TF
222
731
UTATiCN
RAIL ; OTHER EARTHWORK
(AT
I
liliiil
Illliii
GIFT OF
Charles 1,1. Titus
UNIVERSITY FARM
1 i
TABLES
FOR THE COMPUTATION
OF
Railway and Other Earthwork.
V
COMPUTED BY
C. L. CRANDALL, C. E.,
Associate Professor of Civil Engineering, Cornell University
Member of American Society of Civil Engineers.
SECOND EDITION.
NEW YORK:
JOHN WII,EY & SONS,
53 EAST TENTH STREET.
1893-
UN-IV '* CALIFORNIA
LIBRARY
BRANCH OF THE
COPYRIGHT, 1893,
BY CHARTS L. CRANDAU,.
ANDRUS & CHURCH,
PRINTERS,
ITHACA, N. Y.
Preface to the First Edition.
THE object of these tables is to present a convenient aid in the Com-
putation of Railway Earthwork. The volume is first found by the
approximate method of averaging end areas, to which can be added, if
desired, a correction due to the strict prismoidal formula, or a correc-
tion due to a modification of it for irregular ground. This separation
reduces the labor, if both parts are computed at once, while it allows
of computing only the approximate value at first, which will answer
for all purposes except the final estimate, leaving the prismoidal cor-
rection to be computed at leisure on those volumes only where it is
large enough to be appreciable.
The Table for "Volume by Averaging End Areas," is a table for
triangular prisms, and it will give the approximate volume of triangu-
lar prismoids in cubic yards for a length of one hundred feet, between
the parallel triangular ends, by taking the sum of the two numbers
found with the base and altitude of each of the two end triangles as
arguments. The table is thus independent of side slope, or width of
roadbed, and will apply to borrow pits, trenches, etc., by dividing up
the cross sections into triangles and trapezoids.
The method of dividing up the sections for three-level, five-level
and irregular ground, while not essential, is believed to give the least
labor in computing, either with or without tables.
The method for irregular ground is due to J. W. Davis, and was pub-
lished in his Formula for R. R. Earthwork, 1876.
The Table for "Prismoidal Correction" is a table for triangular
prismoids, and it will give the exact correction to be applied to the ap-
proximate volume found above for all solids which can be divided into
triangular prismoids, having plane parallel ends, and planes or hyper-
bolic paraboloids for side faces. By using A. M. Wellington's method
for irregular ground, that for the purposes of the prismoidal correction
only, the ground may be called regular, the correction is easily and
quickly applied, to three-level, five-level, and irregular ground. It
may be noted that when either the bases or altitudes of the end trian-
gles are equal, the correction disappears.
While quantities are only given for full feet the corrections for tenths
are easily taken out from the margin without multiplication and can
be added mentally with but slight increase of labor.
The Tables have been carefully recomputed in print, and it is confi-
dently believed they are free from errors.
PREFACE TO THE SECOND EDITION.
The first edition was not electrotyped, and advantage has been taken
of this to entirely recast the text, adding a proof of the prismoidal for-
mula and a comparison with others in use, a formula for correction for
curvature, and rules for cross sectioning.
ITHACA, N. Y., August, 1893.
ZZ 76,
Contents.
DERIVATION OF PRISMOIDAL FORMULA AND COMPARI-
SON WITH OTHERS IN USE.
SEC. PAGE.
1. Prisinoidal Formula i
2. Averaging End Areas 2
3. Middle Areas 3
4. Equivalent Mean Heights 3
5. Triangular Prismoid with Curved Center Line 4
VOLUME FOR RAILWAY EARTHWORK IN CUBIC YARDS BY
AVERAGING END AREAS.
6. Formula for Three-level Ground 7
7. Five-level Ground 7
8. Irregular Ground 8
9. Tables 10
10. Example, Three level Ground n
11. Example, Five-level Ground n
12. Example, Irregular Ground 12
13. Borrow Pits 13
14. Sidehill Work 13
15. End Sections . . 14
16. Correction for Curvature 14
17. Example, Correction for Curvature 16
PRISMOIDAL CORRECTION IN CUBIC YARDS.
18. Three level Ground 17
19. Five-level Ground 17
20. Irregular Ground, Borrow Pits, etc. . 17
21. Table of Prisinoidal Corrections 18
22. Example, Three- level Ground 18
23. Example, Five-level Ground 19
24. Example, Irregular Ground 19
25. Example, Borrow Pits 20
COMPUTATION, END AREAS WITH PRISMOIDAL CORRECTION.
26. Three-level Ground 21
27. Irregular Ground 21
28. Rules for cross-sectioning 23
Table of Triangular Prismoids for Railway Earthwork, in cubic yards,
by Averaging End Areas.
Table of Prismoidal Corrections for Triangular Prismoids for Rail-
way Earthwork.
Derivation of Prismoidal Formula and
Comparison with Others in Use.
Section x x>
First End'
cU(d
i. Prismoidal Formula. In computing railway earthwork, it is
customary to measure sections perpendicular to the center line at sta-
tions 100 feet apart, and at as many
intermediate points as may be found
necessary.
To find the volume between two
such parallel sections :
Take at first the simple case shown
in Fig. i, where the section is straight
from the center to the slope stake on
each side, and where straight lines
joining corresponding points of the
upper lines of the sections lie in the
ground surface. The ground surface
thus formed is a hyperbolic parabo-
loid, a characteristic property of
which is that all planes parallel to
the end sections cut the surface in
straight lines, whose increments for
decrements) in length, starting from
either end, are proportional to their
distances from that end.
With the notation of Fig. i, the
area of the right-hand side of the first
section (including the grade triangle
under the roadbed),
A = $(a + c)d.
At the distance z, since all the linear dimensions of the section can
be interpolated,
*. ~1 *1
(a)
Plan
Section
Second
Fig ..
Elementary volume of length dz = area into length
Integrating between the limits z =o to z = /,
Volume = C 1 A Z dz
J o
PRISMOIDAL AND OTHER FORMULAS. [| 2,
-c) + (c'-c) (d'-#)
Volume =
26
-L <*+0 / 5 , a+c d'l c'c d'l a+cdl^ </ cd_l
2 6 + 23" 2622+26
Vol. =
where A m is the area of the mid section found by making /-?- 2
in (a).
1. e., the true volume equals the sum of the areas of the end sections
and four times that of the mid section, multiplied by one-sixth the
perpendicular length between sections.
This is the prismoidal formula, and the first proof of its application
to an earthwork solid, as above, is due to Professor Gillespie.*
It is evident that an earthwork solid with polygonal end sections
lying in parallel planes can be divided into solids with triangular bases
as above, when there are the same number of sides in the end poly-
gons and lines joining corresponding points of these sections lie in the
bounding surfaces ; therefore the prismoidal formula will give the true
volume for all such solids. The solid thus described will be called a
prismoid ; i. e., a prismoid will be defined as a solid having two plane
polygonal parallel bases with the same number of sides in each, joined
by either planes or hyperbolic paraboloids for side faces. With trian-
gles for bases, the prismoid becomes a triangular prismoid.
2. Averaging End Areas. This is a method in common use on
account of its simplicity. Applying it to the triangular prismoid of
$ i, and including the grade triangle for simplicity,
By averaging end areas,
2 2
True volume by the prismoidal formula,
Vol - d > d >\ 2l \ c + c ' d + d ' 2/
>p .2 6^2 6^2 2 34 2 3
Subtracting Vol. e from the true value, Vol. p ,
Vol. -Vol =-^di-^d>L + W a l_
23 2 3*23
* See Roads and Railroads, New York, 1847.
Bq-3]
EQUIVALENT MEAN HEIGHTS.
(2)
where Cor. e is the correction to be applied to the volume found by aver-
aging end areas in order to obtain the true volume.
Usually in passing from lighter to heavier work c' > c and d r > d
(or conversely from heavier to lighter), making the above correction
usually negative; i. <?., the method by averaging end areas gives on
the average too large a volume.
3. Middle Areas. The linear dimensions of the mid section will
be the means of those of the end sections, giving
Subtracting this from the true value,
Vol. p -Vol. m = -
2
___ _ (d+d/)
3 24 24 82 3
24
(3)
where Cor. m is the correction to be applied to the volume found by mid-
dle areas in order to obtain the true volume.
This correction is one-half that found by averaging end areas, and
has the opposite sign, i. e., the method on the average gives results too
small, but the error is only one-half as great as by the preceding
method.
4. Equivalent Mean Heights. In this method the center height
of a level section having an equal area is found for each end section,
Fig. 2.
and the volume of the reduced solid is then accurately found by the
prismoidal formula.
Applying this to two-level ground, with slope stake heights * ^ r ,
slope ratio s, reduced center height c lt and distances out to slope stakes
** 8 J *
From the reduced area,
or;
PRI&MOIDAL AND OTHER FORMULAS. [g 5,
The true volume of the reduced solid,
Vol. eim . hl =
The true volume of the original solid,
Subtracting the volume of the reduced solid from that of the origi-
nal one,
Vol. p -Vol. einihi = [/, */
or,
Cor.
e . m . hi
(4)
The correction is thus always positive, *, <?., the method always gives
too small a result for two-level ground, except when ^ // = // t r ;
hence it will usually give too small results for ordinary ground.
5. Triangular Prismoid -with Curved Center Line. In curving
the center line of the roadbed to the radius J?, let the dimensions of
Fig-3-
Eq. 4] CURVED CENTER LINE. 5
all the cross sections remain unchanged. This will give the solid
easiest to approximate in the field, and the one most like the prismoid,
since all lineal dimensions of the cross section can be interpolated.
In Fig. 3, let B C be the center line of the roadbed ; e, e<)>, e 1 ', the
eccentricities of the sections, or the distances of their centers of grav-
ity from the center of the roadbed ; A, AQ, A / , the areas of the trian-
gular sections, with bases b, b<j>, b f , and altitudes a, a<j>, a' ; $ the cen-
tral angle from B to any point F; I the length B Cof the solid along
the center line of the roadbed, Rty f .
