Fremling Field Notebooks Summer 1974 & 1980-1981

TANCES FROM SIDE STAKES FOR Cnoss-SECIONING SLoPE 1 TO 1. ROADWAY OF ANY WmTH 0

.1

.2

0 1 2

.3

0.00 0.10 0.20 0.30 1.00 1.10 1.20 1.30 2.00 2.10 2.20 2.30 3.30 3 3.00 3.10 3.20 4.30 4 4.00 4.10 4.20 5o.20 5.30 S6.00 5.10 2 6.00 6.10 6.20 6.30 7 70.00 70.10 70.20 70.30 8 8,.00 81.10 81.20 38.30 9 39.00 39.10 19.20 39.30 10 10.00 10.10 10.20 10.30 11 1 .00 11.10 11.20 11.30 12 12.00 12.10 12.20 12.30 18 13.00 13.10 13.20 13.30 14.20 14.30 14 14.00 14.10 15 16.00 15.10 15.20 15.30 16.30 18 16.00 16.10 16.20 21 17.00 17.10 17.20 17.30 43 18.00 18.10 18.20 18.30 19 19.00 19.10 19.20 19.30 .20.2020.30 0 20.00 20.10 1 21.00 21.10 21.20 21.30 2 22.00 22.10 22.20 22.30 23 23.00 23.10 23.20 23.30 24 24.00 247.10 24.20 24.30 26 25.00 25.10 25.20 25.30 26 28.00 26.10 26.20 26.30 27 27.00 27.10 27.20 27.30 28 28.00 28.10 28.20 28.30 29 29.00 29.10 29.20 29.30 0 30.00 30.10 30.20 30.30 31 31.00 31.10 31.20 31.30 32 32.00 32.10 32.20 32.30 S33.00 33.10 33.20 33.30 34.30 34 34.00 34.10 34 20 25 35.00 35 10 35.20 35.30 88 38600 36.10 36.20 38.30 37 37.00 37.10 37.20 37.30 38 38.00 38.10 38.20 38.30 39 39.00 39.10 39.20 39.30 40 40.00 40.10 40.20 40.30 41.10 41.20 41.30 4 1 41.00 42.20 42.30 42.10 42 42.00 43.10 43.20 43.30 43 43.00 44.20 44.30 44.10 44 44.00 45.10 45.20 45.30 45 45.00 46 46.00 46.10 46.20 46.30 47.20 47.30 47.10 47 47.00 48.10 48.20 48.30 48 48.00 49.20 49.30 49.10 49 49.00 50.30 50.20 0.00o0 50.10

.4

.5

0.40 1.40 2.40 3.40 4.40 5.40 6.40 70.40 81.40 9 .40 10.40 11.40 12.40 13.40 14.40 15.40 16.40 17.40 18.40 19.40 20.40 21.40 22.40 23.40 24.40 25.40 28.40 27.40 28.40 29.40 30.40 31.40 32.40 33.40 34.40 35.40 36.40 37.40 33.40 39.40 40.40 41.40 42.40 43.40 44.40 45.40 46.40 47.40 48.40 49.40 50.40

0.50 1.50 2.50 3.50 4.50 5.50 6.50 70.50 8.80 91.50 10.50 11.50 12.50 13.50 14.50 15.50 16.50 17.50 18.50 19.50 20.50 21.50 22.50 23.50 24.50 25.50 26.50 27.50 28.50 29.50 30.50 31.50 32.50 33.50 34.50 35.50 386.50 37.50 38.50 39.50 40..50 41.50 42.50 43.50 44.50 45.50 46.50 47.50 48.50 49.50 50.50

.6

.7

0.60 0.70 1.60 1.70 2.60 2.70 9.60 3.70 4.60 4.70 5.60 5.70 6.60 5.70 70.60 70.70 8,.60 81.70 39.60 39.70 10.60 10.70 11.60 11.70 12.60 12.70 13.60 13.70 14.60 14.70 15.60 15.70 16.60 16.70 17.60 17.70 18.60 18.70 19.60 19.70 20.60 2.70 21.60 21.70 22.60 22.70 23.60 23.70 24.60 24.70 25.60 25.70 26.60 26.70 27.80 27.70 28.60 28.70 29.60 29.70 30.60 30.70 31.60 31.70 32.60 32.70 33.60 33.70 34.60 34.70 33.60 35.70 36.60 36.70 37.60 37.70 38.60 .38.70 39.60 39.70 40.60 40.70 41.60 41.70 42.60 42.70 43.70 43.60 44.60 44.70 45.60 45.70 46.60 46.70 47.70 47.60 48.60 48.70 49.70 49.60 50.60 50.70

.8

9

0.80 0.90 1.80 1.90 2.80 2.90 3.90 3.80 4.90 4.80 5.80 5.90 6.80 6.90 70.80 70.90 81.80 1.90 39.80 39.90 10.80 10.90 11.80 11.90 12.80 12.90 13.80 13.90 14.80 14.90 15.80 15.90 16.80 16.90 17.80 17.90 18.80 18.90 19.80 19.90 20.80 20.90 21.80 21.90 22.80 22.90 23.80 23.90 24.80 24.90 '25.80 25.90 28.80 28.90 27.80 27.90 28.80 28.90 29.80 29.90 30.80 30.90 31.80 81.90 32.80 32.90 33.80 33.90 34.80 34.90 35.80 35.90 36.80 36.90 37.80 37.90 38.80 38.90 39.80 39.90 40.80 40.90 41.80 41.90 42.80 42.90 43.80 43.90 44.80 44.90 45.80 45.90 48.80 46.90 47.80 47.90 48.80 48.90 49.80 49.90 50.80 50.90

0 1 2 3 4 2

6

8 39 10 11 12 13 14 15 18 17 18 19 20 21 22 23 24 25 28 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 45 435 48 47 48 49

50

Distance of slope stake from side or shoulder stake for any width roadway, slope 1 to 1. Iti ground is nearly level the cut or f111at 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. It ground is not level estimate the difference in elevation between the side stake and slope stake, lower target by this amount it cut, elevate if ill. Add this amount to cut or fll 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.

I

I -b~c

I

c

4I,4

rr/

w

t,a 4

(0

lu~c~

JI i-

So2

e~ ~ r0L20?r

I

t~r

A

tI

12.0L ij

r't

(2

, ja

°

•

kR F

IMPROVED

TABLES

AND

INFORMATION

'y

f-

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.

