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1. DOUBLE PROPORTIONING USING STATE PLANE COORDINATES JAMES WENGLER, PLS, CFS

2. THE DIRECTION OF LINES 2-17. The direction of each line of the public land surveys is determined with reference to the true meridian as defined by the axis of the earth’s rotation. Bearings are stated in terms of angular measure referred to the true north or south. for station error and curvature do not enter into the ordinary solar transit directions. However, for the purpose of a comparison of the solar transit direction of the chord of a long line, half the value of the convergency of the meridians of the two end stations is applied. For example, a parallel of latitude as run by solar transit methods is a true latitudinal curve, i. e.-a small circle of the earth, everywhere due east or west. The transit line or chord between any two distant points of the parallel is a great circle, whose mean azimuth, or bearing at midpoint, is due east and west. At one end of the chord the forward azimuth is always northeasterly (or northwesterly) ; at the opposite end, the back azimuth will be northwesterly (or northeasterly). At the end stations of the chord, the difference between the forward (or back) azimuth and due east or west, will be equal to half the value of the curvature counting from the two end stations. At the end stations of the chord, the difference between the forward azimuth and the back azimuth +180° will be the full value of the convergency of the meridians of the two end stations. THE GEODESY OFLARGE-SCALE CADASTRAL SURVEYS Transfer of Azimuth, Station Error, and Curvature 2-74. When carrying forward the direction of lines through intermediate transit stations by the method of fore-and-back sights and deflection angles, two corrections become important where the purpose is to maintain accuracy. First, each station setup involves uncertainty in the maintenance of the direction of a line, or in the value of the angle that may be turned, called "station error." Second, if the line is other than a meridian, its direction will have an increment of curvature; this is applied in order to convert from the forward azimuth to the back azimuth of that same line at the next station. By basic law, and the Manual requirements, the directions of all lines are stated in terms of angular measure referred to the true north (or south) at the point of record. As solar transit orientation is designed to give the meridian at each station, thereby avoiding cumulative errors of conventional transit methods, the corrections

3. Double Proportionate Measurement 5-25. The term “double proportionate measurement” is applied to a new measurement made between four known corners, two each on intersecting meridional and latitudinal lines, for the purpose of relating the intersection to both. In effect, by double proportionate measurement the record directions are disregarded, excepting only where there is some acceptable supplemental survey record, some physical evidence, or testimony that may be brought into the control. Corners to the north and south control any intermediate latitudinal position. Corners to the east and west control the position in longitude. One identified original corner is balanced by the control of a corresponding original corner on the opposite side of a particular missing corner which is to be restored. Each identified corner is given a controlling weight inversely proportional to its distance from the lost corner. Lengths of proportioned lines are comparable only when reduced to their cardinal equivalents.The method may be referred to as a “four-way” proportion. The method of double proportionate measurement is generally applicable to the restoration of lost corners of four townships and of lost interior corners of four sections. Cardinal point  : n.   One of the four principal directions on a compass: north, south, east, or west. SIN 85° X 5280.00 = 5259.91 S 85° E 5280.00 COS 3° X 5293.20 = 5285.95 N 3° W 5293.20

4. GLO RECORD MEASUREMENTS GLO =S2°50’E 81.10 CH. GLO =S87°34’W 79.80 CH. GLO =S89°59’E 80.27 CH. GLO =NORTH 80. CH.

5. GLO RECORD MEASUREMENTS CONVERTED TO FEET CHAINS X 66 = FEET GLO =S2°50’E 81.10 CH. GLO=5352.60 GLO=5266.80 GLO=5297.82 GLO =S87°34’W 79.80 CH. GLO =S89°59’E 80.27 CH. GLO =NORTH 80. CH. GLO=5280

6. GLO CARDINAL EQUIVALENTS COS 2°50’ X 5352.60 = 5346.06 SIN 87°34’ X 5266.80 = 5262.05 COS 0° X 5280 = 5280 GLO =S2°50’E 81.10 CH. GLO=5352.60 CE=5346.06 SIN 89°59’ X 5297.82 = 5297.82 CE=5262.05 CE=5297.82 GLO=5266.80 GLO=5297.82 GLO =S87°34’W 79.80 CH. GLO =S89°59’E 80.27 CH. GLO =NORTH 80. CH. CE=5280 GLO=5280

