Fremling Field Notebook 1958

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40 mm.

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DISTANCEs

O3

PROM

SIDE STK

FoR COss-SECTIONING

S8oP: I To 1. ROADnAY or Ax Wmta 0

0.20 1.20 2.20

0.10

1.10

3.00 4.00

5.20 4.20

40

3.00

21.10 12.10

13.30 12.30 14.30 15.

18.10

15.30

16

8.20 9.20 10.20

12:.1080

14.

11.

15.00 18.00 19.08

26.00

26.00 27.0

00

31.00 20 32.00

CO

34.00

41.10

35 00 37.00

42.

39.30 39.30 40.80 463 41.30 47-3

40.20

44.10

44.00

45.00

45.30

60. 00

4$.30

47.00 48.00 49.00

1

60.00

I

t

t from

t

48.40 47.40 48.40 49.40 50.40

42.50

or shld

43.50 44.50 45.50 46.50 47.50 48.60 49.50

50.50 stas

22.90

19.80 20.0

25.90 20.90

16.90 17.90

18.90

21.80

22.70 23.70 24.70 5.70 0.70 26.70 31.70

21.90 81.90 22.90

22.80

18.90 28.90 23.90 24.90 24.90 25.0 27.90 6.00 28.90 29.90 29.90 30.90

23.80

27.70

34.80

28.70

329.70 527.80 40.70 341.70 29.80

84.60 35.60

32.70 4.70

37.60 38.60 39.00

39.70

41.0

It0

42.90

0.80

43.90 .0 E 44.0 45.90 49.9 50.90

47. D

50.60

I

.90

15.80

18.80

40.60 41.60 42.60 43.60 44.60 45.70 48.70 47.60 48.60 40.70 50.70

41.50

t

12.80

19.70

36.60

39.50

14 80

12.70 13.70 14.70 15.70 16.70 17.70 18.70

12.60

33.60

38.10

43.0

8.70 9.70 10.70 11.70

I3.60

87.40 38.40 39.40 40.40 41.40 42.40

4.80

1.90 2.90 4.00 3.90 5.90 6.90

2.80

6.70 5.70

10.60

430.50 25.50 38.50 36.50

4.70

2.70 3.70

3.60 5.60

32.60 22.60 218.50 29.50 t4.50 25.00 320.50 26.60 321.50 26.60 27.60 23.50 28.60 29.60 40.60 31.60 27.50

30.40

35.30

1.70

2.60

13.00 14.60 11.600 16.50 15.60 16.60 17.60 213.50 17.60 2214.50 18.60 2315.50 19.60 124.50 20.60

30.40 34.40 8.40 38.40

32.30 33.30 34.30

38.20

46.10 42.10 37.10

41.

296.30 302.30 31.30

37.20

43.10 39.10 39.10

40.

21.30 22.30 23.30 24.30 5.30

Isi

21.20 16.20 28.10 8.10 22.20 8.10 23.20 24.20 2.10 25.20 26.20 30.20 31.20 28.20 32.20 33.20 47.20 0.10 35.20

9

0.90

0.70

18.50 19.50

18.40 19.40 20.40 21.40 22.40 23.40 24.40 25.40 26.40 27.40 28.40 29.40

20.30

.8

0.80 1.80

.7

0.60

1.60

20.80

15.40

1.30 19.30

20.20 20.20

21.10 17.10

24.00

1.30 130 10.30

11,.9

18.10 I.10

I1.00 2.00 23.00

1.40

2.40 3.40 4.40 7.40 7.40 8.40 9.40 6.40 10.40 11.40 12.40 13.40 14.40

418.10

0.50 S1.50 2.60 3.50 14.50 15.50

0.40

7.:

3.10

28

0.30 1.30 2.30 3.30 45.30

4.20

5.00

.8

.5

.1.4 8

.

.1

0.00

50.80

..i

I

for any width roadway

' slo

' to

1

1e the out or 511at sie stake Is located by the double entry method Infrom row. T number in body of table in same row and column gives distance It ground Is not level estimate the difference In elevaton between the estake. s to amount by this amount If ut, elvate if l. Add ta slope stake, low de s distance in table. et up rod at this point, and line of sight should ut target. cut or 1 a necessary. adjustment the slight e not does It it Distance

is nand kto

y

stake

side

?csLc-Ltr k+ll wJ-lv\cvx

YJ.rnC 1

4

Cee

re4-hr

/

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 1Y2 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 ihe side stake and slope stake, lower target by this amount if cut, elevate if fll. 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 FORMULJ. LB =LPON = LOPL LA = LMOP R = OB = c= 1 sin A sos A tan A=

c b b

=.

C

a b

=

MQ OM

=

NT

ON

1

sec A= OQ

I

OM ON

b= sin B= OL "

1

cot A= NT

csc A=

aa=sos B = LP

1 b b=

=

MQ 1

MQ= cot B-= MQ

NT= tan B= NT OQ = csc B = OQ

T- == OT = sec B = OT

I

vers A = LM= LM = covers B ==

OP

OP-LP .

OP-LP = vers B OP exsec A = PQ = coexsec B coexsec A = PT - exsec B covers A .

sin / A =cos

2

IA = 2 cos 2 A = cos' A-sin' A sin 2 A = 2 sin A cos A sin C' sin B sin A Law of Lines a B C Law of Cosines c'a2-a+b'-2 ab cos C Law of Tangents a+b a-b

tan i2 (A+ ) tan 1/ (A-8)

TABLE II -

Continued

TRIGONOMETRIC FORMUAE (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. as, /(s-a) (s-b) (s-c) Let aLb = s, =r 2 cos, 'sA

tan ' A-

tan 'I, B tan I/s C -

(s- a)

r

S-a

s-b s-b

s-c

Area of a triangle: Area= / ab Sin C

Area

s (s--a) (s-b) (s-c) PRISMOIDAL .FORMULA.

(B+b-4M)

Vol. -

h = altitude; b, B = bases; M = midsection TABLE III

INCHES AND FRACTIONS OF AN INCH IN DECIMALS OF A FOOT 0 =/i .0052 .0104 i .0156 1/4 .0208 b/,.0260 eis .0313 'its .0365 '/i .0417 *sle .0469 5/s .0521 "s/s .0573 a1 .0625 /1 .0677 '/s .0729 ste.0781 1 .0833 0

1

2

.0885 .1719 .0938 .1771 .0990 .1823 .1042 .1875 .1094 .1927 .1146 .1979 .1198 .2031 .1250 .2083 .1302 .2135 .1354 .2188 .1406 .2240 .1458 .2292 ,1510 .2344 .1563 .2396 .1615 .2448 .1667 .2500 1

2

3

4

5

6

7

8

9

10

11

.2552 .204 .2656 .2708 .2760 .2813 .2865 .2917 .2969 .3021 .3073 .3125 .3177 .3229 .3281 .3333

.3385 .3438 .3490 .3542 .3594 .3646 .3698 .3750 .3803 .3854 .3906 .3958 .4010 .4063 .4115 .4167

.4219 .4271 .4323 .4375 .4427 .4479 .4531 .4583 .4635 .4688 .4740 .4292 .4844 .4896 .4948 .5000

.5052 .5104 .5156 .5208 .5260 .5313 .5365 .5417 .5469 .5521 .5573 .5625 .5677 .5729 .5781 .5833

.5885 .5938 .5990 .6042 .6094 .6146 .6198 .6250 .6302 .6354 .6406 .6458 .6510 .6563 .6615 .6667

.6719 .6773 .6823 .6875 .6927 .6979 .7031 .7083 .7135 .7188 "7240 .7292 .7344 .7396 .7448 .7500

.7552 .7604 .7656 .7708 .7760 .7813 .7865 .7917 .7969 .8021 .8073 .8125 .8177 .8229 .8281 .8333

.8385 .8438 .8496 .8542 .8594 .8646 .8698 .8750 .8802 .8854 .8906 .8958 .9010 .9063 .9115 .9167

.9219 .92711 .9323 .9375 .9427. .9479 .9531 .9583 .9635 .9688 .9740 .9792 .9844 .9896 .9948 1.000

3

4

5

6

7

;3.

r

.

