is the area of the figure dfohikd, which is equal to the area of the map. Let bou=Y, urih=L, ric=0, wrc=Z, akw=K, efb=B, and ade=A. I say that Y+Z+B=K+L+A. Y=L+0; add Z to both, then Y+Z=L+0+Z: but Z +0=K, put K instead of Z+O, then Y+Z=L+K; add to both sides the equal triangles B and A, then Y+Z+B=L +K+A. If therefore B+Y+Z be taken from abc, and in lieu thereof we put L+K+A, we shall have the figure dfohikd =abc; but that figure is made up of the meridian distance when east multiplied into the southing, and the meridian distance when west multiplied into the northing less by the meridian distance when west multiplied into the southing. Q. E. D. COROLLARY. Since the meridian distance when west multiplied into the southing is to be subtracted, by the same reasoning the meridian distance when east multiplied into the northing must be also subtracted. SCHOLIUM. From the two preceding theorems we learn how to find the area of the map when the first meridian passes through it; that is, when one part of the map lies on the east and the other on the west side of that meridian. Thus, The merid. east RULE. dist. when west S multiplied into the southings, their sum is the area of the map. The inerid. east dist. when west But, northings, multiplied into the northings, the sum southings, of these products taken from the former gives the area of the map. These theorems are true when the surveyor keeps the land he surveys on his right-hand, which we suppose through the whole to be done; but if he goes the contrary way, call the southings northings and the northings southings, and the same rule will hold good. General Rule for finding the Meridian Distances. 1. The meridian distance and departure both east or both west, their sum is the meridian distance of the same name. 2. The meridian distance and departure of different names, that is, one east and the other west, their difference is the meridian distance of the same name with the greater. Thus, in the first method of finding the area, as in the following field-book, name. The first departure is put opposite the northing or southing of the first station, and is the first meridian distance of the same Thus, if the first departure be east, the first meridian distance will be the same as the departure, and east also, and if west it will be the same way. In the 5th and 11th stations, the meridian distance being less than the departures and of a contrary name, the map will cross the first meridian, and will pass, as in the 5th line, from the east to the west line of the meridian; and in the 11th line it will again cross from the west to the east side, which will evidently appear if the field-work be protracted, and the meridian line passing through the first station be drawn through the map. The field-book cast up by the first method will be evident 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. |