Y,

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 FORMULE. LA = LMOP

LB =LPON

R = OB sin A=

sos A=

= LOPL

c= 1

si c

a = 1

b

b - b = sin B = OL I

b c

tan A=

a b

MQ Q OM

cot A=

NT ON

NT 1

sec A =

OM OT csc A = ON LM vers A = =

OP

a

sos B= LP

MQ = MQ= cot B = MQ I = NT = tan B =

NT

OQ = csc B = OQ I OT OT = OT= sec B OT I OQ

LM = covers B =

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

sin hlA-

S-Cos

sin 2 A= 2 sin A cos A sin A Law of Lines a Law of Cosines Law of Tangents

ICos

A

2

cos 1 A 2 cos 2 A = cos" A-sin' A sin B sin C' C

c'=a'+b'-l2 ab cos C a-b

tan'I(A+)

tan ', (A--B)

A

TABLE II -

Continued

TRIGONOMETRIC FORMULE (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. s, Leta+b+c Let = s, 2

cos i

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

s

=r

s (s--a) A=

be

tan 1/t A

-r

tan'/, B -tan /s, C=

.

s-b s-c

Area of a triangle: Area - /*ab Sin C Area = /s (s--a) (s-b) (s-c) PRISMOIDAL .FORMULA.

Vol. -

(B+b-4M)

h = altitude; b, B = bases; M = midsection TABLE III INCHES AND FRACTIONS OF AN INCH IN DECIMALS OF A FOOT 0

./s.0052

, .0104 .0156 1/4 .0208 a/e .0260 .0313 'l .0365 1I, .0417 hse .0469 "/s.0521 /1s .0573 .0625 ' ~/e .0677 .0729 /s .0781 1 .0833

S/e

tSe

14e

Ve

0

1

2

.0885 .0938 .0990 .1042 .1094 .1146 .1198 .1250 .1302 .1354 .1406 .1458 ,1510 .1563 .1615 .1667

.1719 .1771 .1823 .1875 .1927 .1979 .2031 .2083 .2135 .2188 .2240 .2292 .2344 .2396 .2448 .2500

1

2

3

4

.2552 .3385 .2604 .3438 .2656 .3490 .2708 .3542 .2760 .3594 .2813 .3646 .2865 .3698 .2917 .3750 .2969 .3803 .3021 .3854 .3073 .3906 .3125 .3958 .3177 .4010 .3229 .4063 .3281 .4115 .3333 .4167

3

4

5 .4219 .4271 .4323 .4375 .4427 .4479 .4531 .4583 .4635 .4688 .4740 .47,92 .4844 .4896 .4948 .5000

5

6

7

.5052 1.5885 .5104 .5938 .5156 .5990 .5208 .6042 .5260 .6094 .5313 .6146 .5365 .6198 .5417 .6250 .5469 6302 .5521 .6354 .5573 .6406 .5625 .6458 .5677 .6510 .5729 .6563 .5781 .6615 .5833 .6667

6

8

9

10

11

.6719 .6771 .6823 .6875 .6927 .6979 .7031 .7083 .7135 .7188 7240 .7292 .7344 .I736 .7448 .7500

.7552 .7604 .7656 .7708 .7760 .7813 .7865 .7917 .7969 .8021 .8013 .815 .8177 .8229 .8281 .8333

.8385 .8438 .8490 .8542 .8594 .8646 .8698 .8750 .8802 .8854 .8906 .8958 .9010 .9063 .9115 917

.9219 .9271 .9323 .9375 .9427 .9479 .9531 .9583 .9635 .9688 .9740 .9792 .9844 .9896 .99A8i 1.000

9

10

11

71 8

i $/i

/s

We hs/

/is s/s ai 3/4

i /s 1

TABLE IV

Deg.

00

10' 20 30 40 50

-RADI,

ODINursE8 AND DunooNs

Radius

Mid. Ord.

Tan. Offaet

Def. for t 1 Foo

34377.5 17188.8 11459.2 8594.42 6875.55

.036 .073 .109 .145 .182

.145 .291 .436 .582 .727

0.05'

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

.436 .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

.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

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

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

955.37 929.57 905.13 881.95 859.92

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

1f09.57

1719.12 1637.28 1562 88 1494.95

Deog.

0.10 0.15 0.20 0.25

Note. Chord Deflection =2 times tangent deflection.

Radius

Mid. Ord.

Tan. Offset

Def. for 1 Foot

819.02 781.84 764.49 747.89

1.528 1.600 1.637 1.673

6.105 6.395 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

637.28 614.56 603.80 693.42

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

383.07 370.78 359.27 348.45 338.27 319.62 302.94

3.277 3.387 3.496 3.606 3.716 3.935' 4.155

13.05 13.49 13.92 14.35 14.78 15.64 186.51

4.50 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 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

TABLE V V/ CURVE FORMULE FOR SIMPLE CURVES COMPILED BY J. CALVIN

(2) c =

(1) c =/2Ra

LOCKE, C. E.

/a'+b'

(R+b) (R-b) = fV2R (R(R12R (3) c = (4) c 2 m (2R-m) (6) c= 2T cos LI I (5) c - 2Rsin 1 /sI

(7 e= R exsec Il 1 '/4 I / e = R tan1/tan (10)

b -= a (2R-a

(11)

b

y

V ~-i

1 (9) e = T tan /4

I

c

c

(12) b= R sin I

--b4

"2R)4R1 2R ) (13) b = a cot 1 2 I

(14) R= a

b'= c (15) R c 8m 2a 2m 2a (16) d = YR (2R-(2R+c) 2R-c) = YR (2R- y 4R2-c) (17) d = i

(18) d = 2R sin '/ I

m= R (20)

R-

R+

(19) m =

R

(21) m =R vers /2 1 (22) m = R sin 1 /2 I tan'/4I (24) a =

2R

(25) a = R -

/(R+b) (R--b)

R - ' (23) m =/s c tan ' Il R --

R'-b'

1 (28 a =1 R sin I tan /s (27) a = R vers I (26) a== 2 R (sing 1/2 I)2 /2 I R tan T = I (31) T sin (30) a= I b tan '/s (29) a=

(32) 1=-

X 57.295780 R

(34) L = IR X 0.01745329 (36 Area Seg.2

X 57.295780 IX 8d-c (35) L = -3 3

(33) R

= I 36LR--R Area Seg.sin

LR-Rb

2

2

TABLE VI

SINES, COSINES, TANGENTS, COTANGENTS t

sin

tan

sin

tan

sin

tan

sin

tan

sin

tan

sin

tan

S

0'

0'

10'

10'

20'

20'

30'

30'

40'

40'

50'

50'

c

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

as

0000 0029 0175 0204 349 378 524 552 699 727 875 901 1051 1074 228 248 405 421 584 593

0029 0058 0204 0233 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 317 334 495 507 673 679

