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XII <br />TABLE VIII. -NATURAL TRIGONOMETRICAL FUNCTIONS. <br />Angle <br />Sine. <br />Tan: <br />Cotg. <br />Cosin. <br />Angle <br />Sine. <br />Tan. <br />Cotg. <br />Cosin. <br />or <br />..1 2 1 <br />of <br />4 <br />5 <br />6 1 <br />7 1 8 9 <br />10 <br />32 <br />.5299 <br />.6249 <br />1.600 <br />.84805 <br />58 <br />30 <br />.6225 <br />.7954 <br />1.257 <br />:78261 <br />30 <br />10 <br />.5324 <br />.6289 <br />1.590 <br />.84650 <br />50 <br />40 <br />.6248 <br />.8002 <br />1.250 <br />.78079 <br />20 <br />20 <br />.5348 <br />.6330 <br />1.580 <br />.84495 <br />40 <br />50 <br />.6271 <br />.8050 <br />1.242 <br />.77897 <br />10 <br />30 <br />.5373 <br />.6371 <br />1.570 <br />.84339 <br />30 <br />.22 <br />.30 <br />.37 <br />.44 <br />.52 .59 .67 <br />.74 <br />40 <br />.5398 <br />.6412 <br />1.560 <br />.84182 <br />20 <br />39 <br />.6293 <br />.8098 <br />1.235 <br />.77715 <br />51 <br />50 <br />.5422 <br />.6453 <br />1.550 <br />.84025 <br />10 <br />10 <br />.6316 <br />.8146 <br />1.228 <br />.77531 <br />50 <br />1.22 <br />1.33 <br />1.44 <br />1.55 1.67 <br />7• <br />.13 .26 <br />20 <br />.6338 <br />.8195 <br />1.220 <br />.77347 <br />40 <br />33 <br />.5446 <br />.6494 <br />1.540 <br />.83867 <br />57 <br />30 <br />.6361 <br />.8243 <br />1.213 <br />.77162 <br />30 <br />10 <br />.5471 <br />.6536 <br />1.530 <br />.83708 <br />50 <br />40 <br />.6383 <br />.8292 <br />1.206 <br />.76977 <br />20 <br />20 <br />.5495 <br />.6577 <br />1.520 <br />.83549 <br />40 <br />50 <br />.6406 <br />.8342 <br />1.199 <br />.76791 <br />10 <br />30 <br />.5519 <br />.6619 <br />1.511 <br />.83389 <br />30 <br />.61 <br />'.82 <br />1.02 <br />1.22 <br />1.43 1.63 1.83 <br />2.04 <br />40 <br />.5544 <br />.6661' <br />1.501 <br />.83228 <br />20 <br />40 <br />.6428 <br />.8391 <br />1.192 <br />.76604 <br />50 <br />50 <br />.5568 <br />.6703 <br />1.492 <br />.83066 <br />10 <br />10 <br />.6450 <br />.8441 <br />1.185 <br />.76417 <br />50 <br />2.65 <br />2.89 <br />3.13 <br />3.37 3.61 <br />14 <br />.26 .52 <br />20 <br />.6472 <br />.8491 <br />1.178 <br />.76229 <br />40 <br />34 <br />.5592 <br />.6745 <br />1.483 <br />.82904 <br />56 <br />30 <br />.6494 <br />.8541 <br />1.171 <br />.76041 <br />30 <br />10 <br />.5616 <br />.6787 <br />1.473 <br />.82741 <br />50 <br />40 <br />.6517 <br />.8591 <br />1.164 <br />.75851 <br />20 <br />20 <br />.5640 <br />.6830 <br />1.464 <br />.82577 <br />40 <br />50 <br />.6539 <br />.8642 <br />1.157 <br />.75661 <br />10 <br />30 <br />.5664 <br />.6873 <br />1.455 <br />.82413 <br />30 <br />1.00 <br />1.33 <br />1.67 <br />2.00 <br />2.33 2.67 3.00 <br />3.33 <br />40 <br />.5688 <br />.6916 <br />1 446 <br />.82248 <br />20 <br />41 <br />.6561 <br />.8693 <br />1.150 <br />.75471 <br />49 <br />50 <br />.5712 <br />.6959 <br />1.437 <br />.82082 <br />10 <br />10 <br />.6583 <br />.8744' <br />.1.144 <br />.75280 <br />50 <br />4.07 <br />4.44 <br />4.81 <br />5.18 5.56 <br />21 <br />.39 .78 <br />20 <br />.66041 <br />.8796 <br />1.137 <br />.75088 <br />40 <br />35 <br />.5736 <br />.7002 <br />1.428 <br />.81915 <br />55 <br />30 <br />.6626 <br />.8847 <br />1.130 <br />.74896 <br />30 <br />10 <br />.5760 <br />.7046, <br />1.419' <br />.81748 <br />50 <br />40 <br />.66481 <br />.8899 <br />1.124 <br />.74703 <br />20 <br />20 <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 />.5807 <br />.7133 <br />1.402 <br />.81412 <br />30 <br />1.39 <br />1.85 <br />2,31 <br />2.78 <br />3.24 3.70 4.17 <br />4.63 <br />40 <br />.5831 <br />.7177 <br />1.393 <br />.81242 <br />20 <br />42 <br />.6691 <br />.9004 <br />1.111 <br />.74314 <br />48 . <br />50 <br />.5854 <br />.7221 <br />1.385 <br />.81072 <br />10 <br />10 <br />.6713 <br />.9057 <br />1.104 <br />.74120 <br />50 <br />5.50 <br />6.00 <br />6.50 <br />7.00 7.50 <br />28 <br />.52 1.04 <br />20 <br />.6734 <br />.9110 <br />1.098 <br />.73924 <br />40 <br />36 <br />.5878 <br />.7265 <br />1.376 <br />.W902 <br />54 <br />30 <br />.6756 <br />.9163 <br />1.091 <br />.73728 <br />30 <br />10 <br />.5901 <br />.7310 <br />1.368 <br />.80730 <br />50 <br />40 <br />.6777 <br />.9217 <br />1.085 <br />.73531 <br />20 <br />20 <br />.5925 <br />.7355 <br />1.360 <br />.89558 <br />40 <br />50 <br />.6799 <br />.9271 <br />1.079 <br />.73333 <br />10 <br />30 <br />.5948 <br />.7400 <br />1.351' <br />.80386 <br />30 <br />1.78 <br />2.37 <br />2.96 <br />3.56 <br />4.15 4.74 5.33 <br />5.92 <br />40 <br />.5972 <br />.7445 <br />1.343 <br />.80212 <br />. 20 <br />43 <br />.6820 <br />.9325 <br />1.072 <br />.73135 <br />47 <br />50 <br />.5995 <br />.7490 <br />1.335 <br />.80038 <br />10 <br />10 <br />.6841 <br />.9380 <br />1.066 <br />.72937 <br />50 <br />6.93 <br />7.56 <br />8.18 <br />8.81 9.44 <br />35 <br />.65 1.30 <br />20 <br />.6862 <br />.9435 <br />1.060 <br />.72737 <br />40 <br />37 <br />.6018 <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 />.6041 <br />.7581 <br />1.319 <br />.79688' <br />50 <br />40 <br />.6905 <br />.9545 <br />1.048 <br />.72337 <br />20 <br />20 <br />.6065 <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 />.6088 <br />.7673 <br />1.303 <br />.79335 <br />30 <br />2.17 <br />2.89 <br />3.61 <br />4.33 <br />5.05 5.78 6.50 <br />7.22 <br />40 <br />.6111 <br />:7720 <br />1.295 <br />.79158 <br />20 <br />44 <br />.6947 <br />.9657 <br />1.036 <br />.71934 <br />46 <br />50 <br />.6134. <br />.7766 <br />1.288 <br />.78980 <br />10 <br />10 <br />.6967 <br />.9713 <br />1.030 <br />.71732 <br />50 <br />the outside cuts (or fills) =b, and ji 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 />20 <br />.6988 <br />'..9770 <br />1.024 <br />.71529 <br />40 <br />38 <br />.6157 <br />.7813 <br />1.280 <br />.78801 <br />52 <br />30 <br />.7009 <br />.9827 <br />1.018 <br />.71325 <br />30 <br />10 <br />.6180 <br />.7860 <br />1.272 <br />.78622. <br />50 <br />40 <br />.7030 <br />.9884 <br />1.012 <br />•71121 <br />20 <br />20 <br />.6202 <br />.7907 <br />1.265 <br />.78442. <br />40 <br />50 <br />.7050 <br />..9942 <br />1.006 <br />.70916 <br />10 <br />.7071 <br />.1 <br />.1. <br />.70711 <br />45 <br />� <br />of <br />Cosin. <br />Cotg. <br />Tan. <br />Sine. <br />Angic <br />Cos in. <br />Cotg. <br />Tan. <br />Sine. <br />Angle <br />XIII <br />TABLE IX. -CALCULATION of EARTHWORK. <br />LIF,IG HT <br />v <br />..1 2 1 <br />3 <br />4 <br />5 <br />6 1 <br />7 1 8 9 <br />10 <br />11 1 <br />12113 <br />14 1 15 <br />1 <br />.02 .04 <br />.06 <br />.07 <br />.09 <br />.11 <br />.13 .15 .17 <br />.18 <br />.20 <br />.22 <br />.24 <br />.26 .28 <br />2 <br />.04 ..07 <br />.11 <br />.15 <br />.18 <br />.22 <br />.26 .30 .33 <br />.37 <br />.41 <br />.44 <br />.48 <br />.52 .56 <br />3 <br />.06 .11 <br />.17 <br />.22 <br />.28 <br />.33 <br />.39 .44 .50 <br />.56 <br />.61 <br />.67 <br />.72 <br />.78' .83 <br />4 <br />.07 .15 <br />.22 <br />.30 <br />.37 <br />.44 <br />.52 .59 .67 <br />.74 <br />.81 <br />.89 <br />.96 <br />1.04 1.11 <br />5 <br />.09 .19 <br />.28 <br />.37 <br />.46 <br />.56 <br />.65 .74 .83 <br />.93 <br />1.02 <br />1.11 <br />1.20. <br />1.30 1.39 <br />6 <br />.11 .22 <br />.33 <br />.44 <br />.56 <br />.67 <br />.78 .89 1.00 <br />1.11 <br />1.22 <br />1.33 <br />1.44 <br />1.55 1.67 <br />7• <br />.13 .26 <br />.39 <br />..52 <br />.65 <br />.78 <br />.91 1.04 1.16 <br />1.30 <br />1.42 <br />1.55 <br />1.68 <br />1.81 1.94 <br />8 <br />.15 .30 <br />.44 <br />.59 <br />.74 <br />.89' <br />1.04 1.19 1.33 <br />1.48 <br />1.63 <br />1.78 <br />1.92 <br />2.08 2.22 <br />9 <br />.17 .33 <br />.50 <br />.67 <br />.83 <br />1.00 <br />1.17 1.33 1.50 <br />1.67 <br />1.83 <br />2.00 <br />2.17 <br />2.33 2.50 <br />10 <br />.18 .37 <br />..56 <br />.74 <br />.93 <br />1.11 <br />1.30 1.48 1.67 <br />1.85 <br />2.04 <br />2.22 <br />2.41 <br />2.59 2.78 <br />11 <br />.20 .41 <br />.61 <br />'.82 <br />1.02 <br />1.22 <br />1.43 1.63 1.83 <br />2.04 <br />2.24 <br />2.44 <br />2.65 <br />2.85 3.06 <br />12 <br />.22 .44 <br />.67, <br />.89 <br />1.11 <br />1.33 <br />1.56 1.78 2.00 <br />2.22 <br />2.44 <br />2.67 <br />2.89 <br />3.11 3.33 <br />13 <br />.24 .48 <br />'.72 <br />.96 <br />1.20 <br />1.44 <br />1.68 1.92 2.16 <br />2.41 <br />2.65 <br />2.89 <br />3.13 <br />3.37 3.61 <br />14 <br />.26 .52 <br />.78 <br />1.04 <br />1.30 <br />1.55 <br />1.81 2.08 2.33 <br />2.59 <br />2.85 <br />3.11 <br />3.37 <br />3.63 3.89 <br />15 <br />.28 .56 <br />.83 <br />1.11 <br />1.39 <br />1.67 <br />1.94 2.22 2.50 <br />2.78 <br />3.06 <br />3.33 <br />3.61 <br />3.89 4.17 <br />16 <br />.30 .59 <br />.89 <br />1.18 <br />1.48 <br />1.78 <br />2.07 2.37 2.67 <br />2.96 <br />3.26 <br />3.56 <br />3.85 <br />4.15 4.44 <br />17 <br />.31 .63 <br />.94 <br />1.26 <br />1.57 <br />1.89 <br />2.20 2.52 2.83 <br />3:15 <br />3.46 <br />3.78 <br />4.09 <br />4.41 4.72 <br />18 <br />.33 .67 <br />1.00 <br />1.33 <br />1.67 <br />2.00 <br />2.33 2.67 3.00 <br />3.33 <br />3.67 <br />4.00 <br />4.33 <br />4.67 5.00 <br />19 <br />.35 .70 <br />1.06 <br />1.41 <br />1.76 <br />2.11 <br />2.46 2.82 3.17 <br />3.52 <br />3.87 <br />4.22 <br />4.57 <br />4.92 5.28 <br />20 <br />.37 .74 <br />1.11 <br />1.48 <br />1.85 <br />2.22 <br />2.59 2.96 3.33 <br />3.70 <br />4.07 <br />4.44 <br />4.81 <br />5.18 5.56 <br />21 <br />.39 .78 <br />1.17 <br />1.55 <br />1.94 <br />2.33 <br />2.72 3.11 3.50 <br />3.89 <br />4.28 <br />4.67 <br />5.06 <br />5.44 5.83 <br />22 <br />.41 .81 <br />1.22 <br />1.63 <br />2.04 <br />2.44 <br />2.85 3.26 3..67 <br />4.07 <br />4.48 <br />4.89 <br />5.30 <br />5.70 