Fremling Field Notebook 1966 - 1967

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DSANCEs FaOM

SIDE

TSmTs FOR COsS-SECTIONING

SLOPs 1 TO 1. ROADWAY or ANY WIDT 0

.1

1:00 3.00 4.00

7.00 8.00 10.00 11.00 13.00

16.0 17r.00

1.1o

2.10 3.10 6.10 .10 7.10 8.10 9.10 10.10 11.10 12. 10 13.10 14.10 15.10 16.10

17.10 18.40 19.10 20.00 20.

21.10 22.10 22.00 23.10 23.00 24.00i 24.10 25.10 25.00 26.10 26.00 27.00 27.10 28.10 28.00 29.00 29.10 30.10 31.10 3230 31.

33.10 34.10 35 10 36. 10 33.00 37.10 38.10 39.10 37.00 40.1I0 41.10 42.10 42.00 43.10 43.00 44.10 44.0 45.10 46.10 45.0 48.00 49.00 50.00

47,10 48.10 49.10 50.10

.2

.3

0.20 1.20

0.30 1.30 2.30 3.30 4.30

4.20 8.20

.4 1.40 2.40 3.40 4.40

4.20 8.20 9.20 40.20 11.20 12.20 13.20 14.20 15.20 16.20 17.20 18.20 19.20 20.20 21.20 22.20 23 20 24.20 25.20 26.20 27.20 28.20 29.20 32 20 33.20 34 20 35.20 36.20 39.20 40.20 41.20 42.20 45.20 46.20 47.20 50.20

2.80 85.30 6.30 0.30 11.30 12.30 13.30 14.30 15.30 16.30 17.30 18.30 19.30 20.30 21.30 22,30 23.30 24.30 25.30 26.30 27,30 28.30 29.30 30.30 31.30 3.30 f3jo

7.40 8:40 10.40 11.40 12.40 13.40

18.40

21.40 23.40 25.40

29.40 30.40

34 30 34.40 35.301 35.40 36.30 37.30 3)8.40 38.30 39.30 39.40 40.30 40.40 41.00 41,40 42.30 43.40 43.30 44.30 45.30 45.40 46.30 47.30 48 .30 48.40 49.30 50,30 49.40 50.40

.5 0.50 1.50 42.50 3.50 6.50 4.50 7.50 5.50 8.50 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 34.50

.6 1.80

3:60

1.7

.8

0.70 1.70

0.80

2.70 3:70

4.82

5,70

6.60 7.60

42.50.

43,50 44.50 45.50 46.50 47.50 46.50 49.50 50.50

4.90

3.80

980

6:70 87.70

11.60 12.60

10.70

83.90

13.60

12.705

14.60

14.70

12.8

13.70 14.80

14.9

15.60

15.80'

16.60 17.60 18.60 20 19.60 20.60 21.60 22.60 23.60 24.60

16.80

2560

17.90 18.80

18.70 39.70 20.70 21.7 22.7

16.90

25.7 26.7 27.700 28.70 29.70 30.70

31.60 32.60

.32.70

33.60

33.70 34.70

35.50 36.50 37.50 38.50 39.50

190

j 17.90 18.90 23.80 19.90 24.80 20.90 25.80 26.80 2.80;

28.80 29.80

32.80 33.80

24.9

34.80 35.80 36.80

35.70 36.70

2690

37.70 37.80

38:880

39.60 40.60 41.60 43.60 4560

27.9

3.80 41.7 30.80

43.70

43.80

45.70

46.60 48.60 50.60

46.70

47.80

49.70 50.700

4:80.

50.80+j

I

"

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.t'he number In body of table in same row and column gives stance 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 O11.Add this amount to cut or fli and find distance in table, Bet up rod at this point, and Ulineof sight should cut target.

If it does not make the slight adjustment necessary.

-

~

1111

IIII~IL

i

4,1

1'

-7 )5-

.

1 J

1

\\VzG ': +r r

(

I,

r:"

C^r

f

R

i

4') ' ./ C9;

:4 j ' f

/a

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 13 to 1.

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

of table in same row and column gives distance from side stake to slope stake. If ground is not

level estimate the difference in elevation between the side stake and slope stake, lower target by this amount if cut, elevate if fill. Add this amount to cut or fill and find distance in table.

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

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

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

The distance from a point on the tangent to the curve is very nearly the square of the tangent length divided by twice the radius.

TABLE II

TRIGONOMETRIC FORMULAE. LA

=

LMOP LB =LPON = LOPL R= OB= c 1

sin A

a

sos A

b

I

b

b

c

1

b

a

MQ OM

NT

NT

ON

1

tan A cot A ==

sec A= O OM csc A= ON vers A =

=

c

-

=

a = sos B= LP b = sin B = OL

MQ = MQ = cot B = MQ 1 NT = tan B= NT

Q = OQ = csc B= OQ 1 OT = OT= sec B= OT

1

= LM = covers B=

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

coexsec A = PT = exsec B

sin 1/sA-

/1-CsA

2

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

1 cos = cos '/sA 2A=

cos 2 A = cos yA--sine A sin B sin C'

Law of Cosines

c=a*2-+b-2 ab cos C a-b _tan 1/ (A+B) Tangents of Law a-b

1+Cos

tana'l(A-B)

A

TABLE II -

Continued

TRIGONOMETRIC FORMUILE (continued) IA 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. Let a+b+c = s, (s-a) (s-b) (s-c) 2

cos '/,A tan 1/,

tan

s

V

a)

s r

-

B=

s-b

C= tan '/,

r s-c

Area of a triangle: ='I ab Sin C Area= Area = fs (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

1

/ss .0052 1.0885 '/s.0104 .0938 9/is .0156 .0990 1/4 .0208 .1042 .0260 .1094 'is .0313 .1146 '1n .0365 .1198 /a .0417 .1250 *he .0469 .1302 '/,.0521 .1354 "/1e .0573 .1406 *;,.0625 .1458 1/1s.0677 .1510 9/ .0729 .1563 1s .0781 .1615 1 .0833 .1667

6/1e

0

1

2

3

4

.1719 .1771 .1623 .1875 .1927 .1979 .2031 .2083 .2135 .2188 .2240 .2292 .2344 .2396 .2448 .2500

