Rectangular surveying.

is so small as not to be affected by the earth's curva

ture.

Second Method of Solution. Let ABCEFH (fig. 21) be the field to be measured. Starting from its most easterly or its most westerly point, the point A for instance, measure successively round the field the bearings and lengths of all its sides. Through A draw the meridian NS, on which let fall the perpendiculars BB', CC, EE, FF", and HH'. Also draw CB E", EF, and HF" parallel to NS.

Then the area of the required field is

ACCB+AHFF CCBB' + B'BA + AHH
+HHFF.

+ B'BA + AHH + HHFF];

or doubling and changing a very little the order of the terms,

2 ABCEFH = [2 C'CEE +2 E'EFF'] [2B'BA +2 CC'BB' +2 H'HFF' +2 AHH'].

Again,

2 B'BA = BB'

(277)

X AB'

2 CC'BB' = (BB' + CC'′) × B'C'
2 C'CEE' = (EE' + CC') × EC

(278)

2 E'EFF = (EE + FF) × E'F
2 H'HFF = (HH' + FF) × HF
2 AHH = HH
X AH'.

Rectangular surveying.

So the determination of the required area is now reduced to the calculation of the several lines in the second members of (278). But the rest of the solution may be more easily comprehended by means of the following table, which is precisely similar in its arrangement to the table actually used by surveyors, when calculating areas by this process.

In the first column of the table are the successive sides of the field.

In the second and third columns are the differences of latitude of the several sides, the column headed N, corresponding to the sides running in a northerly direction, and that headed S, corresponding to those running in a southerly direction.

These two columns are calculated by the formula

Diff. lat. dist. X cos. bearing.

In the fourth and fifth columns are the departures of the several sides; the column headed E, corresponding to the sides running in an easterly direction, and that headed W, to those running in a westerly direction.

Rectangular survey.

These two columns are calculated by the formula

Departure

dist. X sin. bearing.

In the sixth column, headed Departure, are the departures of the several vertices, which end each side of the field from the vertex A. This column is calculated from the two columns E and W, in the following manner. The first number in column Departure is the same as the first in the two columns E and W; and every other number in column Departure is obtained by adding the corresponding number in columns E and W, if it is of the same column with the first number in those two columns, to the previous number in column Departure; and by subtracting it, if it is of a different column.

EE' = E'E" + EE" = CC + EE"

FF = FF + FF" = EE' + FF"

HHFF

=FF FF

0 = HH - HH'.

In the seventh column, headed Sum, are the first factors of the second members of (278). This column is calculated from column Departure in the following manner. The first number in column Sum is the same as the first in column Departure; every other number in column Sum is the sum of the corresponding num

Rectangular survey.

ber in column Departure added to the previous number in column Departure, as is evident from simple inspection.

In the eighth and ninth columns are the values of the areas, which compose the first members of (278). These columns are calculated by multiplying the numbers in column Sum by the corresponding numbers in columns N and S, which contain the second factors of the second members of (278). The products are written in the column of North Areas, when the second factors are taken from column N, and in that of South Areas, when the second factors are taken from column S.

If we compare the columns of North and South Areas with (277), we find that all those areas, which are preceded by the negative sign, are the same with those in the column of North Areas; while all those, which are connected with the positive sign, belong to the column of South Areas. To obtain, therefore, the value of the second member of (277), that is, of double the required area, we have only to find the difference between the sums of the columns of North and South Areas. [B. p. 107.]

62. Corollary. The columns N, S, E, and W, are those which would be calculated in Traverse Sailing, if a ship was supposed to start from the point A, and proceed round the sides of the field till it returned to the point A. The difference of the sums of columns N and S is, then, by traverse

Correction of errors.

sailing, the difference of latitude between the point from which the ship starts, and the point at which it arrives; and the difference of columns E and W is the departure of the same two points. But as both the points are here the same, their difference of latitude and their departure must be nothing, or

Sum of column N

Sum of column E

sum of column S;

sum of column W.

But when, as is almost always the case, the sums of these columns differ from each other, the difference must arise from errors of observation. If the error is great, new observations must be taken; but if it is small, it may be divided among the sides by the following proportion.

The sum of the sides each side = whole error :
error corresponding to each side.

(279)

The errors corresponding to the sides are then to be subtracted from the differences of latitude, or the departures which are in the larger column, and added to those which are in the smaller column.

63. EXAMPLES.

1. Given the bearings and lengths of the sides of a field, as in the three first columns of the following table, to find its area.

Solution. The table is computed by $61.