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100,

3, for ac (0+) read ac sin ( + 0). 102, Fig. 2, for C read D; and for D read C. 103, Line 5 from bottom, for ccos ß read ccos y.

PREFACE.

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.

There are many other parts connected with Spherics that might be treated of, but which are not adapted to a Rudimentary Treatise like the present; those, however, who wish to see all the higher departments fully developed, must consult the writings of that distinguished mathematician, Professor Davies, of the Royal Military Academy, Woolwich.

Hutton's Course, the Ladies' and Gentleman's Diaries, (latterly comprised in one), Leybourne's Repository, the Mechanics' Magazine, and various other periodicals, teem with the productions of his fertile mind, both on this and other kindred subjects.

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SPHERIC S.

PRELIMINARY CHAPTER.

1. 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 O be the centre of the sphere, A PBA a section made by a plane passing through it, draw F OC to the cutting plane, and produce it both ways to D and E, and draw the radii of the sphere OA,

OP.

Now, since OCP and OCA are right angles, OA2. OC2 = AC2, and OP2 O C2 =

B

N

D

A

G

M

E

PC2, but OA2 = 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

Since by Euclid XI. 20, any two of the plane angles, which form the solid angle at O, are together greater than the third, hence any two of the arcs which measure those angles must be greater than the third.

6. Since the solid angle at O (see fig. p. 3) is contained by three plane angles, and by Euclid XI. 21, these are together less than four right angles, hence the three arcs of the spherical triangle which measure those angles must be together less than the circumference of a great circle, that is a + b + c > 360, and since any two sides of a triangle is greater than the third, we have a + b > c; b + c > a; a + c > b.

ON THE POLAR OR SUPPLEMENTAL TRIANGLE.

7. If three arcs of great circles be described from the angular points A, B, C, of any spherical triangle A B C, as poles, the sides and angles of the new triangle, D F E, so formed will be the supplements of the opposite angles and sides of the other, and vice versa.

Since B is the pole of D F, then B D is a quadrant, and since C is the pole of D E, C D is a quadrant; therefore the distances of the points B and C from D being each a quadrant, they are equal to each other, hence D is the pole of B C.

DE=180°-C; EF=180°—A;

FD=180°-B; and D=180°-BC;

E=180°-AC; F=180°-AB.

Also, AB=180°-F; BC=180°—D;

AC=180°-E; A=180-FE;

B=180° — FD; C=180° — DE.

F

A

B

E

The sum of the three angles of a spherical triangle is greater than two right angles, and less than six right angles. For if a+b+c' be the sides of the supplemental or polar triangle, A=180°-a'; B=180°-b'; c=180°-c';

hence A+B+C+a' + b′+c′=6× 90=6 right angles; but a' + b' + c' is less than four right angles, by Euclid XI. 21; therefore A+B+C is greater than two right angles; and as the sides a', b', c', of the polar triangle must have some magnitude, the sum of the three angles A, B, C must be less than six right angles.

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