2552 .2604 .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

2

3

4

5

6

.4219 .5052 .4271 .5104 .4323 .5156 .4375 .5208 .4427 .5260 .4479 .5313 .4531 .5365 .4583 .5417 .4635 .5469 .4688 .5521 .4740 .5573 .47,92.5625 .4844 .5677 .4896 .5729 .4948 .5781 .5000 .5833 5

6

7

8

9

10

11

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

.6719 .677 .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 .8490 .8542 .8594 .8646 .8698 .8750 .8802 .8854 .8906 .8958 .9010 .9063 .9115 .9167

.9219 .9271 .9323 .9375 .9427 .9479 .9531 .9583 .9635 .9688 .9740 .9792 .9844 .9896 .9948 1.000

7

8

9

10

11

'/ '/s 1/ */1s 5/ "/s 3/s

"*/1l 'is Is, 1

Deo.

**

10' 20 30 40 50

1 10 20 30 40 50

8 ,

Note.

TABLE

IV

-RADIr,

Radiu

Mid. Ord.

Offet

34377.5 17188.8.

11459.2 8594.42

.036 .073 .109 ,145 .182

5729.65 4911.15 4297.28 3819.83 3437.87 3125.36

.218

Tan.

ORDINATES AND Def. for 1 Foot

.145 0.05' .291 0.10 .436 0.15 .582 0.20 .727 0.25

.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

0.80 0.85 0.70 0.75 0.80 0.85

1910.08 1?09.57 1719.12 1637.28 1562.88 1494.95

.655 .691 .727 764 .800 836

2.818 2.763 2.908 3.054 3.109 3.345

0.90 0.95 1.00 1.05 1.10 1.15

.873 1432.689 1375.40 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

1148.28 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

955.37 929.67 905.13 881.95 859.92

1.309 1.346 1.382 1.418 1.455

5.234 5.3709 5.524 5.669 5.814

1.80 1.85 1.90 1.95 2.00

.255

Chord Deflection

Deo.

-2

times tangent deflection.

DBlu

CHrONS

Radius

Mid. Ord.

Tan. OBeet

Def. for 1 FOOt

819.02 781.84 764.49 747.89

1.528 1.600 1.837 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

637.28 614.56 603.80 593.42

1.965 2.037 2.074 2.110

7.846 8.136 8.281 8.426

2.70 2.80 2.85 2.90

573.89 546.44 521.67 499.06 478.34 459.28 441.68 425.40 410.28 396.20

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 16.51

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

231.01 222.27 214.18 206.68 199.70 193.18

5.476 5.897 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

6.30 6.60 6.90 7.20

STABLE V CURVE FORMULAE FOR SIMPLE CURVES COMPILED BY J. CALVIN LOCKE, C. E.

(1)

c =

42Ra

(2) c- =

a+b'"

R-b)

Y2R (R - 1/(R+b) (R-b) = /2R (R(4) c =2 /m (2R-m) (6) c = 2T cos (5) c 2R sin 1/1 (3) c=

()

e= Rexsec /s 1

(8) e (10) b= -

Rtan1 /sI tan '/I

1a

(9) e-

Ttan =/.I

(2R-a) (1'

b (cC--

c-

(12) b= Rsin I (14) R = (16) d =

(13) b= acot' I4m (15) R =2 -8m 2m 2a 2a R (2R-/(2R+c) 2R-c) = R (2R- y 4R--c

(17) d = 2(R(s (2

m

1) d = 2R sin R+

R T

I

(29) a= btan

(32) I

I

R-

(30) a= T sin I

(33) R =

X 57.295780

(34) L = IRX 0.01745329 (36 Area Seg. =

1/4

(19) m = = R

LR-Rb

R -4-

1 (31) T= R tan /:

I X 57.295780

257.295780 (35) L2 8d-c

LR-R' sin I 2

Rsian

TABLE VI

SINES, COSINES, TANGEN TS, C)TANOENTS

do

v

sin 0'

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

tan sin tan sin tan sin 0' 10' 10' 20' 20' 30'

tan sin 30' 40'

tan sin 40' 50'

tan to 50'

00000029 0175 0204 349 524 552 699 727 875 901 1051 1074 228 24 405 421 584 593

00290058 0204 0233 378 407 553 581 729 756 904 929 1080 1103 257 279 435 449 614 622

0058 0087 000087 0116 0233 0262 262 291 407 436 437465 582 610 612 640 758 785 787 814 934 958 963 987 1110 1132 1139 1161 287 305 317 334 4 478 495 507 644 650 673 679

01160145 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

736 908 2079 250 419 588 756 924 18 3090 19 256

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

793794 974 965 2156 2136 339 306 524 476 711 644 899 812 389 939 281 3145 476311

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

883880 2065 2051 247 221 432 391 617 560 805 728 994 896 3185 3062 378 228 574 393

914 79 2Q95 78 278 77 46276 64875 836 74 3026 73 217 72 411 71 607 70

20 21 22 23 24 25 26 27 28 29

420 584 746 907 4067 226 384 540 695 848

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

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

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

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

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

80569 400668 210 67 417 66 628 65 841 64 5059 63 280 62 505 61 735 60

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

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

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

930 6168 412 661 916 7177 445 720 8002 292

969 59 6208 58 453 57 703 56 95955 7221 54 490 53 766 52 8050 51 342 50

10 11 12 13 14 15 16 17

851 5Q75 5812 5050 6048 200 6088 225 330 5373 289 348 536 495 577 519 787 640 830 664 7046 783 7089 807 355 948 310 925 581 6065 627 6088 860 202 907 225 8146 338 8195 361

5125 275 422 568 712 854 995 6134 271 406

89 88 87 86 85 84 83 82 81 80

40 428 391 450 441 472 491 494 54 517 591 539 642 49 41 561 693 583 744 604 796 626 847 648 899 670 95248 9110 756 9163 777 9217 799 9271 47 42 691 9004 713 9057 734 43 820 325 841 380 862 435 884 490 905 545 926 601 46 44 947 657 967 713 988 770 7009 827 7030 884 7050 942 45 45 7071 1.0000 7092 1.0058 7112 1.0117 133 1.0176 153 1.0235 173 1.0295 44

.4

60' cos

60' 50' cot cos

50' 40' cot cos

40' 30' 30' 20' cot cos cot cos

20' cot

10' cos

10' to cot .

