Fremling Field Notebooks 1979-1980

:

;

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w

;

.: = b

.4

.l Y

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y Q

-

0

,1

SLOPE 1 TO 1. ROAI WAT 0r ANY WIDT .2

.3

.4

.5

.6

.7

.8

0

0.00

0

.10

0.20

0.30

0.40

0.50

0.0

0.70

0.80

S

6.00

.10

.20

.30

.40

.50

.60

.70

.80

2.00 0.00 4.00

2.10 3.10 4.10

2.20 3.20 4.20

5.30 3.30 4.30

540 8.40 4.40

.50 3.50 4.50

5.60 3.60 4.60

5.70 3.70 4.70

5.80 3.80 4.80

5.90 3.90 4.90

82

1 13.00 14 14.00 S15 .00 18 18.00

13.10 3.20 14.10 14.20 5.10 5.20 18.10 1.20

3.30

3.40 4.40 5.40 8.40

3.50 4.50 9.50 1.50

3.60 4.60 5.60 .60

3.70 4.70 5.70 1.70

3.80 4.80 5.80 1.80

3.90 4.90 5.90 .90

3 4

11 2

21.00 13.00 19.00 21 122.00 210.00 13 15.00 16 18.00 20 20.00 41 38 8.00 28.00 23 23.00 24 24.00 25 25.00 26 26.00 27 27.00 28 28.00 29 29.00 30 30.00 31 32.00 31.00 32

12.10 10.10 19.10 12.10 21.10 13.10 1.10 420.10 38.10 2100 1.10 22.10 23.10 24.10 25.10 26.10 27.10 28.10 29.10 30.10 31.10 32.10

1.50 13.50 19.50 16.50 14.50

1.60 13.60 19.60 12.60 14.60

1.50

1.60 19.80

1.70 13.70 1.70 12.70 14.70 1.70 19.70 20.70 38.70 21.70 22.70 23.70 24.70 25.70 26.70 27.70 28.70 29.70 50.70 81.70 32.70

1.80 13.80 19.80 12.80 14.80 1.80 19.80 20.80 38.80 21.80 22.80 23.80 24.80 25.80 26.80 27.80 28.80 29.80 60.80 81.80 32.80

1.90 13.90 19.90 12.90 14.90 1.90 1.90 20.90 38.90 21.90 22.90 23.90 24.90 25.90 26.90 27.90 28.90 29.90 30.90 31.90 32.90

33 34

33.00 34.00

36 37 38 39 40 41 42 43 44 45 4 47

36 00 37.00 38.00 39.00 40.00 41.00 42.00 43.00 44.00 45.00 46.00 47.00

33.10 34.10 35 10 36.10 37.10 38.10 39.10 40.10 41.10 42.10 43.10 44.10 45.10 46.10 47.10

37.20 38.20 39.20 40.20 41.20 42.20 43.20 44.20 45.20 46.20 47.20

1.30 1.40 130.30 13.40 1.30 19.40 12.30 14.30 12.40 14.40 18.30 13.40 19.30 19.40 420.30 20.40 38.30 38.40 21.30 21.40 22.30 22.40 23.30 23.40 24.30 24.40 25.30 25.40 26.30 26.40 27.30 27.40 28.30 28.40 29.30 29.40 30.30 30.40 31.30 31.40 32.30 2.40 33.30 34.30 33.40 34.40 35.30 35.40 38.30 36.40 3620 37.30 37.40 38.30 38.40 39.30 39.40 40.30 41.30 40.40 41.40 42.30 42.40 43.30 43.40 44.30 44.40 45.30 45.40 45.30 46.40 47.30 47.40

50

50.00

50.10

50.20

50.30

2 0

1

25i

3.00

.1

1.20 13.20 .20 12.20 14.20 5.20 19.20 20.20 3820 21.20 22.20 23.20 24.20 25.20 26.20 27.20 28.20 29.20 30.20 31.20 20 32 33.20 34 20 3520

4.30

5.30 1.30

50.40

.9 0.0

0

.90 4

1 13 19 14 15 1 20 38 21

19.50 20.50 38.50 21.50 22.50 23.50 24.50 25.50 26.50 27.50 28.50 29.50 30.50 31.50 32.60

20.60 38.60 21.60 22.60 23.60 24.60 25.60 26.60 27.60 28.60 29.60 30.60 31.60 32.60

33.80 34.50 35.50 36.50 37.50 38.50 39.50 40.50 41.50 42.50 43.50 44.50 45.50 46.50 47.50

33.70 33.60 34.60 34.70 3560 dfrn0 35.70i 36.60 36.70 37.60 37.70 38.60 .38.70 39.60 39.70 40.70 40.60 41.60 41.70 42.60 42.70 43.60 43.70 44.60 44.70 45.60 45.70 46.70 47.60 47.70

33.80 34.80 35.80 o 36.80 37.80 38.80 39.80 40.80 41.80 42.80 43.80 44.80 45.80 46.80 47.80

33.90 34.90 t 36.90 37.90 38.90 39.90 40.90 41.90 42.90 43.90 44.90 4590 46 90 47.90

33 34

50.60

50.80

50.90

50

50.50

46.60

50.70

2 24 25 26 27

28

29 30 31 32

36 37 33 39 40 41 3 43 44 45 48 47

of slope stake from side or shoulder stake for any width roadway, elope 1 to 1. If located Is by the double entry method in left gr is nearly level the cut or fill at side stake ol and to row. The number in body of table in same row and column give distance fr om side stake to sope stake. If ground is no level estimate the difference in elevation between the a'ide stake and slone stake, lower target by this amount if cut, elevate if fill. Add this amount to cut r fill and find distance in table. Set up rod at this point, and line of sight should cut target. l it does not make the slight adjustment necessary.

.. L 0o

oo

a

I

,"

SaJw

Lat

2.

2.3I S. '-o 2 IFs

b( 542j 2.S7

7)

1

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r

L

F, tI

OIL

Tr~1

X1It~e

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FOP

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,a

i:

Ba50~acL

1

-ouwJ Please ca( Olr45in2~tf7a.

I

b 0

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r I

h.

IMPROVED

TABLES

AND

INFORMATION

DIRECTIONS FOR USE OF TABLES TABLE No. XIV Distance of slope stake from side or shoulder stake for any width roadway, slope 1 to 1.

If ground is nearly level, the cut or fill at side stake is located by the double entry method in left column and top row. The number in body of table in same row and column gives distance

from side stake to slope stake. If ground is not level estimate the difference in elevation between the side stake and slope stake, lower target by this amount if cut, elevate if fill. Add this amount to cut or fill and find distance in table.

Set up rod at this point, and line of sight should cut target. If it does not make the slight adjustment necessary.

TABLE No. VIII To find Tangent and External for curve of any other degree, divide by degree of curve and add correction found in column of corrections.

Degree of curve with a given I may be found by dividing tangent, (or external). opposite I by given tangent, (or external).

The distance from a point on the tangent to

the curve is very nearly the square of the tangent length divided by twice the radius.

TABLE II TRIGONOMETRIC FORMUL. LA = LMOP LB =LPON R = OB= c 1

=

LOPL

a

sin A-=

-=

a=sosB=LP I b

C sos A= tan

A=

cot A =

sec A = csc A

vers A =

b

=

b= sin B= OL

-

c

I

a

MQ

MQ

b

OM

I

NT

NT

OM OT = ON

I OT 1

= MQ = cotB=MQ

= NT = tan B= NT

= OOQQ csc B = OQ = OT = sec B = OT

= LM = covers B

covers A = OP-LP -

OP

OP-LP

= =

vers B

exsec A = PQ = coexsec B coexsec A = PT = exsec B sin /, A

1-Cos

VIs2

A

sin 2 A= 2 sin A cos A ssin A Law of Lines a

cos /sA =

1fCos

2

cos2A A= cos A-sin' A sin C' sin B =-B .

Law of Cosines c'=a'+b'-2 ab cos C Law of Tangents a =b =tan', (A+) a-b tanu/I (A-B)

A

TABLE II -

Continued

TRIGONOMETRIC FORMULA (continued) in any triangle: Given a, b, C; to find c, B, A.

Use Law of Lines.

Given A, B, c; to find a, b,C.

Use Law of Lines.

Given" a, b,c; to find A, B, C.

a+b+c

(s-a) (s-b)(s-c)

2

s

/s A

cos

r

= f/s (s-a)

tan 1/ A

r.

s-a

tan /s B =

r

s-b

tan 3Is C =

r s-c

Area of a triangle: Area = 'Is ab Sin C Area = 11s (s--a) (s-b) (s-c) PRISMOIDAL FORMULA.

