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VTT <br />TABLE VIII. -NATURAL TRIGONOMETRICAL FUNCTIONS. <br />Angle I <br />Sine. I <br />Tan. <br />Cotg. <br />Cosin. <br />1 <br />Angle <br />Sine. <br />Tan. <br />Cotg. <br />Cosin.) <br />8 <br />or <br />32 <br />110111 <br />1.600 <br />.84805 <br />58 <br />or <br />30 <br />.6225 <br />.7954 <br />1.257 <br />.78261 <br />30 <br />10 <br />.5299 <br />.6249 <br />1.590 <br />.84650 <br />50 <br />40 <br />.6248 <br />.8002 <br />1.250 <br />.78079 <br />20 <br />20 <br />.5324 <br />.5348 <br />.6289 <br />.6330 <br />1.580 <br />.89495 <br />40 <br />50 <br />.6271 <br />.8050 <br />1.242 <br />.77897 <br />10 <br />30 <br />40: <br />.5373 <br />.6371 <br />1.570 <br />1.560 <br />.84339 <br />.84182 <br />30 <br />20 <br />39 <br />.6293 <br />.W98 <br />1.235 <br />.77715 <br />51 <br />50• <br />.5398 <br />.6412 <br />1.550 <br />.84025 <br />1010 <br />.15 <br />.6316 <br />.8146 <br />1.228 <br />.77531 <br />50 <br />.67 <br />.5422 <br />.6453 <br />.89 <br />.96 <br />1.04 <br />20 <br />.6338 <br />.8195 <br />1.220 <br />.77347 <br />40 . <br />33 <br />.65 <br />.6494 <br />1.540 <br />.63867 <br />57 <br />30 <br />,.6361 <br />.8243 <br />1.213 <br />.77162 <br />30 <br />10 <br />.5446 <br />..44 <br />1.530 <br />.78 <br />50 <br />40 <br />.6383 <br />.8292 <br />1.206 <br />.76977 <br />20 <br />20 <br />.5471 <br />.5495 <br />.6536 <br />.6577 <br />1.520 <br />.83708 <br />.83549 <br />40 <br />50 <br />.6406 <br />.8342 <br />1.199 <br />.76791 <br />10 <br />30 <br />40 <br />.5519 <br />.6619 <br />1.511 <br />1.501 <br />.83389 <br />30 <br />20 <br />40 <br />.6428 <br />.8391 <br />1.192 <br />.76604 <br />50 <br />50 <br />.5544 <br />.6661 <br />1.492 <br />.83228 <br />10 <br />10 <br />.6450 <br />.8441 <br />1.185 <br />.76417 <br />50 <br />.83 1:00 <br />.5568 <br />.6703 <br />1.50 <br />.83066 <br />1.83 <br />20 <br />.6472 <br />.8491 <br />1.178 <br />.76229 <br />40 <br />34 . <br />.56 <br />.74 <br />1.483 <br />.82904 <br />56 <br />30 <br />.6494 <br />.8541 <br />1.171 <br />.76041 <br />30 <br />10 <br />.5592 <br />.6745 <br />'.6787 <br />1.473 <br />'.61 <br />50 <br />40 <br />.6517 <br />.8591 <br />1.164 <br />.75851 <br />20 <br />20 <br />.5616 <br />.5640 <br />.6830 <br />1.464 <br />.82741 <br />.82577 <br />40 <br />50 <br />.6539 <br />.8642 <br />1.157 <br />.75661 <br />10 <br />30 <br />40 <br />.5664 <br />.6873 <br />.6916 <br />1.455 <br />1 446 <br />.62413 <br />.82248 <br />30 <br />20 <br />41 <br />.6561 <br />.8693 <br />1.150 <br />.75471 <br />49 <br />50 <br />.5688 <br />1.92 <br />1.437 <br />.82082 <br />10 <br />10 <br />.6583 <br />.8744 <br />1.144 <br />.75280 <br />50 <br />.52 <br />.5712 <br />.6959 <br />1.30 