At any point D,
Elementary volume = A^ D E = A^ (R -\- e^) dq>
/ R
A. (R 4- ed$
o
Since all cross section dimensions can be interpolated,
Substituting,
Volume =
-- + b (a'-a)
+ [ab
=\l [ab 4- a(b' b} % + b(a'a) k + (
a
+[ ab +a (b'-b) i + d(a'a) i + (a'-a)(b'-b] i ] ^^
= ^\ab + i ab' + i a'b + a'b'
+(ab + \ab' + \ a'b + a'V}
PRISMOIDAL AND OTHER FORMULAS. \_\ 5,
m m -
Volume
(5)
where A m is the area of the mid section, and <?, e' ', are positive for the
convex side of the curve.
The first product is the volume which would be obtained by the
prismoidal formula neglecting curvature, leaving the second for the
correction due to curvature.
Volume for Railway Earthwork
IN CUBIC YARDS BY AVERAGING END AREAS.
6. Formula for Three-level Ground. Dimensions in feet, volume
in cubic yards.
Area of section = (a+^r)- 1 4. (a+c) - r a ~
d,
_ Fig. 4.
At the next section, TOO feet from the first, priming corresponding
parts,
Area' =(+<) ^-"y
Multiplying the average area by the length, and dividing by 27 to
reduce to cubic yards,
/\ 100
Volume =(area + area')
2 X 27
Volume =(a+c] w%% a&%$ + (a+c'} w' f ab f . (6)
/. e., the effect of each cross section on the volume reduces to the
constant times the total width between slope stakes times the sum
formed by adding the center height to the height of the grade trian-
gle ; minus the constant times the base times the altitude of the grade
triangle.
For the next solid,
Volume' = (area'+ area") i$.
The terms comprising area' should therefore be added separately,
for convenience in finding the second volume, etc.
7. Five -level Ground. In five- level ground the intermediate
heights, t { and t t , are taken over (or under) the edges of the roadbed.
By joining the edges of the roadbed with the ground at the center, two
triangles are formed on the left having /, for the common base, rf, for
7
8
VOLUME BY AVERAGING END AREAS.
the sum of the altitudes, and ^ d l for double area ; two on the right with
double area t t d r ; and one at the center with double area be.
Therefore, Area first end
=*L^4.if*f _{_*
Area second end =
*L + **'**' + ^
222
Volume = ^,f |+^rff-h^f l+A^/f f+V^/f t+^^f f . (7)
i. e., the effect of each cross section on the volume reduces to the con-
stant times the distance out to the left slope stake times the left shoul-
der height ; plus the constant times the distance out to the right slope
stake times the right shoulder height ; plus the constant times the
width of roadbed times the center height.
8. Irregular Ground. In Fig. 6, let d\, f\, g\ , and d r , / r , g r
. . . = the distances to the left, and right, to the slope stakes, and in-
termediate breaks in the ground surface ; and r^ u^t\ . . . , and r r , u r ,
t r . . , = the corresponding heights.
H
Fig-. 6.
The area of the section may be found, by taking the two triangles
with the common base a-\-c, and altitudes </, and d rt adding the trian-
i
gle with area GH- -- 1 , and subtracting the triangle with area DE- *J,
and the grade triangle This gives,
Area first end (the right side being regular)
Kq. 8] IRREGULAR GROUND. 9
where IG and FE are to be taken as negative if measured below the
horizontal line FI. Expressing them in terms of measured quantities,
since IG = (r, c] & and FE = (Lc] ,
d \ g\
Area first end
W-,) + etc .
2 2
(a + A) J + ( u \ r i) ^- + (^ A) " -f- etc. +
2 2 '22
where each break or intermediate height //, Z>, etc., adds a term of the
form (u { f\\ If //"is omitted, the corresponding term (, r,)^ 1 dis-
2 2
appears and the other term becomes (c rtf-1 , as may be found by
leaving off the triangle GH&\ if //"and D are omitted both terms
disappear, and (0+^) J becomes (a+c) -i, as may be seen from the ex-
pression for the right side.
If the next cross section, TOO feet distant, is similar to the first, and
the same letters with primes are used for corresponding parts, we shall
have,
Area second end,
= (a + *,') L+K'-^OH- (*'-/ I / ) + etc.
222 22
The volume will be found by multiplying the average area by the
length, and dividing by 27 to reduce to cubic yards.
Volume in cubic yards = (area 4- area 7 )
100
2 X 27
Substituting the value of each end area,
Volume
etc. +(a+c)d r
(8)
Or, the effect of each cross section on the volume reduces :
For the irregular side ; to a constant times the distance to the slope
stake times the sum formed by addiug the height next the slope stake
to that of the grade triangle ; pins the constant times the distance out
to each break, in turn, times the difference formed by subtracting the
first height outside from the first height inside that break.
For the regular side ; to the constant times the distance out to the
slope stake times the sum formed by adding the center height to the
height of the grade triangle-
10 VOL UME BY A VER AGING END AREAS. \_\ 9,
From the total sum must be subtracted the constant times the base
times the altitude of the grade triangle.
For the next solid,
Volume = (area'4- area") _!2?__ .
2 X 27'
the terms comprising area' should therefore be added separately, for
convenience in finding the second volume, etc.
It may be noted that, if the expressions for the areas of the origi-
nal triangles had not been reduced, the constant times the base times
the altitude of each, would appear as its effect upon the volume.
Hence the computer may divide the sections into any triangles he may
wish ; the volume will result by multiplying the constant into the base
times the altitude of each triangle, and adding the products.
9. Tables. The table gives the value of (a-\-c)dk& (or/rf|f), for
a -f- c (height) and d (width) as arguments ; or it gives the effect which
any triangle in the cross section has upon the volume, for a length be-
tween stations of 100 feet, by using the altitude and base of the trian-
gle as arguments. After looking ont the partial volume for each tri-
angle for both ends and adding, if the length be not TOO, multiply the
sum by T ^ of the length.
To look out the partial volume for a triangle : Find the integral
part of the height factor in the line of height headings at the top of the
page ; then move down this column until opposite the integral part of
the width factor in the width column and note the number ; then move
to the left on this line (or the next below if the tenths of width be
greater than 5) and take out the " Correction for tenths of height" ;
them move to the bottom of the page and under the column first used
(or the next to the right if the tenths of height be greater than 5) take
out the " Correction for tenths of width." Take the sum of the three
mentally. Look out the quantity for each of the other terms ; the
algebraic sum will be the total volume. The terms coming from
"area second end," should be kept separate as they will be used in
finding the volume of the next solid.
If one of the arguments cannot be found in the table divide it by
two, look out the number and multiply by two ; if both cannot, divide
each by two and multiply by four.
In order to impress upon the mind the fact that the correction for
tenths of width is placed under the height column, it should be remem-
bered that the differences down the column are constant, so that cor-
rections or proportional parts of these differences can be placed under
it, but the widths are increased as we move down the column, or these
corrections are for tenths of width. Similarly for the lines horizon-
tally, or the corrections for tenths of height.
After becoming familiar with the table, it will often save time to call
height "width," and width "height," and to take the correction for
tenths from the side or bottom of the page as most convenient, pro-
vided always that it is opposite (or under) the heading number to
which it belongs.
Eq. 8] EXAMPLE, FIVE-LEVEL GROUND. n
10. Example, Three-level Ground.
Sta. Center. Left. Right.
20 . . 10.8 .
21 24.2, .... 32-6
19-6 ' 24.1
Hi! 32.6
23-7' ' ' 55-9
21+20 . . 46.7, . .,. 5Q.6
37-0 82.9
a =4.7, 6 = 14. f
Substituting in formula (6), or following the rule $ 6, we shall have,
in connection with \ 9 :
Sta. 20,
a + c = 4.7 + 10.8 = 15.5, w = 19.6 + 24. i = 43.7.
From the Table of Triangular Prismoids,
Under height 15, right of width 43, 597
Left of width 44, correction for 0.5, .......... 20
Under height 15, correction for 0.7, 10
627
Arguments, = 4.7 and b =- 14 (grade prism), 61 566
Sta. 21,
a + ^'=4.7 + 24. 2 = 28.9, ^' = 23.7 + 55.9 = 79.6, . . . 2131
= 4.7, = 14, 6 1 2070
Total in yards, ... 2636
From above, for sta. 21, 2070
Sta. 21 + 20,
a + '=4. 7 + 46. 7 = 51.4, ze/'=37-o + 82.9 =119.9, 5712
a = 4. 7, = 14 (grade prism), 61 5651
7721
Total in yards 7721 X .20= 1544
Grand total, 4180
ii. Example, Five-level Ground.
Sta. Center. Left. Right.
46 . . 10.8, . . IL4 . . !_', . . . 9^ . . A_ 6
7 25.1 7 19.9
47 - - 6.4, . . M . . 1. . . At > .' K
7 17-8 7 14-5
* The numerator is the height above or below grade at the point and
the denominator is its distance from the center.
t The side slope s' is here assumed to be one and one-half horizontal
to one vertical, so that
12 VOLUME BY A VE RAGING END AREAS. [g 12,
Substituting in formula (7), using the Table for the multiplications,
Sta. 46, arguments, 25.1 and 11.4, 265
19.9 " 9.9, 183
14.0 " 10.8, 140 588
Similarly for Sta. 47, 272
Total in yards, 860
12. Example, Irregular Ground.
Sta. Center. Left. Right.
20 . . 10.8, . M, , . IL . . JH
19.6 18 24.1
21 24.2 IM . . ^ . 20 4 32.6
16 23.7' 42 55.9
21+20 . 46.7, . 4 iZ . . 2ao , . . 4*L . i _5 . . 50- 6
20 37.0 20 58 82.9
b = 14, 5 = 1.5 to i, a = 4.7.
Substituting in formula (8), using the Table for the multiplications,
Volume from sta. 20 to sta. 2 1 .
Sta. 20,
a-\-t= 4.74-11.2=15.9, o? r =24.i ; c r r =io.8 11.4= 0.6, - r =i8 ;
#+=15.5, ^,=19.6; #=4.7, =14.
Under height 15, right of width 24, .. . . 333
I/eft of width 24, cor. for 0.9, 20
Under height 16, cor. for o. i, i
354
Arguments, 0.6 and 18, 10
19.6 and 15.5, 281
14 and 4.7 (grade prism), .... 61
564
Sta. 21, Arguments,
20.4+4.7=25.1, and 55.9, .... 1299
24.2 32.6= 8.4, and 42, . . . . 327
18.6+4.7=23.3, and 23.7, .... 512
24.2 11.1=13.1, and 16, 194
' 14 and 4.7 (grade prism), .... 61
1617
Total in yards, 2181
Volume from 21 to 21+20.
