Fremling Field notebook 1956 - 1957

out read 16.7, the 3distance

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.I .2 .3 .4 .5 .6 .7 .8 .9 t Distance out from Side or Shoulde.r Stake'

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4.5 6.0

5 7.5 6 9.0 7 10.5 8 12.0 9 13.5 10 15.0 11 16.5 12 18.0 13 19.5 14 21.0 15 22.5 16 24.0 17 25.5 18 27.0 19 28.5 20 30.0 21 31.5 22 33.0 23 34.5 24 36.0 25 37:5 26 39.0 27 40.5 28 42.0 29 43.5

30

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31

46.5

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48.0

38 34

49.5 51.0

35 52.5 36 37 88

54.0 55.5 7.0 8.5 0.0

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6.3 7.8 9.3 10.8 12.3 13.8 15.3 16.8 18.3 19"8 21.3 22.8

24.3

25.8 27.3 28.8 30.3 31.8 33.3 34.8 36.3 37.8 39.3 40.7 40.8 42.2 42.3 43.7 43.8 45.2 45.3 46.7 46.8 48.2 48.3 49.7 49.8 51.2 51.3 52.7 52.8 54.2 54.3 55.7 55.8 57.2 57.3 58.7 58.8 60.2' 60.3

6.5 8.0 9.5 11.0 12.5 14.0 15.5 17.0

18.5 20.0 21.5 23.0 24.5 26.0 27.5 29.0 30.5 32.0 33.5 35.0 36.5 38.0 39.5 41.0 42.5 44.0 45.5 47.0 48.5 50.0 51.5 53.0

0.6 2.1 3.6 5.1 6.6 8.1 9.6 11.1 12.6 14.1 15.6 17.1 18.6 20.1 21.6 23.1 24.6 26.1 27.6 29.1 30.6 32.1 33.6 35.1 36.6 38.1 39.6 41.1 42.6 44.1

45.6

47.1 48.6 50.1 51.6 53.1 54.5 54.6 56.0 56.1 57.5 57.6 59.0 59.1 60.5 60.6

FEL & ESSER CO., N. Y.

0.8 2.3 3.8 5.3 6.8 8.3 9.8 11.3 12.8 14.3 15.8 17.3 18.8 20.3 21.8 23.3 24.8 26.3 27.8 29.3 30.8 32.3 33.8 35.3 36.8 38.3 39.8 41.3 42.8 44.3 45.8 47.3 48.8

50.3 51.8

0.9 2.4

1.1 2.6

4.1 5.6 6.9 7.1 8.4 8.6 9.9 10.1 11.4 11.6 12.9 13.1 14.4 14.6 15.9 16.1 17.4 17.6 18.9 19.1 20.4 20.6 21.9 22.1 23.4 23.6 24.9 25.1 26.4 26.6 27.9 28.1 29.4 29.6 30.9 31.1 32.4 32.6 33.9 34.1 35.4 35.6 36.9 37.1 38.4 38.6 39.9 40.1 41.4 41.6 42.9 43.1 44.4 44.6 45.9 46.1 47.4- 47.6 48.9 49.1 50.4 50.6 51.9 52.1 53.4 53.6

53.3 54.8 54.9 55.1 56.3 57.8 59.3

60.8

56.4 57.9 59.4 60.9

56.6 58.1 59.6 61.1

1.2 2.7 4.2 5.7 7.2 8.7 10.2 11.7 13.2 14.7 16.2 17.7 19.2 20.7 22.2 23.7 25.2 26.7 28.2 29.7 31.2 32.7 34.2 35.7 37.2 38.7 40.2 41.7 43.2 44.7 46.2 47.7 49.2 50.7 52.2 53.7 55.2 56.7 58.2 59.7 61.2

1.4 2.9 4.4 5.9 7.4 8.9 10.4 11.9 13.4 14.9 16.4 17.9 19.4 20.9 22.4 23.9 26,4 26.9 28.4 29.9 31.4 32.9 34.4 35.9 37.4 38.9 40.4 41.9 43.4 44.9 46.4 47.9 49.4 50.9 52.4 53.9 55.4 56.9 58.4 59.9 61.4

For Curve Tables see end of book.

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CURVE TABLES. Published by KEUFFEL &; ESSER CO.

HOW TO USE CURVE TABLES. Table I. contains Tangents and Externals to a 10 curve. Tan. and Ext..to any other radius may be found nearly enough, by dividing the Tan. or Ext. opposite the given Central Angle by the given degree of curve.

To find Deg. of Curve, having the Central Angle and Tangent: Divide Tan. opposite the given Central Angle by the given Tangent.

To find Deg. of Curve, having the Central Angle and External: Divide Ext. opposite the given Central Angle by the given External.

To find Nat. Tan. and Nat. Ex. See. for any angle by Table I.: Tan. or Ext. of twice the given angle divided by the radius of a 10 curve will be the Nat. Tan. or Nat Ex. Sec.

EXAMPLE. Wanted a Curve with an Ext. of about 12 ft. Angle of Intersection or I. P.=23° 20' to the R. at Station 542+72.

Ext. in Tab. I opposite 23° 20' = 120.87 120.87+12=10.07. Say a 100 Curve.

Tan. in Tab. I opp. 230 20' 1183.1 1183.1 +10=118.31.

Correction for A. 230 20' for a 10° Cur. =0.16 118.31 +0.16 = 118.47 = corrected Tangent.

(If corrected Ext. is required find in same way) Ang. 23* 20'= 23.33 ° 10 = 2.3333 = L. C. ° 2 19 '= def. for sta.

542 I. P. =sta.

542+72 40 49 '= a "a +50 Tan.= 1 .18.47 97 19'= ea a 543 B. C.= sta.

541+53.53 9o49 ' = a a " --50 2 .33.33 L C.= 543+ 11o40'= a a a 86.86 E. C. =Sta. 543+86.86 100--53.53 =46.47X3'(def. for 1 ft. of 100 Cur.)= 139.41'= ° 2 191'=def. for sta. 542.

Def. for 50 ft. =20 30' for a 10° Curve.

Def. for 36.86 ft.= 1° 50' for a 10° Curve. x y

/J.yJ

IO*Cwr

II

TAS Angle

2

a

L -- Tangents and Externals to a 1 Curve.

Chord = 100 ft.