~s

9

~b

0

(

11

/ 1/ /su /4

/e als '/is 1/4

*/is 6/s

"ie 8/4

s/m

je

1, 1

TABLE IV

Deg.

('

Radius

-RAII, OiDmNATSr

Mid. Ord.

Tan. Offset

Def. for 1 Foo t

10' 34377.5 20 17188.8 30 11459.2 40 8594.42 50 6875.55

.036 .073 .109 .145 .182

.145 .291 .4386 .582 .727

0.05' 0.10 0.15 0.20

5729.65 4911.15 4297.28 3819.83 3437.87 3125.36

.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

2864.93 2644.58 2455.70 2292.01 2148.79 2022.41

.438 .473 .509 .545 .582 .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

1910.08 .65 1?00.57 .691 1719.,12 .727 1637.28 764 1562 88 .800 836 1494.95

2.618 2.763 2.908 3.054 3.199 3.345

0.90 0.95 1.00 1.05 1.10 1.16

1432.89 .873 1375.40 .9009 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

1146:28 1109.33 1074.68 1042.14 1011.51 982.64

1.091 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

955.37 29.57 905.13 881.95 859.92

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

Note.

Chord Deflection

AND DznloNs

Deg.

Radius

Mid. Ord.

Tan. Offset

Def. for 1 Foot

819.02 781.84 764.49 747.89

1.628 1.800 1.637 1.673

6.105 8,395 6.540 6.685

2.10' 2.20 2.25 2:30

716.78 688.16 674.69 661.74

1.746 1.819 1.855 1.892

6.976 7.266 7.411 7.556

2.40 2.50 2.55 2.60

637.28 614.58 603.80 593.42

1.965 2.037 2.074 2.110

7.846 8.136 8.281 8.426

2.70 2.80 2.85 2.90

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 4.05 4.20 4.35

388.07 870.78 359.27 348.45 338.27 319.62 302.94

3.277 3.387 3.496 3.606 3.716 3.935' 4.155

13.05 13.49 13.92 14.35 14.78 15.64 10.51

4.80 4.65 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

21.64 22.50 23.35 24.19 25.04 25.88

7.50 7.80 8.10 8.40 8.70 9.00

0.25

=2 times tangent deflection.

6.583

V TABLE V CURVE FORMULIE FOR SIMPLE CURVES COMPILED BY J.CALVIN LOCKE, C. E. (1)

c =1/2Ra

(2) c -= a'+b

_

(3) (4) (5) (7 (8)

c = 2R (R - /(R+b) (R--b'= f2R (R-- Rfb c = 2 m (2-m) c = 2R sin '/sI (6)c = 2T cos 's 1 1/2I e = R exsec 1 == R tan /Itan '/s (9) e -Ttan /l1 (10) b - a (2R-a)

(11) b =

c+

)

(12) b= Rsin I (14) R --a'b' abs

2a

2

j vc - 41(13) b= a cot I14

-c

d' c'+4m' (15) R =--f

2a

2m

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

Sm

R (2R-

(<18) d = 2R sin '/

(20) m= R

(R+

(21) m = R vers

/21

2=R

f 4R'-c) (19) m

-)((R+

R

2R"

R -r-

(22) m = R sin /21 tan '/4 I (23) m = /s c tan'/ I (IR2-b' R (25) a - R - Y(R b) (R--b) =-

2 (24) a 26) a = 2 R (sin' '1/2 1) (27) a = R vers I (28 a = R sin Itan 1/L (31) T = R tan 1 /s I (30) a = T sin I (29) a= b tan 1/sI (32) 1=

L

X 57.295780

(34) L = IR X 0.01745329 (36 Area eg.= LR-R' sin I 2

L

(33) R= - X 57.295780 (35) L = LR-Rb 2

8d-c

TABLE VI

SINES, COSINES, TANGENTS, COTANGENTS sin 0'

tan sin 0 ' 10'

tan sin tan 10' 20'

tan

0 0000 1 175 -2 349 3 523 4 698 5 872 6 1045 7 219 8 392 9 564

00000029 0175 0204 349 378 524 552 699 727 875 901 1051 1074 228 248 405 421 584 593

0029 0058 204 0233 378 407 553 581 729 756 904 929 1080 1103 257 279 435 449 614 622

0058 0087 023 0262 407 436 582 610 758 785 934 958 1110 1132 287 305 465 478 644650

10 11 12 13 14 15 16 17 18 19

736 908 2079 250 419 588 756 924 3090 256

763 765 944 937 2126 2108 309 278 493 447 679 616 867 784 3057 952 249 3118 443 283

793 794 974 965 2156 2136 339 306 524 476 711 644 899 812 3089 939 281 3145 476 311

42 420 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

30 31 32 33 34 35 36 37 38 39

5000 150 299 446 592 736 878 6018 157 293

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

40 41 42 43 44 45

591 539 642 49 54T 517 441 472 491494 428 391 450 952 48 670 847 648 796 626 744 604 561 693 583 9271 47 9163 777 9217 799 9110 756 9057 734 691 9004 713 545 926 601 46 435 884 490 905 325 841 380 862 820 94245 8847050 827030 770 7009 713 988 65 967 947 44 1.0235 173 1.0295 7071 1.0000 7092 1.0058 7112 1,0117 133 1.0176 153

"

4.

S60' cos

sin

tan

30'

30'

tan sin 40' 50'

tan so 50' -a

00870116 262 291 437 465 612 640 787 814 963 987 1139 1161 317 334 495 507 673 679

0116 0145 291 320 466 494 641 669 816 843 992 1016 1169 190 346 363 524 536 703 708

0145 320 495 670 846 1022 198 376 554 733

89 88 87 86 85 84 83 82 81 80

823 822 2004 994 186 2164 370 334 555 504 742 672 931840 3121 3007 314 173 508 338

853 2035 217 401 586 773 962 3153 346 541

883 880 2065 2051 247 221 432 391 617 560 805 728 994 896 3185 3062 378 228 574 393

914 2Q95 278 462 648 836 3026 217 411 607

79 78 77 76 75 74 73 72 71 70

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 9 488 5206 643 430 797 658 950

772 557 973 719 4176 881 383 4041 592 200 806 358 5022 514 243 669 467 823 96 975

805 69 4006 68 210 67 417 66 62865 841 64 5059 63 280 62 505 61 735 60

5812 5050 6048 200 289 348 495 53 787 640 7046 783 310 925 581 6065 860 202 8146 338

851 5975 6088 225 330 5373 577 519 830 664 7089 807 355 948 6276088 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 41 422 422 661 568 916 712 7177 854 445 995 720 6134 8002 271 292 406