From above for sta. 21, 1617
Sta. 21+20, .
50+4.7=54.7, and 82.9, 4200
48.1 50.6= 2.5, and 58, . . . . 134
46.7 50= 3.3, and 20, 62
40.7+4.7=45.4, and 37, 1556
46.7 20=26.7, and 20, 494
14 and 4,7 (grade prism), .... - 61
5993
7610
76lO X. 20= 1522
3703
Eq. 8] SIDE HILL WORK. 13
Any other method of dividing the sections into triangles can be used
if preferred, although it will result in increased labor.
13. Borrow Pits. In borrow pits, ditches or other work, where the
cross sections are made up of triangles or trapezoids, take the altitude
and base of each triangle, or the sum of the parallel sides, and the per-
pendicular distance between them for each trapezoid, as arguments, as
with the two sections below.
/^i"> - '" iX"
^^^~, M5/^
""vi ;' 0-< a- a ! so I/in
s "-'-
Sta. I.50.
Sta. i, 27 and 14 2, .
355
50 " 28.0, .
1296
50 " 26.4,
1223
20 " 12.6, .
233
- 3107
Similarly for sta. 1+50, 2672
5779
5779 X -5<=> = 2890 yards.
14. Sidehill Work. When a "grade point" occurs in a cross sec-
tion, one side being in cut, the other in fill, to compute the cutting, put
in a break at the grade point and call the cuts on the fill side zero, and
the distance to the slope stake b -=- 2. Then proceed as for irregular
sections.
In computing fills the process would be reversed.
EXAMPLE.
Sta. Center. Left. Right.
46 . . -2.4, . . .. ^ - *4
4.3 12.3 16.6
A*J J^ T *7 _*2."
Roadbed 18 feet in cut, 14 feet in fill, slopes \\ to i. Altitude of
ade triangle in cut = 9 -r- 6,
For cutting, the notes would be
grade triangle in cut = 9 -v- ii 6, in fill = 7 -s-ll = 4.7.
46 .
. . O.O, .
0.0
2.2
0.0
An
I 7
4-3
12.3
3-8
o.o
9.0
0.0
' 14.7' '
2.6
9.0
Giving,
Sta. 46, 0.0+6.0= 6, and 12.3, . . 69
0.02.2==: 2.2, and 4.3, . 9
0.0+6.0= 6, and 9, .... 50
Grade triangle, 6 and 18, 100
10
Sta. 47, similarly, . . 60
Total in cu. yards, ... 7
VOLUME BY AVERAGING END AREAS.
16
Or it may be more convenient to leave out the grade prism, thus :
Sta. 46, height=2.2, width= 9 4.3= 4-7, giving -.59
" 47, " =i-7, " =14-7+2.6=17.3, " . . 28
" =3-8, " 9.0. " . . 32
Total in yards, .
For filling, the notes would be
Sta. Center. Left.
o.o o.o
4-3 7-o' '
69
Right.
46 . . 2.4,
47 -
Giving,
Sta. 46,
o.o o.o
7.0' 2.6
0.0+4.7=4.7, and 7, . . 31
2.40.0=2.4, " 4.3, . . 9
2.4+4.7=7.1, " 16.6, . . 109
Grade triangle, 4.7 (< 14, . . 61
Sta. 47, similarly,
-6.4
16.6
- 2-9
11.4
88
n
Total fill in cu. yards, ... 99
15. End Sections. A cross section should be taken wherever
either edge of the roadbed in cutting reaches grade. This will require
two sections in passing from
a cut to a fill, or vice versa,
while a third is sometimes
added where the center line
reaches grade. The solid, in
cut, between the two sections,
is a pyramid with volume
equal to area of base into one-
third the altitude, or it may
be taken under the rule, base
into one-half the altitude, and
afterwards corrected by the
prismoidal formula.
The solid in fill, between
the two sections, can usually
be taken as having the same
altitude, the volume not be-
ing required as accurately as
for cut ; the exact values can
be interpolated when necessary.
16. Correction for Curvature. For a solid corresponding to the
triangular prismoid, as described in 5, we have from (5),
8.
Eq. 13] CORRECTION FOR CUR VA TURE. 1 5
Correction = ^ \_(A -f 2 AJ e + (2 ^ m -f ^4/) */] ( 9 )
''] (approx.) (10)
The latter equation is the one in common use, giving, with less labor,
sufficient accuracy for earthwork solids.
Three-level Ground. To find e, the eccentricity of the center of
gravity, we have,
d, d T
Fig- 9
v ^'
For the right side,
2
Center of gravity from center line = -i
3
(Product), = (a + c) r
6
Similarly, for the left side,
(Product)! =
Hence f (product), + (product), (a-\-c) 6
total area ~ (a-f-^r) d r -\- d\
Or, ,-ftz& (ii)
. 3
Correction = J\_A (d r d l ) -f ^/ 7 (rf/ rf/)] (12)
6 /c
where ^4 and ^ x must include the grade triangle.
Or, Correction y = -^LQ K ~ ^i) + Q f W ~ ^/)] ( 13)
where Q and Q' are the tabular quantities or volumes corresponding to
A and A' (A = f ^ Q) and Correction y is in cubic yards.
Five-level and irregular ground can usually be taken as three-level,
with the same distances to slope stakes, for the purposes of this cor-
rection, unless the intermediate heights show a marked tendency to
lie above, or below, the lines joining the side heights with the center,
when the section can be plotted and an equalizing line drawn, or in
extreme cases each triangular prismoid can be taken separately by (10).
Sidehill Ground, neglecting all breaks between slope stakes. The
center of gravity will be on the line joining the center of the base with
the vertex, at one-third its length from the base.
16 VOLUME BY AVERAGING END AREAS. [ 17,
The distance from the lower end of this line to the center, Fig. 10,
2
Distance from the upper end = d r .
Hence <? = f/_ M-f-i^ r = /_ f/j-|-^ r \ (14)
where d l will be positive if the grade point is on the same side of the
center as the slope stake.
Substituting in (10),
Correction = 1- A ( b - _ rf, -f- d t ) + A'( b -- df + d/)~] (15)
Correction,, = _L f ( * <*, + </ r ) + 0' ( * */ + <*/) 1
2 _^
(16)
17. Example, Correction for Curvature. In the example of \ 12,
let the center line be on a 7 curve to the right, ^ = 5730-1-7 = 819,
and let the intermediate breaks be omitted in finding the correction
for curvature.
Volume from sta. 2 j to sta. 21.
Q, including the grade triangle, = 625 ; Q f 1678 ; d\ = 19.6 ; d r =
241 ; flf/^23.7; d/ = 55.9.
Substituting in (13),
Correction _ [625(24. 1 19 6)4-1678(55.9 23.7)] = 23
3X819
Volume from sta. 21 to sta. 21 +20,
Correction == J-_[[6 7 8(55.923.7)+6o54(82. 9 37.0)] = 135
3 X 819
.2 X 135 27
Total in cu. yards = 50
The correction is negative, the center of gravity and the center of
the curve lying on the same side of the center line.
In the example of \ 1.4, let the center line be on a 7 curve to the right.
Volume from sta. 46 to sta. 47.
Substituting in (16),
Correction = -- L_ [10(9 12.3 4.3) + 60(9 14.7 + 2.6)] = o
3 X T 9
This correction would be positive, the center of gravity and the cen-
ter of the curve being on opposite sides of the center line.
Prismoidal Correction in Cubic Yards.
1 8. Three-level Ground. The prismoidal correction to be applied
to the volume of a triangular prismoid found by averaging end areas
in order to obtain the true volume, has been found to be by (2),
Correction = (c c'}(d' d)~
where c and d are the altitude and base of the triangle at the first end,
and c' and d' those at the second end.
For three-level ground, per station in cubic yards, this would re-
duce to
Pris. correction = (c c / )(zv / w] (17)
3X54
where, as in Fig. 4 and eq. (6), c and c' are the center heights and w
and w' are the total widths.
This correction is to be added to the approximate volume found by
end areas ; it disappears when the center heights, or the widths, are
equal, and it can often be neglected without computation when in-
spection shows one of the factors very small.
19. Five-level Ground. The intermediate heights having been tak-
en over (or under) the edges of the roadbed. In $ 7, see Fig. 5, each
section was divided into three triangles, one having the distance to the
left slope stake, d lt for base, and the height over the left edge of the
roadbed, t lt for altitude, the second the corresponding dimensions,
d r , t r , on the right, and the third the width of roadbed and the center
height. Applying (2), and remembering that since the width of road-
bed is the same at both ends, the correction for the third triangle will
be zero :
Pris. correction =| (-//) (d/-di)+(t r -t') (d'-d r ) IT^TT ( l8 )
20. Irregular Ground, Borrow Pits, etc. Since the areas of the
sections will be made up of triangles, the correction will be made up of
as many terms as there are triangular prismoids in the volume, each
term being of the form
(c c / )(w / w} 5
3X54
This requires that there should be the same number of breaks at each
end of the solid, as already indicated in $ i, in deriving the prismoidal
formula and defining the prismoid. If there are not, the distance
from the center at which a ridge or hollow runs out at the more regu-
lar section should be noted in the field, and the height either noted or
computed.
17
1 8 PRISMOIDAL CORRECTION IN CUBIC YARDS. [22,
A. M. Wellington* recommends that for the purposes of the pris-
moidal correction irregular ground be assumed to be three-level ground
having the center height and total width of the irregular section.
This makes the correction the same as for regular ground, and while
for a single volume it may give a result considerably in error, the sign
is as likely to be plus as minus, so that there is less tendency to accu-
mulation of error than in the various approximate methods described
in || 2-4
21. Table of Frismoidal Corrections. The table gives the value of
(c c'}(w' w] 5 , for c-c f (height height 7 ) and w' w (width'
3X54
width) as arguments ; which is the correction for a length of TOO feet.
If the length be not 100, multiply the result by T ^ of the length.
