975
5862284
.6043475
.6224546
.6405499
27.6586334
.6767050
.6947648
•
.7128129
.7308492
27.7488739
7668868
7848880
.8028775
.8208555
27.8388218
.8567766
.8747197
.8926514
.9105715
Cube
Roots.
8.9628095
9669570
·
.9711007
.9752406
9793766
8.9835089
9876373
.9917620
.9958829
9.0000000
9.0041134
0082229
0123288
•0164309
.0205293
9.0246239
.0287149
.0328021
.0368857
.0409655
9.0450417
.0491142
.0531831
.0572482
.0613098
9.0653677
.0694220
0734726
.0775197
.0815631
9.0856030
.0896392
.0936719
.0977010
1017265
9. 1057485
.1097669
.1137818
.1177931
.1218010
9.1258053
.1298061
.1338034
.1377971
.1417874
9. 1457742
.1497576
.1537375
.1577139
.1616869
•
9. 1656565
1696225
.1735852
.1775445
.1815003
9. 1854527
1894018
•
1933474
.1972897
.2012286
•
Recipro-
cals.
No.
720
721
.001388889
1386963
1385042
1383126 723
1381215 724
722
.001379310
725
1377410 725
1375516 727
1373626 728
1371742 729
.001369863
1367989
1366120
730
731
732
1364256 733
1362398
.001360544
734
735
736
1358696
1356852 737
1355014 738
1353180
739
.001351351
740
1349528
741
1347709 742
1345895
743
1344086 744
.001342282 745
1340483 746
1338688 747
1336898 748
1335113
749
.001333333
750
1331558 751
1329787 752
1328021 753
1326260
754
.001324503
1322751
757
1321004
1319261 758
1317523 759
755
756
.001315789
760
1314060 761
1312336 762
1310616 763
1308901 764
001307190
1305483
765
766
1303781 767
1302083 768
1300390 769
.001298701
770
771
1297017
1295337 772
1293661 773
1291990
774
775
776
.001290323
1288660
1287001
1285347
1283697 779
777
778
652
No.
780
781
782
783
784
00000000000
งงงง
787
788
789
790
791
792
793
794
795
796
797
798
799
807
808
80g
815
816
817
818
819
820
821
800
640000
801
641601
802
643204
644809
803
804 646416
648025
805
806
649636
822
823
824
825
826
TABLE XXVI.—SQUARES, CUBES, SQUARE ROOTS,

Squares. Cubes.
608400
609961
611524
613089
614656
616225
617796
619369
620944
622521
833
834
835
836
624100
625681
627264
628849
837
838
839
630436
632025
633616
635209
636804
638401
651249
652864
65448I
677329
678976
680625
682276
827 683929
828 685584
829 687241
810
656100 531441000
811
812
535387328
657721 533411731
659344
813 660969
814 662596
537367797
664225
665856
667489
669124
670761
672400
67404 I
675684
474552000
476379541
478211768
480048687
481890304
483736625
485587656
487443403
489303872
491169069
493039000
494913671
496793088
498677257
500566184
502459875
504358336
506261573
508169592
510082399
512000000
513922401
515849608
517781627
519718464
521660125
523606616
525557943
527514112
529475129
539353144
541343375
543338496
545338513
547343432
549353259
830 688900 571787000
831
573856191
832
575930368
578009537
551368000
553387661
555412248
557441767
559476224
561515625
563559976
565609283
567663552
569722789
690561
692224
693889
695556
580093704
582182875
697225
698896 584277056
700569 586376253
702244 588480472
703921
590589719
Square
Roots.
27.9284801
9463772
9642629
.9821372
28.0000000
28.0178515
.0356915
.0535203
·
•
0713377
.0891438
28.1069386
.1247222
1424946
1602557
.1780056
28. 1957444
·
·
2134720
.2311884
.2488938
.2665881
•
28.2842712
.3019434
3196045
.3372546
3548938
28.3725219
.3901391
.4077454
.4253408
4429253
•
·
•
28.4604989
.4780617
.4956137
.5131549
5306852
28.5482048
5657137
.5832119
.6006993
.6181760
•
28.6356421
.6530976
.6705424
.6879766
.7054002
28.7228132
.7402157
7576077
7749891
.7923601
28.8097206
8270706
8444102
8617394
8790582
28.8963666
.9136646
•
•
•
• 9309523
9482297
•9654967
Cube
Roots.
9.2051641
2090962
•
.2130250
2169505
.2208726
9.2247914
2287068
2326189
.2365277
.2404333
9.2443355
2482344
•
•
.2599114
9.2637973
2676798
2715592
.2754352
.2793081
•
9.2831777
2870440
•
2521300
2560224
.2909072
2947671
.2986239
9.3024775
.3063278
.3101750
.3140190
.3178599
9.3216975
.3255320
.3293634
• 3331916
• 3370167
9.3408386
.3446575
.3484731
3522857
.3560952
•
9.3599016
.3637049
.3675051
.3713022
.3750963
9.3788873
3826752
.3864600
.3902419
.3940206
9.3977964
.4015691
4053387
•
4091054
.4128690
9.4166297
.4203873
.4241420
.4278936
.4316423
•
Recipre-
cals.