SVol. -

(B+b 4M)

h = altitude; b, B

bases; M= midsection TABLE III

INCHES AND FRACTIONS OF AN INCH IN DECIMALS OF A FOOT 0

1

2

3

4

5

6

s .0052 .0885 .1719 .2552 .3385

.4219 .5052 .0104 .0938 .1771 .2604 .3438 .4271 .5104 .0156 .0990 .123 .2656 .3490 .4323 .5156 1/4.0208 .1042 .1875 .2708 .3542 .4375 .5208 ,/c, .0260 .1094 .1927 .2760 .3594 .4427 .5260 'se .0313 .1146 .1979 .2813 .3646 .4479 .5313 'as .0365 .1198 .2031 .2865 .3698 .4531 .5365 S.0417 .1250 .2083 .2917 .3750 .4583 .5417 'he .0469 .1302 .2135 .2969 .3803 .4635 .5469 '/ .0521 .1354 .2188 .3021 .3854 .4688 .5521 "/ta .0573 .1406 .2240 .3073 .3906 .4740 .5573 a4 .0625 .1458 .2292 .3125 .3958 .47,92 .5625 1/se .0677 ,1510 .2344 .3177 .4010 .4844 .5677 1/s .0729 .1563 .2396 .3229 .4063 .4896 .5729 'a/,e .0781 .1615 .2448 .3281 .4115 .4948 .5781

7

8

9

10

.5885 .5938 .5990 .6042 .6094 .6146 .6198 .6250 .6302 .6354 .6406 .6458 .6510 .6563 .6615

.6719 .6771 .6823 .6875 .6927 .6979 .7031 .7083 .7135 .7188 .7240 .7292 .7344 .7396 .7448

.7552 .7604 .7656 .7708 .7760 .7813 .7865 .717 .7969 .8021 .8073 .8125 .8177 .8229 .8281

.8385 .8438 .849 .8542 .8594 .8646 .8698 .8750 .8802 .8854 .8906 .8958 .9010 .9063 .9115

7

8

9

10

11

1 .06833 .1667 .2500 .3333 .4167 .5000 .5833 .6667 .7500 .8333 .9167 1.000 0

1

2

3

4

5

6

'

.9219 .9271 .9323 .9375' .9427 .9479 .9531 .95831 .9635 /s .9688 .9740 "/s .9792 /s, .9844 i .989 6 j .9948 W'.

11

1

De

0

10' 20 30 40 50

1

TABLE

IV

_ _I_

.

Radiu

34377.5 17188.8 11459.2 8594.42 6875.5

-RAxn,

.036 .073 .109 .145 .182

Offset

ORDINATES AND DmL'arcONs

for

1Foot

.145 .291 .436 .582 .727

0.05' 0.10

40 50

.218 .255 .291 .327 .364 .400

.873 1.018 1.164 1.309 1.454 1.600

0.30 0.35 0.40 0.45 0.50 0.55

10 20 30 40 50

2864.93 2644.58 2455.70 2292.01 2148.79 2022.41

.436 .473 .509 .545 .618

1.745 1.891 2.036 2.181 2.327 2.472

0.60 0.65 0.70 0.75 0.80 0.85

10 20 30 40 50

1910.08 1P.9.57 1719.12 1637.28 1562 88 1494.95

.655 .691 .727 764 .800 836

2.618 2.763 2.908 3.054 3.199 3.345

0.90 0.95 1.00 1.05 1.10 1.15

10 20 30 40 50

1432.69 .873 1375.40 .909 1322.53 .945 1273.57 .982 1228.11 1.018 1185.78 1.055

3.490 3.635 3.718 3.926 4.071 4.217

1.20 1.25 1.30 1.35 1.40 1.45

10 20 30 40 50

1146.28 1100.33 1074.68 1042.14 1011.51 982.64

1.001 1.127 1.164 1.200 1.237 1.273

4.362 4.507 4.653 4.798 4.943 5.088

1.50 1.55 1.60 1.65 1.70 1.75

10 20 30 40

955.37 '929.57 905.13 881.95 859.92

1.309 1.346 1.382 1.418 1.455

5.234 5.379 5.524 5.669 5.814

1.80 1.85 1.90 1.95 2.00

30

2

.582

Radius

Ord.

Offset

819.02 781.84 764.49 747.89

1.528 1.600 1.637 1.673

6.105 8.395 6.540 6.685

2.10' 2.20 2.25 2.30

8

716.78 1.746 688.16 1.819 6874.69 1.855 661.74 1.892

6.976 7.266 7.411 7.556

2.40 2.50 2.55 2.60

9

637.28 614.56 603.80 593.42

1.965 2.037 2.074 2.110

7.846 8.136 8.281 8.426

2.70 2.80 2.85 2.90

10

573.69 546.44 521.67 499.06 478.34 459.28 441.68 425.40 410.28 396.20

2.183 2.292 2.402 2.511.

2.620 2.730 2.839 2.949 3.058 3.168

8.716 9.150 9,585 10.02 10.45 10.89 11.32 11.75 12.18 12.62

3.00 3.15 3.30 3.45 3.60 3.75 3.90

383.07 370.78 359.27 348.45 338.27 319.62 302.94

3.277 13.05 3.387 13.49 3.496 13.92 3.606 14.35 3.716 14.78 3.935 15.64 4.1565 18.51

4.50 4.66 4.80 4.95 5.10 5.40 5.70

287.94 274.37 262.04 250.79 240.49

4.374 4.594 4.814 5.035 5.255

17.37 18.22 19.08 19.94 20.79

6,00 6.30 6.60 6.90 7.20

231.01 222.27 214.18 206.68 199.70 193.18

5.476 5.697 5.918 6.139 6.360 6.583

21.64 22.50 23.35 24.19 25.04 25.88

7,50 7.80 8.10 8.40 8.70 9.00

0.20 0.25

5729.65 4911.15 4297.28 3819.83 3437.87 3125.36

10 20

De.

11 12

S

18 14 18

4

6

16 1?

18 19

1 a2

2s

Note. Chord Deflection

28 9 80

=2 times tangent deflection.

for

1Foot

4.05

4.20 4.35

(1) c =

V TABLE V CURVE FORMUJL FOR SIMPLE CURVES COMPILED BY J. CALVIN LOCKE, C. E. ,2a (2) c =-}fa'+b

(3) c= 2R (R - f(R+b) (R-b) = V2R (R-R- b (4) c = 2 m (2R-m) (6) c = 2T cos l1

(5) c = 2Rsin /lI (7) e= R exsec l/ I

(8) e = R tan s/t Iann /I (2R--a)

(9)e = T tan x/I

(10) ba

(11) b=

-e

c

(12) b== R sin I a b (14) R= a'+b'

c

2a

(13) b= a cot '/, I 2 (15) R = - d

c*+4m' m

2m

2a

(16) d = VR (2R-/(2R+c) 2R-c) = VR (2R(17) d =

(18) d = 2R sin 1/4 I

2f

(20) m=RT

( +-)

(21) m = R vers (24) a=

/2 1

R-)==

8m 4R' -c2

(19) m =

R

R'-

--

(22) m= R sin /2i1 tan '/4 I (23) m = 1/c tan '141 2 (25) a = R- f(R+b) (R-b)= R- 1R' -b'

26) a= 2 R (sin '1 21)2 (27) a = R vers 1 (28 a = R sin Itant'I (29) a==btan '/I (30) a= Tsin 1 (31) T= Rtan1 /,I

(32) 1-

X 57.295780

(33) R =

-

(34) L = IR X 0.01745329 (35) L = s LR-Rb LR-R sin I 2 2

X 57.295780 8d.