To look out the number : Find the integral part of the height fac-
tor in the line of height headings at the top of the page ; then move
down this column until opposite the integral part of the width factor
in the width column, and note the number ; then move to the left on
this line (or the next below if the tenths of the width factor be greater
than 5) and take out the " Correction for tenths of height height' " ;
then move to the bottom of the page and under the column first used
(or the next to the right if the tenths of the height factor be greater
than 5) and take out the "Correction for tenths of width' width."
Take the sum of the three mentally.
If one of the arguments cannot be found in the table, divide it by
two, look out the number and multiply by two ; if both cannot, divide
each by two and multiply by four.
The remarks of | 9 apply to this table the same as to the table of
triangular prismoids.
22. Example, Three-level Ground. See \ 10.
Sta.
Center.
Left.
Right.
20 . . .
. .10.8
8.4
11.4
19.6
24.1
21 . . .
. .24.2, . . . .
ii. i
32.6
23-7
55-9
2I+2O .
. .46.7, . . . .
20. o
50.6
37-0
82.9
Correction for volume from sta. 20 to sta. 21.
c c'= 10.8 24.2= - 13.4, w' w = 79.6 43.7 = + 35.9.
From the Table of Prismoidal Corrections.
Under 13, right of argument 35, = 140
Left 36, corr. for 0.4 = 4
Under 13, corr. for 0.9= 4 148
* Computation from Diagrams of Railway Earthwork.
Eq. 1 8] EXAMPLE, IRREGULAR GROUND. 19
Correction for volume from sta. 21 to sta. 21 -f- 20.
' = 24 2 46.7 = 22.5, w' ze>= 119,9, 79-6 = + 40.3;
22.5 and 40.3 as arguments = 272 -\- 6 -f 2 = 280
280 X .20= 56
Total correction to the volume found by " End areas " = 204
23. Example, Five-level Ground.
Sta. Center. Left. Right.
46 . 10.8, . IM . . "^ 9^9 . M
8 20.1 8 16.6
47 . 6.4, . M . 7^ . ^ . . 1
8 15.2' 8 13.0
fy= 16, side slope ratio 5= i to i. See Fig. 5 and (18).
/, // = n.4 6.8 = + 4.6, <// ^, = 15.2 20.1 = 4. 9.
/ r // = 9.9 5.7 = + 4.2, d/ d r = 13.0 16.6 = 3.6.
4.9 and 4.6 as arguments = 54-1 + 1 = 7
4.2and3-6 =4+1 =5
Correction in yards = 12
By the approximate method of \ 20 : 10.8 6.4 = + 4.4 ; 28.2
36.7= 8.5.
+ 4.4 and 8.5 = 10 -f i + i = 12, = correction in yards.
24. Example, Irregular Ground. See \ 12.
Sta.
Center.
Left.
Right.
20 . .
. 10.8, .
M,
19.6' "
II. 2 11.4
' 18 ' '24.1
18.6
ii. i
20.4 32.6
21
. 24.2, .
' 16 '
w"
42 ' 55-9
Aft 1
40.7
20. o
48.1 50.0 50.6
21 -)-2O
4v-7i
20
37.0
20 58 82.9
= 14, .9 = 1.5 to i, a = 4.7.
The break 16 feet out at left of sta. 21 runs out 10 feet left, at sta. 20.
The break 20 feet out at right of sta. 21+20 runs out at the center
at sta. 21.
By interpolation, the height at 10 feet to the left at sta. 20= 10.8
J^_ [10.8 8.41=9.6.
19.6
Correction for volume from sta. 20 to sta. 21.
Sta. 20 would be written
9-6 8.4
.. . .
I0 '" ' * lo ' ' TP' ' l8 2 4- x
Changing the factors of 12 which were used in finding volume or
the areas of the triangles, to correspond with the interpolated break,
20 PRISMOIDAL CORRECTION IN CUBIC YARDS. [25,
Arguments, 19.623.7= 4.1,23.314.3=+ 9.0, corr.= n
24-1 55-9= 3i-8, 25.1 15.9=+ 9.2, " 90,
10 16 = 6.0, 13.1 2.4=+io.7, " 20
" 18 42 =24.0, 8.4-fo.6= 7.8, " + 58 63
Correction for volume from sta. 21 to sta. 21+20.
Sta. 21 would be written
24.2 l8 - 6 "- 1 24.2 20.4 32.6^
16 ' 23.7' ' o 42 55.9'
giving as above,
Arguments, 23.7 37.0= 13.3, 45.4 23.3=4-22.1, corr.= 90
55-9 82.9= 27.0, 54-7 25- 1=4-29-6, " 247
16 20 = 4.0, 26.7 I3.i=+i3.6, " .17
42 58 =16.0, 2.5+8.4=+ 5.9, " 29
o 20 =2o.o, 3.3 3.8= 7.1, " 4 44
339
339 X. 20= 68
Total correction in yards= 131
By the approximate method of \ 20 the result would be that found
for the first example, or 204 cu. yds.
25. Example, Borrow Pits. See example, \ 13.
Arguments which were used in finding volume ; for sta. i, 27 and
14.2, 50 and 28.0, 50 and 26.4, 20 and 12.6 ; for sta. i 4 5> 34 an d IO -4>
50 and 21.4, 50 and 24.6, 17 and 13.6.
The corrections will all disappear except for the first and last pro-
duct at each station.
Arguments, 27 34 = 7, 10.4 14.2 = 3.8, corr. = + 8
" 20 17 = + 3, 13.6 12.6 = + 1.0, " = + i
Correction in yards = 9 + .50 = 5.
Computation End Areas with Prismoidal
Correction.
26. Three-level Ground.
St
o
.. Center.
7 I
Left.
6-4
Right.
8.0
. 9 8
14.4
7-6
16.0
12.4
. 12.4 .
' 15-6
12.4
20.4
10.3
8 7
20.4
9-8
18.3
8.1
. . IO.2
' 17.8 '
u. 6
19.6
16.1
9.8
' 17.8
4
Roadbed 16 ft., side slopes i to I.
COMPUTATION.
Sta.
Width.
Height,
orh.+a.
Yds. by Knd areas
w w'.
ht' ht.
Yards,
pris'dl.
o
i
2
3
4
30-4
36.0
38.7
33-9
37-4
I5-I
17.8
20.4
16.7
18.2
426
594
M 73 I
525
631
782
1087
1018
918
-5.6
2.7
+4-8
3-5
+2.7
+2.6
3-7
+ 1-5
5
2 '
6
2
Approx. volume = 3805
Prismo'l cor. = 15
True volume = 3790
27. Irregular Ground.
5-2
7-4 3-4
V
13.2 8
11.4
IO .
. 4.9
9-6 4.1
17.6 9 '
3-6
ii. 6
7-2 5.0
8.4 5-2
8.9
10,14 1 *
. 9.0 .
8 12
16.4 7
16.9
16 | 12
12
6.8
11.4
12.8
7-1
16
20.8
15.1
T1
8 6
7-7
7-4
5-2
M *
12
15-4
13.2
*The breaks at the left of n run out 10 ft. and 14 ft. respectively to
the left of 10, etc.
22 END AREAS WITH PRISMDL. CORRECTION.
27,
+47 - 2.o .
4
5-4
0.0
10
13-4
8.0
0.0
4-2
2.0
1
4.o
12.2
IO.O
8|
+2O . 6.O . . .
O.O
4.4
8.0 '
13-6
Roadbed 16 ft, slopes i to I in cut.
Roadbed 14 ft., slopes i^ to i in fill.
Computation on the assumption that, for the purpose of
moidal correction only, the ground is regular.
the pris-
Sta.
Width.
Height,
or h.+a.
Yds. by Bnd areas
w w'.
ht' ht.
Yards,
pris'dl.
9
13.2
148
182
11.4
15-4
163
8
3-4
25
10
17.6
12.9
211
1 1.6
12. 1
130
9
i-3
II
484
-4.6
1.9
+ 2
ii
16.4
13.0
I 9 8
12.
1.2
13
8
+4
30
16.9
13.2
ON 207
7
+0.1
M I
537
4.1
+4-1
5
12
20.8
19.4
1 374
16.0
6.0
-89
15-1
14.8
206
676
2.6
2.2
+ i
13
15.4
15-7
224
12.
1.2
13
13.2
16.6
203
693
+7.3
+ 1.8
+4
+47
13.4
12.0
148
10.
3-4
32
8
IO.O
_ 74
184
+7-2
6.6
7
Pris'l
for 14
r
5-4
o
14
4.0
4.2
1 16
46
+ 12.0
1.2
3
+20
O.O
0.0
oo
3
+ 4.0
4.2
i
Approx. volume = 2623
Prismo'l cor. 9
Corrected volume = 2614
16
9
Eq. 18]
RULES FOR CROSS SECTIONING.
Computation With the true prismoidal correction.
Sta.
Width.
Height,
or h.+a.
Yds. by End areas
w w'.
ht' ht.
Yards,
pris'dl.
9
13.2
14.8
182
11.4
15.4
163
8
3-4
25
10
17.6
12.9
211
4.4
1.9
+2
Pris'l*
Ji7.6
16.6
for ii
1 J ^"
1.9
1 10
3-7
n.6
I2.I
I 3
0.2
3-3
9
1-3
II
484
1.0
2.1
+i
ii
16.4
13.0
I 9 8
+ 1.2
-3-6
i
12
1.2
13
+2.0
+0.7
8
+4.0
30
+2.0
+7-7
+4
16.9
13.2
2O7
5-3
+1.1
2
7
+0.1
I
537
+2.0
1.2
I
12
20.8
19.4
374
4-4
+6-4
8
16
6.0
oT~ 89
Pris'l*
/ J 6
1.4
4.0
O.2
for ii
Ii6
-4.6
H
I
8.0
8.6
+21
15.1
14.8
1 206
Pris'l*
/i5-i
15.0
+1.8
+1.8
+ 1
for ii
1 12
o-3
676
5-0
0.4
+ 1
13
15-4
15.7
224
+5-4
3-7
6
12
1.2
13
+4-o
+7-2
+9
13.2
16.6
203
693
+1-9
+1.8
+47
13-4
12.0
148
+2.0
3-7
i
IO
3-4
32
+ 2.0
4.6
i
8
IO.O
74
184
+5-2
6.6
5
14
12.2
8.0
90
+ 1.2
4.0
i
4
42
16
+6.0
0.8
i
8
8.0
59
45
0.0
2.O
+20
Pris'l*
( 8
1 8
8.0
0.0
+4.2
4.0
O.O
+4.2
i
for 14
I 8
8.0
3
0.0 0.0
Approx. volume + 2622 + 40
Prism'l. Cor. =+12 28
True volume = 2634
28. Rules for Cross Sectioning. In deriving the prismoidal for-
mula, the ground surface of each prismoid was assumed to be a hyper-
bolic paraboloid. Hence if the prismoidal formula, or its equivalent,
is to be used :
* These can usually be avoided in practice by placing the same num-
ber of breaks in adjacent sections as already indicated in $ 20.
