IB^^K ^
TF
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IRLF
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REESE LIBRARY
UNIVERSITY OF CALIFORNIA.
Deceived MAR ...1.5 1893 , /^ .
'Otr, Class No.
PRACTICAL TREATISE
RAILWAY CURVES
LOCATION,
FOR YOUNG ENGINEERS
^OBTAINING A FULL DESCRIPTION OP THE INSTRUMENTS, THE MANNER OP ADJUSTING THEM, AN*
tHE METHODS OF PROCEEDING IN THE FIELD, NEW AND SIMPLE FORMULA FOR <M-
POUND AND REVER8B CURVING, RULES FOR CALCULATING EXCAVATION A>T>
EMBANKMENT, STAKING OUT WORK, &C., TOGETHER WITH TABLES OF
NATURAL SINES AND TANGENTS, RADII, CHORDS, ORDINATES,
AMD OTHERS OF GENERAL USE IN THE PROFESSION.
BY
WILLIAM F. SHUNK,
UNIVERSITY
CIVIL ENGINEER.
PHILADELPHIA:
HENRY CAREY BAIRD & CO.,
INDUSTRIAL PUBLISHERS, BOOKSELLERS AND IMPORTERS,
810 WALNUT STREET.
Entered, according to the Act of Congress, in the year 1854, by
E. H. BUfLER ft CO.,
In the Clerk's Office of the District Court of the United States, in and for the Extern
District of Pennsylvania.
PREFACE.
THE located line for railway is a series of curves and
straight lines, or tangents. These are first plotted to a
large scale from data gathered on preliminary survey. It
is therefore desirable that all explorations should be made
with extreme care, as upon their correctness depend, in no
small degree, the labour and time required in location.
It were better for accuracy that all angles should be made
and recorded from the plates, and the needle used only as
a test, or check. Good chaining is indispensable. Great
attention, too, should be given to the proper use of the
elope instrument. By these means a working map can be
constructed in the office upon which the proposed location,
grade lines, &c., may be traced with tolerable resemblance
to fact. Still many errors attach to both data and map,
and these, together with the unexpected obstacles encoun-
tered in the field, require ready knowledge of the means
for overcoming them.
It has been my design to present this knowledge to my
younger fellows in the profession. I have endeavoured to
(3)
IV PREFACE.
do it lucidly and concisely without supposing unusual cases
without prolix proof or complex figuring. The problems
given are of frequent occurrence, and the tables appended
will be found useful and correct.
To STRICKLAND KNEASS, a gentleman whose professional
abilities are well known, I return thanks for valuable
assistance. I would likewise make my acknowledgments,
for useful suggestions, to CHARLES DELISLE, an engineer
of high mathematical attainment.
I am aware that much more might have been said
much more suggested on the subject of location ; but a
field book being the object, the compact plan precluded
any extensive essay.
If the work with brevity combines clearness, and is
comprehensive withal, it is the work intended.
W. F. SHUNK.
CONTENTS.
1'A.GK
PREFACE ........ 3
Explanations ........ 7
ARTICLE I. Of the Instruments. The Level . . . It
Its adjustment ...... 13
The Rod .. .... 14
Levelling . . . . . 15
The Transit ....... 17
The Vernier . . . . . . 18
Adjustment of Transit . . . . .19
II. Preliminary propositions .... 20
III. To avoid an obstacle in tangent . . . .21
IV. Triangulation ... 22
V. Of calculating tangents to any degree of curvature . 24
VI. To trace a curve with transit and chain ... 25
VII.- To triangulate on a curve .... 2?
VIII. To change the origin of any curve, so that it shall termi-
nate in a tangent, parallel to a given tangent . .31
IX. To change a P. C. C. with similar object. First. When ihe
second curve has the smaller radius . . . 52
X. Second. When the last curve has the larger radius . 34
Synopsis of formulae for compound curving . . 35
XL Having located a compound curve terminating in any tan-
gent, to find the P. C. C. at which to commence another
curve of given radius which shall terminate in the same
tangent. First. When the latter curves have the smaller
radii ....... 3C
XII. Second. When the latter curves have the larger radii . 38
1* (5)
VI CONTENTS.
PA.GJ*
ART. XIII. To change a P. R. C. so that the second curve shall .ermi-
nate in a tangent parallel to a given tangent . . 39
XIV. How to proceed when the P. C. is inaccessible . 41
XV. To avoid obstacles in the line of curve . . 43
XVI. To calculate reverse curves . . 45
XVII. Having givon a located curve, terminating in any given
tangent, to find where a curve of different radius will ter-
minate in a parallel tangent . . . 46
XVIII. Having a curve located and terminating in a given tangent,
to find the P. C. C. whereat to begin another curve of
given radius which shall terminate in a parallel tangent 48
XIX. To locate a Y . . . . . 49
XX. To run a tangent to two curves . . . .61
XXL Ofordinates 52
XXII. To find the radius corresponding to any chord and deflexion
angle. Deflexion and tangential distances . . 54
XXIII. Of excavation and embankment . . . .56
XXIV. Side staking ... 60
TABLES.
Natural Sines and Tangents ...... f>3
Radii ..... . 89
Long Chords ... . 90
Ordinates ..... 91
Squares and Square Roots . 94
Slopes and Distances for Topography . . , 108
EXPLANATIONS
ALL railway curves are parts of circles. They are
designated generally from their character as simple, com-
pound, or reverse ; and specifically from the central angle
subtended by a chord of 100 feet at the circumference,
this being the length of the chain in common use. It is
found that the circle described with radius of 5730 feet
has a circumference of 36,000 feet. Since there are 360
in the circle, the central angle subtended by a chord of
100 feet is, in this case, equal to 1, and the curve is
named a one degree curve. So likewise in a circle with
radius of 2865 feet, half of 5730, the central angle cor-
responding to the chord 100 is 2; the curve is then called
a two degree curve.
The beginning of a curve is called the point of curva-
ture, or simply the P. C., and its termination the point of
tangent, marked P. T.
A compound curve is composed of two curves of different
radii, turning in the same direction, and having a common
tangent at their point of meeting. This point is called the
point of compound curvature, or P. C. C.
(7)
RAILWAY CURVES AND LOCATION.
A reverse curve is composed of two curves turning in
different directions, and having a common tangent at their
point of meeting, which latter is named the point of re-
versed curvature, or the P. II. C.
All sines and tangents made use of in this work are
from the table at the end of the volume. For calculating
curves it is not necessary to use more than four decimals.
A. Bench is a shoulder hewn with the axe on the but-
tressed base of a tree, and so shaped at the top as to afford
footing to the rod. The tree is blazed and the elevation
of the bench marked on it with red chalk. Benches serve
as permanent reference points to the level. They are
placed, where it is possible, about one thousand feet
apart.
Points. The operation termed pointing is the fact of
putting a peg firmly into the ground, and of driving in its
top a tack, or making thereon an indentation whose place
is indicated by cross keel marks, directly in the line of col-
limation of the transit. Thus true lines are traced on the
ground, and angles measured accurately. When the transit
is set over a point it is so posited that the plumb hangs
immediately above the tack head. If the head plate of
the tripod be much inclined the plumb should be examined
after levelling the instrument, as that operation disturbs it
to some extent.
Stations. The line of a survey is marked on the ground
at regular intervals, by stakes two feet in length, blazed,
and numbered from up in arithmetical progression.
These stakes are named stations. On exploration they
are commonly placed two hundred, and on location, one
hundred feet apart.
It is customary, when locating, to drive pegs even with
the surface along the true line, and to place the stations a
couple of feet to the right, numbers facing in, to show their
EXPLANATIONS.
position. The pegs are less liable to disturbance from frost,
animals, &c.
In locating for construction stakes are driven on sharp
curves at intervals of 50, sometimes 25 feet.
The Chain in general use for railway surveys is made
of soft iron. It is 100 feet long, and divided into 100
links, each one foot in length. At every tenth link is
attached a brass drop, toothed so as to indicate its distance
from the end. It presents the advantages of durability,
accuracy, and expedition.
A PRACTICAL TREATISE
ON
RAILWAY CURVES AND LOCATION.
ARTICLE I.
OF THE INSTRUMENTS.
THE LEVEL.
THE level is an instrument used in ascertaining the
undulations of the ground
along the line of a survey,
and of measuring these ir-
regularities accurately in
reference to an assumed
base called the datum. It
consists mainly of the
telescope k i, the spirit-
level and its encasement
6, the Y's c c, the rectan-
gular bar o d, the axis e, the plates and levelling screws
/ m, and the tripod g.
In the tube of the telescope at /*, and at right angles to
its axis, is placed a flat ring, called the diaphragm. To
this ring the cross-hairs are attached two delicate spider
lines stretched over it vertically and horizontally, and inter-
secting at the centre. It isjield jjrposition by four
UNIVERSITY
12 RAILWAY CURVES AND LOCATION.
slightly movable screws, which pierce the tube in the
direction of the "cross-hairs." i is a milled head for
adjusting the focus of the object glass, and k an inserted
tube, containing several lenses, which may be moved out
or in so as to make the spider-lines distinctly visible.
A straight line looked along from the eye glass at k
through the intersection of the cross-hairs is the line of
sight, technically named the line of collimation.
The immediate supports of the telescope are called the
Y's, from their resemblance to that letter. If a small arch
were sprung between the two legs of the Y it would give
a good idea of the clasping pieces which hold the telescope
in place. They are jointed to one leg and secured to the
other by pins which may be withdrawn and the pieces
turned back in order to remove the telescope, or change it
end for end.
The Y's are attached at right angles to the bar d, which,
again, is connected firmly at right angles with the hollow
axis e. This latter fits closely over and is revolvable hori-
zontally around a solid axis s, which, passing through the
plate/, is secured to the head of the tripod by means of a
loose ball-and-socket joint. The plate /has four levelling
screws inserted in it ; with these the instrument may be
brought to a horizontal position even when the lower plate
is considerably inclined.
One of the Y's is movable for a short space up or down
by means of the capstan-head screw o. The spirit-level is
likewise movable both vertically and laterally by means of
screws at either end.
n is a clamp screw, and p a tangent-screw for slightly
turning the telescope in a horizontal direction.
THE LEVEL 13
To adjust the Level.
First. To make the line of collimation coincide with the
of the telescope.
Set the instrument firmly, and direct the telescope
toward some distant, distinct object, such as a nail-head. ^
Clamp fast, and with tangent-screw fix the line of collima-
tion upon the object accurately. Revolve the telescope
half way round in the Y's, i. e. until the bubble is above
it, and if the horizontal spider-line still covers the point, it
requires no adjustment. If it does not, reduce the error
one-half by means of the diaphragm screws, and complete
the reduction with the capstan-head screw. Revolve the
telescope round to its first position, and if the horizontal
line and point do not then coincide, repeat the operation
until they do, in any position of the telescope. In similar
wise the vertical hair may be adjusted, when the line of
collimation should cover the point through an entire revo-
lution of the telescope.
Great care should be taken in this as well as in all other
adjustments of cross-hairs, that the opposite screw of the
diaphragm be loosened before tightening its fellow, or
injury to the instrument must result.
Second. To make the axis of the spirit-level parallel to
the line of collimation.
With levelling screws bring the bubble to the middle of
its tube, reverse the telescope in its Y's, and if the bubble
does not then stand in the middle correct one-half the
deviation with the screw at the left end of the bubble-case,
and the other half with the capstan-head screw. Again
reverse the telescope in its Y's, and, if necessary, repeat
the operation.
Now revolve the telescope a short distance in its Y's, so
as to bring the spirit-level to one side of its lowest position.
If the bubble deviates from the middle, correct the error
2
14 RAILWAY CURVES AND LOCATION.
with the lateral screws at the right end of the bubble-case,
and examine the previous adjustment before lifting the
instrument.
Third. To bring the line of collimation parallel to the
bar.
Turn the telescope until it stands directly over two of
the levelling screws, and with them bring the bubble to the
middle of the tube. Then revolve the telescope horizontally
until it stands over the same screws, changed end for end.
If the bubble does not still stand in the middle of the tube,
correct one-half the deviation with the capstan-head, and
one-half with the levelling screws.
Place the telescope over the other levelling screws and
proceed in a similar manner, and continue the corrections
until the bubble stands without varying during an entire
revolution of the instrument upon its axis.
This completes the adjustment of the level.
THE ROD.
The rod used in levelling consists of a staff and a target,
which latter is so attached to the staff as to be movable
along it from end to end. The rod is commonly seven feet
long, but, being composed of two rectangular pieces fitted
together by means of a sliding groove, it can be extended
to nearly double that length. It is graduated to feet and
tenths of a foot. The target is a circle of wood or iron,
usually four-tenths in diameter, and divided into quadrantal
sectors by a horizontal and vertical line which intersect at
its centre. The sectors are painted alternately red and
white, so that their dividing lines are visible at a consider-
able distance. On the back of the target, where it meets
the graduated side of the rod, is fixed a chamfered brass
edging, whereon the space of one-tenth is graven from the
centre down. This is subdivided into ten spaces marking
LEVELLING. 15
hundredths, and these latter divided into halves, so that the
height of the middle of the target above the base of the
rod may be accurately read to within -005 of a foot.
There is a similar graduated tenth on the standing part
of the rod, to be used for high sights when the sliding
groove comes into play.
Both target and rod are provided with clamp screws.
LEVELLING.
The operation technically called levelling is performed
thus :
Suppose a the starting point, or zero, in reference to
which all the inequalities of the surface along the line of
survey are measured, as at the points <?, ,/. The hori-
zontal line af is called the datum line. This is arbitrarily
assumed. It may be considered, for example, at any dis
tance above the point #, and the irregularities of the ground
measured from an imaginary level line in ether ; but for
convenience of figuring, and other politic reasons, it is cus-
tomary in seaport towns to take high tide as datum. In*
land, the summer surface of the nearest stream, or, when
commencing on a ridge, the highest neighbouring knoll is
issumed.
Well ! suppose a to be zero, and the instrument, for
nstance, set and levelled at b. Stand the rod at , and
nilide the target up until its cross-lines are covered by the
cross-hairs in the telescope ; i. e., until the line of collima-
tion coincides with the centre of the target. The leveller
directs the movements of the target by raising or lowering
16
RAILWAY CURVES AND LOCATION.
his hand. A circular motion of the hand signifies " make
fast." The bubble should always be examined before the
rod is taken down, and the latter should be read twice, or,
if convenient, shown to the leveller, in order to guard
against mistake. If in this case it reads 8 feet, the height
of the instrument is then 8 feet above a. To find the ele-
vation of c above a, take the rod thither and lower tho
target until coincidence results as before. If the rod
reads 2 feet, of course c is 8 2 = 6 feet above a.
If it is necessary to lift the instrument here, a small peg
is driven at c before sighting to that point, to insure firm
footing for the rod. Sighting back from the new position,
d, the rod reads 5 feet ; then 5 -j- t>, the elevation of c
above a, = 11, the height of the telescope at d above a.
If at e the reading is 6, the elevation of that point is 11
6 = 5, and if at / the reading is 8 the elevation of
that point in reference to a is 11 8 = 3, marked -{- 3,
The rule, therefore, in levelling is, at each new stand of
the instrument, to add the reading of the rod sighted back
at, to the discovered elevation of the point at which th<*
rod stands, for the height of the instrument ; and to sub
tract from this height the reading of the rod at any points
observed from the new position in order to find the eleva-
tion of those points. The above is noted in the field-bock
as follows:
Station.
Rod,
Height of
Instrument.
Total, or
Elevation.
a
8-00
8-00
00
C
2-00
+ 6.00
5-00
11-00
e
6-00
-f- 5-00
f
8-00
+ 3-00
i . .
