from the two foregoing theorems, and therefore requires no further explanation; but to find the area by the second method take this RULE. When the meridian distances are east, put the products of north and south areas in their proper columns, but when west in their contrary columns; that is, in the column of south area when the difference of latitude is north, and in north when south: the reason of which is plain from the last two theoThe difference of these two columns will be the area rems. of the map. Construction of the Map from either the first or the second Table. PL. 10. fig. 3. Draw the line NS for a north and south line, which call the first meridian; in this line assume any point, as 1, for the first station. Set the northing of that stationary line, which is 3.54, from 1 to 2, on the said meridian line. Upon the point 2 raise a perpendicular to the eastward, the meridian distance being easterly, and upon it set 13.22, the second number in the column of meridian distances from 2 to 2, and draw the line 1, 2 for the first distance line: from 2 upon the first meridian set the northing of the second stationary line, that is, 9.65, to 3, and on the point 3 erect a perpendicular eastward, upon which set the meridian distance of the second station 16.82, from 3 to 3, and draw the line 2, 3, for the distance line of the second station. And since the third station has neither northing nor southing, set the meridian distance of it 33.02, from 3 to 4, for the distance line of the third station. To the fourth station there is 29.44 southing, which set from 3 to 5; upon the point 5 erect the perpendicular 5, 5; on which lay 13.54, and draw the line 4 to 5. In the like manner proceed to set the northings and southings on the first meridian, and the meridian distances upon the perpendiculars raised to the east or west; the extremities of which connected by right lines will complete the map. The foregoing Field-book, Method II. It is needless here to insert the columns of bearing or distances in chains, they being the same as before. A Specimen of the Pennsylvania Method of CALCULATION; which for its simplicity and ease in finding the Me-. ridian Distances is supposed to be preferable in practice to any thing heretofore published on the subject. Find, in the first place, by the Traverse Table, the lat. and dep. for the several courses and distances, as already taught; and if the survey be truly taken, the sums of the northings and southings will be equal, and also those of the eastings and westings. Then, in the next place, find the meridian distances, by choosing such a place in the column of eastings or westings as will admit of a continual addition of one, and subtraction of the other; by which means we avoid the inconvenience of changing the denomination of either of the departures. The learner must not expect that in real practice the columns of lat. and those of dep. will exactly balance when they are at first added up, for little inaccuracies will arise, both from the observations taken in the field and in chaining; which to adjust, previous to finding the meridian distances, we may observe, that if in small surveys the difference amount to two-tenths of a perch for every station, there must have been some error committed in the field; and the best way in this case will be, to rectify it on the ground by a resurvey, or at least as much as will discover the error.- But when the differences are not within those limits, the columns of northing, southing, easting, and westing may be corrected as follows: Add all the distances into one sum, and say, as that sum is to each particular distance, so is the difference between the sums of the columns of northing and southing to the correction of northing or southing belonging to that distance: the corrections thus found are respectively additive when they belong to the column of northing or southing which is the less of the two, and subtractive when they belong to the greater; if the course be due east or west, the correction is always additive to the less of the two columns of northing or southing. The corrections of easting and westing are found exactly in the same manner. The following example will sufficiently illustrate the manner of applying the rule. also the first distance is 70; say then, 791 70.4: .04, ference between the columns of northing and southing is .4, In this example, the sum of the distances is 791, and the dif to the southing 53.6, from which the correction .04 should be which fourth proportional .04 is the first correction belonging |