TABLE VI

SINES, COSINES, TANGENTS, COTANGENTS

4

sin 0'

tan 0'

sin 10'

tan sin 10' 20'

tan sin 20' 30'

tan sin 30' 40'

tan sin 40' 50'

tan 50'

0 0000 I 175 2 349 3 523 4 698 5 872 6 1045 7 219 8 392 9 564

0000 0029 075 0204 349 378 524 552 699 727 875 901 1051 1074 228 248 405 421 584 593

0029 0058 02040233 378 407 553 581 729 756 904 929 1080 1103 257 279 435 449 614 622

0058 0087 0233 0262 407 436 582 610 758 785 934 958 1110 1132 287 305 465 478 644 650

0087 0116 262 291 437 465 612 640 787 814 963 987 1139 1161 317 334 49550 673 679

0116 0145 291 320 466 494 641 669 816 843 992 1016 119 190 346363 524 536 703 708

0145 89 32088 49587 67086 846 85 1022 84" 198 83 376 82 55481 73380

10 736 11 908 12 2079 13 250 14 419

763 765 944 937 2126 2106 309 278 493 447

793 794 974 965 2156 2136 339 306 524 476

823 822 2004 994 186 2164 370 334 555 504

853 851 2035 2022 217 193 401 363 586 532

883 880 2065 2051 247 221 432 391 617 560

91479 2Q95 78 278 77 46276 64875

16 756 17 924 183090 19 256

867 784 3057 952 249 3118 443 283

899 812 3089 939 281 3145 476 311

931 840 3121 3007 314 173 508 338

962 868 3153 3035 346 201 541 365

994 896 3185 3062 378 228 574 393

3026 73 217 72 411 71 60770

20 420 21 584 22 746 23 907 24 4067 25 226 26 384 27 540 28 695 29 848

640 448 839 611 4040 773 245 934 452 4094 663 253 877 410 5095 566 317 720 543 874

673 475 872 638 4074 800 279 961 487 4120 699 279 913 436 5132 592 354 746 581 899

706 502 906 665 4108 827 314 987 522 4147 734 305 950 4462 5169 617 392 772 619 924

739 529 939 692 4142 854 348 4014 557 173 770 331 986 488 5206 643 430 797 658 950

772 557 973 719 4176 881 383 4041 592 200 806 358 5022 514 243 669 467 823 696 975

805 69 4006 68 210 67 417 66 628 65 841 64 5059 63 280 62 50561 73560

30 5000 31 150 32 299 33 446 34 592 35 736 36 878 37 6018 8 157 39 293

774 5025 6009 175 249 324 494 471 745 016 7002 760 265 901 536 6041 813 180 8098 316

5812 5050 6048 200 289 348 536 495 787 640 7046 783 310 925 581 6065 860 202 8146 338

851 5075 6088 225 330 5373 577 519 830 664 7089 807 355 948 627 6088 907 225 8195 361

890 5100 6128 250 371 398 619 544 873 688 7133 831 400 972 673 6111 954 248 8243 383

930 5125 6168 275 412 422 661 568 916 712 7177 854 445 995 720 6134 8002 271 292 406

969 59 6208 58 453 3 70356 95955 7221 54 490 53 766 52 805051 34250

V

b

491 494 54 517 591 539 642 49 40428 391 450 441 472 95248 744 604 796 626 847 648 899 670 41 561 693 583 9217 799 9271 47 9110 756 9163 777 9057 734 42 691 9004 713 43 820 325 841 380 862 435 884 490 905 545 926 601 46 713 988 770 7009 827 7030 884 7050 942 45 44 947 657 967 45 7071 1.0000 7092 1.0058 7112 1.0117 133 1.0176 153 1.0235 173 1.0295 44

4

S60' cos

60' cot

50' cos I'

50' cot

40' cos

40' cot

30' cos

30' cot

20' cos

20' cot

10' cos

10' cot

TABLE VI (continued)

SINES, COSINES TANGENTS, COTANGENTS (continued)

0

Ssin

0 0'

tan 0'

sin 10'

tan sin 10' 20'

tan sin 20' 30'

tan sin 30' 40'

tan 40'

sin 50

tan 50'

46 7193 1.0355 7214 1.0416 7234 1.0477 7254 1.0533 7274 1.0599 7294 1.0661 43 47 48 49

314 431 547

.0724 .1106 .1504

333 451 566

.0786 .1171 .1571

353 470 585

.0850 .1237 .1640

373 490 604

.0913 .1303 .1708

392 509 623

.0977 .1369 .1778

412 528 642

.1041 42 .1436 41 .1847 40

50 660 1.1918 7679 1.1988 7698 1.2059 7716 1.2131 7735 1.2203 7753 1.276 39 51 771 .2349 790 .2423 808 .249 826 .2572 844 .2647 862 .2123 38 52 880 .2799 898 .2876 916 .2954 934 .3032 951 .3111 969 .3190 37 53 986 .3270 8004 .3351 8021 .3452 8039 .3514 8056 .3597 8073 .3680 36 54 8090 .3764 107 .3848 124 .3934 141 .4019 158 .4106 175 .4193 35 55 192 .4281 208 .4370 225 .4460 241 .4550 258 .4641 274 .4733 34 56 290 .4826 307 .4919 323 .5013 339 .5108 355 .5204 371 .5301 33 57 387 .5399 403 .5497 418 .5597 434 .5697 450 .5798 465 .5900 32 58 480 .6003 4% .6101 511 .6212 526 .6319 542 .6426 557 .6534 31 59 572 .0643 587 .6753 601 .6864 616 .6977 631 .7090 646 .7205 30

60

8732 1.7917 29 816 .8676 28 897 .9486 27 975 2.0353 26 9051 ,1283 25 124 .2286 24 194 .3369 23 261 .4545 22 325 .5826 21 387 .7228 20

70 71 72 73 74 75 76 77 78 79

397 2.7475 9407 2.7725 9417 2.7980 9426 2.8239 9436 2.8502 9446 2.8770 19 455 .9042 465 .9319 474 .9600 483 .9887 492 3.0178 502 3.0475 18 511 3.0777 520 3.1084 528 3.1397. 537 3.1716 546 .2041 555 .2371 17 563 .2709 572 .3052 580 .3402 588 .3759 596 .4124 605 .4495 16 613 .4874 621 .5261 628 .5656 636 .6059 644 .6470 652 .6891 15 659 .7321 667 .7760 674 .8208 681 .8657 689 .9136 696 9617 14 703 4.0108 710 4.0611 717 4.1126 724 4.1653 730 4.2193 737 4.2747 13 744 .3315 750 .3897 757 .4494 763 .5107 769 .5736 775 .6382 12 781 .7046 787 .7729 793 .8430 799 .9152 805 .9894 811 5.0658 11 816 .1446 822 5.2257 827 5.3093 833 5.3955 838 5.4845 843 .5764 10

660 1.7321 8675 1.7437 8689 1.7556 8704 1.7675 8718 1.7797 61 746 8040 760 .8165 774 .8291 788 .8418 802 .8546 62 829 .8807 843 .8940 857 .9074 870 .9210 884 .9347 63 910 .%26 923 .9768 936 .9912 949 2.0057 %2 2.0204 64 988 2.0503 9001 2.0655 9013 2.0809 9026 .0965 9038 .1123 65 9063 .1445 075 .1609 088 .1775 100 .1943 112 .2113 66 135 .2400 147 .2637 159 .2817 171 .2998 182 .3183 67 205 .3559 216 .3750 228 .3945 239 .4142 250 .4342 68 272 .4751 283 .4960 293 .5172 304 .5386 315 .5605 69 336 .6051 346 .6279 356 .6511 367 .6746 377 .6985

80 9848 5.6713 9853 5.7694 9858 5.8708 9863 81 877 6.3138 881 6.4348 886 6.5606 890 82 903 7.1154 907 7.2687 911 7.4287 914 83 925 8.1443 929 8.3450 932 8.5555 936 84 945 9.5144 948 9.7882 51 10.078 954 85 962 11.430 964 11.826 7 12.250 969 86 976 14.300 978 14.924 980 15.605 981 87 986 19.081 988 20.206 989 21.470 990 88 994 28.636 9995 31.242 999 34.368 997 89 9998 57.290 9999 68.750 9999 85.940 9999

60

cos

60' 50' cot

cos

50' 40' cot

cos

40' 30' cot

cos

5.9758 9868 6.0844 9872 6.6912 894 .8209 899 7.5958 918 7.7704 932 8.7769 939 9.0098 942 10.385 957 10.711 959 12.706 971 13.197 974 16.350 983 17.169 985 22.903 992 24.542 993 38.189 997 42.964 9998 114.58 1.000 171.88 1.000

30'

cot

20'

cos

20'

cot

10'

cos

6.1970 9 .9682 8 7.9530 7 9.2553 6 11.059 5 13.727 4 18.075 3 26.432 2 49.104 1 343.77 0

10'

cot

.

TABLE

VII Radius

Deflections for Sub Chords for Short Radius Curves.

M sub chord = sin of idef. angle 20 Ft.

25Ft.

1 59' 206'

17' ° ' 2 25 2 33'

2 58' 3 I '

2 13'

2 41'

3 33

2020'

2049' 2 57' 3 05'

3 44 3 55 4 07'

343' 3 58' 4 2' 4 26' 4'

3

4

sin. def.ang.

12.5 Ft.

3o

193.18

1051

32

181.39 171.o01

Curve °

34

161.8o 153.58

36

38° 40o 42' 44

46

48o

o

°

52°

54 56* 580 6o0°

I

146.19 139.52 133.47

127.97 122.92

100.00o

T

R tan

T"

.Sin. Sin. I D

2 27 ° 2 34' 2 41' 20 48r 2 55'

3022' 3029' ° ' 3 35

°

13,

3 211

3°° 21'

R D = 5o tan I

18'

4 29'

for

ft.

