of his method of doing this, and of his arrangement of the data and the results in a simple, clear, and convenient form, belongs to special works on surveying. This article merely gives some examples which can be solved without any knowledge of professional details. The various rules for finding the area, of a triangle and a trapezoid, are supposed to be known. In solving these problems, the student should make the plotting or mapping an important feature of his work.

The Gunter's chain is generally used in measuring land. It is 4 rods or 66 feet in length, and is divided into 100 links.

An acre = 10 square chains 4 roods=160 square rods or poles. The points of the compass have been explained in Art. 30.

EXAMPLES.

1. A surveyor starting from a point A runs S. 70° E. 20 chains, thence N. 10° W. 20 chains, thence N. 70° W. 10 chains, thence S. 20° W. 17.32 chains

N

D1

C1

D

H

to the place of beginning. What is the area of the field which he has gone around?

Make a plot or map of the field, namely, ABCD. Here, AB represents 20 chains, and the bearing of B from A is S. 70° E. BC represents 20 chains, and the bearing C from B is N. 10° W., and so on. Through the most westerly point of the field draw a north-and-south line. This line is called the meridian. In the case of each line measured, find the distance that one end of the line is east or west from the other end. This easting or westing is called the departure of the line. Also find the distance that one Bend of the line is north or south of the other end. This northing or southing is called the latitude of the line. For example, in Fig. 30 b, the departures of AB, BC, CD, DA, are B1B, BL, CH, DD1, respectively; the latitudes of the boundary lines are AB1, B1C1, C1D1, D1A, respectively. It should be observed (Art. 36) that the algebraic sum of the departures of the boundary lines is zero, and so also is the algebraic sum of their latitudes. The following formulas are easily deduced :

B

FIG. 30b.

L

By means of the departures, the meridian distance of a point (i.e. its distance from the north-and-south line) can be found. Thus the meridian

=

Hence in Fig. 30 b, AB1, B1B,

distance of C is C1C, and C1C D1D+ HC. B1C1, CIC, C1D1, D1D can be computed. Now

area ABCD

=

trapezoid D1DCC1 + trapezoid C1CBB1 — triangle ADDı - triangle ABB1.

The areas in the second member can be computed; it will be found that area ABCD: 26 acres.

=

NOTE. Sometimes the bearing and length of one of the lines enclosing the area is also required. These can be computed by means of the latitudes and departures of the given lines. The formulation of a simple rule for doing this is left as an exercise to the student.

2. In Ex. 1, deduce the length and bearing of DA from the lengths and bearings of AB, BC, CD.

3. A surveyor starts from A and runs 4 chains S. 45° E. to B, thence 5 chains E. to C, thence 6 chains N. 40° E. to D. Find the distance and bearing of A from D; also, the arca of the field ABCD. Verify the results by going around the field in the reverse direction, and calculating the length and bearing of BA from the lengths and directions of AD, DC, CB.

4. A surveyor starts from one corner of a pentagonal field, and runs N. 25° E. 433 ft., thence N. 76° 55′ E. 191 ft., thence S. 6° 41′ W. 539 ft., thence S. 25° W. 40 ft., thence N. 65° W. 320 ft. Find the area of the field. Deduce the length and direction of one of the sides from the lengths and directions of the other four.

5. From a station within a hexagonal field the distances of each of its corners were measured, and also their bearings; required its plan and area, the distances in chains and the bearings of the corners being as follows: 7.08 N.E., 9.57 N. E., 7.83 N. W. by W., 8.25 S. W. by S., 4.06 S.S.E. 7° E., 5.89 E. by S. 310 E.

35. Summary. Chapter II. was concerned with defining and investigating certain ratios inseparably connected with (acute) angles, and attention was directed to the tables of these ratios and their logarithms. In Chap. III. it was shown how these definitions and tables can be used in finding parts of a rightangled triangle when certain parts are known. In Chap. IV. the knowledge gained in Chap. III. was employed in the solution of some of the many problems in which right-angled triangles appear. In Art. 34 it has been seen that this knowledge can serve for the solution of oblique triangles. It follows, then, that it can serve for the solution of problems in which oblique triangles appear, and, accordingly, for the solution of all problems

involving the measurement of straight lines only. Consequently, the student is now able, without any additional knowledge of trigonometry, to solve the numerical problems in Chaps. VII., VIII. It is thus apparent that even a slight acquaintance with the ratios defined in Chap. II. has greatly increased the learner's ability to solve useful practical problems.

