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EARTHWORK MENSURATION,
ON T HE BASIS OF‘ T HE
PRISMOIDAL FORMU LA .
CONT AINING A SIMPLE AND LABOR-sAVING MET HOD OF
OBTAINING PRISMOIDAL CONTENTS DIRECTL! FROMEND AREAS.
ILLU STRATED B ! EXAMPLES,
AND ACCOMPANIED B ! PLAIN RULES FOR PRACTICAL U SE.
C O N W A ! R . H O W A RD,
C IVIL ENG INEER,RICHMOND ,
VA .
NEW ! ORK
D . VAN NOSTRAND,PU BLISHER,
23 MURRA! AND 27 WARREN STREET.
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Entered , according to Act of Congress, in the year 1874, byD . VAN NOSTRAND
In the Oflice of the Librarian of Congress, Washington , D. C.
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PREFA CE .
T HIS work claim s to present a new and systematized method offinding the prism oidal contents of Earthwork by m eans of T ablesaccompanied by Rules so plain and simple of application as to fit itfor the c omm on uses of Engineers.
When the ratios of the side slopes are constant between end sections of which the transverse surface lines are sensibly sim ilar, allordinary cases of thorough cut and fill, term inal pyram ids, side-hillwork
,and borrow pits are covered by Formulae andand the prismoidal contents for all side slopes and bases are
taken from T ables 4 and 5 by Rules andIn the m ethod used, the heights of equivalent level sections are
not involved,nor is any calculation needed for loo-feet lengths
beyond ascertaining the half- sum and the difference of two quantities. For the m ost part T ables do the work of the calculator
,and
any one who can approximate cubic contents by the rough m ethodof Average Areas i s competent to obtain the prism oidal contentsby the Rules given.
T he tables of level cuttings are not needed when areas are given ,and are included chiefly for use in prelim inary estim ates when theonly data are the centre heights and the angles of the transversesurface slopes. With these, the heights of equivalent level sections are readily found by Mr. T rautwine’
s well-known and very ingenions diagram s, than which for the purpose intended probably no bettermeans can be devised. W hen these heights hav e been ascertained
,
the use of the special Correction T ables in connection with those oflevel cuttings will reduce to a minimum the labor of computingthe prismoidal contents . If further tables of level cuttings are
considered necessary, the reader is referred to Mr. T rautwine’s
Excavation and Embankm ent,”or to the example given at the
end of this work, by careful attention to which any required tablemay be written out with entire accuracy in a few hours. Specialcorrections for any side slopes may be . obtained by Rule 12.
Not an inconsiderable advantage of the present m ethod is that, by
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4
giving accurate corrections for the fam iliar approximations in gene .
ral use, the calculator has the elem ent of error constantly beforehim ,
and must speedily learn by practice, if not by theory, the casesin which such corrections become important . But While enough isgiven
,both by rule and example, inPart II. toguide the least theo
retical in the use of the tables, in Part I.
‘
a'
strictly mathematicalinvestigation of principles and derivation of formulae is subm ittedto the careful reader.T he article on Correction of Contents for Curvature was s
‘uggested by that on the same subject in Henck
’s'
FieldBook, but,by the formulae and table of factors given, in ordinary cases thecorrections are much m ore readily obtained In practice .
All of the tables In this work have been Calculated by the writer,and
,as the system used was that of continued additions with special
tests at intervals,it Is believed that they will be found absolutely
correct within the purposed lim its, whether the . last figure of anyam ount given be intended to express the nearest Whole number orthe nearest decimal.
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NOTATION AND SIGNS USED.
A and A’ end areas of earthwork.
M m iddle area.a and a' areas of triangle between road-bed and intersec
tion of side slopes produced.
6 and b’ road-bed widths.c and of centre heights of profile .7L and h’ heights of equivalent level sections.8 and s’ ratios of opposite side slopes to 1 .
d and d’ side distances.h,and 71-
2side heights“
N, N'
, n and n’ correction numbers.
C contents for 100 feet.
Q correction for curvature.
32“ greater or less than.
”
“ the difference between.
Grade triangle triangle between the base and the intersection of the side slopesproduced.
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EARTHW SURATION.
PART I.
AREAS—GROUND SLOPING TRANSVERSEL! . THOROUGH-CU T .
Fig. 1 .
Let area ABCFD A, area DFG a
,centre height BE 2 0 ,
side heights AK and CL and side distances AM and NCd and d
’
, base DE 2 b, and ratios of side slopes to 1 z s and s
'.
CASE 1 .—Side slopes the sam e . 3
’ z 8 . Produce the side slopesuntil they meet in G.
Es g, hence EG
and area (4
But BG c £18 , henceI
area AQG A+ a 0+
62
and A —5
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8
Exmnple.—Given s ’ z 3 z s b 2 18 ft. d 2 cl
’
6
7
(tab . 1 ) z 12,and a (tab . 2) z 108.
2
and A z 108 7845 .
CASE 2 .—Opposite side Slopesunequal . “
32 2 3 .
T he areas of the triangles DAE,EAR, BCE, and ECF are
respectively
x li5
10 0—20 1 7 and 2
2 2 2“
2
(b i c(
'
d d’
)and
,A g
2
Example—s ff ; 3
’ z 1 ; b 1 6 ; c d 82: cl’2
.
20 . 8 ;17t1
752
8A z
2
CASE 3 .-DE greater or less than EF .
Let DE 2 g, and EF 2 2T he triangles DAE,_EAB and BCE have the sam e expressions
bl
for their areas as in case 2, and area ECFQ
- n—2_
—+c
hence,
q
1 3.
7) b'
Example—Double W idth track. 3
Q ; 4
-
1; Q:
-
2z 21
c 2 71,a t
,z: d & d
2
Formula (1) applies only to case 1 formula (2) to cases 1 and 2and form ula (3) is general for all cas es where the whole road-bed
width i s either in cutting or embankm ent, and the surface slopesare sensibly regular between the centre and side stakes.
AREAs .—s IDE HILL CUTTING.
Let . g the hori zontal distance from centre line to grade pointopposite , and A the area of excavation.
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9
CASE 1 .—Both centre and side height in excavation .
T he areas of triangles DAE and EAR are as before, and that
of the triangle running out to grade(fig
777)-
20 (d+q)
hence,
A2
E'xmnpla
—s 1, b 20, c z d and
2‘
CASE 2.—Centre height in embankm ent.
1
o o o o o o o o o .o c o g o o o o o 0 0 0 0 0
9—5 10Example.
—b 18, 7) z 10, q z o . A
AREAS.—GROUND LEVEL TRANSVERSEL! .
Fig. 2.
F B
0 0
CASE 1 .—Side slopes the sam e, or s ’ z s .
AE Z FB Z ILS, and EF z CD
Area ABCD
Exempla
A : 0 : 26 X 20 2 520.
When the field notes are given, this example can, of course, beworked by any one of formulae or
CASE 2 .—Opposite side slopes unequal, or s
’
25 3 .
AE FB' and EF CD.
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1 ;
A : x 20 : 620 .
AREAS—GEO UND BROKEN TRANSVERSEL ! .
Fig . 3 . Q
T o calculate the area abcdefgT he elevations and horizontal distances apart of the points
a , b, 0 , (Z, c,f , 9 , m ust be determ ined in the usual m anner beforethe surface is disturbed, and of b
'
, c'
,(Z’
, f’
, g’
,after the excava
tion is m ade .
Calculate the area A a b 0 (Z3f g B between the surface line andthe assum ed datum plane AB also
T he area Au 6’
c’
d’
e'
f’
g’
yB between the bottom of the pit as
excavated and the sam e datum plane AB.
T he difference between the results so obtained,gives the area
required.
When the cross sections of the line have the surface brokentransversely, if the slope stakes are supposed to be at a and g (fig.
and AB is the plane of the road-bed,calculate
l st zthe area A a b c d cn
2d the triangles of excess WT he difference between the above two results will give the areaof earthwork required.
For side hill work the ‘ process is sim ilar,except that only one
2
triangle of excess 197
-
28is to be deducted.
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1 1
T his of course applies to embankment aswell as excavation.
None of the preceding cases require that the crOss section shallbe drawn before calculating its area.
CONT ENT S.—FRU ST UM FORMU LA.
F ig. 4.
If ABCD and be two consecutive cross Sections withlike surface lines and side slopes but unequal bottom widths, byproducing the side slopes until they meet at E and E'
,the whole
figures ABE and A'B'E
’ are sim ilar as well as the triangles CDEand But the solid being the differencebetween the frustum s ABEA’
B'
E' and CDEC ’
D’E
’ its cubic contents are
ABE A’B'E
’
vABE A’
B’
E’
CDE C’
D’E
’
V'
CDE C’
D'E'
in Which 1 represents the distance between the cross sections.
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12
If areas ABCD,A’B
’C
’D ’
,CDE and C ’
D ’E’ be represented byA, A
’a and a
’ respectively, then taking Ias 100 feet, and repre
senting the contents In cubic yards by C, we have :
3
If CD C’D
’ then a ’a, and the formulabecom es
27”
3
When CD C’D ’
0, a vanishes, and
C3
which is the formula for the frustum of a pyram id.
By formulae and (10) the whole of the formulae for cubiccontents hereafter given m ay be conveniently tested.
As the solid resulting from connecting the hom ologous sides oftwo sim ilar and parallel sections of unequal areas is the frustum of
a pyram id,formula (10) is applicable to any plane solid with such
end sections.
CONTENTS - PRISROIDAL FORMULA .
Rig. 5.
Let ABCDF be a given cross section, with a base FD b, and s
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13
and 3 ’ the ratios of i ts side slopes to 1 also let IHDEbe an equiva
lent cross section with level surface, height MN fl, and with s am e
base and side slopes. Produce the side slopes to their intersection at
E, and from E let fall the perpendicular EL on IE, intersecting the
base in G. Let area ABCDF IKDF A, and FDE z: a.
In the triangle FDE,FG EG x 3 , and GD EG x s
’
,or
FD bFD EC (8 whence EG
3 + 8’and area FDE
FD xEC (7 bF
2 2' X
8 + 3’
7)Sim ilarly In triangle IKE,
EL hm
IK +8’ and areaIKE
b2
sfl —Af +8 +S
r
2+61
2
consequently,
b2
+8)
be
m 7 Em o o o o o o o o o o o o
from which ,
6 b“
For convenience o f calculation,let GE
sis ’be represented
I)”
b2
s s’
3 s’
by g, and EL by H ; then asI t?
“
it "
we have, by substitution
H"
For a second section with corresponding parts b’
,H
’
, 3 and s’
,and
areas A’ and a'
I 12 I28 S
,
A (H -
9 PEand for the area M of a cross section m idway between A and A’
,
M +3’
(12)
The prism oidal formula for the contents C between two endareas A and A’ at a distance apart 2 l
,with an area M m idway be
tween them is
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14
éi lé: A+A’
A+A’
6 2 3
and by substitution in (13)
CA+A
’
3
2 M ZO O O O O O O O O O O O O O O O
and substituting this in.(13)
D :
6l
T he two last expressions for the value of 0 show that the calculation of contents by averaging the end areas requires a minus correc
tion ; and by the m iddle area (or, what is equivalent, taking theamount corresponding to the average
'
of the end heights from a
special table) aplus correction of exactly half as much . T he actualminus correction will now be found. By substituting the values ofA
, A’ and M in the second term of (14) we have
and reducing *
H ZVAA+2 s+s
r and g' and by substitution in (16)
Neglecting the common factors and l, and th e denom inator, the“
second term becom es,9H
(HQ—g? -(H
’9g1 9)__2 2 119—9 ,+H’ 9—
gm
2 2
2119—29 9—2H’ Q—Qg’ Q—I-IQ—zHH ’L
2
H9 H—HOQ—(g—fl’
P2 2
a nd res toring the factors and l, and the denom inator, w e obtain for
m ula
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15
Reducing
a b
1/ VA
m akingZ: 100, dividing by 27 observing that (x (y a - ac)2
(u~ m) and that a, we obtain
(h~5 f
Sd- S
aw
T his is the general formula when the opposite side slopes and endroad-bed widths are both different.When the road-bed widths are the sam e, or 6 6
’ z 0, the lastterm vanishes, and the formula becom es
(18)
T his is the general formula for all slopes and bases where the baseis constant between the two end sections.When bz b' : 0
, az o, and
AzA'
amT his is the general formula for the frustum of a pyramid, ’rsuch as
may be the solid between two sections of side hill excavation .
T he correction in term s of equivalent level heights k and la’
maybe found directly from (16) as followsWhen 6 ’
b,the expression (g -
g’
)2 vanishes and (16) becom es
9In squaring the binom ial of radicals the factor/i becom es
s+s’
s + s’
in the fi rst term , V2 in the second , and (VI )
?
in the third,or
8—1—8’ 34—3 ,
in each8
—3; thus cancelling the factor except in the last term of the
num erator.J[ Formula (10) before given for the frust'
um of a pyram id m ay be trans?
A+ A+VIE? 2A 2A’
+ 2¢AN3 6
form ed into form ula for
N
_7 2 A A 100(K/X
G
W/A XVhen A’=-O
'
in formula (19) it becom es(V
9 7
A A 100 1
7
0
7
0,which i s the formula for the solidity of a pyram id , as
it should Le .
7
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16
A+A’
s+s
2 (7b ut (7 +1 5 ) =(k_and substituting
,m akingZ100,
'
and dividing by 27,
(20)
As the plus‘ correction for calculation by“
m iddle area was found tobe one half of the m inus correction for averaging end areas
, by
m aking the requisite changes in (20)
C
but when b’: b
, from formula we obtain *'
2
8+Sl
2 2
and by substitution7L+7L
'h~ 7t s—l—s ’
100c)0
2°
2 ) (12 ) 2
T his formula is'
for use when the equivalent level heights have beenobtained.
APPLICATION OF T HE PRISMOIDAL FORMULA .
T he prism oidal formula in its ordinary form is applicable to avariety of solids
,regular and irregular
,but requires that the actual
m iddle section shall be previously determ ined and its area known .
In a modified form it can be applied practically by m eans of
tables ; sueh applications, however, a lways involving a value of the
By substituting the values of H,H ’
, g and g’in form ula (12) it becom es
b b’
(Hm?)2
mak ing b’ : b, and squaring
(“ride
-W(“ Eris-
e)(kfz-fwd”w ay—4541
1379
4
M i ) (b ) 1 a )s +s'
a+s’ + 2hh ++
s +s’
s +a’ +2
a+ s’
"H”s + s
’a+s
’s+ s
’
This also results directly from formula (7) by tak ing the area of a secondsection for a. height of and av eraging like parts forM .
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18
the end sections to intersection are H’s A
,and H"
s A’,and forthe area of the m iddle section
,by averaging like parts
2O
which is the same value of M as that before obtained . Substitutingthis in the prism oidal formula
100 and reduemX27
g,
X100
6 27
which is the formula for the frustum of a pyram id, and shows thatthis value of M introduced into the prism oidal formula lim its itsapplication to such solids only as are frustum s of pyram ids. T hiswill be illustrated further from Example 5
,page 36, in which when
carried to the intersection of the side slopes produced, the end sections are sim ilar.T hus carried to intersection, the end areas and the actual m iddle
area are respectively 349 , 2951 , and 1333 , as given page 3 6 .
By Roots and Squares
By equivalent level heights
zQ/A—r
—V2951 x—23
By substituting this value of M in the prism oidal formula
1438 tab . 4 z 5326 cyds.
