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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 substract the less from the greater; the remainder will be double the area of the field in square rods or square chains and links, which ever measure was used in the survey.

[In the preceding explanations, the meridian is supposed to pass through the extreme west angle of the field. It is best always to take the extreme east or west angle.]

Fig. 62.

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Demonstration of the preceding rules.

See Fig. 62. and ExamPLE I.

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51 H 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, 77.15X20.74=1600.0910, which is double the area of the triangle A 2 B, as is evident from the rule to find the area of the triangle, PROB. IX. Rule I. 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.

northing made by the second course, and the line B C, 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., 65.86x31.66=2085.1276, double the area of the trapezoid 2 B C 3. Place this product for the second number in the column of north areas.

To the line 3 C add C D 30.04, that 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 the 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 number 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 give double the area of the trapezoid 1 E D 3. The number thus pro: duced is 7858.1020, which is to be placed for the first num", ber 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. This, being multiplied by 54.10, the length of the line E F, which is the southing of the fifth course as corrected in balancing, and the same as the line 1 4, will give double the area of the parallelogram 1 E F 4, which is 9166.7040, the second number in the column of south areas.

From the line 4F 84.72, substract 39.95, which is a west course and it leaves 4 G 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 substract 23.75, the westing made by the 7th course, and you have 21.02, the length of the line 5 #, which is the meridian distance at the end of the seventh course, and the seventh number in the first departure column. The line 4 G 44.77, added to the line 5 H 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 5, which is the southing of the seventh course, will give double the area of the trapezoid 4 G H 5, which is 21 19.0959, the third number in the column of south areas.

the last number in the second departure column. This being multiplied by 26.65, length of the line 5 A, and the northing of the last course, produces 560.1830, which is double the area of the triangle A 5 H, and the last number in the column of north areas. Note. It will be observed that against the third and sixth courses there

are no areas; the reason is, that these courses being one.cast 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 A 2 B, the trapezoid 2 BC3, and the triangle A 5 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 3 DE 1, the parallelogram 1 E F 4, and the trapezoid 4 G H 5; and that these include not only the field, but also what was included in the north areas.

Therefore the north areas substracted 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 substracted 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 and 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 added to or substracted from the num. ber last set in the latitude 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

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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 continual addition

course.

ning out before you get through the several courses, you may begin at such a course in the field book as will admit of a continual addition and substraction; and when you get to the bottom, go to the top, and you will end in cipher at the course next above that where you began.

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The area standing against the 1st course, is the triangle That against the 2d course is the trapezoid 23ai, lying without the field.

That against the 3d course is the trapezoid 34ac, and it is a south area, lying partly within and partly without the field.

That against the 4th course is the trapezoid 45cm, and it is also a south area.

That against the 5th course is the trapezoid 56mo, and it is also a south area.

That against the 6th course is the trapezoid 67no, and it is a north area, a part of which is already contained in two different products. In this area, and in the two succeeding areas, all of that part of the figure which lies above the line 06, is considered void, and it may again be covered by both north and south products.

The area against the 7th course is the trapezoid 78 en, and it is a north area.

Against the 8th course there is no latitude.

The area against the 9th course is the trapezoid 9,10,ex and it is a south area.

That against the 10th course is the trapezoid 10, 11, xz, and it is a south area:

That against the 11th station is the parallelogram 11, 12,sz, and is a north area.

That against the 12th station is the trapezoid 12, 13,8V, and it is a south area.

That against the 13th station is the triangle 13,1,0, and it is a north area.

When the field is very irregularly shaped, it will often happen that parts of the same area will be contained in several different products in the column of areas. As often as a space is covered by a north product, so often will it be covered by a south produet; but in the final result, the sum of one column being substracted from that of the other, will leave what is included within the boundary lines of the field.

This method of calculating the area of a field by the northings, southings, eastings, and westings, divides the field, with a certain quantity of the adjoining ground, into right angled triangles, right angled trapezoids, parallelograms, or squares, as may be seen by the figures. It may therefore with pro: priety be called RECTANGULAR SURVEYING.*

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