"
OF THE TRANSIT. 17
The advantages of this method of levelling over the old
system of backsights and foresights are, that it affords
readier facilities for testing the correctness of the work,
and it may be carried on more rapidly. By the old plan
each sight at the rod was linked with that which preceded
it, and added one more to a continuous calculation in which
a single error affected all the following work. Here, how-
ever, if haste is required, the calculation of the interme-
diate sights or "cuttings" may be omitted entirely while
in the field, the reading of the rod only being set down ;
the "totals" may be worked from peg to peg, and the lia-
bilities to mistake thus decreased about eighty per cent.
OF THE TRANSIT.
The transit is an instrument for measuring horizontal
angles. It consists of the tele-
scope a c, the Y's d, the compass-
box, &c., e g, and the axis k.
The telescope is furnished like
that of the level, and the instru-
ment is similarly fitted to its tri-
pod. The telescope revolves in
a vertical circle, and is attached
to the Y's by means of a trans-
verse axis whose extremities turn in smooth journals at the
head of the Y's. The body of the instrument at/ contains
a magnetic needle, with its usual circular surrounding,
graduated to degrees and quarter degrees. The flooring of
this box has, on one side, an opening with chamfered edge
upon which the vernier is engraved. This latter, together
with the telescope, Y's, and all the upper part of the
instrument, is made to revolve by means of the screw h,
upon a solid plate beneath, which is likewise graduated
from to 180 each way. Thus angles may be measured
2*
18
RAILWAY CURVES AND LOCATION 7 .
accurately without using the needle at all. It need be
regarded merely as a check, g is a clamp screw for secur-
ing the plates together, and i a screw for fastening the
needle so as to prevent its vibrations while the instrument
is being carried from place to place. A plumb is suspended
from the axis of the transit, by means of which its centre
may be placed over a point on the ground.
THE VERNIER.
The vernier, in the transit, is a graduated index which
serves to subdivide the divisions of the graduated arc on
the lower plate. There are many varieties of the vernier,
but familiarity with one renders easy the acquaintance
with all, since the same general principle is pervading.
The figure represents a common form. Let a b be part
of any graduated arc, and c d the vernier. It will be
observed that the degrees on the limb are divided into
spaces of 15' each. Now if the vernier be made equal in
length to fourteen of those spaces, and be further divided
into fifteen equal parts, it is evident that each of these
parts will contain 14'.
Then, if of the vernier coincides with any division of
the limb, the first line of the vernier to the left will be
just one minute behind the first line of the limb to the
left ; the second vernier line two minutes behind the second
limb line, and so on ; so that if the vernier be moved to
the left over the space of 15' on the limb, the lines from
to 15 of the vernier would coincide successively with lines
TO ADJUST THE TRANSIT 1 J
of the limb, and thus any angle may be read accurately to
minutes.
The vernier in the figure reads 48' to the left. A
vernier graduated decimally is much more convenient on
railway locations than those with the common graduation
to minutes. This is principally on account of its adapted-
ness to running in curves when the 100 feet chain is used.
The work can be done with more ease and rapidity. One
objection to it is that the tables in general use are calcu-
lated for degrees and minutes.
TO ADJUST THE TRANSIT.
Place the instrument firmly at a, level it, clamp all
fast, and with tangent-screw set the cross-hairs on the
point b, at any convenient distance. Reverse the telescope
on its axis, and fix another point in the opposite direction,
.....
of
as nearly as possible equidistant from a. Now loose the
lower clamp and revolve the entire upper part of the
instrument half way round on its axis. Clamp fast, and
having brought the cross-hairs again to coincide with 5,
reverse the telescope. If the sight strikes as before, the
instrument is in adjustment. If not, place another point,
d, where it does strike, and suppose c to be the point pre-
viously fixed : the point e, midway between d and c, is then
in the straight line. With the adjusting pin carefully
place the vertical cross-hair upon /, distant from d one-
quarter of the space d c with tangent-screw set it on e.
and reverse the telescope. If the points have been cor-
rectly placed, and the hair properly moved, the sight will
strike b, and the adjustment is complete.
20
RAILWAY CURVES AND LOCATION.
After finishing this adjustment, the telescope may still
net revolve truly in the meridian. This inaccuracy there
is no method of removing in the field. It should be sent
tc an instrument-maker for repairs.
ARTICLE II.
PRELIMINARY PROPOSITIONS.
1. In any circle the angle o cf at the centre, subtended
by the chord o /, is double the angle o af, at any part of
the circumference on the same side of the chord.
2 The angle fbe, formed by any chord / 6, with a,
tangent at either extremity, is called a tangential angle.
AVOIDING OBSTACLES. 21
and is equal to half the angle / c b at the centre, or is
equal to the angle fa b at the circumference.
. 3. The exterior angle dbf, formed at the circumference
by the two equal chords a 5, b /, is called a deflexion angle,
and is equal to the central angle b of, or double the tan-
gential angle e bf. df is called the deflexion distance,
and e /the tangential distance.
4. The exterior angle p o m of two unequal chords, is
equal to the sum of their tangential angles, or half the sum
of their central angles.
5. The exterior angle i k o, formed by tangents, is equal
to the central angle b c o, subtended by the chord which
connects their points of contact with the curve.
ARTICLE III.
TO AVOID AN OBSTACLE IN THE LINE OF TANGENT.
A GLANCE at the figure will show, that having deflected
to <?, and placed the instrument at that point, the angle
he d must be made equal to twice d be, and the distance
c d equal to the distance b c. Still another deflection at
d, equal to the original angle turned, is necessary in order
to sight again along the tangent.
Should the obstruction be continuous a parallel line
may be run, as from c to /, by deflecting at c an angle
22 RAILWAY CURVES AND LOCATION.
equal to that at 5, and at /, repeating the deflection in
order to strike tangent.
If the angle dbc exceeds 4, and the distance be is
greater than 200 feet, or even with an angle of 2|,
should the distance be greater than 300 feet, be will
differ sensibly in length from b k, and a calculation of the
latter becomes necessary. To effect this, multiply the
natural cosine of the angle k b c by b c. This result
doubled will give b k d, the length proper along tangent.
Thus : Suppose k b c, the angle deflected, to have been
5, and the distance b c 340 feet. Then -9962, the natural
cosine of 5, multiplied by 340, gives 338*7 for the dis-
tance b k. Double this makes bkd = 677-4, and shows
a difference of 2-6 feet between b d and bed.
ARTICLE IV.
SHOULD THE OBSTRUCTION LIE ON THE OPPOSITE BANK
AND it is desirable on any account not to run the line
from d y corresponding to a d, set the
instrument at a, in tangent, and deflect
clear of the obstacle to d. Point at d.
deflect to e, and point also there
marking the angles c a d, d a e. Chain
the base de, and placing the transit at
e, measure the angle dea. Data are
thus obtained sufficient for the calcula-
tion of the line d a. The object now is
to find the point c and the angle dca.
The angle a d e subtracted from 180 will supply the
AVOIDING OBSTACLES. 23
angle c d a, so that in the smaller triangle we have
obtained two angles and their included side. The dis-
tance cd, and angle dca readily follow. The transit
standing at e, c is placed, of course, in the prolongation
of the base d e, and the distance c d is carefully set off
with the rod. Moving the instrument to c, and turning
the angle ecf = 180 dca, we are again in tangent.
Example. Let cad= 6, dae = 35, dea = 42,
and the base d e = 200 feet. Then in the triangle dea
we have
Nat. sine d a e = (-5736) : nat. sine d e a = (-6691) : :
d e = (200) : d ,
f)691 v 200
Wherefore, d a = &>-- = 233-3 feet.
*O ( o\)
Again, the angle cda = 77. The angle d a c being
6,acd is consequently = 97, and in the small triangle
we have, Nat. sin. a c d (-9925) : nat. sin. c a d =
(1045):: da = (233-3): c d.
1045 X 233-3
Therefore c d = - 79925 = 24-564 feet, and d cf
180 97 = 83.
NOTE. A common and convenient plan for triangulat-
ing a creek is as per figure. Set the in-
strument at b, fix a point d on the oppo-
site shore, and making d b c a right angle,
place c at any convenient distance. Now
move to <?, sight to c?, and making d c a a
right angle also, fix a, in the same line
with b and d. a, c, and d are points in
the circumference of a circle whose dia-
meter is a d, a b : b c : : b c : b d, and therefore Id = ~r
24
RAILWAY CURVES AND LOCATION.
ARTICLE V.
HAVING GIVEN THE ANGLE edb, FORMED BY THE INTERSEC-
TION OF TWO STRAIGHT LINES, IT IS REQUIRED TO FIND
THE POINT a OR 6, AT WHICH TO COMMENCE A CURVE OF
<HVEN RADIUS.
Draw the bisecting line d c. Then the angle d c i =
half the angle acb or its
equal e d b ; and in the tri-
angle d c a, the angle d a c
being a right angle, we have
Rad. of 1 : Nat. Tang.
do a : Rad. ca : ad. There-
fore ad = Nat. Tang, d c a
X Rad. e a.
Example 1. Let e d b
= 48 and a c = 1460
feet. Here half the angle
edb or acb = 24, the
Nat. Tang, of which is *4452 ; and multiplying by Rad.
1460, we have 650 feet for the length of a d or d bj the
tangents.
2. If a die given and radius required,
ad 650
Kad. = c^ T A A /--> 1460.
Nat. Tang, a c d -4452
The following rule? are approximate, and sufficiently
correct for all purposes of location.
To find the degree of curvature of a b divide 5730 by
the radius in feet ; and to find the length of the curve in
feet divide the angle a c b (after reducing minutes to hun-
dredths) by the degree of curvature the chord in each
case being 100 feet in length.
TO TRACE A CURVE. 25
ARTICLE VI.
TO TRACE A CURVE WITH TRANSIT AND CHAIN. t
THE legree of curvature and the angle to be turned are
known. If the latter is expressed in degrees and minutes,
reduce the minutes to hundredths, since the 100 feet chain
is used, and divide the whole angle by the degree of cur-
vature. The quotient will be the length of the curve in
feet, and the P. T. is at once ascertained.
Let m a be the tangent, and a the P. C. Place the
transit at a, index reading 0, and direct the sight along
the tangent m a e. The first deflection will be half the
central angle subtended by the chord used, and all the
stakes put in from a will be fixed by similar tangential
deflections. (Prelim. Prop. 1.)
3
26 RAILWAY CURVES AND LOCATION.
When the point d is reached, the angle dab, shown
on the index, will* be half the angle dbe, or its equal
a c d, at the centre. Move the instrument to d, sight
back to a, and turn to double the index angle. The
telescope is now directed along the tangent b d g, and
the angle dbe = acd = dab-{-adb 1 reads on the
index. Note this angle in the column of tangents oppo-
site station d. Continue the curve from this new posi-
tion, precisely as was done at a, and set the point h.
Move to h, see that the vernier has not been disturbed,
and sight back to d. The index now shows the angle
(d b e -j- h d </), and the object is to turn the angle d hf,
i. o. repeat the angle fd A, as was done before at d, and,
at the same time, have the whole angle (d b e -f- hfy)
indicated on the plate. To effect this, merely add this
angle fhd to the present reading. It will be found sim-
pler, in practice, to double the entire angle thus far turned,
and subtract from the product the last tangent, viz. d b e.
The vernier, turned to this resultant angle, will put the
telescope in tangent line to h. And so on.
^Example. u At sta. 24 -f- 50 commence a 4 curve to
the left for 35 12'." Suppose this a required duty. First,
reducing minutes to hundredths, we have 35 -20, which,
divided by 4, gives 880 feet for the length of the curve.
Adding 880 to 24 -j- 50 it is at once seen that sta. 33 + 30
is the P. T
Let a be the P. C., = 24 -f 50. Now the deflexion
angle being 4, the tangential angle is 2, with a chord of
100 feet. With a chord of 50 feet, therefore, the tan-
gential angle is 1, and this deflexion from tangent m a e
fixes station 25. A deflexion from this latter point of 2,
the chord being 100 feet, fixes station 26. And so on.
When you have fixed the point d, = sta. 28, the index
reads 7. .Move up to station 28, sight back to the P. C.,
and turn the index to 14. This throws you on tangent
TO TRACE A CURVE. 27
Proceed as before, with the 2 deflexions, to sta. 31, = h.
Move up, and sight back to sta. 28. The index now reada
20. Multiplying by 2, and subtracting the last tangent,
we have the reading of the tangent at h = 26 : we have
turned 26 of the curve. Continue as before. After
putting in sta. 33, to find the deflexion which shall fix the
P. T., 33 + 30, say, as 100 feet : 30 feet : : 2 : the re-
quired deflexion, = 36'. We may here remark the great
convenience of an instrument graduated to hundredth of
a degree instead of sixtieths. In the present example it
would be seen immediately that the tangential angle for
100 feet being 2, for 1 foot it would be 2 hundredth* of
a degree, and for 30 feet it would be 60 hundredths.
Well ! when the P. T., = 33 -f 30, is fixed, the index
reads 30 36'. Move up, see that the vernier has not been
disturbed, and sight back to sta. 31. Now twice the index
reading, minus the last tangent, = 61 12' 26, =
35 12', the present tangent, which is the final tangent,
which finishes the curve.
The advantage of this manner of running a curve is that
the instrument shows at a glance the work done, and there-
fore errors may be detected with greater facility. By
comparing at the P. T. the total index angle with the
distance run, the work is tested at once.
28 RAILWAY CURVES AND LOCATION.
The above is recorded in the field book as follows
6
03 .
J3 O
o> -M o
3 S oi2
S
g
o S
ctt co
r- I <M <M CO CO
O O O O O O O
O O O O O O CO
In running compound and reversed curves the operation
is quite as simple as the foregoing. A point is fixed at
the P. C. C., or P. R. 0., and turning into tangent at
that point, the second curve is traced from this tangent,
without regard to what precedes. In reverse curving, it
is a good plan to adjust the index in such manner at the
P. R. C., that when we turn into tangent it will read 0.
TO TRIANGULATE ON A CURVE.
29
This saves troublesome work, and it is advisable moreover
to show in the field-book the contained angle of each curve,
as well as the test of the two tangents with the magnetic
course.
ARTICLE VII.
TO TRIANGULATE ON A CURVE.
SET the transit at a, and, as usual, sight back, and turn
into tangent. Estimate the distance to the farther bank
do it liberally and make a deflexion around the curve,
corresponding to your estimated distance. Fix a point b
in this line. Measure any convenient angle, b a c, and sut
the point c. Move to 6, measure the base b <?, the angle
a b c y and, before lifting the instrument calculate the line
b a. If the angle turned from tangent to d exceeds 4,
and the distance is greater than 200 feet, the chord a d
must also be calculated, as per example, and the difference
between this and b a will be the distance b d to the point
d, in the curve, which can be fixed from b.
3*
30 RAILWAY CURVES AND LOCAUON-
Should b fall between d and a the operation is analogous.
Example. Let a be a point in a 6 curve. Having set
the transit, and turned into tangent, the distance to the
farther verge is estimated 400 feet. The tangential angle
for 100 feet is 3, and to fix d, 400 feet distant, is conse-
quently 12. Deflect this angle, fix a point in line, and
complete the triangulation, as previously illustrated in
Art. IV., p. 22. Suppose a b found equal to 472 feet.
Now the tangential angle to d = half the central angle,
= fea, = 12; and to find the length of the chord a d y
we have, in the triangle efa,
Rad. : Sin. a ef : : ea : af, that is
Rad. of 1 : Nat. Sin. 12 : : 9554, the Rad. of the 6
curve : half the chord required. Wherefore a d = twice
the Nat. Sin. 12 X 955-4 = -2079 X 2 X 955-4 ==
397-25 feet. Subtracting this from 472, we have the
distance, 74*75 feet, back to the point in the curve. Move
the instrument to d, set the index at 12, sight back to a,
and turning to 24, the telescope is in tangent. A deflexion
of 3 will fix the next station.