.001282051
1280410
1278772
1277139
1275510
.001265823
1264223
1262626
.001273885 785
1272265 786
1270648 787
1269036 788
1267427
789
No.
780
781
792
1261034 793
1259446 794
.001257862
795
796
.001234568
1233046
1231527
782
783
784
1256281
1254705 797
1253133 798
1251564 799
1230012
1228501
.001226994
1225490
.001250000
800
1248439
801
802
1246883
1245330 803
1243781
804
001242236 805
1240695 806
790
791
1239157 807
1237624 808
1236094
80g
.001219512
1218027
1216545
1215067
1213592
.001212121
.001204819
1203369
810
811
812
813
814
1223990 817
1222494 818
1221001
819
815
816
820
821
822
823
824
825
1210654 826
1209190 827
1207729 828
1206273 829
830
831
832
1201923
I200480
1199041 834
833
.001197605 835
1196172 836
1194743 837
1193317 838
1191895 839
653
No. Squares
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
00.00!
78
879
880
881
882
883
884
885
887
888
889
705600
707281
708964
710649
712336
890
891
714025
715716
717409
719104
720801
722500
724201
725904
727609
*729316
731025
732736
734449
736164
737881
748225
749956
751689
753424
755161
756900
758641
760384
762129
763876
765625
767376
769129
774400
776161
886 784996
CUBE ROOTS AND RECIPROCALS

739600
741321
743044
744769
746496 644972544
647214625
649461896
651714363
653972032
656234909
Cubes.
592704000
594823321
596947688
599077107
601211584
603351125
605495736
607645423
609800192
611960049
792100
793881
614125000
616295051
618470208
892 795664
893 797449
894 799236
895 801025
896 802816
897 804609
898
806404
808201
899
620650477
622835864
625026375
627222016
629422793
631628712
633839779
674526133
770884 676836152
772641
679151439
636056000
638277381
640503928
642735647
658503000
660776311
663054848
665338617
667627624
669921875
672221376
777924
779689
688465387
781456 690807104
783225 693154125
695506456
697864103
681472000
683797841
686128968
786769
788544 700227072
790321 702595369
707347971
709732288
712121957
714516984
716917375
719323136
721734273
Square
Roots.
724150792
726572699
28.9827535
29.0000000
.0172363
.0344623
0516781
29.0688837
.0860791
.1032644
•
.1204396
.1376046
29.1547595
.1719043
1890390
.2061637
2232784
29.2403830
•
·
.2574777
2745623
.2916370
.3087018
•
29.3257566
.3428015
.3598365
.3768616
.3938769
29.4108823
.4278779
4448637
.4618397
.4788059
•
29.4957624
.5127091
.5296461
5465734
5634910
29.5803989
•
5972972
.6141858
.6310648
.6479342
29.6647939
.6816442
.6984848
704969000 29.8328678
.8496231
.8663690
.8831056
.8998328
29.9165506
.7153159
.7321375
29.7489496
7657521
7825452
.7993289
.8161030
•
· 9332591
•9499583
.9666481
.9833287
•
Cube
Roots.
9.4353880
.4391307
.4428704
.4466072
.4503410
9.4540719
.4577999
4615249
.4652470
4689661
•
9.4726824
.4763957
4801061
.4838136
.4875182
•
9.4912200
.4949188
4986147
.5023078
5059980
•
•
9.5096854
5133699
.5170515
.5207303
5244063
9.5280794
• 5317497
.5354172
• 5390818
• 5427437
9.5464027
•5500589
.5537123
•5573630
5610108
9.5646559
5682982
.5719377
•
• 5755745
•5792085
9.5828397
.5864682
• 5900939
•5937169
• 5973373
9.6009548
6045696
6081817
•
•
.6117911
.6153977
9.6190017
.6226030
.6262016
.6297975
.6333907
9.6369812
.6405690
6441542
.6477367
.6513166
•
Recipro-
cals.
840
841
.001190476
1189061
1187648
1186240 843
1184834
842
844
845
.001183432
1182033
846
1180638 847
1179245 848
1177856
849
No.