969 6208 453 703 959 7221 490 766 8050 342

20'

sin 40'

851 2022 193 363 532 700 868 3035 201 365

60' 50'

50' 40'

40' 30'

30' 20'

20' 10'

10'

cot

cot

cot

cot

cot

cot

cos

cos

cos

....

cos

cos

59 58 3 56 55 54 53 52 51 50

.1

TABLE VI (continued) SINES, COSINES, TANENTS COTANGENTS (continued) sin

tan sin

0'

0

10'

ten

sin

tan sin

tan

sin

tan

sin

tan

10' 20'

20' 30'

30'

40'

40'

50'

50' b

46 7193 1.0355 7214 1.0416 7234 1.0477 7254 1.0533 47 314 .0724 333 .0786 353 .0850 373 .0913 48 431 .1106 451 .1171 470 .1237 490 .1303 585 .1640 604 .1708 49 547 .1504 566 .1571

72714 1.0599 392 .0977 509 .1369 623 .1778

7294 1.0661 43 412 .1041 42 528 .1436 41 642 .1847 40

50 660 1.1918 7679 1.1988 7698 1.2059 7716 1.2131 7735 1.2203 844 .2647 51 771 .2349 790 .2423 808 .2497 826 .2572 52 880 .799 898 .2876 916 .2954 934 .3032 951 .3111 53 986 .3270 8004 .3351 8021 .3452 8039 .3514 8056 .3597 158 .4106 54 8090 .3764 107 .3848 124 .3934 141 .4019 55 192 .4281 208 .4370 225 .4460 241 .4550 258 .4641 56 290 .4826 307 .4919 323 .5013 339 .5108 355 .5204 57 387 .5399 403 .5497 418 .5597 434 .5697 450 .5796 58 480 .6003 496 .6101 511 .6212 526 .6319 542 .6426 59 572 .6643 587 .6753 601 .6864 616 .6977 631 .7090

7753 1.2'6 39 862 .2123 38 969 .3190 37 8073 .3680 36 175 .4193 35 274 .4733 34 371 .5301 33 465 .5900 32 557 .6534 31 646 .7205 30

60 660 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 .9626 923 .9768 936 .9912 949 2.0057 962 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 .2460 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 .0051 346 .6279 356 .6511 367 .6746 377 .6985

8732 1.7917 29 816 .8676 28 897 .9486 27 975 2.0353 26 9051 ,1283 25 124 .2286 24 194 .336923 261 .4545 22 325 .5826 21 387 .7228 20

70 71 72 73 74 75 76 77 78 79

9446 502 555 605 652 696 737 775 811 843

2.8770 3.0475 .2371 .4495 .6891 .9617 4.2747 .6382 5.0658 .5764

19 18 17 16 15 14 13 12 11 10

5.9758 9868 6.0844 9872 6.6912 894 .8269 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 .000 171.88 1.000

6.1970 .9682 7.9530 9.2553 11.059 13.727 18.075 26.432 49.104 343.77

9 8 7 6 5 4 3 2 1 0

397 2.7475 9407 2.7725 9417 .7980 9426 2.8239 9436 2.8502 455 .9042 465 .9319 474 .9600 483 .9887 492 3.0178 511 3.0777 520 3.1084 528 3.1397. 537 3.1716 546 .2041 563 .2709 572 .3052 580 .3402 588 .3759 596 .4124 613 .4874 621 .5261 628 .5656 636 .6059 644 .6470 659 .7321 667 .7760 674 .8208 681 .8657 689 .9136 703 4.0108 710 4.0611 717 4.1126 724 4.1653 730 4.2193 744 .3315 750 .3897 757 .4494 763 .5107 769 .5736 805 .9894 781 .7046 787 .7729 793 .8430 799 .9152 8146 822 5.2257 827 5.3093 833 5.3955 838 5.4845

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 951 10.078 954 85 962 11.430 964 11.826 967 12.250 969 86 976 14.300 978 14.92 980 15.05 981 87 986 19.081 988 20.206 989 21.470 990 88 994 28.636 9995 31.242 9996 34.368 997 89 9998 57.290 9999 68.750 9999 85.940 9999

60

60' 50'

cos ....

cot .

cos

.

0

50' 40' cot :8

40' 30'

cos

1r$0

cot cos i~

30' cot

20' cos

20' cot

10'

cos

10'

cot

a

TABLE VII

Deflections for Sub Chords for Short Radius Curves. sub chord

I def.

sin of

Length

angle

Degree of

Radius

Curve

sin.jdef.ang.

12.5 Ft.

15 Ft.

20 Ft.

25 Ft.

for 100ft.

1051

20 17''

2 58' 3 10'

3 43 3 8' 4012' 4o26' 4 40'

101..15 101o.33

360 380°

193.18 181.39 171.01 161.8o 153.58

40

146.19

42 44°

o

30 32"

°

34

R

1

59'

2

° 2 5

20 33

3021

2 41 0 49' 2 57 3 0o5 3 13

333

139.52 133.47

2006' 2 131 2 20' 2o 27 2 34 20 41

460 480

127.97 122.92

20 48' 2o 55

3 21' 329'

4029

°

118-31

3 02' 3°09

338' 3 0 46 1

4 5' 5 01 02'

3 54

5 0.

13 , 1

50 ° 52

540 56°0

114.-6 iIO110.11

56

58o ° 60

o6.5

103.14 l00.00

16

3 44

4 54'

4 07' 4 18'

' 5 5 22' 5 6'

4

4

5

3 29

4

o'

5

3035'

418'

5

02

T

R tang I

R = T cot.

S

Si"

R

Sin.

D = o R tE D = 5 t

Sin.

3 55

322'

Sin.

=

z

40 '

101.48 Ioi.66 1o.85 102.o6 102.29 102.53 102.76 13.00oo

550

23

60041 6- 17' 6°3' -0o

6044'

104.14

44

70

I

104.72

de

cChord hords

103.24

I0 103.54

I

R

D

II

103.84 4.43

o50 R ex. sec ID

E = T tan

'

of are

No.chords=Tan. def.= j 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 1z for I ft. see Table II.), and divide given deflection by the product.

Rule 2. Multiply given deflection by 57.3, and divide the product by 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. 1o.o+2oo00=.5. 1oo+.5= 1oo.5 hyp.

Given Hyp. loo, Alt. 25.252+20oo3.125 100-3.125=96.875 =Base.

Error in first example, .002; in last, .045.

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 0.574ds, where d is the distance in miles.

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

PROBABLE ERROR. If dg, de, d5 , etc. are the discrepancies of various results from the mean, and if 7Id=the sum of the squares of these differences and n=the number of observations, then the probable error of the mean= : 0.6745 )

TABLE VIII I 3*10

II T t'E

50.00 58.34 66.67 75.01 83.34 40'.