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

10 736 11 908 12 2079 13 250 14 419 15 588 16 756 17 924 18 3090 19 256

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

823 822 2004 994 186 2164 370 334 555 504 742 672 931 840 3121 3007 314 173 508 338

853 851 2035 2022 217 193 401 363 586 532 773 700 962 868 3153 3035 346 201 541 365

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

20 420 21 584 22 746 23 907 24 4067 25 226 26 384 217540 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

772 973 4176 383 592 806 5022 243 467 696

805 4006 210 417 628 841 5059 280 505 735

69 68 67 66 65 64 63 62 61 60

30 5000 31 150 32 299

774 5025 6009 175 249 324

5812 5050 6048 200 289 348

*

851 5075 6088 225 330 5373

890 5100 6128 250 371 398

930 5125 6168 275 412 422

969 59 6208 58 453 51

34 592 35 736 36 878 37 6018 38 157 39 293

745 016 7002 760 265 901 536 6041 813 180 8098 316

787 640 7046 783 310 925 581 6065 860 202 8146 338

830 664 7089 807 355 948 627 6088 907 225 8195 361

873 688 7133 831 400 972 673 6111 954 248 8243 383

916 712 7177 854 445 995 720 6134 8002 271 292 406

959 55 7221 54 490 53 766 52 8050 51 342 50

557 719 881 4041 200 358 514 669 823 975

40 428 391 450 441 472 491 494 541 517 591 539 642 49 41 561 693 583 744 604 796 626 847 648 899 670 952 48 42 691 9004 713 9057 734 9110 756 9163 777 9217 799 9271 47 43 820 325 841 380 862 435 884 490 905 545 926 601 46 44 947 657 967 713 988 770 7009 82. 7030 884 7050 94245 45 7071 1.0000 7092 1.0058 7112 1.0117 133 1.0176 153 1.0235 173 1.0295 44

60'

60'

50'

50'

40'

40' 30'

30' 20'

20' 10' .10'

.d cos

cot

cos

cot

cos

cot

cot

cot

cos

cos

cos

cot

t .1

TABLE VI (continued) SINES, COSINES, TANGENTS, COTANGENTS (continued) sin S O'

tan sin O' 10'

tan sin 10' 20'

tan sin 20' 30'

tan sin 30' 40'

tan 40'

sin 50'

tan tb 50' O

46 7193 1.0355 7214 1.0416 7234 1.0477 7254 1.0533 7274 1.0599 47 314 .0724 333 .0786 353 .0850 373 .0913 392 .0977 48 431 .1106 451 .1171 470 .1237 490 .1303 509 .1369 623 .1778 49 547 .1504 566 .1571 585 .1640 604 .1708

7294 1.0661 43 412 .1041142 528 .1456 41 642 .1847 40

50 660 1.1918 7679 1.1988 7698 1.2059 7716 1.2131 7735 1.2203 51 771 .2349 790 .2423 808 .249r 826 .2572 844 .2647 951 .3111 52 880 .2799 898 .2876 916 .2954 934 .3032 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 355 .5204 56 290 .4826 307 .4919 323 .5013 339 .5108 450 .5798 57 387 .5399 403 .5497 418 .5597 434 .5697 58 480 .6003 496 .6107 511 .6212 526 .6319 542 .6426 59 572 .6643 587 .6753 601 .6864 616 .6977 631 .7090

7753 1.2176 39 862 .2723 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 18675 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 112 .2113 65 9063 .1445 075 .1609 088 .1775 100 .1943 66 135 .2460 147 .2637 159 .2817 171 .2998 182 .3183 .4142 250 .4342 228 .3945 239 67 205 .3559 216 .3750 68 272 .4751 283 .4960 293 .5172 304 .5386 315 .5605 69 336 .6051 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 .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 19 492 3.0178 502 3.0475 18 455 .9042 465 .9319 474 .9600 483 .9887 555 .2371 17 511 3.0777 520 3.1084 528 3.1397. 537 3.1716 546 .2041 596 .4124 605 .4495 16 563 .2709 572 .3052 580 .3402 588 .3759 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 13 775 .6382 12 769 .5736 744 .3315 750 1.3897 757 .4494 763 .107 811 5.0658 11 781 .7046 787 .7729 793 .8430 799 .9152 805 .9894 816 .1446 822 5.2257 827 5.3093 833 5.3955 838 5.4845 843 .5764 10

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 051 10.078 954 85 962 11.430 964 11.826 967 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 9996 34.368 997 89 9998 57.290 9999 68.750 9999 85.940 9999

60 cos

60' 50' cocos cot cos

50' 40' os

40' 30' cot 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'

10' cos

6.1970 .9682 7.9530 9.2553 11.059 13.727 18.075 26.432 49.104 343.77

10' cOtott

9 8 7 6 5 4 3 2 1 0

TABLE Degree of

VII

Deflections for Sub Chords for Short Radius Curves. sub chord = sin of

Radius

R

50

J def. angle

Length ofare

Curve

sin. I def.ang.

12.5 Ft.

15 Ft.

20 Ft.

30

193.18 181.39

*51

2 171

20 58'

I 59'

20 25

310'

171.01 161.80

20 6' 20 13'

2033 2 41'

30 2' 3033'

4012

153.58 146.19 139.52

2 20o'

2 49'

20 27 2° 34'

27'

3 ' 3 55

4° 40' 4° 54' 53 8'

102.06

133.47

2

3

13'

4

5

102.53

127.97 122.92 118.31 I14.06

2 4 ' 2 55' 3 02' 3 09'

2

4 29, 4 4 4 51' 2' 5

3

16'

3 3 3 3 3

54'

°

Io6.5o

3

22'

4

02

5

580

103. 14 00.oo0

3 29' 30 35

4 4

10'

534

18'

50 44

°

32 340 368

380 40 40

440 °

46

48 ° 0 ° 52

110.11

540 56

60

T = Rtan j I S5

8

D =S

Sin. Sin.

50 tan D Dx =

°

°

22

o101.15

101o.33 O101.48 I.66

.85 102.29

5 36' 5 50' 6004 6 17'

102.76 103.oo0 103.24 Io3.54

60 44

104.14

60570 7 I'

104.43 104.72

6

31'

1o3.84

Chord def. = chord

R

50

I

4 26'

23

for 100 ft.

43

58'

O 13'

Sin. ID

D

Sin.

29' 38' 46'

3

,18

R = T cot. J I

tan j I

Sin.

41

25 Ft.

E = R ex. sec J 1 E = T tan j I

No..chords = Tan. def.= I 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

°

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 .0o745, and the product by the distance.

