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each other, it is evident that one being added and the other subtracted, there will in the end be no Remainder.
5. The next step in the process is to form a second departure Column, the numbers in which show the Sum of the Meridian Distances at the end of the first and second, second and third, third and fourth Courses, &c.
The first number in this Column will be the first in the other Departure Column; to which add the second number in that Column for the second in this ; for the third add the second and third ; and for the fourth the third and fourth; and so on till the Column be completed. See EXAMPLE I.
The first number to be placed in the second Departure Column is 20.74; to this add 45.12 and it makes 65.86 for the second number; to 45.12 add 75.16 and it makes 120.28 for the third number; to 75.16 add 84. 72 and it makes 159.88 for the fourth number; to 84. 72 add 84.72 and it makes 169.44 for the fifth number; to 84.72 add 44.77 and it makes 129.49 for the sixth number; to 44.77 add 21.02 and it makes 65.79 for the seventh number; to 21.02 add 0.0 and it makes 21.02 for the eighth number.
6. When the work is thus far prepared, multiply the several numbers in the second Departure Column, by the Northings or Southings standing against them respectively ; place the Products of those multiplied by the Northings in the Column of North Areas, and of those multiplied by the Southings in the Column of South Areas; add up these two Columns and subtract the less from the greater ; the Remainder will be double the Area of the Field, in Square Rods or Square Chains and Links, whichever measure was used in the Survey.
Demonstration of the preceding Rules. See PLATE III. Fig 63. and EXAMPLE I. The
dotted Line A 2 represents the Northing, and the Line 2 B the Easting made by the first Course : These multiplied together, that is, 77.15 X 20.74
angle A 2 B, as is evident from the Rule to find the Area of a Triangle, Prob. IX. Rule 1. This number is to be placed for the first number in the Column of North Areas. The Line 3 C represents the Sum of the Eastings made by the first and second Courses, which is 45.12, the second number in the first Departure Column; if to this you add 20.74 the length of the Line 2 B you have 65.86, which is the second number in the second Departure Column, and which represents the Sum of the two Lines 3 C and 2 B. These two Lines with the Line 2, 3 which represents the Northing made by the second course, and the Line BC, one of the sides of the Field, form a Right Angled Trapezoid. Now, by the Rule to find the Area of such a Trapezoid, See PROB. X. 65.86 X 31.66 = 2085.1276, double the Area of the Trapezoid 2 BC 3. Place this Product for the second number in the Column of North Areas.
To the Line 3 C add CD 30.04 the Easting made by the third Course, and you have 75.16, which is the Sum of the Eastings made by the three first Courses, and the third number in the first Departure Column. To this add 9.56 the Easting of the fourth Course, and you have 84.72 the length of the line 1 E, which represents the Sum of the Eastings made by the four first Courses, and is ihe fourth number in the first Departure Column. These two, viz. the Lines 3 D 75.16 and 1 E 84.72 added together make 159.88 the fourth num. ber in the second Departure Column; which being multiplied by 49.15 the length of the Line 3, 1 which represents the Southing made by the fourth Course, will gill give double the Area of the Trapezoid 1 ED 3. The number thus produced is 7858.1020, which is to be placed for the first number in the Column of South Areas.
The fifth Course being due South, it is evident the Sum of the Eastings will remain the same as at the end of the fourth Course: That is, the Line 4 F equals the Line 1 E, which is 84.72. These added make 169.44 the fifth number in the second Departure Column.
EF, which is the Southing of the fifth Course, as corrected in balancing, and the same as the Line 1, 4will give double the Area of the Parallelogram 1EF4, which is 9166.7040 the second number in the Column of South Areas.
From the Line 4F 84.72 subtract 39.95 which is a West Course, and it leaves 4G 44.77 the Sum of the Eastings, or the Meridian Distance, at the end of the sixth Course, and the sixth number in the first Departure Column. From this subtract 23.75 the Westing made by the seventh Course, and you have 21.02 the length of the Line 5H, which is the Meridian Distance at the end of the seventh Course, and the seventh number in the first Departure Column. The Line 4G 44.77 added to the Line 5H 21.02 make 65.79 the seventh number in the second Departure Column. This being multiplied by 32.21 the length of the Line 4, 5which is the Southing of the seventh Course, will give double the Area of the Trapezoid 4GH5, which is 2119.0959 the third number in the Column of South Areas.