101.15

101.33 101.48 o1.66

101.85

4 54 o8'

102.29

5o 22'

102.53

5 36'

102.76

2.6

30 29' 30 38' 3 46'

4 40'

5 50'

103.00

4

1o3.24

3 54'

5

6004 6 17' 31

4 02' 4 o10' 4018'

52'

6

5°34 5 44'

6° 57 70I

104.14 104.43

50 E= R ex. sec E = T tan I I

5I'

52'

R = T cot. a I D R =Sin

I

Sin. tD = Sin.

2

3002' 118.31 14.o6 30 09' ° 16' I10.11o.I 3 Io6.5o 1o3.14

15 Ft.

'

3

103.54 1o3.84 104.72

Chord de.chordP R I

No.chords = Tan.def.

I

D chord def.

The square of any distance, divided by twice the radius, will equal the distance from tangent to curve, very nearly.

To find angle for a given distance and deflection.

Rule I. Multiply the given distance by .01745 (def. for to for z ft. see Table II.), and divide given deflection by the product.

Rule 2. Multiply given deflection by 57.3, and divide the productby the given distance.

To find deflection for a given angle and distance. Multiply the angle by .01745, and the product by the distance.

GENERAL DATA RIGHT ANGLE TRIANGLES. Square the altitude, divide by twice-the base. Add quotient to base for hypotenuse.

Given Base loo, Alt. Io.Io2 2oo=.5. 00-oo.5=100.5 hyp.

Given Hyp. Too,Alt. 25.252+200=3.125. 100-3.125

96.875=Base.

Error in first example, .oo2; in last, .o45.

To find Tons of Rail in one mile of track: multiply weight per yard

by II, and divide by 7.

LEVELING. The correction for curvature and refraction, in feet and decimals of feet is equal to 0.574d, where ° d is the distance in miles.

The correction for curvature alone is closely, id2. The combined correction is negative.

PROBABLE ERROR. If d,, d3, do,etc. are the discrepancies of various results from the mean, and if Id2=the sum of the squares of thesedifferences and n=the number of observations, then the probable error of the mean=- 0.6745 n (n-1)

TANOENTS AND EXTERNALS TO A 1 CURVE.

TABLE VIII ST 10

50.00 58.34 66.67 75.01 83.34 91.68 100.01 10' 108.35 20' 116.68 30' 125.02 40' 133.36 50' 141.70 150.04 So 10' 158.38 20' 166.72 30' 175.06 40' 183.40 50' 191.74 200.08 40 10' 208.43 20' 216.77 30' 225.12 40' 233.47 50' 241.81 250.16 50 10' 258.51 20' 266.86 30' 275.21 40' 283.57 50' 291.92 300.28 go 10' 308.64 316.99 20' 30' 325.35 333.71 40' 50' 342.08 350.44 70 10' ]358.81 To 20' 367.17 30' 375.54 383.91 40' 50' 392.28 400.66 s 10' 409.03 417.41 20' 30'i 425.79 40' 434.17 50' 442.55 450.93 10' 459.32 20' 467.71 30' 476.10 40' 484.49 50' 492.88 501.28 10o 10' 509.68 20' 518.08 30: 526.48 534.89 50 543.29 20' 30' 40'.

E

=10

~~;'---"

no .218 10' .297 20' .388 30' .491 5o C. T 40' .606 50' .733 .873 .001, 120o 10, 1.024 20' 1.188 30' 1.364 40' 1.552 1.752 50' 1.964 100 C. 18o 10' 2.188 T 2.425 .06 20' 2.674 30' E 2.934 .003 40' 50' 3.297 140o 3.492 10' 3.790 20' 4.099 4.421 30' 40' 4.755 5.100 5Io C. 50' 150 5.459 .09 T 5.829 10' E 6.2111 .004 20' 6.606 30' 7.013 40' 7.432 50' 7.863 16o 8.307 10' 8.762 20' 9.230 20 C. 30' 9.710 40' T 10.202 .13 50' E 10.701 170 11.224 10' 11.753 20' 12.294 30' 12.847 40' 13.413 50' 13.991 180 14.582 25 C,. 10' 15.184 T 20' 15.799 .16 30' 16.426 40' E 17.065 .007 50' 17.717 18.381 10' 19.058 20' 19.746 30' 20.447 40' 21.161 300 C 50' T 21.887 200 22.624 .19 10' E 23.375 20' 24.138 .008 30' 24.913 40' 25.700 50'

T=

Rtan

't

200

T

E

551.70 560.11 568.53 576.95 585.36 593.79 602.21 610.64 619.07 627.50 635.93 644.37 652.81 661.25 669.70 678.15 686.60 695.06 703.51 711.97 720.44 728.90 737.37 745.85 754.32 762.80 771.29 779.77 788.26 796.75 805.25 813.75 822.25 830.76 839.27 847.78 856.30 864.82 873.35 881.88 890.41 898.95 907.49 916.03 924.58 933.13 941.69 950.25 958.81 967.38 975.96 984.53 993.12 1001.7 1010.3 1018.9 1027.5 1036.1 1044.7 1053.3

26.500 27.313 28.137 28.974 29.824 30.686 31.561 32.447 33.347 34.259 35.183 36.120 37.070 38.031 39.006 39.993 40.992 42.004 43.029 44.066 45.116 46.178 47.253 48.341 49.441 50.554 51.679 52.818 53.969 55.132 56.309 57.498 58.699 59.914 61.141 62.381 63.634 64.900 66.178 67.470 68.774 70.091 71.421 72.764 74.119 75.488 76.869 78.264 79.671 81.092 82.525 83.972 85.431 86.904 88.389 89.888 91.399 92.924 94.462 96.013

I

210 10' 20' 30' T 40' .06 50' E -006 22o0 10' 20' 30' 40' 50' 100 C. 23s 10' T 20' .13 30' E011 40' .011 50' 240 10' 20' 30' 40' 150 C. 50' T 250 .19 10' E1 20' .017 30' 40' 50' o 26 10' 20' 200 C. T

+ so C.

.26 E

.022

250 C

T .32 E .028

300 C.

T .39 E .034

T

E

I'

=300

1061.9 97.577 1070.6 99.155 1079.2 100.75 1087.8 102.35 50 C. T 1096.4 103.97 1.105.1 105.60 .10 E 1113.7 107.24 1122.4 108.90 1131.0 110.57 1139.7 112.25 1148.4 113.95 1157.0 115.66 .01325 1165.7 117.38 100 C. 1174.4 119.12 1183.1 120.87 1191.8122.63 1200.5 124.41 1209.2 126.20 1217.9 128.00 T 1226.6 129.82 1235.3 131.65 E 1244.0 133.50 1252.8 135.35 1261.5 137.23 1270.2 139.11 1279.0 141.01 1287.7 142.93 .038 1296.5 144.85 1305.3 146.79 1314.0 148.75 1322.8 150.71 1331.6 *152.69 1340.4 154.69 1349.2 156.70 1358.0 158.72 1366.8 160.76 o 1375.6 162.81 27 0 10' 1384.4 164.86 20' 1393.2 166.95 30' 1402.0 169.04 40' 1410.9 171.15 50' 1419.7 173.27 1428.6 175.41 280 10' 1437.4 177.55 250 C. T 20' 1446.3 179.72 30' 1455.1 181.89 .49 184.08 E 40' 1464.0 50' 1472.9 186.29 .065 188.51 1481.8 290 10' 1490.7 190.74 20' 1499.6 192.99 30' 1508.5 195.25 40' 1517.4 197.53 50' 1526.3 199.82 300 C o T 30 0 1535.3 202.12 .59 10' 1544.2 204.44 E 20' 1553.1 206.77 30' 1562.1 209.12 .018 40' 1571.0 211.48 50' 1580.0 213.86

E =R exsec12

TABLE VIII TANGENTS AND EXTERNALS TO A 10 CURVE

i 310

10' 20' 30' 40" 50' 3gg 10' 20' 30' 40' 50' $So 10' 20' 30' 40' 50' 34o 10' 20' 30' 40' 50'

850

10' 20' 30' 0' 50' 360 10' 20' 30'

40'

10' 30' 40' 50' 880 10' 20' 30' 40' 50' 390 10' 20' 30' 40' 50' 400 10' 20' 30' 40' 50'

T 1589.0 1598.0 1606.9 1615.9 1624.9 1633.9 1643.0 1652.0 1661.0 1670.0 1679.1 1688.1 1697.2 1706,3 1715.3 1724.4 1733.5 1742.6 1751.7 1760.8 1770.0 1779.1 1788.2 1797.4 1806.6 1815.7 1824.9 1834.1 1843.3

1852.5 1861.7 1870.9 1880.1 1889.4 1898.6 1907.9 1917.1 1926.4 1935.7 1945.0 1954.3 1963.6 1972.9 1982.2 1991.5 2000.9 2010.2 .2019.6 2029.0 2038.4 2047.8 2057.2 2066.6 2076.0 2085.4 2094.9 2104.3 2113.8 2123.3 2132.7

