Elements of trigonometry, plane and spherical. Adapted to the present state of analysis. To which is added, their application to the principles of Navigation and nautical astronomy. With logarithmical, trigonometrical, and nautical tables

1838

PART	1

Data by which triangles are determined	2

Geometrical construction of triangles	3

Scales of equal parts	4

MEASUREMENT OF ANGLES 5 By the sexagesimal division of the circumference	5

By the centesimal division	6

Engineers method	7

Protractors	8

18	109

Astronomical example	112

20	113

APPLICATION OF PLANE AND SPHERICAL TRIGONOMETRY	123

PRINCIPLES OF NAVIGATION ARTICLE Page 96 Definitions	125

Plane sailing	129

Traverse sailing	132

Parallel sailing	138

Use of the protractor	9

SOLUTIONS BY CONSTRUCTION 10 Finding the distance of inaccessible objects 11 Measurement of heights	11

General view of Analytical Trigonometry	12

Their use	13

THE SINE 14 Its definition	14

Sine of the supplement	15

Variations in the value of the sine	16

Sines of other quadrants derived from the first	17

Algebraic sign of the sine THE TANGENT	18

Its definition and changes of value	19

Recapitulation of the values of the tangent THE SECANT	20

Definition and variations of value of the secant	21

Algebraic sign of the secant	22

Definition of the complement of an arc or angle	23

Its definition	24

Its Algebraic sign	25

Its variations of value	26

Negative arcs THE COSINE	27

Values of the cosecant	29

ARTICLE	31

Corollaries from these	36

Advantage of logarithms	42

Rule to find the logarithm of any number between 1 and 10000	49

Rule to find from the table the logarithmic sine tangent c	56

Use of the arithmetical complement	62

and 70 Derivation of formulæ for the sine and cosine of the	69

Derivation of a formula for the sine of half an angle in terms of	73

Definitions of celestial circles	79

14	80

15	84

Three angles being given to find the sides	85

16	93

17	101

RIGHT ANGLED TRIANGLES	105

Middle latitude sailing	141

Mercators sailing	145

Example of the last	147

NAUTICAL ASTRONOMY 103 Introductory	149

Definitions	150

Corrections to be applied to the observed altitudes of celestial objects	152

Dip or depression of the horizon	153

Refraction	155

Parallax	157

Examples of corrections	158

Method of determining the latitude at sea by the meridian altitude	161

By two altitudes	166

On finding the longitude by the lunar observations	171

Variation of the compass	177

PART V	180

Value of the sine and cosine of 45	181

Sine of 30 and secant of 60	182

Construction of table of sines c	183

PART VI	185

Introduction	189

Quadrantal triangles	195

Formula to be employed instead of Napiers rules in certain cases where great accuracy is required	199

Two sides and the included angle of a spherical triangle being given to find the third side directly	201

Two angles and the interjacent side being given to find the third angle	202

Rules relative to ambiguous cases	203

Additional formula where three sides or three angles of a spherical triangle are given	204

Determination of the effect of minute errors in data	205

222	3

20	4

72	5

21	7

2233	9

67	11

Title	Elements of trigonometry, plane and spherical. Adapted to the present state of analysis. To which is added, their application to the principles of Navigation and nautical astronomy. With logarithmical, trigonometrical, and nautical tables
Author	Charles W. HACKLEY
Published	1838
Original from	The British Library
Digitized	7 Sep 2015

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