CURVE FORMULAS T = R tan I/2 T = 50 tan I/2 / Sin. D/2 Sin. D/2 = 50/R Sin. D/2 = 50 tan I/2 / T R = T cot. I/2 R = 50 / Sin. D/2 E = R ex. sec I/2 E = T tan I/4 Chord def. = chord²/R No. chords = I/D Tan. def. = 1/2 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 1. Multiply the given distance by .01745 (def. for 1° for 1 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 .01745, 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 100, Alt. 10. 10²÷200=.5. 100+.5=100.5 hyp. Given Hyp. 100, Alt. 25.25²÷200=3.125. 100-3.125=96.875=Base. Error in first example, .002; in last, .045. To find Tons of Rail in one mile of track: multiply weight per yard by 11, and divide by 7. LEVELING. The correction for curvature and refraction, in feet and decimals of feet is equal to 0.574d², where d is the distance in miles. The correction for curvature alone is closely, 3/8d². The combined correction is negative. PROBABLE ERROR. If d₁, d₂, d₃, etc. are the discrepancies of various results from the mean, and if Σd²=the sum of the squares of these differences and n=the number of observations, then the probable error of the mean= ±0.6745√Σd²/n(n-1) SOLAR EPHEMERIS. Attention is called to the Solar Ephemeris for the current year, published by Keuffel & Esser Co., and furnished free of charge upon request, which is 3¼x5⅝ 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' .0167 | 11' .1833 | 21' .3500 | 31' .5167 | 41' .6833 | 51' .8500 2 .0333 | 12 .2000 | 22 .3667 | 32 .5333 | 42 .7000 | 52 .8667 3 .0500 | 13 .2167 | 23 .3833 | 33 .5500 | 43 .7167 | 53 .8833 4 .0667 | 14 .2333 | 24 .4000 | 34 .5667 | 44 .7333 | 54 .9000 5 .0833 | 15 .2500 | 25 .4167 | 35 .5833 | 45 .7500 | 55 .9167 6 .1000 | 16 .2667 | 26 .4333 | 36 .6000 | 46 .7667 | 56 .9333 7 .1167 | 17 .2833 | 27 .4500 | 37 .6167 | 47 .7833 | 57 .9500 8 .1333 | 18 .3000 | 28 .4667 | 38 .6333 | 48 .8000 | 58 .9667 9 .1500 | 19 .3167 | 29 .4833 | 39 .6500 | 49 .8167 | 59 .9833 10 .1667 | 20 .3333 | 30 .5000 | 40 .6667 | 50 .8333 | 60 1.0000 TABLE V. — Inches in Decimals of a Foot. 1-16 .0052 | 3-32 .0078 | ½ .0104 | 3-16 .0156 | ¼ .0208 | 5-16 .0260 | ⅜ .0313 | ½ .0417 | ⁵⁄₈ .0521 | ¾ .0625 | ⅞ .0729 1 .0833 | 2 .1667 | 3 .2500 | 4 .3333 | 5 .4167 | 6 .5000 | 7 .5833 | 8 .6667 | 9 .7500 | 10 .8333 | 11 .9167