6.11 <br />23 <br />.43 :85 <br />1.28 <br />1.70 <br />2.13 <br />2.56 <br />2.98 3.41 3.83 <br />4.26 <br />4.68 <br />5.11 <br />5.54 <br />5.96 6.39 <br />24 <br />.44 .89 <br />1.33 <br />1.78 <br />2.22 <br />2.67 <br />3.11 3.56 4.00 <br />4.44 <br />4.89 <br />5.33 <br />5.78 <br />6.22 6.67 <br />25 <br />.46 .92 <br />1.39 <br />1.85 <br />2,31 <br />2.78 <br />3.24 3.70 4.17 <br />4.63 <br />5.09 <br />5.56 <br />6.02 <br />6.48 6.94 <br />26 <br />.48 .96 <br />1.44 <br />1.92 <br />2.41 <br />2.89 <br />3.37 3.85 4.33 <br />4.82 <br />5.30 <br />5.78 <br />6.26 <br />6.74 7.24 <br />27 <br />.50 1.00 <br />1.50 <br />2.00 <br />2.50 <br />3.00 <br />3.50 4.00 4.50 <br />5.00 <br />5.50 <br />6.00 <br />6.50 <br />7.00 7.50 <br />28 <br />.52 1.04 <br />1.55 <br />2.07 <br />2.59 <br />3.11 <br />3.63 4.15 4.67 <br />5.18 <br />5.70 <br />6.22 <br />6.74 <br />7.26 7.78 <br />29 <br />.54 1.07 <br />1.61 <br />2.15 <br />2.68 <br />3.22 <br />3.76 4.30 4.83 <br />5.37 <br />5.91 <br />6.44 <br />6.98 <br />7.52 8.06 <br />30 <br />.56 1.11 <br />1.67 <br />2.22 <br />2.78 <br />3.33 <br />3.89 4.44 5.00 <br />5.55 <br />6.11 <br />6.67 <br />7.22 <br />7.78 8.33 <br />31 <br />.57 1.15 <br />1.72 <br />2.30 <br />2.87 <br />3.44 <br />4.02 4.59 5.17 <br />5.74 <br />6.32 <br />6.89 <br />7.46 <br />8.04 8.61 <br />32 <br />.59 1.18 <br />1.78 <br />2.37 <br />2.96 <br />3.56 <br />4.15 4.74 5.33 <br />5.92 <br />6.52 <br />7.11 <br />7.70 <br />8.30 8.89 <br />33.61 <br />1.22 <br />1.83 <br />2.44 <br />3.05 <br />3.67 <br />4.28 4.89 5.50 <br />6.11 <br />6.7'2 <br />7.33 <br />7.94 <br />8.55 9.17 <br />34 <br />.63 1.26 <br />1.89 <br />2.52 <br />3.15 <br />3.78 <br />4.40 5.04 5.67 <br />6.29 <br />6.93 <br />7.56 <br />8.18 <br />8.81 9.44 <br />35 <br />.65 1.30 <br />1.94 <br />2.59 <br />3.24 <br />3:89 <br />4.53 5.18 5.83 <br />6.48 <br />7.13 <br />7.78 <br />8.42 <br />9.08 9.72 <br />36 <br />.67 1.33 <br />2.00 <br />2.67 <br />3.33 <br />4.00 <br />4.66 5.33 6.00 <br />6.67 <br />7.33 <br />8.00 <br />8.67 <br />9.33 10.00 <br />37 <br />.68 1.37 <br />2.06 <br />2.74 <br />3.42 <br />4.11 <br />4.79 5.48 6.17 <br />6.85 <br />7.54 <br />8.22 <br />8.91 <br />9.59 10.28 <br />38 <br />.70 1.41 <br />2.11 <br />2.82 <br />3.52 <br />4.22 <br />4.92 5.63 6.33 <br />7.03 <br />7.74 <br />8.44 <br />9.15 <br />9.85 10.56 <br />39 <br />.72 1.44 <br />2.17 <br />2.89 <br />3.61 <br />4.33 <br />5.05 5.78 6.50 <br />7.22 <br />7'.95 <br />8.67 <br />9.39 <br />10.11 10.83 <br />40 <br />.74 1.98 <br />2.22 <br />2.96 <br />3.70 <br />4.44 <br />5.18 5.92 6.67 <br />7.41 <br />8.15 <br />& 89 <br />9.63 <br />10.37 11.11 <br />Table gives cu. yds. in I•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 tog as tenths. <br />Thus i f w=16.2 and h=5.3, cu. yds,=1.48+.018+.089=1.587 cu. yds. or nractic•nlly 159 <br />cu. yds. per 100 ft. , If w exceeds 40 ft., use one hal rand multiply result by 2, i f both <br />wand hare large use one half of each and multiply result by 4. Any cross-section <br />may be divided into triangles by the following rule. To the triangle of the sum of <br />the outside cuts (or fills) =b, and ji 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 />