TAtBLE VI (continued)

SINES, COSINES. TANGOENTS COTANOENTS (continued) sin

tan

sin

ten

sin

tan

O'

0'

10'

10'

20'

20' 30'

sin

tan

sin

tan

sin

tan

30' 40'

40'

50'

50'

46 7193 1.0355 7214 1.0416 7234 1.0477 7254 1.0533 7274 1.0599 47314 .024333 .0786 353 .0850 373 .0913 392 .0977 48 431 .1106 451 .71 470 .1237 490 .1303 . 509 1369 49 547 .1504 566 ..1571 585 .1640 604 .1708 623 .1778 50 600 1 1.1876 51 771 .2349 790 52 880 .2799 898 53 986 .3270 8004 54 800 .3764 107 55 192 .4281 208 56 290 .4826 307 57 387 .5399 403 58 480 .6003 496 59 572 .6643 587

1988 .2423 .2876 .3351 .3848 .4370 .4919 .5497 .6107 .6753

8718 1.7797 802 .8546 884 .9347 962 2.0204 9038 .1123 112 .2113 182 .3183 250 .4342 315 .5605 377 .6985

397 2.7475 9407 2.7725 9417 2.7980 9426 2.8239 9436 2.8502 492 3.0178 455 .9042 465 .9319 474 .9600 483 .9887 511 3.0777 520 3.1064 528 3.1397 537 3.1716 546 .2041 596 .4124 563 .2709 572 .3052 580 .3402 588 .3759 644 .6470 613 .4874 621 .5261 628 .5656 636 .6059 659 .7321 667 .7700 674 .8208 681 .8657 689 .9136 703 4.0108 710 4.0611 717 4.1126 724 4.1653 730 4.2193 769 .5736 744 .3315 750 .3897 757 .4494 763 .5107 805 .9894 781 .7046 787 .7729 793 .8430 799 .9152 816 .1446 822 5.2257 8271 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 14.924 980 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

60' 50'

50'

40'

cos

cot

cot

cos

cos

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

7698 1.205 7716 1.2131 7735 1.2203 7753 1.26 3 808 .2497 826 .2572 844 .2647 862 .2723 38 916 .2954 934 .3032 951 .3111 969 .3190 37 8021 .3452 8039 .3514 8056 .3597 8073 .3680 36 124 .3934 141 .4019 158 .4106 175 .4193 35 225 .4460 241 .4550 258 .4641 274 .4733 34 371 .5301 33 355 .5204 323 .5013 339 .5108 418 .5597 434 .5697 450 .5798 465 .5900 32 511 .6212 526 .6319 542 .6426 557 .534 31 601 .6864 616 .0977 631 .7090 646 .7205 30

60 660 1.7321 8675 1.7437 8689 1.7556 8704 1.7675 788 .8418 61 746 .8040 760 .8165 774 .8291 62 829 .8807 843 .8940 857 .9074 870 .9210 63 910 .9626 923 .9768 936 .9912 949 2.0057 64 988 2.0503 9001 2.0655 9013 2.0809 9026 .0965 65 9063 .1445 075 .1609 088 .1775 100 .1943 06 135 .2460 147 .2637 159 .2817 171 .2998 67 205 .3559 216 .3750 228 .3945 239 .4142 283 .4960 293 .5172 304 .5386 68 272 .4751 346 .6279 356 .6511 367 .6746 69 336 .6051 70 71 72 73 74 75 76 77 78 79

s

40' 30' cot cos

8732 1.717 29 816 .8676 28 897 .9486 21 975 2.0353 26 9051 ,1283 25 124 .2286 24 194 .3369 23 261 .4545 22 325 .5826 21 387 .7228 20 9446 502 555 605 652 696 737 775 811 843

2.8770 19 3.0475 18 .2371 17 .4495 16 .6891 15 .9617 14 4.2747 13 .6382 12 5.0658 11 .5764 10

5.9758 9868 6.0844 9872 6.1970 9 899 .9682 8 6.6912 894 .82M9 7.5958 918 7.7704 932 7.9530 7 8.7769 939 9.0098 942 9.2553 6 959 11.059 5 10.385 957 10.711 914 13.727 4 12.706 971 13.197 985 18.075 3 16.350 983 17.169 22.903 992 24.542 993 26.432 2 38.189 997 42.964 9998 49.104 1 114.58 1.000 171.88 1.000 343.77 0

30'

20'

20'

10'

10'

cot

cos

cot

cos

cot

TABLE VII Degree

Deflections for Sub Chords for Short Radius Curves.

Radius

sin. I def. ags.

30

32.°

34° 36 38

400 ° 42

460

10 51'

2 17 2°° 5' 2 33

2058'

3043'

o01.15

30 0' 3°3o 3° 33 3 4

30 ° 58'

t01.33 101.48 Iol.66 zox.85

46.19

2 271 2° 34'

3

4

30oo.o00

T

2

D

Sin.

D

=

°

2 41 °

49

'

2° 57 ° 3 05' °

41

2 48' 20 55 '

3002 3o o09 3o 16'

4

30 29 308' 3 46'

4

4

4

I8'' 40'

5

5 36'

29

50

02.o06 02.29 102.53

o02.76 to3.00 103.24

60 7'

103.54

o 8'

5° 44

7

50 Rex.sec =T tan I

6 31' 60 44' 60 571 ° II'

ChorId def. de-

R =Sin. E E

54

5 8' 5°22' ° 6004 f

R= T cot.

Rin. 50 tn

55' °

12'

40 26' 440'

for 100 ft.

'405 5 0 5 03' 5 23' 5 34'

4°02'

4°

4

407'

3 13' 3 21

3 54'

3°22' 3 29 3°35'

R tan 5otan I Sin. J D

Sin.

ofare

zo 59, 2 06 ' 2 13 ° 2 20'

60

T

angle_ _

181.39 71.01 161.8o 153.58

58

T

ef.