7. FIELD MEASURED VALUES N=462520.90E=830910.54 CE=5346.06 S 85° 24’ 47.227” W 10435.3221 N=457336.62E=825365.99 CE=5297.82 CE=5262.05 N 3° 03’ 53.047” E 10611.9576 N=456502.10E=835767.89 CE=5280 N=451924.12E=830343.18

8. PROPORTIONING THE LATITUDE N=462520.90E=830910.54 1.) TOTAL THE CARDINAL EQUIVALENTS 5280 + 5346.06 = 10626.06 2.) DETERMINE THE PROPORTION OF THE SOUTHERLY PORTION OF THE LINE. 5280 = ____X____ 10626.06 10611.9576 X=5272.9926 CE=5346.06 5338.9650 457189.57099 N 830625.09642 E 48°12’ 19.56880” N 3.) DETERMINE THE PROPORTION OF THE NORTHERLY PORTION OF THE LINE. 5346.06 = ____Y____ 10626.06 10611.9576 Y=5338.965 CE=10626.06 N 3° 03’ 53.047” E 10611.9576 4.) TOTAL THE TWO PROPORTIONS FOR A CHECK: 5272.9926 + 5338.9650 = 10611.9576 5272.9926 5.) COMPUTE THE COORDINATE OF THE TEMPORARY LATITUDINAL POSITION AND CONVERT TO LATITUDE AND LONGITUDE. NOTE YOU ONLY NEED THE LATITUDE OF THIS POSITION TO DETERMINE THE FINAL POSITION. CE=5280 N=451924.12E=830343.18

9. PROPORTIONING THE DEPARTURE 1.) TOTAL THE CARDINAL EQUIVALENTS 5297.82 + 5262.05 = 10559.87 4.) TOTAL THE TWO PROPORTIONS FOR A CHECK: 5235.3351 + 5199.9870 = 10432.3221 2.) DETERMINE THE PROPORTION OF THE WESTERLY PORTION OF THE LINE. 5297.82 = ____X____ 10559.87 10435.3221 X=5235.3351 5.) COMPUTE THE COORDINATE OF THE TEMPORARY DEPARTURE POSITION AND CONVERT TO LATITUDE AND LONGITUDE. NOTE YOU ONLY NEED THE LONGITUDE OF THIS POSITION TO DETERMINE THE FINAL POSITION. 3.) DETERMINE THE PROPORTION OF THE EASTERLY PORTION OF THE LINE. 5262.05 = ____Y____ 10559.87 10435.3221 Y=5199.9870 456917.94659 N 830584.55745 E 124°09’ 19.43155” W CE=10559.87 N=456502.10E=835767.89 5235.3351 5199.9870 CE=5297.82 N=457336.62E=825365.99 CE=5262.05 N 85° 24’ 47.227” W 10435.3221

10. COORDINATE VALUES OF THE LOST CORNER POSITION N=462520.90E=830910.54 CONVERT THE LATITUDE FROM THE NORTH-SOUTH PROPORTION AND THE LONGITUDE FROM THE EAST-WEST PROPORTION TO STATE PLANE COORDINATES. N 3° 22’ 25” E 5339.34 N 88° 24’ 11” W 5232.39 S 82° 24’ 52” E 5217.20 N=457336.62E=825365.99 457190.81294 N 830596.34367 E N=456502.10E=835767.89 S 2°45’ 07” W 5272.77 48°12’ 19.56880” N 124°09’ 19.43155” W N=451924.12E=830343.18

11. COORDINATE VALUES OF THE LOST CORNER POSITION COMPUTED USING CARDINAL OFFSETS. N=462520.90E=830910.54 1.) COMPUTE THE COORDINATES AND CONVERGENCE (MAPPING ANGLE) AT BOTH TEMPORARY PROPORTIONED POSITIONS. 5338.9650 N 3° 03’53.047” E 2.) COMPUTE A BEARING-BEARING INTERSECT USING CARDINAL (TRUE) DIRECTIONS CONVERTED TO GRID BEARINGS. 457189.57099 N 830625.09642 E CONV. = -02°28’ 23.72” 457190.81295 N 830596.34381 E N=457336.62E=825365.99 5235.3351 5199.9870 N 85° 24’ 47.227” W 456917.94659 N 830584.55745 E CONV. = -02°28’ 24.04” N=456502.10E=835767.89 5272.9926 3.) N 87°31’ 36” W FROM TEMPORARY NORTH-SOUTH PROPORTIONED POSITION AND N 2° 28’ 24” E FROM TEMPORARY EAST-WEST PROPORTIONED POSITION. 4.) NOTE: THE EAST-WEST CONVERGENCE SHOULD BE COMPUTED AT A MIDPOINT BETWEEN THE TWO PROPORTIONED POSITIONS. N=451924.12E=830343.18