GENERAL DATA Square the altitude, divide by twice- the Add quotient to base for hypotenuise.

RIGHT ANGLE TRIANGLES.

base.

Given Base Ioo, Alt. 1o.1o2 +200=.5.

I00+.5=oo.5 hyp.

Given Hyp. Too, Alt. 25.252+2oo=3.25. 00oo--3.125 =96.875 = Base.

Error in first example, .oo2; 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 0.574 d ,where d is the distance in miles.

The correction for curvature alone is closely, id*. The combined correction is negative.

PROBABLE ERROR. Ifd d, d,,etc. are the discrepancies of various 2 results from the mean, and if =the sum of the squares of these differences and n=the number of observations, then the probable eror of the mean= -. 0.67451

TABLE VIII TANGENTS AND EXTERNALS TO A I

10 go 10' 20' 30' 40'

50' go 20" 10' 20' 30' 40' 50'

so

30

10' 20' 30' 40' 50'

o40

10' 20' 30' 40' 50' 50.

10' 20' 30' 40' 50' 60 10' 20' 30' 40' 50' 70

10' 20'

30' 40'

T 50.00 58.34 66.67 75.01"83.34 91.68 100.01 108.35 116.68 125.02 133.36 141.70 150.04 158.38 166.72 175.06 183.40 191.74 200.08 208.43 216.77 225.12 233.47 241.81 250.16 258.51 266.86 275.21 283.57 291.92 300.28 308.64 316.99 325.35 333.71 342.08 350.44 358.81 367.17 375.54 383.91 392.28 400.66

10' 409.03 20' 417.41 30' 425.79 40' -434.17 442.55 90 450.93 10' 459.32 20' 467.71 30' 476.10 40' 484.49 50' 492.88 501.28 tO10 10' 509.68 20' 518.08 30' 526.48 40' 534.89 50' 543.29

I=101

E

I

T

551.70 560.11 568.53 .388 576.95 .491 T 585.36 .606 593.79 .733 .001 602.21 .873 610.64 1.024 1.188 20' 619.07 30' 627.50 1.364 40' 635.93 1.552 1.752 50' 644.37 10 C. 1.964 o30 652.81 2.188 10' 661.25 20' 669.70 2.425 .06 30' 678.15 2.674 E 40' 686.60 2.934 .00 50' 695.06 3.207 o 703.51 3.492 140 10' 711.97 3.790 20' 720.44 4.099 4.421 30' 728.90 4.755 40' 737.37 50' 745.85 5.100 15o C T 5.459 .09 50o 754.32 5.829 10' 762.80 E 6.211 *00 20' 771.29 6.606 30' 779.77 40' 788.26 7.013 50' 296.75 7.432 805.25 7.863 10' 8.307 8.762 9.230 200 C 30' 830.76 9.7102 40'" 839.27 10.202 .13 50' 847.78 E 856.30 10.707 70 11.224 .006 10' 864.82 11.753 873.35 12.294 30' 881.88 12.847 40' 890.41 13.413 50' 898.95 I0 13.991 907.49 o 10 25 C 916.03 14.582 20' 924.58 T 15.184 15.799 .16 30' 933.13 16.426 40' 941.69 E 17.065 50' 950.25 17.717 go90 958.81 18.381 10' 967.38 19.058 20' 975.96 19.746 30' 984.53 20.447 40' 993.12 21.161 30o C. 50' 1001.7 T 200 1010.3 21.887 22.624 .19 10' 1018.9 E 23.375 20' 1027.5 24.138 .008 30' 1036,1 24.913 40' 1044.7 25.700 50' 1053.3 .218 .2917

T=

no

10' 20' 30' 40' 50' o20

Rtm =puI_

E II=20o 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 79.671 181.092 82.525 83.972 85.431 86.904 88.389 89.888 91.399 92.924 94.462 96.013

I

T

10 CURVE. E

1 I=30

210

1061.9 97.577 10' 1070.6 99.155 1079.2 100.75 20' 50+ Soc.* 30' 1087.8 102.35 T 40' 1096.4 103.97 .06 50' 1.105.1 105.60 E 1113.7 107.24 X006 220 10' 1122.4 108.90 20' 11131.0110.57 30' 1139.7 112.25 40' 1148.4 113.95 50' 1157.0 115.66 1165.7 117.38 100 C. 230 10' 1174.4 119.12 T 1183.1 120.87 20' .13 30' 1191.8122.63 E 40' 1200.5 124.41 .011 50' 1209.2 126.20 1217.9 128.00 240 10' 1226.6 129.82 20' 1235.3 131.65 30' 1244.0 133.50 40' 1252.8 135.35 150 C. 50' 1261.5137.23 1270.2 139.11 T 250 .19 10' 1279.0 141.01 E 1287.7 142.93 20' .017 30' 1296.5 144.85 40' 1305.3 146.79 50' 1314.0 148.75 1322.8 150.71 260 10' 1331.6.152.69 20' 1340.4 154.69 200 C. 30' 1349.2 156.70 40' 1358.0 158.72 T 50' 1366.8 160.76 .26 E 270 11375.6 162.81 .022 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 1428.6 175.41 280 250 C. 10' 1437.4 177.55 T 20' 1446.3 179.72 .32 30' 1455.1 181.89 E 40' 1464.0 184.08 .028 50' 1472.9 186.29 1481.8188.51 290 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 30 T 1535.3 202.12 .39 10' 1544.2 204.44 E 20' 1553.1 206.77 .034 30' 1562.1 209.12 40' 1571.0 211.48 50' 1580.0 213.86

E = R exsec Ij I

+ 5°T T .10 .E .013

10 C. .19 E .025

15 C. T .29 E .038

200 C. .39 E T

.0625

T

.49

E .065

300 C

T .59 E

.08

TABLE VIII TANGENTS AND EXTERNALS TO A I

T

E

I=4001

I

" 1589.0 1598.0 1606.9 1615.9 1624.9 1633.9 1643.0 10' 1652.0 20' 1661.0 30' 1670.0 40' 1679.1 1688.1 50' So30 1697.2 10' 1706.3 20' 1715.3 30' 1724.4 40' 1733.5 50' 1742.6 340 .1751.7 6 10' 17 9.8 20' 1770.0 30' 1779.1 40' 1788.2 50' 1797.4 850 1806.6 10' 1815.7 20' 1824.9 30' 1834.1 40' 1843.3 50' 1852.5 88360 1861.7 10' 1870.9 20' 1880.1 30' 1889.4 40' 1898.6 50' 1907.9 1917.1 $70 10' 1926.4 20' 1935.7 30' 1945.0 40' 1954.3 50' 1963.6 1972.9 380 10' 1982.2 20' 1991.5 30' 2000.9 40' 2010.2 50' .2019.6 2029.0 890 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 10' 20' 30' 40' 50'