For.
calculation by equivalent level heights as table 15 has abase of 14 feet
,and the above heights are taken to intersection of
side slopes, x 14 x }29;2must be deducted from contents
taken from tables.
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19
By Rule 4
table 15 .
table 17 .
100Deduct 29.8x 14x
27table 4 .
cyds.
By m ean proportional or frustum formula
table 4 . cyds.
By deducting the grade prism 32. 7 x —O7
0
the sam e result as that given on page 36 is obtained.
Another case in which the area of the actual m iddle section canbe deduced from the end areas directly
,is when each of the latter
can be expressed by two surface dim ensions,one of which is the
sam e for both end sections, as in solids whose end sections areparallelogram s or triangles with the sam e base and different heights
,
or vice versa . T hus if biz A and blt' A’ represent the end areas
of a solid of which the end sections are triangles with the sam ebase and different heights, as may be the case in side hill cuttingwhere the transverse surface slope increases regularly between theend sections
,by averaging like parts the m iddle area is
bh—i—eh’A+A
’
2 2
And as the prism oidal formula is applicable here, by substituting
this value of M
121 cyds practically
‘A+A 1“ 4100 A+A
’
100
6 27 2X27
which is the average area form ula, in this case giving the prismoidalcontents . As an example, suppose the ' triangular end sections ofthe solid to have a base of 20 feet and heights of 10 and 40 feetrespectively. T hen A z 10 x 10 z 100
‘
A'
10 x 40 z 400 and1o+ 4o A+A
'
2M : 250
By the prism oidal formula
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20
Calculated by Roots and Squares M
and this substituted in the prism oidal formula gives
100+4OO
6
+4X 225
£270
table 4 864 cyds.
Here the average area formula gives the prismoidal contents, and ’
the prism oidal formula applied by its m odification of Roots andSquares gives a very rough approxim ation. T he sam e inaccuracy isof course involved in the m ethod by equivalent level heights, whatever m ay be the shape of the equivalent and sim ilar end sections Ofwhich the level heights are obtained. For instance
,if the side hill
work i s excavated at rock slope, the level heights, if carried tovertex
, may be taken for sections“with any other side slopes, as 1 to
1 , or 1 1} to 1 .
At 1 to 1 carried to vertex H1
30
10 ; H’ V43
0
20, and to calculate by table 12, with side slopes 1 x 1 and base 18feet
15 table 12
10~ 2o 10 table 121
100Deduct 15 x 18 x -
2—7~ 270 table 4
864 cyds .
at 14 to 1 carried to vertex H 100 x 451r H’
V400
and to calculate by table 15, with side slopes 14 to 1 , andbase 14 feet.
12.245’ table 15 . . I 1468
table 17 .
Deduct 12 .245 x 14 x1
2
0
7
0table 4 —635
864 cyds.
T he two last examples show the sam e error of 62 cyds. obtained byEquivalent level heights, as before by Roots and Squares .
By m ean proportlonals or frustum formula
table 4 864 cyds.
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21
By Rule 2,10025400
250 table 4
18 table 5 62
864 cyds.If the above sections were sim ilar
,as for instance with dim ensions
10 x 10 and 20 x 20, the first m ethod’ by average areas would givetoo much by 62 cyds, whilst by the others the true prism oidal con-f
tents would be obtained.
If both the heights and bases are different and the sections areA+A
2and greaternot sim ilar, the m iddle area will be less than
and cannot be obtained directly from the end
areas. In such cases, the exact contents can be determ ined by theprism oidal formula only by first obtaining the dim ensions of theactual m iddle section and calculating its area
Practically in railroad earthwork it is only when the transversesurface lines of the end sections are very dissim ilar and the areasdiffer greatly in size that the resulting errors becom e important, andas at such points the cross sections are usually taken nearertogether
,it is very rarely the case that the m ethods of Roots and
Squares and Equivalent level heights fail of practical correctness.In cases of doubt, however, especially when the surface is warpedbetween the end sections, it is safer and better to obtain the area ofthe actual middle section before calculating the contents.
CORRECTION OF CONTENTS FOR CURVATURE .
T he following article was suggested by that given in Henck’s
Field Book,
”page 110.
In excavation on curves,although the cross sections are ac tually
staked out in the direction of the radii at the extrem ities of thechords, the calculation of contents is m ade as if these cross sectionswere perpendicular to the chords. In som e cases, especially wherethe transverse surface slope is considerable, this is the occasion of a
sensible error requiring a corresponding correction, the am ount ofwhich is determ ined as follows
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22
Suppose A, B, and C to be three consecutive 100 feet.
stationson a curve of radius OB and BF and BH the side distances atstationB .
T he calculation of contents between A and B,and B and C
m ade as if the cross sections at these points were on the lines K Land KL, and K
’
L' and K
2L2 ,
or perpendicular to the cho1ds AB
and BC, requires at each station a correction sim ilar to that at B ,
which will now be considered. It 18 evident that the correction isthe dif‘ference between the masses KEK’ and L'
EL, on oppositesides of the centre line
,and between the two vertical planes KL and
K’
L’ these m asses having for their cross sections respectively the
half-breadths BF and BH . T he angle KBK’ being very sm all, thearcs KEK'and L’
HL will be considered as straight lines ; and, asthe angle KBF L
’
BH 1} KBK’
T BA D,the deflection
angle of the curve,the distance KF BF x sin D or, generally
for small angles,any horizontal line as KK’
or‘
L’
L m easured perfpendicularly to the radius OB
,and term inated by the planes KL
and K’
L’
,is practically equal to BF or BH (the corresponding
horizontal distance from the centre line) multiplied by 2 sin D .
Consequently, the masses KEK'and L ’
BLbeing considered as truncated prism s with the areas of the half-breadths BF and BH asbases, their heights at any given ‘points are equal to the horizontaldistances of these points from the centre line, multiplied into twicethe sine of the deflection angle.
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23
F ig . 7 .
Let FEET represent the cross sectionat B (FigT o simplify calculations, the equal prisms MPT and BT N are
added.
T he area FET (BP+PT )F
; 0 -1—326
2; Q’and the heights
of the prism corresponding are (Zx 2 sin D at F ,and z 0 at B
and T . Its contents therefore r: Sim ilarly
the contents of prism HBT and the
correction - required,which is the difi erence of their volum es,
6 d2 2 si b d”
2 sin D
72?“
fi x 3
b
and if Q represents the required correction in cub ic yards,
0 0 0 0 0 0 0 0 0 0 0 0 0
But, from formula A-Ht, the area carried
to intersection of side slopes also sin D fig, and as R 2
Conditions—Single width road-bed
and opposite side
slopes the sam e .
T ransverse surfaceslopes regular.
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24
which C o represents the degree of curve, 2 sin D 50 x 2 x
00
T herefo1e,
(d d’
)O X o
Q
. 0 u s
In side hill wor ,k as
'
shown by Mr. Henck, the general formula
for the correction in cubic feet 1s 39-73(d—l—b—w)3__R
9’ m which
70 represents the width of excavation at the road-bed. But as f?A
, the area of earthwork, in this case the correction in cubicyards is
5 1 3 x 3 x 27Values of the last factor in formulae (23) and (24) are given in
T able 18 .
In excavation the correction for curvature as obtained by formulee (23) and (24) is to be added when the curve is convex, andsubtracted when it is concave toward the higher ground
,and in cm
bankm ent these conditions are reversed. It is supposed to be
applied at the m iddle one of three cross sections at intervals of 100
feet, and all on the same
'
curve.
If the distance to either of the cross sections next the one underconsideration differs from 100 feet, the correction found as above isto be multiplied by the half sum of the two distances and divided by100.
At points of curve or tangent one of these distances of coursebecom es nothing.
Whether the side slopes, or the widths from the centre line tothe edge of the road bed, are different 0 1 not, if the transverse surface lines are broken
,the cross sections should be drawn to scale,
the two half-breadths divided into triangles,and the horizontal dis
tances from the centre line to the corne1 s of each subdividingtriangle m easured on the drawing. T he sum of the three distances
2 sin D
3
contents ln cub ic feet of the prism corresponding. It is not material how the sides of the subdividing triangles are drawn, providedthat the whole of each triangle is on the sam e side of the centre line.
T he difference of the masses whose cress sections are the half
for each triangle multiplied by its area and by willgive the
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26
PART II.
PLAIN INST RU CT IONS
FOR OBTAINING THE PRISMOIDAL CONTENTS OF EARTHWORK,W ITH
PRACTICAL RULES AND EXAMPLES SHOW ING THE USES OF THEACCOMPAN ! ING TABLES IN SIMPLIF ! ING COMPU
TAT IONS B ! THE FORMULA?) OF PART I .
T HE following Rules for computation of Cubic Contents are basedon the condition that the transverse surface lines of the end sectionsshall be sensibly sim ilar ; but it will b e observed that 1 , 2, and
‘
S
together cover all cases to which the m ethod of Roots and Squares,
or of Equivalent level heights, can be correctly applied,and that
the practical lim it of their application may be indefinitely extendedby increasing the proxim ity of the cross sections in rough ground.
T o fi nd the prismoidal contents of thorough- cut orfi ll when road- bed
w idth and side slepes are constant between end sections.
Given areas,side slopes
,and base (A and A
'
, s and s’
,and b) .
RULE 1 .
—(FORMULAEnter table 2 with the given road-bed width (b) , and the half
sum of the ratios of the side s and take out the corre
spending area a . Add this to each of the given end areas and the
square roots of the resulting quantities (MA-Ri and MPH-Ht) fromtable 3 are N and N
'
,the correction numbers.
Enter table 4 with the average of the end areas
table 5 with the difference of the correction numbers andtake out the corresponding quantities. T he difference of the quantities taken from tables 4 and 5 is the contents in cubic yards for a
length of 100 feet.For a different length multiply by the length in feet and divide
by 100.
Example—Given A r : 974 ; A
’ z: 87 3 32 ; s’ z 53 ; h 20 .
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27
From table 2 when 6 20 and {_gi : the area of the gra de
triangle (a) 1 60
A12
“,
table 4
N~N' table 5 —200
C ontents for 100 feet 1765 cyds.For a different length as 80 feet, 1 765 x 1412 cyds.
Nora—If the square roots of the areas to the intersection of theside slopes are obtained and recorded when the areas are calculated,as will ordinarily be found m ore convenient, the
' data are A and A ’
and N and N’
,and only the two last steps of Rule 1 are necessary.
T ofi nd theprismoidal contents of side hill work, pyramids , and any
solid w ith'
sim ilar end sections .
Given end areas (A and
RULE 2 (FORMULA
T ake the square roots of the end areas (VA and (A? ) fromtable 3 z n and n ’
.
Enter table 4 with the average of the end areas
table 5 with the difference of the correction numbers andtake out the corresponding quantities. T he difference between thequantities taken from tables 4 and 5 is the ~contents in cubic yardsfor 100 feet.For a different length multiply by the length in feet and divide
by 100.
Example—Given end areas A z: 41 and A’ z 185 .
VA r: 41 table 3 z 2 n ; 1/A’
185 table 3 2 2 n’
.
A+A'
41+i s5
2 2113 table 4
n~ n' table 5
C ontents for 100 feet cyds.
For a different length,as 25 feet
,cyds.
Em mple.—Pyram id. Given end areas A z 104 and A’ z 0 .
VA 2 104 table 3 r 2 : n ; VP? 2 0 n'
.
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28
rei
jo
52 table 4
n~ n’ table 5
Contents for 100 feet 128. 4 cyds.
For a different length, as 60 feet, 128. 4 x 0 . 6 : 77 cyds.NOTE.
—Examples under Rule 1 can be readily tested by Rule 2,the difference in the working being that thew rade prism is fi 1 st
included and then deducted. For instance, inc
the example givenunder Rule the end areas to intersection of s ide slopes are 1 134and 247, and the square roots corresponding and —thc11
11 341f table 4 2558
table 5 - 200
Contents to intersection of side slopes .
Less grade prism 1 60 table 4 .
Contents of earthwork for 100 feet 1765 cyds.
T ofi nd theprismoidal contents of thorough cut orfi ll when the end9 cad- bed widths are different.
Given end areas,side slopes, and end road-bed widths (A andA
’
;
s and s’ Z) and
RULE 3 (FORMULA.
Enter table 2 with and b, h’
and h~ b’ respectively
,and
2
take out the corresponding areas a,a
’
and a”. From table 3 take
out the square roots of the end areas to intersection !VA-Hi N,
and MA ’
+a’ _N’
.
Enter table 4 withAi _ a
and table 5 with N~ N’
,and the
difi erence between the corresponding quantities taken from tables 4and 5 is the contents in cubic yards for 100 feet. For a differentlength multiply by the length in feet and divide by 100.
Example—Given hz w ; b
’: 40 ; 3
’: 1565 ;
A’: 253 .
Here it 2 128,a
’_800 a”—288 N 1 : and N’
A+A a” 288
2 6 2_6
table 4'
N~ N’ table 5
Contents for 100 feet
For a different length , as 50 feet 1 749 cyds .
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29
T he example under Rule 3 is of a case where averaging the endareas gives less than the prism oidal contents. It m ay be tested byFormula 8 , page 12, as also Rules 1 and 2 by Formulae 9 and 10 .
T ofi nd theprismoidal contents when the ground is level transversely ,or where the heights of equivalent level sections have been obtained.
Given level heights,base and half-sum Of ratios of side slopes
h and“ h’h and
842
—8
RULE 4 (FORMULA
Enter the table of level cuttings for the proper base and side
slopes with the half- sum of the end height and the table
of special plus corrections for the sam e side slopes w ith the difference of the end heights and take out the correspondingquantities. T he sum of these quantities is the contents for 100 feet .For a different length, multiply by the length in feet and divide
by 100.
Example—Given b 14 h z h
’
1
h+h’ ‘8
2 2table 15
h~ h’
8 . 8 28 .2 table 17 +368
Contents for 100 feet 4408 cyds.
For adifferent length,as 85 feet, 4408 x 3747 cyds.
T o fi nd the Correction for Curvature in single w idth thorough-cut
when the transverse surface slope is regular.
Given : area to intersection of side slopes,degree of curve
,and
difference of side distances (A—a, C °
, and
RULE 5 (FORMULA
Enter table 18 with d~ d’ and take out the corresponding factormultiply this into the product of A+a by C °
,and the result is Q.
the correction in cub ic yards, to be applied at the middle one of
three stations, all on the sam e curve and 100 feet apart. If the distance to either of the other two stations from the m iddle one differsfrom 100 feet, multiply by the half-sum of the two distances anddivide by 100.
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30
T his correction is to be added or subtracted accordingly as thecurve i s convex or concave toward the higher ground .
.Eaaniple.—Given c 28 ; h1 40 ; h2 1 6 ; d 2 74 ; d
’
38
21 z 28 ; R 1400 ; or A+a 2090 ; C°
2 d~ d’
3 6 .
.3 6 table 18 :
and 2090 x 4 .09 x cyds.
If the distances to the two adjacent stations are 50 and 40 feet
-
566.1 espectively, the correction 1 equired is
200
cyds.
T ofi nd the correctionfor curvature in side-hill worh° when the trans
verse su‘
rface slope is regular.
Given : area ; degree of curve ; side distance ; road-bed width ;and width Of excavationat road-bed (A C
°d 5
RULE 6 (FORMULAEnter table 18 with d+h—w and take out the corresponding
factor : multiply this by the product of A by C°
, and the result is
Q the cor1 ection 111 cubic yai ds, to be applied in all l espects as inRule 5 .
Example—Given w 1 7 h 30 d 5 1 h
l24 R
1 600 or A r: 204 C°
d+h—w 64 .