850
.001176471
1175088 851
1173709 852
853
854
1172333
1170960
.001169591
1168224
1166861
1165501
1164144
860
.001162791
1161440
861
1160093 862
1158749
1157407
863
864
001156069
1154734
1153403 867
1152074
868
1150748 869
855
856
857
858
859
.001149425 870
1148106
1146789
I145475
1144165
871
872
873
874
875
.001142857
1138952
1137656
865
866
1141553 876
.001129944
1128668
1127396
1126126
1124859
26 78
1140251 877
878
879
880
.001136364
1135074
1133787 882
881
1132503
883
1131222
884
885
886
887
888
889
.001123596
890
1122334
891
1121076 892
1119821 893
1118568 894
.001117318 895
1116071 896
1114827 897
1113586 898
1112347 899
654
No. Squares. Cubes.
goo
gor
902
903
904
905
906
907
908
gog
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
937
938
939
TABLE XXVI.—SQUARES, CUBES, SQUARE ROOTS,

945
946
810000
811801
813604
815409
817216
947
948
949
819025
820836
822649
824464
826281
828100
829921
831744
833569
835396
837225
839056
840889
842724
844561
846400
848241
850084
851929
853776
930
864900
931 866761
932
868624
933
870489
934
872356
935
874225
936
876096
877969
855625
857476
859329
861184
863041
940 883600
941 885481
942
887364
943
889249
944 891136
879844
881721
893025
894916
896809
898704
950
951
952 906304
953
908209
954 910116
902500
904401
729000000
731432701
733870808
736314327
738763264
955
912025
956 913936
957 915849
958
917764
959 919681
741217625
743677416
746142643
748613312
751089429
*753571000
756058031
758550528
761048497
763551944
766060875
768575296
771095213
773620632
776151559
778688000
781229061
783777448
786330467
788889024
791453125
794022776
796597983
799178752
801765089
804357000
806954491
809557568
812166237
814780504
843908625
846590536
849278123
851971392
900601 854670349
817400375
820025856
822656953
825293672
827936019
830584000
833237621
835896888
838561807
841232384
857375000
860085351
862801408
865523177
868250664
870983875
873722816
876467493
879217912
881974079
Square
Roots.
30.0000000
.0166620
.0333148
0499584
0665928
30.0832179
.0998339
.1164407
1330383
1496269
·
•
•
•
30.1662063
.1827765
1993377
.2158899
•
.2324329
30.2489669
2654919
2820079
2985148
.3150128
•
·
•
30, 3315018
.3479818
.3644529
3809151
.3973683
30.4138127
.4302481
4466747
.4630924
.4795013
·
30.4959014
5122926
5286750
5450487
.5614136
30.5777697
•
•
.5941171
.6104557
.6267857
.6431069
30.6594194
.6757233
.6920185
.7083051
7245830
•
30.7408523
.7571130
7733651
.7896086
.8058436
30.8220700
.8382879
8544972
8706981
.8868904
•
•
30.9030743
.9192497
.9354166
•9515751
.9677251
Cube
Roots.
9.6548938
.6584684
.6620403
.6656096
6691762
9.6727403
.6763017
•
9.6905211
6940694
.6976151
.7011583
7046989
9.7082369
.7117723
•
•
.7153051
7188354
.7223631
9.7258883
.7294109
6798604
6834166
6869701
.7329309
•7364484
.7399634
9.7434758
7469857
.7504930
•
.7539979
.7575002
9.7610001
.7644974
.7679922
.7714845
•
.7749743
9.7784616
7819466
.7854288
7889087
7923861
•
9.7958611
.7993336
8028036
8062711
•
.8097362
9.8131989
.8166591
.8201169
.8235723
.8270252
9.8304757
.8339238
.8373695
.8408127
.8442536
9.8476920
.8511280
.8545617
.8579929
.8614218
Recipro-
cals.
IIIIII100°
goo
1109878 901
1108647 902
1107420 903
1106195
904
905
1103753 906
1102536 907
908
1101322
I100110
gog
.001104972
.001098901
1097695
1096491
910
911
912
1095290 913
914
1094092
.001092896
1091703
1090513
1089325
1088139
001081081
No.
920
.001086957
1085776
1084599
1083424 923
921
922
1082251 924
.001063830
1062699
1061571
915
916
917
918
919
1079914
1078749 927
1077586
928
1076426
929
.001058201
.001075269
930
1074114
931
1072961 932
1071811
933
934
935
1070664
.001069519
1068376
1067236 937
936
1066098 938
1064963
939
925
926
.001047120
942
1060445 943
1059322 944
945
1057082 946
1055966 947
1054852 948
1053741 949
940
941
.001052632
950
1051525
951
1050420 952
1049318 953
1048218 954
955
1046025 956
1044932 957
1043841 958
1042753 959
655
No. Squares.
៩៩៩៩៩៩៦៩៩ ៩៩៩
960
962
963
964
970
933156
968
935089
937024
969 938961
972
973
974
975
976
977
978
Foto 1 of 10 of oo oo
980
983
984
985
986
988
979 958441
989
995
996
997
998
921600
923521
925444 890277128
927369
929296
893056347
895841344
931225 898632125
954529
956484
1005
1006
990
991
992
984064
993
986049
994 988036
IOIO
IOII
1012
976144
978121
940900 912673000
942841 915498611
944784 918330048
946729
948676
921167317
950625
952576
980100
982081
994009
996004
999 998001
990025
992016
1000
1001
1002
1004004
1003 1006009
1004 1008016
1010025
1012036
1014049
1007
1008 1016064
1009
Ιοιδοδι
950400
941192000
962361 944076141
964324 946966168
966289 949862087
968256 952763904
970225 955671625
972196
974169
1000000
1002001
CUBE ROOTS AND RECIPROCALS

Cubes.