410 216.3 10' 218.7 + 20' 221.1 So C. T 30' 223.5 40' 226.0 .13 E 50' 228.4 .023 230.9 420 10' 233.4 20' 235.9 30' 238.4 241.0 40' 243.5 50' 246.1 100 C, 430o 10' 248.7 T 20' 251.3 .26 253.9 30' E 256.5 .046 40' 259.1 50' 44o0 261.8 264.5 10 267.2 20' 269.9 30' 272.6 40' 275.3 150C. 50' T 278.1 .40 450 10' 280.8 E 283.6 .070 20' 286.4 30' 289.2 40' 292.0 50' 294.9 460 297.7 10' 300.6 20' 303.5 20 C,. 30' 306.4 40' T 305.3 50' .53 470 312.21 E .093 315.2 10' 318.1 20' 30' 321.1 324.1 40' 327.1' 50' 330.2 480 333.2 25 0 C, 10' 336.3 20' T 339.3 .67 30' 342.4 40' 345.5 .117 50' 348.6 490 351.8 10' 354.9 20' 358.1 30' 361.3 40' 364.5 50' C. 800 367.7 300 T 371.0 10' .80 374.2 20' E 377.5 .141 30' 380.8 40' 384.1 50' T=Rtan11E

T

E

1=50o

10 CURVE

T

E

2732.9 2743.1 2753.4 2763.7 2773.9 2784.2 2794.5 2804.9 2815.2 2825.6 2835.9 2846.3 2856.7 2867.1 2877.5 2888.0 2898.4 2908.9 2919.4 2929.9 2940.4 2951.0 2961.5 2972.1 2982.7 2993.3 3003.9 3014.5 3025.2 3035.8 3046.5 3057.2 3067.9 3078.7 3089.4 3100.2 3110.9 3121.7 3132.6 3143.4 3154.2 3165.1 3176.0 3186.9 3197.8 3208.8 3219.7 3230.7 3241.7 3252.7 3263.7 3274.8] 3285.8 3296.9 3308.0 3319.1 3330.3 3341.4 3352.6 3363.8

618.4 622.8 627.2 631.7 636.2 640.7 645.2 649.7 654.3 658.8 663.4 668.0 672.7 677.3 682.0 686.7 691.4 696.1 700.9 7.05.7 710.5 715.3 720.1 725.0 729.9 734.8 739.7 744.6 749.6 754.6 759.6 764.6 769.7 774.7 779.8 784.9 790.1 795.2 800.4 805.6 810.9 816.1 821.4 826.7 832.0 837.3 842.7 848.1 853.5 858.9 864.3 869.8 875.3 880.8 886.4 892.0 897.5 903.2 908.8 914.5

i

I=t o

" 510 i2142.2 387.4 2151.7 390.7 + 10' 2161.2 394.1 5C. 20' 2170.8 397.4 T 30' 2180.3 400.8 .17 40, 2189.9 404.2 E 50' .037 2199.4 407.6 520 2209.0 411.1 10' 20' 2218.6 414.5 2228.1 418.0 30' 2237.7 421.4 40' 2247.3 425.0 50' 2257.0 428.5 100 C. 5830 2266.6 432.0 10' T. 2276.2 435.6 20' .34 2285.9 439.2 30' E 2295.6 442.8 .075 40' 2305.2 446.4 50' 2314.9 450.0 540 2324.6 453.6 10' 2334.3 457.3 20' 2344.1 461.0 30' 2353.8 464.6' 40' 2363.5 468.4 15oC. 50' T 2373.3 472.1 .51 550 10' 2383.1 475.8 E 2392.8 479.6 .116 20' 2402.6 483.4 30' 2412.4 487.2 40' 2422.3 491.0 50' 2432.1 494.8 5860 2441,9 498.7 10' 2451.8 502.5 20' 2461.7 506.4 20 C,. 30 2471.5 510.3 40' 2481.4 514.3 .68 50' E 570 2491.3 518.2 .151 2501.2 522.2 10' 2511.2 526.1 20' 2521.1 530.1 30' 2531.1 534.2 40' 12541.0 538.2 50' 2551.0 542.2 580 2561.0 546.3 250 C. 10' 2571.0 550.4 20' y 2581.0 554.5 30' .85 2591.0 558.6 40' 2601.1 562.8 .189 50' 2611.2 566.9 590 2621.2 571.1 10' 2631.3 575.3 20' 2641.4 579.5 30' 2651.5 583.8 40' 2661.6 588.0 50' 0 2671.8 592.3 30 C. 60 .2681.9 596.6 1.02 10' 2692.1 600.9 20' E 2702.3 605.3 .227 30 2712.5 609.6 40' 2722.7 614.0 50'

= R exsec '/2 1

+ 5 OC. T .21 E .056

100 C. T .42 E .112

150 C. T .63 E .168

0

20 C. .81 E .225

25 C. T 1.05 E 2.83

300 C. T E .340

TABLE VIII TANGENTS AND EXTERNALS TO A 10 CURVE.

S 61*

10' 20' 30' 40' 50' 62o 10' 20' 30' 40' 50' 680 10' 20' 30' 40' 50' 64o 10' 20'

30' 40' 50' 680 10', 20' 30' 40' 50' 660 10' 20' 30' 40' 50' 6to 10' 20' 30' 40' 50'

680 10' 20' 30' 40' 50'

48

10' 20'

30' 40' 50' 70 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 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

E1 I=700 920.2 925.9 931.6 + 937.3 SoC. 943.1 948.9 .25 E 954.8 .080 960.6 966.5 972.4 978.3 984.3 990.2 100 C. 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 T6 1070.2 E 1076.6 .240 1082.9 1089.3 1095.7 1102.2 1108.6 1115.1 1121.7200 C 1128.2 T 1134.8 1.02 E 1141.4 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 1286.5 .485 1293.6 1300.9

710 10' 20' 30' 40' 50' 720 10' 20' 30' 40' 50'

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 4239.7 730 10' 4252.6 20' 4265.6 30' 4278.5 40' 4291.5 50' 4304.6 740 4317.6 10' 4330.7 20' 4343.8 30' 4356.9 40' 4370.1 50' 4383.3 7'50 4396.5 10' 4409.8 20' 4423.1 30' 4436.4 40' 4449.7 50' 4463.1 4476.5 760 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 40P 4612.2 50' 4626.0 o 78 4639.8 10' 4653.6 20' 4667,4 30' 4681.3 40' 4695.2 50' 4709.2 790 4723.2 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