64 table 181nd cyds.
If both intervals are 50 feet, the correction required is
x x cyds.
For cor1 ection for curvatui e when the transve1 se su1facc slope i sb1 oken, or for double-width thorough cut, see page 24 .
Rules 5 and 6 apply to excavation only. For embankment thecorrection is to be added or subtracted accordingly as thecurve isconcave or convex toward the higher ground.
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31
MISCELLANEOU S EXAMPLES .
EXAMPLE 1 .
(8) (9)
End Ce t t ’u Corr
'n
Areas. sq . roots.
199 6
90 3
3
8
1
9
1
31
4 6
1
0
10“100 .0lhoot
1 0
Example 1,as above
,is of the railroad cut given in Morris’s
pp . 47—54, with contents computed by Rules1,2, and 4, and the auxiliary tables of the present work . As
here used, the areas are supposed to belong to sections which , whencarried to the intersection Of the side slopes in thorough-cut
,are ren
dered sensibly sim ilar,and the examples as here given are ,
intended“ Easy Rules for the Measurem ent Of Earthworks by m eans of the Pris
luc ida1 Formula. By Ellwood Morris , C .E.
” Philadelphia : 1872.
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32
to show Only the comparative facility of arriving at'
the prismoidalcontents by Mr. Morris
’s m ethods and those of the preceding rules
when the above condition of similarity is fulfilled, and not to
endorse the application of the m ethod of Roots and Squares (or°
of
the rules of this work) in cases where the hypothetical m iddle aream aterially differs from the actual onc .
*
Except by trial with the actual m iddle sectioir and the prismoidalformula
,it seem s almost impossible in cases of dissim ilar end sections
to know when the application of the m ethod of Roots and Squares,or
of the preceding rules, begins to fail of practical correctness, but itm ay safely b e assum ed that if the ground is properly and sufficientlycross sectioned , the 1 esults obtained by them will be practically theprism oidal contents.
T he above tabulated example shows all the steps necessary infinding the prism oidal contents in cubic yards when the areas are
given . Columns and (3) being written out, (4) is deriveddirectly from (3) by averaging (5 ) from (3 ) by adding area Of gradetriangle in thorough- cut (6) from (5) by table 3 (7) from (6) bysub traction (8 ) from (4) by table 4 (9) from (7) by table 5 and
(10) from (8) and (9) by subtraction .
Column (4) gives the average end areas throughout the out,
including the term inal pyram ids,and the only break in the routine
of adding the area of‘ the grade triangle in colum n (5 ) is at the point
where the cutting runs out on the lower side . ‘At such points twoareas have to be used, the one of earthworkplus the grade triangle,for computationof thorough- cut by Rule 1 , and the other of earthwork alone
,for the calculation Of the pyram id or side-hill work
into which the thorough-cut changes, and of which the computationof contents falls under Rule 2 .
Column (8) gives the contents between each two stations roughedout by the comm on m ethod of average areas
,column (9) the cor
responding error,and column (10 ) the prism oidal contents , all in
cubic yards.It is not strictly necessary to write out all of the columns given
above , but errors are so m uch more readily detected when all of thesteps are shown, that ordinarily tim e and labor will be saved byadopting som e system of tabulating sim ilar to the above
,both as
regards the number of columns and the arrangem ent by which thefigures referring to each two stations may be recorded on a linebetween them .
See art icle on the appl ication Of the prismoidal formula, page 16.
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34
W ith equivalent level heights given,the above tabulated example
“
shows all the steps required in finding the approxim ate prism oidalconten ts in cub ic yards. Columns and (3 ) being writtenout
, (4 )- is derived
-
directly from (3 ) by averaging, and (5) from (3)by subtracting. T he table of level cuttings for a base of 20 feet andslopes 1 to 1 , from which column (6) should be taken, i s not published in this volum e, but its place may readily
'
be supplied by adding1 . to each of the helgh
‘
ts of and taking 70 from each of
the‘ corresponding quantities in table '12L. Such remainders are theam ounts in colum n Column (7) is derived from (5) by table14
,and (8) from (6 ) .and (7) by addition.
In ordinary ground sloping transversely,the area of earthwork of
the term inal pyram id at the point where the centre height isnothing
,is about one-fourth of the area of the section where
.the
pyram id begins and practically, as only sm all quantities are concorned
,the equivalent level height corresponding m ay be taken as
one- fourth of that corresponding -to l
ithe area of the base Of thepyram id.
T he calculation of contents by equivalent level heights and
tables is well suited for prelim inary or approx imate estim ates,espe
cially if, as in the present case, when the sum of the tenths Of theend heights is uneven, the average is always
'
taken as the tenth next
greater than the actual half- sum .
T he variation between the conten ts of the thorough- cut from 1 to
1 7, as given in Examples 1 and 2, is due to the fact that the equivalent levcl heights are carried out to tenths only. In the presentcase, at a height of 20 feet the increm ent 1s ovor two cubic yards foreach of a foot
,and in embankment at the sameheight it 18
still greater. As in p1actice neither equivalent level heights northose of the tables of level cuttings are carried out to hundredths
,
one cause of the greater accuracy of the previous m ethod by Rules1 and 2 is evident . It may be replied that errors as important areinvolved
.in the field work
,the cross section stakes being set only
approxim ately but that an elem ent of error'
should voluntarily beintroduced into the calculations because another such alreadyexists in the data, is a position that will not be contended for
eriously.
Example 3 .—In a cutting with road bed width 1 6 feet, and oppo
site side slopes 2 to 1,the given a1 cas of two consecutive cross
sections with sim ilar transverse surface lines and at a distance apart
of 100 .
feet,
.are 100 and 1000 square feet l espectively : requi1 ed
the prism oidal contents. Here the area of the grade triangle (table 2)
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35
is 102, and consequently the whole areas to intersection are 202 and1 102.
T ofi nd the correction numbers JVand .N’
.
909 table 3 1
1 109 table 3 N’
. T ofind the contents in cubic y ards .
1 0015
1000550 table 4 9037
table 5 .
Contents for 100 feet
T est by Formula 9 .
V202 >< 1 102 472 m ean area to intersection .
10°599—1
490 table 4 1815 cyds.
Example 4 .—Given 1 00 and 1000 square feet respectively as the
areas of two similar cross sections . 100 feet apart,irrespective of
shape or number Of sides in perim eter r equired the prism oidal contents.
T ofi nd the correction numbers n and n’
100 table 3 n
1000 table 3 n'
T ofi nd the contents in cubic yards .
10015
1 000550 table 4 9037
table 5 L.
Contents for 100 feet
T est by Formula 10 .
V100 x 1000 316 m ean area.
472 table 4 1748 cyds.
Example 5 .—At two stations 100 feet apart with base
i
b 14 feet,and side slopes s 12 to 1 , given the notes of the cross section at
the first station,centre height C : side heights h
1and
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36
and and side distances d and d' and and atsecond station, centre height side heights and andside distances and
Calculation of areas A and A’
,and correction numbers N and 1V
’
.
For the grade triangle corresponding to b c : 14 and$
58
142 ,
the height table 1 and thearea table 2 z 33 a .
By Formula (1) and Rule 1 .
correction number N and 349 33 z 316 z A.
correction number N and 2951 33 2918 zA’.
Calculation of Contents—Formula (18) Rule 1 .
316—22918 —1 617 table 4 5989 cyds.
table 5
Contents for 100 feet
T est by Formula 13 .
From the preceding data the notes of the m iddle area would givecentre height and side distances and and by For
mula (1 )
byFormula (13)X 4 100
T ofi nd the equivalent level heights—(Rule316 table 4 . . 11 70 table -10 equiv. lev. ht.2918 table 4 . table 10
T est by T rautwine’s method, with level heights .
table 10 1 1 74
table 10table 10
6‘
l_3 1 ,263Contents for 100 feet cyds.
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37
By Formula Rule w ith level heights.
table 10
table 15 +393 0
Contents for 100 feet cyds.
By Formula w ith end areas and level heights .
1 61 7 table 4
table 1 7
Contents for 100 feet
Approximation by Formula w ith centre heights of profile sub
stitutedfor level heights .
1617 table'
4
table 1 7
Approximate contents for 100 feet cyds.
T his approximation is for an extrem e case, as in practice thedifference between two consecutive centre heights is rarely as muchas one-half of the difference above taken. In ord1nary cases thisapproximation gives results very nearly correct.It will be Observed that by T rautwine’
s m ethod, as given above ,
three quantities are taken from the tables, and that it involves anaddition of three quantities
,a multiplication
,and a
'
division whilstby Rule 4, which with the sam e data gives the sam e result, the sumof two quantities taken from the tables is the required contents .Example 6 .
—Correction of Contents for Curvature—If thesecond cross section of Example 5 i s at the m iddle one of threestations 100 feet apart, and all of them on a 6
° curve which is concave toward the higher ground, the correction for curvature to bededucted at the station under consideration is Obtained as follows byRule 5From the above C °
6, and from the nOtes of Exam ple 5 ,A+a 2951 , and d~ d
’ But table 1 8and Q 2951 x 6 x cyds.
T est by Henclc’
s Formula .
C {2c(d sin D,in which d and d’
are side distances, h and h’
side heights, c the centre height, and D
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38
the.deflection angle .hence from the above and the notes of
Example 5 ,4 x 2 2 x 136 .4
O2
X 36 . 64
139 . 6 cyds. In practice d~ d’ i s required to then earest foot only.
x 0 . 05934 2 : cu . feet
REMARKS ON EST IMAT ING CONT ENT S
PROFILE EART IIWORK.
In addition to the cross sections at the regular stations, othersare necessary where changes begin in the character of the transversesurface slope, as well
'
as at all points where the surface line of theprofile changes its direction and all of the formulae and rules heretofore given for finding the contents suppose the solid to be betweentwo consecutive cross sections taken at such points.In passing from cutting into embankment, cross sections should
always be taken at the two points on opposite sides of the road-bedwhere the cutting runs out.” T his will obviate the necessity forstaking out the P.P.
”except with a zero point on the centre line,
as, in addition to accurate data for calculation of the pyram ids ofcut and bank which lie between the two cross sections thus taken
,
two m ore zero points, one on each side of the road-bed,will be
given. For like reasons, in passing from thorough into side hillcutting
,the point on the lower side where the excavation runs out
should be cross- sectioned.Where the original quantities of excavation
.
and embankm enthave been calculated, and the work is being done according to theslope-stakes and fi eld-notes, probably the simplest m ethod 01 Obtaining the quantities m oved in an unfinished cutting Or
l
embankment
is to take the average heights above or below the road-bed at eachof the several stations of that portion which has been worked upon
,
and then,. w ith Formula Rule 4, and tables, to calculate by
these heights the quantities rem aining to be done . T he latter subtracted from the original quantities between the sam e stations will,of course
,give the desired am ount.
When the material lies in strata, a sim ilar m eans may be usedfor determ ining the respective quantities of the diiferent kinds of
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39
excavation. For example, a’
cutting may be composed of earth attop, loose rock below the earth, and solid rock at bottom : theamounts then calculated by the loose rock heights, and deductedfrom the original quantities giving the earth, and the solid rocksim ilarly calculated and deducted from the am ounts obtained by theloose rock heights giving the loose rock. When the necessary ave
rage heights have been obtained, the quantities corresponding m aybe found very rapidly by Rule 4 and the propertables.
Forapproxim ate estimates, when the centre heights and transverse surface slopes only are given, the shortest m ethod is to find theequivalent level heights by T rautwine
’s diagram s, and then take out
the contents by Rule 4 .
When the work is carried on irregularly, no general rules forascertaining the true contents can be given. When the cross sectionsare very irregular and dissim ilar, the best practical rule is to takethem at very short intervals. T his in all cases reduces the error inthe calculation of contents to a m inimum .
A very careful and thorough investigation of the m athem aticalm ethods of calculating irregular earthwork is given in the article onEarthwork”in Henck
’s Field-Book,
”and to that the theoreticalreader is referred .
BORROW PITS.
For obtaining the contents of extensive borrow pits,the follow
ing will be found to be about as simple a m ethod as is consistentw ith correctness . Before the excavation is comm enced
,lay off the
surface in squares, rectangles, or triangles, small enough to be considered as plane surfaces
,and take elevations with the Level at all of
the corners. T hese elevations must be referred to a base which willbe below the bottom of the borrow pit when the work is finished.
A plan of the ground as laid ofi should then be m ade,and the
elevations above the base recorded on i t at . the corners. When an
estimate of the quantities excavated is to be m ade during the progress of the work, the horizontal lim its of the pit as then excavatedshould be taken, and inside of these lim its the whole of the groundagain divided into rectangles and triangles without reference to theform er surface divisions, the elevations above the base plane againbeing taken at all corners, including those on the surf ace at theedges of the pit.The original quan tity inside .oi the pi t lim its and down to t he
base plane, taken as a series of truncated prism s, should then be
calculated, and next the quantity rem aining inside of the pit lim its
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40
and above the base plane . The difference between’
these am ountsgives the quantity excavated.
T he advantage of using an independent m ethod .of dividing upthe ground after the original surface has been rem oved is that itrarely happens that the best arrangem ent of these subdivisions forreducing to plane surfaces will agree accurately, either in size or
position, with those originally' taken on the ground surface . If,
however,the same divisions can be taken in the bottom of the pit as
originally on the surface, the difi erences of the elevations at eachcorner taken before and after the excavation is m ade will give theheights of the prism s, of which the contents may be obtained by a
single calculation.
In order to prevent the necessity for recalculating the finishedportions at each estimate, when any portion of the pit will not againbe disturbed, its lim its should be referenced on the ground and indicated on the plan, and its contents recorded separately.
RU LES FOR VARIOU S U SES OF T ABLES .
T o’
jincl the height of an equivalent level section.
* Given areas, side slopes, and base.
RULE 7.
Enter table 4 with the given area, and take out the corresponding
quantity find the quantity nearest to this in t he body of table oflevel cuttings with the given side slopes and base
,and the index
number corresponding is the equivalent level height to the nearesttenth.
W hen centre heights and transverse surface slopes only are given, if rratio to 1 of surface slepe cotangent of surface angle, and 8 ’
8,then the
equivalent level height
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42
T o extend the Correction T ables, general or special.
RULE 11 .
When the difference of the correction numbers,or of the level
heights, is too large to enter the table with , take one-half of it andwith this enter and take out the corresponding quantity
,which mul
tiplied by 4 gives the correction required for a length of 100
feet.Examples.
—In table 5 the correction corresponding to 32 iswhich multiplied by 4 gives the correction correspondingto 64 .
In table 17, the correction corresponding to 1s whichmultiplied by 4 gives the correction corresponding to
T ofi nd the special corrections for any given side slopes from the
general correction table.
RULE 12.
Enter table 5 with h~ h’
,and take out the corresponding quan
tity ; for the'
specialplus corrections m ultiply this by the quarter
sum of the ratios of the side slopes for the special minus
correction multiply by the half- sum T he corrections so
obtained are for lengths of 100 feet .Exanqolcs.
—From table 5 the general m inus correction corre
spending to is and the plus correction for 1 7} is
x a corresponding to table 17. T he m inus cor
rection for 4 is x 4 r: corresponding to
s+s'
table 14 . In like manner with7
lg the plus correction for
x table 8 and with 1 , the m inus cor
rections,general and special, are the sam e, as are N~N
’ and h~h'.‘
(See table 5, and example s 1 and 2, pages,31 and
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43
EXPLANAT IONS OF TABLES .
