THE ELEMENTS OF SPHERICAL TRIGONOMETRY. BY JAMES HANN, gi ...

THE ELEMENTS OF

SPHERICAL TRIGONOMETRY.

BY JAMES HANN, J,\TE MATHEMATICAL MASTER OF KING'S COLLEGE 8CHOOL, LONDON.

gi |Ufo ^bition, mrb Corwcttb, BY CHAELES H. BOWLING, C.E.,

LATE OF TBINITY COLLEGE, DUBLIN.

LONDON: v VIRTUE BROPTAHTEEIRtNSOS&TECItOH.,OW26. ,' IVY LA' NE,

18GG.

PREFACE TO THE

SECOND EDITION.

The first edition of Mr. Hann's " Elements of Spherical Trigonometry" has been carefully revised, and corrected, and several practical problems from the prize questions of different universities have been added to the work. The author states, in his preface to the former edition:--

"In the compilation of this work, the most esteemed writers, both English and foreign, have been consulted, but those most used are De "Fourcy and Legendre.

"Napier's ' Circular Parts' have been treated in a manner somewhat different to most modern .writers. The terms conjunct and adjunct, used by- Kelly and others, are here retained, as they appear to be more conformable to the practical views of Napier himself."

This edition is recommended with increased confidence to the judgment of all mathematical teachers and students, especially naval instructors and students of naval colleges.

0. H. DOWLINGr, C.E.

p CONTENTS.

Pag?

Definitions

1

Polar Triangle

2

Fundamental Formula

4

Relations between the Sides and Angles of Spherical Triangles . . 5, 7

Napier's Analogies

8

Right-angled Triangles

10

Napier's Circular Parts

11

Solution of Oblique-angled Spherical Triangles

18

Ambiguous Cases of Spherical Triangles . . . ' 23

Table of Results from the Ambiguous Cases

25

To reduce an Angle to the Horizon

26

Numerical Solution of Right-angled Spherical Triangles

28

Quadrantal Triangles

38

Oblique-angled Triangles

35

Area of a Spherical Triangle .

44, 48

Girard's Theorem

51

Legendre's Theorem

52, 55

Solidity of a Parallelepiped

55

LexelFs Theorem

57, 59

Polyhedrons

61, 67

Examples and Problems

67

SPHEEIGS. r.

PRELIMINARY CHAPTER.

]. A sphere is a solid determined by a surface of which all

the points are equally distant from an interior point, which is

called the centre of the sphere.

2. Every section of a sphere

made by a plane cutting it, is the

arc of a circle. Let 0 be the centre of the

c----

sphere, A P B A a section made by

a plane passing through it, draw

1

00 to the cutting plane, and pro? duce it both ways to D and E, and

V N M

draw the radii of the sphere 0 A,

OP. '

Now, since OOP and OCA are right anales, O A2 -- 0 C2 = A C2. and OP2-OC2= PC2, but 0A2= OP2;

AC2 = PC2

or AC = PC; hence the section APBA is a circle.

If the cutting plane pass through the centre, the radius of

the section is evidently equal to the radius of the sphere, and

such a section is called a great circle of the sphere.

3. The poles of any circle are the two extremities of that

diameter or axis of the sphere which is perpendicular to the

plane of that circle ; and therefore either pole of any circle is

equidistant from every part of its circumference, and, if it be

a great circle, its pole is 90? from the circumference. A

spherical triangle is the portion of space comprised between

three arcs of intersecting great circles.

4. The angles of a spherical triangle are those on the surface

of the sphere contained by the arcs of the great circles which

form the sides, and are the same as the inclinations of the

planes of those great circles to one another.

5. Any two sides of a spherical triangle are greater than the

third side.

B

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download