E I

= &80

1308.2 1315.6 1322.9 1330.3 O C. 1337.7 1345.1 .30 1352.6 1360.1 T 1367.6 1375.2 1382.8 1390.4 1398.0 100 C 1405.7 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 E 1508.6 .332 1516.7 1524.9 1533.1 1541.4 1549.7 1558.0 1566.3 200 C. T 1574.7 1583.1 1.22 1591.6 E 1600.1 ,445 1608.6 1617.1 1625.7 1634.4 1643.0 1651.7250 C, 1660.5 T 1669,2 1.53 1678.1 E58 1686.9 .558 1695.8 1704.7 1713.7 1722.7 1731.7 1740.8 300 C. 1749.9 T 1759.0 1.84 E 1768.2 1777.4 .671 1786.7 1796.0

1 81o 10' 20' 30' 40' 50' 820 10' 20' 30' 40' 50' 830 10' 20' 30' 40' 50' 840 10' 20' 30' 40' 50' 850 10' 20' 30' 40' 50' 8860 10' 20' 30' 40' 50'

870 10' 20' 30' 40' 50' 880 10' 20' 30' 40' 50' 10' 20' 30' 40' 50' 900 10' 20' 30' 40' 50'

-'RtanI/sI

1805.3 1814.7 1824.1 1833.6 1843.1 1852.6 1862.2 1871.8 1881.5 1891.2 1900.9 1910.7 1920.5 1930.4 1940.3 1950.3 1960.2 1970.3 1980.4 1990.5 2000.6 2010.8 2021.1 2031.4 2041.7 2052.1 2062.5 2073.0 2083.5 2094.1 2104.7 2115.3 2126.0 2136.7 2147.5 2158.4 2169.2 2180.2 2191.1 2202.2 2213.2 2224.3 2235.5 2246.7 2258.0 2269.3 2280.6 2292.1) 2303.5 2315.0 2326.6 2338.2 2349.8 2361.5 2373.3 2385.1 2397.0 2408.9 2420.9 2432.9

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

5359.6 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

E I1=90o

T

E! R ex c1/2

I

50 C.

T .36 E .149

100 C. T .72 E .299

150 C. T 1.09 .E .450

200.C. 1.45 E .603

250 C.

T 1.83 E .756

300 C.

T

2.20 E .910

1

TANGENTS AND EXTERNALS TO A Io CURVE

TABLE VIII i

T

E

11=1

910 10' 20' 30' 40' 50' 920 10' 20' 30' 40' 50'

j1

5830.5 2444.9 5847.5 2457.1 -t5864.6 2469.3 s5 C. T 5881.72481.5 .43 5898.8 2493.8 E 5916.0 2506.1 5933.2 2518.5 .200 5950.5 2531.0 5967.9 2543.5 59853 2556.0 6002.7 2568.6 6020.2 2581.3 930 6037.8 2594.0 10o C. 10' 6055.4 2606.8 T 20' 6073.1 2619.7 .86 30' 6090.8 2632.6 E 40' 6108.6 2645.5 .401 50' 6126.4 2658.5 2671.6 940 6144.3 10' 6162.2 2684.7 20' 6180.2 2697.9 30' 6198.3 2711.2 40' 6216.4 2724.5 50' 6234.6 2737.9 150 C. T 9So0 6252.8 2751.3 1.30 10' 6271.1I 2764.8 E 20' 6289.4 2778.3 .604 30' 6307.9 2792.0 40' 6326.3 2805.6 *50' 6344.8 2819.4 960 6363.4 2833.2 10' 6382.1 2847.01 20' 6400.8 2861.0 30' 6419.5 2875.0 200 C. T 40' 6438.4 2889.0 50' 6457.3 2903.1 1.74 E 6476.2 2917.3 .809 to 10' 6495.2 2931.6 20' 6514.3 2945.9 30' 6533.4 2960.3 40' 6552.6 2974.7 50' 6571.9 2989.2 980 6591.2 3003.8 10' 6610.6 3018.4 250 c.

20' 6630.1 3033.1 T 30' 6649.6 3047.9 2.18 40' 6669.2 3062.8 E 50' 6688.8 3077.7 1.02 990 6708.6 3092.7 10' 6728.4 3107.7 20' 6748.2 3122.9 30' 6768.1 3138.1 40' 6788.1 3153.3 50' 6808.2 3168.7 300 C. 1000 6828.3 3184.1 10' 6848.5 3199.6 2.62 20' 6868.8 3215.1 E 30' 6889.2 3230.8 1.22 40' 6909.6 3246.5 50' 6930.1 3262.3 T

I

1010 10' 20' 30' 40' 50' 102o0 10' 20' 30' 40' 50' 10o0 10' 20' 30' 40" 50' 1040 10' 20' 30' 40' 50' 1050 10' 20'

30' 40' 50' 1060 10' 20' 30' 40' 50' 107o0 10' 20' 30' 40' 50' 1080 10' 20' 30' 40' 50' 1090 10' 20' 30' 40' 50' 1100 10' 20' 30' 40' 50'

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 3919.4 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

R tan 'iuI

1=1llo1

.4.

so C. T .51 E .268

I

T

E!-

1110 10' 20' 30' 40' 50'

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 8684.0 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 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 4983 1 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

if2 0

10' 20' 30' 40' 50' 10 C., 1130 10' T 20' .103 30' E 40' .536 50'

1140

10' 20' 30' 40' 150 C. 50' T 1.56 1150 10' E 20' .806 30' 40' 50" 1160 10' 20' 200 C, 30' 40' T 50' 2.08 E 1170 1.08 10' 20' 30' 40' 50' 1180 10' 250 C. 20' T 30' 2.61 40' E 50' 1.36 o 119 10' 20' 30' 40' 50' 300 C. 1200 10' 3.14 E1.63 20' 30' 1.5 40' 50'

E= R exsec/2

1

120tzo

+ 5 C

T .62 E .360

I0 r T 1.25 721

150 C. 1.93 E 109

200

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)

0'C

R

Feet

0-20 17189 0-40 1-0 1-20 1-40 2 -0 2-20 2-40 3-0 3-20 3-40 4-0 4-20 4-40 5 6 7

8594 5730 4297 3438 2865 2456 2149 1910 1719 1563 1433 1323 1228 1146 955.3 819.0

301 28 Inch Inch .08

.07

.16 .14 .24 .20 .31 .271 .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 1.41 1.23 1.65 1.44

26

Inc

1-24 as 2o1c1R h InchInch 0

.061 .05 .041.03

.12 .10 .18 .15 .23 .20 .29 .25 .35 .30 .41 .35 .47 .40 .53 .45 .59 .50 .65 .55 .71 .60 .77 .65 .83 .70 .89 75 1.06 .90 1.24 1.05

.08 .07 .13 .10 .17 .13 .21 .17 .25 .20 .29 .23 .33 .27 .38 .31 .42 .35 .46 .38 .50 .42 .55 .45 .59 .48 .63 .52 .76 .62 .89 .73

30j

22 20 28 26 24 Inch

h inh Inch Inch

8 716.8 1.88 1.64 1.42 1.20 1.01

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

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

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 6.071

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

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

1.35 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 3.88 3.29

.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

TABLE X

SHORT RADIUS CURVES Radius Feet

Chord Feet

Central Angle

Deflection Angle

35 45 50 60 75 100 120 150 190 200 225 240 250 275 288 300 350 376 400 410

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

Deflection for 1 Foot 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 antle from P. C. to P. T. by central angle of chord, and multiply by length of chord.