50' 91.68 go 100.01 10' 108.35 20' 116.68 30' 125.02 40' 133.36 50' 141.70 3o 150.04 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 so 10' 258.51 20' 266.86 30' 275.21 40' 283.57 50' 291.92 Ao 300.28 10' 308.64 316.99 10' 325.35 40' 333.71 50' 342.08 350.44 10' 358.81 20' 367.17 30' 375.54 40' 383.91 50' 392.28 400.66 10' 409.03 20' 417.41 30' 425.79 40' 434.17 50' 442.55 450.93 go 10' 459.32 20' 467.71 30' 476.10 40' 484.49 50' 492.88 501.28 100 10' 509.68 20' 518.08 30' 526.48 40' 534.89 50' 543.29 10' 20' 30'

TANGENTS AND EXTERNALS TO A 1 CURVE. 1=100

.218 .297 +1 .388 .491 So C. T .606 .733 .03 E .873 .001 1.024 1.188 1.364 1.552 1.752 1.964 100 C. 2.188 T 2.425 .06 2.674 E 2.934 .003 3.207 3.492 3.790 4.099 4.421 4.755 5.100 350 C. T 5.459 .09 5.829 E 6.211 6.606 .004 7.013 7.432 7.863 8.307 8.762 9.230 200 C. 9.710 T 10.202 .13 E 10.707 11.224 .006 11.753 12.294 12.847 13.413 13.991 14.582 25 C 15.184 T 15.799 .16 16.426 E 17.065 .0071 17.717 18.381 19.058 19.746 20.447 21.161 300 C, T 21.887 22.624 .19 E 23.375 24.138 .008 24.913 25.700

I

T

110 10' 20' 30' 40' 50'

551.70 560.11 568.53 576.95 585.36 593.79 602.21 610.64 619.07 627.50 635.93 644.37 652.81 661.25 669.70 678.15 686.60 695.06 703.51 711.97 720.44 728.90 737.37 745.85 754.32 762.80 771.29 779.77 788.26 796.75 805.25 813.75 822.25 830.76 839.27 847.78 856.30 864.82 873.35 881.88 800.41 898.95 907.49 916.03 924.58 933.13 941.69 950.25 958.81 967.38 975.96 984.53 993.12 1001.7 1010.3 1018.9 1027.5 1036.1 1044.7 1053.3

20 40' 50' 180 10' 20" 30' 40' $0'

140 10' 20' 30' 40' 50' 1o 10' 20' 30' 40' 50' to 1010' 20' 30' 40' 50' 170

10' 30' 40" 50'

18o

10' 20' 30' 40' 50'

190 10' 20' 30' 40' 50' 10' 20' 30' 40' 50'

ft

T=Rtan

/uI

E

I

I=200

T

E

I1=30

1061.9 97.577 1070.6 99.155 1079.2 100.75 1087.8 102.35 T 1096.4 103.97 1105.1 105.60 .10 E 1113.7 107.24 .013 1122.4 108.90 1131.0 110.57 1139.7 112.25 1148.4 113.95 1157.0 115.66 1165.7 117.38 100 C. 1174.4 119.12 T 1183.1 120.87 .19 1191.8122.63 E 1200.5 124.41 .025 1209.2 126.20 1217.9 128.00 1226.6 129.82 1235.3 131.65 1244.0 133.50 1252.8 135.35 1261.5 137.23 150 C. T 1270.2 139.11 .29 10' 1279.0 141.01 E 20' 1287.7 142.93 .038 30' 1296.5 144.85 40' 1305.3 146.79 1314.0 148.75 50' o 1322.8 150.71 260 10' 1331.6 152.69 1340.4 154.69 20' 30' 1349.2 156.70 200 C. 61.141 T 40' 1358.0 158.72 62.381 .26 50' 1366.8 160.76 .39 o E 270 1375.6 162.81 E 63.634 .051 64.900 .022 10' 1384.4 164.86 1393.2 166.95 66.178 20' 1402.0 169.04 67.470 30' 68.774" 40' 1410.9 171.15 70.091 50" 1419.7 173.27 71.421 1428.6 175.41 280 72.764 250 C. 10' 1437.4 177.55 250 C. T 74.119 T 20' 1446.3 179.72 75.488 .32 30' 1455.1 181.89 .49 1464.0 184.08o E 76.869 E 1472.9 186.29 .065 78.264 .028 1481.8 188.51 79'.671 81.092 1490.7 190.74 1499.6 192.99 82.525 1508.5 195.25 83.972 85.431 40' 10' 1517.4 197.53 86.904 300 C. 50' 1526.3 199.82 300 .

50' 20' T 30' 1535.3 202.12 T 88.389 290 40' 1544.2 204.44 .59 89.888 .39 E E 91.399 1553.1 206.77 92.924 .034 1562.1 209.12 .078s 211.48 94.462 1571.0 50' 1580.0 213.86 96.013 1 26.500 27.313 28.137 28.974 So C. T' 29.824 30.686 .06 E 31.561 ,006 32.447 33.347 34.259 35.183 36.120 37.070 10o C. 38.031 T 39.006 .13 39.993 E 40.992 .011 42.004 43.029 44.066 45.116 46.178 47.253 48.341 150 C. T 49.441 .19 50.554 E 51.679 52.818 .017 53.969 55.132 56.3009 57.498 58.699 59.914 200 C.

5 7

210 10' 20' 30' 40' 50' 220 10' 20' 30' 40' 50' 230o 10' 20' 30' 40' 50' 240 10' 20' 30' 40' 50'

fr

E =Rxsecgit

I-

1

TABLE VIII TANGENTS AND EXTERNALS TO A 10 CURVE i

T

E

I=400

'~---~~--' 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 1751.7 340 10' 1760.8 20' 1770.0 30' 1779.1 40' 1788.2 50' 1797.4 1806.6 350 10' 1815.7 20' 1824.9 30' 1834.1 40' 1843.3 50' 1852.5 1861.7 10' 1870.9 20' 1880.1 30' 1889.4 40' 1898.6 50' 1907.9 o7 1917.1 10' 1926.4 20' 1935.7 30' 1945.0 40' 1954.3 50' 1963.6 o8째 1972.9 10' 1982.2 20' 1991.5 30' 2000.9 40' 2010.2 50' 2019.6 $90 2029.0 10' 2038.4 20' 2047.8 30' 2057.2 40' 2066.6 50' 2076.0 2085.4 400 10' 2094.9 2104.3 30' 2113.8 40' 2123.3 50' 2132.7 810 10' 20' 30' 40' 50' 820 10' 20' 30' 40' 50' 380 10' 20' 30' 40' 50'

I

T

E I=50o

" 216.3 218.7 221.1 223.5 226.0 228.4 230.9 233.4 235.9 238.4 241.0 243.5 246.1 248.7 251.3 253.9 256.5 259.1 261.8 264.5 267.2 269.9 272.6 275.3 278.1 280.8 283.6 286.4 289.2 292.0 294.9 297.7 300.6 303.5 306.4 305.3 312.2 315.2 318.1 321.1 324.1 327.1" 330.2 333.2 336.3 339.3 342.4 345.5 348.6 351.8 354.9 358.1 361.3 364.5 367.7 371.0 374.2 377.5 380.8 384.1

41

째

10' 20' So C. 30' T .13 40' E 50' .02 420 10" 20' 30' 40" 50' 0

10 C. T .26 E .046

43o

10' 20' 30' 40' 50' 440 10' 20'

째

30 40

15 C. 50' T .40 450 10' E 20' .070 30' 40' 50' 468 10' 20' 30' C. 20 40' T 50' .53 E 470 10' 20' 30'

40' 50' 480

250 C. T .67 .117

T

I

E

I=to

"

10' 20' 30' 40' 50' 480

10' 20' 30' 40' 50' 30oC. 500o T 10' .80 20' E 30' .141 40' 50'