193.18

Io6.50 103.14

°

de. aLength 25 Ft.

Io10.II

54 56*

of

20 Ft.

I18.31 3146

52

sn_

15 Ft.

122.92

480 50s

in of

R

12.5 Ft.

139.52 133.47 I27.97

440

sub chord

50 Sof

Curve

I

I03.84 104.14 o4.43 14.72

chord ---D R

No.chords= I D 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 z ° 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 .oz745, and the product by the distance.

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

Given Base loo, Alt. Io.o2+2o0o=.5. Ioo+.5= oo.5 hyp.

Given Hyp. Ioo, Alt. 25.25+2oo=00 3.125. 0oo-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 I1,and divide by 7.

LEVELING. The correction for curvature and refraction, in feet ,

and decimals of feet is equal to 0.574 d8 where d is the distance in miles.

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

PROBABLE ERROR. If d,, d, d, etc. are the discrepancies of various results from the mean, and ifId*=the sum of the squares of these differences and n=the number of observations, then the probable error of the mean=

Id

TABLE VIII TANOENTS AND EXTERNALS TO A 16 CURVE I o

10' 20' 40'.

50' go 10' 230" 30' 40' 50' So

10" 20' 30' 40' 50' 40

10'

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

30' 40' 233.47 50' 241.81 20' s0 250.16 10' 258.51 20' 266.86 30' 275.21 40' 283.57 50' 291.92 300.28 go 10' 308.64 20' 316.99 30' 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 s0 10' 409.03 20' 417.41 30' 425.79 40' 434.17 50' 442.55 90 450.93 10' 459.32 20' 467.71 30' 476.10 40' 484.49 50' 492.88 501.28 10' 509.68 518.08 30' 526.48 40. 534.89 50. 543.29

I1

T

E

I=10o

.218 .297 .388 .491 .606

551.70 560.11 568.53 5o+C. 576.95 30' + 585.36 40' 50' 593.79 602.21 .001 120 10' 610.64 20' 619.07 30' 627.50 40' 635.93 50' 644.37 652.81 100 C. 180 10' 661.25 T 20' 669.70 .06 30' 678.15 E 686.60 40' .003 695.06 703.51 140 10' 711.97 20' 720.44 728.90 30' 40' 737.37 50' 745.85 754.32 150 10' 762.80 20' 771.29 30' 779.77 40' 788.26 50' 796.75 805.25 160 10' 813.75 20' 822.25 200 C. 30' 830.76 40' 839.27 T" 50' 847.78 .13 E 856.30 17050 .006 10' 864.82 20' 873.35 30' 881.88 40' 890.41 50' 898.95 907.49 180 25e C. 10' 916.03 924.58 T i-: ! 30' 933.13 .16 .C7 40' 941.69 50' 950.25 .007 19o 958.81 10' 967.38 20' 975.96 30' 984.53 40' 993.12 50' 1001.7 300o C T 1010.3 200 .19 10' 1018.9 E 20' 1027.5 .008 30' 1036.1 40' 1044.7 50' 1053.3

.873 1.024 1.188 1.364 1.552 1.752 1.964 2.188 2.425 2.674 2.934 3.207 3.492 3.790 4.099 4.421 4.755 5.100 5.459 5.829 6.211 6.606 7.013 7.432 7.863 8.307 8.762 9.230 9.710 10.202 10.707x 11.224 11.753 12.294 12.847 13.413 13.991 14.582 15.184 15.799 16.426 17.065 17.717 18.381 19.058 19.746 20.447 21.161 21.887 22.624 23.375 24.138 24.913 25.700 T

110 10' 20'

Rtan

itI

E

11=200 11

T

26.500 210 10' 27.313 28.137 20' 28.974 50 C. 30' T 40' 29.824 50' 30.686 .06 E 31.561 220 10' 32.447 20' 33.347 34.259 30' 40' 35.183 50' 36.120 37.070 100 C. 2830 10' 38.031 T 39.006 .13 20' 30' 39.993 E 40.992 .011 40' 42.004 50' 43.029 240 44.066 10' 20' 30' 40' 150 C. 50' T .19 250 50.554 10' E 51.679 .017 20' 52.818 30' 53.969 40' 55.132 50' 56.309 260 57.498 10' 7 20' 58.699 59.914 200 C. 30' 61.141 40' T 62.381 .26 50' E 63.634 270 .021 64.900 10' 66.178 20' 67.470 30' 68.774 40' 70.091 50' 71.421 280 72.764 250 C. 10' 74.119 T 20' 75.488 .32 30' 76.869 E 40' 78.264 .028 50' 79.671 290 10' 81.092 20' 82.525 30' 83.972 40' 85.431 86.904 30 0 C. 50' T 88.389 300 89.888 .39 10' E 20' 91.399 92.924 .034 30' 40' 94.462 96.013 50'

~006

1

E

I=30s

1061.997.577 1070.6 99.155 1079.2 100.75 + 1087.8 102.35 50 C., T 103.97 1096.4 1.105.1 105.60 .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 7 1183.1 120.87 .19 1191.8122.63 E 1200.5 124.41 .025 1209.2 126.20 1217.9 128.00 129.82 131.65 1244.0 133.50 1252.8 135.35 1261.5 137.23 150 C. T 1270.2 139.11 .29 1279.0 141.01 1287.7 142.93 .038 1296,5 144.85 1305.3 146.79 1314.0 148.75 1322.8 150.71 1331.6 *152.69 1340.4 154.69 1349.2 156.70 20 GC. 1358.0 158.72 T 1366.8 160.76 .39 E 1375.6 162.81 1384.4 164.86 .051 1393.2 166.95 1402.0 169.04 1410.9 171.15 i-1419.7 173.27 1428.6 175.41 1437.4 177.55 250 C. 1446.3 179.72 T 1455.1 181.89 .49 1464.0 184.08 E 1472.9 186.29 .065 1481.8 188.51 1490.7 190.74 1499.6 192.99 1508.5 195.25 1517.4 197.53 1526.3 199.82 300 C. T 1535.3 202.12 1544.2 204.44 .59 E 1553.1 206.77 1562.1 209.12 .078 1571.0 211.48 1580.0 213.86