1020100
1022121
884736000
887503681
1013
1024 144
1026169
1028196
1014
1015
1030225
1016 1032256
1017
1018 1036324
1019
1038361
901428696
904231063
907039232
909853209
924010424
926859375
929714176
932574833
935441352
938313739
958585256
961504803
964430272
967361669
970299000
973242271
976191488
979146657
982107784
985074875
988047936
991026973
994011992
997002999
1000000000
1003003001
100601 2008
1009027027
1012048064
1015075125
1018108216
1021147343
1024192512
1027243729
1030301000
1033364331
1036433728
1039509197
1042590744
1045678375
1048772095
1034289 1051871913
1054977832
1053089859
Square
Roots.
30.9838668
31.0000000
.0161248
.0322413
0483494
•
31.0644491
.0805405
0966236
1126984
.1287648
•
·
31.1448230
1608729
.1769145
.1929479
2089731
•
•
31. 2249900
2409987
.2569992
•
2729915
.2889757
•
31.3049517
• 3209195
3368792
•3528308
.3687743
31.3847097
·4006369
.4165561
·
.4324673
•4483704
31.4642654
•4801525
.4960315
.5119025
• 5277655
31.5436206
•5594677
• 5753068
5911380
6069613
•
31.6227766
.6385840
.6543836
.6701752
.6859590
31.7017349
• 7175030
•7332633
.7490157
7647603
•
31.7804972
.7962262
.8119474
.8276609
8433666
31.8590646
.8747549
.8904374
9061123
.9217794
•
•
Cube
Roots.
9.8648483
.8682724
.8716941
.8751135
.8785305
9.8819451
.8853574
.8887673
.8921749
.8955801
9.8989830
•9023835
.9057817
.9091776
.9125712
9.9159624
.9193513
.9227379
.9261222
.9295042
9.9328839
.9362613
.9396363
.9430092
· 9463797
•
9.9497479
9531138
9564775
.9598389
.9631981
•
9. 9665549
.9699095
.9732619
.9766120
.9799599
9.9833055
.9866488
.9899900
9933289
.9966656
•
10. 0000000
0033322
0066622
.0099899
.0133155
10.0166389
.0199601
•
•
.0232791
.0265958
.0299104
10.0332228
.0365330
.0398410
.0431469
.0464506
10.0497521
.0530514
.0563485
.0596435
0629364
Recipro-
cals.
960
.001041667
1040583 961
1039501 962
1038422
963
964
1037344
.001036269
1035197
1034126
1033058
1031992
.001030928
1029866
.001025641
No.
970
971
1028807 972
1027749 973
1026694
974
975
976
.001020408
1019368
1018330
1017294
1016260
.001015228
1014199
1013171
1012146
1011122
965
966
967
968
969
1024590
1023541
977
1022495 978
1021450
979
ΙΟΙΟΙΟΙ00
980
981
982
983
foot of 60 60 00 00
984
985
986
987
988
989
990
1009082 991
1008065 992
1007049 993
1006036 994
.001005025 995
1004016 996
1003009 997
1002004 998
1001001
999
.001000000
1000
1001
0999001
0998004 1002
0997009 1003
0996016 1004
.000995025
1005
0994036 1006
0993049 1007
0992063 1008
0991080 1009
.000990099
1010
0989120 IOII
0988142 1012
0987167 1013
0986193 1014
.000985222
1015
0984252 1016
0983284 1017
0982318 1018
0981354 1019
APPENDIX
PARTIAL THEORY OF THE SPIRAL
PARTIAL THEORY OF THE SPIRAL
Derivation of Coordinates X and Y.-From Fig. 5-3,
When the cosine and the sine
dx=dl cos 8 and dy=dl sin 8.
are expressed as infinite series,
and
But dô=dl,
p
integrating gives:
and
and
dx=dl[1–2171 +
Therefore, dx=dl 1
APPENDIX
But
dy = d[ 3 - 1 1 +
5,729.58
d
dy=dl a l2
M
k l2
1,145,916
x=i[1–
[
2 = 1 [3/3
tan a=
2!' 4!
dx=di[1- +
87
3!' 5! 7!
83
3 7(3!)
=
and d=
84 86
6!
85
a³ 16
8371803
3!
(for simplicity) a 12
a² 14. a4 18 a6 (12
2! 4!
+
i+...]
...]
a5 710
5!
82
84
+
5(2!) 9(4!)
The result obtained by integrating and substituting & for
a l² is
kl
100'
85
+
11(5!)
Substituting and
a01² + ...]
6!
a7 714
7!
86
13(6!)
....]
+
s+...]
+...]