T able 1 is for obtaining the height of the grade triangle . T o
use it,find the half-sum of the ratios of the given side slopes at the
top, and the number vertically below, and on the sam e line w ith thegiven road-bed width in the left column, i s the height required .
T hus with b 1 6 and §j9
2§ g the height correspondingis
T able 2 contains the area of the sam e triangle . It i s used withthe sam e data and entered in the sam e way. T hus with hz 18 and342
- 31{thearea corresponding a z 1 62 .
T able 3 gives square roots to tenths, - or correction numbers ofareas. T o use i t
,find in the body of the table the number nearest
to that which expresses the area under consideration,and the figures
on the sam e horizontal line ln the left column are the whole numbers
,and that imm ediately above it, at the top of the table
,the
tenths of the correction number required. T hus if the area tointersection of side slopes is the correction number N is
if one of sim ilar end areas is 230, the correction number n i s
T able 4 is for finding the contents for 100 feet corresponding toa g1ven area. T he left column contains the tens, and the top theunits
, of the area. In the body of the table are the correspondingcontents in cubic yards for lengths of 100 feet. In the short tableof two lines prefixed, the contents corresponding to the tenths of thearea are given
,and these when requ ired are to be added to the con
tents taken from the m ain table . T hus the contents correspondingto the area are cubic yards .
T able 5 is for obtaining the corrections for computations by average areas. T he arithm etical difference between the correctionnumbers is to be found in whole numbers and tenths respectively, inthe left column and at the top of . the table, and the number correspending in the body of the table is the correction in cubic yards fora length of 1 00 feet. T hus if the difference of the correctien num
hers is the correction corresponding is cyds . T hiscorrection is always to be subtracted.
T he T ables of Level Cuttings for special side slopes and road-bedwidths give the cub ic yards for lengths of 100 feet corresponding tothe different heights, of which the whole numbers are in the leftcolumn and the tenths at top.
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44
T he special tables of plus corrections give the correction forcomputation by averaging equivalent level heights. T he differencesof the end heights in feet and tenths respectively are in the leftcolumn and at top
’
,and the corresponding corrections for lengths
of 100 feet in the body of the table . Care must be taken to use thecorrection table with the half sum of the side slopes the sam e as thatof the table of level cuttings of which the contents are to be corrected.
T he special tables of m inus corrections give the corrections foraverage areas when entered with the heights of equivalent level sections. T he side slopes of the table must be the sam e as those of theend sections, between which the contents are to be corrected .
When the tables of minus corrections for special slopes areentered with the differences of the centre heights of the profileinstead of those of the equivalent level heights, in ordinary ground3. close approxim ation to the true c orrection is obtained.
For the special plus correction tables the half-sum of the sideslopes is indicated at the tap. For the special m inus correctionsthe slopes are indicated at the bottom of the sam e tables.
T able 18 contains factors for calculation of the corrections forcurvature. Its use is explained 111 Rules 5 and 6.
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45
T ABLE N0 . 1 ;
Roadbed Width in Left Column ; half - sum of ratios of Side S lopes
at T ap Height of Grade T riangle in body of T able.
TABLE N0 . 2 .
Roadbed Width in Left Column ; half -sum of ratios ofat T op Area of Grade T riangle in body of T able.
i e e e
33 -3
784 6 5-3
5 5 5 i
5
73-I
1
2 5
36
49648 1
1 0 0
_1_18
43-5
11
15 11
15 15
72 -7
32 -7
4 2 -7
54 -0
15 15
2
9 -3
i
6 -9
1 7 1
l
6
78
91 0
I I
1 2
1 3I41 5
It 11
4 -4 4 -0
5-3 4 -8
1 1 2
1 2 0
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‘46
TABLE No . 3 .
Areas in body of T able ; Correction JVos . , in feet and tenths,in left
column and at top.
. 2 .3 .4 .6 .7 .8
4 -4
1 613
2 6 . 2 7.
“
37-2
50 4 16 513
82 13
1 0 2 .
146 1 41 7 1 13
1 9813
2 2 8 .
2 591 2
2 92 1 4
3 2 713
36413
40 4 .
445-2
4881 1
580 13
630 .
6 8 1 1 2
734 5 1
789131 84613
90 6 .
9671 2
(1 2
513
1 1 13
1 9 1 129 1 2
4 1
70 13
88 1 11 0 81 2
1 30 .
1 5313
1 7913
2 0 71 12 371 2
2 69 .
30 2 13
33813
4 1 61 2
458
50 1 13
5955 164 51 2
697.
750 13
80 613
864 1 4
9 241 2
986
1 1 2 1 41 3413
1 5813
1 85.
2 1 31 2
2 43 1 42 7513
346 .
3841 2
4 24¢t
46613
5 1 0 13
557.
6 0 51 2
70 713
76 1 13
8 1 8.
876 1 2
99813
1 0 30
1 0 96
1 1 6 31 2 32
1 30 31 376
1452
1 52 9
1 60 8
1 6891 772
1 858
1 9452 0 342 1 2 5
2 2 1 8
2 3 1424 1 1
2 51 0
1 0 50
1 1 1 6
1 1 831 2 531 32 51 399I47S1 552
1 632
1 7 141 798
1 8841 97 1
2 0 6 1
2 1 53
2 2472 3432 440
2 540
-4
1 0 631 1 2 91 1 971 2 6 71 340
14 141490
1 568
1 648
1 73 1
1 8 1 51 90 1
19892 0 792 1 72
2 266
2 36 2
2460
2 560
.6
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447
TABLE No . 3—CON0 LU DED .
Areas in body of T able ; Correction Nos . , in feet and tenths,in left
column and at top.
.6
2 6632 76 72 8732 981
30 9 I
3 2 0 4
33 1 8
3434
3552
36 72
3 795
39 1 9
40 45
4 1 73
430 3
4436
4570
470 6
4844
4984
5 1 2 7
SZ7I54 1 7
556 5
57 1 5
5868
60 2 2
6 1 78
6336
6496
6659682 36989
7 1 57
73 2 7
750 0
76 74
7850
80 2 8
82 0 8
839 1
8575876 1
8949
9 1 39. 9332
952 6
9 72 2
99 2 0
1 0 1 2 0
-7
2 6732 7772 8842 992
3 1 0 2
3 2 1 5
332 9
3446
3564
3684
380 7
393 1
4 0 58
4 1 86
43 1 6
4449
4583
472 0
4858’
4998
5 14 1
52 85
5432
5580
5730
588360 3 76 1 946352
6 5 1 2
66 756 839
70 0 6
7 1 74
7344
75 1 7
769 I
7868
80 46
82 2 6
840 985938780
8968
9 1 58
9351
9 545
974 2
9940
1 0 140
.8
2 683'
2 788
2 894
30 0 3
3 1 1 4
3 2 2 6
334 1
3457
3576
3697
38 1 9
3944
40 70
4 1 99
4330
446 2
4 597
4733
4872
50 1 3
5 1 55
INHJ Or
(1 0 5
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48
TABLE N0 . 4 .
Areas T ens in left Column and U nits at top. Contents for 100 feetin cubic yards in body of Table.
0
3 . 4 . 5 . 6 . 7 . 8 .
IS9-3
2 333
30 7-4
344-4
455-0
603-7
751 -9
863 .
9379 74-I
I 1 59 -31 1 96 -3
1 344-4
1455-6
I751 -9
1 863
51 -9
1 63 .
2 0 0 .
2 37 .
459 -3
496 -3
533-3
570 -4
755-6
90 3-7
940 -7
9 77-8
1 163 .
1 2 0 0 .
1 2 37 .
I4S9 -3I496 -3IS33-3IS7O-4
1 644 4
I755-6
351 -9
4 2 5-9
463 .
50 0 .
537
574 -I
759 -3
796 -38333
90 7-4
9 44 -4
I351 -9
1 463 .
1 50 0 .
IS371 574 -I
I7S9 -31 796 -31 8333
59 -3
96 -3I33-3
244 4
355-0
50 3 -7
540 -7
577-8
72 5-9
763 .
80 0 .
837.
IOS9 -3
1 1 33-3
I3SS-0
IS4O-7I577-8
1 763 .
1 80 0 .
1 837 .
63 .
1 0 0 .
1 37.
359 -3
396 -3
433-3
470 -4
50 7-4
544 -4
1 0 63 .
1 1 0 0 .
1 1 37.
I359 -3I396 -3I433 -3I47O-4
1 544 -4
1 80 3 7
IO3-7
3 2 5-9
363
40 0 .
437
4 74-I
659 3
733 -3
770 -4
8444
9 55-0
1 363 .
140 0 .
I4371474 -I
1 5 1
1 659 3
I733 -3I 77O-4
33-3
70 -4IO7-4
40 3-7
44 0 -7
477-8
551 -9
663 .
70 0 .
737
774 -I
9 2 2
9 59 -3
99631 0 33-3
I I44-4
I44O-71 477-8
ISSI-9
1 663 .
1 70 0 .
I737I774 -I
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50
TABLE No .
'
4—'
CONTINUED .
Areas T ens in left Column and U nits at tap. Contents for 1 00 feetin cubic yards in body of T able.
1 0 1
1 0 2
1 0 31 0 41 0 51 0 6
1 0 71 0 8
1 0 91 1 0
1 1 1
1 1 2
1 141 1 51 1 6
1 1 71 1 8
1 1 9
1 2 0
1 2 1
1 2 2
1 2 3
1 2 4
1 2 51 2 6
1 2 71 2 8
1 2 91 3 0
1 3 1
1 3 2
I33I34I3S1 36
1 371 38
I3 9140
14 1
142
I431 44145146
147148
1491 50
3740 -7
3777-8
392 5-9
3963
40 0 0 .
40 37
40 74 -I
4 2 59 -3
4 296 -3
4333-3
4370 -4
440 7-4
4444-4
451 8-5
4555-6
459 2 -6
4 70 3-7
4740 -7
4 777-8
49 2 5-9
4963
50 0 0 .
50 37
50 74-1
52 59 -3
5290 -3
5333-3
5370 -4
540 7-4
5444 -4
5555-0
3744-4
3966 -7
4 00 3-7
40 40 -7
40 77-8
4263 .
430 0
4337
4374 -1
4448-1
4559 -3
4596 -3
4633-3
4 70 7-4
4 744 -4
478 1 -5
48 1 8 . 5
4855-6
4966 -7
50 0 3-7
50 77-8
5 1 51 -9
52 2 5-9
52 63 .
530 0
5337
5374 -1
5559 -3
2 .
3 748-1
3785-2
3859 -3
3896 -3
3933 -3
3970 -4
40 0 7-4
40 44 -4
4 1 55-6
430 3-7
4340 -7
4377-8
44 51 -9
452 5-94 563
46 0 0 .
4637
46 74 -1
4748-1
4785 2
482 2 2
4859 -3
4896 -3
4933-3
4970 -4
50 0 7-4
50 44 4
51 55-0
530 3 -7
5340 -7
5377-8
5451 -9
54889
552 5-9
5563
2 .
3 .
3751 -9
37889
382 5-9
3863 .
390 0
3937
3974 -1
4 1 59 -3
4 1 96 -3
4 2 33-3
42 70 -4
430 7-4
4344 -4
438 1 -5
44 1 8-5
44 55-6
4 566-7
46 0 3 -7
4640 -7
46 77-8
4751 -9
4788-9
482 5-9
4863 .
490 0
4937
4974-1
5 1 59 -3
5 1 96 -3
52 33 -3
52 70 -4
530 7-4
5344 4
538 1 -5
54 1 8-5
5455-6
549 2 -6
552 9 -6
5566 -7
3 .
4 .
3755-6
3792 -6
390 3-7
3940 -7
3977-8
40 51 -9
4 1 2 5-9
4 163 .
42 0 0 .
4 2 37
4 2 74 -1
4348-1
4385-2
4459 -3
4496 -3
4533-3
4 570 -4
46 0 7-4
4644 -4
47 1 8-5
4755-6
4 792 -6
490 3 -7
4940 -7
49 77-8
5 1 2 5-95 1 63 .
52 0 0 .
52 37
52 74 -1
5348-1
5385-2
5459 -3
5496 -3
5533 -3
5570 -4
4 .
5 .
3759 -3
3796 -3
38333
3870 -4
390 7-4
3944 -4
398 1 -5
40 55-6
40 9 2 -6
42 0 3-7
4 2 77-8
43 14 -8
4351 9
4388-9
442 59
4463
4 50 0
4537
4 574 -I
4 759 -3
4 796 -3
4833-3
4870 -4
490 7-4
4944-4
52 0 3-7
52 40 -7
5351 -9
5388-9
542 5-9
5463
550 0 .
5537
5574 -I
5 .
6 .
0
3763
380 0 .
3837
3874 -1
3948-1
3985-2
40 59 -3
40 96 -3
4 1 33-3
0 7 4
44 48 1 5
43 1 8-5
4355-6
439 2 -6
4466 -7
4 50 3 -7
4540 -7
4 577-8
4 72 5-9
4763
480 0 .
4837
4874 -1
4948-1
50 59 -3
50 96 -3
5 1 33-3
52 0 7-4
5244 4
53 1 8-5
5355-6
550 3-7
5540 -7
5577-8
6 .
7 .
3766 -7
380 3 -7
3877-8
39 51 -9
3988-9
40 2 5-9
40 6 3 .
4 1 0 0 .
4 1 37
4 1 74 -I
4359 -3
4396 -3
4433 -3
4470 -4
4 50 74
4 544-4
458 1 -5
4655-6
4766 -7
480 3 -7
4877-8
49 14 -8
4951 -9
49889
50 2 5-950 63 .
5 1 0 0 .
51 37
5 1 74-I
5359 -3
5396 -3
5433-3
5470 -4
550 7-4
5544 -4
7 .
3770 -4
380 7-4
3844 -4
39 18-5
3955-6
4 1 0 3-7
4 140 -7
4 1 77-8
432 5-9
4363
4437
46 59 -3
4696 -3
4733-3
4770 -4
480 7-4
4844 4
4955-6
4992 -6
51 0 3-7
51 77-8
532 5-95363
540 0
5437
5474 -1
5548-1
5585-2
8 .
3774 -I
3959 -3
3996 -3
40 33-3
40 70 4
4 1 0 7-4
4 144-4
41 8 15 .
42 55-6
4329 -6
4366 -7
44 0 3-7
4440 -7
4477-8
4551 -9
4 588-9
46 2 5-9
4663 .
470 0
4737
4774 -I
4959 -3
4990 -3
50 33-3
51 0 7-4
5 144 -4
540 3 -7
5440 -7
54 77-8
551 4 8
555 1 -9
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5592 -6
592 5-9
596 3
60 37 .
6 2 59 -3
6333-36370 -4
6 555-6
6 740 -76 7 77-8
6963 .
70 37 “
70 74 -I
7 2 59 -3
7 2 96 -3
7333 -3
7370 -4
740 7-4
5596 -3
5633-3
5670 -4
570 7-4
5744 -4
5855-6
6 2 63 .
630 0 .
63376374-1
6 559 -36 596 -3
6633-3
6 744 -4
70 0 3 -7
70 77-8
72 2 5-9
72 63 .
730 0
7337
7374 -I
74 1
51
TABLE NO . 4—CONTINUED .
in cubic yards in body of T able.
2 .
56 0 0 .
56 37
5674-1
5748-1
5859 -3
5896 -3
5933-3
5970 -4
6 1 55 6
6340 -76377-8
64 51 -9
6 563 .
66 37 .