24 RULES FOR CROSS SECTIONING. [ 28,
Place cross sections at the regular stations (usually 100 feet apart),
treating the ground as three-level if the surface is very regular ; five-
level if at all broken, as the increased labor in field and office is slight ;
and irregular if more broken. The object being to obtain points
enough in each cross section so that straight lines joining them will
lie in the surface, or will equalize the surface, and to adopt the meth-
od which will secure this at most of the sections without extra breaks,
even if it gives more points than are necessary at some of the more
regular stations.
An occasional extra break can often best be treated by itself, as add-
ing or subtracting a computed amount to or from the regular quantity
which would be obtained without it.
It should also be noted that the formula requires the same number
of sides or points in adjacent cross sections (dividing the solid into tri-
angular prismoids), requiring that when a ridge or hollow runs out its
vanishing point in the more regular section be taken and the section
be made up with the break for the solid on one side and without it for
the solid on the other side of the section. The height of this vanish-
ing point can be computed, but it is usually better to measure it in the
field.
Intermediate cross sections should be added whenever straight lines
joining corresponding points of the regular sections will not lie in the
surface or equalize the surface, and enough should be added to fulfill
this condition. When the center line is curved, the lines joining cor-
responding points of adjacent cross sections should be estimated at the
same curvature in judging of their coincidence with the ground sur-
face.
A cross section should also be placed wherever a grade point occurs
on either edge of the roadbed (usually limited to the edges in cutting)
and one is often placed where a grade point occurs on the center line.
In light work and where the transverse slope is small, a section is
sometimes taken where the center line comes to grade and the other
two sections omitted. See $ 15.
The plus stations are preferably placed at 50 feet, at multiples of 10,
or at whole feet (except grade points) from the last full station, for
convenience of computing grades in the field and quantities in the
office. Breaks on a cross section should be put at multiples of 10, or
at whole feet, from the center, when possible without a sacrifice of ac-
curacy.
When the method by averaging end areas is used, without the pris-
moidal correction, additional sections should be taken whenever the
change in transverse slope or in area between adjacent sections is
great. The reason for this can be seen by referring to (2) ; halving
the distance between sections will halve both factors (c c f } and
(w' w], or quarter the product, so that when the two half length
solids are added together the correction or error will be reduced one-
half. The error due to averaging end areas can thus be made as small
as desirable, but the method if extended far would soon involve more
extra labor than to apply the prismoidal correction.
The same remarks will apply to the method by middle areas.
TABLE
OF
Triangular Prismoids for Railway Earthwork
IN CUBIC YARDS,
BY AVERAGING END AREAS.
(See Prismoidal Correction.)
[Width 1-50: Ht. i-io.J
VOL UME BY A VERA GING
.7
2
3
4
5
6
7
8
9
WIDTH.
1
2
345
678
9 10
i
i
i
i
I
i
2
3
4 5
6
6 7
8| 9
I
i
i
i
i
i
2
2
2
4
6
7 9
ii
13 15
17
19
i
i
i
2
2
2
3
3
3
6
8 ii
14
17
19 22
25
28
i
i
2
2
3
3
3
4
4
7
n
15
19
22
26 30
33
37
i
2
2
3
3
4
4
5
5
9
H
19
23
28 32 37
42
46
2
2
3
3
4
4
5
6
6
n
17
22
28
33
39
44
'50
56
2
3
3
4
5
5
6
7
6
13
19
26
32
39
45
52
58
65
2
3
4
4
5
6
7
8
7
15
22
30
37
44
52
59
67
74
2
3
3
4
5
6
7
8
9
8
17
25
33
42
50
58
67
75
83
2
3
4
5
6
6
7
8
IO
9
28
37
46
56
65
74
83
93
2
3
4
5
6
7
8
9
ii
IO
20
31
41
5I
61
7i
8 1
9 2
IO2
2
3
4
6
7
8
9
IO
12
ii
22
33
44
56
67
78
89
IOO
III
2
4
5
6
7
8
10
n
13
12
24
36
48
60
72
84
96
108
120
3
4
5
6
8
9
10
12
14
13
26
39
52
65
78
91
104
117
130
3
4
6
7
8
10
II
12
15
H
28
42
56
6 9
83
97
in
125
139
I
3
4
6
7
9
10
12
13
16
15
30
44
59
74
89
104
119
133
148
2
3
5
6
8
9
ii
13
17
16
31
47
63
79
94
no
126
142
157
2
3
5
7
8
10
12
13
15
18
17
33
50
67
83
IOO
117
133
150
I6 7
2
4
5
7
9
ii
12
H
16
19
18
35
53
70
88
1 06
123
141
158
I 7 6
2
4
6
7
9
n
13
15
17
20
19
37
56
74
93
in
130
148
167
2
4
6
8
10
12
M
16
18
21
Ig
39
58
78
97
117
136
156
175
194
2
4
6
8
10
12
14
16
18
22
20
61
81
102
122
'43
163
183
2O4
2
4
6
9
ii
13
15
17
19
23
21
43
64
85
1 06
128
149
170
192
213
2
4
7
9
n
13
16
18
20
24
22
44
67
89
III
133
156
178
200
222
2
5
7
9
12
16
19
21
25
2 3
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[Width 51-100: Ht. i-io.]
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VOL UME BY A VER AGING
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0.4
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10
10
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10
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of width.
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(6)
END AREAS.
[Width 51-100: Ht. 21-30.]
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5
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WIDTH.
21
22
23
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26
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[Width 1-50 : Ht. 31-40.]
VOL UME BY A VERA GING
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2394
2444
2495
2546
5
IO
16
21
26
3i
36
4i
47
56
2126
2178
2230
228l
2333
2385
2437
2489
2541
2593
5
5
II
II
16
16
21
21
26
27
32
32
37
38
42
43
48
48
57
58
2l64
2202
2217
2256
2269
2309
2322
2363
2375
2417
2428
2470
2481
2524
2533
2578
2586
2631
2639
2685
5
II
16
22
27
33
38
44
49
59
2240
2294
2349
2404
2458
2513
2568
2622
2677
2731
6
II
17
22
28
33
39
44
50
60
2278
2333
2389
2444
2500
2556
2611
2667
2722
/ \j
2778
6
II
17
23
28
34
40
45
5i
61
2316
2372
2429
2485
2542
2598
2655
2711
2768
2824
6
II
17
23
29
34
40
46
52
62
2354
2411
2469
2526
2583
2641
2698
2756
2813
2870
6
12
1 8
23
29
35
4i
47
53
63
2392
2450
2508
2567
2625
2683
2742
2800
2858
2917
6
12
18
24
3
36
4i
47
53
64
2430
2489
2548
2607
2667
2726
2785
2844
2904
2963
6
12
18
24
30
36
42
48
54
65
2468
2528
2588
2648
2708
2769
2829
2889
2949
3009
6
12
18
24
3i
37
43
49
55
66
2506
2567
2628
2689
2750
2811
2872
2933
2994
3056
6
12
19
25
31
37
43
5o
56
67
2544
2606
2668
2730
2792
2854
2916
2978
3040
3102
6
13
J 9
25
3 1
38
44
50
57
68
2581
2644
2707
2770
2833
2896
2959
3022
3085
3148
6
13
T 9
26
32
3
45
5i
58
69
2619
2683
2747
28ll
2875
2939
3o3
3067
3*3!