87
15(7!)+
(A-2)
Equations A–1 and A-2, in which ▲ is substituted for 8, are
used in calculating X and Y for insertion in spiral tables, such
as Tables XI and XII in Part III.
Derivation of Correction C in a = 38 — C.—Dividing equa-
tion A-2 by equation A-1 gives:
y d 83 26 85
;+: +
X 3 105 155,925
a=tan a- tan³ a+ tan³ a- ...
(A-1)
17 87
3,378,375
(A-3)
(A-4)
659
660
APPENDIX
When the value of tan a from equation A-3 is substituted
in equation A-4 and like terms are collected, the result is
32 85 128 87
83,284,288
δ 883
(A-5)
3
or
a
2,835 467,775
a= 8-a small correction C
3/3/3
Radians being converted to minutes, the value of the correc-
tion C is:
C (in minutes)=516 (10)-7 8³+381 (10)−12 85+ . . . (A-6)
The corrections in Table XVI-C were computed from
equation A-6.
Source of the Corrections Marked * for Conversion to
A.R.E.A. Spiral.-In the case of a spiraled curve, changing
the definition of the degree of the simple curve from Da to De
affects only the coordinates of the offset T.C.
In Table XI the * corrections to be subtracted from the
coordinates of the offset T.C. based upon De, in order to
obtain those based upon Dc, are:
(X−Ra sin A)−(X− Rc sin A)=(R – Ra) sin A (A-7)
and (Y-Ra vers ▲) — (Y - Re vers A) = (R. - Ra) vers ▲ (A-8)
In Table XII the coefficients come from the following
relations:
Total correction to X。 = (R.- Rɑ) sin ▲=* times D
Total correction to o= (R. - Ra) vers ▲
(Rc-Ra) vers A=* times D
INDEX
Aerial photography; see also
Photogrammetry
advantages of, 339, 340
in canal surveys, 318, 339
in highway location, 309-11
miscellaneous uses of, 337,
338
in pipe-line surveys, 317
in route surveying, 321-42
Aerotriangulation
INDEX
analytic, 362, 363
instrumental, by bridging,
362
Alignment design
horizontal, on highways,
193-243, 252
by paper-location, 271-73
vertical, on highways, 243-
254
American Association of State
Highway Officials, 298,
351
design policies of, 187, 306
American Railway Engineer-
ing Association
formula for track superele-
vation of, 278, 280
recommendation for spiral
length by, 218, 280
recommendation for ver-
tical curves by, 69
ten-chord spiral of, 109,
660
trackwork plans of, 291
American Road Builders'
Association, 298
Analytic geometry, solution
of curve problems by,
179-81, 353
Areas
of
cross-sections;
Cross-sections
see
of pavement on curves, 240
Auscor; see Stereomat
Automation
in field
343-48
in location and design,
343-65
in plotting, 357-59
in recording field data, 355
terminology in, 348
measurements,
Balance lines on mass dia-
gram, 145-52
Balance points, 143, 144, 152
Ball bank indicator, 200–203
Balplex plotter, 325
Base line, 266, 302
Binary numbers, 350
Body roll, 199–203
Borrow pits, 138-41
Broken-back curve, 171-73,
252
Bureau of Public Roads, 297,
299, 300, 305, 351-55
Canal surveys, 318, 339
Central angle
of circular curve, 14
of spiral, 86, 87, 102
Change-of-location problems.
examples of, 165–70
hints for solving, 165-67
Checking
of computations, 40, 41
of curve by middle ordi-
nate, 27
of field work on curves, 21.
26, 27
Chord; see also Subchords
long (L.C.), of simple
curve, 14
Chord-gradient method of
calculating vertical
curves, 69-72
663
664
INDEX
Chord lengths on circular
curves, 27, 28
Chord offsets, 32-36
Circular curve; see Simple
curve
Clothoid, 85
Compound curve
calculation of, 58, 59
completely-spiraled, 103-5
definition of, 50
location of, by trial, 61, 62
notation for, 50, 51
notes and field work for,
61. 62
with P.I. inaccessible, 160,
161
problems on, 63, 64
rigid solution of, 50
solution of
by construction, 52, 53
by traverse, 53-57
vertex triangle, 51, 52
summary of methods of
solving, 57, 58
three-centered, 59
use of, 50
Computation
methods of, 39-41
rounding off numbers in,
42-45
significant figures in, 42-45
Construction survey
methods for
location,
in highway
307, 308
in railroad location, 276-
278
relation of, to other sur-
veys, 6
Contour map
field sketches for, 269, 270
by photogrammetry, 327-
330
purpose and limitations of,
8,9
Contours; see Topography
Controlling points, 3, 4, 264
Controls
for horizontal alignment,
252
for vertical alignment, 254
Coordinatograph, 357
Cross-section leveling, 124,
125, 301, 303
Cross-sectioning
on highways, 301, 303
methods of, 124–29
on Pennsylvania Turnpike,
320
Cross-sections
areas of
by calculating machine,
121
by formulas, 119-121
by graphic methods, 124
digital, 361
location of, 117–19
types of, 116, 117, 123
Crown
adverse, 196, 197
favorable, 196, 197
Curvature
aesthetic importance of, 3
correction for, in earth-
work, 136-38
use of, in route location,
3
Curve
broken-back, 171–73, 252
by-passing obstacles on,
30, 31, 37
circular, length of, 19
compound; see Compound
curve
degree of, 15-17
easement, 83-85
even-radius, 31, 107, 108
highway; see Highway
horizontal;
Simple
curve
with inaccessible P. I. or
T.C., 160-63
measurements along, 17,
18, 26-30
metric, 31, 32
see
INDEX
665
Curve (Cont.)