E=Rexsec J1I

TABLE

VIII TANGENTS AND EXTERNALS TO A 1l CURVE I400T

I 1589.0 1598.0 1606.9 1615.9 1624.9 1633.9 1643.0 1652.0 1661.0 1670.0 1679.1 1688.1 $880 1697.2 10' 1706.3 20' 1715.3 30' 1724.4 40' 1733.5 50' 1742.6 1751.7 340 10' 176Q.8 20' I~I.0 30' 1779.1 40' 1788.2 50' 1797.4 350 1806.6 10' 1815.7 20' 1824.9 30' 1834.1 10' 1843.3 1852.5 50' 1861.7 860 10' 1870.9 20' 1880.1 $o30' " 1889.4 40' 1898.6 50' 1907.9 1917.1 870 10' 1926.4 20' 1935.7 30' 1945.0 40' 1954.3 50' 1963.6 880o 1972.9 10' 1982.2 20' 1991.5 30' 2000.9 40' 2010.2 50' .2019.6 $00 2029.0 10' 2038.4 20' 2047.8 30' 2057.2 40' 2066.6 50' 2076.0 2085.4 400 10' 2094.9 20' 2104.3 30' 2113.8 40' 2123.3 50' 2132.7 810 10' 20' 30' 40' 50' 30 10' 20' 30' 40' 50'

ggo

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

-

T

E

I=50o

1

2142.2 387.4 510 2151.7 390.7 10' + 2161.2 394.1 So C. 20' 2170.8 397.4 T 30' 2180.3 400.8 .17 40' 2189.9 404.2 50' E .037 2199.4 407.6 520 2209.0 411.1 10' 2218.6 414.5 20' 2228.1 418.0 30' 421.4 2237.7 40' 2247.3 425.0 50' 2257.0 428.5 100 C 530 2266.6 432.0 10' 2276.2 435.6 .34 20' 2285.9 S439.2 30' E 442.8 .075 2295.6 40' 2305.2 446.4 50' 450.0 2314.9 540 440 10' 2324.6 453.6 10' 20' 2334.3 457.3 20' 30' 2344.1 461.0 30' 40' 2353.8 464.6 40' 150 C. 50' 2363.5 468.4 15 C 50' T T 472.1 2373.3 .i51 550 40 450 10' 2383.1 475,8 10' E .E0 479.6 20' 20' 2392.8 '.070 30' 2402.6 483.4 .116 30' 40' 2412.4 487.2 40' 292.0 491.0 50' 2422.3 50' 294.9 460 2432.1 494.8 860 498.7 297.7 10' 2441.9 10' 300.6 20' 2451.8 20' 303.5 20o C. 30' 2461.7 506.4 200 C 30 306.4 510.3 40' 40' 2471.5 T 305.3 50' 2481.4 514.3 .68 50' .53 E 312.2 518,2 570 E093 470 2491.3 *09 .151 315.2 522.2 10' 2501.2 10' 318.1 526.1 20' 2511.2 20' 321.1 530.1 30' 2521.1 30' 324.1 534.2 40' 2531.1 40 327.1' 538.2 50' 2541.0 50' 542.2 330.2 480o 2551.0 580 333.2 250 C. 546.3 250 C. 10' 2561.0 10' 336.3 550.4 T 20' 2571.0 20' T 2581.0 339.3 554.5 30' 30' .85 .67 558.6 342.4 40' 2591.0 40' E 345.5, .117 50' 2601.1 562.8 .189 50' 590 2611.2 348.61 566.9 490 571.1 351.8 10' 2621.2 10' 354.9 20' 2631.3 575.3 20' 358.1 30' 2641.4 579.5 30' 583.8 40' 2651.5 361.3 40' 50' 2661.6 588.0 364.5 50' C 30 0 C 300 2671.8 367.7 592.3 500 600 T T 371.0 10' 2681.9 596.6 10' 374.2 .80 20' 2692.1 600.9 1.02 20' E E 2702.3 605.3 377.5, .141 30 30' 40' 40' 2712.5 609.6 .227 380.8 384.1 50' 2722.7 614.0 50' R exsec ~it E Rtan'IuI T= R tan 'l I E &R execs1e 410

10' 20' 30' T .13 40' E 50' .023 420 10' 20' 30' 40' 50' 100 C. 430 10' T 20' .26 30' E 40' .046 50'

+ 50o C.

T 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

I

I

E 618.4 622.8 627.2 631.7 636.2 640.7 645.2 649.7 654.3 658.8 663.4 668.0 672.7 677.3 682.0 686.7 691.4 696.14 700.9 705.7 710.5 715.3 720.1 725.0 729.9 734.8 739.7 744.6 749.6 754.6 759.6 764.6 769.7 774.7 779.8 784.9 790.1 795.2 800.4 805.6 810.9 816.1 821.4 826.7 832.0 837.3 842.7 848.1 853.5 858.9 864.3 869.8 875.3 880.8 886.4 892.0 897.5 903.2 908.8 914.5

I=too

So c.

T .21 E .056

10t C, T .42 E .112

150 C.

E .168

200 C. T .81 .225

250 C,

1.05 E 2.83

300 C.

1.27 E

TABLE VIII TANGENTS AND EXTERNALS TO A 1 CURVE. i

T

E

I=700I

610 10' 20' 30' 40' 50' 620 10' 20'

30' 40' 50'

$68

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

50' 10' 20'

30' 40' 50' 10' 30' 40' 50' 670

10' 20' 30' 50' 10' 20' 30' 40' 40' 50' .9-1 10' 20' 30'

50' 700 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

T

E

"

-- "

920.2 925.9 + 931.6 937.3 So C. 943.1 948.9 .25 E 954.8 .080 960.6 966.5 972.4 978.3 984.3 990.2 10 C. 996.2 1002.3 .51 1098.3 E 1014.4 159 1020.5 1026.6 1032.8 1039.0 1045.2 1051.4 1057.7 150 C T 1063.9 .76 1070.2 1076.6 1082.9 1089.3 1095.7 1102.2 1108.6 1115.1 1121.7 1128.2 1134.8 1.02 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 E. 1208.9 .403 1215.8 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