6933 -3
70 0 7-4
70 44 -4
7 1 55-0
730 3 -7
7340 -7
73 77-8
3 .
560 3 -7
57 14 -8
5751 -9
5863 .
5937
5974-I
6 1 59 3
6 2 33-3
6344 -4
64 1 8 56455-6
6 75 1 -9
6 863 .
690 0 .
693769 74 -1
7 1 59 3
7 196 -3
72 33 -3
72 70 -4
730 7-4
7344-4
4.
560 7-4
5644 -4
57 1 8-5
5755-6
5792 -6
590 3-7
5940 -7
5977-8
6 1 63 .
6 2 0 0 .
6 2 37 .
6459 -36496 -36 533-36 570 -4
6 755-6
690 3-76940 -76977-8
7 1 2 5-9
7 1 63 .
72 0 0 .
7 2 37
72 74 -I
73 1
7348-1
7385-2
5 .
5759 -3
5796 -3
5833-3
5870 -4
590 7-4
5944 -4
598 1 -5
64 2 5-96463 .
6 50 0 .
6 537
6 759 -36 796 -36833 3
690 7-469 44 -4
70 55 -0
7 2 0 3 -7
7 240 -7
735 1 -9
742 5-9
56 51 -9 .
572 5-9
5703
580 0 .
5837
5874-1
60 96 -36 1 33-3
6355-6
6 540 -76 577-8
6 763 .
680 0 .
6 837.
70 59 -3.
70 96 -3
7 1 33-3
7 1 70 -4
72 0 7-4
72444
73 1 8 -5
7355-0
739 2 -6
7 .
5766 -7
580 3-7
5840 -7
5877-8
5951 -9
5988 -9
60 63 .
6 1 0 0 .
6 1 37 .
6 359 -36 396 -36433 -36470 -4
6 544 -4
6951 -9
70 2 5-9
70 6 3 .
7 1 0 0 .
7 1 37
7 1 74 I
7359 -3
7396 -3
7433 3
8 .
56 59 -3
5733 -3
5770 4
580 7-4
5844-4
59 1 8-5
5955-6
6363 .
640 0 .
64376474 -1
6659 -3
6 733 -36 770 -4
0844-4
69 55-6
7 1 0 3 -7
7 1 77 -8
72 14 8
72 5 1 -9
73 2 5 9
7363
740 0
7437
Areas T ens in left Column and U nits at top. Contents for 1 0 0 feet
9 .
56 2 5-9
5663 .
5737
5774-I
59 59 -3
5996 -369 33-3
6440 76477-8
655 1 9
6663 .
6 70 0 .
6 7376 774-1
6959 -36996 -3
70 33 -3
7 1 0 7 -4
7 144 4
72 55-6
72 9 2 6
7366 -7
740 3 7
7440 -7
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52
TABLE N0 . 4—CONTINUED .
- Ircas T ens in left -Column and Units at tog) . Contents for lQOfeet
‘
2 0 2
2 0 32 0 42 0 52 0 6
2 0 7
42 0 8
$2 0 92 1 0
2 1 1
2 1 2
2 1 32 1 42 1 52 1 6
! 2 l 7I2 1 8
2 1 92 20
2 2 1
2 2 2
2 232 2 4,2 2 5i2z62 2 72 2 8
2 2 92 30
2 3 1
2 3 2
2 332 342 352 3 6
2 372 38
2 392 40
24 1
2 42
2 432 442 452 46
2 472 48
.
2 492 50
7444 -4
748 1 -5
751 8-5
7555-0
759 2 -6
770 3 -7
7740 -7
7777 -8
79 2 5-9796380 0 0 .
80 37 .
82 59 382 96 38333 383 70 4840 7 48444 4848 1 5
8555-6
8592 6
86 2 9 6
8666 7
8740 78777 8
88 14 8
885 1 98888 989 2 5 98963
90 0 0
90 37
90 74 I
9 1 1 1 1
9 2 59 -3
7448-1
7559 -3
7596 -3
2033 376 70 4
770 7 -4
7 744 -4
7855-0
7966 -780 0 3 -7
82 63 .
830 0 .
833 78374 -1
8559 3.
8596 -38633 -3
87444
90 0 3-7
90 40 -7
90 77-8
9 1 5I 9
9 2 2 5-9
9 2 63 .
in cubic yards in body l
of T able.
2 .
745 1 9
7488 9
752 5 9
7563
7637
76 74 -1
7748-1
7859 -3
7896 -3
7933 -3
7970 -4
80 44 -4
8 1 55 6
8 1 9 2 6
82 2 9 6
82 66 7830 3 78340 78377 8
84 14 8
S4SI 98488 9852 5-98563 .
860 0 .
8637"
86 74 1
87 1 1 1
8748 1
8785 2
882 2 2
8859 38896 38933 -3
89 70 -4
90 0 7 4
90 44 4
90 8 1 5
91 1 8 5
9 1 55 6
2 .
3 .
7455-6
749 2 6
752 9 6
7566 7
760 3 7
7640 7
77 14-8
7751 -9
782 5-9
7863 .
790 0
7937
7974 -I
8 1 59 3
8233 -3
83444
8455-6
8751 -9
8863 .
890 0 .
89378974 -1
9 1 59 -3
9 1 96 -3
9 2 33 -3
92 70 -4
3 .
4 .
7459 -3
7496 -3
7533 -3
7570 -4
760 7-4
7044-4
77 1 8-5
7755-0
779 2 -6
790 3-7
7940 -7
79 77-8
8 163 .
82 0 0 .
8 2 37 .
8459 38496 -38533-38570 4
8644-4
8755-0
879 2 6
8829 6
8866 7890 3 78940 78977-8
90 51 -9
9 1 2 5-9
9 1 63 .
9 2 0 0 .
9 2 37
9 2 74 -I
4 .
5 .
7463
750 0 .
7537
7574 -I
7648 1
7685 2
772 2 2
7759 -3
7796 -3
7833-3
790 7-4
7944-4
798
80 1
80 580 98 1 2
8 1 6
835 1 -9
8463 .
850 0 .
85378574-1
8759 -38796 -3
890 7-48944-4
9 2 0 3 -7
9 240 -7
7470 -4
750 7 4
7544 4
758 1 5
76 1 8 5
7655 6
769 2 6
7766-7
780 3 -7
7840 7
9 14 8
9SI 9
988 9
80 63 .
8 1 0 0 .
8 1 37.
82 85 2
832 2 2
8359 38396 38433-38470 4850 7 48544 4858 1 5
869 2 6
6
8766 7880 3 7
895 1 -9
90 2 5-9
90 63 .
9 1 0 0 .
9 1 37
9 1 74 1
7 .
7474-1
7548-1
7659 -3
7696 -3
7733 -3
7770 -4
780 7-4
7844 -4
79 1 8-5
7955-6
7992 -6
832 5-98363 .
840 0 .
84378474-1
8659 -3
8733-38770 4
88444
8955-6
9 1 77 -8
9 2 51 -9
8 .
7477-8
755 1 -9"
76 2 5-9
7663 .
770 0 .
7737
7774 -I
7959 -3
7996 -380 33-3
8 144-4
8440 -78477-8
855 1 9
8663 .
870 0 .
87378774 -1
8959 -38996 -3
90 33 -3
90 70 -4
9 1 0 7-4
9 144 -4
9 2 55 -0
9 .
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53
TABLE N0 . 4 -10 1-30 .
Areas : T ens in q t Column and Units at tap. Contents for 10 0 feetin cubic yards in body of .T able.
2 . 3 . 4. 5 . 6 . 7 . 8 . g.
2 5 1 930 0 930 3 -7 930 74 93 1 1 -1 93 14 -8 93 1 8-5 932 2 -2 932 5-9 93 2 9 -6
2 52 9337 9340 -7 9344-4 9 348- 1 935 1 -9 9355-6 9359 -3 9 363 9 366 -72 53 9370 9 374-1 9 377-8 9 38 1 -5 9 385-2 9388-9 939 2 -6 9396 -3 940 0 940 3-72 54 94 1 1 -1 94 14 -8 94 1 8-5 94 2 2 -2 942 5-9 94 2 9 6 9433 -3 9437 9440 -72 55 9444 -4 9448-1 9451 -9 9455-6 9459 -3 9463 9466 7 9470 -4 9474 -1 947 7-8
2 56 94 8 1 -5 9485-2 9488 -9 949 2 -6 9496 -3 9 50 0 9 50 3-7 9 50 7 -4 9 51 1 -1 9 51 4-8
2 57 9 51 8-5 952 2 -2 9 52 5-9 9 52 9 -6 9 533 -3 9 537 9540 -7 9 544 -4 9 548-1 9551 -92 58 9555-6 9 559 -3 9 563 9 566 -7 9570 -4 9 574-1 9 577-8 9 58 1 -5 9 585-2 9 588-92 59 960 0 .
2 6 0 9637. 9663 .
2 6 1 970 0 .
2 6 2 9 70 3 -7 9 70 7-4 9 7 1 1 -1 9 7 14-8 9 71 8-5 972 2 -2 9 72 5-9 972 9 -6 9 733 -3 97372 63 9 740 -7 9 744 -4 9 748-I 9751 -9 9 755-0 9 759 3 9 703 9 700 -7 9 770 -4 9 774 -I
2 64 9 777-8 9 78 1 -5 9 785-2 9 788 -9 979 2 -6 9 796 -3 980 0 980 3-7 980 7-4 98 1 1 -1
2 65 9 837 .
2 66 985 1 9 9863 .
2 67 9888 9 9892 -6 9896 -3 990 0 990 3-7 990 74 99 1 1 -1 99 14 -8 99 1 8-5 99 2 2 -2
2 68 992 5-9 99 2 9 -0 9933 -3 9937 9940 -7 9944 -4 9948-1 99 51 -9 9955-0 9959 -3
2 59 9963 9966 -7 9970 4 9974-1 9977-8 998 1 -5 9985-2 9988-9 999 2 -6 9996 -3
2 7 01 0 0 14 8 1 0 0 1 8 5 1 0 0 2 2 2 1 0 0 2 59 1 0 0 2 9 6 1 0 0 33 3
2 7 11 0 0 37. 1 0 0 44 4 1 0 0 48 1 1 0 0 5 1 9 1 0 0 59 3 1 0 0 63 . 1 0 0 66 7 1 0 0 70 4
2 721 0 0 74 1 1 0 0 85 2 1 0 0 889 1 0 0 92 6 1 0 0 96 3 1 0 1 0 0 . 1 0 1 0 3 7 1 0 1 0 7 4
2 73 1 0 1 14 8 1 0 1 1 8 5 1 0 1 2 2 2 1 0 1 2 59 1 0 1 2 9 6 1 0 1 33 3 1 0 1 37 . 1 0 140 7 1 0 14442 74
1 0 148 1 1 0 1 51 9 1 0 1 556 1 0 1 59 3 1 0 1 63 . 1 0 1 66 7 1 0 1 70 4 1 0 1 74 1 1 0 1 77 8
2 75 1 0 1 889 1 0 1 9 2 6 1 0 1 96 3 1 0 2 0 0 . 1 0 2 14 8 1 0 2 1 8 52 75
1 0 2 2 2 2 1 0 2 2 59 1 0 2 2 9 6 1 0 2 33 3 1 0 2 37. 1 0 240 7 1 0 244 4 1 0 2 48 1 1 0 2 51 9 1 0 2 55 6
2 7710 2 59 3 1 0 2 63 . 1 0 2 66 7 1 0 2 70 4 1 0 2 74 1 1 0 2 77 8 1 0 2 85 2 1 0 2 889 1 0 2 92 6
2 78 1 0 2 96 3 1 0 30 0 . 1 0 30 3 7 1 0 30 7 4 1 0 3 1 4 8 1 0 3 1 8 5 10 32 2 2 1 0 32 59 1 0 329 6
2 79 1 0 333 3 1 0 337 . 1 0 340 7 1 0 344 4 1 0 348 1 1 0 35 1 9 1 0 355 6 1 0 359 3 1 0 363 .1 0 366 7
2 30 1 0 370 4 1 0 377 8 1 0 383 2 1 0 3889 1 0 39 2 6 1 0 396 3 1 0 40 0 .
2 3 ; 1 0 4 148 1 0 4 1 8 5 1 0 42 2 2 1 0 4 2 59 1 0 429 6 1 0 433 3 1 0 437 .1 0 440 7
2 3 2 1 0 448 1 1 0 451 9 1 0 455 6 1 0 459 3 1 0 463 . 1 0 466 7 1 0 4 70 4 Io477 .8
2 33 1 0 4889 1 0 49 2 6 1 0 496 3 1 0 50 0 . 1 0 50 7 4
2 84 1 0 51 8 5 10 52 2 2 1 0 52 59 1 0 52 9 6 1 0 533 3 1 0 537. 1 0 540 7 1 0 544 4 1 0 548 1 1 0 551 92 85 1 0 553 6 1 0 559 3 1 0 563 . 1 0 566 7 1 0 570 4 1 0 577 8 1 0 583 2 1 0 58892 35 1 0 59 2 6 1 0 596 3 1 0 6 0 0 . 1 0 60 3 7 1 0 6 0 7 4 1 0 6 14 8 1 0 6 1 8 5 1 0 6 2 2 2 1 0 6 2 592 37 1 0 6 29 6 1 0 6 33 3 1 0 6 37. 1 0 640 7 1 0 644 4 1 0 648 1 1 0 6 51 9 1 0 6 55 6 1 0 6 59 3 1 0 663 .
2 33 1 0 666 7 1 0 6 70 4 1 0 6 77 8 1 0 685 2 1 0 6 889 1 0 69 2 6 1 0 696 3 1 0 70 0 .
2 89 1 0 70 3 7 1 0 70 7 4 1 0 7 14 8 1 0 7 1 8 5 1 0 72 2 2 1 0 72 59 1 0 72 9 6 1 0 733 3 1 0 737 .
2 90 1 0 740 7 1 0 744 4 1 0 748 1 1 0 751 9 1 0 75 1 0 759 3 1 0 763 . 1 0 766 7 1 0 770 .
2 9 1 1 0 777 8 1 0 783 2 1 0 788 9 1 0 792 6 1 0 796 3 1 0 80 0 . 1 0 80 3 7 1 0 80 7 42 92 1 0 8 148 1 0 8 1 8 5 1 0 82 2 2 1 0 82 5 9 1 0 82 9 6 1 0 833 3 1 0 837 . 1 0 840 7 1 0 844 4 1 0 848 1
2 931 0 851 9 1 0 855 6 1 0 859 3 1 0 863 . 1 0 866 7 1 0 870 4 1 0 874 1 1 0 877 8
2 94 1 0 888 9 1 0 892 6 1 0 896 3 1 0 90 0 . 1 0 90 3 7 1 0 90 7 4 1 0 9 1 8 5 1 09 2 2 2
2 951 0 9 2 59 1 0 9 29 6 1 0 933 3 1 0 937. 1 0 940 7 1 0 944 4 1 0 948 1 1 0 95 1 9 1 0 9 53 6 1 0 9 59 3
2 96 1 0963 . 1 0 96 6 7 1 0 9 70 4 1 0 9 77 8 1 0 9 83 2 1 0 988 9 1 0 99 2 6 1 0 9 96 32 97 1 1 0 0 0 . 1 1 0 0 3 7 1 1 0 0 7 4 1 1 0 1 8 5 1 1 0 2 2 2 1 1 0 2 59 1 1 0 2 9 6 1 1 0 33 32 98 1 1 0 37. 1 1 0 40 7 1 1 0 48 1 1 1 0 5 1 9 1 1 0 53 6 1 1 0 59 3 1 1 0 6 3 . 1 1 0 66 7 1 1 0 70 42 99 1 1 0 77 8 1 1 0 85 2 1 1 0 88 9 1 1 0 9 2 6 1 1 0 9 6 3 1 1 1 0 0 . 1 1 1 0 3 7 1 1 1 0 7 4
30 0 1 1 1 148 1 1 1 1 8 5 1 1 1 2 2 2 1 1 1 2 59 1 1 1 2 9 6 1 1 1 33 3 1 1 1 37 . 1 1 140 7 1 1 1 444
2 . 3 . 4 . 5 . 6 . 7 .
l8 . g.