E

T

I=40

216.3 218.7 221.1 SoC. 223.5 T 226.0 .13 E 228.4 230.9 .023 233.4 235.9 238.4 241.0 243.5 246.1 100 C. 248.7 T. 251.3 .26 253.9 E 256.5 .046 259.1 261.8 264.5 267.2 269.9 272.6 275.3 15o C. 278.1 280.8 .40 E 283.6 .070 286.4 289.2 292.0 294.9 297.7 300.6 303.5 20' C. 306.4 T 305.3 .53 E 312.2 315.2 .093 318.1 321.1 324.1 327.1' 330.2 333.2 250 C, 336.3 T 339.3 .67 342.4 345.5 .117 348.6 351.8 354.9 358.1 361.3 364.5 367.7 300 C. 'F 371.0 .80 374.2 E 377.5 380.8 .141 384.1

41 o 10' 20' 30' 40'

50' 42' 10' 20' 30' 40"

50'

4,' 10' 20' 30' 40' 50' 440o

10' 20' 30' 40' 50' 45' 10' 40' 50' 460 10' 20' 30' 40' 50' 47o 10' 20' 30' 40' 50' 480 10' 20' 30' 40' 50' 490

10' 20' 30' 40' 50' 500 10' 20' 30' 40' 50'

1 T ='R tan1121

2142.2 2151.7 2161.2 2170.8 2180.3 2189.9 2199.4 2209.0 2218.6 2228.1 2237.7 2247.3 2257.0 2266.6 2276.2 2285.9 2295.6 2305.2 2314.9 2324.6 2334.3 2344.1 2353.8 2363.5 2373.3 2383.1 2392.8 2402.6 2412.4 2422.3 2432.1 2441.9 2451.8 2461.7 2471.5 2481.4 2491.3 2501.2 2511.2 2511.1 2531.1 2541.0 2551.0 2561.0 2571.0 2581.0 2591.0 2601.1 2611.2 2621.2 2631.3 2641.4 2651.5 2661.6 2671.8 2681.9 2692.1 2702.3 2712.5 2722.7

R

T

1=5

387.4 390.7 394.1 397.4 T 400.8 .17 404.2 E 407.6 *13 411.1 414.5 418.0 421.4 425.0 428.5 100 C 432.0 435.6 .34 439.2 E 442.8 .075 446.4 450.0 453.6 457.3 461.0 464.6 468.4 150 C 472.1 T .51 475.8 E 479.6 .116 483.4 487.2 491.0 494.8 498.7 502.5 506.4 200 C 510.3 514.3 .68 E 518.2 .151 522.2 526.1 530.1 534.2 538.2 542.2: 546.3 250 C. 550.4 554.5 .85 558.6 E 562.8 .189 566.9 571.1 575.3 579.5 583.8 588.0 592,3 30T C. 596.6 1.02 600.9 E 605.3 .227 609.6 614.0

E

=(o

°

2732.9 618.4 2743.1 622.8 2753.4 627.2 2763.7 631.7 2773.9 636.2 2784.2 640.7 2794.5 645.2 10' 2804.9 649.7 2815.2 654.3 20' 658.8 30' 2825.6 663.4 40' 2835.9 668.0 50' 2846.3 2856.7 672.7 530 677.3 10' 2867.1 682.0 20' 2877.5 686.7 30' 2888.0 691.4 40' 2898.4 696.1 50' 2908.9 700.9 540o 2919.4 10' 2929.9 705.7 20, 2940.4 710.5 715.3 30' 2951.0 2961.5 720.1 2972.1 725.0 2982.7 729.9 10' 2993.3 734.8 20' 3003.9 739.7 30' 3014.5 744.6 40' 3025.2 749.6 50' 3035.8 754.6 3046.5 759.6 860 10' 3057.2 764.6 20' 3067.9 769.7 30 3078.7 774.7 40' 3089.4 779.8 50' 3100.2 784.9 3110.9 790.1 570 10' 3121.7 795.2 20' 3132.6 800.4 30' 3143.4 805.6 40' 3154.2 810.9 50' 3165.1 816.1 3176,0 821.4 580 10' 3186.9 826.7 20' 3197.8 832.0 30' 3208.8 837.3 40' 3219.7 842.7 50' 3230.7 848.1 853.5 880 3241.7 10' 3252.7 858.9 20' 3263.7 864.3 30' 3274.8 869.8 40' 3285.8 875.3 880.8 50' 3296.9 600 3308.0 886.4 892.0 10' 3319.1 20' 3330,3 897.5 30 3341.4 903.2 908.8 40' 3352.6 914.5 50' 3363.8 51 10' 20' 30' 40' 50'

E= Rexsec ll1

T .21 E .056

100 C. T .42

E .112

15 c.

T

.63 E .168

200 C. T .81 E .225

250 C. T 1.05 E 2.83

300

C.

1.27

.340

TABLE VIII TANGENTS AND EXTERNALS TO A 10 CURVE. I

10' 20' 30' 40' 50' 620 10' 20' 30' 40' 50' 630

10' 20' 30' 40' 50'

640o 10' 20' 30' 40' 50'

$50

10' 20' 30' 40' 50' 660 10' 20' 30' 40'

50' 10' 20' 30' 40' 50' 10' 20' 30' 40' 50' 10' 20' 30' 40' 50' 700 10' 20' 30' 40' 50'

--

T 3375.0 3386.3 3397.5 3408.8 3420.1 3431.4 3442.7 3454.1 3465.4 3476.8 3488.3 3499.7 3511.1 3522.6 3534.1 3545.6 3557.2 3568.7 3580.3 3591,9 3603.5 3615.1 3626.8 3638.5 3650.2 3661.9 3073.7 3685.4 3697.2 3709.0 3720.9 3732.7 3744.6 3756.5 3768.5 3780.4 3792.4 3804.4 3816.4 3828.4 3840.5 3852.6 3864.7 3876.8 3889.0 3901.2 3913.4 3925.6 3937.9 3950.2 3962.5 3974.8 3987.2 3999.5 4011.9 4024.4 4036.8 4049.3 4061.8 4074.4

--

E

I=700

1

920.2 71o 10' 925.9 931.6 20' + 937.3 Soc.* 30' T 943.1 40' .25 50' 948.9 E 954.8 720 960.6 10' 966.5 20' 972.4 30' 978.3 40' 984.3 50' 990.2 730 996.2 10' 1002.3 20' .5i 30' 1008.3 E 1014.4 40' 159 1020.5 50' 1026.6 748 1032.'8 10' 1039.0 20' 1045.2 30' 1051.4 40' ° 1057.7 15 C, 50' T' 75o 1063.9 .76 1070.2 10' E 1076.6 .240 20 1082.9 30" 1089.3 40' 1095.7 50' 1102.2 760 10' 1108.6 20' 1115.1 1121.7200 C 30' 40' 1128.2 T 1134.8 1.02 50' E 1141.4 .321 10' 1148.0 1154.7 1161.3 30' 40' 1168.1 1174.8 50' 1181.6 780 1188.4 250 C, 10' T 1195.2 20' 1202.0 1.28 30' E 40' 1208.9 50' 1215.8 .403 1222.7 10' 1229.7 20' 1236.7 30' 1243.7 1250.8 40' 50' 1257.9 30 o C T 1265.0 800 10' 1272.1 1.54 E 1279.3 20' 1286.5 .485 40' 1293.6 1300.9 50' 11~ R tan1/!1 E=

T 4086.9 4099.5 4112.1 4124.8 1137.4 4150.1 4162.8 4175.6 4188.5 4201.2 4214.0 4226.8 4239.7 4252.6 4265.6 4278.5 4291.5 4304.6 4317.6 4330.7 4343.8 4356.9 4370.1 4383.3 4396.5 4409.8 4423.1 4436.4 4449.7 4463.1 4476.5 4489.9 4503.4 4516.9 4530.4 4544.0 4557.6 4571.2 4584.8 4598.5 4612.2 4626.0 4639.8 4653.6 4667.4 4681.3 4695.2 4709.2 4723.2 4737.2 4751.2 4765.3 4779.4 4793.6 4807.7 4822.0 4836.2 4850.5 4864.8 4879.2

