when two sides and the angle opposite to one of them are given, 66;
Case III., when two sides and the included angle are given, 71; Case
IV., when the three sides are given, 74; area of a triangle, 78-79.
CHAPTER V. MISCELLANEOUS EXAMPLES :
Plane Trigonometry, 82-99; goniometry, 99–105.
EXAMINATION PAPERS, 106-116.
CHAPTER VI. CONSTRUCTION OF TABLES :
Logarithms, 117; exponential and logarithmic series, 120; trigo-
nometric functions of small angles, 125; Simpson's method of con-
structing a trigonometric table, 127; De Moivre's theorem, 128;
expansion of sin x, cos x, and tan x, in infinite series, 132.
CHAPTER VII. THE RIGHT SPHERICAL TRIANGLE:
Introduction, 135; formulas relating to right spherical triangles,
137; Napier's rules, 141. Solutions: Case I., when the two legs are
given, 142; Case II., when the hypotenuse and a leg are given, 142;
Case III., when a leg and the opposite angle are given, 143; Case IV.,
when a leg and an adjacent angle are given, 143; Case V., when the
hypotenuse and an oblique angle are given, 144; Case VI., when the
two oblique angles are given, 144. The isosceles spherical triangle, 149.
CHAPTER VIII. THE OBLIQUE SPHERICAL TRIANGLE :
Fundamental formulas, 150; formulas for half angles and sides,
152; Gauss's equations and Napier's analogies, 154. Solutions: Case
I., when two sides and the included angle are given, 156; Case II.,
when two angles and the included side are given, 158; Case III., when
two sides and an angle opposite to one of them are given, 160; Case
IV., when two angles and a side opposite to one of them are given,
162; Case V., when the three sides are given, 163; Case VI., when
the three angles are given, 164. Area of a spherical triangle, 166.
CHAPTER I. DEFINITIONS. INSTRUMENTS AND THEIR USES:
Definitions, 135; instruments for measuring lines, 136; chaining,
136; obstacles to chaining, 138; the surveyor's compass, 141; uses
of the compass, 143; verniers, 145; the surveyor's transit, 149; uses
of the transit, 150; the theodolite, 150; the railroad compass, 150;
plotting, 153.
CHAPTER II. LAND SURVEYING:
Determination of areas, 155; rectangular surveying, 159; field
notes, computation, and plotting, 160; supplying omissions, 164;
irregular boundaries, 164; obstructions, 164; modification of the
rectangular method, 167; variation of the needle, 168; methods of
establishing a true meridian, 170; dividing land, 173; United States
public lands, 176; Burt's solar compass, 177; laying out the public
lands, 179; Plane-table surveying, 181; the three-point problem, 186.
Definitions, 190; the Y level, 191; the levelling-rod, 191; differ-
ence of level, 192; levelling for section, 195; substitutes for the
Y level, 198; topographical levelling, 200.
TRIGONOMETRIC FUNCTIONS OF ACUTE
As lengths are measured in terms of various conventional units, as the foot, meter, etc., so different units for measuring angles are employed, or have been proposed.
In the common or sexagesimal system the circumference of a circle is divided into 360 equal parts. The angle at the centre subtended by each of these parts is taken as the unit angle and is called a degree. The degree is subdivided into 60 minutes, and the minute into 60 seconds. A right angle is equal to 90 degrees.
NOTE. The sexagesimal system was invented by the early Babylonian astronomers in conformity with their year of 360 days.
In the circular system an arc of a circle is laid off equal in length to the radius. The angle at the centre subtended by this arc is taken as the unit angle and is called a radian.
The number of radians in 360° is equal to the number of times the length of the radius is contained in the circumference. It is proved in Geometry that this number is 2T (T=3.1416) for all circles; therefore the radian is the same angle in all circles.
Since the circumference of a circle is 2π times the radius,
2π radians=360°, and π radians = 180°;
By the last two equations the measure of an angle can be changed from radians to degrees or from degrees to radians.
Thus, 2 radians = 2 × = 2 × (57° 17' 45")=114° 35' 30".
The circular system came into use early in the last century. It is found more convenient in the higher mathematics, where the radians are simply expressed as numbers. Thus the angle means π radians, π and the angle 3 means 3 radians.
On the introduction of the metric system of weights and measures at the close of the last century, it was proposed to divide the right angle into 100 equal parts called grades, which were to be taken as units. The grade was subdivided into 100 minutes and the minute into 100 seconds. This French or centesimal system, however, never came into actual use.
1. Reduce the following angles to circular measure, express
ing the results as fractions of π. 123° 45', 37° 30'.
60°, 45°, 150°, 195°, 11° 15',
3. What decimal part of a radian is 1°? 1'?
4. How many seconds in a radian ?
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