NOTE. In this case, if preferred, a third proportional
might be formed with the chord of crossing, as shown in
the note to Art. IV.
TO CHANGE THE ORIGIN OF A CURVE.
31
ARTICLE VIII.
TO CHANGE THE ORIGIN OF ANY CURVE, SO THAT IT SHALL
TERMINATE IN A TANGENT PARALLEL TO A GIVEN TAN-
GENT.
LET df/be the located curve, terminating in a tangent/^,
and the nature of the ground requires that it should ter-
minate in the tangent e i, parallel to fk. At /, the tele-
scope being directed along the tangent fk, turn to the
right an angle equal to the central angle d bf, previously
turned to the left on the curve. This will direct the tele-
scope along ef y parallel to d I. Measure <?/, and go back-
on the tangent, d I, a distance, c d, equal to it. The curve,
retraced from c and consuming the same angle, will termi
nate tangentially in e i. An example in this case is not
necessary.
82
RAILWAY CURVES AND LOCATION.
ARTICLE IX.
TO CHANGE A P. C. C. SO THAT THE SECOND CURVE SHALL
TERMINATE IN A TANGENT PARALLEL TO A GIVEN TAN-
GENT.
LET a b d be the compound curve, located and terminat-
ing in the tangent d h. Continue the larger curve to e,
and from e, with radius e I = k 6, describe the curve ?/,
terminating tangentially in / g, parallel to d h. From c,
the centre of the larger curve, let fall upon fg the perpen-
dicular eg, and fill up the figure as above. Call the radii
respectively R and r, the angle b k d, or its equal, k e m,
x, and the angle e If, or its equal, lcn,y. Let the dis-
tance if, or h g, be named D. Now the line c g is made
up of the lines c m -f- m h -{- kg, i. e., eg = cosin. x
( R r) -f r -f D. c g is also made up of the lines c n
+ ng, i. e., eg = cosin. y (R r) + r. Therefore cosin.
x (R r ) _|_ r -f_ D = cosin. y (R r) + r, and reducing,
TO CHANGE A ? P. C. C. 33
cosin. x (R r) 4- D
cosm. y == T^ ; so that the distance z/,
or Jig, measured rectangularly between the two tangents,
being added to the nat. cosin. x 9 will give the nat. cosin. of
the angle e If, to be turned on the smaller curve. The
angle y, subtracted from the angle x, gives of course the
angle b c e, to be advanced on the larger curve ; or, divid-
ing this angle by the degree of curvature of a b, we find
the distance from b to e the P. C. C. proper.
If ef be the second curve located, and the tangent to be
touched lies within, it is evident that we must retreat upon
the large curve, and, by subtracting D from the cosine of
the angle y, we obtain the cosine of the angle x.
^Example. Suppose a b a 3 curve located, and com-
pounding, at b, into a 6 curve, which latter is continued
to the right through an angle of 42. At the P. T. we
discover that the proper tangent is 64 feet to the left.
We must throw our curve out, then we must advance on
the 3 curve a certain distance. How to find this distance :
The radius of a 3 curve = 1910 ; the radius of a 6 curve
= 955-4; R r, therefore, = 954-6. The nat. cosin.
42 = -7431. Now, by the formula just obtained, we
cos. x (R r) + D (-7431 X 954-6) 4- 64
- i '_ -
, -j \ r\r * rt ' ~~~-
(R r) 9o4-6
8101 = nat. cosin. 35 53'. Subtracting this from 42*,
we have 6 07', the angle to be advanced on the 8 curve ;
or, reducing minutes to hundredths, and dividing by 3, we
find 204 feet, the distance from b to the correct P. C. G
34
RAILWAY CURVES AND LOCATION.
ARTICLE X.
SHOULD THE SECOND CURVE BE ONE OF LONGER RADIUS
THAN THE FIRST,
OUR illustration takes simpler form, and the application
of D varies vice versa.
See figure, analogous to that of the previous problem.
Here of, eg are equal and parallel radii; fh a perpen-
dicular connecting them. Draw its fellow, c n. Then,
nf = c h, and, consequently, h g = o n. Again, letting
k m fall, perpendicular to eg, we have c m = n I, and o I
= c m -j- o n', i. e., cosin. y (R r) = cosin. x (R r)
-f-D-
We observe that, with a curve of this nature, in order
tc throw the line farther out, it is necessary to go back,
toward b ; or, having located to y, if the object tangent lie
within, we must advance toward e.
Example. Suppose a b a 5 curve, b the P. C. C., and
bg a 2 curve. Setting the instrument at g, the P. T.,
and turning into tangent, we find that we are a distance
Jig, = 53 feet, too far to the left. The first question is,
what angle have we turned on the second curve. Let it
TO CHANGE A P. C. 0. 35
be 28. Now we know, that, in order to strike farther to
the right, we must advance on the 5 curve. Consequently,
D must be added to the cosine of 28, to give us the cosine
of the proper angle for the 2 curve ; and the difference
between 28 and this newly found angle will be the angle
we are to advance on the 5 curve. Thus : the rad. of a
5 curve = 1146 feet, that of a 2 curve = 2865 feet
and their difference = 1719 feet. The nat. cos. of 28'
= 8829. Then =. 9137, = nat.
cos. 23 58*. This, subtracted from 28, leaves 4 02',
80 feet, from b to the correct P. C. C.
Synopsis of the preceding formulce.
Call D the distance between tangents as before, a the
angle of the second curve located, and b the angle of the
same curve to be substituted for it.
FIRST, when the second curve has the smaller radius
Tangent falling within the point, cosine b =
cos, a (R -r) J-J)
(R-rj"
Tangent falling without the point, cosine b =
cos. a (R r) D
SECOND, when the second curve has the larger radius
Tangent falling within the point, cosine b =
cos, a (R r) P
(R r )
Tangent falling without the point, cosine b =
cos. a (R r) + D
Very little attention will familiarize these formulae, and
render the field practice easy.
36
RAILWAY CURVES AND LOCATION.
ARTICLE XL
HAVING LOCATED THE COMPOUND CURVE a b d, TERMINAT-
ING IN THE TANGENT df, IT IS REQUIRED TO FIND THE
P. C. C. 5, AT WHICH TO COMMENCE ANOTHER CURVE OF
GIVEN RADIUS, WHICH SHALL ALSO TERMINATE TAN-
GENTIALLY IN df.
PLOT the curves as per figure. From c let fall c g, per-
pendicular to the tangent, df. From k and , the lesser
centres, drop Jc m, i Z, perpendicular to c g. Call the great
radius R, the smaller radius r, and the intended radius of
the second curve /. Likewise name h k d, the angle of
the small curve located, x, and b i e the angle to be found
for the proposed curve, y. Now, the tangent df, and the
curve a b, lying unstirred, the line c g is an unvarying dis-
tance, and it is made up of the lines c m -f m g, i. e., c g
= (R r) cosin. x + r. It also consists of the lines c I
COMPOUND CURVES 37
-f- lg, i. e., eg = (R /) cosin. y -}- /, and, reducing,
(R r) cosin. x -f- r /
nat. cosin. y = ^ ,r . This, there
fore, is the formula by means of which we can ascertain
the point #, as follows :
^Example. Imagine a 2 curve, a A, compounding into
a 6 curve, h d, which terminates at c?, in the tangent df.
The tangent lies well ; the curve a h likewise ; but it is
desired to throw the line to the left, on better ground,
between d and A, by means of an intercalary 4 curve.
We wish, then, to know the distance, h b, back to the new
P. C. C.
The radius of a 2 curve = 2865 feet, of a 6 curve =r
955-4 feet, and their difference (R r) = 1909-6. The
radius of a 4 curve = 1433, and the difference (R /)
is. therefore, 1432 feet. Let h k d, the angle turned on
the 6 curve, be 41, the nat. cos. of which = -7547.
T , (-7547 X 1909-6) + 9554 1433
Then, * 1432~ ~ = ' ^
nat. cosin. 47 43'. Subtracting 41, we have 6 43', the
angle h c b. Reducing minutes to hundredths, and divid-
ing by 2, we find 336 feet to bo the distance from h to 6.
A 4 curve of 47 43', traced from this latter point, will
terminate in the tangent ej.
RAILWAY CURVES AND LOCATION.
ARTICLE XII.
IF THE LATTER CURVES HAVE LARGER RADII THAN THE
THE solution retains its shape and simplicity.
Draw the figure as above, and, for the sake of uniformity,
name the radii as before. The curve a 6, and tangent df,
being constant, the distance m g, or d w, is here constant
Call it A. Now A, in the first place, is equal to d i n i,
i. e., = r (r R) cosin. x\ and, in the second place,
it is equal to g c m <?, = / (/ R) cosin. y ; where
fore r (r R) cosin. x = / (/ R) cosin. ^, and
(r R) cosin. x -}- / r
consequently cosin. y = - > ' , , __ -p \
Example. Suppose a b a T c curve, compounding, at ,
into a 5 curve, b d, which latter subtends an angle of 38,
and terminates in the tangent df. We wish to substitute
a terminal 2 curve, Jig, and to know the position, A, of
the new P. C C.
The radius of a 7 curve = 819 feet = R. r and r',
TO CHANGE A P. R. C.
39
the radii respectively of 5 and 2 curves, are equal to
1146, and 2865 feet, r R, therefore, = 327, and
/ R = 2046 feet. The nat. cosin. of 38 = -788.
mi k ^ r i ('788 X 327) + 2865 1146
Then, by the formula, * on/ia - ~ ==
9661, = the nat. cosin. of 14 57', the angle to be turned
on the 2 curve. Subtracting this from 38, we have 23
03', the angle to be continued on the 7 curve. Reducing
minutes to hundredths, and dividing by the degree of curva-
ture, 7, we find 329 feet, the distance from b to the new
P. C. C. h.
ARTICLE XIII.
TO CHANGE A P. R. C., SO THAT THE SECOND CURVE SHALL
TERMINATE IN A TANGENT PARALLEL TO A GIVEN TAN-
GENT.
LET a d I be the reverse curve, located and terminating
in the tangent I h. Call the radius c J, R, and the radius
b e, r. Suppose ig the given tangent. At a distance
40 RAILWAY CURVES AND LOCATION.
from it equal to i e, the radius of the second curve, draw
the parallel line, op. With c as a centre, and radius of,
= R -j- r, describe the integral curve, /e, cutting op in e.
d then, is the centre of the curve adjusted.
Application.
Place the transit at the P. T., Z, and turn into a tan-
gent, Im, parallel to d n, the common tangent of the two
curves at d. Unless some wide mistake has been made,
the distance Z&, measured along this line to ig, the tan-
gent proper, will be about equal to the distance ef, and
we shall have the proportion, cf :fe : : c d : db, i. e., R
+ r : ef : : R : d b, which gives -^5 , as a simple
formula for finding the distance back from d to 5, the cor-
rect P. R. C. This rule, though sufficiently true for most
cases, is not mathematically justifiable. It will be seen
that ef, or its equal i I, the distance we wish to measure,
is a curving distance, part of the circumference of a -circle
concentric with a b. Its radius is (R -f- r), therefore its
5730*
degree of curvature = ,^ ., or, more simply, equals
the product of the degrees of curvature of the curves com-
posing the reverse, divided by their sum. To be strictly
accurate, then, set the instrument at Z, turn into tangent
I m as before, and trace the curve i Z, until it strikes the
tangent ig. The angle which il subtends, being divided by
the degree of curvature of a 6, will give the distance, d 6,
to the P. R. C. proper. The curve retraced from 6, will
terminate tangentially in ig, and its angle, bei, will be
equal to df I deb.
Example. Let a d I be a reverse curve, composed of a
3 curve, a d, and a 6 curve, d 1. Let the angle df I be
equal to 52, and suppose the distance If to have been
WHEN P. C. IS INACCESSIBLE.
41
found 34 feet. Being part of a 2 curve, it therefore sub-
tends a central angle of 41'. This corresponds to a dis-
tance of 23 feet, to be gone back on the 3 curve, and
52-00 41' = 51 19', the angle to be turned from h
on the 6 curve, in order to strike the tangent ig.
ARTICLE XIV.
HOW TO PROCEED WHEN THE P. C. IS INACCESSIBLE.
IN the figure, drawn to illustrate this case, let c be the
point of curvature, c a the tangent, and eke the curve.
Now the angle d c e, included between the tangent and
any chord, as c e, fixing the point e, is known. Make c b
along tangent, equal to c e, and connect be. If a circle
were now described from c as a centre, with radius c e or
c b, d, e, and 5, would be points injiajurcjimference, and
42 RAILWAY CURVES AND LOCATION.
the angle dbe at once proven equal to half the angle dot.
With proof precisely similar, d c e = half of d g e, and,
consequently, d b e is equal to one-fourth of the central
angle subtended by the chord c e.
Example. Suppose c to be the inaccessible point of
curvature of a 6 curve, eke. It is concluded to run to
the third station, e. First we must calculate the length
of the chord c e. The angle d c e = 9, and from Art.
VII. we have
Rad. of 1 : 9554 : : nat. sin. 9 = -1564 : ^, whereby
e G is shown equal to 298-8. Place the transit then at 6,
298-8 feet distant from the P. C., and deflect to the left
an angle of 4 30', equal to half the angle dee. This is
in line to e, and b e must likewise be calculated as follows :
In the triangle b c h we have
Rad. of 1 : nat. cosin. 4 30' = -9969 : : b c = 298-8 : b h
b e
, whereby b e is shown equal to 595-7 feet. Arriving
at e, the index reads 4 30'. Sight back to 6, turn to 18,
and the telescope will be in tangent. Suppose, however,
that having reached /, 100 feet from e, this latter point is
also found inaccessible. We find k a different point in the
curve, thus: The angle feg = 18 4 30' = 13 30',
and the tangential angle g e k = 3. Consequently the
angle fek = 10 30', and, drawing the bisecting line e i,
we have, in the triangle efi,
Rad. of 1 : nat. sin. 5 15' = -0915 : : ef = 100 : fi
= 9-159 feet. Therefore fk = 18-318 feet, and the
angle efk = 90 5 15' = 84 45'. At / deflect this
angle to the right, and measure the distance fk carefully
*vith the rod. At k, sighting back to /, and turning the
equal angle fke, the telescope will be directed to e, and
the curve may be continued.
If it is inconvenient to run the line b e, the point e may
TO AVOID OBSTACLES IN THE LINE OF CURVE. 43
be reached thus : Fix the P. C. Find the tangential
distance d e, corresponding to the angle dee. Carefully
with the rod lay off b I, equal to it, at right angles to b e
Set the transit at Z, and, in line with c, put in e.
The distance b I should not exceed 10 or 12 feet.
NOTE. The foregoing illustrations will apply when the
P. T. is likewise inaccessible.
ARTICLE XV.
TO AVOID OBSTACLES IN THE LINE 0"F CURVE.
LET b Jc h be the curve. We can either follow the tan-
gents h d, db, or trace a parallel curve, g a, within the
first ; which tracing is effected thus : Set the instrument
at A, the P. C., and offset any distance Jig, at right angles
to the tangent h d. It will be observed that as the dis-
tance h g increases, the distance g a decreases, whilst the
angle subtended by g a remains equal to that subtended
44 RAILWAY CURVES AND LOCATION.
by h b ; i. e., our deflexions on the offset curve stand
unchanged, but the corresponding chords, g /, f.e, &c.,
are less than their equivalents, h i 9 i k, &c., along h b.