E

I

I=80

1308.2 1315.6 1322.9 + 1330.3 T 1337.7 .30 1345.1 E 1352.6 .110 1360.1 1367.6 1375.2 1382.8 1390.4 1398.0 10o C. 7O 1405.7 T 1413.5 .61 1421.2 E 1429.0 .220 1436.8 1444.6 1452.5 1460.4 1468.4 1476.4 1484.4 150 C. T 1492.4 .91 1500.5 1508.6 .332 1516.7 1524.9 1533.1 1541.4 1549.7 1558.0 1566.3 200 C. 1574.7 T 1583.1 1.22 E 1591.6 1600.1 .445 1608.6 1617.1 1625.7 1634.4 1643.0 1651.7 25° C. T 1660.5 1669.2 1.53 E 1678.1 1686.9 .558 1695.8 1704.7 1713.7 1722.7 1731.7 1740.8 300 C. T 1749.9 1759.0 1.84 E 1768.2 .671 1777.4 1786.7 1796.0

~

''

T

4893.6 4908.0 4922.5 4937.0 40" 4951.5 50' 4966.1 4980.7 820 10' 4995.4 20' 5010.0 30' 5024.8 40' 5039.5 50' 5054.3 5069.2 830 10' 5084.0 20' 5099.0 30" 5113.9 40' 5128.9 50' 5143.9 5159.0 840 10" 5174.1 20" 5189.3 30' 5204.4 40' 5219.7 5234.9 5250.3 850 10' 50' 5265.6 5281.0 20' 5296.4 30' 5311.9 50' 5327.4 88040' 5343.0 10' 5359.6 5374.2 30' 5389.9 40' 5405.6 20' 50' 5421.4 5437.2 870 10' 5453.1 5469.0 20' 30' 5484.9 40' 5500.9 50' 5517.0 5533.1 880 10' 5549.2 20' 5565.4 30' 5581.6 40' 5597.8 50' 5614.2 5630.5 10' 5646.9 20' 5663.4 30' 5679.9 40' 5696.4 50' 5713.0 5729.7 900 10' 5746.3 20' 5763.1 30' 5779.9 40' 5796.7 50' 5813.6 81o

10' 20' 30'

E=R exsec 11:

E

=90

1805.3 1814.7 1824.1 1833.6 50 C. T 1843.1 .36 1852.6 1862.2 .149 1871.8 1881.5 1891.2 1900.9 1910.7 1920.5 C. 1930.4 100 T 1946.3 .72 1950.3 E 1960.2 .299 1970.3 1980.4 1990.5 2000.6 2010.8 2021.1 2031.4 150 C. 2041.7 T 2052.1 2062.5 2073.0 .450 2083.5 2094.1 2104.7 2115.3 2126.0 2136.7 200 C. 2147.5 T 2158.4 1.45 E 2169.2 2180.2 .603 2191.1 2202.2 2213.2 2224.3 2235.5 2246.7 250 C. 2258.0 T 2269.3 1.83 E 2280.6 2292.0 .756 2303.5 2315.0 2326.6 2338.2 2349.8 2361.5 300 C. T 2373.3 2.20 E 2397.0 2408.9 .910 2420.9 2432.9

I'

T

5830.5 5847.5 5864.6 5881.7 5898.8 5916.0 5933.2 5950.5 5967.9 5985r3 6002.7 6020.2 6037.8 6055.4 6073.1 6090.8 6108.6 40' 50' 6126.4 6144.3 940 10' 6162.2 20' 6180.2 30' 6198.3 6216.4 40' 50' 6234.6 950 6252.8 10' 6271.1 20' 6289.4 30' 6307.9 40' 6326.3 50' 6344.8 o 960 6363.4 10' 6382.1 20' 6400.8 30' 6419.5 40' 6438.4 50' 6457.3 6476.2 $70 10' 6495.2 20' 6514.3 30' 6533.4 40' 6552.6 50' 6571.9 980 6591.2 10' 6610.6 20' 6630.1 30' 6649.6 40' 6669.2 50' 6688.8 990 6708.6 10' 6728.4 20' 6748.2 30' 6768.1 40' 6788.1 50' 6808.2 1000 6828.3 10' 6848.5 20' 6868.8 30' 6889.2 40' 6909.6 50' 6930.1 10" 10' 20' 30' 40' 50' 920 10' 20' 30' 40' 50' 980 10' 20' 30'

L=11(1 I T E11

TANGENTS AND EXTERNALS TO A 1o CURVE

TABLE VIII E

T

E

6950.6 6971,3 6992.0 7012.7 7033.6 7054.5 7075.5 7096.6 7117.8 7139.0 7160.3 7181.7 7203.2 7224.7 7246.3 7268.0 7289.8 7311.7 7333.6 7355.6 7377.8 7399.9 7422.2 7444.6 7467.0 7489.6 7512.2 7534,9 7557.7 7580.5 7603.5 7626.6 7649.7 7672.9 7696.3 7719.7 7743.2 7766.8 7790.5 7814.3 7838.1 7862.1 7886.2 7910.4 7934.6 7959.0 7983.5 8008.0 8032.7 8057.4 8082.3 -8107.3 8132.3 8157.5 8182.8 8208.2 8233.7 8259.3 8285.0 8310.8

3278.1 3294.1 3310.1 3326.1 3342.3 3358.5 3374.9 3391.2 3407.7 3424.3 3440.9 3457.6 3474.4 3491.3 3508.2 3525.2 3542.4 3559.6 3576.8 3594.2 3611.7 3629.2 3646.8 3664.5 3682.3 3700.2 3718.2 3736.2 3754.4 3772.6 3791.0 3809.4 3827.9 3846.5 3865.2 3884.0 3902.9 3921.9 3940.9 3960.1 3979.4 3998.7 4018.2 4037.8 4057.4 4077.2 4097.1 4117.0 4137.1 4157.3 4177.5 4197.9 4218.4 4239.0 4259.7 4280.5 4301.4 4322.4 4343.6 4364.8

1110 10' + 20' Soc. T 30' .51 40' E 50' .268 112o0 10' 20' 30' 40' 50'

E

R exsec =/s

I=1

2444.9 -2457.1 2469.3 5o C. T 2481.5 .43 2493.8 E 2506.1 2518.5 .200 2531.0 2543.5 2556.0 2568.6 2581.3 2594.0 o10o C. 2606.8 T 2619.7 .86 2632.6 E 2645.5 .401 2658.5 2671.6 2684.7 2697.9 2711.2 2724.5 2737.9 150 C. T 2751.3 1.30 2764.8 E 2778.3 .604 2792.0 2805.6 2819.4 2833.2 2847.0 2861.0 2875.0 200 C. 2889.0 T 2903.1 1.74 E 2917.3 .809 2931.6 2945.9 2960.3 2974.7 2989.2 3003.8 3018.4 250 C. 3033.1 T 3047.9 2.18 3062.8 E 3077.7 1.02 3092.7 3107.7 3122.9 3138.1 3153.3 3168.7 300 C. 3184.1 T 3199.6 2.62 3215.1 E 3230.8 1.22 3246.5 3262.3

o

101 0 10' 20' 30' 40'

50' 1020 10' 20' 30' 40' 50' o 103 10' 20' 30' 40'

50' 104o 10' 20' 30' 40' 50' 1050 10"

20'

30' 40' 50' 106o0 10' 20' 30' 40'

50'

1070 10' 20' 30' 40' 50'

1080 10' 20' 30' 40' 50' 1090 10' 20' 30' 40' 50' o 110 10' 20' 30' 40'

50'

T= Rtan

2

1

100 C. T .103 E

.536

150 C. T 1.56 E .806

200 C. T 2.08 E 1.08

250

0

T 2.61 E 1.36

300+C.

3.14

E

1.63

8336.7 8362.7 8388.9 8415.1 8441.5 8468.0 8494.6 8521.3 8548.1 8575.0 8602.1 8629.3 8656.6 1180 10' 8684.0 20' 8711.5 30' 8739.2 40' 8767.0 50' 8794.9 8822.9 1141, 10' 8851.0 20' 8879.3 30' 8907.7 40' 8936.3 50' 8965.0 8993.8 1150 10' 9022.7 20' 9051.7 30' 9080.9 40' 9110.3 50' 9139.8 9169.4 1160 10' 9199.1 20' 9229.0 30' 9259.0 40' 9289.2 50' 9319.5 1172. 9349.9 10' 9380.5 20' 9411.3 30' 9442.2 40' 9473.2 50' 9504.4 t180 9535.7 10' 9567.2 20' 9598.9 30' 9630.7 40' 9662.6 50' 9694.7 Stoo 9727.0 1190 9759.4 10' 20' 9792.0 30' 9824.8 40' 9857.7 50' 9890.8 9924.0 1200 10' 9957.5 9991.0 20' 30' 10025.0 40' 10059.0 50' 10093.0

