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

*This is not the first station in the actual survey, but only the most westerly point of the survey as calculated by the foregoing method from the field-notes, which, for convenience' sake, I call the first station in making out this table.

The meridian distances in this column are the sum of two adjacent meridian distances; but at the most westerly point the meridian distance is nothing, hence the first dep. is the first meridian distance, and, in like manner, the last dep. is the last meridian distance.

+ Demonstration. Let us consider that every tract of land has an extreme southerly 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

8

N 39 W 7.50

5.82

2

2

.03 .02

.03

g།༅།༅།༄

༅ |༅།རྩ

96

2

59.8539

.2128

42.4402

172.7846

19.23

9.52

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 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+2.65=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. N.; hence, the third point is the most northerly point of the survey.

=

=

H, 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 contents of the figure IABCDEFK, the contents of the figure FKIAHG be subtracted, the remainder will be the area of the survey. The multangular figure IABCDEFK is composed of all these trapezoids, viz. IABL, BCML, CDNM, EOND, and FKOE; but (by Prob. 10) (IA+LB)×IL= 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 colunn areas, that is, (IA+BL)XIL+(BL+CM)×LM+(CM+DN)× MN+(DN+EO)×N0+(E0+FK) OK = twice the arca of the figure IABCDEFK the sum of the east area column. And in like manner we demonstrate that (FK+PG)×PK: twice the area of the trapezoid FKPG; 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 PGX PH=twice the area of the triangle PGH, and IA×IH = twice the area of the triangle IAH; hence (FK+PG)×PK+PG×PH+IA×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)×NO +(E0+FK)×OK—[(FK+PG)×PK+PG×PH+IA×ÌH]=twice the area of the survey; consequently, the sum of the east area columnthe sum of the west area column - twice the area of the survey. Q. E. D.

=

Now, by calling the most southerly point of the survey the first station, and proceeding 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.

.03 .02 3.69

3.33

.02 .02

30

.02 3.98

18.92 50.7159

75.2220

.02.02 3.93

22.81

13.9141

26,74

2

24.72
22.70

127.8024

༄ | ༅ །༅
༄།།༄

89

60

Ans. 13A. OR. 11P.

13.069695 Acres.

10)130.69695

2)261.3939

Each of the numbers in the column of latitudes is twice the mean latitude 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 foregoing method from Table I.

SECTION IV.

OF OFFSETS.

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 offsets and intersections, and with equal certainty; especially where creeks, &c. bound the survey.

Offsets are perpendicular lines drawn or measured from the angular points of the land, that lie on the right or left-hand to the stationary distance, 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 4ch. 251., as well as the measure of the offset ad, which admit to be 1ch. 127., thus by the side of your field-book, in a line with the first station, say at 4ch. 251. L. 1ch. 12l., that is, at 4ch. 251. there is an offset to the left-hand of 1ch. 127.

This done, proceed on your distance line to e opposite to the angle b, and measure eb; supposing then le to be 7ch. 40l., and eb 3ch. 40l., say (still in a line with the first station in your field-book) at 7ch. 401. L. 3ch. 40l., that is, at 7ch. 407. there is an offset to the left of 3ch. 401.; proceed then with your distance line to f opposite to the angle c, and measure fc; suppose then 1f to be 13ch. and fc 1ch. 25l., say, in the same line as before, at 13ch. L. 16h. 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 offsets, it is necessary to have a perch chain, or a staff of half a perch, divided into links for measuring them; for by this 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 offsets 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 distance line, stop at 7 opposite g; admit 21 to be 4ch. 10l., and the offset lg 1ch. 201., then in a line with the second station in your field-book, say at 4ch. 107. R. 1ch. 20l., that is, the offset is a right-hand one of 1ch. 201. Again, at m, which suppose to be 10ch. 251. from 2, take the offset mh of 1ch. 15l., and in a line with the second station, say at 10ch. 251. R. 1ch. 157. In the same line, when you come to the boundary at i, insert the distance 2i, 13ch. 10l., thus, at 13ch. 107. 0; that is, at 13ch. 10l. there is no offset. At n, which is 15ch. from 2, take the offset nk 45l., and still opposite to the second station say at 15ch. L. 45l.

Let the line 3, 6 represent the boundary which by means of water, briers, or any other impediment, cannot be measured. In this case make one or more stations within or without the land, where the distances may be measured, and draw a line from the beginning of the first to the end of the last distance, thus: make stations at 3, 4, and 5, take the bearings, and measuring the distances as usual, which insert in your field-book, and draw a mark like one side of a parenthesis, from the third to the fifth station, to show that a line drawn from the third station to the farthest end of the fifth stationary line will express the boundary. Thus,

Suppose the point p of the boundary to be inaccessible by means of the lines 6p or p7 being overflowed, or that a quarry, furze, &c. might prevent your taking their lengths: in this case take the bearing of the line 6, 7, which insert opposite to the sixth station in your field-book with the other bearing; then direct the index to the point p, and insert its bearings on the left side of the field-book, opposite to the sixth station, annexing thereto the words Int. for boundary; and having measured and inserted the distance 6, 7, set the index in the direction of the line 7p, and insert its bearing on the left of the seventh station