To find their length, h i, i k, &c., being equal to 100
feet, we have the proportion, c h : eg :: hi: x; i. e.,
R, : rad. h g : : 100 feet : x, where x symbols the un-
known chord. Now set the transit at g, turn into tangent
parallel to h d, and with the shortened chord, fix fea.
Rectangularly to the tangents at these points, and distant
h g, will be i, k, b of the curve proper.
Example. Let b h be a 4 curve, and the offset distance
85 feet. The radius then is 1433, and
1433 : 1348 : : 100 : 94, the short chord.
To follow the tangents, suppose the angle b c h = 42.
Then by Art. V. we find the tangent h d = 550 feet,
which distance we duly measure, and, at d, deflecting 42,
lay oft an equal distance to b, the point of tangency.
FIND THE RADII OF REVERSE CURVE.
45
ARTICLE XVI.
HAVING GIVEN THE ANGLES d b k, mkl, AND THE DISTANCE
b &, IT IS REQUIRED TO FIND THE RADII C 6, ef OF THE
EASIEST REVERSE CURVE WHICH SHALL UNITE a d, ~k
THE angle d b e is equal to the angle a c e, half of which
is b c e. So likewise ef k is equal to half of I km.
Then, [nat. tang, b c e -f nat. tang, efk] : nat. tang,
b c e : : b k : b e, and b k b e = e k. Wherefore rad. c e
be ek
= v and rad. ef = 7 ?-.
nat. tang, bee nat. tang, kfe
Example. Suppose the angle d b e = 54 30', the
angle lkg = W 20', and the distance Ik = 832 feet.
Therefore the angle b c e = 27 15', the nat. tang, of
which is -5150, and the angle efk = 16 40', the nat.
tang, of which is -2994. The sum of the tangents = -8144.
Then, to find b e, we have
As -8144 : -5150 : : 832 : 526, and subtracting this from
b k, we have e k = 306 feet.
526 306
Again, the radius c e = ^TH, and the radius ej = .2994?
= 1022 feet.
46 RAILWAY CURVES AND LOCATION.
ARTICLE XVII.
HAVING GIVEN THE CURVE /(/, LOCATED AND TERMINATING
IN THE TANGENT g m, IT IS REQUIRED TO FIND WHERE
A CURVE OF DIFFERENT RADIUS WILL TERMINATE IN A
TANGENT PARALLEL TO g m.
be the curve located, and fh the curve proposed,
commencing at the common point /, and terminating in
the parallel tangents, gm,hl. We wish to find the length
und direction of the line g h, connecting the points of tan-
Call the radii respectively R and r, and the central
angle, fdg orfch, x. Now g k = d e, = sine of x, c d
being radius ; i. e., = nat. sin. a; X (R r). Again, the
%
angle kg h = fh b = -, and g k = cosin. kg h, g h being
SY*
radius, i. e., g k = nat. cos. - X g h, and, consequently.
nat. cos. - X g h = nat. sin. x X (R r), wherefore
(R r) nat. sin. x
gh= .
nat. cos. <r
It will be observed that the angle leg h, included between
TERMINATION OP A CURVE. 47
the tangent to a curve at any point g, and the line g h
connecting </ with an equivalent point h in any other curve
fh commencing at the same P. C.,/, and turning in the
same direction, is invariably equal to half the common
central angle, fdg orfch.
Example. Let fg be a 7 curve, subtending a central
angle, fdg, of 44 26'. Having arrived at g, the P. T.,
and turned into tangent, g m, it is desired to fix the P. T.,
A, of a 4 curve which shall likewise subtend an angle of
44 26'. Here the radii are, respectively, 819 and 1433
feet, and E, r = 614. The nat. sine of 44 26' = -700,
and the nat. cosine of half this angle, viz. : 22 13' =
614 y *7
9258. Then, by the formula, g h = --- = 464-2.
Deflecting, therefore, to the left, an angle of 22 13', and
laying off the distance 464-2 feet, we arrive at the point h.
Move to A, sight back to g, and a deflexion of 22 13' to
the right will direct the telescope along the tangent h L
If h were the P. T. located, and g the point required,
the same angle and distance would apply.
48 RAILWAY CURVES AND LOCATION.
ARTICLE XVIII.
HAVING THE CURVE nfg LOCATED, AND TERMINATING IN
THE TANGENT g W, IT IS REQUIRED TO FIND THE POINT
/, WHEREAT TO COMPOUND WITH ANOTHER CURVE 01
GIVEN RADIUS, WHICH SHALL TERMINATE IN i I, PARAL-
LEL TO g m.
[See previous figure.]
NAME the radii and angle as before. Measure the dis-
tance g i, between the tangents, and call it D. Then c h
is equal to c e -j- e k -f- g i or k h ; i. e., R = (R r)
R_( r _D)
cosm. x -f- r -\- D, or, transposing, cosm. x = j^ \
Thus discovering the angle f d g^ divide it by the curvature
nfg, and we have the distance, <//, to the P. C. C./. The
second curve, traced from this point, will terminate tan-
gentially in h I.
^Example. Let the curvatures equal those of the last
problem, and suppose the distance g i to be 175-6 feet.
Then, by the formula, cosin. x = gTT~~ ~ == '7143
= cosin. 44 26'. Reducing minutes to hundredths, and
dividing by 7, we have 635 feet, the distance back to the
P. C. C.
COMMENCEMENT OF A CLRVE
49
ARTICLE XIX.
HAVING GIVEN A TANGENT a b, AND A CURVE b k, LOCATED,
IT IS REQUIRED TO FIND THE POINT d, OR /, AT WHICH
TO COMMENCE A CURVE OF GIVEN RADIUS, WHICH SHALL
BE TANGENT TO BOTH.
DRAW the radius b c, and call it R. Name the radius
of the other curve r. Make b e equal to r, and, through
e. draw the line e I, parallel to the tangent a b. From c
as a centre, with eg, = (R -|- r), as radius, sweep the arc
of a circle ; which arc will intersect # I at g, and, from the
equidistances, prove g the centre of the other curve touch-
ing a b y b k, tangentially, at the points d and /.
Now, to find the point / for purposes of location, we
must know the angle b cf. Call it x. In the triangle
eg e, we have eg : c e : : radius : cosin. g c e, or b cf\ i, e*,
(R -f r) : (R r) : : 1 : nat. cosin #, wherefore nat. cosin.
x = TO" r { So that, having divided the difference of
^rt -f- r)
the radii by their sum, we shall find opposite to the quo-
tient, in the table of nat. cosines, the angle b cf required.
This angle, divided by the degree of curvature of b k, will
5
60 RAILWAY CURVES AND LOCATION
give the distance from b to the P. R. C., /; and 180 x
will equal the angle f g d, to be turned on the other curve.
Another plan, which may be preferred, for finding the
point d, is as follows :
Make c g the diameter of a semicircle, ctg. This semi-
circle is tangent to b d at t, and tf, perpendicular to eg,
is a common tangent to the curves b k, df. We have then,
/ X /^ = tf\ i. e., R X r = tang. 2 1. Multiplying the
radii together, and extracting the square root of the product,
we find the distance tf, or its equal t b, which, doubled, is
the distance b d, from b to the point of curvature, d.
Example. Suppose k b a 3 curve, tangent to the line
a b. We wish to know the point /, at which to begin a 5
curve, which shall be tangent to both. Here R = 1910,
and r = 1146, wherefore (R + r) = 3056, (R r) =
764, and = = - , = -25, = nat. cosh, 75 31'.
Reducing minutes to hundredths, and dividing by 3, we
have 2517 feet, the distance from b to the P. R. C.,/.
If the point d be required, we have ^/1910 X 1146 =
1478-9. Doubling this result we find 2957'8 feet, the dis-
tance from b to the P. C., d.
If the radii are equal, of course the distance b d is equal
to their sum, and the angle # is a right angle.
TO RUN A TANGENT TO TWO CURVES. 51
ARTICLE XX.
TO RUN A TANGENT TO TWO CURVES.
LET g , e /r, be the two curves. First plot them
carefully to a large scale, and, finding from this plot an
approximate P. T., say 6, run the tangent ef. Now the
chord in the curve g b, to which ef is parallel, is known,
and consequently the shortest distance, fg, between the
tangent and the curve may be measured. Then ef : f g
:: rad. : tang, feg, and nat. tang, feg = -^> Practi-
cally, this angle feg will be found equal to g a b or dc e,
so that, dividing it by the degree of curvature of k <*, we
shall have the distance, e d, to the proper P. T.
Strictly speaking, the angle g ef is too great by the
small angle bhg. There are two modes of calculating
this latter, but they are both complex, and in any but a
most unusual case, the above rule is sufficiently accurate.
Example. Let k e be a 5 curve. Suppose the tangent
ef = 1632 feet, and the distance gf = 42 feet,
Then 9 = -0257 = nat. tang, of 1 28'. Reducing
52
RAILWAY CURVES AND LOCATION.
minutes to hundredths, and dividing by 5
feet, the distance back to the correct P. T.
we have 29
Should the curves turn in the same direction, or should
e be a fixed point from which to run a tangent to g b, the
above illustration is still applicable.
ARTICLE XXI.
OBDINATES.
TO FIND THE MIDDLE ORDINATE TO ANY GIVEN CHORD, IN
A CURVE OF ANY GIVEN RADIUS.
1st. (See figure in Art. XVL) Ic c = */ a e 2 a F, and
a c or be k e = the ordinate required. That is, from
the square of the radius subtract the square of half the
chord, and take the square root of the remainder from
radius, for the middle ordinate.
Example. The radius a c being 819 feet, and the chord
a/ 100 feet, to find the middle ordinate, Ik.
Here a <? a t? = 670761 2500, = 668261, the
square root of which is c k, = 817-5, which taken from
radius 819, leaves 1-5, the required middle ordinate.
ORT>I NATES. 63
2d. Subtract the nat. cosine of the tangential ar/gle
from 1, and multiply the remainder by radius.
Example. Suppose a b a 7 curve. Here the nat.
cosine of 3 30', the tangential angle, is -9981, which, sub-
tracted from 1, leaves *0019. Multiplying this latter by
radius 819, we have 1*5, the middle ordinate as before
HAVING GIVEN THE MIDDLE ORDINATE, TO FIND ANY OTHER.
1st. eg = \/ c e 2 eg*, and ed = eg g d or clt.
Example. Suppose the distance Jed or eg to be 20
feet. Thence e e 2 c g 2 = 670761 400 = 670361. the
square root of which is eg, = 818*76. Taking from this
the distance gd = 817 '5, we have the ordinate ed =
1-26.
*2d. Multiply the ordinates of a 1 curve by the deflexion
angle of the curve whose ordinates are required. This is
an approximation sufficiently exact for railwa-y curves.
54
RAILWAY CURVES AND LOCATION,
ARTICLE XXII.
TO FIND THE RADIUS CORRESPONDING TO ANY DEFLEXION
ANGLE, AND TO EQUAL CHORDS OF ANY GIVEN LENGTH.
HERE the deflexion angle, d b a, is of course equal to
the central angle d c b, subtended by the given chord d 6,
SIA-*'- i ,jiv 180 deb
and the triangle c d b being isosceles we have ~
A
= c d b, or d b c.
Then nat. sin. deb : nat. sin. c b d :: db : c d 9 the
radius.
Example. Let the deflexion angle be 5, and the chord
100 feet. Required the radius. The nat. sine of 5 =
-I OAO CO
0872 and = 87 30', the nat. sine of which is
z
9990.
Then -0872 : -9990 : : 100 : radius = 1146 feet.
An approximate result, sufficiently accurate for all pur-
poses of railway location, may be found by simply dividing
5730 by the deflexion angle.
To find the deflexion angle corresponding to any given
RADIUS CORRESPONDING TO DEFLEXION ANGLE. .>
radius and chord. In this case, we have c d : df : : rad
of 1 : nat. sin. of half the deflexion angle ; therefore, divide
half of the chord by the radius of the curve ; the quotient
will }yc the nat. sine of half the deflexion angle.
JExample. Radius as before 1146 feet, and chord 100
feet. Then j||g = -0436, the nat. sin. of 2 30'. Dou-
bling this result, we have the deflexion angle, viz. 5 00'.
To find the deflexion distance with chord of 100 feet and
any radius. Divide the constant number 10000 by the
radius in feet ; the quotient will be the deflexion distance :
for the deflexion distance with a radius of 10000 ft-et is
1 foot, and the deflexion distances for other radii increase
inversely as the radii.
Example. What is the deflexion distance for a 5 curve,
the chord being 100 feet? Here ^^ = 8-72 feet, the
deflexion distance.
To find the deflexion distance with any given chord and
radius. The deflexion distance is equal to twice the
natural sine of half the deflexion angle, multiplied by the
chord. Thus, the chord being 100 feet, and the deflexion
angle 5, we find the nat. sine of 2 30' equal to -0436,
which doubled, and multiplied by 100, gives 8*72 feet, as
above.
The tangential distance, with any radius and chord, is
in like manner equal to twice the nat. sine of half the tan-
gential angle multiplied by the chord. Thus, the tangential
angle being 2 30' and the chord 100 feet, we find the nat.
S'ne of 1 15' = -0218. Multiplying by 2, and 100, we
have 4*36 feet, the tangential distance.
For all curves under 11, the tangential distance is equal
to half the deflexion distance.
J>6 RAILWAY CURVER AND LOCATION
ARTICLE XXIII.
OF EXCAVATION AND EMBANKMENT.
A RAILWAY line having been located, the first office dutiea
are to map it down, to make a continuous profile, and an
approximate estimate of the cost of grading, &c. The " ele-
vation" of each stake above or below a certain assumed
base being fixed, the profile is drawn, on paper prepared
for the purpose, to a horizontal scale of 400 feet, and a
vertical scale of 40 feet, to the inch. This distorted pic-
ture presents at a glance, in compact shape, the undula-
tions of the surface, and from it grades are adopted, to
balance as nearly as possible the excavation and embank-
ment. By these grades, noted in the record, the cutting
or filling is ascertained at each station. Suppose, for
instance, that the elevation of station 40 is 12 feet, and
that of station 100 is 72 feet. The distance between 40
and 100 is 6000 feet, and the difference between 12 and
72 is 60 feet. Consequently, to connect those points, we
require an ascending grade of 1 in 100 feet. Grade at
station 54 is therefore 26 feet, and at station 60, 32 feet.
If the elevation at station 60 be 38 feet, of course we have
6 feet of excavation, and this is marked in our estimate
sheets -f 6. If the elevation at that point should be 27
feet, o feet of embankment is the consequence, which is
marked accordingly, 5 ; plus indicating excavation, and
minus embankment.
The usual slope for embankment is 1 feet horizontal
to 1 foot vertical, making an angle of about 34 with the
OF EXCAVATION AND EMBANKMENT. 57
horizon ; that for earth cut, 1 to 1, or 45, and for rock
cut J to 1, or 76. The slopes of the ground surface at
each station being known, together with the breadth of the
intended roadway, we are prepared to calculate the cross-
sectional areas, and from them to determine the cubic yards
of excavation and embankment.
To facilitate these operations the following rules were
prepared. With Trautwine's common diagram, and the
table of squares and square roots appended to the volume,
they will be found an observable assistance.
Suppose li the road bed, and fh the depth of an ordi
nary clay cut. Produce the side slopes until they meet in
the point k. Then the angle being 45, ef=fk, and
ef 2 = twice the triangle efk, = the triangle e kg. For
like reasons I h = h k, and I h 2 = triangle I k i. But the
triangle Iki, taken from the triangle ekg, leaves the area
elig, to find which we have therefore the following
Rule. To half the breadth of the roadway, add the
depth of the cut. From the square of this sum, subtract
the square of half the breadth of the roadway for the
area.
