And the latitude of any

multiplied by the length of the course. course is equal to the cosine of the bearing of the course into its length. If the forward bearing of a course is northward, its latitude is called a northing, and is given a + sign in the com putations (see bB, cC, and Ed above); if the forward bearing is southward, the latitude is called a southing, and is given a sign (see Aa and eA above). Having a table of natural cosines, find in it the cosine of the bearing of the course and multiply it by the length of the course and the product will be the latitude of the course, to which give the + or sign according as it is a northing or southing; or having tables of logarithmic cosines and logarithms of numbers, find the logarithm of the cosine of the bearing and the logarithm of the length of the course, add them together and look in the table of logarithms for the number corresponding to the sum, which will be the latitude of the course.

A due east-and-west course has no difference of latitude. Likewise the surveyor in passing from B to C departs a distance east from B equal to 6C. This distance bC is called the departure of the course BC, and is equal to the sine of the bearing of the course multiplied by the length of the course, as the departure of any course is equal to the sine of its bearing into its length.

If the forward bearing of a course is east, its departure is called an easting and is given a + sign; if west, its departure is called a westing and is given a → sign. Having a table of natural sines, the value of the sine for the bearing of any course can be taken out, and multiplied by the length of the course will give the departure of that course, to which the proper sign must be given; or with tables of logarithmic sines and logarithms of numbers, the departures may be found similarly to the method of determining latitudes. A due north-and-south course has no departure. The meridian distance of B is the distance aB; the meridian distance of Cis the distance g C; and so the meridian distance of any point is its perpendicular distance from the reference meridian NS. The meridian distance

of a point, as A, on the reference meridian is 0. The meridian distance of the course BC is the perpendicular distance of its middle point x from the reference meridian NS, midway between a and g, and the meridian distance of any course is the meridian distance of its middle point.

The double meridian distance (D. M. D.) of any course is, therefore, equal to twice its meridian distance, or equal to the sum of the meridian distances of its two extreme points.

The D. M. D. of the course AB is, therefore, equal to aB, which is also its departure; and of EA is equal to eE, which is its departure; and so the D. M. D. of any course, one extremity of which is on the reference meridian, is equal to its departure.

Calling AB the first course in the computation of the area, and BC the second, and so on around, it is seen that the D. M. D. of the first course, AB, is equal to its departure aB. That the D. M. D. of the second course, BC, [equal to aB+(gb +b)] is equal to the D. M. D. of the first course (aB), plus the departure of the first course (aB=gb), plus the departure of the course itself (bC). That the D. M. D. of the second or any other course (except the two as AB and EA meeting on the reference meridian) is equal to the D. M. D. of the preceding course, plus the departure of that course, plus the departure of the course itself. Of CD it is equal gC+fD=(gC+aB)+ bC+cD, since (aB=ƒk)+(bC=kc)+cD=fD›

Therefore, comparing the quantities, in the equation beginning 2ABCDEA=, with those in the preceding explanations, it will be seen that twice the area of the field is equal to the algebraic sum of the products of the double meridian distance if each course into its own latitude, being careful to observe the sigus. In stating that north latitudes and east departures are positive and south latitudes and west departures are negative, these terms are used in their algebraic sense, so that if a positive D. M. D. and a negative departure are added together their algebraic sum will be the numerical difference, with the sign of the greater. And so the algebraic sum of the products

above will be the numerical difference between the sum of the positive products and the sum of the negative products.

A check on the correctness of the work is, that the D. M. D. of the last course should be equal to its departure.

Had the surveyor gone around the field with it on his right instead of on his left, the signs of the latitudes and departures would be changed, but the numerical values would be the same.

HOW TO COMPUTE THE AREA.—Understanding the above, the surveyor can now proceed to compute his area from his field notes as follows:

Rule a FORM B and enter therein the Stations, as given by the letters on the plot, Fig. 217, and the Bearings and Dis tances of the courses; then with a table of natural cosines and sines proceed to find the latitudes and departures of each course as explained, and enter them in the FORM A, north latitudes in the + column, and south latitudes in the-column; and east departures in the + column, and west departures in the column. Thus, for the first course AB, bearing S 41° 45' E, length 5.06 chains.

The course being southward, the lat. is negative; hence enter 3.78 in the lat. column of FORM A.

The course being eastward, the dep. is positive; hence enter 3.37 in the + dep. column of FORM B. Proceed in the same manner for all the other courses.

If the bearings of the courses are read only to the nearest 15' of arc, then the work of computing the latitudes and departures may be much simplified by using the Traverse Table, in which is given the latitude and departure corresponding to bearings that are expressed in degrees and quarters of a degree, from 0° to 90°, and for every length of course

from 1 to 100 computed to two places of decimals. The latitudes and departures so tabulated were obtained by using the formulas and method just explained; viz., lat.=course X cos. of bearing; dep.=course X sine of bearing. When the courses are longer than 100, as 175, take out the lat. and dep. for 100 and then for 75 and add them together. If the length of a course is expressed decimally, as 5.06, take out the lat. and dep. for 5, then for 6 and move the decimal point, in the latter, two places to the left and add to those for 5.

As the compass can be read to closer than 15', and if the mean is taken when fore and back sights differ only slightly, the results will seldom be expressed in quarter degrees, and when the transit is used, the azimuths will be read to the nearest minute, and the Traverse Table could not then be used in all cases.

If using logarithmic tables, rule a FORM A, enter the courses, at tops of columns noting whether northing or southing for latitudes, and whether easting or westing for departures, with their proper signs; then from the log. tables take the log. cos, and log. sin. of the first course and the log. of its length and enter them in FORM A as shown*. Do this for each course. Then add each pair of logarithms for the log. lat. and log. dep., and enter the table of logarithms of natural numbers and find the number corresponding to each log. sum and write it under it. Having found them all, enter them in their proper columns opposite their respective courses in FORM B.