4386.1 4407.6 4429.2 4450.9 4472.7 4494.6 4516.6 4538.8 4561.1 4583.4 4606.0 4628.6 4651.3 4674.2 4697.2 4720.3 4743.6 4766.9 4790.4 4814.1 4837.8 4861.7 4885.7 4909 9 4934.1 4958.6 49831 5007.8 5032.6 5057.6 5082.7 5107.9 5133.3 5158.8 5184.5 5210.3 5236.2 5262.3 5288.6 5315.0 5341.5 5368.2 5395.1 5422.1 5449.2 5476.5 5504.0 5531.7 5559.4 5587.4 5615.5 5643.8 5672.3 5700.9 5729.7 5758.6 5787.7 5817.0 5846.5 5876.1

+

50oC. T .62 E .360

100 C. T 1.25 *E 721

150 T C.

1.93

E 109

0

20 C. T 2.52 E 1.46

250 C.

T 3.16 E 1.83

300 C. T 3.81 E 2.20

TABLE IX MIDDLE ORDINATES OF RAILS Length of Rail (feet) C

O'

R 130128

Feet

0-20 17189

0-40 8594 1-0 5730 1-20 4297 1- 40 3438 2-0 2865 2-20 2456 2-40 2149 3-0 1910 3-20 1719 3-40 1563 4-0 1433 4-20 1323 4-40 1228 5 1146

6

7

26

24

22 20 C

.08 .07 .16 .14 .24 .20 .31 .27 .39 .34 .47 .41 .55 .48 .63 .55 .71 .62 .78 .68 .86 .75 .94 .82 1.02 .89 1.10 .96 1.18 1.03

.061 .05 .04 .03 .12' .18 .23 .29 .35 .41 .47 .53 .59 .65 .71 .77 .83 .89

RI

01 Feet

Inch Inch lach Inchn .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 75

.08 .13 .17 .21 .25 .29 .33 .38 .42 .46 .50 .55 .59 .63

.07 .10 .13 .17 .20 .23 .27 .31 .35 .38 .42 .45 .48 .52

819.0 1.65 1.44 1.24 1.05

.89

.73

8 9 10 11 12 13 14 15 16 17 18 19 20 22 24

0

28

26

124

122

IchInch Inchn Inch ch l

716.8 1.88 1.64 1.42 1.20 1.01 637.3 573.7 521.7 478.3 441.7 410.3 383.1 359.3 338.3 319.6 302.9 287.9 262.0 240.5

2.12 2.36 2.59 3.83 3.05 3.30 3.54 3.76 4.00 4.21 4.45 4.70 5.16 5.64

955.3 1.41 1.23 1.06 .90 .76 .62 26 222.3 6.0

1.84 2.05 2.26 2.47 2.66 2.87 3.08 3.28 3.48 3.67 3.89 4.09 4.44 4.92

1.60 1.78 1.95 2.15 2.30 2.48 2.68 2.83 3.02 3.18 3.36 3.55 3.84 4.20

1.35 1.50 1.65 1.81 1.96 2.10 2.26 2.40 2.57 2,70 2.86 3.00 3.30 3.59

1.14 1.27 1.39 1.54 1.66 1.78 1.91 2.04 2.16 2.28 2.41 2.54 2:80 3.04

20

Inch

.84

.94 1.04 1.15 1.26

1.36

1.46 1.57 1.67 1.78 1.87 1.98 2.09 2.29 2.50

5.29 4.58 3.88 3.29 2.70

TABLE X SHORT RADIUS CURVES Radius

Chord

Central

Deflection

Deflection

Feet

Feet

Angle

Angle

for 1 Foot

35 45 50 60 75 100 120 150 190 200 225 240 250 215 288 300 350 376 400 410

10 10 15 15 15 20 20 20 25 25 25 25 25 25 50 50 50 50 50 50

16-26 12-46 17-16 14-22 11- 30 11-30 9-34 7-39 7-32 7-10 6-25 5 58 5 44 5-12 9-58 9-32 8-12 "-40 7-10 7-00

8-13 6-23 8-38 7-11 5-45 5-45 4-47 3-49 3-46 3-35 3-12 2-59 2-52 2-.36 4 -59 4-46 .4-06 3-50 3-35 3-30

49.3 38.3 34.5 28.8 23.0 17.3 14.3 11.5 9.15 8.6 7.7 7.2 6.9 6.2 6.0 5.7 4.9 4.6 4.3 4.2

To find length of curve divide anIe from P. C. to P. T. by central angle of chord, and multiply by length of chord.

TABLE

XI

Z

.

INCLINED DISTANCE OF 100 FT. REDUCED TO HORIZONTAL Slope

Horizntal

Correction

Rise

Slope

0-00' 15' 30' 45' 1 00 15 30 45 2 00 15 30 45 3 00 15 30 45 4 00 15 30 45 5 00 15 30 45 6 00 15 30 45 7 00 15 30 45

100.000 99.99 99.996 99.991 99.985 99.976 99.966 99.953 99.939 99.923 99.905 99.885 99.863 99.839 99.813 99.786 99.756 99.725 99.692 99.657 99.619 99.580 99.540 99.497 99.452 99.406 99.357 99.307 99.255 99.200 99.144 99.087

0.000 0.001 0.004 0.009 0.015 0.024 0.034 0.047 0.061 0.077 0.095 0.115 0.137 0.161 0.187 0.214 0.244 0.275 0.308 0.343 0.381 0.420 0.460 0.503 0.548 0.594 0.643 0.693 0.745 0.800 0.856 0.913

0.000 0.004 0.009 0.013 0.017 0.022 0.026 0.031 0.035 0.039 0.044 0.048 0.052 0.057 0.061 0.065 0.070 0.074 0.078 0.083 0.087 0.092 0.096 0.100 0.105 0.109 0.113 0.118 0.122 0.126 0.131 0.135

8000' 15' 30' 45' 900 15 30 45 10 00 15 30 45 11 00 15 30 45 12.00 15 30 45 13 00 15 30 45 14 00 15 30 45 15 00 15 30 45

H

ta 99.027 98.5 98.902 98.836 98.769 98.700 98.629 98.556 98.481 98.404 98.325 98.245 98.163 98.079 97.992 97.905 97.815 97.723 97.630 97.534 97.437 97.338 97.237 97.134 97.030 96.923 96.815 96.705 96.593 96.479 96.363 96.246

Correction 0.973 1.035 1.098 1.164 1.231 1.300 1.371 1.444 1.519 1,596 1.675 1.755 1.837 1.921 2.008 2.095 2.185 2.277 2.370 2.466 2.563 2.662 2.763 2.866 2.970 3.077 3.185 3.295 3.407 3.521 3.637 3.754

Rise 0.139 0.143 0.148 0.152 0.156 0.161 0.166 0.169 0.174 0.178 0.182 0.181 0.191 0.195 0.199 0.204 0.208 0.212 0.216 0.221 0.225 0.229 0.233 0.238 0.242 0.246 0.250 0.255 0.259 0.263 0.267 0.271

For each foot take one one-hundredth of each reading.

TABLE XII

MINUTES IN DECIMALS OF A DEGREE. 20'30" .34167 30'30" .50833 40'30" .67500 50'30" 683335100 21 00 .3500031 00 .51667 4100 30 .35833 30 .52500 30.69167 30 22 00 .36667 32 00 .53333 42 00 .70000 52 00 30 .37500 30 .54167 30 .70833 30 23 00 .38333133 00 .55000 43 00 .71667 53 00 30 30.55833 30.39167 54 00 24 00 0 .40000 40833 34 00 30 .56667 .57500 30 .741675 30 .75000 5500 .5333450 30 0 .75833 .41667 30 30 .59167 .25000 25 00 30 30 .42500 .25833

'30" I 00 30 2 00 30 3 00 30 4 300

.0083310'30" .17500 .01667 11 00 .18333 .02500 30 .19167 .03333 12 00 .20000 .04167 30 .20833 .05000]13 00 .21667 30.22500 05833 .23333 .06667 4 3 .07500 .2467 .2 3

500 30

.08333 15 300 .09167

6 00 30 700 30 8 00 30 9 00 30 10 00

.10000 .10833 .11667 .12500 .13333 .14167 .15000 .15833 .16667

1600 .266 26 00 .43333 30.27500 30.44167 1700 .283332700 .45000 30 .45833 30.29167 18 00 .30000 .46667 308 33 28 00 30 .

30 .47500 19 00 .31667 29 00 .48333 30 .32500 30 .49167 2000 .33333 30 00 .50000

36 00 .600004600 30 .0833 30 3700 .61667 4700 30 30.62500 38 00 .63333 48 00 30 30 .64167 39 00 .65000 49 00 30 .65833 30 40 00 .66667 50 00

.84167 .85000 .85833 .86667 .87500 .88333 .89167 90000 .90833 .91667 .92500

.76667 5600 .93333 .7500 30 .94167 .78333 57 00 .95000 .95833 5.79167 30 .80000 58 00 .96667 .97500 .80833 30 .98333 .8166759 00 .82500 30 .99167 83 . 333 0 00 1.00000

TABLE

XIII--CRRECTIONS

FOR TANGENTS AND EXTERNALS

These corrections are to be added to the approximate values, found by dividing the tangent,or external for a 10 curve (Table VIIl) by the degree of curve, in order t obtain the true tangents, or externals. Intermediate values may be obtained by interpolation.