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54
TABLE N0. 4
Areas T ens in left Column and Units at top. Contents for . 100 feetin cubic yards in body of Table.
2 . 3 . 4 . 5 . 7 . 8 . 9 .
1 1 148 1 1 1 1 51 9 1 1 1 55 6 1 1 1 59 3 1 1 1 63 . 1 1 1 66 7 1 1 1 70 4 1 1 1 74 1 1 1 1 77 8 1 1 1 8 1 51 1 1 85 2 1 1 1 88 9 1 1 1 9 2 6 1 1 1 96 3 1 1 2 0 0 . 1 1 2 0 3 7 1 1 2 0 7 4 1 1 2 14 8 1 1 2 1 8 51 1 2 2 2 2 1 1 2 2 5 9 1 1 2 2 9 6 1 1 2 33 3 1 1 2 37. 1 1 240 7 1 1 244 4 1 1 248 1 1 1 2 51 9 1 1 2 55 6
1 1 2 59 3 1 1 2 63 . 1 1 2 66 7 1 1 2 70 4 1 1 2 74 1 1 1 2 77 8 1 1 2 8 1 5 1 1 2 85 2 1 1 2 88 9 1 1 2 9 2 6
1 1 2 96 3 1 1 30 0 . 1 1 30 3 7 1 1 30 7 4 1 1 3 14 8 1 1 3 1 8 5 1 1 32 2 2 1 1 3 2 5 9 1 1 32 9 6
1 1 333 3 1 1 337 . 1 1 340 7 1 1 344 4 1 1 348 1 1 1 35 1 9 1 1 355 6 1 1 3 59 3 1 1 363 . 1 1 366 71 1 3 70 4 1 1 374 1 1 1 377 8 1 1 385 2 1 1 3889 1 1 392 6 1 1 396 3 1 140 0 . 1 140 3 71 1 40 7 4 1 14 14 8 1 14 1 8 5 1 14 2 2 2 1 142 59 1 14 29 6 1 1 433 3 1 1437 . 1 1440 71 1 444 4 1 1448 1 1 14 51 9 1 1455 6 1 1 459 3 1 1 463 . 1 1466 7 1 1 470 4 1 1474 1 1 1 4 77 8
1 1485 2 1 1 4889 1 149 2 6 1 1 496 3 1 1 50 0 . 1 1 50 3 7 1 1 50 7 4 1 1 5 1418
1 1 51 8 5 1 1 52 2 2 1 1 52 5 9 1 1 52 9 6 1 1 533 3 1 1 537“
1 1 540 7 1 1 544 4 1 1 548 1 1 1 551 91 1 555 6 1 1 559 3 1 1 563 . 1 1 566 7 1 1 570 4 1 1 574 1 1 1 577 8 1 1 585 2 1 1 58891 1 592 6 1 1 596 3 1 1 60 0 . 1 1 60 3 7 1 1 60 7 4 1 1 6 14 8 1 1 6 1 8 5 1 1 6 2 2 2 1 1 6 2 5 91 1 6 2 9 6 1 1 633 3 1 1 637 . 1 1 640 7 1 1 644 4 1 1 648 1 1 1 6 51 9 1 1 655 6 1 1 659 3 1 1 663 .
1 1 666 7 1 1 6 79 4 1 1 6 74 1 1 1 6 77 8 1 1 685 2 1 1 688 9 1 1 692 6 1 1696 3 1 1 70 0 .
1 1 70 3 7 1 1 70 7 4 1 1 7 14 8 1 1 71 8 5 1 1 72 2 2 1 1 72 59 1 1 72 9 6 1 1 733 3 1 1 737 .
1 1 740 7 1 1 744 4 1 1 748 1 1 1 751 9 1 1 755 6 1 1 759 3 1 1 763 . 1 1 766 7 1 1 770 4 1 1 774 1
1 1 777 8 1 1 785 2 1 1 788 9 1 1 792 6 1 1 796 3 1 1 80 0 . 1 1 80 3 7 1 1 80 7 41 1 8 14 8 1 1 8 1 8 5 1 1 82 2 2 1 1 82 5 9 1 1 82 9 6 1 1 833 3 1 1 837 . 1 1 840 7 1 1 844 4 1 1 848 1
1 1 85 1 9 1 1 855 6 1 1 859 3 1 1 863. 1 1 866 7 1 1 870 4 1 1 874 1 1 1 877 8 1 1 885 2
1 1 888 9 1 1 89 2 6 1 1 896 3 1 1 90 0 . 1 1 90 3 7 1 1 90 7 4 1 19 14 8 1 19 1 8 5 1 1 9 2 2 2
1 1 9 2 5 9 1 1 9 2 9 6 1 1 933 3 1 1 937 . 1 1 940 7 1 1 944 4 1 1 948 1 1 1 951 9 1 1 955 6 1 1 959 31 1 963 . 1 1 966 7 1 1 9 70 4 1 1 974 1 1 1 977 8 1 1 985 2 1 1 9889 1 1 99 2 6 1 1 996 31 2 0 0 0 . 1 2 0 0 3 7 1 2 0 0 7 4 1 2 0 14 8 1 2 0 1 8 5 1 2 0 2 2 2 1 2 0 2 59 1 2 0 2 9 6 1 2 0 33 31 2 0 37 . 1 2 0 40 7 1 2 0 44 4 1 2 0 48 1 1 2 0 5 1 9 1 2 0 55 6 1 2 0 59 3 1 2 0 63 . 1 2 0 66 7 1 2 0 70 41 2 0 74 1 1 2 0 77 8 1 2 0 85 2 1 2 0 889 1 2 0 9 2 6 1 2 0 96 3 1 2 1 0 0 . 1 2 1 0 3 7 1 2 1 0 7 4
1 2 1 14 8 1 2 1 1 8 5 1 2 1 2 2 2 1 2 1 2 5 9 1 2 1 2 9 6 1 2 1 33 3 1 2 1 37 0 1 2 1 40 7 1 2 144 41 2 1 48 1 1 2 1 51 9 1 2 1 55 6 1 2 1 59 3 1 2 163 . 1 2 1 66 7 1 2 1 70 4 1 2 1 74 1 1 2 1 77 8
1 2 1 85 2 1 2 1 88 9 1 2 1 92 6 1 2 1 96 3 1 2 2 0 0 . 1 2 2 0 3 7 1 2 2 0 7 4 1 2 2 14 8 1 2 2 1 8 51 2 2 2 2 2 1 2 2 2 59 1 2 2 29 6 1 2 2 33 3 1 2 2 37. 1 2 240 7 1 2 244 4 1 2 248 1 1 2 2 51 9 1 2 2 55 6
1 2 2 59 3 1 2 2 63 . 1 2 2 66 7 1 2 2 70 4 1 2 2 74 1 1 2 2 77 8 1 2 2 85 2 1 2 2 889 1 2 2 9 2 6
1 2 2 96 3 1 2 30 0 . 1 2 30 3 7 1 2 30 7 4 1 2 3 14 8 1 2 3 1 8 5 1 2 3 2 2 2 1 2 3 2 5 9 1 2 32 9 6
1 2 333 3 1 2 337 . 1 2 340 7 1 2 344 4 1 2 348 1 1 2 351 9 1 2 355 6 1 2 359 3 1 2 363 . 1 2 366 71 2 370 4 1 2 374 1 1 2 377 8 1 2 385 2 1 2 3889 1 2 39 2 6 1 2 396 3 1 240 0 . 1 240 3 71 240 7 4 1 24 14 8 1 24 1 8 5 1 24 2 2 2 1 242 5 9 1 242 9 6 1 2 433 3 1 2437. 1 2 440 7
1 2444 4 1 2448 1 1 2 451 9 1 2 455 6 1 2459 3 1 2463 . 1 2466 7 1 2470 4 1 2474 1 1 2477 8
1 2485 2 1 24889 1 249 2 6 1 2496 3 1 2 50 0 . 1 2 50 3 7 1 2 50 7 4 1 2 514 8
1 2 51 8 5 1 2 52 2 2 1 2 52 5 9 1 2 52 9 6 1 2 533 3 1 2 537. 1 2 540 7 1 2 544 4 1 2 548 1 1 2 551 91 2 555 6 1 2 559 3 1 2 563 . 1 2 566 7 1 2 570 4 1 2 574 1 1 2 577 8 1 2 58 1 5 1 2 585 2 1 2 588 91 2 59 2 6 1 2 596 3 1 2 60 0 . 1 260 3 7 1 2 60 7 4 1 2 6 14 8 1 2 6 1 8 5 1 2 6 2 2 2 1 2 6 2 5 91 2 6 2 9 6 1 2 633 3 1 2 637. 1 2 640 7 1 2 644 4 1 2 648 1 1 2 651 9 1 2 655 6 1 2 659 3 1 2 663 .
1 2 666 7 1 2 670 4 1 2 674 1 1 2 677 8 1 2 685 2 1 2 6889 1 2 69 2 6 1 2 696 3 1 2 70 0 .
1 2 70 3 7 1 2 70 7 4 1 2 7 14 8 1 2 7 1 8 5 1 2 72 2 2 1 2 72 5 9 1 2 72 9 6 1 2 733 3 1 2 737.
1 2 740 7 1 2 744 4 1 2 748 1 1 2 751 9 1 2 755 6 1 2 759 3 1 2 763 . 1 2 766 7 1 2 770 4 1 2 774 1
1 2 7 77 8 1 2 785 2 1 2 7889 1 2 792 6 1 2 796 3 1 2 80 0 . 1 2 80 3 7 1 2 80 7 41 2 8 14 8 1 2 8 1 8 5 1 2 82 2 2 1 2 82 5 9 1 2 0
82 9 6 1 2 833 3 1 2 837 . 1 2 840 7 1 2 844 4 1 2 848 1
1 2 851 9 1 2 855 6 1 2 859 3 1 2 863 . 1 2 866 7 1 2 870 4 1 2 874 1 1 2 877 8 1 2 885 2
1 2 8889 1 2 89 2 6 1 2 896 3 1 2 90 0 . 1 2 90 3 7 1 290 7 4 1 2 9 14 8 1 2 9 1 8 5 1 2 92 2 2
1 2 9 2 59 1 292 9 6 1 2 933 3 1 2 937. 1 2 940 7 1 2 9 44 4 1 2 948 1 1 2 9 51 9 1 2 955 6 1 2 959 3
1 2 963 . 1 2 966 7 1 2 970 4 1 2 974 1 1 2 977 8 1 2 985 2 1 2 9889 1 2 99 2 6 1 2 996 3
2 . 3 . 4 . 5 . 5 . 7 . 8 . 9 .
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56
TABLE NO . 4—CON0LU DED .
Areas : T ens in left Column and Units at top. Contents for 1 00 feetin cubic yards in body of Table.
2 . 3 . 4 . 5 . 7 . 8 . 9 .
1 30 0 0 . 1 30 0 3 7 1 30 0 7 4 1 30 14 8 1 30 1 8 5 1 30 2 2 2 1 30 2 59 1 30 2 9 6 1 30 33 31 3637 . 1 30 40 7 1 30 44 4 1 30 48 1 1 30 5 1 9 1 30 55 6 1 30 59 3 1 30 63 . 1 30 66 7 1 30 70 41 30 74 1 1 30 77 8 1 30 85 2 1 30 889 1 309 2 6 1 30 96 3 1 3 1 0 0 . 1 3 1 0 3 7 1 3 1 0 7 4
1 3 1 1 4 8 1 3 1 1 8 5 1 3 1 2 2 2 1 3 1 2 5 9 1 3 1 2 9 6 1 3 1 33 3 1 3 1 37 . 1 3 140 7 1 3 1 44 41 3 148 1 1 3 1 5 1 9 1 3 1 55 6 1 3 1 59 3 1 3 1 6 3 . 1 3 1 66 7 1 3 1 70 4 1 3 1 74 1 1 3 1 77 8
1 3 1 85 2 1 3 1 889 1 3 19 2 6 1 3 196 3 1 32 0 0 . 1 3 2 0 3 7 1 32 0 7 4 1 32 14 8 1 32 1 8 51 3 2 2 2 2 1 32 2 5 9 1 32 2 9 6 1 32 33 3 1 32 37. 1 3 240 7 1 3244 4 1 3 248 1 1 3 2 5 1 9 1 32 55 6
1 32 59 3 1 3263 . 1 3 266 7 1 3 2 70 4 1 32 74 1 1 32 77 8 1 32 85 2 1 32 889 1 3 292 6
1 3 296 3 1 330 0 . 1 330 3 7 1 330 7 4 1 33 14 8 1 33 1 8 5 1 332 2 2 1 332 5 9 1 33 2 9 6
1 3333-3 1 3337 1 3340 -7 1 3344-4 1 3348-1 1 335 1 -9 1 3355-6 1 3359 -3 1 3363 1 3366 71 3370 4 1 3374 1 1 3377 8 1 3385 2 1 33889 1 3392 6 1 3396 3 1 340 0 . 1 340 3 71 340 7 4 1 34 14 8 1 341 8 5 1 342 2 2 1 342 59 1 342 9 6 1 3433 3 1 3437 . 1 3440 71 3444 -4 1 3448-1 1 3451 -9 1 3455-6 1 3459 -3 1 3463 1 3466 -7 1 3470 4 1 3474 -1 1 3477 -8
1 3485 2 1 34889 1 3492 6 1 3496 3 1 350 0 . 1 350 3 7 1 350 7 4 1 35 14 8
I351 8-5 I3S2 S-9 1 352 9 -6 I3S33 -3 1 3537 I3540 17 1 3544 -4 I3S4S-I I3SSI -91 3555-6 1 3559 -3 1 3563 1 3566 -7 1 3570 4 1 3574 -1 1 3 577-8 1 358 1 -5 1 3585-2 1 3588 91 359 2 6 1 3596 3 1 360 0 . 1 360 3 7 1 360 7 4 1 36 14 8 1 36 1 8 5 1 36 2 2 2 1 36 2 5 91 36 2 9 6 1 3633 3 1 3637 . 1 3640 7 1 3644 4 1 3648 1 1 36 51 9 1 3655 6 1 36 59 3 1 3663 .
1 3666 7 1 36 70 4 1 3674 1 1 3677 8 1 3685 2 1 36 88 9 1 369 2 6 1 3696 3 1 370 0 .
1 370 3 7 1 37 14 8 1 37 1 8 5 1 372 2 2 1 372 5 9 1 372 9 6 1 3733 3 1 373 7 .