-- "

E.R

710 10' 20' 30'

40' 50' 72o 10' 20' 30' 40' 50' 730

10' 20' 30' 40' o

74 10' 50' 20'

30' 40' 50' 75o

10' 20' 30' 40' 50' 76o 10' 20' 30'

40' 50' 0o

10' 20'

30' 40' 780 10' 20' 30' 40' 50' 790

10' 20' 30' 40' 50' 800 10' 20' 30' 40' " 50' 11

.4tnisn'

40869 4099.5 4112.1 4124.8 1137.4 4150.1 4162.8 4175.6 4188.5 4201.2 4214.0 4226.8 4239.7 4252.6 4265.6 4278.5 4291.5 4304.6 4317.6 4330.7 4343.8 4356.9 4370.1 4383.3 4396.5 4409.8 4423.1 4436.4 4449.7 4463.1 4476.5 4489.9 4503.4 4516.9 4530,4 4544.0 4557.6 4571.2 4584.8 4598.5 4612.2 4626.0 4639.8 4653.6 4667.4 4681.3 4695.2 4709.2 4723.2 4737.2 4751.2 4765.3 4779.4 4793.6 4807.7 4822.0 4836.2 4850.5 4864.8 4879.2

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

1I--80o

I

+ 50C. T

.30 E .110

100 C T .61 E .220

150o C. T .91

.332

20o C. T 1.22 1E

.445

250 C T 1.53 E .558

300 C.

T 1.84 E .671

T

E

1=90*

" 310 10" 20' 30' 40" 50' 820 10' 20'

4893.6 4908,0 4922.5 4937.0 4951.5 4966.1 4980.7 4995.4 5010.0 5024.8 30' 40' 5039.5 50' 5054.3 5069.2 830 10' 5084.0 20' 5099.0 30' 5113.9 40' 5128.9 5143.9 50' 5159.0 840 10' 5174.1 5189.3 30' 5204.4 40' 5219.7 50' 5234.9 5250.3 850 10' 5265.6 20' 5281.0 30' 5296.4 40' 5311.9 50' 5327.4 5343.0 86o 10' 5358.6 20' 5374.2 30' 5389.9 40' 5405.6 50' 5421.4 5437.2 88 10' 5453.1 20' 5469.0 30' 5484.9 40' 5500.9 50' 5517.0 5533.1 880 10' 5549.2 20' 5565.4 30' 5581.6 40' 5597.8 50' 5614.2 5630.5 890 5646.9 20' 5663.4 30' 5679.9 10' 5696.4 40' 50' 5713.0 5729.7 900 10' 5746.3 5763.1 30' 5779.9 5796.7 50' 5813.6

"

=Rexsec 'is1

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 2292.0 2303.5 2315.0 2326.6 2338.2 2349.8 2361.5 2373.3 2385.1 2397.0 2408.9 2420.9 2432.9

o

0 C.

T .36

.149

100 C. .72 E .299

15o C. T 1.09 .E .450

20* C. T

1.45 .E .603

250 C. T 1.83 E .756

300 C. T 2.20 E .910

TABLE VIII I

T

TANGENTS AND EXTERNALS TO A 1 CURVE

IE

I

_

-

T _0(

t=t1

E

---'~~--

5830.5 2444.9 5847.5 2457.1 5864.6 2469.3 5881.72481.5 5898.8 2493.8 50' 5916.0 2506.1 5933.2 2518.5 40' 5950.5 2531.0 10' 20' 5967.9 2543.5 30' 5985r3 2556.0 40, 6002.7 2568.6 50' 6020.2 2581.3 6037.8 2594.0 10' 6055.4 2006.8 20' 6073.1 2619.7 30' 6090.8 2632.6 40' 6108.6 2645.5 6126.4 2658.5 6144.3 2671.6 9o 10' 6162.2 2684.7 50' 6180.2 2697.9 20' 30' 6198.3 2711.2 40' 6216.4 2724.5 50' 6234.6 2737.9 6252.8 2751.3 10' 6271.1 2764.8 20' 6289.4 2778.3 30' 6307.9 2792.0 40' 6326.3 2805.6 *50' 6344.8 2819.4 6363.4 2833.2 6o 10' 6382.1 2847.0 20' 6400.81 2861.0 30' 6419.5 2875.0 40' 6438.4 2889.0 50' 6457.3 2903.1 6476.2 2917.3 970 10' 6495.2 2931.6 20' 6514.31 2945.9 30' 6533.4 2960.3 40' 6552.6 2974.7 50' 6571.9 2989.2 6591.2 3003.8 80 10' 6610.6 3018.4 20' 0630.1 3033.1 30' 6649.6 3047.9 40' 6669.2 3062.8 50' 6688.8 3077.7 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 1000 6828.3 3184.1 10' 6848.5 3199.6 20' 6868.8 3215.1 30' 6889.2 3230.8 40' 6909.6 3246.5 50, 6930.1 3262.3 1io 10' 20' 30'

- -

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

10o 10' 5@ C. 30' T 30' 40' .43 50' E .200 1020 10' 20' 30' 40'

50' 100 C. 1030 10' T 20' .86 30' E 40' .401 50' 1040o 10' 20'

150 C. 1.30 E .604

30' 40' 08' 20' 30' 40'

50' 106o 10'

20 C

20'

30' 40' 50' 1070 10' 20' 30' 40' 50' 1080 10' 250 C 20' 30' 2.18 40' E 50' 1.02 1090 10' T 1.74 E

7814.3

30' 40' 50' 300 C 1100 10' 2.62 20' E 30' 1.22 40' 50'

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

T=cR taa YsI

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

I

T

E

1..20

" 10' + 20' 5 C. 30' T 40' .51 E 50' .268 112o 10' 20' 30' 40' 50' 100 C. 1130 10' T 20' .103 30' E 40' .536 50' 1140 10' 20' 30' 40' 150 C. 50' 1.T6 1150 10' E 20' .806 30' 40' 50' 1160 10' 20' 30' 200 C. 40' T 50' 2.08 ° E 117 1.08 10' 30' 40' 50' 1180 257 C 10' 20'