3 J 94
6
13
19
26
3 2
39
45
52
58
70
2657
2722
2787
2852
2917
2981
3046
3111
3176
3241
7
!3
20
26
33
39
46
53
59
7i
2695
2761
2827
2893
2958
3024
3090
3156
3221
3287
7
J 3
20
27
33
40
47
53
60
72
2733
2800
2867
2933
3000
3067
3i33
3200
3267
3333
7
14
20
27
34
4r
47
54
61
73
2771
2839
2906
2974
3042
3109
3i77
3244
33 12
338o
7
14
21
27
34
4i
48
55
62
74
2809
2878
2946
3015
3083
3152
3220
3289
3357
3426
7
14
21
28
35
42
49
56
63
75
2847
2917
2 9 86
3056
3125
3i94
3264
3333
3403
3472
7
14
21
28
35
42
49
56
63
76
2885
2956
3026
3096
3167
3237
3307
3378
3448
3519
7
14
21
2 9
36
43
50
57
64
77
2923
2994
3066
3137
3208
3280
335i
3422
3494
3565
7
14
22
29
36
43
5i
58
65
78
2961
3033
3106
3178
3250
3322
3394
3467
3539
3611
7
15
22
2 9
37
44
5i
59
66
79
2999
3072
3145
3219
3292
3365
3438
35ii
3584
3657
7
15
22
30
37
44
52
59
67
80
3037
3 III
3185
3259
3333
3407
348i
3556
3630
3704
8
15
23
30
37
45
53
60
68
81
3075
3150
3225
3300
3375
3450
3525
3600
3675
3750
8
15
23
30
38
46
53
6r
68
82
3 J i3
3^9
3265
3341
3417
3493
3569
3644
3720
379 6
8
15
23
31
38
46
54
61
69
83
3 J 5i
3228
3305
3381
3458
3535
3612
3689
3766
3843
8
16
23
31
39
47
54
62
70
84
3189
3267
3344
3422
35oo
3578
3656
3733
3811
3889
8
16
24
31
39
47
55
63
7i
85
3227
336
3384
3463
3542
3620
3699
3778
3856
3935
8
16
24
32
40
48
56
64
72
86
3265
3344
3424
3504
3583
3663
3743
3822
3902
398i
8
16
24
32
40
48
56
64
73
87
3303
3383
3464
3544
3625
3706
3786
3867
3947
4028
8
16
24
33
4i
49
57
65
73
88
334i
3422
3504
3585
3667
3748
3830
39 11
3993
4074
8
16
25
33
4i
49
58
66
74
89
3379
346i
3544
3626
3708
379 1
3873
3956
4038
4120
8
17
25
33
42
50
58
67
75
90
3417
35oo
3583
3667
3750
3833
3917
4000
4083
4167
8
17
25
34
42
5i
59
67
76
91
3455
3539
3623
3707
3792
3876
3960
4044
4129
4213
9
17
26
34
43
5i
60
68
77
92
3493
3578
3663
3748
3833
3919
4004
4089
4174
4259
9
17
26
34
43
52
60
69
78
93
353 i
3617
3703
3789
3875
3961
4047
4133
4219
4306
9
17
26
35
44
52
61
70
78
94
3569
3656
3743
3830
3917
4004
4091
4178
4265
4352
9
18
26
35
44
53
62
70
79
95
3606
3694
3782
3870
3958
4046
4134
4222
43 10
4398
9
18
27
36
44
53
62
7i
80
96
3644
3733
3822
39"
4000
4089
4178
4267
4356
4444
9
18
27
36
45
54
63
72
81
97
3682
3772
3862
3952
4042
4I3 1
4221
43 11
4401
449 i
9
9
CO CO
27
28
36
37
45
46
54
55
64
64
73
73
82
83
98
99
3720
3758
3811
3850
3902
3942
3993
4033
4083
4125
4174
4217
4265
4308
4356
4400
4446
4492
4537
4583
9
r 9
28
37
46
56
65
74
83
IOO
3796
3889
398i
4074
4167
4259
4352
4444
4537
4630
.7
.2
-3
-4
-5
.6
7
.8
9
41
42
43
44
45
46
47
48
49
50
O.I
0.2
8
8
8
8
8
4
9
4
9
4
9
5
9
5
9
-3
ii
12
12
12
13
13
13
13
14
14
Correction for tenths
0.4
15
16
16
16
17
17
17
18
18
19
0.5
19
19
20
20
21
21
22
22
23
2 3
of width.
0.6
23
23
24
24
25
26
26
27
27
28
0.7
27
27
28
29
29
30
30
31
32
32
0.8
3
3 1
3 2
33
33
34
35
36
36
37
0.9
34
35
36
37
38
38
39
40
41
42
(II)
TABLE
OF
Prismoidal Corrections for Triangular
Prismoids for Railway Earthwork,
IN CUBIC YARDS,
To be applied to the approximate volume found
by averaging end areas.
[w x w 1^50 : h h' i-io.]
PRISMOIDAL
.1
2
j
#
5
<5
7
8
9
w' w.
1
2
3
4
5
6
7
8
9
10,
I
i
i
i
2
2
2
2
3
3
i
2
i
i
2
2
3
4
4
5
6
6
i
i
i
i
3
i
2
3
4
5
6
6
7
8
9
i
i
i
i
4
i
2
4
5
6
7
9
IO
ii
12
I
i
i
i
i
5
2
3
5
6
8
9
ii
12
14
15
i
i
i
i
2
6
2
4
6
7
9
ii
13
15
17
19
i
i
2
2
2
7
2
4
6
9
ii
13
15
17
19
22
i
i
2
2
2
8
2
5
7
10
12
15
17
2O
22
25
i
i
i
2
2
2
2
9
3
6
8
ii
14
17
19
22
25
28
I
i
2
2
2
2
3
10
3
6
9
12
15
19
22
25
28
31
i
2
2
2
3
3
ii
3
7
10
14
17
20
24
27
31
34
i
2
2
3
3
3
12
4
7
ii
15
19
22
26
30
33
37
i
2
2
2
3
3
4
13
4
8
12
16
20
24
28
32
36
40
i
2
2
3
3
3
4
14
4
9
J 3
17
22
26
3
35
39
43
i
2
2
3
3
4-
4
15
5
9
14
19
23
28
32
37
42
46
i
2
2
3
3
4
4
16
5
10
15
20
25
30
35
40
44
49
I
2
2
3
3
4
4
5
17
5
10
16
21
26
31
37
42
47
52
I
2
2
3
3
4
4
5
18
6
ii
17
22
28
33
39
44
50
56
I
2
2
3
4
4
5
19
6
12
18
2 3
29
35
41
47
53
59
I
2
2
3
4
4
5
6
20
6
12
19
25
3 1
37
43
49
56
62
I
2
3
3
4
5
5
6
21
6
13
19
26
32
39
45
52
58
65
I
2
3
3
4
5
5
6
22
7
14
20
27
34
4i
48
54
61
68
I
i
2
^
4
4
5
6
6
23
7
u
21
28
35
43
50
57
64
7i
I
i
2
3
4
4
5
6
7
24
15
22
30
37
44
52
59
67
74
I
2
2
3
4
5
5
6
7
25
8
15
23
31
39
46
54
62
69
77
I
T
2
2
2
3
^
4
5
5"
6
6
6
7
26
27
8
8
16
24
32
40
48
CQ
56
;8
64
6?
72
7c
80
83
I
2
3
2
4
c
6
7
8
W
28
9
^7
26
35
43
J^
52
D u
60
w
69
/.
78
86
I
2
3
A
4
6
7
8
29
9
18
27
36
45
54
63
72
81
90
I
2
4
5
6
6
7
8
30
9
19
28
37
46
56
6 5
74
83
93
I
2
4
5
6
7
8
9
3i
10
19
29
38
48
57
67
77
86
96
I
2
4
5
6
7
8
9
32
10
20
30
40
49
59
6 9
79
89
99
I
2
4
5
6
7
8
9
33
10
20
31
4i
5i
61
71
81
9 2
102
I
2
4
5
6
7
8
9
34
10
21
31
42
52
63
73
84
94
105
I
2
4
c
6
8
9
10
35
ii
22
32
43
54
65
76
86
97
108
I
2
4
6
7
8
9
10
36
ii
22
33
44
56
67
78
89
100
in
I
2
6
7
8
9
10
37
ii
23
34
46
57
69
80
9i
103
114
I
2
3
6
7
8
9
ii
38
12
23
35
47
59
70
82
94
106
117
I
2
4
6
7
8
10
ii
39
12
1L
36
48
60
72
84
96
108
1 20
I
4
6
7
9
10
ii
40
12
25
37
49
62
74
86
99
in
123
I
4
6
8
9
10
ii
4i
13
25
38
5i
63
76
89
IOI
114
127
I
4
6
8
9
10
2
42
J
26
39
52
65
78
9 1
IO/
117
130
I
4
7
8
9
ii
2
43
fl
27
40
53
66
80
93
106
119
133
I
4
7
8
10
n
2
44
L
27
41
54
68
81
95
109
122
136
I
4
6
7
8
10
ii
'
45
L
28
42
56
69
83
97
in
!25
139
I
4
6
7
9
10
ii
,
46
i
28
43
57
7J
85
99
ii/
128
142
I
4
6
7
9
10
12
\
47
I
29
44
58
73
87
102
116
131
H5
I
4
6
7
9
10
12
\
48
:
3
44
59
74
89
IO/
119
133
148
r-
6
8
9
n
12
Iz
49
(
3
45
60
76
9i
1 06
121
136
151
2
c
w
6
8
9
ii
12
\i
50
I
3i
46
62
77
93
108
123
139
154
.7
.^
J
4
5
.6
7
.8
9
1
2
3
4
5
6
7
8
9
10
O.I
0.2
i
]
0.3
i
I
i
I
Correction for tenths
0.4
i
i
I
i
I
0-5
i
i
I
i
2
of width 7 width.
0.6
i
i
i
I
2
2
0.7
i
i
2
2
2
2
0.8
]
i
2
2
2
2
0.9
i
i
i
2
2
2
2
(14)
CORRECTIONS.
[ w /_ w j_ 50: h h x 11-20.]
.t
.2
3
-4
5
.6
7
.8
9
w' w.