1 3740 7 I3744 -4 1 3748-1 1 3751 -9 1 3755-6 1 3759-3 1 3763 1 3766 -7 1 3770 4 1 3774 -1
1 3777 8 1 3 785 2 1 37889 1 3792 6 1 3796 3 1 380 0 . 1 380 3 7 1 380 7 41 38 14 8 1 38 1 8 5 1 382 2 2 1 382 59 1 382 9 6 1 3833 3 1 3837 . 1 3840 7 1 3844 4 1 384 8 1
1 3851 9 1 3855 6 1 3859 3 1 3863 . 1 3866 7 1 3870 4 1 3874 1 1 3877 8 1 3885 2
1 38889 1 389 2 6 1 3896 3 1 390 0 . 1 390 3 7 1 390 7 4 1 39 1 4 8 1 39 1 8 5 1 39 2 2 2
1 39 2 5-9 1 39 2 9 -6 1 3933-3 1 3937 1 3940 -7 1 3944-4 1 3948-1 1 395 1 -9 1 3955-6 1 3959 -31 3963 . 1 3966 7 1 3970 4 1 3974 1 1 3977 8 1 3985 2 1 39889 1 399 2 6 1 3996 31 40 0 0 . 140 0 3 7 140 0 7 4 1 40 14 8 1 40 1 8 5 140 2 2 2 1 40 2 5 9 1 40 2 9 6 140 33 3140 37 . 140 40 7 140 44 4 140 48 1 140 51 9 140 55 6 140 59 3 14 0 63 . 1 40 66 7 140 70 41 40 74 1 140 77 8 140 85 2 140 88 9 140 92 6 140 96 3 14 1 0 0 . 14 1 0 3 7 1 4 1 0 7 4.
1 4 1 1 4 8 1 4 1 1 8 5 1 4 1 2 2 2 1 4 1 2 59 14 1 29 6 1 4 1 33 3 14 1 37 4 1 4 140 7 1 4 144 414 148 1 14 1 51 9 1 4 1 55 6 14 1 59 3 14 1 63 . 14 1 66 7 14 1 70 4 14 1 74 1 14 1 77 8
14 1 85 2 1 4 1 88 9 14 1 9 2 6 14 1 96 3 1 42 0 0 . 142 0 3 7 142 0 7 4 1 42 14 8 1 4 2 1 8 5142 2 2 2 1 42 2 5 9 142 2 9 6 142 33 3 142 37 . 1 4240 7 1 4244 4 1 4248 1 1 4 2 5 1 9 142 55 6
142 59 3 1 42 63 . 1 42 66 7 1 4 2 70 4 1 42 74 1 142 77 8 1 4 2 85 2 1 4 2 889 142 92 6
142 96 3 1430 0 . 1 430 3 7 1430 7 4 143 14 8 143 1 8 5 1 432 2 2 1432 5 9 1 432 9 6
14333 3 1 4337 14340 -7 1 4344 -4 14348-1 14351 -9 14355-6 14359-3 14363 14366 714370 4 1 4374 1 14377 8 14385 2 14388 9 1 4392 6 14396 3 1 440 0 . 1440 3 71440 7 4 144 14 8 1 44 1 8 5 144 2 2 2 144 2 59 1 44 2 9 6 1 4433 3 1 4437 . 14440 714444 -4 14448-1 14451 -9 14455-6 14459 -3 1 4463 1 4466 -7 1 4470 -4 1 4474-1 14477 -8
14485 2 144889 14492 6 14496 3 1 450 0 . 1450 3 7 1 450 7 4 14514 8
1451 8 5 14 52 2 -2 1452 5-9 1452 9 -6 14533-3 14 537 14540 -7 14544 4 1 4548-1 14551 -914 555-6 1 4559 -3 14563 14566 -7 14570 4 14574-1 14577-8 1458 1 -5 14585-2 14588 91 4592 6 14 596 3 1 460 0 . 1460 3 7 1460 7 4 146 14 8 146 1 8 5 14 6 2 2 2 146 2 5 9146 2 9 6 14633 3 14637 . 14640 7 14644 4 1 4648 1 1 46 5 1 9 1 4655 6 1 4659 3 14663 .
14666 7 1 46 70 4 14674 1 14677 8 14685 2 1 46889 1 4692 6 1 4696 3 1 470 0 .
14 70 3 -7 1470 7 4 1 47 1 1 -1 147 14-8 147 1 8-5 1472 2 -2 1 472 5-9 1472 9 -6 1 4733-3 1473714 740 -7 1 4748-1 14751 -9 14755-6 14759 -3 14763 14766 -7 1 4770 4 1 4774-1
14777 8 1 4785 2 14788 9 14 792 6 14796 3 1480 0 . 1480 3 7 1480 7 4148 1 4 8 148 1 8 5 1482 2 2 1482 5 9 1482 9 6 14833 3 14837 . 14840 7 14844 4 14848 1
2 . 3 . 4 . 6 . 7 . 8 . 9 .
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56
TABLE NO . 5 .
Minus Corrections corresponding to N~ N'
,or n~ n
'
,and generalj
'
or
a ll side slopes . F or computation by average Areas.
D if erence of Correction numbers in f eet and ten t/is in lef t column anda t top Correction in cubic y ards f or 100 f t. in body of T able.
2 . 3 . 4 . 6 . 7 .
0 0
1 . 2 0
3 6 4 8
6 3 7 1 8 9
9 9 1 0 9 1 2 0 13 1 14 2
1 6 7 2 0 8
2 3 7 2 4 5 2 8 5
3 2 0 37 6
39 -5 40 -5 4 1 5 4 2 -5 43 -6 4 5-7 46 -7 4 7 8 48-9
59 366 8 72 0
80 2
94 91 14 2
1 2435
1 54 1
2 0 0 0
2 32 3 244 52 56 9
3 2 6 -5 3 2 9 -4 33 2 -2 335-1. 338-0 340 -9 343 -8 346 -7 349 7 3 52 -6
3 55-6 358-5 36 1 -5 364 -5 36 7-5 3 70 -5 3 73 -6 376 -6 3 79 7 382 -7
3858
4 1 73 4 2 0 -5 4 2 3 7 4 2 7-0 430 -2 433 -5 436 -8 440 -1 443 -4 446 -7
4 84 -0 4 87 -4 490 -9 494-4 49 7-9 50 1 -4 50 4 -9 50 8-5 5 1 2 5 1 5-6
5 1 9 -1 52 2 -7 52 6 -3 529 -9 533-6 537-2 540 -8 544 -5 548 55 1 -9
555-6 559 -3 563 -0 566 -7“
570 -5 574-2 578-0 585 589 46 24 2
664 1
70 5 2
7 1 3 -6 7 1 7-8 72 2 -0 726 -2 730 -5 734 -7 739 -0 743 -3 747 6 751 9
764 8 79 1 1 795 6
8 1 836 0
882 0
938-9 943-7 948-5 9 53-4 9 58-2 963-1 968 -0 9 72 -9 9 77-8 9 82 7
1 0 73 4 1 0 78 5
1 1 6 2 71 2 1 6 9
1 3 1 7 6
1 375 2
o . 2 . 5 . 6 . 7 . 8 . 9
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10
co
x:
own
-ra
w
w
H
Q
l
Ft
451 51
52 L 5
593 -3
66647
74 1 458 1 713
89513
66A:
1 2 79 1
1 90 13
2 55-4
3ZL S389 1 )
458A:
52 813
6 74A:
749 -0‘
82 5 5
90 3 4 1
. 2
1 1 1;
72 1 1
1 34 1 )
1 9 71 1
2 6 2 1 )
3 2 81 2
39549
465JZ53537
6 8L 5
7561 )
833 1 1
9 1 1 43
-3
r7 8
78431 40 1 :
2 0 313
2 68
334
40 2
472 J I
542 4)6 1 51 :
6 89
76484 1
9 1 9
555
T ABLE NO . 6 .- LEVEL CUTTING S .
-4
2 315
84 441 46 52 1 0 1 )
2 7511
34 1 6
40 9 6
479 I
550 0
6 2 2 5696 4
77 1 8
848 7
92 7 1
-5
2 913
1 52 13
2 1 64;2 8 1 7
348 3
4 1 64;
486 1 :
5571 2
6 29 13
70 34)779 4 1856 4;
935-0
35 8
963 1
ISQJZ2 2 2 1;2 8843
3554
42 3 4 4
49342
564 2 46 371 :
7 1 1 2 4
7871
86443
942 4)
-9
534)1 1 541I78J I
242 2 1
30 81 :
3752 4
444 -52
5 14 2 4
5S6 JI
65943
733498 1 0 4 :
88737
96613
9 7413
1 0 56
1 1 38
1 2 2 1
1 30 7I3931 482
1 57 1
1 66 2
I7SS1 849
2 0 4 1
2 1 40
2 240
2 34 1
2 5492 6 552 76 2
2 87 1
2 98 1
32 0 7
33 2 1
3438
3556
3675
3796
39 1 8
40 4 1
4 1 6 7
4 2 93
455 1
4682
48 1 5
4949
50 84
52 2 1
5360
564 1
5784
592 96 0 756 2 2 2
982 15
1 0 641 1 46
1 2 30
1 3 1 5140 2
1490
1 580
1 6 7 1
1 7641 858
I9S42 0 5 1
2 1 50
2 2 50
2 352
2 4552 5592 6652 7732 882
2 993
3 1 0 5
32 1 8
3333
3449
356 7
3687
380 8
3930
40 54
430 6
4434
4564
4695
4962
50 98
52 35
5374
5514
5656
5799
594360 896 2 37
990 15
1 0 72
H 541 2 38
1 32414 1 1
I4991 5891 68 1
I7741 868
2 1 60
2 36 2
2 570
2 676
2 7842 893
3 1 1 6
32 30
3345
346 1'
3579
3699
382 0
3942
4 1 92
43 19
4577
470 9
4842
4976 .
51 I2
5249
5388
552 8
56 70
58 1 3
5958
6 2 52
998
0 80
1 1 631 247I333142 0
1 50 8
1 598
1 7831 8771 973
2 372
2 581
2 6872 795
30 1 5
3 1 2 7
324 1
3356
3473
358 1
371 1
3832
3955
40 79
42 0 5
4332
4460
4590
472 2
4855
4989
51 2 5
52 63
540 2
0 0 70 88
1 1 7 1
2 5
34
42
1 51 71 60 7I6991 792
1 8871 9832 0 81
2 1 80
2 2 80
2 382
2 486
2 59 1
2 69 8
2 80 6
2 9 1 5
30 2 6
3 1 38
32 52
3368
3485
3593
3 72 3
3844
396 7
42 1 7
460 3
4 735
4868
50 0 3
5 1 39
52 77
54 1 6
5556
5698
5842
59876 1 346 2 82
1 0 1 51 0 96
I I 791 2 641 350
I4371 52 6
1 6 1 6
1 70 8
1 80 2
1 896
1 993
2 1 9 0
2 2 9 1
2 3932 496
2 60 2
2 70 8
2 8 1 6
2 92 6
0 3IS
2 6
3379
3496
360 5
3735
3856
3979
4 1 0 4
42 30
4357
4486
46 1 6
4748
4882
50 1 6
51 53
SZ9 I
0 2 31 0 51 1 88
1 2 72
1 358
1446
I5351 6 2 6
1 7 1 8
1 8 1 1
1 90 6
2 0 0 2
2 2 0 0
2 30 1
240 32 50 72 6 1 2
2 7 1 92 82 72 937
30 48
3 1 6 1
32 75
339 I
350 8
36 1 7
3747
3869’
3992
4 1 1 6
4242
4370
4630
476 2
4895
50 30
5166
530 4
5444
5585
572 7
587 1
60 1 6
6 1 63 .
feet.
.7 .8
4 1 -8 474)1 0 2 1; 1 0 9 1 :
1 65 4
2 949 30 1 5
430 -3 437-2
50 0 1 : 50 743
57813
65 1 1)
794 -7 80 2 -4
950 9 958-8
1 0 3 1 1 0 391 1 1 3 1 1 2 1
1 1 96 1 2 0 51 2 8 1
1 36 7 1 376
1455 14641 553
1 6351 72 7 1 736
1 82 1 1 830
1 9 1 6 1 92 52 0 1 2 2 0 2 2
2 1 1 0 2 1 2 0
2 2 1 0 2 2 2 0
2 3 1 1 2 32 1
2 4 1 3 2 4242 51 7 2 52 8
2 6 2 3 2 6342 730 2 74I
2 838 2 8492 948 2 959
30 7 1
3 1 73 3 1 84
3 2 87 3 298
340 3 34 14
352 0 353 2
36 2 9 364 1
3759 377I
388 1 3893
4 1 2 9 4 142
42 55 42 68
4383 4396
451 2 4 52 5
4643 . 46 56
4775 4788
490 9 492 2
50 44 50 57
51 80 5 1 94
53 1 8 5332
5458 5472
5599 56 1 3
S74 I 5756
5885 590 0
60 46
6 1 78 6 1 93632 6 634 1
.8
I O471 1 2 91 2 1 31 2 98
1 385I4731 56 2
1 6531 7451 8391 9352 0 32
2 1 30
2 2 30
2 33 1
2 4342 538
2 751
2 970
30 82
3 1 95
33 1 0
342 6
3653
3783
40 29
4 1 54
42 81
4538
480 1
4935
50 7 1
52 0 8
5346
5486
56 2 7
5770
59 1 4
6 2 0 76356
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.
59
TABLE N0 . 7—LEVEL 0 111 1 111 0 5 .
344V
5O
.7
73 0
1 784 42 854 4
39313
50 313
6 1 54 2
7 2 851
842 4 4
9 5843
2- =l ° (7 2 28 feet.
.8
83 4 41 89 4:
2 96 4 2
40 413
SLL8
6 2 6 4 4
739854
970
934)I99 7
30 71 )
4 1 597
52 54)6 371 )
750 13
8654;
98 1 15
1 0 76
I I941 3 1 5I4371 560
1 6851 8 1 1
1 938
2 0 6 8
2 1 98
2 330
24642 5992 736
2 874
30 1 3
3 1 54
32 97
344 I
3586
3733
3882
40 3 2
4 1 83
4336
4490
4646
480 4
496 2
5 1 2 3
52 85
5448
56 1 3
5779
5946
6 1 1 6
6 2 86
6458
6632
680 76984
7 1 6 2
7 34I
752 2
770 5
788980 7482 6 1
8450
8640
883 1
90 24
~7
1 0 871 2 0 6
1 32 7I4491 572
1 697
1 82 31 951
2 0 8 1
2 2 1 1
2 344
2478
2 6 1 32 750
2 888
30 2 7
3 1 69
33 1 1
3455
360 1
3748
389 7
40 47
4 1 98
43SI
450 6
466 2
481 9
4978
51 39
530 1
5464
56 2 9
5795
59636 1 33630 36476
6650
682 5
70 0 2
7 1 80
7359
7S4 I
772 3
790 780 9 3
82 80
846986598850
90 43
.8
1 2 1 8
I3391 46 1
1 5851 7 1 0
1 836
1 964
2 2 2 52 3572 49 1
2 6 2 6
2 76 3
3O4 I3 1 83
332 6
3470
36 1 6
3763
39 1 2
4 2 14
436 7
452 1
46 78
4835
4994
5 1 55
53 1 7
548 1
5646
581 2
5980
6 1 50
632 1
666 76842
70 1 9
7377
7559
7 742
792 6
8 1 1 2
82 998488
86 78
8870
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"
60
TABLE NO . 8 .
Plus Corrections for —5
o . 2 . 4 ° 5 0
Now —The quantities in the above fable
corrections for =5°
6 . 7 .
multiplied by 2 give the minus
9 1 5
73-978-7
9 2 ‘s0
9 73
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61
TABLE No . 9 .—LEVEL 0 1111 1110 0 5 .
