forgottenbooks.com€¦ · PREFA CE. T HIS work claims to present a new and systematized method of ﬁnding the prismoidal contents of Earthwork by m eans of T ables accompanied by

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.

Entered , according to Act of Congress, in the year 1874, byD . VAN NOSTRAND

In the Oflice of the Librarian of Congress, Washington , D. C.

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

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.

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.

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

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.

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.

10

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.

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.

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

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

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

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

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 .

https://www.forgottenbooks.com/join

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.

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

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.

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

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.

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.

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


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 .

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'

.

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 .

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.

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.

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.

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.


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)

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 .


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.



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

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



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.

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


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

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

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

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


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

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.

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.

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

‘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

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

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


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

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

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 .

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.

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 .

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 .

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


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

.

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

"

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

61

TABLE No . 9 .—LEVEL 0 1111 1110 0 5 .

{gig -é; b 16 feet

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

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

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


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

67

TABLE NO . 15 .—LEVEL CUTTINGS .

b 14feet.

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

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 .

TABLE NO . 18 .

Factors for Correction of Contents on Curves.

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

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

Documents

forgottenbooks.com€¦ · PREFA CE. T HIS work claims to present a new and systematized method of ﬁnding the prismoidal contents of Earthwork by m eans of T ables accompanied by