Example. Suppose the breadth of the roadway to be
32 feet, and the depth hf, 7 feet. Half of 32 = 16, the
square of which, viz., 256, becomes a constant subtrahend.
16 -{- 7 = 23, and a reference to the table shows the
square of 23 equal to 529. Therefore 529 256 = 273,
the area required in square feet.
If there be a regular slope, as eg (p. 58), we can plot it on
the diagram, and read at once the side cuts, g k, en.
58 RAILWAY CURVES AND LOCATION.
the area is equal to that of the trapezoid e n k g, minus
the two triangles e n h, mk g.
Calling e ft, or its equal n li, a, m k or g k, 5, and half
the breadth of the roadway, c, we have the triangle mk g
= b X 77 = - The triangle e n h is in like manner = ,
22 2
and the area of the trapezoid = a-\-b-{-2cX J .
2
Then the area required = a-j-5-|-2tfX - ~
2
2 ab -f 2 ac -f 2bc
= (a -f b) c + a.5.
Should the slopes be J to 1, as in rock cut, the area
might, in similar-wise, be shown equal to (a -f- b) c -\ ^-,
and that of embankment, where the slopes are 1J to 1,
equal to (a -f- #) c -j ^-. Wherefore, we arrive at the
following general rule for finding cross-sectional areas,
where the ground surface is a regular declivity :
Multiply the sum of the side cuttings by half the breadth
of the roadway, and mark the product. Multiply the pro-
duct of the side cuttings by the ratio of the side slopes to
1, and add this result to the previous product marked for
the area.
Example 1. Earth excavation road bed 32 feet, side
OF EXCAVATION AND EMBANKMENT. 59
slopes 1 to 1, right cutting 12 feet, left cutting 3 feet.
Required the cross-sectional area. Here (a -f- b) c -\- a b
= (12 + 3) 16 -f 12 X 3 = 240 + 36 = 276 square
feet, the area required.
Example 2. Rock cut road bed 28 feet, side
slopes J to 1, cuttings as before. Required the area.
Here (a -f b) c + t-*J. = 210 +^ = 219 square
feet, the area.
Example 3. Embankment roadway 27 feet, side
slopes 1| to 1, cuttings as before. Here ( + #)<? +
3 ^ 2 X ^ = 202-5. + 54 = 256-5 square feet, the cross-
sectional area.
Other formulae might be given for varying surface slopes,
but they would involve a simple matter, and require more
time for calculation than a division of the diagram area
into triangles.
To find the cubic content of an excavation or embank-
ment :
Multiply half the sum of the two end areas by the
distance between them ; thus : supposing 282 and 310
to be the end areas, and 100 feet the distance, we have
(282 + 310)
yards. Or we may multiply the sum of the end areas by
100, and divide the product by 6 and 9, the factors of 54,
which is the number of cubic feet contained in two cubic
yards ; thus, 5?|?? = 9866-6, and this divided by 9 =
1096-3, as above.
60
RAILWAY CURVES AND LOCATION.
PRISMOIDAL FORMULA.
To the two end areas add four times the middle area,
and multiply the sum by one-sixth of the length of the pris-
moid. Thus : from the foregoing example, the sum of the
end areas, 592, added to four times their mean, 1184,
gives 1776, and 1776 X 16-7 = 29659-2 cubic feet, =
1098'5 cubic yards. The former rule is approximate
sufficiently so for rough preliminary calculations, but where
strict correctness is required, as in a final field estimate,
the prismoidal formula should be usefl. It applies to all
solid bodies with plane faces and parallel ends.
ARTICLE XXIV.
SIDE STAKING.
THE object in side staking is to find the point where
the surface of the ground intersects the slope of the road
formation. For performing this work with level and rod,
place the instrument in such a position as to command as
many stations as possible, whether they be on the upper
SIDE STAKING 61
or lower side, and ascertain by transfers from the bench
the height of your instrument with reference to grade.
This will facilitate operations by giving usually small num-
bers to work with. Knowing your rate of grade, subtract
or add, as the case may be, as you change from station to
station.
Having the level in place, we will suppose on an exca-
vation, our object is first to place the lower side stake.
Let the width of the road bed be 26 feet, and the side
slopes 1 to 1. Suppose the instrument to be 11 feet above
grade and the centre cut 5 feet, which latter is already
recorded in the field book for construction duties. Look-
ing at the fall of the hill we judge that at the distance of
17 feet from the centre stake the descent is 1'5 feet, which
would give us 3-5 feet cutting. We take an observation
at that point and find that the rod reads 10 feet, which,
subtracted from our height above grade, leaves us only one
foot cutting, and shows that our judgment has been at
fault. The distance measured is too much, for were we to
increase the distance we would reduce the cutting, and it
is therefore evident that we are too far from the centre
Let us then make the distance from the centre stake 15
feet, and take another sight, when the rod reads 9, show-
ing 2 feet cutting, which, added to half the road formation,
slope being 1 to 1, equals the distance measured. The
stake is therefore correct. The same process applies to
the upper side.
The only difference in staking out for banks is that you
add 1J times the height of the bank to the distance mea-
sured, or, as a general rule, the height of the bank multi-
plied by the ratio of the side slopes to unity.
Some judgment is required to be expeditious in this
work, which is obtained only by experience.
6
62
RAILWAY CURVES AND LOCATION.
The following is a common form of field record for con
struction :
,*
jji8t.
Left Side.
Centre.
i
Right Sido
Course.
Mag.
Course.
Elevation
Grade.
Dist.
AorB
A
B
Dist.
AorB
S-ll A
530
100
N20OOW
N 20-05 W
+ 560-4
+ 555-4
15-0
2-0 A
5-0
21-0
VTUEAL SINES AND TANGENTS
(63)
|l NAT. SINE. 65 '
' '
O o [o 2
3 t 4
5 6
70 : /
, o
000 OOCO 017 4524 034 8995
052 3360 069 7565
087 1557
104 5285
121 8693! 60
i
29091 7432,035 1902
6264 070 0467
4455
R178
1 221 581 59
'2
581b 018 0341
4809
91691 3368
7353 105 1070
4468 58
3
8727
3249
7716
053 2074 6270
0880251
3963
7355i 57
4
001 1636
6158
036 0623
4979 9171
3148
6856
1230241
56
; 5
4544
9066
3530
7883,071 2073
6046
9748
312-
55
6
7453019 1974
6437
054 0788J 4974
8943
106 2G41
6015
54
i 7
C02 0362
48S3
9344
3693| 7876
089 1840
5533
8901
53
I 8
3271
7791
037 2251
6597
072 0777
4738
8425
124 1768
52
9
6180 020 0699
5158
9502
3678
7635
107 1318
4674
51
10
9089 3608
8065
055 2406
6580
090 0532
4210
7560
11
003 1998 6516
0380971
5311
9481
3429
7102
125 0446
49
1 12
4907J 9424
3878
8215
073 2382
6326
9994
3332
48
13
7815
021 2332
6785
Oo6 1119
5283
9223
108 2885
6218
47
14
004 0724
5241
9692
4024
8184
091 2119
5777
9104
46
i 15
3633
8149
039 2598
6928
074 1085
5016
8669
126 1990
45
16
6542
022 1057
5505
9832
3986
7913
109 1560
4875
44
17
9451
3965
8411
057 2736
6887
392 0809
4452
7761
43
18
005 2360
6873
040 0318
5640
9787
3706
7343
127 0646
42
19
5268
9781
4224
8544
075 268S
6602
1100234
3531
41
20
8177
023 2690
7131
058 1448
5589
9499
3126
6416
40
21
006 1086
5598
041 0037
4352
8489
093 2395
6017
9302
39
22
3995
8506
2944
7256
076 1390
5291
8908
128 2186
;$)
23
6904
024 1414
5850
059 0160
4290
8187
111 1799
5071
37
24
9813
4322
8757
3064
7190
094 1083
4689
7956
36
25
007 2721
7230
042 1663
5967
0770091
3979
7580
1290841
35
2G
5630
025 0138
4569
8871
2991
6875
1120471
3725
34
i 27
8539
3046
7475
060 1775
5891
9771
3361
6609
33
28
008 1448
5954
043 0382
4678
8791
0952&66
6252
9494
32
29
4357
8662
3288
7582
078 1691
5562
9142
130 2378
31
30
7265
026 1769
6194
061 0485
4591
8458
1132032
5262
30
31
009 0174
4677
9100
3389
7491
^96 1353
4922
8146
20
32
3083
7585
044 2006
6292
079 0391
4248
7812
131 1030
28
33
5992
027 0493
4912
9196
3290
7144
1140702
3913
27
34
8900
3401
7818
062 2099
6190
097 0039
3592
6797
2G
35
010 1809
6309
045 0724
5002
9090
2034
6482
. 9681
25
36
4718
9216
3630
7905
080 1989
5829
9372
132 2564
24
37
7627
028 2124
6536
063 0808
4880
8724
1152261
5447
23
38
Oil 0535
5032
9442
3711
7788
098 1619
5151
8330
22
39
3444
7940
046 2347
6614
081 0687
4514
8040
133 1213
21
40
6353
029 0847
5253
9517
3587
7408
1160929
4096
20
41
9261
3755
8159
064 2420
6486
099 0303
3818
6979
HI
42
012 2170
6662
047 1065
5323
9385
3197
6707
9S62
Ifc
43
5079
9570
3970
8226
082 2284
6092
9596
134 2744
17 ,
44
7987
030 2478
6876
0651129
5183
8986
1172485
5627
in i
45
013 0896
5385
9781
4031
8082
100 1881
5374
8509
H
1 46
3805
8293
048 2687
6934
033 0981
4775
8263
136 1392
u i
i 47
6713
031 1200
5592
9836
3880
76'19
118 1151
4274
13 '
48
9622
4108
8498
066 2739
6778
101 0563
4040
7156
12
49
014 2530
7015
049 1403
5641
9677
3457
6928
136 0038
U
50
5439
9922
4308
8544
084 2576
6351
9816
291
10
51
8348
032 2830
7214
067 1446
5474
92451192704
5801
9
52
0151256
5737
0500119
4349
8373
102 2138
5503
8683
8
53
4165
8644
3024
7251
085 1271
5032
8481
137 1564
7
54
7073
033 1552
5929
068 0153
4169
7925 120 1368
4445
A
55
9982
4459
8835
3055
7067
1030819
4256
7327
5
1 56
016 2890
7366
051 1740
5957
9966
3712
7144
138 0208
4
i 57
5799
034 0274
4645
8859
086 2864
6605 121 0031
3089
3
58
8707
3181
7550
069 1761
5762
9499 2919
5970
2
59
0171616
6088
052 0455
4663
8660
104 2392 5806
8850
1
60
4524
8995
3360
7565
087 1557
5285; 8693
139 1731
(1
89
88
87
86
85
84 1 83
8-JQ
'
NAT. COSINE.
06 NAT. TAN.
/
GO
1
2
3 [ 4
5
6 7
000 0000
017 4551
034 9208
052 40781069 9268
087 4887
05 1042
22 7841
60
1
2909
7460
035 2120
69951070 2191
7818
3983
23 079&
59
2
5818
018 0370
5033
9912
5115
088 0749
6925
3752
58
3
8727
3280
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NAT. COTAN.
NAT. SINE. 73
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529 9193
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NAT. COSINE.
74 NAT. TAN
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9925 462
9945 219
9961 947
9975 641
9986 295
60
1
582
,)877 333
9903 035
816
523
9962 200
843
447 59
2
9849 086
792
489
9926 169
825
452
9976 015
598
58
3
589
:)878 245
891
521
9946 127
704
245
743
57
4
9850 091
697
9904 293
873
428
954
4451 898
56
5
593
J879 148
694
9927 224
729
9963 204
645
9987 046
55
6
9351 093
599
9905 095
573
9947 026
453
843
194
54
7
593
;)380 048
494
922
327
701
9977 040
340
53
! 8
9852 092
497
893
9928271
625
948
237
486
52
9
590
945
9906 290
618
921
9964 195
433
631
51
10
9853 087
9881 392
687
965
9948 217
440
627
775
50
11
583
838
9907 083
9929 310
513
685
821
919
49
12
9854 079
9882 284
478
655
807
929
9978 015
9988 061
43
13
574
728
873
999
9949 101
9965 172
207
203
47
14
9855 068
9833 172
9908 266
9930 342
393
414
399
344
46
15
561
615
659
685
685
655
589
484
45
16
9856 053
9884 057
9909051
9931 026
976
895
779
623
44 '
i 17
544
493
442
367
9950 266
9966 135
968
761
43 \
13
9857 035
939
832
706
556
374
9973 156
899
42
19
524
9835 378
9910 221
9932 045
844
612
343
9989 035
41 '
20
9858013
817
610
334
9951 132
849
530
171
40
21
501
9836 255
997
721
419
9967 085
716
306
39
22
988
692
9911 384
3933 057
705
321
900
440
33
23
9859 475
9887 128
770
393
990
555
998C 084
573
37
24
960
564
9912 155
728
9952 274
789
267
706
36
25
9860 445
998
540
9934 062
557
9963 022
450
837
35
26
929
9838 432
923
395
840
254
631
968
34
27
9361412
865
9913 306
727
9953 122
485
811
9990 098
33
28
894
9889 297
683
9935 058
403
715
991
227
32
29
9362 375
723
9914 069
389
683
945
9981 170
355
31
30
856
9890 159
449
719
962
9969 173
348
482
30
31
9863 336
588
823
9936 047
9954 240
401
525
609
29
32
815
9891 017
9915 206
375
518
628
701
734
26
33
9864 293
445
584
703
795
854
877
859
27
34
770
872
961
9937 029
9955 070
9970 080
9982 052
933
26
35
9865 216
3892 298
9916 337
355
345
304
225
9991 106
25 |
36
722
723
712
679
620
528
398
228
24 1
37
9866 196
9893 148
9917 036
9938 OU3
893
750
570
350
23 1
38
670
572
459
326
9956 165
972
742
470
22
39
9867 143
994
832
648
437
9971 193
912
590
21
40
615
9894416
9918 204
969
708
413
9983 082
709
20
41
9868 087
833
574
9939 290
978
633
250
827
19
42
557
9395 258
944
610
9957 247
851
418
944
18 '
43
0869 027
677
9919314
923
515
9972 069
585
9992 060
17
44
496
9396 096
682
9940 246
783
286
751
176
16
45
964
514
9920 049
563
9958 049
502
917
290
15 1
46
9370 431
931
416
830
315
717
9984 031
404
14 |
47
897
9897 347
782
9941 195
580
931
245
517
13
48
9371 363
762
9921 147
510
844
9973 145
408
629
12
49
827
9393 177
511
823
9959 107
357
570
740
11
50
9872 291
590
874
9942 136
370
569
731
851
10
51
T54
9399 003 9922 237
448
631
780
891
960
9
52
9373 216
415
599
760
892
990
9985 050
9993 069
8
53
678
826
959
9943 070
9960 152
9974 199
209
177
7
54
9874 138
9900 237
9923 319
379
411
408
367
284
6
55
598
646
679
688
669
615
524
390
5
56
9875 057
9901 055 9924 037
996
926
822
680
195
4 ''
57
514
462 394
9944 303
9961 183
9975 023
835
600
3
58
972
869 751
609
438
233
989
704
2
59
9876 428
9902 275 9925 107
914
693
437
9936 143
806
1
60
833
681
462
9945 219
947
641
295
908
'
9 3
8
70
6
5
40
3
2
' !
NAT. COSTIVE.