Oblique triangles can sometimes be solved in a more elegant manner than that pointed out in Art. 34. In order to show this, further consideration of angles and the trigonometric ratios is necessary. Consequently, in Chap. V. some important additions are made to the idea of a straight line and the idea of an angle; the trigonometric ratios are defined in a more general way, namely, for all angles, instead of for acute angles only, and the principal relations of these ratios are deduced. Chapter VI. treats of the ratios of two angles in combination. While it is necessary to consider these matters before proceeding to the solution of oblique triangles given in Chap. VII., it should be said that the knowledge that will be gained in Chaps. V., VI., VII., is necessary and important for other purposes besides the solution of triangles. In fact, the latter is one of the least important of the results obtained in these chapters.

N.B. Questions and exercises suitable for practice and review on the subject-matter of Chap. IV. will be found at page 184.

CHAPTER V.

TRIGONOMETRIC RATIOS OF ANGLES IN GENERAL.

a line unlimited in length in Suppose that a point starts at

36. Directed lines. Let MN be the directions of both M and N. P and moves along this line for some given distance. In order to mark where the point stops, it is necessary to know, not only this distance, but also the direction in which the point has moved from P. This direction may be indicated in various ways; by saying, for instance, that the point moves toward the right from P, or toward the left from P; that the point moves toward N, or toward M; that the point moves in the direction of N, or in the direction of M; and so on. Mathematicians, engineers, and others

have agreed to use a particular method (and this practically comes to the adoption of a particular rule) for indicating the two opposite directions in which a point can move along a line, or in which distances along a line can be measured. This convention, or rule which has been adopted for the sake of convenience, is as follows:

Distances measured along a line, or along parallel lines, in one direction shall be called positive distances, and shall be denoted by the sign+; distances measured in the opposite direction shall be called negative distances, and shall be denoted by the sign

The convenience of this custom, fashion, or rule, will become apparent in the examples that follow. In Fig. 31 let distances

* Advances in mathematics have often depended upon the introduction of a good custom which has at last been universally adopted and made a rule. Thus, for example, the custom of using exponents to show the power to which a quantity is raised, which was first introduced in the first half of the sixteenth century, and made gradual progress until its final establishment in the latter half of the seventeenth century, has been of great service in aiding the advances of algebra.

measured in the direction of N be taken positively; then distances measured in the direction of M will be taken negatively. On directed lines the direction in which a line is measured, or in which a point moves on a line, is indicated by the order of the letters naming the line. Thus, for example, if a point moves from B to C, the distance passed over is read BC. In this reading, the starting point is indicated by the first letter B, and the stopping point, by the last letter C. After the same fashion, CB means the distance from C to B. If, for instance, there are 3 units of length between B and C, then BC

=

+ 3, CB = − 3.

EXAMPLES.

1. Suppose a point (Fig. 31) moves from P to B, thence to C, thence to D, thence to F. Let the number of units of length between P and B, B and C, C and D, F and D, be 2, 3, 2, 10, respectively. The point starts at P and stops at F; hence the distance from the starting point to the stopping point is PF. In this case the point's trip from P to F is made in several steps as indicated above. That is, on properly indicating the lines passed over,

- 10.]

= 2 + 3 + 2 − 10 [·.· FD=+10, then DF =
==

== 3.

This shows that the final position of the moving point is three units to the left of P. This example also shows one great convenience of the rule of signs in measurement, namely, that by attending to this rule and to the proper naming of the lines passed over by a moving point, one immediately obtains the result of the successive movements.

NOTE. In the following examples, in lines that lie east and west, let measurements toward the east be taken positively; in lines that lie north and south, let measurements toward the north be taken positively.

2. A man travelling on an east and west line goes east 20 mi., then east 16 mi., then west 18 mi., then east 30 mi. What is his final distance from the starting point? [Draw a figure, and indicate the successive trips by letters.]

3. A man travelling on an east and west line goes west 20 mi., then east 10 mi., then east 25 mi., then east 30 mi., then west 45 mi. Do as in Ex. 2.

4. A man travelling on a north and south line goes north 100 mi., then south 60 mi., then south 110 mi., then north 200 mi., then north 15 mi., then south 247 mi. Do as in Ex. 2.