{gig -é; b 16 feet
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1D
GPQ
own
-p
cp
N
H
0
‘
1 0 51 3
2 L4 8
3 2 7 8
56413
6 884;8 1 647
. I
1 0 1 41 1 6432 2 51;
33943
4 5643
577» I
70 1 58 2 913
. 2
2 0 13
1 2 74‘
2 374
350 13
46842
58943
7L4 I
842 47
-3
3 1 431 374)2 4843
36 2 4 4
480 4 2
60 1 13
72 6 8
855-7
132
TABLE NO . 10 .
—LEVEL CUTTINGS .
-4
4L 8
1 4813
2 5913
492 4 :
6 1 4 1 )
86813
-5
52 43IS9 72 70 13
38513
50 442
6 2 6 4 4
752 4388 1 1;
948 4 96 L5 9 7449 988431 0 831 2 2 2
1 3651 5 1 1
1 66 1
1 8 1 51 9 72
2 1 332 2 98
2 46 72 6392 8 1 52 994
3 1 78
336 5
3556
3 750
3946
4 1 50
4 356
4 56 5
4 778
4994
52 1 5
5439
5667
5898
6 1 33
6 3 72
6 6 1 56 86 1
7 1 1 1
7365
76 2 2
78838 1 48
84 1 786 89896 5
9 244
9 52 8
9 8 1 5
1 2 36
I3791 52 6
1 676
1 988
2 1 50
2 3 1 52 4842 6 56
2 833
30 1 3
3 1 96
3384
3575
3 770
3968
4 1 70
4376
4586
4 799
50 1 6
52 37
546 2
5690
59 2 2
6 1 576 396
66 396 886
7 1 36
7390
7648
79 1 0
8 1 7584448 7 1 6
8993
9 2 73
9 556
9844
1 1 1 1
1 2 50
I3941 54 1
1 69 2
1 846
2 0 0 42 1 66
2 332
2 50 1
2 6 742 850
3O3 I3 2 1 5
340 3
3594
3789
3988
4 1 9 1
4397
460 7
482 1
50 38
52 59
5484
57 1 3
59456 1 8 1
642 0
6 66469 1 1
7 1 6 2
74 1 6
7674
7936
82 0 2
8471
8744
90 2 0
93O I9 585
9873
1 1 2 51 2 6 51 40 8
1 556
1 70 71 86 2
2 0 2 0
2 1 82
2 348
2 5 1 8
2 69 1
2 868
30 49
3 2 34
34 2 2
36 1 4
380 9
40 0 8
4 2 1 1
44 1 8
46 2 8
484 2
50 60
52 82
550 7
5736
5968
6 2 0 564456 688
6936
7 1 87
7442
770 0
796 2
82 2 8
8498
877 1
90 48
932 9
96 1 4
990 2
1 0 0 2
1 1 38
1 2 79I42 3157 11 72 2
1 8772 0 36
2 1 992 3652 5352 70 92 886
3 0 6 7
3 2 52
344 I
3633
382 9
40 2 8
42 32
4439
46 50
4 864
50 82
530 4
5530
5759
599 2
6 2 2 8
64696 7 1 3696 1
72 1 2
746 7
772 6
798982 55852 58 799
90 76
9357
9642
993 I
1 0 1 51 1 52
I3931438
1 586
1 738
1 893
2 0 52
2 2 1 52 382
2 552
2 72 6
2 90 4
30 86
32 7I
3460
3652
3849
40 49
4 2 52
446 0
4 67 1
4 886
5 1 0 4
532 6
5552
5782
6 0 1 56 2 52
64936 738
6986
72 38
7493
7752
80 1 582 82
8552
882 6
9 1 0 4
9 386
967 1
9960
1 0 1 0 6
1 0 40 0
1 0 698
1 1 0 0 0
1 1 30 6
1 1 6 1 51 1 9 2 8
1 2 2 441 2 56 51 2 889
.0
1 0 1 351 0 430
1 0 72 8
1 1 0 30
1 1 336
I I9 591 2 2 76
1 2 59 71 2 9 2 2
. I
1 0 1 64IO4S91 0 758
1 1 0 6 1
1 1 36 71 1 6 771 1 99 1
1 2 30 8
1 2 6 2 9I2 9S4
. 2
1 0 1 941 0 4891 0 788
1 1 0 9 1
1 1 398
1 1 70 8
1 2 0 2 2
1 2 340
1 2 66 2
1 2 987
-3
1 0 2 2 31 0 51 91 0 8 1 8
1 1 1 2 2
1 142 91 1 740
1 2 0 541 2 372
1 2 6941 30 2 0
.4
1 0 2 52
I0 5491 0 8491 1 1 52
1 1460
1 1 77 1
1 2 0 86
1 240 41 2 72 6
1 30 52
-5
z } ; I) 28f eet.2
.7 .8 .9
84 -1
3 1 6 -3
790 -3 80 3-7
1 0 4 2 1 0 56 1 0 70
1 1 80 1 1 941 3 2 2 1 336 1 350
146 7 1 482
1 6 1 6 1 63 1 1 646
1 768 1 784 1 7991 92 5 1 940 1 956
2 0 85 2 1 0 1 2 1 1 72 246 2 2 6 5 2 2 82
2 4 1 6 2 433 2450
2 587 2 60 42 76 2 2 779 2 7972 940 2 958 2 976
3 1 2 2 3 14 I 3 1 59
330 8 332 7 3346
3498 351 7 3536
369 1 37 I I 3730
3888 390 8 392 8
40 89 4 1 0 9 4 1 30
4 2 94 43 1 4 4335
4 50 2 4 52 3
47 14 4 735 4 756
49 2 9 49 51 4973
5 148 51 70 51 93
537 1 5394 54 1 6
5598 56 2 1 5644
582 8 5852 58756 0 6 2 6 0 86 6 1 1 0
630 0 6324 6 348
6 54 2 6 566 6 590
6 787 6 8 1 2 6 836
70 36 70 6 1 70 86
72 88 73 1 4 7339
7545 7570 7596
780 5 783 1 7857
80 6 8 80 9 58336 8363860 7 8634 866 2
8882 890 9 8937
9 1 60 9 1 88 9 2 1 6
9442 94 71 9499
9 72 8 9 757 9 786
1 0 0 1 8 1 0 0 47 1 0 0 76
1 0 3 1 1 1 0 34 1 1 0 370
1 0 60 8 1 0 638
1 0 90 9 1 0 939 1 0 9 70
1 1 2 14 1 1 244 1 1 2 751 1 52 2 1 1 553 1 1 5841 1 834 1 1 86 5 1 1 896
1 2 149 1 2 1 8 1 1 2 2 1 31 246 8 1 2 50 0 1 2 5331 2 79 1 1 2 824 1 2 856
1 3 1 1 8 1 3 1 51 1 3 1 84
.7 .8
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63
TABLE NO . 1 1 .
8’
Plus Correctw ns for 31;
2 3 4 .5 6 8 9
7
2 3 4 5 .5 7 8 g
I1
41 7 i f:Lwas C
'orrectwns foi 2 4
.
NOT E .-T he quantities from the above table div ided by tw o giv e the plus
8 18,
correction s for2 4
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64
TABLE NO . 12 .—LEVEL CUTTINGS.
855
5313 1 ; feet.
—nn —h “
631 )
1 40 1 )
2 244;3 1 643
4 1 513
52 2 436 3643
757 8
886471 0 2 3
1 1 6 71 3 1 8
14 76
1 642
1 8 1 6
1 996
2 1 84
2 5832 793
30 1 1
3236
3956
4 2 1 1
4473
4 743
50 2 0
530 4
5596
5896
6 2 0 2
6 51 6
6838
7 1 6 7
750 3
78478 198
8556
89 2 2
92 96
9676
1 0 0 641 0 460
1 0 8631 1 2 731 1 69 1
1 2 1 1 6
1 2 5491 2 9891 3436
1 389 1
I43S3148231 530 0
1 57841 6 2 76
1 6 776
1 72 82
1 7796
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4-9
30 -9
37-3
444
79 0
89 . 2
I49 .4
445-7
4694
493-8
3 1 -5
45-2
53-0
70 -4
90 3
I137-4
I79 3I944
47 1 9
496 -3
TABLE NO . 14 .
Plus Corrections for
-4
47-5
55-4
934
30 43
344 -3
43 1 7
455-1
4 79 -I
.4
Mum s Corrections for
NOT E—For m inus corrections for2
s+2 _1
.5 .6 .7
34 -0 34-7 3 5-3
94 -5 95 6
1 0 6 .8
1 1 8 .6
369 5
389 -0
4340 . 436 3
459 9
484 .0
50 8 .8
.5 .6 .7
1
2 2 °
1 , see T able 5.
4 5
43 -0
77-0
1 33 5
373 -8
.8
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67
TABLE NO . 15 .—LEVEL CUTTINGS .
b 14feet.
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8
TABLE No . 16 .
—LEVEL C UTTINGS . 15 ; b 26feet.
-5
49451 56 1 )2 75-5
40 51 :
54513
860 13
1 0 351 2 2 0
1 4 1 6
1 6 241 84 2
2 0 72
2 3 1 2
2 5642 82 7
3 1 0 1
3387
3683
3990
430 9
4638
4979
S33 I
569460 68
64536 850
72 57
76 76
8 1 0 58546
8998
946 1
99351 0 42 0
1 0 9 1 6
114241 1 942
1 2472
1 30 1 2
1 356414 1 2 71470 1
1 52 871 58831 6490
1 7 1 0 91 7738
1 83 79190 3 1
I96942 0 368
2 1 0 532 1 750
2 24572 3 1 76
2 390 524646
2 5398
2 6 1 6 1
-5
8+2
.6
59 -8
1 68432 871)
4 1 837
560 45
7 1 3877 .
I0 531 2 391 436
1 6451 8652 0 952 3372 590
2 854
3 1 2 9
34 1 6
37 1 3
40 2 2
4S4I
4672
50 14
5367
573 I
6 1 0 6
6492
6 890
72 98
77 1 8
8 1 49859 1
90 44
950 8
99831 0 4691 0 966
I I475I I9951 2 52 51 30 6 71 36 2 0
14 1 84
I47S91 5346
IS9431 6 552
1 7 1 7 1
1 780 2
1 8444I90 971 976 1
2 0 436
2 1 1 2 2
2 1 82 0
2 2 52 8
2 32 48
2 39792472 1
2 54742 6 238
.6
-7
70 1 :
1 7913
30 0 45
432 4 4
57543
89413
1 0 71
1 2 58
I4S71 666
1 8872 1 1 92 36 2
2 6 1 6
2 88 1
3 1 58
3445
3 744
40 53
4374
4 70 6
50 49
540 3
5768
6 1446 53 2
6930
7340
7760
8 1 92
8635
90 89
95551 0 0 3 1
1 0 51 8
1 1 0 1 71 1 52 6
1 2 0 47IZS791 3 1 2 2
1 36 76
1424 1
148 1 8
IS4OS1 60 0 41 66 1 3I72 341 7866
1 850 91 9 1 631 982 8
2 0 50 42 1 1 92
2 1 890
2 2 60 0
2 332 0
240 52
247952 55492 63 1 5
-7
.8
80 13
1 9 1 43
3 1 3 1 2
446 1 :
590 1 2
7451 4
9 IL 71 0 891 2 78
I4771 688 .
1 9 1 0
2 1 432 3872 64 2
2 90 8
3 1 86
3474
3 774
40 85
440 7
4 740
50 84
5439
580 56 1 82
657 1
69 71
738 1
780 38 2 36
8680
9 1 35
960 2
1 0 0 791 0 568
1 1 0 6 71 1 578
1 2 1 0 0
1 2 633I3 1 77I37SZ14 2 98
14876
1 54641 6 0 641 6675I72 97I7930
1 85741 9 2 2 9I989S2 0 572
2 1 2 6 1
2 1 96 1
2 26 71
2 339324 1 2 6
24870
2 56 2 52 639 2
9 1 1 2
2 0 3 1 )
3 2 6 1 )
460 1:
6 0 51 2
76 1 5
9 2 81)1 1 0 71 2 971498
1 71 0
I9332 1 6 724 1 2
2 668
2 936
3 2 1 4
3504380 4
4 1 1 6
4439
4 773
51 1 9
5475
5842
6 2 2 1
6 6 1 0
70 1 1
74 2 3
7846
82 80
872 5
9 1 82
96491 0 1 2 8
1 0 6 1 71 1 1 1 8
1 1 630
1 2 1 531 2 6871 32 3 2
1 3788
14356
I493 4ISS241 6 1 241 6 736
I73S9I79931 8639I9Z9S1 996 2
2 0 64 1
2 1 330
2 2 0 3 1
2 2 74323466
24 20 0
249452 570 2
2 6469
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69
TABLE NO . 17
3’
Plus Uorrootzous for £451 112
1 2 3 4 .5 6 7 8 9
. 1 .2 .3 .4 .5 .6 .7 .8 .9
Minus Corrootzous for8+—8 : 2
NOTE .—T he quantities from above table div ided by tw o give theplus correc
3t ions for
4 .
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TABLE NO . 18 .
Factors for Correction of Contents on Curves.
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T he Construction of T ables of Contents of Level Cuttings.
Base 2 1) half sum of side slopes 3 .
For each of height, the second difference
Between heights and first difference E—s{ c
1064—8
( 11064—S
5-427
C ontents for a height of10
22 . 7 1064- 27 >< s
5 .4 _2054—27x 43T o write out a table of level cuttings progressing in height by tenths,rule five columns car1 ied to heights of 2 . 7 when s 1 or one of itsmultiples
,and to heights of when s z 4 or one of its oddmultiples.
Example—(See portion of table given below) o z 28 s . z:
'
1 .
Here the second difference 1 ; first difference betweenheights and between andPlace the heights from to in the first column ; then put
first diiference in third column opposite in first, andsecond difi erence imm ediately above the first difference .
As a test for the continued addition of the second difference, putthe first difference in its place in third column
,opposite
in first. Now add for each of height up totaking care to 1 ecord the repeating fractions correctly, and see that thelast addition gives 12 .407407+ opposite 2 .7. T hen add each am ountin third column to the am ount on its left ln second, recmding eachsum in the next line below
,and keeping the repeating fractions cor
rect. T he contents in second column opposite should be 2
1064—278 2
Now repeat the am ounts in the second column to the nearest tenth ,placing them in the fourth column, and as before with regard . to theheights in the first. From the fourth column
,by subtraction
,write
the first differences anew,to the nearest tenth
,in the fifth column ,
and opposite their respective positions in the third.
For the rem ainder of the table, rule columns in sets of threes thefirst of each set to contain respectively the heights
'
from to
to to etc. T hen increase each o f the first differences inthe 5th column by 23 z and the first difi erences from to
are obtained for the eigh th column. T hese again increased by give
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72
the first differences from to for the eleventh column, etc. Inthis way the first differences for the whole table m ay be written to oneplace of decim als. .Each first difference is to be added to the contentsopposite in the next column on the left, and the sum recorded in thefirst line below. With contents calculated by Formula C (1) -His)
100
27; at intervals for tests, m istakes a1 s almost
'impossible.
T o ca1 ry out the table to whole numbers only, repeat the secondcolumn to the nearest whole number, get the first differences to wholenumbei s by subtraction, and proceed 111 all 1 espects as above dir .ected
(I ) (2 ) (3) (4) (5) (6) (8) (9) (1 0 ) (II)
7t ><
q
Contents .
0 0 740 74 Di
ff
Hei
ghts
Contents.
Dl
fi'
g l
3 1 9 40 740 7
In case the second colum n does not give a whole number at the height o fit should be carried out to or to the requ isite m ultiple . of