T = R tan 's I

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 2521.1 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

387.4 390.7 394.1 397.4 400.8 404.2 407.6 411.1 414.5 418.0 421.4 425.0 428.5 432.0 435.6 439.2 442.8 446.4 450.0 453.6 457.3 461.0 464.6 468.4 472.1 475.8 479.6 483.4 487.2 491.0 494.8 498.7 502.5 506.4 510.3 514.3 518.2 522.2 526.1 530.1 534.2 538.2 542.2 546.3 550.4 554.5 558.6 562.8 566.9 571.1 575.3 579.5 583.8 588.0 592.3 596.6 600.9 605.3 609.6 614.0

510

10' soC. 20' T 30' 40' .17 E 50' .037 52째 10' 20'

30' 40'6 50' o

100 C, 5S 10' T 20' .34 30' E 40' .015 50' 540

15c C. T .51 E .116

200 C,

.68 E .151

10' 20' 30' 40' 50' o

55

10' 20' 30' 40' 50' 560 10' 20' 30 40' 50' 570

10' 20' 30' 40'

50' 580 10' 250 C. 20' T 30' .85 40' E 50' .189 590

10' 20' 30' 40' 50' 0 30 C. 600 T 10' 1.02 20' E 30 .227 40' 50'

E = R exsec

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 2804.9 649.7 2815.2 654.3 2825.6 658.8 2835.9 663.4 2846.3 668.0 2856.7 672.7 2867.1 677.3 2877.5 682.0 2888.0 686.7 2898.4 691.4 2908.9 696.1 2919.4 700.9 2929.9 705.7 2940.4 710.5 2951.0 715.3 2961.5 720.1 2972.1 725.0 2982.7 729.9 2993.3 734.8 3003.9 739.7 3014.5 744.6 3025.2 749.6 3035.8 754.6 3046.5 759.6 3057.2 764.6 3067.9 769.7 3078.7 774,7 3089.4 779.8 3100.2 784.9 3110.9 790.1 795.2 3121.7 3132.6 800.4 3143.4 805.6 3154.2 810.9 3165.1 816.1 3176.0 821.4 3186.9 826.7 3197.8 832.0 3208.8 837.3 3219.7 842.7 3230.7 848.1 3241.7 853.5 3252.7 858.9 3263.7 864.3 3274.8 869.8 3285.8 875.3 880.8 3296.9 3308.0 886.4 892.0 3319.1 3330,3 897.5 903.2 3341.4 3352.6 908.8 914.5 3363.8

112 1

5oC. T .21 E .056

100 C. T .42

E .112

15O C.

T .63 E .168

200 C. T .81 .225

0

25 C. T 1.05 E 2.83

300 C. T 1.27 .340

TABLE VIII TANGENTS AND EXTERNALS TO A 1 CURVE. I

T

1

I=70o

E

T

I=-0o

E

T

I

EI=0

" 1* 10' 20' 30' 40' 50' 620 10' 20' 30' 40' 50'

680 10' 20" 30' 40' 50' 64o 10' 20' 30' 40' 50' 6o0 10 20' 30' 40, 50' 0 10' 20' 30' 40' 50' 60 10' 20' 30" 40' 50" 10' 20' 30' 40' 50' 10, 20' 30, 40' 50'

10 20' 30' 40' 50'

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 3673.7 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

920.2 925.9 931.6 +o 937.3 So C. 943.1 948.9 .25 954.8 960.6 966.5 972.4 978.3 984.3 990.2 996.2 1002.3 .51 1008.3 E 1014.4 159 1020.5 1026.6 1032.'8 1039.0 1045.2 1051.4 1057.7 150 C. 1063.9 .76 1070.2 E 1076.6 .240 1082.9 1089.3 1095.7 1102.2 1108.6 1115.1 1121.720 ° C. 1128.2 T 1134.8 1.02 1141.4 E 1148.0 .321 1154.7 1161.3 1168.1 1174.8 1181.6 1188.4 250 C. T 1195.2 1202.0 1.28 1208.9 E 1215.8 .403 1222.7 1229.7 1236.7 1243.7 1250.8 1257.9 300 C. T 1265.0 1272.1 1.54 E 1279.3 .485 1286.5 1293.6 1300.9 E

4086.9 4099.5 4112.1 4124.8 1137.4 4150.1 4162.8 4175.6 4188.5 4201.2 4214.0 4226.8 4239.7 730 10' 4252.6 20' 4265.6 30' 4278.5 40' 4291.5 50' 4304.6 4317.6 740 10' 4330.7 20' 4343.8 30' 4356.9 40' 4370.1 50' 4383.3 4396.5 750 10' 4409.8 20' 4423.1 30' 4436.4 40' 4449.7 50' 4463.1 4476.5 780 10' 4489.9 20' 4503.4, 30' 4516.9 40' 4530.4 50' 4544.0 770 4557.6 10' 4571.2 20' 4584.8 30' 4598.5 40' 4612.2 4626.0 50' 4639.8 780 10' 4653.6 20' 4667.4 30' 4681.3 40' 4095.2 50' 4709.2 4723.2 790 10' 4737.2 20' 4751.2 30' 4765.3 40' 4779.4 50' 4793.6 4807.7 800 10' 4822.0 20' 4836.2 30' 4850.5 40' 4864.8 50' 4879.2 10' 20' 30' 40' 50' 720 10' 20' 30' 40' 50'

Rtan 1/a'I

1308.2 1315.6 1322.9 1330.3 1337.7 1345.1 1352.6 1360.1 1367.6 1375.2 1382.8 1390.4 1398.0 1405.7 1413.5 1421,2 1429.0 1436.8 1444.6 1452.5 1460.4 1468.4 1476.4 1484.4 1492.4 1500.5 1508.6 1516.7 1524.9 1533.1 1541.4

1549.7 1558.0 1566.3 1574.7 1583.1 1591.6 1600.1 1608.6 1617.1 1625.7 1634.4 1643.0 1651.7 1660.5 1669,2 1678.1 1686.9

81o 10' 20' + 30' T 40' .30 50' E o .110 82 10' 20' 30' 40' 50' 830 10' 20' 30' 40' 50' 840 10' 20' 30' 40' 150 C. 50' 850 .91 10' E 20' .332 30" 40' 50' 88o0 10' 20' 200 C. 30' 40' T 1.22 50' E 870 .445 10' 20 30' 40' 50' 880 250 C. 10' T 20' 30" 1.53 40' .558 50'

1695.8 1704.7 1713.7 1722.7 1731.7 1740.8 300 C. 1749.9 1759.0 1768.2 1777.4 1786.7 1796.0

T 1.84 E .671

10' 20' 30' 40' 50' 900 10' 20' 30' 40' 50

4893.6 4908,0 4922.5 4937.0 4951.5 4966.1 4980.7 4995.4 5010.0 5024.8 5039.5 5054.3 5069.2 5084.0 5099.0 5113.9 5128.9 5143.9 5159.0 5174.1 5189.3 5204.4 5219.7 5234.9 5250.3 5265.6 5281.0 5296.4 5311.9 5327.4

5343.0

5358.6 5374.2 5389.9 5405.6 5421.4 5437.2 5453.1 5469.0 5484.9 5500.9 5517.0 5533.1 5549.2 5565.4 5581.6 5597.8 5614.2 5630.5 5646.9 5663.4 5679.9 5696.4 5713.0 5729.7 5746.3 5763.1 5779.9 5796.7 5813.6