30' 40' 50' 1190 10' 20' 30' 40' 50' 300 C, 1200 T 10' 3.14 20' 30' 1.63 40' 50' 2.61 E

1.36

L =Rexsec rp1

8336.7 4386.1 8362.7 4407.6 0 8388.9 4429.2 $ C. 8415.1 4450.9 8441.5 4472.7 .6Z 6468.0 4494.6 E 8494.6 4516.6 :360 8521.3 4538.8 8548.1 4561.1 8575.0 4583.4 8602.1 4606.0 8629.3 4628.6 8656.6 4651.3 10 C. 8684.0 4674.2 T 8711.5 4697.2 1.25 8739.2 4720.3 8767.0 4743.6 721 8794.9 4766.9 8822.9 4790.4 8851.0 4814.1 8879.3 4837.8 8907.7 4861.7 8936.3 4885.7 8965.0 49099 150C. 8993.8 4934.1 1.93 9022.7 4958.6 9051.7 4983 1 109 9080.9 5007.8 9110.3 5032.6 9139.8 5057.6 9169.4 5082.7 9199.1 5107.9 9229.0 5133.3 9259.0 5158.8 200 C. 9289.2 5184.5 T 9319.5 5210.3 2.52 E 9349.9 5236.2 9380.5 5262.3 1.46 9411.3 5288.6 9442.2 5315.0 9473.2 5341.5 9504.4 5368.2 9535.7 5395.1 9567.2 5422.1 250 C. 9598.9 5449.2 9630.7 5476.5 3.16 9662.6 5504.0 E 9694.7 5531.7 1.83 9727.0 5559.4 9759.4 5587.4 9792.0 5615.5 9824.8 5643.8 9857.7 5672.3 9890.8 5700.9 9924.0 5729.7 300 C. 9957.5 5758.6 9991.0. 5787.7 3.81 E 10025.0" 5817.0 10059.0 5846.5 2.20 10093.0 5876.1

TABS

IX

MIDDLE ORDINATES OF RAILS Length of Rail (feet) R

Feet 0-20 17189 0-40 8594 5730 1-0 1-20 4297 1-40 3438 2865 '2 -0 2-20 2456 2-40 2149 3-0 1910 3-20 1719 3-40 1563 1433 4-0 4-20 1323 4-40 1228 1146 5 955.3 6 7 819.0

30

281 26

I1nch lachh In

R I

24212011C

fnch ch

8 .08 .07 .06 .05.041.03 .14 .12 .10 .08 .07 9 .16 .24 .20 .18 .15 .13 .10 10 .27 .23 .20 .17 .13 11 .31 .39 .34 .29 .25 .21 .17 12 .47 .41 .35 .30 .25 .20 13 .35 .29 .23 14 .55 .48 .41 .63 .55 .47 .40 .33 .27 15 .62 .53 .45 .38 .31 16 .71 .78 .68 .59 .50 .42 .35 17 .86 .75 .65 .55 .46 .38 118 .60 .50 .42 19 .94 .82 .71 1.02 .89 .77 .65 .55 .45 20 1.10 .96 .83 .70 .59 .48 22 1.18 1.03 .89 75 .63 .52 24 1.41 1.23 1.06 .90 .76 .62 26 1.65 1.44 1.24 1.05 .89 .73

128

20 nch InchInch

ench t Ich I nch inch

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 1.64

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 6.07

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

1.42 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

i

1.20 1.35 1.50 1.65 1.81 1.96 2.10 2.26 2.40 2.57 2.70 2.86 3.00 3.30 3.59 3.88

1.01 1.14 1.27 1.39 1.54 1.66 1.78 1.91 2.04 2.16 2.28 2.41 2.54 2.80 3.04 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 Radiuset Feet

--

Chord Feet

"-

Central Angle 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

Deflection

Deflection

Angle

for 1 Foot

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 angle 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 I

Slope 0o00' i5' 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

ta 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

Correction 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

Rise

Slope

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

800Q' 15' 30' 45' 9 00 15 30 45 10 00 15 30 45 11 00 15 30 45 1200 15 30 45 13 00 15 30 45 14 00 15 30 45 15 00 15 30 45

H

on

Correction

Rise

99.027 98.965 98.902 986.836 98.769 98.700 98.629 98.556 98.481 98.404 98.325 98.245 98.163 98.0719 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

0.973 1.035 1.098 1.164 1.231 1.300 1.371 1.444 1.519 1.596 1.675 1.755 1.837 1.921 2.008 2.095 2.185 2.277 2.370 2.466 2.563 2.662 2.763 2.866 2.970 3.077 3.185 3.295 3.407 3.511 3.637 3.754

0.139 0.143 0.148 0.152 0.156 0.161 0.166 0.109 0.174 0.178 0.182 0.185.

0.191 0.195 0.199 0.204 0.208 0.212 0.216 0.221 0.225 0.229 0.233 0.238 0.242 0.246 0.250 0.255 0.259 0.263 0.267 0.271

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

TABLE XII MINUTES IN DECIMALS OF A DEGREE. 0'30" 1 00 30 2 00 30 3 00 30

.00833 .01667 .02500 .03333 .04167 .05000 .05833

400

.06671400

30 5 00 30 6 00 30 700 30 8 00 30 900 30

.07500 30 .24167 30 .40833 .08333 15 00 .2500025 00.41667 .00167 30 .25833 30 .42500 .10000 1600 .26667 2600 .43333 .10833 30 .27500 30 .44167 .11667 1700 .28333 27 00 .45000 .12500 30 .29167 30 .45833 .13333 1800 .300002800 .46667 .14167 30 .30833 30 .47500 .150001900.316672900 .48333 .15833 30 .32500 30 49167

1000

10'30" 11 00 30 12 00 30 13 00 30

.16667 20 00

.17500 .18333 .19167 .20000 .20833 .21667 .22500

20'30" 21 00 30 22 00 30 2300 30

.233332400

.3333330

00

.34167 .35000 .35833 .36667 .37500 .38 .39167

30'30" 31 00 30 32 00 30 13300 30

.400003400

30 3500 30 36 00 30 37 00 30 38 00 30 39 00 30

.50000 40 00

.50833 40'30" .51667 41 00 .52500 3 .533342 00 .54167 30 .550004300 .55833 30

.67500 168333 .69167 .70000 .70833 .71667 72500

.566674400

.7333354

.9009000

.57500 30 .583334500 .59167 30 .60000 46 00 .60833 30 .61667 47 00 .62500 30 .63333 48 00 .64167 30 .65000 49 00 .65833 30