Tangent Eternal

lgle

10' 20 3o 40 50

50.00 58.34 66.67 75.01 83.34 91.68

10 20 30 40 50

100.01 108.35 116.68 125.02 133.36 141.70

.87 1.02 1.19 1.36 1.55 1.75

9

150.04 158.38 166.72 175.06 183.40 191.74

1.96 2.19 2.43 2.67 2.93 3.21

10

10 20 30 40 50

200.08 208.43 216.77 225.12 233.47 241.81

3.49 3.79 4.10 4.42 4.76 5.10

11

10 20 30 40 50

5.46 5.83 6.21 6.61 7.01 7.43

12

40 50

250.16 258.51 266.86 275.21 283.57 291.92 300.28 308.64 316.99 325.35 333.71 342.08

7.88 8.31 8.76 9.23 9.71 10.20

18

10 20 30 40 50

350.44 358.81 367.17 375.54 383.91 392.28

10.71 11.22 11.75 12.29 12.85 13.41

5

10 20 30

.22 .30 .39 .49 .61 .73

Tangent External

e

10' 20 30 40 50

400.66 409.03 417.41 425.79 434.17 442.55

13.99 14.88 15.18 15.80 16.43 17.07

15*0

10 20 30 40 50

450.93 459.32 467.71 476.10 484.49 492.88

17.72 18.38 19.06 19.75 20.45 21.16

16

10 20 30 40 50

501.28 509.68 518.08 526.48 534.89 543.29

21.89 22.62 23.38 24.14 24.91 25.70

17

10 20 30 40 50

551.70 560.11 568.53 576.95 585.36 593.79

26.50 27.31 28.14 28.97 29.82 30.68

18

602.21 610.64 619.07 627.50 635.93 644.37

31.56 32.45 33.35 34.26 35.18 36.12

19

10 20 30 40 50

652.81 661.25 669.70 678.15 686.60 695.06

37.07 38.03 39.01 39.99 40.99 42.00

20

703.51 711.97 720.44 728.90 737.37 745.85

43.03 44.07 45.12 46.18 47.25 48.34

21

80

10 20

30 40 50

Tangent External

10' 20 30 40 50

754.32 762.80 771.29 779.77 788.26 796.75

49.44 50.55

10 20 30 40 50

805.25 813.75 822.25 830.76 839.27 847.78

56.31 57,50 58.70 59.91 61.14 62.38

10 20 30 40 50

856.30 864.82 873.35 881.88 890.41 898.95

63.63 64.90 66.18 67.47

10 20 30 40 50

907.49 916.03 924.58 933.13 941.69 950.25

71.42 72.76 74.12 75.49 76.86 78.26

10 20 30 40 50

958.81 967.38 975.96 984.53 993.12 1001.7

79.67 81.09 82.53 83.97 85.43 86.90

10 20 30 40 50

1010.3.

1018.9 1027.5 1036.1 1044.7 1053.3

88.39 89.89 91.40 92.92 94.46 96.01

10 20 30 40 50

1061.9 1070.6 1079.2 1087.8 1096.4 1105.1

97.57 99.16 100.75 102.35 103.97

52.82 53.97 5868 55.13

70.09

105.60

Corrections to be Added (T= Tangent. E= External.)

Ange S

10o 15o 20| 25f

um5 T=.02

E=.000 T=.03 E .001 T=.04 E= .003 T=.06 E= .006 T=.08 SE=.009

10

.03 .000 .06 .003 .10 .007 .13 .011 .16 .018

1 15

.05 .001 .09 .004 .14 .010 .19 .017 .24 .027

200

25'

30

350

400

.06

.08 .0022 .16 .007 .24 .018 .32 .028 .40 .046

.10 .002 .19 .008 .29 .023 .39 .034 .49 .056

.11 .002 .22 .009 .34 .027 .45 .038 .58 .065

.18

.001 .13 .006 .19 .014 .26 .022 .33 .036

450

5W

.16 .003.003 .00 .004 .25 .28 .31 .011 .012 .014 .39 .45 .51 .029 .032 .035 .51 .58 .65 .045 .051 .051 .67 .75 .83 .074 .083 .093 .15

550

600

650

70

.18

.20

.21

.23

.004 .84 .015 .53 .039 .72 .063 .90 .106

8 .017 .58 .043 .79 .070 .99 .120

5 .005 .42 .4 .018 20 .63 .68 .047 .051 .84 .90 .076 .083 1.06 1.14 .127 .135

Tangents and Externals to a 10 Curve.

Chord =100 ft.

L-

ToAmi S

-

Tangent External

ne

Agle

1481.8 1490.7 1499.6 1508.5 1517.4 1526.3

188.51 190.74 192.99 195.25 197.53 199.82

10 20 30 40 50

1535.3 1544.2 1553.1 1562.1 1571.0 1580.0

202.12 204.44 206.77 209.12 211.48 213.86

1589.0 1598.0 1606.9 1615.9 1624.9 1633.9

216.3 218.7 221.1 223.5 226.0 228.4

88

10 20 30 40 50

1643.0 1652.0 1661.0 1670.0 1679.1 1688.1

230.9 233.4 235.9 238.4 241.0 243.5

89

10 20 30 40 50

150.71 152.69 154.69 156.70 158.72 160.76

1697.2 1706.3 1715.3 1724.4 1733.5 1742.6

246.1 248.7 251.3 253.9 256.5 259.1

40

10 20 30 40 50

1375.6 1384.4 1393.2 1402.0 1410.9 1419.7

162.81 164.86 166.95 169.04 171.15 173.27

10 20 30 40 50

1751.7 1760.8 1770.0 1779.1 1788.2 1797.4

261.8 264.5 267.2 269.9 272.6 275.3

1428.6 1437.4 1446.3 1455.1 1464.0 1472.9

175.41 177.55 179.72 181.89 184.08 186.29

10 20 30 40 50

1806.6 1815.7 1824.9 1834.1 1843.3 1852.5

278.1 280.8 283.6 286.4 289.2 292.0

107.24 108.90 110.57 112.25 113.95 115.66

290

1165.7 1174.4 1183.1 1191.8 1200.5 1209.2

117.38 119.12 120.87 122.63 124.41

30

10 20 30 40 50

1217.9 1226.6 1235.3 1244.0 1252.8 1261.5

128.00 129.82 131 65 133.50 135.35 137.23

381

10 20 30 40 50

1270.2 1279.0 1287.7 1296.5 1305.3 1314.0

139.11 141.01 142.93 144.85 146.79 148.75

32

10 20 30 40 50 10 20 30 40 50

1322.8 1331.6 1340.4 1349.2 1358.0 1366.8

10 20 30 40 50 10 20 30 40 50

SlO

24

25

27

28

Angle

10' 20 30 40 50

1113.7 1122.4 1131.0 1139.7 1148.4 1157.0

10', 20 30 40 50

Tangent External

126.20

)

35

IL

Tangent External

50

1861.7 1870.9 1880.1 1889.4 1898.6 1907.9

294.9 297.7 300.6 303.5 306.4 309.3

10 20 30 40 50

1917.1 1926.4 1935.7 1945.0 1954.3 1963.6

312.2 315.2 318.1 321.1 324.1 327.1

10 20 30 40 50

1972.9 1982.2 1991.5 2000.9 2010.2 2019.6

330.2 333.2 336.3 339.3 342.4 345.5

10 20 30 40 50

2029.0 2038.4 2047.8 2057.2 2066.6 2076.0

348.6 351.8 354.9 358.1 361.3 364.5

10 20 30 40 50

2085.4 2094.9 2104.3 2113.8 2123.3 2132.7

367.7 371.0 374.2 377.5 380.8 384.1

2142.2 2151.7 2161.2 2170.8 2189.9

387.4 390.7 394.1 397.4 400.8 404.2

2199.4 2209.0 2218.6 2228.1 2237.7 2247.3

407.6 411.1 414.5 418.0 421.4 425.0

860

10' 20 30 40

lO 10 20 30 40 50 42

10 20 30 40 50

2180.3

Corrections to be Added (T = Tangent. E = External.)