TABLE

XI

INCLINED DISTANCE OF 100 FT. REDUCED TO HORIZONTAL. Slope

Horizontal Distance

Correction

Rise

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 99.999 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

"

I

Horionta Distance

Slope

99.027 98.965 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

800Q' 15' 30' 45' 9 00 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

Correction

Rise

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 2.006 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 3.401 3.521 3.637 3.754

0.139 0.143 0.148 0.152 0.156 0.161 0.165 0.169 0.174 0.178 0.182 0.18% 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 0.211

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

TABLE XI I MINUTES IN DECIMALS OF A DEGREE. 0'30" .00833 10'30" .17500 20'30" .34167 30'30" .50833 40'30" .67500 50'30" .84167 .85000 0 .350 31 00 .516671 41 00 ,68333 5100 1oo00 .01667 1100 .183332.18333 30 2 00 30 300 30 4 00 30 500 30 600

30

30 8 00 30 9 00 30 1000

30 2200 30 23 00 30 2400 30 25 00 30 2600

.35833 30 .36667 3200 .37500 30 .38333133 .39167 30 .4000034 00 .40833 30 .41667 35-00 .42500 30 .433333600

.52500 30 .69167 30 .533342 oo .70000 52 00 30 .54167 30.70833 5000 43 00 .71667 5300 30 .55833 30 .72500 733334 0 .56667 44 00 .57500 30 .741671 30 .58333 45 00 .75000 55 00 30 30 .75833 .59167 .600004600 .766675600 .60833 30 .77500 30

.12500 30 .29167 30 .133331800 .30000 2800 .14167 30 .30833 30 .15000 19 00 '.31667 29 00 .15833 30 .32500 30 .16667 2000 .33333 3000

.45833 30 .466673800 .47500 30 .48333 39 00 30 .49167 .50000 4000

.62500 30 .63333 48 00 .64167 30 .65000 4900 .65833 30 .666675000

.02500 30 .03333 1200 30 .04167 .05000 1300 .05833 30 .0666714 00 .0750011 30 .08333 15 00 .09167 30 .1000016 00

.10833

.19167 .20000 .20833 .21667 .22500 .23333 .24167 .25000 .25833 .26667

30 .27500

30 .44167

30

.85833 .86667 .87500 .88333 .89167 .90000 .90833 .91667 .92500 .93333

.94167

.79167 30 .95833 .80000 58 00 .96667 .80833 30 .97500 .81667 59 00 .98333 .82500 30 .99167 .833336000 1.00000

TABLE XIII--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 V III) by the degree of curve, in order to obtain the true tangents, or externals. Intermediate values may be obtained by interpolation.

FOR TANGENTS ADD

Angle

5 S

100

15e

°

20

25

°

03 .06

09

13

16

160°

04

14

19

24.29

20

06

.13

S6o

.08

16

80

10 11

S

10

19

.26 .32

24 .33

19 .29 22 34

40

.26 .40 .53 .67 60 76 .30 44

13 15

00 866 600 66o

17 19 .21 23

100 ° T6 800 865

76 1.02 25 .51 .27 .56 .83 1.12 30 ,61 .91 1.22 33 66 1.00 1.33

.51 .57 .63 .69

°

35

°

4501 50.

.34

.45 .51

49 .588 67

.75 .83

°

60

°

65

3438

.39451.5 .58 .65

.39

°

55

19.22 .25.28.31

42

700

,4

58 .63 .68

3 .72

79

.84 .90

90 .99 1.06 1.14

.80 .93 1.06 1.20 1.34 1.49 1.64 1.79 1.94 91 1.06 1.21 1.37 1.52 1.70 1.87 2.04 2.21

1.28 1.40 1.53 1.68

62 1.25 1 93 2.52 3.16

1.54 1.69 1.84 2.02

1.80 1.98 2.15 2.36

2.20 2.57 2.40 2.80 2.62 3.06 3.143.67

3.814

1.36 1.52 1.71 1.88

1.54 1.72 1.94 2.13

1.72 1.92 2.17 2.38

2.0 2.27 2.46 2 7

2.33 2.60 2.57 2.87 2 78 3.10 3.053.40

1.912.10 2.14 2.35 2.38 2.60 2.63 2.88

2.29 2.56 2.83 3.13

3.16 3.44 3.47 3.78 3.78 4.12 4.1414.55

2.88 3.16 3.44 3.77

2.94 3.32 3.70 4.10 4.50 3.20 3.61 4.02 4.40 4.98 3.503.954.404.885.37 4.21 4 76 5 88 6.43

5.31

4.91 5.38 5.85 7 01

2.48 2.77 3.07 3.39 3.72 4.09 4.46 4 8Q 5.32 5.83 6.34 7 60

455.11 5.776 447 127.80 8.50 9 22

FOR EXTERNAL Central Angle

°

40

.68 .85 1.02 1.19 76 .95 1.14 1.32 .84 1.05 1.27 1.49 .93 1.16 1 401.64

36 .72 1.09 1.45 1.83 39 791.19 1.55 2.00 .43 .86 1.30 1.74 2.18 51 1.03 1.56 2.08 2..61

900 60 1000 110

°

39 49 .59 .69 79 .89 .99 1.091.20 1.29 1.39 47 .58 .69 70 .81 .92 1041.291.421.541.6688

40 460

.34 .38 42 46

°

30

10°

1S

or Cuava

DEOaRE

Central

ADD

Danasz or Cuonv

5

°

10

°

15

20

°

250

°

30 135

°

40

45

°

50

°

55

°

o

60

65

°

70

°

10 16 200

.001 .003 .004 .006 .007 .008 .009 .003.007 .010 .014 .018 .023 .027 .006.011 .017 .022 .028 .034 .038