86 NAT. TAN.
'
79
80
81
8-2
83
84^
85
/
5-1445540
5-6712818
6-3137515
7-1153697
8-1443464
9-5143645
11-430052
60
, 1
525557
809446
256601
304190
639786
410613
468474
59
2
605813
906394
376126
455308
837041
679068
5071541 58
3
686311
5-7003663
496092
607056
8-2035239
949022
546093
57
4
767051
101256
616502
759437
234384
9-6220486
585294
56
! 5
848035
199173
737359
912456
434465
493475
624761
55
6
929264
297416
858665
7-2066116
635547
766000
664495
54
7
5-2010738
395988
980422
22C422
837579
9*7044075
704500
53
8
092459
4:4889
6-4102633
3753788-3040536
321713
744779
52
9
174428
594122
225301
530987
244577
600927
785333
51 ,
10
256647
693688
348428
687255
449558
881732
820107
50 I
11
339116
793588
472017
844184
655536
9-8164140
867282
49
: 12
421836
893825
596070
7-3001780
862519
448166
908682
98 ,
13
504309
994400
720591
160047
8-4070515
733823
950370
47
14
588035
5-8095315
845581
318989
279531
9-9021125
992349
46
i 15
671.517
196572
971043
478610
489573
310088
12-034622
45
! 16
755255
298172
6-5096981
638916
700651
600724
077192
44 i
17
S39251
400117
223396
799909
912772
893050
120062
43 ,
18
923505
502410
350293
961595
8-5125943
10-018708
163236
42
19
5-3008018
605051
477672
7-4123978
340172
043283
206716
41
20
092793
708042
605538
2S7064
555468
078031
250505
40
21
177830
811386
733892
450855
771838
107954
294609
39
22
263131
915084
862739
615357
989290
138054
339028
38
23
348696
5-9019138
992080
780576
8-6207333
168332
383768
37
24
434527
123550l6-6121919
946514
427475
198789
428831
36
25
520626
228322
252258
7-5113178
648223
229428
474221
35
26
606993
333455
383100
280571
87008P
260249
519942
34
27
693630
4:48952
514449
44S699
8-7093077
291*55
565997
33
28
780538
544815
646307
617567
317198
322447
612390
32
29
867718
651045
778677
787179
542461
353827
659125
31 ,
i 30
955172
757644
911562
957541
768S74
385397
706205
30
31
5-4042901
864614
6-7044966
7-6128657
996446
417158
753634
29
32
130906
971957
178891
300533
8-8225186
449112
801417
28
33
219188
6:0079676
313341
473174
455103
481261
849557
27
34
307750
187772
448318
646584
686206
513607
898058
26
35
396592
296247
583826
820769
918505
546151
P46924
25
36
485715
405103
719867
995735
8-9152009
578895
996160
24
i 37
575121
514343
856446
7-7171486
386726
611841
13-045769
23 j
38
664812
623967
993565
348028
622668
644992
095757
22
39
754788
733979
6-8131227
525366
859843
678348
146127
21
40
845052
844381
269437
703506
9-0098261
711913
196883
20
41
935604
955174
408196
882453
337933
745687
248031
19
42
5-5026446
6-1066360
547508
7-8062212
578867
779673
299574
18
43
117579
177943
687378
242790
821074
813872
35151P
17
44
209005
289923
827807
424191
9-1064564
848288
403867
16
45
300724
402303
968799
606423
309348
882921
45662E
15
46
392740
515085
6-9110359
789489
555436
917775
59979?
14 '
47
485052
628272
252489
973396
802838
952850
563391
13 ,
48
577663
741865
395192
7-9158151
9-2051564
988150
617409
12
49
670574
855867
538473
343758
301627
11-023676
671856
11
50
763786
970279
682335
530224
553035
059431
726738
10 ;
51
857302
(5-2085106
826781
717555
805802
C95416
782060
9 !
52
951121
200347
971806
905756
9-3059936
131635
837827
8
53
5-6045247
316007
7-0117441
8-0094835
315450
168CS9
894C45
7
54
139680
432086
263662
284796
572355
204780
950719
6
1 55
234421
548588
410482
475647
830663
241712
14-007856
5
56
329474
665515
557905
667394
9-4090384
278885
065459
4
57
424838
782868
705934
860042
351531
316304
123536
3
58
520516
900651
854573:8-1053599
614116
353970
182092
2
i 59
616509
6-301886G
7-1003326
24S071
878149
391885
241134
1
60
712818
137515
153697
443464
9-5143645
43C052
3COC6G
10
9
8
7 6
5
40
'
NAT. OOT/VN.
NAT. SINK. NAT. TAN. 87
i '
88
89
'
'
8G
87 3
88
89
' i
9993 908
9998 477
60
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14 300666
19-081137
28-636253
57-289962
60 '
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52759
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877089
58-261174
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110
57758
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295922
29-122005
59-265872
58 ;
3
209
625:57
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482273
405133
371106
60-305820
57 ,
4
308
673 56
4
543833
515584
624499
61-382905
56
5
405
72055
5
605916
627296
882299
62-499154
55
6
502
766 54
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668529
740291
30-144619
63-656741
54
7
598
812 53
7
731.579
854591
411580
64-858008
53
8
693
85652
8
79*372, 970219
683307
66-105473
52
9
788
900 51
g
85U616 23-087199
959928
67-401854
51
10
881
942 50
10
924417
205553
31-241577
68-750087
50
11
974
98449
11
989784
325308
528392
70-153346
49
1 12
9995 066
9999 025 48
12
kd-055723
446486
820516
71-615070
48
13
157
065!47
13
122242
569115
32-118099
73-138991
47
14
247
10546
14
189349
693220
421295
74-729165
46
15
336
143 45
15
257052
818828
730264
76-390009
45
16
424
18144
16
325358
945966
33-045173
78T26342
44
17
512
21843
17
394276
21-074664
366194
79-943430
43
18
1 19
599
684
254 42
289 4 1
18
19
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533981
204949
336851
693509
34-027303
81-847041 42
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10
20
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367771
85-939791
40
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21
676233
605630
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88-143572
39 I
22
937
389
22
748337
742569
35-069546
90-463336
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421
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8-21105 881251
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92-908487
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101
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24
894545 22-021710
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95-489475
36
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182
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35
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963667
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36-177596
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26
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16-043492
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101-10690
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349855
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114-58865
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32
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668
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213666
39-056771
122-77396
28
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798
602
27
33
587396
371777
505695
127-32134
27
34
871
714
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34
668112
532052
965460
132-21851
26
35
943
736
25
35
749614
694537
40-435837
137-50745
25
36
9997 015
756
24
36
831915
859277
917412
143-23712
24
37
086
776
23
37
915025124*026320
41-410588
149-46502
23
38
156
795
22
3H
998957
195714
915790
156-25909
22
39
224
813
21
39
17-083724
367509
42-433464
163-70019
21
40
292
831
20
40
169337
541758
964077
171-88540
20 ,
41
360
847
19
41
255809 719512
43-508122
180-93220 I 19
42
426
863
19
42
343155 897826
44-066113
190-98419 18
43
492
878
17
43
431385 25-079757
638596
202-21875
17 !
44
556
892
16
4-1
5205161 264361
45-226141
214-85762
16 i
45
620
905
15
45
6105591 451700
829351
229-18166
15
46
683
917
14
46
701529 641832
46-448862
245-55198
14
47
i 48
43
745
807
867
928
939
949
13
12
11
47
49
49
793442) 834823:47-085343
88631026-030736 739501
980150 229638,48-412084
264-44080
286-47773
312-52137
13
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13-074977
43160049-103891
343-77371
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101
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491-10COO
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Infinite.
'
IQ
'
/
3 1 2 1
NAT. COSINK. NAT. COT AN.
89
TABLE OF RADII, &c.~- Chord 100 Feet.
An<j;l<> of
Radius
Def. dis
Angle o
Radius
Def. dis
Angle ol
' Radius Def. dist
LL'tk-.vn
in feet.
in feet
Detiex'n
in feet.
in feet
Deflex'n
in feet.
in feet.
O /
/
/
5 |68760-0 -145
3 25
1677-0
5-962
12 U
468-7
21.360
, 10
34380-0 -291
30
1637-0
6-108
3C
459-3
21-790 !
15
22920-0 -436
35
1599-0
6-253
o 45
450-3 22-210
20 117190-01 -581
40
1563-0
6-398
13
441-7 22-640
25 13752-0] -727
45
1528-0
6-544
15
433-4 23-070
30
35
11460-01 -872
9823-0 1-017
50
o 55
1495-0
1463-0
6-689
6-835
3C
o 45
425-5 23-510
417-7 i23-940 i
40
8595-0 1-163
4 .
1433.0! 6-980
14
410-3 24-370 ,
45
7640-0 1-308
15
1348-0 7-416
15
403-1 J24-810 i
! 50
6876-0 1-453
30
1274-0 7-853
30
396-2 ;25-24</ '
i o 55
6251-0 1-600
o 45
1207-0 8-289
o 45
389-6 ;25-670
1
5730-0 1-745
5
1146-0
8-722
15
383- U26-1 10
5
5289-0 1 1-890
15
1092-0
9-159
15
376-9 126-520
10
4912-0
2-036
30
1042-0
9-595
30
370-8 J26-940
15
4584-0
2-181
o 45
996-8
10-030
o 45
365-0 27-370
20
4298-0
2-327
6
955-4
10-470
16
359-3 27-830
25
4045-0 2-472
15
917-0
10-900
o 30
348-4 28-700
30
35
3820-0 2-618
3619-0 2-763
30
o 45
882-0
849-3
11-340
11-780
17
o 30
338-3
328-7
29-560
30-430 i
! 40
3438-0
2-908
7
819-0
12-210
18
319-6
31-290
45
3274-0
3-054
15
790-8
12-640
o 30
311-0
32-500
50
3125-0
3-199
30
764-5
13-080
19
302-9
33010
o 55
2990-0
3-345
o 45
739-9
13-510
o 30
295-3
33-870
2
2865-0
3-490
8
716-8
13-950
20
287-9
34-730
5
2750-0
3-635
15
695-1
14-380
21
274.4
36-440
10
2644-0
3-781
30
674-6
14-810
22
262-0
38-150
15
2547-0
3-926
o 45
655-5
15-250
23
250-8
39-870
20
2456-0
4-072
9
637-3
15-680
24
240-5
41-580
25
2371-0
4-217
15
620-2
16-120
25
231-0
43-280
30
2292-0
4-363
30
603-8
16-550
26
222-3
44-980
35
2218-0
4-508
o 45
588-4
16-990
27
214-2
46-680
40
21490
4-653
10
573-7J17-430
28
206-7
48-380 !
45
2084-0
4-799
15
559-7 17-870
29
199-7
50-070
50
2023-0! 4-9 14
30
546.4
18-300
30
193-2
51-760
! o 55
1965-0 5-090
o 45
533-8J18-730
31
187-1
53-450
3
1910-0
5-235
11
521-719-170
32
181-4
55-130
5
1859-0 5-380
15
510-119-610
33
176-0
56-800
10
1810-0 5-526
30
499-120-050
34
171-0
58/470
15
1763-0! 5-671
o 45
488-520-500
35
166-3
60-140"
20
1719-0
5-817
12
478-3j20-940
36
.161-8
61-800 j
8*
90
TABLE OF LONG CHORDS.
Raclics in feet.
Angle of
Deflection.
Length
1 Station.
of Chord in f<
2 Stations.
jet required tc
3 Stations.
subtend
4 Stations.
5730-0
1
100
200-0
300-0
400-0
4584-0
*
100
200-0
300-0
399-9
3820-0
|
100
200-0
300-0
399-9
3274-0
100
200-0
300-0
399-8
2865-0
2 f
100
200-0
299-9
399-7
I 2547-0
*
100
200-0
299-9
399-6
2292-0
100
200-0
299-8
399-5
2084-0
f.
100
200-0
299-8
399-4
1910-0
3
100
200-0
299-7
399-3
1763-0
i
100
200-0
299-7
399-2
. 1637-0
|
100
200-0
299-6
399-1
1528-0
100
200-0
299-6
399-0
1433-0
4 f
100
199-9
299-6
398-9
1348-0
\
100
199-9
299-5
398-7
1274-0
100
199-9
299-4
398-5
1207-0
I
100
199-9
299-3
398-3
1146-0
5
100
199-9
299-2
398-0
1092-0
k
100
199-8
299-1
397-8
1042-0
100
199-8
299-0
397-6
1 996-8
!
100
199-7
298-9
397-5
955.4
6
100
199-7
298-8
397-3
917-0
i
100
199-7
298-7
397-0
882-0
100
199-7
298-6
396-7
849-3
1
100
19^fc
298-5
396-5
819-0
7
100
199-6
298-4
396-2
790-8
i
100
199-6
298-3
396-0
764-5
*
100
199-6
298-2
395-7
739-9
1
100
199-6
298-1
395-4
1 716-8
8
100
199-6
298-0
395-1
695-1
i
100
199-5
297-9
394-8
674-6
100
199-5
297-8
394-5
655-5
3.
100
199-4
297-7
394-3
637-3
9
100
199-4
297-5
394-1
620-2
i
100
199-4
297-4
393-7
603-8
100
199-3
297-3
393-2
588-4
1
100
199-2
297-2
392-8
573-7
10
100
199-2
297-0
392-4
:
TABLE OF ORDINATES.
Ordinates 10 feet apart. Chord 100 feet.
Distances of the Ordinates from the end of the 100 feet Chord.
Deflexion Angle
in Degrees and
50 feet.
40 feet.
30 feet.
20 feet.
10 feet.
Lengths of Ordinates in feet.
o /
5
018
017
015
012
006
10
036
035
031
023 ' -013
15
054
052
046
035
019
20
073
070
061
047
026
25
091
087
076
058
032
30
109
105
092
070
039 i
35
127
123
108
082
045
40
145
140
123
093
052 i
45
163
157
137
105
058
50
182
175
153
117
065
o 55
200
192
168
128
071
1
218
209
183
140
078
5
236
226
198
152
085
10
254
244
214
163
091
15
273
261
229
175
098
20
291
279
244
187
104 i
25
309
296
259
198
111
30
327
314
275
210
117
35
345
331
290
*221
124
40
364
349
305
233
130
45
382
366
321
245
137
50
400
'-384
336
256
144
o 55
418
401
351
268
150
2
436
419
366
280
157
5
454
436
382
291
163
10
473
454
397
303 . -170
15
491
471
412
315 -17'
20
509
489
428
326 1.
25
527
506
443
338
190
30
545
524
458
350
196
35
564
541
474
361
203
40
582
559
489
373
209
45
600
576
504
384
216 ;
.50
618
594
519
396
222
o 55
636
611
535
408
229 i
3
654
629
550
419
235
5
673
646
565
431
242
10
691
664
581
443
249
15
709
681
596
454
255
! 20
727
699
611
466
262
; 25
j -
745
716
627
478
268
1 1
92
TABLE OF ORDINATES. CONTINUED.
Ordinates 10 feet apart. Chord 100 feet.
Distances of the Ordinates from the end of the 100 feet Chord.
Deflexion Angle
in Degrees and
Minutes.
60 feet.
40 feet.
30 feet.
20 feet.
10 feet
Lengths of Ordinates in feet.
o /
3 30
764
734
642
489
275
35
782
751
657
501
281
! 40
800
769
673
512
288
45
818
786
688
524
294
50
836
804
703
536
301
o 55
854
821
718
547
308
4
873
839
734
559
314
15
927
891
780
594
334
30
981
944
825
629
354
o 45
1-036
996
871
664
373
5
1-091
1-048
917
699
393
15
1-146
1-100
963
734
413
30
1-200
1-153
1-009
769
432
o 45
1-255
1-205
1-055
804
452
6
1-309
1-258
1-100
839
472
15
1-364
1-310
1-146
874
492
30
1-419
1-362
1-192
909
511
o 45
1-473
1-415
1-238
944
531
7
1-528
1-467
1-284
979
551
15
1-582
1-520
1-330
1-014
570
30
1-637
1-572
1-375
1-048
590
o 45
1-692
1-624
1-421
1-083 .