parabolic; see Parabolic
curves, Vertical curve
parallel to simple curve,
37-39
parallel to spiral, 111, 238–
240
primary purpose of, 12
reverse; see Reverse curve
simple; see Simple curve
spiraled; see Spiraled
curve
stakes on, 28-30
staking
by deflection angles, 22-
24
by offsets, 32–37
stationing on, 18
through fixed point, 175-
177
transit set-ups on, 24-26
vertical; see Vertical curve
Curve problems
in highway design, 185–261
special
examples of, 159–184
methods of solving, 159,
160
Curve tables [see tables in
Part III of book]
Data
plotter, 357
presentation, 348, 362
processing, 348
procurement, 348
recording, 348, 355, 356
reduction, 348
transmission, 348, 356, 357
Deflection angle
from set-up on spiral, 97-
101
to spiral, 89-93
Deflection-angle method for
circular curves, 22-24
Degree of curve
arc definition of, 16
chord definition of, 16
Degree of curve (Cont.)
formulas for, 16
selection of, 21
Design; see Office studies
Development, 3, 5
Digital
computer; see Electronic
computer
terrain data, 359-62
terrain model, 359–62
Distribution of earthwork,
142-52
Earthwork
analysis of
by mass diagram, 145–52
by station-to-station
method, 144
balance lines in, 145–52
balance points in, 143, 144,
152
basis of payment for, 115,
123
calculation of, 131-34
classification of, 115, 123
correction of, for curva-
ture, 136-38
cross-sections of; see
Cross-sections
estimate of, 271, 272, 334
from aerial photographs,
334
operations included under,
115
practice problems in, 152-
155
shrinkage, swell, and set-
tlement of, 141, 142
slope staking for, 126-29
tables for computation of,
131-34
volume of; see Volume of
earthwork
Easement curves, 83-85; see
also Spirals
Edge lengths on highway
curves, 236-38
666
INDEX
Electronic computer
advantages of, 352, 353
earthwork quantities by,
354
fundamentals of, 349-51
horizontal alignment by,
354, 355
programs in route loca-
tion, 351-55
traverse computation by,
354
vertical alignment by, 355
Electroplotter, 357
Electrotape, 347
Elevations; see Leveling
Excavation; see Earthwork
External distance
of simple curve, 14
of spiral, 91
Field notes
for by-passing obstacles,
275, 276
for compound curves, 61
for cross-section leveling,
124, 125
for simple curve, 23, 24
for slope staking, 127, 130
for spiraled curve, 94, 95
Field work [see type of sur-
vey desired]
Floating mark, 325–27
Free haul, 142-44, 147-51
Galileo-Santoni Stereocarto-
graph, 325
General Motors Proving
Ground, 197, 199, 203,
206
Geodimeter, 343–45
Grade
balance between curvature
and, 3
Grade contour, 271, 318
Grade rod, 128, 129
Ground control
by bridging, 327, 362
in photogrammetric map-
ping, 323-27, 337, 362
by radar, 324
Hand level
in cross-sectioning, 125
topography by, 268-70
Headlight sight distances,
247-49, 251
Highway
cross-sectioning on, 301,
303
curves in
edge lengths on, 236-38
pavement areas on, 240
vehicle operation on,
215-21,
197-99, 213,
227, 228
design elements of, in re-
lation to speed, 186,
187
design speed on, 187
location of, examples of,
308, 309
over-all travel speed on,
187
preparation of plans for,
298-301, 305–7
relocation of, survey meth-
ods in, 301-303
running speed on, 187
side friction factor on; see
Side friction factor
sight distances on; see
Sight distance
speed signs on, 203
superelevation of; see Su-
perelevation
surveys for, 297-312
test track design, 199
widening, on curves; see
Widening of highway
Highway projects, magni-
tude of, 11
Highway Research Board,
298, 312, 341, 342, 363–
365
INDEX
667
Inaccessible points on
curves, 160-63
Integrated mapping system,
359
Integration of photogramme-
try and computer, 359–
363
Intersection
of line and curve, 177, 178
point of, for horizontal
curve, 13
Kelsh plotter, 325, 360
L-line, 273, 274
Length of curve, definition
of
on horizontal curve, 19
on vertical curve, 66
Leveling
bench, 266, 267
on
on location survey, 274
preliminary surveys,
266, 267
profile, 267, 274
Limit of economic haul, 143,
149
Location controls, 3, 4, 264
Location survey
function of, 9
methods of, in railroad lo-
cation, 273–75
relation of, to other sur-
veys, 5, 6
Maps
accuracy of, 328, 337, 340
base, 322
photogrammetric, costs of,
335, 336
planimetric, 322, 324, 333
scales of, in photogram-
metry, 328-30, 333, 334,
336
topographic, by photo-
grammetric methods,
322, 327-30
Mark sense cards, 348, 355
Mass diagram, 145–152
Merritt Parkway, 253