S00 850

.013.025 .038 .051 .065 .078 .090 .103 .116 .129 .149 .170 .179 .188 .018.035 .054 .072 .086 .109 .131153 .175 .197 .213 .230 .247 .264

400° 45

.023 .046 .070 .093 .117 .030 .060 .093 .119 .153

500 50 60 ° 66

.037 .075 .116 .151 .189 .227 .266

700° T 80 ° 85

.080 .159 .095.182 .110.220 .128 .259

.240 .286

90° 100 110

.149 .174 .200 .268

.450 .522 .604 .806

120

.360 .721 1.08

826

95

.009 .018 .027

.046 .03 .142 .056 .112 .168 .067 135 .204

.299 .350 .401 .536

.011 .012 .014 .029 .032 .035 .045 .051 .057

.015 .017 .018 .039 .043 .047 .063 .070 .076

036 .046 .056 .065 .074 .083 .093 .106

.188 .236 .225 .283 .273 .343

.321 .403 .383 .480 .332 .445 .568 .391 .524 .657 .603 .756 .706 .985 .809 1.01 1.08 1.35

.141 .184

.172 .216

.203 .254

.568 .678 .787 .926

.265 .277 .325 .351

127

.290 .315 .378. .411

135

.341 .445

.305 .345 .34 .425 .467 .508 .550

.283 .332 .381 .340 .398 .457 .412 .483 .554 .485 .578 .671 .790

.234 .289

120

.020 .051 .083

.420 .479 .516 .575 .625 .697

.652 .735 .777 .877 .903 1.02 1.06 1.20

.530 .638 .711

.582 .697 .845

.819 .906 .977 1.07 1.13 1.25 1.34 1.47

.994 1.18 1.38 1.62

.641 .700 .774 .851 .922 1.01 1.08 1.29 1.50 1.76

.9101.07 1.22 1.38 1.54 1.70 1.87 2.03 1.06 1.25 1.43'1.62 1.80 1.99 2.18 2.38 1.22 1.43 1.64 1.85 2.06 2.28 2.50 2.73 1.63 1.91 2.20 2.48 2.78 3.05 3.35 8.66

1.17 1.39 1.62 1.91 2.20 2.58 2.96 3.96

1.45 1.82 2.19 2.57 2.95 3.33 3.72 4.11 4.50 4.91 5.32

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DISTANCES FROM SIDE STAKES FOR CROSS-SECTIONING.

S

SLOPE 1I/o TO 1. ROADWAY OF ANY WIDTH.

01

I r

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 St 25 26 27 28 29

9

31 32 33 34 36 37 398 40 41

42 43 44 45 46

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0

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0.00 1.50 3.00 4.50 6.00 7.50 9.00 10.50 12.00 13.50 15.00 16.50 18.00 19.50 21.00 22.50 24.00 25.50 27.00 28.50 30.00 31.50 33.00 34.50 36.00 37.50 39.00 40.50 42.00 43.50 45.00 46.50 48.00 49.50 51.00 52.50 54.00 55.50 57.00 58.50 60.00 61.50 63.00 64.50 66.00 67.50 69.00 70.50 72.00 73.50 75.00

0.15 1.65 3.15 4.65 ,6.15 7.65 9.15 10.65 12.15 13.65 15.15 16.65 18.15 19.65 21.15 22.65 24.15 25.65 27.15 28.65 30.15 31.65 33.15 34.65 36.15 37.65 39.15 40.65 42.15 43.65 45.15 46.65 48.15 49.65 51.15 52.65 54.15 55.65 57.15 58.65 60.15 61,65 63.15 64.65 66.15 67.65 69.15 70.65 72.15 73.65 75.15

0.30 1.80 3.30 4480 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

.3

0.45 1.95 3.45 4.95 6.45 7.95 9.45 10.95 12,45 13.95 15.45 16.95 18.45 19.95 21.45 22.95 24.45 25.95 27.45 28.95 30.45 31.95 33.45 34.95 36.45 37.95 39.45 40.95 42.45 43.93 45.45 46.95 48.45 49.95 51.45 51.30 52.80 52.95 54.30 54.45 55.80 15.95 57.30 57.45 58.80 58.95 60.30 60.45 61.80 61.95 63.30 63.45 64.80 64.95 66.30 66.45 67.80 67.95 69.30 69.45 70.80 70.95 72.30 72.45 73.80 73.95 75.30 75.45

THE NATIONAL HOLYOKE NEW YORK

mi"

.41

.5

0.60

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 70.75

2.10 3.60 5.10 6.60 8.10 9.60 11.10 12.60 14.10 15.60 17.10 18.60 20.10 21.60 23.10 24.60 26.10 27.60 29,10 30.60 32.10 33.60 35.10 36.60 38.10 39.60 41.30 42.60 44.10 45.60 47.10 48.60 50.10 51.60 53.10 54.60 56.10

57.60

59.10 60.60 62.10 63.60 65.10 66.60 68.10 69.60 71.10 72.60 74.10 75.60

BLANK

CHICAGO

1 .6

.7

1

.8

1.20 2.70 4.20 5.70 7.20 8.70 10.20 11.70 13.20 14.70 16.20 17.70 19.20 20.70 22.20 23.70 25.20 26.70 28,20 29.70 31.20 32.70 34.20 35.706 37.20 38.70 40.20 41.70 43.20 44.70 46.20 47.70 49.20 50.70 52.20 53.70 55.20 "56.70 58.20 59.70 61.20 62.70 64.20 65.70 67.20 68.70 70.20 71.70 73.20 74.70 76.20 76.20 BOOK COMPANY MASSACHUSETTS

BOSTON

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 29.40 30.90 32.40 33.90 35.40 36.90 38.40 39.90 41.40 42.90 44.40 45.90 47.40 48.90 50.40 51.90 53.40 54.90 56.40 57.90 59.40 60.90 62.40 63.90 65.40 66.90 68.40 69.90 71.40 72.90 74.40 75.90

1.05 2.55 4.05 5.55 7.05 8.55 10.05 11;55 13.05 14.55 16.05 17.55 19.05 20.55 22.05 23.55 25.05 26.55 28.05 29.55 31.05 32.55 34.05 35.55 37.05 38.55 40.05 41.55 43.05 44.55 46.05 47.55 49.05 50.55 52.05 53.55 55.05 56.55 58.05 59.55 61.05 62.55 64.05 65.55 67.05 68.55 70.05 71.55 73.05 74.55 76.05

SAN FRANCISCO

1.9 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

64.36 65.85 70.35 71.85 73.35 74.85

76.35 76.35

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 24

27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

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