Angle 20 25o 300 350 °

40

°

°

Curve 5

100

T=.06 E=.000 T=.08 E=.009 T=.10 E=.013 T=.11 E=.018 T=.13 E=.023

.18 .19 .011.017 .16 .24 .018 .027 .19 .29 .025 .038 .22 .34 .035 .054 .26 .40 .046 .070

T=.15 E=.030

30 .060

15

°

°

40

450

°

50

550

200

25°

30

350

.26 .022 .83 .036 .39 .051 .47 .072 .53 .093

.32 .028 .40 .046 .49 .065 .58 .086 .67 .117

.39 .034 .49 .056 .59 .078 .69 .109 .80 .141

.45 .51 .088 .045 .58 .67 .065 .074 .69 .79 .090 .103 .80 .93 .131 .153 .93 1.06 .172 .203

.60 .119

.70 .153

1.70 1.37 1.52 1.21 .91 1.06 .184 .216 .254 .289 .325 .351

.58 .65 72 79 .051 .057 .063 .070 .75 .83 .90 .99 .083 .093.106.120 .89 .99 1.091.20 .116 .129 .149 .170 1.05 1.17 1.29 1.42 .175 .197 .213 .230 1.20 1.34 1.49 1.64 .234 .265 .277 .290

o

45

.44 .093

°

60

650

.84 .076 1.06 .127 1.29 .179 1.54 .247 1.79 315

1.87 2.0 .378 .411

700

.90 .083 1.14 .185 1.39 .188 1.66 .264 1.94 .341 2.21 .445

IV

TABLE I. Int. e

e

TangentExternal

Tangents and Externals to a 10 Curve.

Chord =100 ft.

Int.

Angle gle

2257.0 2266.6 2276.2 2285.9 2295.6 2305.2

428.5 432.0 435.6 439.2 442.8 446.4

500

2314.9 2324.6 2334.3

51

2353.8 2363.5

450.0 453.6 457.3 461.0 464.6 468.4

10 20 30 40 50

2373.3 2383.1 2392.8 2402.6 2412.4 2422.3

10 20 30 40 50

47

592.3 596.6 600.9 605.3 609.6 614.0

10 20 30 40 50

2732.9 2743.1 2753.4 2763.7 2773.9 2784.2

618.4 622.8 627.2 631.7 636.2 640.7

472.1 475.8 479.6 483.4 487.2 491.0

2794.5 2804.9 2815.2 2825.6 2835.9 2846.3

645.2 649.7 654.3 658.8 663.4 668.0

59

2432.1 2441.9 2451.8 2461.7 2471.5 2481.4

494.8 498.7 502.5 506.4 510.3 514.3

2856.7 2867.1 2877.5 2888.0 2898.4 2908.9

672.7 677.3 682.0 686.7 691.4 696.1

60

10 20 30 40 50

2491.3 2501.2 2511.2 2521.1 2531.1 2541.0

518.2 522.2 526.1 530.1 534.2 538.2

2919.4 2929.9 2940.4 2951.0 2961.5 2972.1

700.9 705.7 710.5 715.3 720.1 725.0

61

10 20 30 40 50

2551.0 2561.0 2571.0 2581.0 2591.0 2601.1

542.2 546.3 550.4 554.5 558.6 562.8

2982.7

729.9 734.8 739.7 744.6 749.6 754.6

62

3003.9 3014.5 3025.2 3035.8

2611.2 2621.2 2631.3 2641.4 2651.5 2661.6

566.9 571.1 575.3 579.5 583.8 588.0

3046.5 3057.2 3067.9 3078.7 3089.4 3100.2

10 20 30 40 50

nt

4 500 s 6_P

65

a I nt.

IAnge

2671.8 2681.9 2692.1 2702.3 2712.5 2722.7

10' 20 30 40 50

46

Tagnnl

10' 20 30 40 50

430

44

Tangent External

a

Tangent 3110.9 3121.7 3132.6 3143.4 3154.2 3165.1

790.1 795.2 800.4 805.6 810,9 816.1

3176.0 3186.9 3197.8 3208.8 3219.7 3230.7

821.4 826.7 832.0 837.3 842.7 848.1

10 20 30 40 50

3241.7 3252.7 3263.7 3274.8 3285.8 3296.9

853.5 858.9 864.3 869.8 875.8 880.8

10 20 30 40 50

3308.0 3319.1 3330.3 3341.4 3352.6 3363.8

886.4 892.0 897.5 903.2 908.8 914.5

10 20 30 40 50

3375.0 3386.3 3397.5 3408.8 3420.1 3431.4

920.2 925.9 931.6 937.3 943.1 948.9

10 20 30 40 50

3442.7 3454.1 3465.4 3476.8 3488.3 3499.7

954.8 960.6 966.5 972.4 978.3 984.3

759.6 764.6 769.7 774.7 779.8 784.9

3511.1 3522.6 3534.1 3545.6 3557.2 3568.7

990.2 996.2 1002.3 1008.3 1014.4 1020.5

Corrections to be Added (T = Tangent.

E= External.)

2344.1

2993.3

570

10' 20 30 40 50

Curve0

100

150

20

250

300

350

T=.13

.26

.40

.53

.67

.80

.93

E=.023

.046

.070

.003

.117

.141

.172

.203

.234

.265

.277

.290

.15

T=.15 E=.030 T=.17 E=.037 T=.19 E=.046 T=.21 E=.056 T.23 E=.067

.30 .060 .34 .075 .38 .093 .42 .112 .46 .135

.44 .093 .51 .116 .57 .142 .63 .168 .69 .204

.60 .119 .68 .151 .76 .188 .84 .225 .93 .273

.76 .153 .85 .189 .95 .236 1.05 .283 1.16 .343

.91 .184 1.02 .227 1.14 .283 1.27 .340 1.40 .412

1.06 .216 1.19 .266 1.42 .332 1.49 .398 1.64 .483

1.21 .254 1.36 .305 1.52 3.81 1.71 .457 1.88 .554

1.37 .289 1.54 .345 1.72 .420 1.94 .516 2.13 .625

1.52 .325 1.72 .384 1.92 .479 2.17 .575 2.38 .697

1.70 .351 1.91. .425 2.14 .530 2.38 .636 2.63 .711

1.87 .378 2.10 .467 2.35 .582 2.60 .697 2.88 .

2.04 .411 2.29 .508 2.56 .41 2.83 .774 3.1 .922

400

450

50°

550

600

650

70°

.106 1.20 1.34 1.49 1.64 1.79 1.9 .