610
8
1-746
1-677
1-467
1-118
629
15
1-801
1-729
1-513
1-153
649
30
1-855
1-782
1-559
1-188
609
o 45 -
1-910
1-834
1-605
1-223
689
9
1-965
1-886
1-651
1-258
708
15
2-019
1-939
1-696
1-293
728
30
2-074
1-991
1-742
1-328
748
I o 45
2-128
2-044
1-788
1-363
767
10
2-183
2-096
1-834
1-398
787 [
15
2-238
2-148
1-880
1-433
807
30
2-292
2-201
1-926
1-468
827
o 45
2-347
2-254
1-972
1-503
846
11
2-401
2-306
2-018
1-538
866
15
2-456
2-359
2-064
1-574
886
30
2-511
2-411
2-110
1-609
906
o 45
2-566
2-464
2-156
1-644
926
12
2-620
2-516
2-203
1-680
946
15
2-675
2-569
2-249
1-715
966
30
2-730
2-621
2.295
1-750
985
if
93
TABLE OF ORDINATES. CONTINUED.
Ordinates 10 feet apart. Chord 100 feet.
Distances of the Ordinates from the end of the 100 feet Chord. ;
Deflexion Angle
iu Degrees and
50 feet.
40 feet.
30 feet.
20 feet.
10 feet.
Minutes.
Lengths of Ordinates in feet.
o /
12 45
2785
2-674
2-341
1-785
1-005
13
2-839
2-726
2-387
1-820
1-025 1
15
2-894
2-779
2-433
1-855
1-045
30
2-949
2-832
2-479
1-891
1-065
o 45
3-000
2-884
2-525
1-926
1-085
14
3-058
2-937
2-571
1-961
1-105
15
3-113
2-989
2-618
1-996
1-124
30
3-168
3-042
2-664
2-031
1-144
o 45
3-222
3-094
2-710
2-067
1 164
15
3-277
3-147
2-756
2-102
1-184
15
3-332
3-200
2-802
2-137
1-204
30
3-387
3-252
2-848
2-172
1-224
o 45
3-442
3-305
2-895
2-208
1-244
16
3-496
3-358
2-941
2-243
1-264
o 30
3-606
3-463
3-033
2-314
1-304
17
3-716
3-569
3-125
2-384
1-344
o 30
3-826
3-674
3-218
2-455
1-384
i 18
3-935
3-779
3-310
2-525
1-424
i o 30
4-045
3-885
3-403
2-596
1-464
19
4-155
3-990
3-495
2-666
1-504
o' 30
4-265
4-096
3-588
2-737 1-544
20
4-375
4-201
3-680
2-808
1-583
94
A TABLE OF THE SQUARES AND SQUARE ROOTS
OF NUMBERS.
From 1 to 1000.
No.
Squares.
Square Roots.
No.
Squares.
Square Roots.
1
1
1-0000
44
1936
6-6332
2
4
1-4142
45
2025
6-7082
i 3
9
1-7320
46
2116
6-7823
4
16
2-0000
47
2209
6-8556
5
25
2-2360
48
2304
6-9282 '
i 6
36
2-4495
49
2401
7-0000
7
49
2-6457
50
2500
7-0711
8
64
2-8284
51
2601
7-1414
9
81
3-0000
52
2704
7-2111
10
100
3-1623
53
2809
7-2801
11
121
3-3166
54
2916
7-3485
12
144
3-4641
55
3025
7-4162
13
169
3-6055
56
3136
7-4833
14
196
3*7416
57
3249
7-5498
15
225
3-8730
58
3364
7-6158
16
256
4-0000
59
3481
7-6811
17
289
4-1231
60
3600
7-7460
18
324
4-2426
61
3721
7-8102
19
3G1
4-3589
62
3844
7-8740
20
400
4-4721
63
3969
7-9372
21
441
4-5826
64
4096
8-0000
22
484
4-6904
65
4225
8-0628
, 23
529
4-7958
66
4356
8-1240
24
576
4-8990
67
4489
8-1853
25
625
5-0000
68
4624
8-2462
26
676
5-0990
69
4761
8-3066
27
729
5-196L
70
4900
8-3666
28
784
5-2915
71
5041
8-4261
29
841
5-3852
72
5184
8-4853
30
900
5-4772
73
5329
8-5440
31
961
5-5678
74
5476
8-6023
32
1024
5-6568
75
5625
8-6603
33
1089
5-7446
76
5776
8-7178
34
1156
5-8309
77
5929
8-7750
35
1225
5-9161
78
6084
8-8318
36
1296
6-0000
79
6241
8-8882
37
1369
6-0828
80
6400
8-9443 ;
38
1444
6-1644
81
6561
9-0000
39
1521
6-2450
82
6724
9-0554 i
40
1600
6-3246
83
6889
9-1104
41
1681
6-4031
84
7056
9-1651
42
1764
6-4807
85
7225
9-2195
43
1849
6-5574
86
7396
9-2736 i
i
95
A TABLE OF THE SQUARES AND SQUARE ROOTS
OF NUMBERS. CONTINUED.
From 1 to 1000.
No.
Squares.
Square Roots.
No.
Squares.
Square Roots.
! 87
7569
9-3274
130
16900
11-4017
88
7744
9-3808
131
17161
11-4455
89
7921
9-4340
132
17424
11-4891
90
8100
9-4868
133
17689
11-5326
91
8281
9-5394
134
17956
11-5758
92
8464
9-5917
135
18225
11-6189
93
8649
9-6436
136
18496
11-6619
94
8836
9-6954
137
18769
11-7047
95
9025
9-7468
138
19044
11-7473
96 i 9216
9-7979
139
19321
11-7898
97
9409
9-8488
140
19600
11-8322
98
9604
9-8995
141
19881
11-8743
99
9801
9-9499
142
20164
11-9164
100
10000
10-0000
143
20449
11-9583
101
10201
10-0499
144
20736
12-0000
102
10404
10-0995
145
21025
12-0416
103
10609
10-1489
146
21316
12-0830
104
10816
10-1980
147
21609
12-1244
105
11025
10-2469
148
21904
12-1655
106
11236
10-2956
149
22201
12-2065
107
11449
10-3440
150
22500
12-2474
108
11664
10-3923
151
22801
12-2882
109
11881
10-4403
152
23104
12-3288
110
12100
10-4881
153
23409
12-3693
111
12321
10-5356
154
23716
12-4097
112
12544
10-5830
155
24025
12-4499
113
12769
10-6301
156
24336
12-4900
114
12996
10-6771
157
24649
12-5300
115
13225
10-7238
158
24964
12-5700
116
13456
10-7703
159
25281
12-6095
117
13689
10-8166
160
25600
12-6491
118
13924
10-8628
161
25921
12-6886 !
119
14161
10-9087
162
26244
12-7279 ,
120
14400
10-9544
163
26569
12-7671
i 121
14641
11-0000
164
26896
12-8062 |
122
14884
11-0454
165
27225
12-8452
123
15129
11-0905
166
27556
12-8841
124
15376
11-1355
167
27889
12-9228
125
15625
11-1803
168
28224
129615
126
15876
11.2250
169
28561
13-0000
127
16129
11-2694
170
28900
13-0384
128
16384
11-3137
171
29241
13-0767
129
16641
11-3578
172
29584
13-1149 1
96
A TABLE OF THE SQUARES AND SQUARE ROOTS
OF NUMBERS. CONTINUED.
From 1 to 1000.
No.
Squares.
Square Roots.
No.
Squares.
Square Koots.
173
29929
13-1529
216
46656
14-6969
174
30276
13-1909
217
47089
14-7309
175
30625
13-2287
218
47524
14-7648
I 176
30976
13-2665
219
47961
14-7986
177
31329
13-3041
220
48400
14-8324
178
31684
13-3417
221
48841
14-8661
179
32041
13-3791
222
49284
14-8997
180
32400
13-4164
223
49729
14-9332
181
32761
13-4536
224
50176
14-9666
182
33124
13-4907
225
50625
15-0000
183
33489
13-5277
226
51076
15-0333
184
33856
13-5647
227
51529
15-0665
185
34225
13-6015
228
51984
15-0997
186
34596
13-6382
229
52441
15-1327
187
34969
13-6748
230
52900
15-1657
188
35344
13-7113
231
53361
15-1987
189
35721
13-7477
232
53824
15-2315
190
36100
13-7840
233
54289
15-2643
191
36481
13-8203
234
54756
15-2970
192
36864
13-8564
235
55225
15-3297
193
37249
13-8924
236
55696
15-3623
194
37636
13-9284
237
56169
15-3948
195
38025
13-9642
238
56644
15-4272
196
38416
14-0000
239
57121
15-4596
197
38809
14-0357
240
57600
15-4919
198
39204
14-0712
241
58081
15-5242
199
39601
14-1067
242
58564
15-5563
200
40000
14-1421
243
59049
15-5885
201
40401
14-1774
244
59536
15-6205
202
40804
14-2127
245
60025
15-6525
203
41209
14-2478
246
60516
15-6844
204
41616
14-2828
247
61009
15-7162
205
42025
14-3178
248
61504
15-7480
206
42436
14-3527
249
62001
15-7797
207
42849
14-3874
250
62500
15-8114
208
43264
14-4222
251
63001
15-8430
209
43681
14-4568
252
63504
15-8745
210
44100
14-4914
253
64009
15-9060
211
44521
14-5258
254
64516
15-9374
212
44944
14-5602
255
65025
15-9687
213
45369
14-5945
256
65536
16-0000
214
45796
14-6287
257
66049
16-0312
215
46225
14-6629
258
66564
16-0624
97
A TABLE OF THE SQUARES AND SQUARE ROOTS
OF NUMBERS. CONTINUED.
From 1 to 1000.
No.
Squares.
Square Roots.
No.
Squares.
Square Roots.
259
67081
16.0935
302
91204
17-3781
260
67600
16-1245
303
91809
17-4069
261
68121
16-1555
304
92416
17-4356
262
68644
16-1864
305
93025
17-4642
263
69169
16-2173
306
93636
17-4928
264
69696
16-2481
307
94249
17-5214
265
70225
16-2788
308
94864
17-5499
266
70756
16-3095
309
95481
17-5784
267
71289
16-3401
310
96100
17-6068
268
71824 '
16-3707
311
96721
17-6352
269
72361
16-4012
312
97344
17-6635
270
72900
16-4317
313
97969
17-6918
271
73441
16-4621
314
98596
17-7200
272
73984
16-4924
315
99225
17-7482
273
74529
16-5227
316
99856
17-7764
274
75076
16-5529
317
100489
17-S045
275
75625
16-5831
318
101124
17-8325
276
76176
16-6132
319
101761
17-8606
277
76729
16-6433
320
102400
17-8885
278
77284
16-6733
321
103041
17-9165
279
77841
16-7033
322
103684
17-9444
280
78400
16-7332
323
104329
17-9722
281
78961
16-7630
324
104976
18*0000
282
79524
16-7928
325
105625
18-0277
283
80089
16-8226
326
106276
18-0555
284
80656
16-8523
327
106929
18-0831
285
81225
16-8819
328
107584
18-1108
286
81796
16-9115
329
108241
18-1384
287
82369
16-9411
330
108900
18-1659
288
82944
16-9706
331
109561
18-1934
289
83521
17-0000
332
110224
18-2209
290
84100
17-0294
333
110889
18-2483
291
84681
17-0587
334
111556
18-2757
292
85264
17-0880
aas
112225
18-3030
293
85849
17-1172
336
112896
18-3303
294
86436
17-1464
337
113569
18-3576
295
87025
17-1756
338
114244
18-3848
296
87616
17*2046
339
114921
18-4119
297
88209
17-2337
340
115600
18-4391
298
88804
17-2627
341
116281
18-4662
299
89401
17-2916
342
116964
18-4932
300
90000
17-3205
343
117649
18-5203
301
90601
17-3493
344
118336
18-5472
98
A TABLE OF THE SQUARES AND SQUARE ROOTS
OF NUMBERS. CONTINUED.
From 1 to 1000.
No.
Squares.
Square Roots.
No.
Squares.
Square Roots.
345
119025
18-5742
388
150544
19-6977
346
119716
18-6011
389
151321
19-7231
347
120409
18-6279
390
152100
19-7484
348
121104
18-6548
391
152881
19-7737
349
121801
18-6815
392
153664
19-7990
350
122500
18-7083
393
154449
19-8242
351
123201
18-7350
394
155236
19-8494
352
123904
18-7617
395
156025
19-8746
353
124609
18-7883
396
156816
19-8997
354
125316
18-8149
397
157609
19-9248
355
126025
18-8414
398
158404
19-9499
356
126736
18-8680
399
159201
19-9750
357
127449
18-8944
400
160000
20-0000
358
128164
18-9209
401
160801
20-0250
359
128881
18-9473
402
161604
20-0499
360
129600
18-9737
403
162409
20-0749
361
130321
19-0000
404
163216
20-0997
362
131044
19-0263
405
164025
20-1246
363
131769
19-0526
406
164836
20-1494
364
132496
19-0788
407
165649
20-1742
365
133225
19-1050
408
166464
EO-1990
366
133956
19-1311
409
167281
20-2237 -
367
134689
19-1572
410
168100
20-2485
368
135424
19-1833
411
168921
20-2731
369
136161
19-2094
412
169744
20-2978
370
136900
19-2354
413
170569
20-3224
371
137641
19-2614
414
171396
20-3470
372
138384
19-2873
415
172225
20-3715
373
139129
19-3132
416
173056
20-3961
374
139876
19-3391
417
173889
20-4206
375
140625
19-3649
418
174724
20-4450
376
141376
19-3907
419
175561
20-4695
377
142129
19-4165
420
176400
20-4939
378
142884
19-4422
421
177241
20-5183
379
143641
19-4679
422
178084
20-5426
380
144400
19-4936
423
178929
20-5670
381
145161
19-5192
424
179776
20-5913
382
145924
19-5448
425
180625
20-6155
383
146689
19-5704
426
181476
20-6398
384
147456
19-5959
427
182329
20-6640
385
148225
19-6214
428
183184
20-6882
386
148996
19-6469
429
184041
20-7123
387
149769
19-6723
430
184900
20-7364
11
99
A TABLE OF THE SQUARES AND SQUARE ROOTS
OF NUMBERS. CONTINUED.
From 1 to 1000.
No.
Squares.
Square Roots.
No.
Squares.
Square Roots.
431
185761
20-7605
474
224676
21-7715
432
186624
20-7846
475
225625
21-7945
433
187489
20-8086
476
226576
21-8174
434
188356
20-8327
477
227529
21-8403
435
189225
20-8566
478
228484
21-8632
43G
190096
20-8806
479
229441
21-8861
437
190969
20-9045
480
230400
21-9089
438
191844
20-9284
481
231361
21-9317
439
192721
20-9523
482
232324
21-9545 I
440
193600
20-9762
483
233289
21-9773 1
441
194481
21-0000
484
234256
22-0000
442
195364
21-0238
485
235225
22-0227
443
196249
21-0476
486
236196
22-0454
444
197136
21-0713
487
237169
22-0689
445
198025
21-0950
488
238144
22-0907
446
198916
21-1187
489
239121
22-1133
447
199809
21-1424
490
240100
22-1359
448
200704
21-1660
491
241081
22-1585
449
201601
21-1896
492
242064
22-1811
450
202500
21-2132
493
243049
22-2036
451
203401
21-2368
494
244036
22-2261
452
204304
21-2603
495
245025
22-2486 j
453
205209
21-2838
496
246016
22-2711
454
206116
21-3073
497
247009
22-2935
455
207025
21-3307
498
248004
22-3159
456
207936
21-3542
499
249001
22-3383
457
208849
21-3776
500
250000
22-3607
458
209764
21-4009
501
251001
22-3830
459
210681
21-4243
502
252004
22-4054
460
211600
21-4476
503
253009
22-4277
461
212521
21-4709
504
254016
22-4499
462
213444
21-4942
505
255025
22-4722
463
214369
21-5174
506
256036
22-4944 i
464
215296
21-5407
507
257049
22-5167
465
216225
21-5639
508
258064
22-5388 1
466
217156
21-5870
509
259081
22-5610
467
218089
21-6102
510
260100
22-5832 I
1 468
219024
21-6333
511
261121
22-6053
469
219961
21-6564
512
262144
22-6274
470
220900
21-6795
513
263169
22-6495
471
221841
21-7025
514
264196
22-6716 |
472
222784
21-7256
515
265225
22-6936
473
223729
21-7486
516
266256
22-7156
100
A TABLE OF THE SQUARES AND SQUARE ROOTS
OF NUMBERS. CONTINUED.