Metric curves, 31, 32
Middle-ordinate method of
staking curve, 32, 36
Middle ordinate of simple
curve, 14
Mosaic
controlled, 322, 323
uncontrolled, 322, 323
Multiplex plotter, 325, 326
National map-accuracy
standards, 328, 329
Nistri
Photocartograph, 325
Photostereograph, 325
Notes; see Field notes
Obstacles, by-passing
on curves, 30, 31, 37, 162-65
on tangents, 274-76
Office studies
in highway design, 305–307
in paper location, 271–73
relation of, to field work, 5
Offset curve
parallel to simple curve,
37-39
parallel to spiral, 111, 238-
240
Offset T. C., 87
Offsets
formulas for, 33, 34
staking curves by, 32-37
Ordinates from a long chord,
32, 37
Orthophotography, 330
Osculating circle, 95–97
Overhaul, 142-44, 147-52
Oversteering, 198, 199
P-line, 266, 273, 274
668
INDEX
Paper location procedure,
271-73
Parabolic curves
laying out, by taping, 79,
80
uses of, 65
Parallel curve; see Offset
curve
Pennsylvania Turnpike
aerial surveys on, 329
cross-sectioning on, 320
high-speed tests on, 199,
204, 219, 228
run-off design on, 225, 226
sight distances on, 248
spirals on, 219, 226
Photo-contour map, 330
Photogrammetry; see also
Aerial photography
definitions in, 321, 322
for detailed location
studies, 309-11, 333,
334
integration of, with com-
puter, 359-63
limitations of, 338, 339
plotting instruments in,
325
for reconnaissance, 310,
330-33
Photographs, aerial, types of,
321
Photronix, Inc., 359, 360
Pipe-line surveys, 316, 317,
333
Point of intersection, 13
Preliminary survey
earthwork estimate from,
272, 273
leveling on, 266, 267
plotting map of, 270
purposes of, 7, 8
in railroad location, 266-70
relation of, to other sur-
veys, 5
topography on, 267–70
traverses on, 265, 266
Prismoidal formulas, 134-36
Property surveys, 278
Radar altimetry, 348
Radius of circular arc, 13-16
Railroad location, economics
of, 294
Railroad relocations, 293-95
Railroad surveys, 262-96
Railroad track
layouts for, 290-93
realignment problems
with, 289, 290
string lining of, 281-88
superelevation of, 278-80
Railroad turnout and cross-
over, 291 [see also
Table XXIV in Part
III of book]
Railway; see Railroad
Reaction time, 188–90
Reconnaissance
aerial, 301, 302, 310, 330,
331
for highway location, 302,
310, 311
importance of, 6, 7
instruments for, 262
for railroad location, 262,
264
relation of, to other sur-
veys, 5
by stadia method, 265
Relocation
by field location method, 8
problems on, examples of,
170-75
railroad, 293-95
Reverse curve
definition of, 50
limitations and use of, 59,
60
non-parallel-tangent, 61
notes and field work for, 61
parallel-tangent, 60
practice problems on, 63,
64
INDEX
669
Reverse curve (Cont.)
replacement of, 173, 174
Rounding off, 42–45
Route location; see
Route surveying
controlling points in, 3, 4,
264
also
cross-drainage line in, 5
importance of curves and
grades in. 3
influence of terrain on, 4, 5
influence of type of project
on, 4
proper use of topography
in, 8, 9
ridge line in, 4
side-hill line in, 4
use of development in, 3, 5
valley line in, 4
Route surveying; see also
Route location
accuracies required in, 10,
11
aerial photography in, 321–
342
definition and purposes of,
1
and design, 2
and economics, 2
and engineering, 9
sequence of field and office
work in, 5, 6
Rule of offsets, 67
Runoff; see Superelevation
Runout, 226
Seismic studies, 304
Semi-final location, 9
Settlement of embankment,
142
Shrinkage of earthwork, 141,
142
Side friction factor, 197
recommended values of,
205
research on, 199–206
Sight distance
headlight, 247, 248–51
on horizontal curves, 240-
243
at interchanges, 251, 252
passing, 188, 190-94
stopping, 188-90
at underpass, 251, 252
on vertical curves, 243-51
Significant figures, 39-45
Simple curve; see also Curve
computations for, 20, 23
field work in staking, 21–31
formulas for, 32–34
methods of superelevating,
226-28
methods of widening, 228-
235
notation for, 13, 14
parallel offset to, 37-39
practice problems on, 45-49
with spirals, 90-94
Slip angles, 198, 199
Slope-staking, 126-29
Soil surveys, 303-305
Special curve problems
examples of, 159-84
methods of solving, 159,
160
Speed on highways;
Highway
Spiral; see
Curve
see
also Spiraled
A.R.E.A. ten-chord, 109,
660
between arcs of compound
curve, 101-103
calculation of, 91-93, 95
without special tables,
109
central angle of, 86, 87, 102
with compound curve, 103–
105
deflection angles to, 89, 90.