2.21 .44 2.48 .550, 2.77 .700 3.07 .851 3.39 1.01

Tasza L -

Tangents and Externals to a

Curve.

V

Chord = 100 ft. nge

Tangent External

g.

1308.2 1315.6 1322.9 1330.3 1337.7 1345.1

780 10' 20 30 40 50

4639.8 4653.6 4667.4 4681.3 4695.2 4709.2

1643.0 1651.7 1660.5 1669.2 1678.1 1686.9

10 20 30 40 50

4162.8 4175.6 4188.5 4201.2 4214.0 4226.8

1352.6 1360.1 1367.6 1375.2 1382.8 1390.4

78

4723.2 4737.2 4751.2 4765.3 4779.4 4793.6

1695.8 1704.7 1713.7 1722.7 1731.7 1740.8

783 10 20 30 40 50

4239.7 4252.6 4265.6 4278.5 4291.5 4304.6

1398.0 1405.7 1413.5 1421.2 1429.0 1436.8

4807.7 4822.0 4836.2 4850.5 4864.8 4879.2

1749.9 1759.0 1768.2 1777.4 1786.7 1796.0

10 20 30 40 50

4317.6 4330.7 4343.8 4356.9 4370.1 4383.3

1444.6 1452.5 1460.4 1468.4 1476.4 1484.4

4893.6 4908.0 4922.5 4937.0 4951.5 4966.1

1805.3 1814.7 1824.1 1833.6 1843.1 1852.6

50

4396.5 4409.8 4423.1 4436.4 4449.7 4463.1

1492.4 1500.5 1508.6 1516.7 1524.9 1533.1

4980.7 4995.4 5010.0 5024.8 5039.5 5054.3

1862.2 1871.8 1881.5 1891.2 1900.9 1910.7

10 20 30 40 50

4476.5 4489.9 4503.4 4516.9 4530.4 4544.0

1541.4 1549.7 1558.0 1566.3 1574.7 1583.1

10 20 30 40 50

5069.2 5084.0 5099.0 5113.9 5128.9 5143.9

1920.5 1930.4 1940.3 1950.3 1960.2 1970.8

10 20 30 40 50

4557.6 4571.2 4584.8 4598.5 4612.2 4626.0

1591.6 1600.1 1608.6 1617.1 1625.7 1634.4

10 20 30 40 50

5159.0 5174.1 5189.3 5204.4 5219.7 5234.9

1980.4 1990.5 2000.6 2010.8 2021.1 2031.4

1026.6 1032.8 1039.0 1045.2 1051.4 1057.7

3650.2 3661.9 3673.7 3685.4 3697.2 3709.0

1063.9 1070.2 1076.6 1082.9 1089.3 1095.7

72

10 20 80 40 50

3720.9 3732.7 3744.6 3756.5 3780.4

1102.2 1108.6 1115.1 1121.7 1128.2 1134.8

3792.4 3804.4 3816.4 3828.4 3840.5 3852.6

1141.4 1148.0 1154.7 1161.3 1168.1.

1174.8

3864.7 3876.8 3889.0 3901.2 3913.4 3925.6

1181.6 1188.4 1195.2 1202.0 1208.9 1215.8

75

3937.9 3950.2 3962.5 3974.8 3987.2 3999.5

1222.7 1229.7 1236.7 1243.7 1250.8 1257.9

78

4011.9 4024.4 4036.8 4049.3 4061.8 4074.4

1265.0 1272.1 1279.3 1286.5 1293.6 1300.9

77

10 20 30

84

Corrections to be Added (T ITangent. °

Int

Curve 50

100

150 20

800

T=.21 E=.056

.42

.63

6o

T=.23 E=.067 T.25 E=.080 T=.27

.40 .1 .51 .159 .56

.69 .204 .76 .240 .83

T=.30 S =.110 TI=.33 E=.128

.61 .220 ,66 .259

0o J75 80 o

E=.095

Tangent External

4086.9 4099.5 4112.1 4124.8 4137.4 4150.1

3580.3 3591.9 3603.5 3615.1 3626.8 3638.5

10 20 30 40 50

Angle

10' 20 30 40 50

10' 20 30 40 50

3768.5

Tangent Extea

250

300

350

10 20 30 40 50

E = External.)

400 450

500

550

600

650

700

.84 1.05 1.27 1.49 1.71 1.94 2.17 2.38 2.60 2.83 3.07

.112 .168 .225 .283 .340 .398 .457 .516 .575 .836 .697 .774 .851

.182

.286

.93 1.16 1.40 1.64 .273 .343 .412 .483 1.02 1.28 1.54 1.80 .321 .403 .485 .568 1.12 1.40 1.69 1.98 .383

.480 .578 .678

1.88 .554 2.06 .652 2.27 .777

.91 1.22 1.53 1.84 2.15 2.46 .332 .445 .558 .671 .787 .903 1.00 1.33 1.68 2.02 2.36 2.70 .391 24 .657 790 1.06

2.13 .625 2.33 .735 2.587

2.38 .697 2.60 .819

2.78 1.02 3.05 1.20

3.10 1.13 8.40 1.34

.877

2.63 .711 2.88 .906 3.16

2.88 3.13 .845 .922 3.16 3.44 .994 1.08 3.47 3.78

3.39 1.01 3.72 1.17 4.09

3.44 1.25 3.77 1.47_

3.78 1.38 4.14 162

4.46 162 4.89 1.1

.977 1.07

1.18 1.29

4.12 1.50 4.55 1.76.