The Line H5, 21.02 is the Westing of the last Course, and the last number in the second Departure Column. This being multiplied by 26.65 the length of the Line 5A, and the Northing of the last Course, produces 560.1830, which is double the Area of the Triangle A5H, and the last number in the Column of North Areas. Wole. It will be observed that against the third and sixth Cour
ses there are no Areas ; the reason is that these Courses being one East and the other West, there is no Northing or Southing to be multiplied into them ; regard can therefore be
had to them only in forming the Departure Columns. By inspecting the Figure, and attending to the preceding illustrations, it will be seen that the three North Areas represent double the Area of the Triangle A2B, the Trapezoid 2BC3, and the Triangle A$H, all of which are without the boundary Lines of the Field; Also, that the three South Areas represent double the Area of the Trapezoid 3DEI, the Parallelogram IEF4, and the Trapezoid 4GH5; and that these in
clude not only the Field but also what was included in the North Areas. Therefore the North Areas subtracted from the South, the Remainder will be double the Area of the Field, contained within the black Lines.
Additional Directions and Explanations. The Northings and Southings may be added and subtracted instead of the Eastings and Westings; then there will be two Latitude Columns instead of Departure Columns; and the numbers in the second Latitude Column must be multiplied into the Eastings and Westings, and you will have East and West Areas.
When the Course is directly North or South, the Distance must be set in the North or South Column; when East or West, in the East or West Column. There will therefore sometimes be no number to be add. ed to or subtracted from the number last set in the Lat. itude or Departure Column ; then the number last placed in the Column must be brought down and set against such Course; as in ExAMPLE I. at the 5th Course. It may also sometimes be the Case that there will be no number to multiply into the number in the second Latitude or Departure Column ; then that number must be omitted, and against such Course there will be no Area; as in ExAMPLE I. at the 3d and 6th Courses.
When the Northings or Southings, Eastings or Westings, beginning at the top, will not admit of a
ontinual addition of the one and subtraction of the other, without running out before you get through the several Courses, you may begin at such a Course as will admit of a continual addition and subtraction; and when you get to the bottom go to the top, and you will end in Cypher at the Course next above that where you began ; as in ExAMPLE II. which begins at the 9th Course to add the Eastings and subtract the Westings.
1 dep2 dep North South No. Courses. Rods N. S. E.W. Col. Col. Areas Areas 1 N.750 0'E. 54.8|14.2
144.1/235.3 3341.26 2 N.20 30 E. 41.238.6 1144 1158.5 302.6 11680.36 3 jEast
164.8 223.3381.8. 4 s. 33 30 W. 141.2. 117.7...177 9 145.4/368 7
43395.99 5 S 76 0 W. 64 15.5 162.183.3228.7
3544.85) 6 North 36 36
83.3 166.6 5997.60 S. 84 0W. 46.4). 4.9 46.1 37.2 120.5
590.45 8 N.53 15 W. 46 4 27.8
37.2 00 37.2 1034.16 9 N.36 45 E. 76.8161.5 46 46 46 2829 110
56 (51.7 21.4) 67.4. 113.4) 5862.78
1996.50 12 s. 150W. 43.4 41.9 11.2 102.9|217
9092.30 13 S. 16 45 W.) 40,5 38.8. 11.7| 91.2|194.11
Area 110 Acres, 2 Roods, 23 Rods.
Note. In the above EXAMPLE you might begin at the 4th Course
to add the Westings and subtract the Eastings; or at the 6th Course to add the Northings and subtract the Southings; or at the 11th Course to add the Southings and subtract the Northings. So in every Survey same place may be found where you may begin to add and subtract, without running out before you get through all the Courses. When a Field is very irregularly shaped, it will often happen that parts of the same Area will be contained in several different Products in the Columns of Areas; but in the final result, one Column being subtracted from the other will leave what is included within the boundary Lines of the Field.
DEMONSTRATION. See PLATE III. Fig. 64. and EXAMPLE II.
The Area standing against the 9th Course, which is where the Calculation begins, is the Triangle 12K, all without the Field.
The Area against the 10th Course is the Trapezoid 2KL3, also without the Field.
The Area against the 11th Course is the Trapezoid 4ML3. This is a South Area, and contains a part of the Field and also part of the preceding North Area.
The Area against the 12th Course is the Trapezoid 5NM4, part within and part without the Field.