E=R exsec / 1

1805.3 1814.7 + 1824.1 1833.6 5o C. T 1843.1 1852.6 .36 E 1862.2 .149 1871.8 1881.5 1891.2 1900.9 1910.7 1920.5 10o C. 1930.4 T 1940.3 .72 1950.3 E 1960.2 .299 1970.3 1980.4 1990.5 2000.6 2010.8 2021.1 2031.4 150 C. T 2041.7 1.09 2052.1 2062.5 .450 2073.0 2083.5 2094.1 2104,7 2115.3 2126.0 2136.7 200.C. 2147.5 T 2158.4 1.45 2169.2 .603 2180.2 2191.1 2202.2 2213.2 2224.3 2235.5 2246.7 250 C. T 2258.0 2269.3 1.83 E 2280.6 2292.0 .756 2303.5 2315.0 2326.6 2338.2 2349.8 2361.5 30o C. T 2373.3 2385.1 2.20 2397.0 2408.9 .910 2420.9 2432.9

TABLE VIII

TANGENTS AND EXTERNALS TO A 10 CURVE

IT E

tI T E I=201 10o 10' 20' 30' 40' 50' 20" 10' 20' 30' 40' 50' 10' 20' 30' 40' 50' 040 10' 20' 30' 40' 50' 950 10' 20' 30' 40' *50' 0 " 10' 20' 30' 40' 50' 970

10' 20' 30' 40' 50' 980 10' 20' 30' 40' 50' 990

10' 20' 30' 40' 50' 1000 10' 20' 30' 40' 50'

5830.5 5847.5 5864.6 5881.7 5898.8 5916.0 5933.2 5950.5 5967.9 5985:3 6002.7 6020.2 6037.8 6055.4 6073.1 6090.8 6108.6 6126.4 6144.3 6162.2 6180.2 6198.3 6216.4 6234.6 6252.8 6271.1 6289.4 6307.9 6326.3 6344.8 6363.4 6382.1 6400.8 6419.5 6438.4 6457.3 6476.2 6495.2 6514.3 6533.4 6552.6 6571.9 6591.2 6610.6 6630.1 6649.6 6669.2 6688.8 6708.6 6728.4 6748.2 6768.1 6788.1 6808.2 6828.3 6848.5 6868.8 6889.2 6909.6 6930.1

2444.9 2457.1 2469.3 2481.5 2493.8 2506.1 2518.5 2531.0 2543.5 2556.0 2568.6 2581.3 2594.0 2606.8 2619.7 2632.6 2645.5 2658.5 2671.6 2684.7 2697.9 2711.2 2724.5 2737.9 2751.3 2764.8 2778.3 2792.0 2805.6 2819.4 2833.2 2847.0 2861.0 2875.0 2889.0 2903.1 2917.3 2931.6 2945.9 2960.3 2974.7 2989.2 3003.8 3018.4 3033.1 3047.9 3062.8 3077.7 3092.7 3107.7 3122.9 3138.1 3153.3 3168.7 3184.1 3199.6 3215.1 3230.8 3246.5 3262.3

1010 10' 20' 30' 40' 50' .20 ,1020 10' 20' 30' 40' 50' 100 C. 10830 10' T 20' .86 30' E 40' .401 50' 1040 10', 20, 30' 40' 150 C.

-t

5o C. T .43 E

1.30 E .604

10" 20' 30' 40' 50' 1060 10' 20'

200 C. T 40' 30" 1.74 50' E .809 1070 10' 20' 30' 40' 50' 1080 25 C. 10' 20' T 30' 2.18 40' 50' 1.02 109o 10' 20' 30' 40' 50' 300 C. 1100 10' 2.62 20' E 30' 1.22 40' 50' T= R tan

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

21

l=1104 I T E 1-t20

3278.1 3294.1 + 3310.1 S5 C. "3326.1 T 3342.3 .51 3358.5 E .268 3374.9 3391.2 3407.7 3424.3 3440.9 3457.6 3474.4 100 C 3491.3 T 3508.2 .103 3525.2 E 3542.4 .536 3559.6 3576.8 3594.2 3611.7 3629.2 3646.8 3664.5 150 C T 3682.3 1.56 3700.2 E 3718.2 .806 3736.2 3754.4 3772.6 3791.0 3809.4 3827.9 3846.5 200 C. 3865.2 T 3884.0 2.08 3902.9 3921.9 1.08 3940.9 3960,1 3979.4 3998.7 4018.2 4037.8 250 C 4057.4 T 4077.2 2.61 4097.1 E 4117.0 1.36 4137.1 4157.3 4177.5 4197.9 4218.4 4239.0 300 C. 4259.7 T 4280.5 3.14 4301.4 4322.4 1.63 4343.6 4364.8

1 10 10' 20' 30' 40' 50' 1120 10' 20' 30' 40' 50' 1130 10' 20' 30' 40' 50'

1140

10' 20'

30' 40' 50' 1150

10' 20' 30' 40' 50' 1160 10' 20' 30' 40' 50' 11710' 30' 40' 50' 1180 10' 20' 30' 40' 50' 10' 20' 30' 40'

50' 1200 10' 20' 30'

40' 50'

E= R exsec 1121

8336.7 8362.7 8388.9 8415.1 8441.5 6468.0 8494.6 8521.3 8548.1 8575.0 8602.1 8629.3 8656.6 8634.0 8711.5 8739.2 8767.0 8794.9 8822.9 8851.0 8879.3 8907.7 8936.3 8965.0 8993.8 9022.7 9051.7 9080.9 9110.3 9139.8 9169.4 9199.1 9229.0 9259.0 9289.2 9319.5 9349.9 9380.5 9411.3 9442.2 9473.2 9504.4 9535.7 9567.2 9598.9 9630.7 9662.6 9694.7 9727.0 9759.4 9792.0 9824.8 9857.7 9890.8 9924.0 9957.5 9991.0.

10025.0 10059.0 10093.0

4386.1 4407.6 + 4429.2 50 C. 4450.9 T 4472.7 .62 4494.6 E 4516.6 .360 4538.8 4561.1 4583.4 4606.0 4628.6 4651.3100C 4674.2 T 4697.2 1.25 4720.3 'E 4743.6 721 4766.9 4790.4 4814.1 4837.8 4861.7 4885.7 4909 9 4934.1 4958.6 4983 1 5007.8 5032.6 5057.6 5082.7 5107.9 5133.3 5158.8 20 C. 5184.5 5210.3 2.52 E 5236.2 5262.3 1.46 5288.6 5315.0 5341.5 5368.2 5395.1 5422.1 250 C. 5449.2 T 5476.5 3.16 5504.0 E 5531.7 1.83 5559.4 5587.4 5615.5 5643.8 5672.3 5700.9 5729.7 300 C. T 5758.6 5787.7 3.81 E 5817.0 5846.5 2.20 5876.1

TABLE

IX

MIDDLE ORDINATES OF RAILS Length of Rail (feet) C

o'

R

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 6 955.3 7 819.0

30

28

26

24

.08 .07 .061 .05 .16 .14 .12 .10 .24 .20 .18 .15 .31 .27 27 .23 .20 .39 .34 .29 .25 .47 .41 .35 .30 .55 .48 .41 .35 .63 .55 .47 .40 .71 .62 .53 .45 .78 .68 .59 .50 .86 .75 .65 .55 .60 .94 .82 .71 .77 .65 1.02 .89 .83 .70 1.10 .96 75 1.18 1.03 .89 1.41 1.23 1.06 .90 1.65 1.44 1.24 1.05