12. COMPARISON OF COORDINATES N=462520.90E=830910.54 N 3° 22’ 25” E 5339.34 N 88° 24’ 11” W 5232.39 S 82° 24’ 52” E 5217.20 N=457336.62E=825365.99 457190.81295 N 830596.34381 E N=456502.10E=835767.89 S 2°45’ 07” W 5272.77 457190.81294 N 830596.34367 E N=451924.12E=830343.18

13. COORDINATE VALUES OF THE LOST CORNER POSITION COMPUTED NEGLECTING CARDINAL EQUIVALENTS. N=462520.90E=830910.54 N 3° 23’ 55” E 5342.62 N 3° 22’ 25” E 5339.34 N 88° 24’ 11” W 5232.39 S 82° 24’ 52” E 5217.20 N=457336.62E=825365.99 N 88° 22’ 05” W 5229.96 S 82° 27’ 08” E 5219.29 457187.67548 N 830593.82368 E N=456502.10E=835767.89 S 38°46’ 07” W 4.02 S 2° 43’ 35” W 5269.52 S 2°45’ 07” W 5272.77 457190.81294 N 830596.34367 E N=451924.12E=830343.18

14. WHY PROPORTIONING USING ONLY THE LATITUDES AND LONGITUDES WILL NOT WORK! 464803.66761 N 832833.03761 E 48°13’ 35.57930” N 124°08’ 51.27973” W 461070.14380 N 829099.51380 E N 45° 00’ 00” E 5280 48°12’ 57.17888” N 124°09’ 43.98584” W N 45° 00’ 00” E 5280 457336.62 N 825365.99 E 48°12’ 18.77169” N 124°10’ 36.67009” W

15. WHY PROPORTIONING USING ONLY THE LATITUDES AND LONGITUDES WILL NOT WORK! 48°13’ 35.57930” N 124°08’ 51.27973” W 173615.57930” N 446931.27973” W CONVERT THE NORTHERLY AND SOUTHERLY LATITUDE AND LONGITUDE POSITIONS TO DECIMAL DEGREES AND MULTIPLY BY 3600. N 45° 00’ 00” E 5280 48°12’ 57.17888” N 124°09’ 43.98584” W 173577.17888” N 446983.98584” W N 45° 00’ 00” E 5280 48°12’ 18.77169” N 124°10’ 36.67009” W 173538.77169” N 447036.67009” W

16. WHY PROPORTIONING THE LATITUDES AND LONGITUDES WILL NOT WORK! 173615.57930” N 446931.27973” W AVERAGE THE LATITUDES AND LONGITUDES BY ADDING THEM TOGETHER AND DIVIDING BY 2 AND CONVERTING BACK TO DD-MM-SS.SSSSS (173615.57930” + 173538.77169”) / 2 = 173577.17550” = 48° 12’ 57.17550” N (447036.67009” + 446931.27973”) / 2 = 446983.97491” = 124° 09’ 43.97491” W N 45° 00’ 00” E 5280 CONVERT THE LATITUDE AND LONGITUDE BACK TO STATE PLANE COORDINATES 48°12’ 57.17888” N 124°09’ 43.98584” W 461069.76937 N 829100.23824 E 48°12’ 57.17550” N 124°09’ 43.97491” W N 45° 00’ 00” E 5280 173538.77169” N 447036.67009” W

17. WHY PROPORTIONING USING ONLY THE LATITUDES AND LONGITUDES WILL NOT WORK! 464803.66761 N 832833.03761 E N 44° 59’ 30” E 5280.75 461070.14380 N 829099.51380 E S 62°40’ 02” E 0.82 N 45° 00’ 00” E 5280 461069.76937 N 829100.23824 E N 45° 00’ 30” E 5280.25 N 45° 00’ 00” E 5280 457336.62 N 825365.99 E