1.55 <br />1:81 <br />2'.08 <br />20 <br />.6604 <br />.8796 <br />1.137 <br />.75088 <br />40 <br />35 <br />15 <br />.28' <br />1.428 <br />.81915 <br />55 <br />30 <br />:6626 <br />.8847 <br />1.130 <br />.74896 <br />30 <br />10 <br />.5736 <br />.7002 <br />.7046 <br />1.419 <br />.81748 <br />50 <br />40 <br />.6648 <br />.8899 <br />1.124 <br />.74703 <br />20 <br />20 <br />.5760 <br />.5783 <br />.7089 <br />1.411 <br />.81580 <br />40 <br />50 <br />.6670 <br />.8952. <br />1.117 <br />.74509 <br />10 <br />30 <br />40 <br />.5607 <br />.7133 <br />':7177 <br />1.402 <br />1.393 <br />.81412 <br />.81242 <br />30 <br />20 <br />42 <br />.6691 <br />.9004 <br />1.111 <br />.74314 <br />48 <br />50 <br />.5831 <br />1.33 <br />1:385 <br />.81072 <br />10 <br />10 <br />.6713 <br />.9057 <br />1.104 <br />.74120 <br />50 <br />5.00 <br />.5854 <br />.7221 <br />.70 <br />1.06 <br />1.41 <br />20 <br />.6734 <br />.9110 <br />1.098 <br />.73924 <br />40 <br />36 <br />4.57 <br />4.92 <br />1.376 <br />.80902 <br />54 <br />30 <br />.6756 <br />.9163 <br />1.091 <br />.73728 <br />30 <br />10 <br />.5878 <br />.7265 <br />1.368 <br />.80730 <br />50 <br />40 <br />.6777 <br />.9217 <br />1.085 <br />.73531 <br />20 <br />20 <br />.5901 <br />.5925 <br />.7310 <br />1.360 <br />.89558 <br />40 <br />50 <br />.6799 <br />.9271 <br />1.079 <br />.73333 <br />10 <br />30 <br />40 <br />.5948 <br />..7355 <br />.7400 <br />1.351 <br />1.343 <br />.80386 <br />.80212 <br />30 <br />20 <br />43 <br />.6820 <br />.9325 <br />1.072 <br />.73135 <br />47 <br />50 <br />.5972 <br />.7445 <br />1.335 <br />.80038 <br />10 <br />10 <br />.6841 <br />.9380 <br />1.066 <br />.72937 <br />50 <br />5.11 <br />.5995 <br />.7490 <br />6.39 <br />,24 <br />.44 <br />20 <br />.6862 <br />.9435 <br />1.060 <br />.72737 <br />40 <br />37 <br />4.44 <br />'.7536 <br />1.327 <br />.79864 <br />53 <br />30 <br />.6884 <br />.9490 <br />1.054 <br />.72537 <br />30 <br />10 <br />.6018 <br />3,70 <br />1.319 <br />.79688 <br />50 <br />40 <br />.6905 <br />.9545 <br />1.048 <br />.72337 <br />20 <br />20 <br />.6041 <br />.6065 <br />.7581 <br />.7627 <br />1.311 <br />.79512 <br />40 <br />50 <br />.6926 <br />.9601 <br />1.042 <br />.72136 <br />10 <br />30 <br />40 <br />.; 6088 <br />".6111 <br />.7673 <br />1.303 <br />1.295 <br />.79335 <br />.79158 <br />30 <br />20 <br />44 <br />.6947 <br />.9657 <br />1.036 <br />.71934 <br />46 <br />50 <br />6.50 <br />.7720 <br />1.288 <br />'.78980 <br />10 <br />10 <br />.6967 <br />.9713 <br />1.030 <br />.71732 <br />50 <br />4.67 <br />.6134 <br />.7766 <br />6.22 <br />6.74 <br />7.26 <br />20 <br />.6988 <br />.9770 <br />1.024 <br />.71529 <br />40 <br />38 <br />3:76 <br />4.30 <br />1.280 <br />.78801 <br />52 <br />30 <br />.7009 <br />.9827 <br />1.018 <br />.71325 <br />30 <br />10 <br />.6157 <br />.7813 <br />1.272 <br />.78622 <br />50 <br />40 <br />:7030 <br />.9884 <br />1.012 <br />.71121 <br />20 . <br />20 <br />.6180 <br />.6202 <br />.7860 <br />.7907 <br />1.265 <br />.78442 <br />40 <br />50 <br />.7050 <br />.9942 <br />1.006 <br />.70916 <br />10 <br />6.89 <br />7.46 <br />8.04 <br />8.61 <br />32 <br />..59 <br />1.18 <br />.7071 <br />.1 <br />1. <br />.70711 <br />45' <br />0, <br />5.33 <br />Cosin. <br />Cotg. <br />Tan. <br />Sine. <br />Angl <br />8.89 <br />Cosin. <br />Cotg. <br />Tan. <br />Sine. <br />Angle <br />3.05 3.67 <br />{ <br />El <br />XIII <br />TABLE IX. -CALCULATION OF EARTHWORK. <br />y <br />HEIGHT <br />1 <br />2' <br />3 1 <br />4 <br />5 6 <br />7 <br />8 <br />9 <br />110111 <br />12 <br />13 <br />14 <br />15 <br />1. <br />.02 <br />.04 <br />.O6 <br />,07 <br />,09 .11 <br />.13.15 <br />.17 <br />.18 <br />.20 <br />,22 <br />.24 <br />.26 <br />.28 <br />2 <br />,04..07 <br />.11 <br />.15 <br />.18 .22 <br />.26 <br />.30 <br />.33 <br />.37 <br />.41 <br />.44 <br />.48 <br />.52 <br />.56 <br />3 <br />.06 <br />.11 <br />.17 <br />.22 <br />.28 .33 <br />.39 <br />.44 <br />.50 <br />.56 <br />.61 <br />.67 <br />.72 <br />.78 <br />.83 <br />4 <br />.07 <br />.15 <br />.22 <br />.30 <br />..37 .44 <br />.52 <br />.59 <br />.67 <br />.74 <br />.81 <br />.89 <br />.96 <br />1.04 <br />,1.11 <br />5 <br />.09 <br />.19 <br />.28 <br />.37. <br />.46 .56 <br />.65 <br />.74 <br />.83 <br />.93 <br />1.02 <br />.1.11 <br />1.20 <br />..1.30 <br />1.39 <br />6.11 <br />.22 <br />.33 <br />..44 <br />.56 .67 <br />.78 <br />.89 <br />1.00 <br />1.11 <br />1.22 <br />1.33 <br />1.44 <br />1:55 <br />-1.67 <br />.13 <br />.26 <br />,39 <br />.52 <br />.65 .78 <br />.91 <br />1.04 <br />1.16 <br />1.30 <br />1.42 <br />1.55 <br />1.68 <br />1.81 <br />1.94 <br />.15 <br />.30 <br />.44 <br />.59 <br />.74 .89 <br />1.04 <br />1.19 <br />1.33 <br />1,48 <br />1.63 <br />1:78 <br />1.92 <br />2.08 <br />2.22 <br />9 <br />.17 <br />.33 <br />.50 <br />.67 <br />.83 1:00 <br />1.17 <br />1.33 <br />1.50 <br />1.67 <br />1.83 <br />2.00 <br />2.17 <br />2.33. <br />2.50 <br />10 <br />.18. <br />.37. <br />.56 <br />.74 <br />.93 1'.11 <br />1.30 <br />1.48 <br />1.67 <br />1.85 <br />2,04 <br />2.22 <br />2.41 <br />2.59 <br />2.78' <br />11 <br />.20, <br />..41 <br />'.61 <br />'r82 <br />1.02 1.22 <br />1.43 <br />1.63 <br />1.83 <br />2.04 <br />2,24 <br />2.44 <br />2,65 <br />2.85 <br />3.06 <br />12 <br />.22 <br />.44 <br />.67 <br />.89 <br />'1.11 1.33 <br />1.56 <br />1.78 <br />2.00 <br />2.22 <br />2.44 <br />2.67 <br />2.89 <br />3.11 <br />3.33 <br />13 <br />.24, <br />-.48 <br />.72 <br />:96 <br />1.20 1.44 <br />1.68 <br />1.92 <br />2.16 <br />2.41 <br />2.65 <br />2.89 <br />3.13 <br />3.37 <br />3.61 <br />14 <br />' .26 <br />.52 <br />: 78 <br />1.04 <br />1.30 1.55 <br />1:81 <br />2'.08 <br />2.33 <br />2.59 <br />2.85 <br />3.11 <br />3.37 <br />3.63 <br />3.89 <br />15 <br />.28' <br />e, ,56 <br />..83 <br />1.11 <br />1.39 1.67 <br />1.94 <br />2.22 <br />2.50 <br />2.78 <br />3.06 <br />3.33 <br />3.61 <br />3.89 <br />4.17 <br />16 <br />.30 <br />:59 <br />:89 <br />1:18 <br />1.48 1.78 <br />2.07 <br />2:37 <br />2.67 <br />2.96 <br />3.26 <br />3.56 <br />3.85 <br />4.15 <br />4.44 <br />17 <br />.31 <br />.63 <br />.94 <br />1.26 <br />1.57 1.89 <br />2.20 <br />2.52 <br />2.83 <br />3:15 <br />3.46 <br />3.78 <br />4.09 <br />4.41 <br />4.72 ' <br />18 <br />.33 <br />.67 <br />1.00 <br />1.33 <br />1.67 2.00 <br />2.33 <br />2.67 <br />3.00 <br />3.33 <br />3.67 <br />4.00 <br />4.33 <br />4.67 <br />5.00 <br />19 <br />.35 <br />.70 <br />1.06 <br />1.41 <br />1.76 2.11 <br />2.46 <br />2.82 <br />3.17 <br />3.52 <br />3.87 <br />4.22 <br />4.57 <br />4.92 <br />5.28' <br />20 <br />.37 <br />.74 <br />1.11 <br />1.48 <br />1,85 2.22 <br />2.59 <br />2.96 <br />3.33. <br />3.70 <br />4.07 <br />4.44 <br />4.81 <br />5,18 <br />5.56 <br />.21 <br />`.39 <br />.78 <br />1.17 <br />1:55 <br />1.94 2.33 <br />2.72 <br />3.11 <br />3.50 <br />3.89 <br />4.28 <br />4.67 <br />5.06 <br />5.44 <br />5.83 <br />22 <br />' .41 <br />'.81 <br />1.22 <br />1.63 <br />2.04 2:44 <br />2.85 <br />3.26 <br />3.67 <br />4:07 <br />4.48 <br />4.89 <br />5.30 <br />5.70 <br />6.11 <br />'23 <br />.43 <br />.85. <br />1.28 <br />1.70 <br />2.13 2.56 <br />2.98 <br />3.41 <br />3.83 <br />4.26 <br />4.68 <br />5.11 <br />5.54 <br />5.96 <br />6.39 <br />,24 <br />.44 <br />.89 <br />1.33 <br />1.78 <br />2.22 2.67 <br />3.11 <br />3.56 <br />4.00 <br />4.44 <br />4.89 <br />5.33 <br />5.78 <br />6.22 <br />6.67 <br />25 <br />.46 <br />.92 <br />1.39 <br />1.85 <br />2.31 2.78 <br />3.24 <br />3,70 <br />4.17 <br />4.63 <br />5.09 <br />5.56 <br />6.02 <br />6.48 <br />6.94 <br />'26 <br />.48 <br />.96 <br />1.44 <br />1.92 <br />2.41 2.89 <br />3.37 <br />3.85 <br />4.33 <br />4.82 <br />5.30 <br />5.78 <br />6.26 <br />6.74 <br />7.24 <br />-27 <br />.50.. <br />1.00 <br />1.50 <br />2.00 <br />2.50 3.00 <br />1.50 <br />4.00 <br />4.50 <br />5.00 <br />5.50 <br />6.00 <br />6.50 <br />7.00 <br />7.50 <br />'28 <br />.52 <br />1.04 <br />1.55 <br />2.07 <br />2.59 3.11 <br />3.63 <br />4.15 <br />4.67 <br />5.18 <br />5.70 <br />6.22 <br />6.74 <br />7.26 <br />7.78 <br />29 <br />.54 <br />1.07 <br />1.61 <br />2.15 <br />2.68 3.22 <br />3:76 <br />4.30 <br />4.83 <br />5.37 <br />5,91 <br />6.44 <br />6.98 <br />7.52 <br />8.06 <br />30. <br />'.56 <br />1.11 <br />'1.67 <br />2.22 <br />2.78 3.33 <br />3.89 <br />4.44 <br />5.00 <br />5.55 <br />6.11 <br />& 67 <br />7.22 <br />7.78 <br />8.33 <br />31 <br />.57 <br />1.15 <br />1.72 <br />2.30 <br />2.87 3.44 <br />4.02 <br />4.59 <br />5.17 <br />5.74 <br />6.32 <br />6.89 <br />7.46 <br />8.04 <br />8.61 <br />32 <br />..59 <br />1.18 <br />1.78 <br />2.37 <br />2.96 3.56 <br />4.15 <br />4.74 <br />5.33 <br />5.92 <br />6.52 <br />7.11 <br />7.70 <br />8.30 <br />8.89 <br />33 <br />.61' <br />1:22 <br />1.83 <br />2.44 <br />3.05 3.67 <br />4.28 <br />4.89 <br />5.50 <br />6.11 <br />6.72 <br />7.33 <br />7.94 <br />8.55 <br />9.17 <br />34 <br />.63 <br />1.26 <br />1.89, <br />2.52 <br />3.15 3.78 <br />4.40 <br />5.04- <br />5.67 <br />6.29 <br />6.93 <br />7.56 <br />8.18 <br />8.81 <br />9.44 t <br />35 <br />'.65 <br />1.30 <br />1.94 <br />2.59 <br />3.24 3.89 <br />4.53 <br />5.18 <br />5.83 <br />6.48 <br />7:13 <br />7.78 <br />8.42 <br />9.08 <br />9.72 <br />36 <br />.67 <br />1.33 <br />2.00 <br />2:67 <br />3,33 4.00 <br />4.66 <br />5.33 <br />6.00 <br />6.67 <br />7.33 <br />8.00 <br />8.67 <br />9.33 <br />10.00 <br />37.68 <br />1.37 <br />2.06 <br />2.74 <br />3.42 4.11 <br />4.79 <br />5.48 <br />6.17 <br />6.857.54 <br />22 <br />8.91 <br />9.59 <br />10.28 <br />38 <br />:70 <br />1.41 <br />2.11 <br />2.82 <br />3.52 4.22 <br />4.92 <br />5.63 <br />6.33 <br />7.03 <br />7.74 <br />8.8. 44 <br />9,15 <br />9.85 <br />10.56 <br />89 <br />:72 <br />1.44 <br />2:17 <br />2.89 <br />3.61 4.33 <br />5.05 <br />5.78 <br />6.50 <br />7.22 <br />7:95 <br />& 6719,39 <br />10.11 <br />10.83 <br />40 <br />.74 <br />1..48 <br />2.22 <br />2.96 <br />3.70 4.44 <br />5.18 <br />5.92 <br />6.67 <br />7.41 <br />8.35 <br />8.89 <br />4:63 <br />LQ:37 <br />11.11 <br />Table gives cu. ydg. in 1'ft, of a triangle of given width -and height., Corrections <br />for tenths of width are one tenth the values found under -each height considering <br />the widths from 1 to 9 as -tenths and similarly the corrections for tenths of height are <br />one tenth the figures opposite width considering the'heights from 1 to 9 as tenths. <br />Thus i f w=16.2 and h=5.3, cu. yds.=1.48+:U18+.089=1.587 cu, yds: or practically 159 <br />cu. yds: per 100 ft..,lf w_exceeds 40 ft., use one half and multiply result by 2, if both <br />w and h are large use one half of each -and multiply result by 4: -Any cross-section <br />may be divided intotriangles by the following rule. To the triangle of the sum of <br />the onside cuts (or fills) =h, and M the roadbed =w, add the triangles formed by <br />taking the distance out to each break in turn,( -W's) by the difference between the . <br />cuts (or fills) on- each side' of it ( =h's) always subtracting the outer from'the inner. <br />�_ . <br />