NXE+SxW. Now, 2FEXGH or AD=2 area ZGIN=YZGEa+QEHNa,

And 20PLN or BX=2MLNQ=2CLNB, since COM=OQB. But, 2CLNB=2CRaB+2CLR+20NB. Hence,

20P x LN=2CRaB+2CLR+20NB. But. 2FEX GH=

2ZGEa+QEHNAa. Their sum =2CRaB+2CLR+2ZGEa+QEHB, or 2EAG.

=2CRaB+2CLR+2ZAa. But, 2ST XLZ=2RTVZ+2LWTR... Hence, the sum of the three products=2CRaB+QRTVZ+2CWT+2ZAa.

=2CRaB+2RTVZ+2TAV+2ZAa.
=2CRaB+2RAZ+2ZAa.

=2 area of the triangle ABC. In this figure there is no northing which has a western meridian distance, nor southing having an eastern meridan distance, and therefore there is but one set of products.

The demonstration would, however, be similar for every position of the assumed meridian, and the general theorem would be equally true.

166. It remains to illustrate these principles by an example. The area, therefore, is 10 acres, 0 roods, 4.15520 perches.

0 In this example, the work on the field is begun at station 1. (Pl. 6, Fig. 4). The bearings are taken at all the stations, 1, 2, 3, 4, 5, 6, and 7; the distances are measured, and the entries made in the first three columns of the table. The distances are estimated in chains and links. The latitudes and departures are then found in the traverse table, and entered in their proper columns. These columns are then added up, and it is found that the northings exceed the southings by 6 links, and the westings the eastings by 8 links; but these sums should be equal. (160.)

The inequality arises from the inaccuracy of the measurement of the lines and bearings; but if the differences do not exceed four links for each station, it is deemed unnecessary to make a remeasurement. The corresponding columns are thus rendered equal. Divide the difference between the northings and so things by 2. Now, if this half difference be added in the smaller column, and subtracted in the larger, the amounts in the two columns become equal. But, to be strictly accurate, the half difference should be apportioned among the lines in proportion to their lengths. That is, we should make the proportions, as the sum of the northings, is to the half difference to be distributed, so is each northing to its portion of such half difference: and, as the sum of the southings, is to half the difference, so is each southing to its portion of the half difference; the half difference being thus distributed among the

: northings and among the southings, their sums will be equal. The sums of the eastings and westings are made equal in the same way. This is called balancing the work, and the columns under balanced are the balanced latitudes and departures. It is, in general, hardly necessary to make the proportions; and, if the differences are not very unequal, the half difference may be distributed equally.

It is evident, from an inspection of the columns of departures, that 6 is the most westerly station ; through this point let the assumed meridian be drawn.

The column D. M. D. is the column of double meridian dis

columns of balanced departures; the mode of calculation has been too fully explained to need further illustration.

The columns NXW+SRE, NXE+SXW, contain the several products named in the general theorem.

167. It is now required to make a plot of the ground, whose dimensions and area are expressed in the table.

Draw any line on the paper, as NS, (Pl. 6, Fig. 4,) to represent the assumed meridian, which passes through station 6. The line which is run from station 6 lies on the east of the meridian, and makes with it an angle of 16°; its latitude in the balanced column is 3.37, and its departure.97. There are two ways of plotting this line. First, with a protractor, or scale of chords, make the angle SFG equal to 16', and lay off, from a convenient scale of equal parts, the line FGʻ, equal to 3.50, its length, taken from the column of distances. Or, secondly, from the scale of equal parts, lay off FG, equal to 3.37, the latitude : draw the perpendicular GGʻ, and make it equal to .97, the departure, and join the points F and G. The latter method of plotting is preferred.

Through the station 7, draw a meridian, A'G, and lay down the line GA. Then draw the meridian AB, and lay down the line AB, and thus until all the lines are drawn: FGABCDE is the plot required.

Example 2.-To find the area of a piece of land, of which the following are the notes. Station 1, S. 40° W., dist. 70 rods; Station 2, N. 45° W., dist. 89 rods; 3, N. 36° E., 125; 4, North 54; 5, S. 81° E., 186; 6, S. 8° W., 137; 7, W., 130.

Answer.-207 acres, 3 roods, 22.69 perches.

Example 3.-Given the bearings and distances as follows: Station 1, S. 40; E., dist. 31.80 ch.; 20, N. 54° E., 2.08 ch.; 3d, N. 29° E., 2.21 ch. ; 4th, N. 28° E., 35.35 ch.; 5th, N. 57° W., 21.10 ch.; and 6th, S. 47° W., 31.30. Required the area.

Answer.-92 acres, 3 roods, 30 perches.

168. It was remarked, in article 159, that the field notes would be used, under the supposition of the land being kept

0

in example 1, we had gone round the land the other way, the bearing from A to G would have been B'AG, or N. the angle B'AG, W. But B'AG=AGA', which was the bearing from G to A, and was S. 65o E. Now, both the meridional and longitudinal letters are different, according as the line is run one way or the other, but the bearing and the distance remain the same.

We may therefore conclude, that, a back sight, or reverse bearing, is equal to the corresponding forward sight; and that if the land be kept on the right hand, the notes will be the same as though it were kept on the left, excepting that the meridional and longitudinal letters, for the same course, will be different.

169. As the needle is sensibly affected by the presence of ferruginous substances, it is best to take both the forward and back sights. If they agree they may be relied upon, if not, the needle is influenced by local attraction. It then becomes necessary to ascertain at which of the stations at the extremities of the line, the local attraction exists. This is done by comparing the bearings of the line with the bearings of other lines of the survey.

170. In the operations of surveying it is sometimes necessary to know the angles which the lines make with each other. These are readily found from the field notes. At station A, (Pl. 6, Fig. 4,) for example, the bearing AB is B'AB, N. 34° E.; the bearing of GA, from 7 to 1, is A'GA, S. 65° E.

But A'GA=GAB ; hence GAB=GAB+B AB=65° +34 =99°; and, in a similar manner, the angle of any two of the lines may be computed.

171. It is often necessary to ascertain the distance and bearing of two points, one of which cannot be seen from the other.

From either point measure a line lying as nearly as convenient in the direction of the second point, and take its bearing. From the extremity of this line, as a second station, take the bearing and measure the distance of a second line leading in the direction of the point, and so on, until the point is reached. Then arrange the notes in a tabular form, and

a