.74167 .75000 .75833 .76667 .77500 .78333 .79167 .80000 .80833 .81667 .82500

.90833 .91667 .92500 .93333 .94167 .95000 .95833 .96667 .97500 .98333 .99167

.60667 50 00

50'30" 51 00 30 52 00 30 5300 30 30 5500 30

5600

30 57 00 30 5800 30 59 00 30

.84167 .85000 .85833 .86667 .87500 .88333 .89167

.83333 60 00 1.00000

TABLE XIII--

RRECTIONS FOR TANGENTS AND EXTERNALS

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

FOR TANGENTS ADD Duoas or CUnva

Central Angle 50

200

165

10°

100 18.

200 S2i

03 .06 04 10 06 .13 .08 16

09 14 19 24

80e 85 40 41° 60

10 19 11 22 13 .26 15 .30 17 .34 19 .38 .21 42 23 46

.29 .34 .40 44 .51 .57 .63 .69

50

60 6bo

70 76

800

586

13 19 .26 .33

36 .72 1.09 39 79 1.19 .86 1.30 51 1.03 1.6

35

30I

16 19 24 .29 .32 .39 40 49

400

.22 .34 .46 .58

450

50.

600

550

65

700

.25 .28 .31 34 38 42 .46 .39 45 .51 .53 68 .68 .68 .61 .58 .65 .72 79 .84 .90 67 .75 .83 90 .99 1.06 1.14

79 .89 .991.091.20 1.29 1.39 39 49 .59 .69 47 .58 .69 .70 .81 .92 1.04 1.29 1.42 1.54 1.6 .53 .67 .80 .93 1.06 1.20 1.34 1.49 1.64 1.79 1.94 60 76 91 1.06 1.21 1.37 1.52 1.70 1.87 2.04 2.21 .68 .85 1.02 1.19 1.36 1.54 1.72 1.91 2.102.292.48 76 .951.141.32 1.52 1.72 1.92 2.14 2.35 2.56 2.77 .841.051.271.491.711.94 2.172.382.602.833.07 93 1.16 1 40 1.64 1.88 2.13 2.38 2.63 2.88 3.13 3.39

25 .51 76 1.02 .27 .56 .831.12 30 .61 .91 1.22 33 66 1.00 1.33

S

25i

1.45 1.55 1.74 2.08

1.28 1.54 1.80 2.06 2.33 2.60 2.88 1.40 1.69 1.98 2.27 2.57 2.87 3.16 1.53 1.842.152.462 783.103.44 1.682.02 2.36 2 7 3.05 3.40 3.77

3.163.44 3.72 3.47 3.78 4.09 3.784.124.46 4.14 4.66 4 80

2.57 2.94 3.32 2.803.20 3.61 3.06 3.50 3 2 3.95 .674. 1 4 76

4.91 5.38 5.85 7 01

1.83 2.00 2.18 2.61

2.20 2.40 2.62 3.14

960 1000 1100

.43

1800

62 1.25 1 93 2.52 3.16 3.81 4

3.70 4.10 4.50 4.02 4.404.98 4.404.885.37 5.31 5 86 6.43

455.11 5.77 6

5.32 5.83 6.34 7 60

44 7 127.80 8.50 9 22

FOR EXTERNALS ADD 1tral Angle

DOREE

O

5o 10

°

OF

CURVE 50.

55b

.004 .010 .017 .027

.006 .014 .022 036

.007 .018 .028 .046

.008 .023 .034 .056

.009 .027 .038 .065

.011 .029 .045 .074

.012 .032 .051 .083

.014 .035 .057 .093

.01 .017 .018 .039 .043 .047 .063 .070 .076 .106 120 127

.013.025 .018.035 .023.046 .030 .060 .037 .075 .046 .093 .056 .112 .067.135

.038 .054 .070 .093 .116 .142 .168 .204

.051 .072 .093 .119 .151 .188 .225 .273

.065 .086 .117 .153 .189 .236 .283 .343

.078 .109 .141 .184 .227 .283 .340 .412

.090 .131 .172 .216 .266 .332 .398 .483

.103 .153 .203 .254 .305 .381 .457 .554

.116 .175 .234 .289 .345 .420 .516 .625

.129 .197 .265 .325 .384 .479 .575 .697

.149 .213 .277 .351 .425 .530 .636 .711

700 75 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

.588 .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

9 95e° 10 110°

.149.299 .174.350 .200 .401 .268 .536

.450 .522 .604 .806

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

1300

.360 .721

200 S5°

800° 85 400 458

50 550

60 60o

20

25°

30*

350

40.

60°865

450

.001 .003 .003.007 .006 .011 .009 .018

100 15

15

.603 .756 .910 1.07 1.22 .706 .985 1.06 1.25 1.43' .809 1.01 1.22 1.43 1.64 1.08 1.35 1.63 1.91 2.20

.170 .230 .290 .378. .467 .582 .697 .845

700

.020 .051 .083 135

.179 .188 .247 .264 .315 .341 .411 .445 . .550 .641 .700 .774 .851 .922 1.01

1.08 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 CROSSSECTIONING. SLOPE Vl/s TO 1. ROADWAY OF ANY WIDTH.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

0

.1

.2

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.4

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 4.80 6.30 7.80 9.30

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 43.95 45.45 46.95 48.45 49.95 51.45 52.95 54.45 55.95 57.45 58.95 60.45 61.95 63.45 64.95 66.45 67.95 69.45 70.95 72.45 73.95 75.49

4.60 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 41.10 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

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 57.30 58.80 60.30 61.80 63.30 64.80 66.30 67.80 69.30 70.80 72.30 73.80 75.30

.5 1 .6 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

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

.7

.8

.9

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

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 35.70 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

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

AMERICAN BLUE PRINT CO., INC.--NEW YORK

II