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RULE II.*

The difference of latitude and departure being found and corrected as in the preceding rule.

As, beginning at the most northerly or most southerly point of the survey, admits of a continual addition of the one and subtraction of the other; make choice of either of these points in order to calculate the area of the survey.

1. It is necessary to calculate the several latitudes in order to find the most northerly or most southerly point of the survey, which may be done from Table I. thus:

The first Lat. is .02 south, which is the difference of latitude between the second point of the survey and the first, when the survey is corrected,

* Demonstration. Let us consider that every tract of land has an extreme sourtherly point, as H; and we reckon so much as any other point is distant from the East and West line IK, (Pl. 14. Fig. 11.) that passes through I, to be its Latitude North, or the difference of Latitude between the points H and A; BL the Lat. of B; CM the Lat. of C; &c.

Thus, if from the content of the figure IA BCDEFK, the content of the figure FKIAHG be subtracted, the remainder will be the area of the survey. The multangular figure 1ABCDEFK is composed of all these trapezoids, viz. IABL, BCML, CDNM, EOND, and FKOE; but (by Prob. 10.) (IA+LB)XIL twice the area of the trapezoid IABL, and (LB + CM) XLM twice the area of the trapezoid BCML, and so for the rest; and LA+LB the sum of the Northings of the points A and B from the line IK, and IL the Easting of the point B from the point A. In like manner the area of every other trapezoid is found; but these are the East column areas, that is, (IA+BL)× IL+ (BL+CM) ×LM+(CM+DN) × MN° +(DN+EO)× NO + (E0+FK) × OK=twice the area of the figure IABCDEFK the sum of the East area column. And, like manner, we demonstrate that (FK+PG)×PK= twice the area of the trapezoid FK PG; but FK+PG the Lat. of F+ the Lat. of G, and PK the Dep. / or Westing of the point G from the point F, and PG× PH=twice the area of the triangle PGH, and 1AX IH twice the area of the triangle 1AH; hence (FK+PG)×PK+PG×PH+1A× IH twice the area of the figure FKIAGH the sum of the West area column. Therefore (IA+ BL)XIL+ (BL+CM) × LM + (CM+DN) ×MN+(DN+EO)X NO +(EO +FK) ×OK— ((FK+PG)×PK÷PG×PH+IA×IH) twice the area of the survey, consequently the sum of the East area column -the sum of the West area column twice the area of the survey. Q. E. D.

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from the next departure 3.93 which is N. subtract .02, and their difference 3.91 is equal to the difference of latitude between the third point and the first which is N. and 3.91-2.02=1.89=the difference of lat. between the fourth point and the first; which is also N. But as the next difference of lat. is south, therefore 5.71-1.89 3.82 the difference of lat. S. between the fifth point and the first; and 3 82+2.99=6.81 the difference of lat. S. between the sixth point and the first; and 6.81 +265-9.46 the difference of lat, S. between the seventh point and the first; and 9.46-5.77= 3.69 the difference of lat. S. between the eighth point and the first; and 3.69-3.69-0; hence it is evident that 9.46 is the greatest lat. S the difference of lat. between the seventh point and the first; therefore, the seventh point of the survey is the most southerly point; and, in like manner, 3.91 the difference of lat. between the third point and the first, is the greatest lat, north; hence, the third point is the most northerly point of the survey.

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Now, by calling the most southerly point of the ́ survey the first station, and proceding to find the latitudes for the several lines in the order in which they were surveyed; that is, the first difference of lat, will be the first lat, which place in the column of latitudes, opposite the said difference of latitude; to the same lat. add the said difference of lat, to which sum add the next difference of lat. if it be of the same name, but subtract if of a different name, and place it in the column of latitudes, in like manner continue to add or subtract the difference of lat. twice, and the last lat. comes out nothing, if the additions and subtractions are rightly performed. Multiply each of the upper numbers in the column of latitudes by the corresponding dep. and place the products in the column

of east or west area, according as the dep. is E. or W. The difference of these columns will be equal to twice the area, half of which will give the area of the survey: as in the following Table.

TABLE III.

18(Cour ss. [D.C.)N. |S. | E. W. c.s.)c.w. N. S. E., W., Lat. E. A. W. A.

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3.35

.03.02 3.69

3.33

50.7159

18.92

18.90

4.00

.02.02

.02 3.98

75.2220

18.88

22.81

0.63

.02.02 3.93

0.61

13.9141

26.74

24.72

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22.70 127.8024

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*Each of the numbers in the column of latitudes is twice the mean lati. tude of two adjacent latitudes; but at the most southerly point the latitude is nothing; hence the first difference of latitude is the first lat. and in like manner, the last difference of lat. is the last latitude. It is also to be remarked that the first station used in this Table is not the first station in the actual survey; but the most southerly point of the survey, as calculated by the fore going method from Table I

SECTION IV.

.

OF OFF-SETS.

IN taking surveys it is unnecessary and unusual

to make a station at every angular point, because the field work can be taken with much greater expedition, by using off-sets and intersections, and with equal certainty; especially where creeks, &c. bound the survey.

Off-sets are perpendicular lines drawn or measured from the angular points of the land, that lie on the right or left hand to the stationary dis tance, thus:

PL. 11. fig. 2.

Let the black lines represent the boundaries of a farm or township: and let 1 be the first station; then if you have a good view to 2, omit the angular points between 1 and 2, and take the bearing and length of the stationary line 1, 2, and insert them in your field-book: but in chaining from 1 to 2, stop at d opposite the angular point a, and in your field-book insert the distance from 1 to d, which admit to be 4C. 25L. as well as the measure of the off-set ad, which admit to be IC. 12L. thus by the side of your field-book in a line with the first station, say at 4C. 25L. L. 1C. 12L. that is, at 4C. 25L. there is an off-set to the left. hand of 1C. 12L.

This done, proceed on your distance line to e opposite to the angle b, and measure cb, supposing then le to be 7C. 40L. and eb 3C. 40L. say (still in a line with the first station in your field-book) "at 7C. 40L. L. 3C. 40L." That is, at 7C. 40L. there is an off-set to the left of 3C. 40L. proceed then with your distance line to f opposite to the angle c, and measure fc; suppose then 1 f to be 13C. and fe 1C. 25 L. say in the same line as before, at 13C. L. 1C. 25L. Then proceed from f to 2, and you will have the measure of the entire stationary line 1, 2, which insert in its proper column by the bearing.

In taking off-sets, it is necessary to have a perch chain, or a staff of half a perch, divided into links for measuring them; for by these means the chain in the stationary line is undisturbed, and the number of chains and links in that line from whence, or to which, the off sets are taken, may be readily known.

Having arrived at the second station, if you find your view will carry you to 3, take the bearing from 2 to 3, and in measuring the distacce line, stop at l opposite g; admit 27 to be 4C. 10L. and the off-set lg 1C. 20L. then in a line with the second station in your field-book, say at 4C. 10L. R. 1C. 20L. that is, the off-set is a right hand one of 1C. 20L. Again at m, which suppose to be 10C. 25L. from 2; take the off-set mh of 1C, 15L. and in a line with the second station, say at 10C. 25 L. R. 1C. 15L. In the same line when you come to the boundary at i, insert the distance 2i, 13C. 10L. thus, at 13C. 10L. 0; that is, at 13C. 10L. there is no off-set. At n, which is 15C. from 2, take the off-set nk 45L. and still opposite to the second station say at 15C. L. 45. L.

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