11
12
13
14
15
16
17
18
19
20
I
3
4
4
4
5
5
5
6
6
6
i
2
7
7
8
9
9
10
10
ii
12
12
i
i
i
3
10
ii
12
13
14
15
16
18
19
i
i
i
i
4
14
15
16
17
19
20
21
22
23
-7
2 5
i
1
i
i
i
5
17
19
20
22
23
25
26
28
29
31
i
i
i
i
i
2
2
2
2
6
7
20
24
22
26
24
28
26
30
28
32
30
35
31
37
33
39
35
37
43
i
2
2
2
8
27
30
32
35
37
40
42
44
47
i
i
2
2
2
2
9
31
33
36
39
42
44
47
50
53
56
i
2
2
2
2
3
10
34
37
40
43
46
49
52
56
59
62
2
2
2
3
3
ii
37
41
44
48
5I
54
58
61
65
68
2
2
3
3
3
12
41
44
48
52
56
59
63
67
70
74
2
2
2
3
3
4
13
44
48
52
56
60
64
68
72
80
2
2
3
3
3
4
14
48
52
56
60
65
69
73
78
82
86
2
2
3
3
4
4
15
5 1
56
60
65
6 9
74
79
83
88
93
i
2
3
3
3
4
4
16
54
59
64
69
74
79
84
89
94
99
2
2
3
3
4
4
5
17
58
63
68
73
79
84
89
94
100
105
2
2
3
3
4
4
5
18
61
67
72
78
83
89
94
100
106
in
2
2
3
4
4
5
5
19
65
7o
76
82
88
94
100
106
in
117
2
2
3
4
4
5
6
20
68
74
80
86
93
99
105
in
117
123
2
3
3
4
5
5
6
21
71
78
84
9I
97
104
no
117
123
130
2
3
3
4
5
5
6
22
75
81
88
95
102
109
115
122
129
136
2
3
4
4
5
6
6
23
78
85
92
99
106
114
121
128
142
2
3
4
4
5
6
7
24
81
89
96
104
in
119
126
T 33
141
148
2
2
3
4
5
5
6
7
25
85
93
loo
108
116
123
J3 1
139
147
154
I
2
2
3
4
5
6
6
7
26
88
96
104
112
120
128
136
144
152
160
I
2
3
3
4
5
6
7
8
27
92
TOO
108
117
125
133
142
150
158
167
I
2
3
3
4
5
6
7
8
28
95
104
112
121
130
138
147
156
164
173
I
2
3
4
4
5
6
7
8
29
98
107
116
125
134
143
152
161
170
179
I
2
3
4
5
6
6
7
8
30
102
III
1 20
130
139
148
157
167
176
185
I
2
3
4
5
6
7
8
9
31
105
U5
124
134
144
153
163
172
182
191
I
2
3
4
5
6
7
8
9
32
I0 9
119
128
138
148
158
168
178
188
198
I
2
3
4
5
6
7
8
9
33
112
122
132
143
153
163
173
183
194
204
I
2
3
4
5
6
7
8
9
34
115
126
136
147
157
168
178
189
199
210
I
2
3
4
5
6
8
9
10
35
II 9
130
140
162
173
184
194
205
216
I
2
3
4
6
7
8
9
10
36
122
133
144
156
I6 7
178
189
200
211
222
I
2
3
5
6
7
8
9
IO
37
126
137
148
160
171
183
194
206
217
228
I
2
4
5
6
7
8
9
II
38
I2 9
141
152
164
176
188
199
211
223
235
I
2
4
5
6
7
8
10
II
39
I 3 2
144
156
169
181
193
205
217
229
241
I
2
4
5
6
7
9
10
II
40
136
148
160
173
185
198
2IO
222
235
247
I
3
4
5
6
8
9
10
II
41
139
152
165
177
190
202
215
228
24O
253
I
3
4
5
6
8
9
10
12
42
143
156
169
181
194
207
22O
233
246
259
I
'3
4
5
7
8
9
ii
12
43
146
159
173
186
199
212
226
239
252
26 5
I
3
4
5
7
8
10
ii
12
44
149
163
177
190
204
217
231
244
258
272
I
3
4
6
7
8
10
ii
13
45
153
I6 7
181
194
208
222
2 3 6
250
264
2 7 8
I
3
4
6
7
9
10
ii
13
46
156
170
185
199
213
227
241
256
270
284
I
3
4
6
7
9
10
12
13
47
160
174
189
203
218
232
247
26l
2 7 6
290
I
3
4
6
7
9
10
12
13
48
163
I 7 8
193
207
222
237
252
267
28l
2 9 6
2
3
5
6
8
9
II
12
49
166
181
197
212
227
242
257
272
287
302
2
3
5
6
8
9
II
12
14
50
170
185
201
216
231
247
262
2 7 8
293
309
.7
.2
3
.*
5
.(5
7
.8
9
11
12
13
14
15
16
17
18
19
20
O.I
i
i
i
i
0.2
i
i
i
i
i
i
i
i
i
i
0.3
i
i
i
i
i
i
2
2
2
2
Correction for tenths
0.4
i
i
2
2
2
2
2
2
2
2
o-5
2
2
2
2
2
2
3
3
3
3
of width ' width.
0.6
Qfj
2
2
2
3
3
3
3
3
4
4
i
0.8
3
3
3
3
4
4
4
4
5
5
0.9
3
3
4
4
4
4
5
5
5
6
(15)
[ w / w 1-50: h h' 21-30.]
PRISMOIDAL
.1
.2
3
4
5
.6
7
.8
9
w 7 w.
21
22
23
24
25
26
27
28
29
30
I
6
7
7
7
8
8
8
9
9
9
i
2
13
14
14
15
15
16
17
17
18
19
i
i
i
i
3
19
20
21
22
23
24
25
26
27
28
i
i
i
i
4
26
27
28
30
3i
32
33
35
36
37
i
i
1
i
i
5
32
34
35
37
39
. 40
42
43
45
46
i
i
i
i
2
6
39
4i
43
44
46
48
50
52
54
56
i
i
2
2
2
7
45
48
50
52
54
56
58
60
63
65
i
i
2
2
2
8
52
54
57
59
62
64
67
69
72
74
i
2
2
2
2
9
58
61
64
67
69
72
75
78
81
83
I
2
2
2
2
3
10
65
68
7i
74
77
80
83
86
90
93
i
2
2
2
3
3
ii
7i
75
78
81
85
88
92
95
98
102
i
2
2
3
3
3
12
78
81
85
89
93
96
IOO
104
107
III
2
2
2
3
3
4
13
84
88
92
96
IOO
104
108
112
116
120
2
2
3
3
3
4
14
9 1
95
99
104
1 08
112
117
121
125
130
2
2
3
3
4
4
15
97
102
106
in
ii6
120
125
I 3
134
139
i
2
3
3
3
4
4
16
104
109
114
119
123
128
133
138
143
148
I
2
2
3
3
4
4
5
17
no
"5
121
126
J 3!
136
142
H7
152
157
I
2
2
3
3
4
4
5
18
117
122
128
133
139
144
150
156
161
I6 7
I
2
2
3
4
4
5
5
19
123
129
135
141
M7
152
158
164
170
I 7 6
I
2
2
3
4
4
5
6
20
130
136
142
148
154
1 60
167
173
179
185
I
2
3
3
4
5
5
6
21
136
143
149
156
162
I6 9
175
181
1 88
194
I
2
3
3
4
5
5
6
22
H3
149
156
163
170
177
183
190
197
204
I
2
3
4
4
5
6
6
23
149
156
163
170
177
I8 5
192
199
206
213
I
2
3
4
4
5
6
7
24
156
I6 3
170
178
185
193
200
207
215
222
I
2
2
3
4
5
5
6
7
25
162
170
177
185
193
201
208
216
224
2 3 I
I
2
2
3
4
5
6
6
7
26
169
177
185
193
20 1
209
217
225
233
241
I
2
3
3
4
5
6
7
8
27
175
183
192
200
208
217
225
233
242
250
I
2
3
3
4
5
6
7
8
28
181
190
I 99
207
216
225
233
242
251
259
I
2
3
4
4
5
6
7
8
29
188
I 9 7
206
215
224
233
242
251
260
269
I
2
3
4
5
6
6
7
8
30
194
204
2I 3
222
231
241
250
259
269
2 7 8
I
2
3
4
5
6
7
8
9
3i
20 1
2IO
220
230
239
249
258
268
277
287
I
2
3
4
5
6
7
8
9
32
207
217
227
237
247
257
26 7
277
286
2 9 6
I
2
3
4
5
6
7
8
9
33
214
224
234
244
255
265
275
285
295
306
I
2
3
4
5
6
7
8
9
34
220
231
241
252
262
273
283
294
34
315
1
2
3
4
5
6
8
9
10
35
227
238
248
259
270
28l
2 9 2
302
3i3
324
I
2
3
4
6
7
8
9
10
36
233
244
256
267
278
289
300
3H
322
333
I
2
3
5
6
7
8
9
10
37
240
251
263
274
285
297
308
320
33i
343
I
2
4
5
6
7
8
9
II
38
246
258
270
28l
293
305
317
328
340
352
I
2
4
5
6
7
8
10
II
39
253
265
277
28 9
301
313
325
337
349
36i
I
2
4
5
6
7
9
10
II
40
259
272
284
296
309
321
333
346
358
370
I
3
4
5
6
8
9
10
II
4i
266
2 7 8
2 9 I
3^4
3i6
329
342
354
367
380
I
3
4
5
6
8
9
10
12
42
272
285
298
311
324
337
350
363
376
389
I
3
4
5
7
8
9
II
12
43
279
2 9 2
305
319
332
345
358
372
385
398
I
3
4
5
7
8
10
II
12
44
285
299
312
326
340
353
367
380
394
407
I
3
4
6
7
8
10
II
*3
45
29?
306
319
333
347
36i
375
389
403
417
I
3
4
6
7
9
10
II
13
46
298
312
327
34i
355
369
383
398
412
426
I
3
4
6
7
9
10
12
13
47
305
319
334
348
363
377
392
406
421
435
I
3
4
6
7
9
10
12
13
48
3ii
326
34i
356
370
385
400
415
430
444
2
3
5
6
8
9
II
12
H
49
3i8
333
348
363
378
393
408
423
439
454
2
3
5
6
8
9
II
12
14
50
324
340
355
370
386
401
417
432
448
463
.7
.2
3
*
5
.6
7
.8
9
21
22
23
24
25
26
27
28
29
30
O.I
i
i
i
i
i
i
i
i
i| i
0.2
i
i
i
i
2
2
2
2
2
2
0-3
r
2
2
2
2
2
3
3
3
3
Correction for tenths
0.4
c\ c
\
3
3
3
3
3
3
3
4
i
i
of width / width.
Ut o
0.6
i.
4
4
A
5
^
i
5
5
6
0.7
;
5
5
5
5
6
(
6
6
6
0.8
c
5
6
6
e
6
7
7
7
7
0.9
6
6
6
7
7
7
g
8
8
i
(16)
CORRECTIONS.
[ W /_ W 1-50: h h' 31-40.]
.?
4 -5
.5
w' w
31
32
33
34
35
36
37
38
39
40
i
1C
) 10 I
1C
i
i
i
i
i
12
2
H
) 2
2
2
2
2
2
2
2;
25
3
2C
! 3
3
3
3
3
3'
3
3
37
4
3*
* 4
4
43
4
4
4
4
4
49
5
4*
* 4
5
5'
5
5
5
5
6c
6
57
5
6
6;
6
6
6
7
7
7
7
67
6
7
7:
7
7
8
8
/ *
86
8
11
I
8
s^
8
8
9
94
9
99
9
86
8
9
9'
9
IOC
10
106
ii
10
9
9
10
ii
1 1/
ii
J2C
12
ii
10
IOC
ii
HP
ii
12
12
I2C
I3
136
12
13
ii
ii
12
12
13
I2(
I4C
13
144
13
14
14
15
I*
14
13-
13
14
147
15
15
160
164
169
173
15
144
14
15
157
16
16
17
176
18
185
i
16
i 53
15*
163
i6
173
17
18
188
I93
198
2 ;
t
17
163
1 6
173
IT?