1.39

Vt

Ange Aaigle

j TligXEr

tenll

Angfl

/Ete fl1a

Taiiee

Angie

j

T

xter

10' 20 30 40 50

5250.3 5265.6 5281.0 5296.4 5311.9 5327.4

2041.7 2052.1 2062.5 2073.0 2083.5 2094.1

5933.2 5950.5 5967.9 5985.3 6002.7 6020.2

2518.5 2531.0 2543.5 2556.0 2568.6 2581.3

6708.6 6728.4 6748.2 6768.1 6788.1' 6808.2

3092.7 3107.7 3122.9 3138.1 3153.3

10 20 30 40 50

5343.0 5358.6 5374.2 5389.9 5405.6 5421.4

2104.7 2115.3 2126.0 2136.7 2147.5 2158.4

6037.8 6055.4 6073.1 6090.8 6108.6 6126.4

2594.0 2606.8 2619.7 2632.6 2645.5 2658.5

6828.3 6848.5 6868.8 6889.2 6909.6 6930.1

3184.1 3199.6 3215.1 3230.8 3246.5 3262.3

2169.2 2180.2 2191.1 2202.2 2213.2 2224.3

6144.3 6162.6 6180.2 6198.3 6216.4 6234.6

2671.6 2684.7 2697.9 2711.2 2724.5 2737.9

6950.6 6971.3 6992.0 7012.7 7033.6 7054.5

3278.1

50

5437.2 5453.1 5469.0 5484.9 5500.9 5517.0

10 20 30 40 50

5533.1 5549.2 5565.4 5581.6 5597.8 5614.2

2235.5 2246.7 2258.0 2269.3 2280.6 2292.0

6252.8 6271.1 6289.4 6307.9 6323.3 6344.8

2751.3 2764.8 2778.3 2792.0 2805.6 2819.4

7075.5 7096.6 7117.8 7139.0 7160.3 7181.7

3374.9 3391.2 3407.7 3424.3

5630.5 5646.9 5663.4 5679.9 5696.4 5713.0

2303.5 2315.0 2326.6 2338.2 2349.8 2361.5

10 20 30 40

50

6363.4 6382.1 6400.8 6419.5 6438.4 6457.3

2833.2 2847.0 2861.0 2875.0 2889.0 2903.1

7203.2 7224.7 7246.3 7268.0 7289.8 7311.7

3474.4 3491.3 3508.2 3525.2 3542.4 3559.6

5729.7 5746.3 5763.1 5779.9 5796.7 5813.6

2373.3 2385.1 2397.0 2408.9 2420.9 2432.9

10 20 30 40 50

6476.2 6495.2 6514.3 6533.4 6552.6 6571.9

2917.3 2931.6 2945.9 2960.3 2974.7 2989.2

7333.6 7355.6 7377.8 7399.9 7422.2 7444.6

3576.8 3594.2 3611.7 3629.2 3646.8 3664.5

5830.5 5847.5 5864.6 5881.7 5898.8 5916.0

2444.9 2457.1 2469.3 2481.5 2493.8 2506.1

6591.2 6610.6 6630.1 6649.6 6669.2 6688.8

3003.8 3018.4 3033.1 3047.9 3062.8 3077.7

7467.0 7489.6 7512.2 7534.9 7557.7

3682.3

88

S6 10 20

80 40

88

TIwx I. -- Tangentsandxternalstoa1P Cure.

Chord =100 ft.

T n Et I Ira E ltIn I a

10

20 30 40

50 10 20 30 40

50

31 10 20

30 40 80

97

3168.7

3294.1 3310.1 3326.1 3342.3

3358.5

3440.9 3457.6

3700.2 3718.2 3736.2 3754.4

3772.6

7580.5

Corrections to be Added (T =Tangent. E. = External.) l

Curve So

100

150

200 250

300

35

T=.39 E=.174

.66 1.00 1.33 1.68 2.02 2.36 .259 .391 .524 .657 .790 .926 .72 1.09 1.45 1.83 2.20 2.57 .299 .450 .603 .756 .910 1.07 .79 1.19 1.55 2.00 2.40 2.80 .350 .522 .706 .985 1.06 1.25

100

T=.43 E=.200

.86 1.30 1.74 2.18 .401 .604 .809 1.01

105.

T=.46

.94

850 900 950

T=.33 E=.128 T=.36 E=.149

E=.230

400 2.70 1.06 2.94 1.22

3.20 1.43 2.62 3.06 3.50 1.22 1.43 1.64 1.90 12.38 2.87 3.34 3.84

450 3.05 1.20 3.32 1.38 3.61 1.62 3.95 1.85 4.35

500 3.40 1.34 3.70 1.54 4.02 1.80 4.40 2.06 4.84

550

600

65.

70

3.77 1.47 4.10 1.70 4.49 1.99 4.88 2.28 5.35

4.14 1.62 4.50 1.87 4.98 2.18 5.37 2.50 5.87

4.55 1.76 4.91 2.03 5.38 2.38 5.85 2.73 6.40

4.89 1.01 5.33 2.20 5.83 2.55 6.34 2.90 6.93

1.42 .470 .700 .938 1.17 1.42 1.65 1.90 2.14 2.39 2.64 2.90 3.18 3.41

TABLE I. - Tangents and Externals to a 1° Curve.

TA

Chord = 100 ft. e 4 Tangent External

Tangent External

A

°

5082.7 5107.9 5133.3 5158.8

4516.6 4538.8 4561.1 4583.4 4606.0 4628.6

9349.9 9380.5 9411.3 9442.2 9473.2 9504.4

5236.2 5262.3 5288.6 5315.0 5341.5 5368.2

8656.6 8684.0 8711.5 8739.2 8767.0 8794.9

4651.3 4674.2 4697.2 4720.3 4743.6 4766.9

9535.7 9567.2 9598.9 9630.7 9662.6 9694.7

5395.1 5422.1 5449.2 5476.5 5504.0 5531.7

8822.9 8851.0 8879.3 8907.7 8936.3 8965.0

4790.4 4814.1 4837.8 4861.7 4885.7 4909.9

9727.0 9759.4 9792.0 9824.8 9857.7 9890.8

5559.4 5587.4 5615.5 5643.8 5672.3 5700.9

8993.8 9022.7 9051.7 9080.9 9110.3 9139.8

4934.1 4958.6 4983.1 5007.8 5032.6 5057.6

9924.0 9957.5 9991.0 10025.0 10059.0 10093.0

5729.7 5758.6 5787.7 5817.0 5846.5 5876.1

Corrections to be Added (T = Tangent.

E = External.)