.04 .08 .13 .17 .21 .25 .29 .33 .38 .42 .46 .50 .55 .59 .63 .76 .89

.03 .07 .10 .13 .17 .20 .23 .27 .31 .35 .38 .42 .45 .48 .52 .62 .73

8 9 10 11 12 13 14 15 16 17 18 19 20 22 24 26

TABLE

30 128nch

R

221 20 C

Inch Inch Inchncch nhnch

I

716.8 637.3 573.7 521.7 478.3 441.7 410.3 383.1 359.3 338.3 319.6 302.9 287.9 262.0 240.5 222.3

1.88 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 6.017

1.64 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 5.29

26

24

22

20

Inch Inch Ich Inch 1.42

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 4.58

1.20 1.01 1.35 1.14 1.50 1.27 1.65 1.39 1.81 1.54 1.96 1.66 2.10 1.78 2.26 1.91 2.40 2.04 2.57 2.16 2.70 2.28 2.86 2.41 3.00 2.54 3.30 2:80 3.59 3.04 3 ,883.29

.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 2.70

X

SHORT RADIUS CURVES Radius Feet

Chord Feet

Central Angle

Deflection Angle

Deflection for 1 Foot

35 45 50 60 75 I00 120 150 190 200 225 240 250 275 288 300 350 376 400 410

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 an le from P. C. to P. T. by central angle of chord, and multipl y length of chord.

TABLE XI INCLINED DISTANCE OF 100 FT. REDUCED TO HORIZONTAL Distan HrDistance

Slope

Correction

Rise

0

Correction

Rise

8 0Q' 15' 30'

99.027 98.965 98.902

0.973 1.035 1.098

0.139 0.143 0.148

0.013

45'

98.836

1.164

0.152

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

9 00 15 30 45 10 00 15 30 45 1100 15 30 45 12.00 15 30 45 13 00 15 30 45 14 00 15 30 45 15 00 15 30 45

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

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

0.156 0.161 0.166 0.169 0.174 0.178 0.182 0.1871 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

0000' 15' 30'

100.000 99.999 99.996

0.000 0.001 0.004

0.000 0.004 0.009

45'

99.991

0.009

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

0.015 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

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

oritane I Slope oDistance

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

TABLE

XII

MINUTES IN DECIMALS OF A DEGREE. '30" 1 00 30 0 2000

.00833 10'30" .016671100 .02500 30 .03333 12 00 00

30 .04167

.17500 20'30" .34167 030" .18333 21 00 .35000 31 00 .19167 30 .35833 30 .20000 2200 3200 00 .36667 1.

00

.50833 .50833 4030" 40'30".67500 .6730'30" 50'30" .51667 4100 ~68333 51 00 .52500 30 .69167 30 00 .700005200 .53333 42 43008833

.21667 . .23333 .24167 .25000 .25833 .26667 .27500 .28333 .29167 .30000 .30833 .31667 .32500

.55000 43 00 30 .55833 .

44 .57500 30 .583334500 .59167 30 .60000 46 00 .60833 30 .61667 4700 .62500 30 .63333 4800 .64167 30 .650004900 .65833 30 .66675000

30.2033

300 .05000 13 00 0 .05833 3 30 .07500 30 5 00 .08333 1500 30 .00167 30 6 00 .10000 16 00 30 .10833 30 7oo00 .11667 17 00 30 .12500 30 8 00 .13333 18 00 7 3 0 9 00 .15000 19 00 30 .15833 30 1000 .16667 2000

30 .37500

23 00 30 2500 30 26 00 30 2700 30 2800 30 29 00 30

.33333 30 000

30.54167

.383333300 .39167 30 34 00 .40833 30 .41667 3500 .42500 30 .43333 36 00 .44167 30 .45000 3700 .45833 30 .46667 38 00 .47500 30 .483333900 49167 30 00

0 40

30

.70833

30

.717 53 00 .72500 30 7333 00 74167 03 .7500055 00 .75833 30 .76667 56 00 .77500 30 .78333 5700 .79167 30 .8000058 00 .80833 30 .81667 59 00 .82500 30 .83333 60 00

.84167 .85000 .85833 .86667

.87500 .8833 .89167 .90833 .91667 .92500 .93333 .94167 .95000 .95833 .96667 .97500 .98333 .99167 1.00000

TABLE

XIII I-CORRECTIONS

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 VIII) by the degree of curve, in order to obtain the true tangents, or externals. Intermediate values may be obtained by interpolation.

FOR TANGENTS ADD or Cuav:

Damons

Central Angle

°

°

5 1 150 200 350

10

°

15

20

°

°

25

03 .0609 o0 13 04 10 14 19 06 .13 19 .26 .08 16 24 .33

80

10 11 13 15

350

0e 45

S"0 °

19 .29 39 22 .34 47. .26 .40 .53 .30 44 60

17 .34 19 38 .21 42 23 46

5 60 ° 65 °

T0 V5 00 850

16 24 .32 40

350 40

30

19 .29 .39 49

.22 .34 .45 .58

°

.25 .39 .51 67

°

°

50

45

.28 45 .58 .75

55

60

°

°

65

°

70

.31 .34 38 42 46 .51 .53 58 .63 .68 .65 .72 79 .84 .90 .83 90 .991.06 1.14

49 .59 .69 79 .89 .99 1.09 1.20 1.29 1.39 .58 69 .70 .81 .92 1.04 1.29 1.42 1.54 1.66 .67 .80 .93 1.06 1.20 1.34 1.49 1.64 1.79 1.94 76 91 1.061.211.371.521.701.872.042.21

.51 .68 .85 1.02 .57 76 .951.14 .63 .84 1.05 1.27 .69 .93 1.16 1 40

25 .51 761.02 .27 .56 .831.12 30 .61 .91 1.22 33 66 1.00 1.33

°

°

1.19 1.32 1.49 1.64

1.36 1.52 1.71 1.88

1.54 1.72 1.721.92 1.94 2.17 2.13 2.38

1.91 2.14 2.38 2.63

2.10 2.29 2.48 2.3 2.562.77 2.60 2.83 3.07 2.88 3.13 3.39

1.281.4 1.80 2.06 2.332.622.60 2.88 3.16 3.44 3.72 1.401.691.98 2.272.572.87 3.163.47 3.78 4.09 1.53 1.84 2.152.46 2 78 3.10 3.443.784.124.46 1.68 2.022.362 70 3.05 3.40 3.774.144.554

00 956 100° 110

86 .72 1.091.451.832.202.57 2.94 3.323.70 4.10 4.504.91 5.32 39 791.191.55 2.00 2.402.803.203.614.024.404.98 5.385.83 .43 .86 1.301.74 2.182623.063.503.95 4.4 .88 5.375.856.34 51 1.03 1.56 2.08 2..61 3.14 3.67 4.21 4 76 5.31 5 86 6.43 7 01 7 60

10

8621.25 1 93 2.52 3.16 3.81 4 4565.115.77 6 4417 12 7.80 8.50 9 22

FoR EXTERNALS ADD or C

Dzau

Central Angle 5°

10

°

15

20

°

25

°

30

35

°

40

°

a 450

500

°

55

60

65

70

°

10 16 20 26

.001 .003 .003 .007 .006.011 .009 .018

.004 .006 .007 .008 .009 .010 .014 .018 .023 .027 .017 .022 .028 .034 .038 .027 036 .046 .056 .065

.011 .029 .045 .074

.012 .032 .051 .083

.014 .015 .017 .018 .020 .035 .039 .043 .047 .051 .057 .063 .070 .076 .083 .093.106 120 127 135

30 85e

.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 .230 .290 .378.