FOR TANGENTS ADD Central Angle 10 15

DaaM or CU 5°

100 15

03 .06 04 10 06 .13 .08 16

200 S5' 8085

200

°

°

35

25 30

400 45

450

°

55.

600 65 700

09 13 16 19 .22 .25 .28 .31 .34 38 42 .46 14 19 24 .29 .34 .39 45 .51 .53 58 .63 .68 19 .26 .32 .39 .45 .51 .58 .65 .72 79 .84 .90 24 .33 40 49 .58 67 .75 .83 90 .991.06 1.14

10 19 .29 39 49 .59 .69 79 11 22 .34 47 .58 .69 .70 .81 13 .26 .40 .53 .67 .80 .93 1.06 15 .30 44 60 78 91 1.06 1.21

400

50

.89 .99 1.09 .92 1.04 1.29 .20 1.34 1.49 1.37 1.52 1.70

1.20 1.42 1.64 1.87

1.29 1.54 1.79 2.04

1.39 1.8 1.94 2.21

1.54 1.72 1.91 2.102.29 1.,721.922.142.352.586 1.94 2.17 2.382.602.83 2.13 2.382.632.88 3.13

2.48 2.77 3.07 3.39

50°° 65° 60

17 .34 .51 .68 .85 1.02 19 38 .57 76 .951.14 .21 42 .63 .84 1.05 1.27 23 46 .69 .93 1.16 1 40

T70 T65 80 S

25 .51 76 1.02 1.28 1.4 1.80 2.06 2.33 2.60 2.883.16 3.44 3.72 .27 .56 .83 1.12 1.40 1.69 1.98 2.27 2.57 2.873.16 3.473.784.09 30 .61 .91 1.221.531.84 2.152.462 783.103.443.784.124.46

9w 100 1100

36 .721.09 1.45 1.83 2.20 20 2.57 2.94 3.32 3.70 4.10 4.504.91 5.32 39 79 1.19 1.55 2.00 2.40 2.80 3.20 3.61 4.024.404.985.385.83 .43 .86 1.30 1.74 2.18 2.62 3.06 3.50 3.95 4.404.88 5.37 5.85 6.34 511.03 1.562.082.61 3.14 3.67 4.21 4 76 5.315 86 6.43 7 01 7 60

180

62 1.25 1 9312.523.16 3.814 45 5.11 5.77 6 44 7 12 7.80 8.509 22

o60

o

33

1.19 1.32 1.49 1.64

1.36 1.52 1.71 1.88

66 1.00 1.33 1.68 2.02 2.36 2 7 3.05 3.40 3.774.144.5 4 8

Foa ExTERNAL ADD DoRarz Or CuRvz

Central 50 100 15 100 150

°

°

°

200

25

300

35

40

450

50*

550

600

°

65

700

.001.003 .003 .007 W00 .006 .011 86o .009.018

.004 .010 .017 .027

.006 .014 .022 036

.007 .018 .028 .046

.008 .023 .034 .0

.009 .027 .038 .06

.011 .029 .045 .074

.012 .032 .051 .08

.014 .035 .057 .093

.015 .017 .018 .020 .039 .043 .047 .051 .063 .070 .076 .083 .106 120 127 135

S0 °

.013 .025 .018 .035 .023 .046 .030 .060

.038 .054 .070 .093

.051 .072 .093 .119

.065 .086 .117 .153

.078 .109 .141 .184

.090 .131 .172 .216

.103 .153 .203 .254

.116 .175 .234 .289

.129 .197 .265 .325

.149 .213 .277 .351

.170 .179 .230 .247 .290 .315 .378, .411

.037 .075 .046.093 .056 .112 .067.135

.116 .142 .168 .204

.151 .189 .188 .236 .225 .283 .273 .343

.227 .283 .340 .412

.266 .332 .398 .483

.305 .381 .457 .554

.345 .420 .516 .625

.384 .479 .575 .697

.425 .530 .636 .711

.467 .582 .697 .845

.080.159 .095 .182 .110.220 .128.259

.240 .286 .332 .391

.321 .383 .445 .524

.485 .578 .671 .790

.568 .652 .735 .819 .906 .994 1.08 1.17 .678 .777 .877 .977 1.07 1.18 1.29 1.39 .787 .903 1.02 1.13 1.25 1.38 1.50 1.62 .926 1.06 1.20 1.34 1.47 1.62 1.76 1.91

90 950 1000 110

.149.299 .174 .35 .200.401 .268 .536

.450 .603 .756 .910 1.07 1.22 1.38 1.54 1.70 1.872.03 2.20 .522 .706 .985 1.06 1.25 1.43 1.62 1.80 1.99 2.18 2.38 2.58 .604 .809 1.01 1.22 1.43 1.64 1.852.06 2.28 2.50 2.73 2.96 .806 1.08 1.35 1.63 1.912.20 2.48 2.76 3.05 3.35 3.66 3.96

1100

.360.721 1.08 1.45 1.82 2.19 2.57 2.95 3.33 3.72 4.11 4.50 4.91 5.32

35

40 ° 4b

00 68 60 60 10 T70

800 85

°

.403 .480 .558 .657

.188 .264 .341 .445

.50 .550 .641 .700 .774 .851 .922 1.01

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SMTWTFS

SMTWTFS

JAN 7 14 21 28

12 8 9 15 16 22 23 29 30

3 45 10 11 12 17 18 19 24 25 26 31

APR 6 13 20 27

FEB

1234567 8 9 10 11 12 13 14 15 16 17 18 19 20 21 2223 24 25 26 27 28 2930

MAY

123 12345 4 5 6 7 8 9 10 6 7 8 9 1011 12 11 12 13 14 15 16 17 1314 15 16 17 18 19 18 19 20 21 22 23 24 20 21 22 23 24 25 26 25 26 27 28 27 28 29 30 31

MAR 123 45678910 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

JUNE 12 3456789 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

5 M

T W

T

F

S M

JULY 1 2 3 4 8 9 10 11 15 16 17 18 22 23 24 25 29 30 31

5 12 19 26

T W

T F S

OCT 6 13 20 27

7 12 14 7 8 9 21 14 15 16 28 21 22 23 28 29 30

3 4 10 11 17 18 24 25 31

5 12 19 26

6 13 20 27

AUG

NOV

1234 5 6 7 8 91011 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

123 4 5 6 7 8 910 11 12 13 14 15 16 17 18 1920 21 22 23 24 25 26 27 28 29 30

SEPT

DEC

2 3 9 10 16 17 23 24 30

4 5 6 7 11 12 13 14 18 19 20 21 25 26 27 28

1 8 15 22 29

2 3 4 5 9 10 11 12 16 17 18 19 23 24 25 26 30 31

1 6 7 8 13 14 15 20 21 22 27 28 29

;

St06r

Z

STEv sr.

xe

;1 1

DISTANCES FROM SIDE STAKES FOR CROSS-SECTIONING. SLOPE 1I/ TO 1.ROADWAY OF ANY WIDTH. I O1 0 I .1 .2 I .3 .5 .6 .-7 t.8 ! . .

10.00 0.151 0.30 0.45 4.60 0.75 0.90 1.05 1.20 1,

6 I-

I

.

I.4

n I 1.95 I 2.10

1.An i

I

2.25 I 2.40

2.55

2.70

2,

5.70

5,

64.20 65.70 67.20 68.70 70.20 71.70 73.20 74.70 76.20

64.

65.

67.

68.

70.

71.

73.

74.

4.20

4,

.V, , 55.05

56.55 58.05 64.50 64.65

64.

67.50 67.65 69.00 69.15 70.50 70.65 72.00 72.15 73.50 73.65 - -" 75.00 75.15

67.

69.

70.

72.

73.

75.

66.00 66.15 66.

63.45 64.95 66.45 67.96 69.45 70.95 72.45 73.96 75.45

THE NATIONAL

63.60 65.10 66.60 68.10 69.60 71.10 72.60 74.10 75.60

63.75 65.25 66.75 68.25 69.75 71.20 72.75 74.25 75.75

BLANK

BOOK

HOLYOKE NEW YORK

63.90 64.05 65.40 65.55 66.90 67.05 68.40 68.55 69.90 70.05 71.40 71.55 72.90 73.05 74.40 74.55 75.90 76.05

COMPANY

MASSACHUSE CHICAGO

BOSTON

TS

SAN FRANCISCO

176.

u ZO