184
18
19;
199
20
210
2 ^
i
18
172
178
183
189
194
200
206
21
217
222
2 ^
t
19
182
188
I 9 4
199
205
21
217
223
229
2 35
2 3
'
'
5
1
20
J 9
198
204
2IC
211
22
22
235
241
247
^J
2
^ ^
;
5
6
21
20
207
214
220
227
233
240
246
253
259
b/
i
2
'. i
j
6
1
1
22
23
2IO
22O
217
227
22;
234
231
241
238
248
244
256
25
263
258
270
265
277
III
*
2
; ;
;
6
7
24
230
237
244
2 5 2
259
267
274
28l
289
296
1
I 2
c
6
i
2 5
2*Z<
247
ore
262
O*7 1
285
^
5/-|i
2
3 4
1
6
6
7
26
249
*T/
257
265
273
28l
289
297
305
o*-* 1
313
321
5
I 2
i
3 ^
c
6
*
1
27
258
267
275
283
292
300
308
317
325
333
4-i
I 2
J
3 ^
c
6
7
c
28
268
277
285
294
302
3 11
32O
328
337
346
01
I 2
;
4 ^
c
6
*
8
29
277
286
295
304
313
322
331
340
358
5
I 2
4
6
6
f
8
30
287
2 9 6
306
315
3 2 4
333
343
352
361
370
!
I 2
4
6
7
8
9
3i
297
306
316
3 2 5
335
344
354
364
373
383
1
2
4
4
6
6
7
8
8
9
9
32
33
306
316
326
336
336
346
346
356
356
367
365
377
ill
385
397
395
407
I
2
4
6
7
8
9
34
325
336
346
357
367
378
388
399
409
420
2
4 ,
6
8
9
35
335
346
356
367
378
389
400
410
421
432
y
2
2
* 6
8
9
36
344
356
367
378
389
400
411
422
433
444
o
2
5 6
8
9
o
37
354
365
377
388
400
411
4 2 3
434
445
457
/
4
5 6
8
9
i
38
3 6 4
375
387
399
410
422
434
446
457
469
2
4 ,
5 6
8
i
39
373
385
397
409
421
433
445
457
469
481
2
4 ,
5 6
9
i
40
383
395
407
420
432
444
457
469
481
494
3
4 .
5 6
8
9
o
i
41
392
405
418
430
443
456
468
481
494
506
i
4 ,
> 6
8
9
o
2
42
402
4 I 5
428
441
454
467
480
493
506
519
3
4 .
> 7
8
9
i
2
43
411
425
438
465
478
491
504
3
4 .
5 7
8
o
i
2
44
421
435
448
462
475
489
502
530
543
3
4 <
> 7
8
o
i
3
45
43i
444
458
472
486
500
528
542
556
3
4 <
> 7
9
i
3
46
440
454
469
483
497
5"
525
540
554
568
3
4 <
> 7
9
2
3
47
450
464
479
493
508
522
537
566
580
3
4 <
> 7
9
2
3
48
459
474
489
504
519
533
548
563
578
593
3
5 <
i 8
9
I
2
4
49
469
484
499
5 r 4
529
544
560
575
590
605
5 3
5 *
> 8
9
I
2
4
50
478
494
509
525
540
556
586
602
617
.2
3 -4
.j
<5
7
^
p
31
32
33
34
35
36
37
38
39
40
O.I
0.2
2
2
2
2
2
2
2
2
2
2
Correction for tenths
o 3
3
3
3
3
3
3
3
c
4
c
4
r
4
K
0-5
5
5
5
5
5
6
6
6
6
6
of width'' width.
0.6
6
6
6
6
6
7
7
7
7
7
0.7
7
7
7
7
8
8
8
8
8
9
0.8
8
8
8
8
9
9
9
9
10
10
0.9
9
9
9
9
10
10
10
ii
ii
ii
(17)
[ w / _ w ^50 : h h' 41-50.]
PRISMOIDAL CORRECTIONS.
./
.2
3
-4
5
.6
7
.8
9
w' w.
41
42
43
44
45
46
47
48
49
50
I
13
13
13
14
14
14
15
15
15
15
i
2
25
26
27
27
28
28
29
30
30
31
i
i
i
i
3
38
39
40
41
42
43
44
44
45
46
i
i
i
i
4
51
52
53
54
56
57
58
59
60
62
i
i
1
i
i
5
63
65
66
68
69
7i
73
74
76
77
i
i
i
2
6
76
78
80
81
83
85
87
89
9 1
93
i
2
2
2
7
89
9i
93
95
97
99
102
104
1 06
1 08
i
2
2
2
8
101
104
106
109
in
114
116
119
121
123
2
2
2
2
9
114
117
119
122
125
128
131
133
136
139
2
2
2
2
3
10
127
130
133
136
139
142
145
148
151
154
2
2
2
3
3
ii
139
H3
146
149
153
156
1 60
163
1 66
170
2
2
3
3
3
12
152
156
159
I6 3
167
170
174
178
181
185
2
2
2
3
3
4
13
165
169
173
177
181
185
189
193
197
20 1
2
2
3
3
3
4
14
177
181
186
190
194
199
203
207
212
216
2
2
3
3
4
4
15
190
194
199
2O4
208
213
218
222
227
231
i
2
3
3
3
4
4
16
202
207
212
217
222
227
232
237
242
247
2
2
3
3
4
4
5
17
2I 5
220
226
2 3 I
2 3 6
241
247
252
257
262
2
2
3
3
4
4
5
18
228
233
239
244
250
256
261
267
272
278
2
2
3
4
4
5
5
19
240
246
252
258
264
270
276
28l
287
293
2
2
3
4
4
5
6
20
253
259
265
272
2 7 8
284
290
2 9 6
302
309
2
3
3
4
5
5
6
21
266
272
279
285
2 9 2
298
305
311
318
324
2
3
3
4
5
5
6
22
2 7 8
285
292
299
306
312
319
326
333
340
2
3
4
4
5
6
6
23
291
298
35
312
319
327
334
341
348
355
I
2
3
4
4
5
6
7
24
304
311
3!9
326
333
34i
348
356
363
37o
2
2
3
4
5
5
6
7
25
316
324
332
340
347
355
363
370
378
386
2
2
3
4
5
6
6
7
26
329
337
345
353
36i
369
377
385
393
401
2
3
3
4
5
6
7
8
27
342
350
358
367
375
383
392
400
408
417
2
3
3
4
5
6
7
8
28
354
363
372
380
389
398
406
415
423
432
2
3
4
4
5
6
7
8
29
367
376
385
394
403
412
421
430
439
448
2
3
4
5
6
6
7
8
30
380
389
398
407
417
426
435
444
454
463
2
3
4
5
6
7
8
9
31
392
402
411
421
43i
440
450
459
469
478
2
3
4
5
6
7
8
9
32
405
415
425
435
444
454
464
474
484
494
2
3
4
5
6
7
8
9
33
418
428
438
448
458
469
479
489
499
509
2
3
4
5
6
7
8
9
34
430
441
45i
462
472
483
493
504
5H
525
2
3
4
5
6
8
9
10
35
443
454
465
475
486
497
508
519
529
540
2
3
4
6
7
8
9
10
36
456
467
478
489
500
5ii
522
533
544
556
2
3
5
6
7
8
9
10
37
468
480
491
502
5H
525
537
548
560
57i
I
2
4
5
6
7
8
9
II
38
481
493
504
5i6
528
540
55i
563
575
586
I
2
4
5
6
7
8
10
II
39
494
506
5i8
530
542
554
566
578
590
602
I
2
4
5
6
7
9
10
II
40
506
5i9
53i
543
556
568
580
593
605
617
3
4
5
6
8
9
10
II
4i
519
531
544
557
569
582
595
607
620
633
3
4
5
6
8
9
10
12
42
531
544
557
570
583
596
609
622
635
648
3
4
5
7
8
9
II
12
43
544
557
57i
584
597
610
624
637
650
664
3
4
5
7
8
10
II
12
44
557
570
584
598
611
625
638
652
665
679
3
4
6
7
8
10
II
13
45
569
583
597
611
625
639
653
667
681
694
I
3
4
6
7
9
10
II
T 3
46
582
596
610
625
639
653
667
68 1
696
710
I
3
4
6
7
9
10
12
13
47
595
609
624
638
653
667
682
696
711
725
I
3
4
6
9
10
12
13
48
607
622
637
652
667
681
696
711
726
741
2
3
5
6
8
9
II
12
14
49
620
635
650
665
68 1
696
711
726
74i
756
2
3
5
6
8
9
II
12
14
50
633
648
664
679
694
710
725
741
756
772
./
.2
v?
-4
5
.6
7
.8
-9
41
42
43
44
45
46
47
48
49
50
O.I
i
i
i
i
i
i
i
i
2
2
0.2
3
3
3
3
3
3
3
3
3
3
0.3
4
4
4
4
4
4
4
4
5
5
Correction for tenths
0.4
5
5
5
5
6
6
6
6
6
6
-5
6
6
7
7
7
7
7
7
8
8
of width' width.
0.6
8
8
8
8
8
9
9
9
9
9
0.7
9
9
9
10
10
10
10
10
ii
ii
0.8
10
10
ii
ii
ii
ii
12
12
12
12
0.9
ii
12
12
12
T 3
13
13
13
14
14
(18)
THIS BOOK IS DUE ON THE LAST DATE
STAMPED BELOW
AN INITIAL FINE OF 25 CENTS
WILL BE ASSESSED FOR FAILURE TO RETURN
THIS BOOK ON THE DATE DUE. THE PENALTY
WILL INCREASE TO SO CENTS ON THE FOURTH
DAY AND TO $1.OO ON THE SEVENTH DAY
OVERDUE.
DEC 1 4 HI
SEP 1 o '
LIBRARY, BRANCH OP THE COLLEGE OP AGRICULTURE
5m-9,'35(s)
22769
Crandall
Tables
fcr
and other e
Q 1 4 '41
SEP 1 Q '56
L-.
the
OT222
12.
1893
TFZZZ
C7
1833
LIBRARY, BRANCH OF THE COLLEGE OF AGRICULTURE, DAVIS
UNIVERSITY OF CALIFORNIA