.179 .247 .315 .411

50 66o 60 6o

.037.075 .046 .093 .056.112 .067 .135

.116 .142 .168 .204

.151 .188 .225 .273

.189 .236 .283 .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

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

70 To 80 85

.080 .159 .095 .182 .110 .220 .128 .259

.240 .286 .332 .391

.321 .383 .445 .524

.403 .480 .558 .657

.485 .578 .671 .790

.568 .678 .787 .926

.652 .777 .903 1.06

.735 .877 1.02 1.20

.819 .977 1.13 1.34

.906 1.07 1.25 1.47

.994 1.18 1.38 1.62

1.08 1.29 1.50 1.76

1.17 1.39 1.62 1.91

90 98* 10 110

.149 .174 .200 .268

.450 .522 .604 .806

.603 .706 .809 1.08

.756 .985 1.01 1.35

.910 1.06 1.22 1.63

1.22 1.43 1.64 2.20

1.38 1.62 1.85 2.48

1.54 1.80 2.06 2.76

1.70 1.99 2.28 3.05

1.87 2.18 2.50 3.35

2.03 2.38 2.73 3.66

2.20 2.58 2.96 3.96

100

.360 .721 1.08 1.45 1.82 2.19 2.57 2.95 3.33 3.72 4.11 4.80 4.91 5.32

40

45e °

.299 .350 .401 .536

1.07 1.25 1.43 1.91

.188 .264 .341 .445

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DISTANCES FROM SIDE STAKES FOR CROSS-ECTIONING, SLOPE

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0.00 0.15 1.65 1.50 3.15 3.00 4.65 4.50 6.15 6.00 7.50 7.65 9.15 9.00 10.50 10.65 12.00 12.15 13.50 13.65 15.00 15.15 16.50 16.65 18.00 18.15 19,50 19.65 21.00 21.15 22.50 22.65 24.00 24.15 25.50 25.65 27.00 21.15 28.50 28.65 30.00 30.15 31.50 31.65 33.00 33.15 34.50 34.65 36.00 36.15 37.50 37.65 39.00 39.15 410.5040.65 42.00 42.15 43.50 43.65 45.00 45.15 46.50 46.65 48.00 48.15 49.50 49.65 51.00 51.15 52.50 52.65 54.00 54.15 55.50 55.65 57.00 57.15 58.50 5.65 60.00 60.15 61.50 61.65 63.00 63.15 64.50 64.65 66.00 66.15 67.50 67.65 69.00 69.15 70.50 70.65 72.00 72.15 73.50 73.65 75.00 75.15 THE

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1 .5

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0.30 1.80 3.30 4.80 6.30 7.80 9.30 10.80 12.30 13.80 15.30 16.80 18.30 19.80 21.30 22.80 24.30 25.80 27.30 28.80 30.30 31.80 33.30 34.80 36.30 37.80 39.30 40.80 42.30 43.80 45.30 46.80 48.30 49.80 51.30 52.80 54.30 55.80 k57.30 58.80 60.30 61.80 63.30 64.80 66.30 67.80 69.30 70.80 72.30 73.80

.75.30

.. 60 0.45 1.95 2.10 3.60 3.45 4.95 5.10 6.45 0.60 7.95 8.10 9.45 9.60 10.95 11.10 12,45 12.60 13.95 14.10 15.45 15.60 16.95 17.10 18.45 18.60 19.95 .20.10 21.45 21.60 22.95 23.10 24.45 24.60 25.95 26.10 27.45 27.60 28.95 29,10 30.45 30.60 31.95 32.10 33.45 33.60 34.95 35.10 36.45 36.650 37.95 38.10 39.45 39.60 40.95 41.10 42.45 42.60 43.95 44.10 45.45 45.60 46.95 47.10 48.45 48.60 49.95 50.10 51.45 51.60 52.95 53.10 54.45 54.60 s55.9556.10 57.45 57.60 58.95 59.10 60.45 60.60 61.95 62.10 63.45 63.60 64.95 65.10 66.45 66.60 67.95 68.10 69.45 69.60 70.95 71.10 72.45 72.60 73.95 74.10 75.49 75.60

NATIONAL

0.75 2.25 3.75 5.25 6.75 8.25 9.75 11.25 12.75 14.25 15.75 17.25 18.75 20.25 21.75 23.25 24.75 26.25 27.75 29.25 30.75 32.25 33.75' 35.25 36.75 38.25 39.75 41.25 42.75 44.25 45.75 47.25 48.75 50.25 51.75 53.25 54.75 56.25 57.75 59.25 60.75 62.25 63.75 65.25 66.75 68.25 69.75 71.20 72.75 74.25 75.75

BLANK

BOOK

HOLYOKE NEWYORK

1.05 1.20 2.55 2.70 4.05 4.20 5.70 5.55 7.05 7.20 8.70 8.55 10.05 10.20 11.55 11.70 13.05 13.20 14.55 14.70 16.05 16.20 17.55 17.70 19.05 19.20 20.55 20.70 22.05 22.20 23.55 23.70 25.05 25.20 26.55 26.70 28.05 28.20 29.55 29.70 30.90 31.05 31.20 32.40 32.55 32.70 33.90 34.05 34.20 35.40 35.55 35.70 36.90 37.05 37.20 38.40 38.55 38.70 39.90 40.05 40.20 41.40 41.55 41.70 42.90 43.05 43.20 44.40 44.55 44.70 45.90 46.05 46.20 47.40 47.55 47.70 48.90 49.05 49.20 50.40 50.55 50.70 51.90 52.05 52.20 53.40 53.55 53.70 54.90 55.05 55.20 56.40 56.55 56.70 57.90 58.05 58.20 59.40 59.55 59.70 60.90 61.05 61.20 62.40 62.55 62.70 63.90 64.05 64.20 65.40 65.55 65.70 66.90 67.05 67.20 68.40 68.55 68.70 69.90 70.05 70.20 71.40 71.55 71.70 72.90 73.05 73.20 74.40 74.55 74.70 75.90 76.05 76.20 0.90 2.40 3.90 5.40 6.90 8.40 9.90 11.40 12.90 14.40 15.90 17.40 18.90 20.40 21.90 23.40 24.90 26.40 27.90.

COMPANY

MASSACHUSETTS CHICAGO

BOSTON

SAN FRANCISCO

1.35 2.85 4.35 5.85 7.35 8.85 10.35 11.85 13.35 14.85 16.35 17.85 19.35 20.85 22.35 23.85 25.35 26.85 28.35 29.85 31.35 32.85 34.35 35.85 37.35 38.85 40.35 41.85 43.35 44.85 46.35 47.85 49.35 50.85 52.35 53.85 55.35 56.85 58.35 59.85 61.35 62.85 64.35 65.85 67.35 68.85 70.35 71.85 73.35 74.85 76.35

0

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