7603.5 7626.6 7649.7 7672.9 7696.8 7719.7

3791.0 3809.4 3827.9 3846.5 3865.2 3884.0

11

7743.2 7766.8 7790.5 7814.3 7838.1 7862.1

3902.9 3921.9 3940.9 3960.1 3979.4 3998.7

112

10 20 30 40 50

7886.2 7910.4 7934.6 7959.0 7988.5 8008.0

4018.2 4037.8 4057.4 4077.2 4097.1 4117.0

113

10 20 30 40 60

8032.7 8057.4 8082.3 8107.3 8132.3 8157.5

4137.1 4157.3

114

10 20 80 40 50 10 20 30 40 50

8182.8 8208.2 8233.7 8259.3 8285.0 8310.8

4259.7 4280.5 4301.4 4322.4 4343.6 4364.8

110

Angle

1000 1050 1100 115 120o

°

9169.4 9199.1 9229.0 9259.0 9289.2 9319.5

°10' 20 80 40 50

109

Tangent External

An e

°

Curve 5

T=.43 E=.200 T=.46 E=.230 T=.50 E=.260 T=.54 _8=.307 T=.1 =.8339

4177.5

4197.9 4218.4 4239.0

10

°

.86 .401 .94 .470 1.03 .535 1.13 .624 1.25

150

1.30 .604 1.42 .700 1.55 .808 1.70 .939 1.89 1.08

°

20

1.74 .809 1.90 .938 2.08 1.08 2.29 1.26 2.52 1.45

10' 20 30 40 50

8336.7 8362.7 8388.9 8415.1 8441.5 8468.0

4386.1 4407.6 4429.2 4450.9 4472.7 4494.6

10 20 30 40 50

8494.6 8521.3 8548.1 8575.0 8602.1 8629.3

10 20 30 40 50 10 20 30 40 50

25

°

2.18 1.01 2.38 L17 2.60 1.36 2.86 1.57 3.16 1.82

30

°

2.62 1.22 2.87 1.42 3.14 1.63 3.45 1.89 3.81 2.20

350

3.06 1.43 3.34 1.65 3.66 1.91 4.03 2.21 4.44 2.56

400

3.50 1.64 3.84 1.90 4.21 2.19 4.63 2.54 5.11 2.95

116

10( 20 30 40 50

450 50

3.95 1.85 4.35 2.14 4.76 2.49 5.23 2.87 5.78 3.33

°

4.40 2.06 4.84 2.39 5.31 2.61 5.83 3.20 6.44 3.72

55

°

4.88 2.28 5.35 2.64 5.86 3.05 6.44 3.53 7.11 4.10

°

5184.5

5210.3

°

60

65

700

5.37 2.50 5.87 2.90 6.43 3.35 7.07 3.88 7.80 4.50

5.85 2.73 6.40 3.16 7.01 .3.65 7.70 4.23 8.51 4.91

6.34 2.96 6.93 3.41 7.59 3.95 8.35 4.58 .21 .32

TABL IL Deg

Radii, Ordinates and Deflections. Chord =100 ft.

Def. Tan. IDist.

Mid. IDist. usSlOrd.

It. 0*10' 34377.

20 17189.

30 11459.

40 8594.4 50 6875.5 5729.6 1 10 4911.2 20 4297.3 3o 3819.8 40 3437.9 50 3125.4 I 2864.9 10 2644.6 20 2455.7 30 2292.0 40 2148.8 50 2022.4 3.

1910.1 10 1809.6 20 1719.1 30 1637.3 40 1562.9 50 1495.0 1432.7 10 1375.4 20 1322.5 80 1273.6 40 1228.1 50 1185.8 1146.3 10 1109.3 20 1074.7 380 1042.1 40 1011.5 50 982.6 6 955.4 10 929.6 20 905.1 30 881.9 40 859.9

It.

.036 .073 .109 .145 .182 .218 .255 .291 .327 .364 .400 .436 .473 .509 .545 .582 .618 .655 .691 .727 .764 .800 .836 .873 .909 .945 .982

Def

ift 1.1

De

Radius

It.

it.

.145 .291 0.05 .291 .582 0.10 .436 .873 0.15 .582 1.164 0.20 .727 1.454 0.25 .873 1.745 0.30 1.018 2.036 0.35 1.164 2.327 0.40 1.309 2.618 0.45 1.454 2.909 0.50 1.600 3.200 0.55 1.745 3.490 0.60 1.891 3.781 0.65 2.036 4.072 0.70 2.181 4.363 0.75 2.327 4.054 0.80 2.472 4.945 0.85 2.618 5.235 0.90 2.763 5.526 0.95 2.908 5.817 1.00 3.054 6.108 1.05 3.199 6.398 1.10 6.689 6.980 7.271 7.561 7.852 8.143 8.433 8.724 9.014 9.305 9.596 9.886 10.18 10.47 10.76 11.05 11.34 11.63

3.345

3.490 3.635 3.718 3.926 1.018 4.071 1.055 4.217 1.091 4.362 1.127 4.507 1.164 4.653 1.200 4.798 1.237 4.943 1.273 5.088 1.309 5.234 1.346 5.379 1.382 5.524 1.418 5.669 1.455 5.814

20' 30 40

8

20 30 40

9

20 30 40 3o 30 30 30

14

30

15

1.15

1.20 1.25 1.30 1.35 1.40.

1.45 1.50

Ls

30 30o

1.55 1.60

1.65 1.70 1.75 1.80 1.85 1.90 1.95

2.00

It. 819.0 781.8 764.5 747.9 716.8 688.2 674.7 661.7 637.3 614.6 603.8 593.4 573.7 546.4 521.7 499.1 478.3 459.3 441.7 425.4 410.3 396.2 383.1 370.8 359.3 348.5 338.3 319.6 302.9 287.9 274.4 262.0 250.8 240.5 231.0 222.3 214.2 206.7 199.7 193.2

Mid Ord. 0r4 it.

1.528 1.600 1.637 1.673 1.746 1.819 1.855 1.892 1.965 2.037 2.074 2.110 2,183 2.292 2.402 2.511 2.620 2.730 2.839 2.949 3.058 3.168 3.277 3.387 3.496 3.606 3.716 3.935

4.155 4.374 4.594 4.814 5.035

n. ist. Dist.Ft Dist.

D Iit. It.

ft.

6.105 12.21 2.10

6.395 6.540 6.685 6.976 7.266 7.411 7.556 7.846 8.136

8.281 8.426 8.716 9.150 9.585 10.02 10.45 10.89 11.32 11.75 12.18 12.62 13.05 13.49 13.92 14.35 14.78 15.64 16.51 17.37 18.22 19.08 19.94 20.79 21.64 22.50

5.255 5.476 5.697 5.918 23.35 6.139 24.19 6.360 25.04 6.583 25.88

12.79 13.08 13.37 13.95 14.53 14.82 15.11 15.69 16.27 16.56 16.85 17.43 18.30 19.16 20.04 20.91 21.77 22.64 23.51 24.37 25.24 26.11 26.97 27.84 28.70 29.56 31.29 33.01 34.73 36.44 38.16 39.87 41.58 43.28 44.99 46.69 48.38 50.07 51.76

2.20 2.25 2.30 2.40 2.50 2.55 2.60 2.70 2.80 2.85 2.90 3.00 3.15 3.30 3.45 3.60 3.75 3.90

4.05 4.20

4.35 4.50 4.65 4.80

4.95 5.10 5.40 5.70 6.00 6.30 6.60 6.00 7.20 7.50 7.80 8.10 8.40 8.70 9.00

The middle ordinate in inches for any cord of length (C) is equal to..0012 OC multiplied by the middle ordinate taken from the above table. Thus, if it desired to bend a 30 ft. rail to fit a 10 degree ourve, its middle ordinate should be .0012X900X2.183 or 2.36 inches.

TABLE IIL Deflections for Sub Chords for Short Radius Curves.

Degree Radius of 50 Curve sin. I def. ang. °

12.5 Ft. 5

sub chord R 15 Ft.

sin of

I def.

20 Ft.

angle of 25 Ft.

Length arc for 100 ft.

193.18

I°

1

2017

2

58'

3043'

101.15

181.39

10 59

2025

3

3

II

171.01

36 380

I6I.80 153.58

2°

Io'

20 4 2 49

3 33 3 44'

40

I146.19

20 o' 2 27

2

3

42

139.52

2 34'

44

133.47

20

460 480

127.97 122.92

2* 48' 2. 55

30 3 29'

4 29' 4 40'

50 520

118.31 114.o6

3 o2' ' 309

4

103.14

3 16' 30 22' 3 29

30 38' 3 46' 3 54' 4 02 4 10'

I60o.oo

3 35

4

30

32

°

34°

° 54 °

56 580

110.11

1o6.5o

2

06'

3'

I

°

2 33' 57'

30 5' °

3

°

3'

18'

21'

4.