92, 93
double, 108
Euler, 85
external distance of, 91
670
INDEX
Spiral (Cont.)
field work for, 91, 92, 110
geometry of the, 85-90
importance of the, 228
laying out, by taping, 110,
111
length of, 213-21, 280
locating any point on, 93,
94
long chord for, 90
long tangent for, 90
minimum cuvature for use
of, 221, 222
notation for, 83, 87-91
offsets to, 89, 110
osculating circle to, 95-97
parallel offset to, 111, 238-
244
on
Pennsylvania Turn-
pike, 219, 226
problems on, 111–14
on railroad track, 280, 281
short tangent for, 90
with simple curve, 90-94
tables on [see tables in
Part III of book]
tangent distance of, 90, 91
theory of the, 659, 660
throw of, 88
transit set-ups on, 97-101
types of, 85
Spiral angle, 86
Spiraled curve
to fit given Es or Ts, 105–
107
formulas for, with radius
as parameter, 107, 108
methods of superelevating,
223-26
methods of widening, 230-
233
with unequal spirals, 91
Stadia, topography by, 267,
268
Stadia traverse, 265, 266
Stakes
construction, 277, 307
slope, 126–29, 307
Stakes (Cont.)
station, 29, 266, 277, 302,
307
Station equation, 165
Station-to-station method of
earthwork analysis,
144
Station-yard, 142
Stationing
on curves, 17, 18
of tangent points, 21
Stereomat, 348, 358
Stereomodel, 323, 326
Stereoscopic
fusion, 323, 325
overlap, 322
pairs, 321, 323, 325, 327,
331, 335
plotting instruments, 325
vision, importance of, 323
String lining of railroad
track, 281-288
Subchords
formulas for, 19, 34
nominal, 18
subdeflections for, 22
true, 18
Subdeflections for subchords,
22
Superelevation
effect of, on spiral length,
217-19
maximum rates of, 206-11
methods of attaining, 223-
228
railroad-track, 278–80
rate of, over range of
curvature, 211
rates of, for design, 211–13
theory of, 193–97
Superelevation runoff, length
of, 222, 223
Survey [see type of survey
desired]
relation of, to engineering,
9, 11
swell of earthwork, 141
INDEX
671
Tables, use of, 19, 20 [see
also tables in Part III
of book]
Tangent
back (initial),
forward, 14
Tangent distance
14
of simple curve, 14
of spiral, 90, 91
Tangent offsets
for horizontal curve, 32, 36
to spiral, 89
for vertical curve, 72, 74
Tangent points
on simple curve, 14, 21
on spiraled curve, 83
Tellurometer, 344, 346, 347
Terrain
influence of, on route loca-
tion, 4, 5
types of, 4, 5
Throw of spiral, 88
Ties between P-line and L-
line, 273, 274
Topography
by field-sketch method,
269, 270
by hand level, 268, 269
in highway relocation, 303
by photogrammetry, 327,
333
plotting, 268, 270
proper use of, 8, 9
by stadia, 267, 268
Transit set-ups
rule for, on circular curve,
25
on spiral, 97-101
Transmission lines
location of, 4
surveys for, 314–16, 333
Traverse solution of curve
problems, 53-57, 161,
167, 168
stadia, 265, 266
transit-and-tape, 266
Trilateration, 344
Tunnel surveys, 319, 320
Underpass, sight distances
at, 251, 252
Understeering, 198
Vertex of horizontal curve,
13
Vertical curve
by chord gradients, 69–72
drainage requirements on,
247, 251
equal-tangent, 65–68
length of, 66, 243–251
lowest or highest point
(turning point)
75, 76
on,
to pass through fixed point,
76-79
problems, 80-82
reversed, 79
sight distances on, 243-51
by tangent offsets, 73, 74
at underpass, 251
unequal-tangent, 73, 75, 353
Volume of earthwork
by average end areas, 129-
131
from borrow pits, 138-41
corrected for curvature,
136-38
by prismoidal formula,
134-36
by tables, 131–34
Wellington, A.M., 6
Widening of highway
formulas for, 229
methods of attaining, 230-
235
reasons for, 228, 229
Wild Autograph, 325
Zeiss Stereoplanigraph, 325
2
200
Patt
400

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