From 1 to 1000.
No.
Squares.
Square Roots.
No.
Squares.
Square Roots. !
i 517
267289
22-7376
560
313600
23-6643
1 518
268324
22-7596
561
314721
23-6854
519
269361
22-7816
562
315844
23-7065
520
270400
22-8035
563
316969
23-7276
521
271441
22-8254
564
318096
23-7487
522
272484
22-8473
565
319225
23-7697
523
273529
22-8692
566
320356
23-7907
524
274576
22-8910
567
321489
23-8118
525
275625
22-9129
568
322624
23-8327
526
276676
22-9347
569
323761
23-8537
527
277729
22-9565
570
324900
23-8747
528
278784
22-9782
571
326041
23-8956
529
279841
23-0000
572
327184
23-9165
1 530
280900
23-0217
573
328329
23-9374
531
281961
23-0434
574
329476
23-9583
532
283024
2a-0651
575
330625
23-9792
533
284089
23-0868
576
331776
24-0000
534
285156
23-1084
577
332929
24-0208
535
286225
23-1301
578
334084
24-0416
536
287296
23-1517
579
335241
24-0624
537 288369
23-1733
580
336400
24-0832
538 289444
23-1948
581
337561
24-1039
539 290521
23-2164
582
338724
24-1247
540
291600
23-2379
583
339889
24-1454
541
292681
23-2594
584
341056
24-1661
542
293764
23-2809
585
342225
24-1868
543
294849
23-3021
586
343396
24-2074
544
295936
23-3238
587
344569
24-2281
545
297025
23-3452
588
345744
24-2487
546
298116
23-3666
589
346921
24-2693
547
299209
23-3880
590
348100
24-2899
548
300304
23-4094
591
349281
24-3105
549
301401
2, 4307
592
350464
24-3310
550
302500
i3-4521
593
351649
24-3516
551
303601
23-4734
594
352836
24-3721
552
304704
23-4947
595
354025
24-3926
553
305809
23-5159
596
355216
24-4131
554
306916
23-5372
597
356409
24-4336
555
308025
23-5584
598
357604
24-4540
556
309136
23-5796
599
358801
24-4745
557
310249
23-6008
600
360000
24-4949
558
311364
23-6220
601
361201
24-5153
559
312481
23-6432
602
362404
24-5357
101
A TABLE OF THE SQUARES AND SQUARE ROOTS
OF NUMBERS. CONTINUED.
From 1 to 1000.
No.
Squares.
Square Roots.
No.
Squares.
Square Roots.
603
363609
24-5560
646
417316
25-4165
604
364816
24-5764
647
418609
25-4362
605
366025
24-5967
648
419904
25-4558
606
867236
24-6171
649
421201
25-4755
607
368449
24-6374
650
422500
25-4950
1 608
369664
24-6576
651
423801
25-5147
609
370881
24-6779
652
425104
25-5343
610
372100
24-6982
653
426409
25-5539
611
373321
24-7184
654
427716
25-5734
612
374544
24-7386
655
429025
25-5930
613
375769
24-7588
656
430336
25-6125
614
376996
24-7790
657
431649
25-6320
615
378225
24-7992
658
432964
25-6515
616
379456
24-8193
659
434281
25-6710
617
380689
24-8395
660
435600
25-6905
618
381924
24-8596
661
436921
25-7099
619
383161
24-8797
662
438244
25-7204
620
384400
24-8998
663
439569
25-7488
621
385641
24-9199
664
440896
25-7682
622
386884
24-9399
665
442225
25-7876
623
388129
24-9600
666
443556
25-8070
624
389376
' 24-9800
667
444889
25-8263
625
390625
25-0000
668
446224
25-8457
626
391876
25-0200
669
447561
25-8650
627
393129
25-0400
670
448900
25-8844
628
394384
25-0600
671
450241
25-9037
629
395641
25-0799
672
451584
25-9230
630
396900
25-0998
673
452929
25-9422
631
398161
25-1197
674
454276
25-9615
632
399424
25-1396
675
455625
25-9808
633
400689
25-1595
676
456976
26-0000
634
401956
25-1794
677
458329
26-0192
635
403225
25-1992
678
459684
26-0384
636
404496
25-2190
679
461041
26-0576
637
405769
25-2389
680
462400
26-0768
638
407044
25-2587
681
463761
26-0960
639
408321
25-2785
682
465124
26-1151
640
409600
25-2982
683
466489
26-1343
641
410881
25-3180
684
467856
26-1534
642
412164
25-3377
685
469225
26-1725
643
413449
25-3574
686
470596
26-1916
644
414736
25-3772
687
471969
26-2107
645
416025
25-3968
688
473344
26-2297
! !
i !
102
A TABLE OF THE SQUARES AND SQUARE ROOTS
OF NUMBERS. CONTINUED.
From 1 to 1000.
No.
Squares.
Square Roots.
No. Squares.
Square Roots.
689
474721
26-2488
7S2
535824
27-0555
690
476100
26-2678
733
537289
27-0740
691
477481
26-2869
734
538756
27-0924 i
692
478864
26-3059
735
540225
27-1109
693
480249
26-3249
736
541696
27-1293
j 694
481636
26-3439
737
543169
27-1477
695
483025
26-3629
738
544644
27-1662
696
484416
26-3818
739
546121
27-1846
697
485809
26-4008
740
547600
27-2029
698
487204
26-4197
741
549081
27-2213
699
488601
26-4386
742
550564
27-2397
700
490000
26-4575
743
552049
27-2580
701
491401
26-4764
744
553536
27-2764
702
492804
26-4953
745
555025
27-2947
703
494209
26-5141
746
556516
27-3130
704
495616
26-533C
747
558009
27-3313
705
497025
26-5518
748 I 559504
27-3496
706
498436
26-5707
749
561001
27-3679
707
499849
26-5895
750
562500
27-3861
708
501264
26-6083
751
564001
27-4044
709
502681
26-6271
752
565504
27-4226
710
504100
26-6458
753
567009
27-4408
711
505521
26-6646
754
568516
27-4591
712
506944
26-6833
755
570025
27-4773
713
508369
26-7021
756
571536
27-4955
714
509796
26-7208
757 i 573049
27-5136
715
511225
26-7395 758 i 574564
27-5318
716
512656
26-7582 759 ! 576081
27-5500
717
514089
26-7769 760 577600
27-5681
718
515524
26-7955 761 579121
27-5862
719
516961
26-8142 ! 762 580644
27-6043
720
518400
26-8328
763 582169
27-6225
721
519841
26-8514
764
583696
27-6405
722
521284
26-8701
765
585225
27-6586
723
522729
26-8887
766
586756
27-6767
724
524176
26-9072
767 588289
27-6948
725
525625
26-9258
768 589824
27-7128
726
527076
26-9444
769 591361
27-7308 i
727
528529
26-9629
770 592900
27-7489 j
728
529984
26-9815
771 594441
27-7669
729
531441
27-0000
772 ! 595984
27-7849
730
532900
27-0185
773
597529
27-8029
731
534361
27-0370
774
599076
27-820S
103
A TABLE OF THE SQUARES AND SQUARE ROO'i'S
OF NUMBERS. CONTINUED.
From 1 to 1000.
No.
Squares.
Square Roots.
No.
Squares.
Square Roots.
775
600625
27-8388
818
669124
28-6007
776
602176
27-8568
819
670761
28-6182
777
603729
27-8747
820
672400
28-6356
778
605284
27-8926
821
674041
28-6531
! 779
606841
27-9106
822
675684
28-6705
780
608400
27-9285
823
677329
28-6880
781
609961
27-9464
824
678976
28-7054
782
611524
27-9643
825
680625
28-7228
783
613089
27-9821
826
682276
28-7402
784
614656
28-0000
827
683929
28-7576
785
616225
28-0179
828
685584
28-7750
786
617796
28-0357
829
687241
28-7924
787
619369
28-0535
830
688900
28-8097
788
620944
28-0713
831
690561
28-8271
789
622521
28-0891
832
692224
28-8444
i 790
624100
28-1069
833
693889
28-8617
791
625681
28-1247
834
695556
28-8791
792
627264
28-1425
835
697225
28-8964
793
628849
28-1603
836
698896
28-9137
794
630436
28-1780
837
700569
28-9310 i
795
632025
28-1957
838
702244
28-9482
! 796
633616
28-2135
839
703921
28-9655
797
635209
28-2312
840
705600
28-9828
798
636804
28-2489
841
707281
29-0000
799
638401
28-2666
842
708964
29-0172
800
640000
28-2843
843
710649
29-0345
801
641601
28-3019
844
712336
29-0517
802
643204
28-3196
845
714025
29-0689
803
644809
28-3373
846
715716
29-0861
804
646416
28-3549
847
717409
29-1033
1 805
648025
28-3726
848
719104
29-1204
806
649636
28-3901
849
720801
29-1376
807
651249
28-4077
850
722500
29-1548
808
652864
28-4253
851
724201
29-1719
809
654481
28-4429
852
725904
29-1890
810
656100
28-4605
853
727609
29-2062
811
657721
28-4781
854
729316
29-2233
812
659344
28-4956
855
731025
29-2404
813
660969
28-5132
856
732736
29-2575
814
662596
28-5307
857
734449
29-2746
815
664225
28-5482
858
736164
29-2916
816
665856
28-5657
859
737881
29-3087
817
667489
28-5832
860
739600
29-3258
r=-- _ -
104
A TABLE OF THE SQUARES AND SQUARE ROOTS :
OF NUMBERS. CONTINUED.
From 1 to 1000.
No.
Squares.
Square Roots.
No.
Squares.
Square Roots.
! 861
741321
29-3428
904
817216
30-0666
862
743044
29-3598
905
819025
30-0832
863
744769
29-3769
906
820836
30-0998
864
746496
29-3939
907
822649
30-1164
865
748225
29-4109
908
824464
30-1330
866
749956
29-4279
909
826281
30-1496
867
751689
29-4449
910
828100
30-1662
868
753424
29-4618
911
829921
30-1828
869
755161
29-4788
912
831744
30- 1993
870
756900
29-4958
913
833569
30 2159
871
758641
29-5127
914
835396
30-2324
872
760384
29-5296
915
837225
30-2490
873
762129
29-5466
916
839056
30-2655
874
763876
29-5635
917
840889
30-2820
875
765625
29-5804
918
842724
30-2985
876
767376
29-5973
919
844561
30-3150
877
769129
29-6142
920
846400
30-3315
878
770884
29-6311
921
848241
30-3480
879
772641
29-6479
922
850084
30-3645
880
774400
29-6648
923
851929
30-3809
881
776161
29-6816
924
853776
30-3974
882
777924
29-6985
925
855625
30-4138
883
779689
29-7153
926
857476
30-4302
884
781456
29-7321
927
859329
30-4467
885
783225
29-7489
928
861184
30-4631
886
784996
29-7658
929
863041
30-4795
887
786769
29-7825
930
864900
30-4959
888
788544
29-7993
931
866761
30-5123
889
790321
29-8161
932
868624
30-5287
890
792100
29-8329
933
870489
30-5450
891
793881
29-8496
934
872356
30-5614
892
795664
29-8664
935
874225
30-5778 j
893
797449
29-8831
936
876096
30-5941
894
799236
29-8998
937
877969
30-6105
895
801025
29-9166
938
879844
30-6268
896
802816
29-9333
939
881721
30-6431
897
804609
29-9500
940
883600
30-6594
1 898
806404
29-9666
941
885481
30"i757
999
808201
29-9833
942
887364
30-6920
900
810000
30-0000
943
889249
30-7083
901
811801
30-0167
944
891136
30-7246
' 902
813604
30-0333
945
893025
30-7409
903
815409
30-0500
946
894916
30-7571
105
A TABLE OF THE SQUARES AND SQUARE ROOTS
OF NUMBERS. CONTINUED.
From 1 to 1000.
No.
Squares.
Square Roots.
No.
Squares.
Square Roots.
947
896809
30-7734
974
948676
31-2090
948
898704
30-7896
975
950625
31-2250
949
900601
30-8058 '
976
952576
31-2410
950
902500
30-8221
977
954529
31-2570
951
904401
30-8383
978
956484
31-2730
952
906304
30-8545
979
958441
31-2890
953
908209
30-8707
980
960400
31-3050
954
910116
30-8869
981
962361
31-3209
955
912025
30-9031
982
964324
31-3369
956
913936
30-9192
983
966289
31-3528
957
915849
30-9354
984
968256
31-3688
958
917764
30-9516
985
970225
31-3847
959
919681
30-9677
986
972196
31-4006
960
921600
30-9839
987
974169
31-4166
961
923521
31-0000
988
976144
31-4325
962
925444
31-0161
989
978121
31-4484
963
927369
31-0322
990
980100
31-4643
964
929296
31-0483
991
982081
31-4802
965
931225
31-0644
992
984064
31-4960
966
933156
31-0805
993
986049
31-5119 !
967
935089
31-0966
994
988036
31-5278 ,
968
937024
31-1127
995
990025
31-5436
969
938961
31-1288
996
992016
31-5595
970
940900
31-1448
997
994009
31-5753 i
971
942841
31-1609
998
996004
31-5911
972
944784
31-1769
999
998001
31-6070
973
946729
31-1929
1000
1000000
31-6228
106
TABLE OF SLOPES, &c. For TOPOGRAPHY.
Vertical Rise
Horizontal
Vertical Rise
Horizontal
Degrees.
in 100 feet
horizontal.
Distance to a
rise of 10 feet.
Degrees.
in 100 feet
horizontal.
Distance to a
rise of 10 feet.
1
1-75
572-9
o
19
34-43
29-0
2
3-49
286 : 4
20
36-40
27-5
3
5-24
190-8
21
38-40
26-0
4
6-99
143-0
22
40-40
24-7
5
8-75
114-3
23
42-45
23-5
6
10-51
95-1
24
44-52
22-4
7
12-28
81-4
25
46-63
21-4
8
14-05
71-2
26
48-77
20-5
9
15-83
63-1
27
50-95
19-6
10
17-63
56-7
28
53-17
18-8
11
19-44
51-4
29
55-43
18-0
12
21-25
47-0
30
57-73
17-3
13
23-09
43-3
35
70-02
14-2
14
24-93
40-1
40
83-91
11-9
15
26-79
37-3
45
100-00
10-0
16
28-67
34-9
50
119-17
8-4
17
30-57
32-7
55
142-81
7-0
]8
32-49
30-7
60
173-20
5-7
; 1^ !
R. B. WEARS, STEREOTYPER.
AN INITIAL FINE OF 25 CENTS
OVERDUE.
p
YB 5360!
2/y
W~ -
_