58'

2'

40 26' 4 40

33

iol.48 Ior.66 101.85

5

54 8'

102.06

4007' 4

5

22'

I02.53

5 36' 50' °

102.76 o103.oo

6 04' 617' ' 60 31

lo3.24 10o3.54

6044 6057'

o104.14 104.43

70 11

104.72

55 18'

51' 02' 13 50 23' 5 34' 5 5

5 44'

4

102.29

I3.84

CURVE FORMULAS R = T cot.

T Rtan I T=5 tanR I Sin. D Sin. Sin.

R

So D =Sin. R 5o tan D

I

Chord def.

IhR

I

50

Sin. DD

R

No. chords = I

E = R ex. sec # ID E T tan I

I

=chord

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 I ° 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 .oI745, 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 00oo,Alt. Io.o10 200 =.5. Ioo+.5= 100.5 hyp.

Given Hyp. 1oo, Alt. 25.25+20oo0=3.125. 100-3.125 =96.875= Base.

Error in first example, .0o2; in last, .o045.

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, Jd S . The combined correction is negative.

PROBABLE ERROR. If d, d d., etc. are the discrepancies of various results from the mean and if dA=the sum of the squares of these difi'ences and n=the number of observations, then the probable error of the. mean= 4 0.674a -1) SOLAR EPHEMERIS. Attention is called to the Solar Ephemerdsor the current year, published by Keuffel & Esser Co., and furnished free of charge upon request, which is 8~x5J in., with about 90 pages of data very useful to the Surveyor; such as the adjustments of transits, levels and solar attachments; directions and tables for determining the meridian and the latitude from observations on the sun and Polaris; stadia measurements; magnetic declination; arithmetic constants; English and Metric conversions; trigonometric formulas; Natural and Logarithmic Functions; and Logarithms of Numbers.

TABLE IV. - Minutes in Decimals of a Degree.

1' 2

8

6

7

8 9 18

.0167 .0333 .0500 .0667 .0833 .1000 .1167 .1333 .1500

11' 12 13 14 15 16 17 18 19

.1833 .2000 .2167 .2333 .2500 .2667 .2833 .3000 .3167

.1667

21' 22 23 24 25 26 27 28 29

20

.3500 .3667 .3833 .4000 .4167 .4333 .4500 .4667 .4833

.3333

30

31' 32 33 34 85 36 37 88 39

.5000

.5167 .5333 .5500 .5667 .5833 .6000 .6167 .6333 .6500

40

.6667

41' 42 43 44 45 46 47 48 49 50

.6833 .7000 .7167 .7333 .7500 .7667 .7833 ,8000 .8167 .8333

51' 52 53 54 85 56 57 58 59 00

.8500 .8687 .8838 .9000 .9167 .9333 .9500 .9667 .9833

1.0000

,

TABL V. -- Inches in Decimals of a Foot. .0052 1-16 1

.0888

.0078 3-82 2

.1,67

. 0 3

.2500

.056 3-16 4

.333

.0208 0

.0260 5-16

.4167

.5000

6

.0313 4 7

.5833

.0417 8

.6667

.521 ,g 9

.0625 % 10

%

11

.7500- .8333 -.9167

Natural Trigonometrical Functions --Angle. Sn. Tan. S.

AnlI&SiL Tan. See. Cor. Ct. C*en.

1.

1.0001 1.0001 1.0002 1.0002 1.0003 1.0003 1.0004 1.0005 1.0006 1.0007 1.0008 1.0010 1.0011 1.0012 1.0014 1.0015 1.0017 1.0019 1.0020 1.0022 1.0024 1.0027 1.0029 1.0031 1.0033 1.0036 1.0038 1.0041 1.0048 1.0046 1.0049 1.0052 1.0055 1.0058 1.0061 1.0085 1.0068 1.0072 1.0075 1.0079 1.0082 1.0086 1.0090 .0094

1.

Case. Com. CIin.

1.0098 1.0102 1.0107 1.0111 1.0115 1.0120 1.0125 1.0129 1.0134 1.0139 1.0144 1.0149 1.0154 1.0160 1.0165 1.017 1.0176 1.0181 1.0187 1.0193 1.0199 1.0205 1.0211 1.0217 1.0223 1.0280 1.0230 1.0243 1.0249 1.0256 1.0263 1.0270 1.0277 1.0284 1.0291 1.0299 1.0306 1.0314 1.0321 1.0329 1.0337 1.0345 1.0353 1.0361 1.0369 1.0377 1.0386 1.0394

1.9 .9999 .99996 .9985 .99979 .99973 .99986 .99958 .99949 .99939 .999209 .99917 .99905 .99892 .99878 .99863 .99847 .99831 .99813 .99795 .99778 .99756 .99736 .99714 .99892 .90968 .99644 .99619 .99594 .99567 .99540 .99511 .99482 .99452 .99421 .99390 .99357 .99324 .99290 .99255 .99219 .99182 .99144 .99106 .99067

.99021 .9898( .98944 .9890 .9885E .98814 .98769 .98723 .9867 .9862 .98580 .98531 .98481 .98430 .98378 .98325 .98272 .98218 .98163 .98107 .98050 .97992 .97934 .97875 .97815 .97754 .97692 .97630 .97566 .97502 .97437 .97371 .97304 .97237 .97169 .97100 .97030 .96959 .96887 .96815 96742 .96667 .96593 .96517 .96440 .96363 .986285 .96208

"o

On. Cit

Cam Sea. Tan.

ShL

cosi.

An I

Cot. Cased. sec.

Tan. Sin. AN

Natural Trigonometrical Functions Anle.Sin. Tan. Sea. Cose. Cotg. Cosin.

1.040 1.0412 1.0422 1.0429 1.0438 1.0448 1.0457 1.0466 1.0476 1.0486 1.049 1.0505 1.0515 1.0525 1.0535 1.0545 1.0555 1.0566 1.0576 1.0587 1.0598 1.0608 1.0619 1.0631 1.0642 1.0653 1.0665 1.0676 1.0688 1.0700 1.0711 1.0723 1.0736 1.0748 1.0760 1.0773 1.0785 1.0798 1.0811 1.0824 1.0837 1.0850 1.0864 1.0877 1.0891 1.0904 1.0918 1.0932

Angle. Sin. Tan. Sea. Cosec. Cot. Cosk.

1.0946 1.0961 1.0975 1.0989 1.1004 1.1019 1.1034 1.1049 1.1064 1.1079 1.1095 1.1110 1.1126 1.1142 1.1158 1.1174 1.1190 1.1207 1.1223 1.1240 1.1257 1.1274 1.1291 1.1308 1.1326 1.1343 1.1361 1.1379 1.1397 1.1415 1.1434 1.1452 1.1471 1.1490 1.1509 1.1528 1.1547 1.1566 1.1586 1.1606 1.1626 1.1646 1.1666 1.1687 1.1707 1.1728 1.1749 1.1770

.9612E .96041 .95964 .95882 .9579 .95715 .95630 .95451 .95372 .95284 .95195 .95101 .95015 .94924 .94832 .94740 .94646 .94552 .94457 .94361 .94264 .94167 .94068 .93969 .93869 .93769 .93667 .93565 .93462 .93358 .93253 .93148 .93042 .92935 .92827 .92718 .92609 .92499 .92388 .92276 .92164 .92050 .91936 .91822 .91706 .91590 .91472

.91 .9123 .9111 .90875 .90753 .90631 .90507 .90383 .90259 .90133 .90007 .89879 .89752 .89623 .89493 .89363 .89282 .89101 .88968 .88835 .88701 .88566 .88431 .88295 88158 .88020 .87882 .87743 .87603 .87462 87321 .87178 87036 .86892 .86748 86603 .86457 .86310 .86163 .86015 .85866 .85717 .85567 .85416 .85264 .85112 .84959 oC

Cosin. Cotg. Coses. Sea. Tan. Sin. Angle

---

Coe*l Cotg.

Cose.

See. Tan. StI. An

---

-Natural Anke.SIn. Tan. Sea.

Trigonometrical Functions

Coota.Co. C

.

Anle. S.

Tan. Sea. Cases. Cot. Cas

o,

1.1792 1.8871.600.84805 1.1813 1.878 1.5900 .84650 1.1835 1.87C 1.580.84495 1.1857 1.861 1.570 .84339 1.1879 1.853 1.561. .84182 1.1901 1.84 1. .84025 1.1924 1.830 1.54 .83867 1.1946 1.82 1.53 .83708 1.1969 1.82 1.520 .83549 1 .1992 1.812 1.511 .83389 40 1.2015! 1.804 1.5011.83228 50 1.203 1.796 1.4921.83066 1.206 1.788 1.4831.82904 134 1.2086 1.7811.473 .82741 20 1.211 1.773 1.464 .82577 30 1.2134 1.766 1.455 .82413 40 1.2158 1.758 1.446182248 50 1.2183 1.751 1.437 .82082, 1.2208 1.743 1.428 .81915 35 10 1.2233 1.7361.419 .81748 1.2258 1.72 1.411 .81580 1.2283 1.7221.402 .81412 1.2309 1.71 1.398 .81242 1.2335 1.70 1.385 .81072 1.2361 1.7011.376 .80902 1.2387 1.695 1.368 .80780 10 1.2413 1.688 1.3601.80558 80 1.2440 1.6811 .351.80386 3730 1.2460 1.67 1.3431 .80212 40 SO 1.2494 1.668 1.335 .80038 50 1.2521 1.662 1.327 .79864 1.2549 1.655 1.39.79688 10 20 1.2577 1.649 1.311 .79512 1.260 1.6431.303 .79335 30 1.2633 1.636 1.295 .79158 40 1.26601 1.63 1.2881.78980 .50 1.2690 1.624 1.280 .78801 38 10 1.2719 1.6181.272 .78622 1.2748 1.612 1.265 .78442 20 30.6 .7 1.2778 1.606 1.257 .78261 30 40.824 .8001.28081.601 1.250.78079 20 5 .27 .8051.2838 1.595 1.242 77897 10 32 10 20 30 40 50 33 10 20

Co.

Cot Ca.

sec.Sea. Tan. Sin. Angle

1.2868 1.2898 1.2929 1.2959 1.2991 1.3022 1.3054 1.3086 1.3118 1.3151 1.3184 1.3217 1.3251 1.3284 1.8318 1.3352 1,386 1.3421 1.3456 1.3492 1.3527 1.3563 1.3600 1.3636 1.8673 1.3711 1.3748 1.3786 1.3824 1.3863 1.3902 1.3941 1.3980 1.4020 1.4061 1.4101 1.414

.77715 51 .77531 50 .7734740 .77162 30 .76977 20 .76791 10 .76604 50 .76417 50 .76229 .7604130 .7585120 .7566110 .75471 49 .75280 50 .75088.40 .7489630 .74703 20 .74509 10 48 .74314 .7412050 .73924 40 .7372830 .7353120 .7333 10 .73135 47 .72937 50 .72737 40 .7253730 .7233720 .72136 10 .71934 46 .71732 50 °.71529 40 .71325 30 .71121 20 .70916 10 45 .70711

Coain. Cotg. Cases. Sea. Tan. Sin. Angle

*PO~ '*4AW4{o

D.o,

WO4.I

ca.

Qbt

Asc~a

.. CF,& A

4

C. 4493

C4"~~cof'~.

f

Lt VA

s~

~)Gr

m JL

54 R

-

~I--°

a

""

W~~cl~e~JzvA el_~P;r~I, Bo~w9~\

r~~r

i?,d?.

0

~

,

i .r':-

.

TRIGONOMETRIC

b Right Triangle

Angl.

a e

co- e 't

A

I

, a, b

cotB, a o =

= aI

B=90 -A,

b =

B = 90 0-A,

a = b tan A, -

acotAc=

a

sin A.

.

cos A.

C

O = 180 0-- (A

sin A'

sinB

b sin4 in a

A+B=1800-

= 180 --

0

+

A, B, C

6,

e

Area

, area = %/(s-a)

af b

as , =180-(.A+B) (s-b) (s-o)

be sin A

2 A, B,

,aI Area

c sin (7 A sin A

0, tan (A-B)= (a--b)tan z (A+B)

,sin #A=

s= a

sin A

-(A + B), a =

sin jB=d (s --

Area

a sin a

B), a

a sin C sin A

b,a

021--

sin A -=cosB b =N(5_+a(oa

b

B, a

a, 6, a

-

t

a

B= 90°-A, a = sin A, b= ocos A, Solution of Oblique Triangles

b,c, i e,

a

,cot = ,see =;, cosec

4

= bi=

tan A

, B, b

;

6

Oblique Triangles Solution of Right Triangles o a b a b

0

u,

a

b

=

Required A, B,a

Given a,b

FORMULAE

as sin B sin &7 area = 2 sin Al

REDUCTION TO HORIZONTAL Horizontal distance=Slope distance my cosine of the verticalangle. Thus: slope di St " Vert. angle =5 10'. From Table, Page] .e9959. Horizontal distance=319.4X.9959= I 1 Horizontal distance also= Slope distan distance times (1-cosine of vertical ani same figures as in the preceding examr ing result is obtained Cosine 50 10'=.9959.

Horizontal distance 319.4X.0041=1.31. 319.4-1.31=318.09 ft.

When the rise is known, the horizontal distance is approximatelyance less the square of the rise divided by twice the slope distance. Tb slope distance=302.6 ft. Horizontal distance=302.14 X 14 =302.6-0