PART II.

SPHERICAL

TRIGONOMETRY

AND

PRACTICAL ASTRONOMY.

77. A spherical triangle is formed by three arcs of great circles. The planes of these arcs produced, form a trihedral angle, the vertex of which is at the centre of the sphere. (Spherical Geom. Prop. 7.)

The angle which two planes make with each other, is called a diedral angle. This angle is equal to the angle formed by two lines, drawn one in each plane and perpendicular to the common intersection of the two planes at the same point. (Geom. of Planes, Def. 6.)

The angles of a spherical triangle are the angles formed by the planes of the arcs. (Spher. Geom., Def. 7.) These are the diedral angles of the trihedral angle mentioned above.

The sides of the spherical triangle are the arcs of the great circles, by which it is bounded. The arcs subtend the plane angles of the trihedral angle, and consequently measure them. The arcs are given in degrees, and since they contain the same number as the plane angles which they subtend, these plane angles may be employed in a demonstration, instead of the arcs, or sides of the spherical triangle; and for a like reason, the diedral angles of the trihedral angle may be employed instead of the angles of the spherical triangle.

If we suppose the trihedral angle, which has its vertex at the centre of a small sphere, to be produced so as to cut out a triangle upon a larger concentric sphere, the sides of the triangle upon the larger sphere, con

* Having the same centre.

taining respectively the same number of degrees as the plane angles of the trihedral angle, will contain the same number of degrees respectively as the sides of the spherical triangle cut out by the trihedral angle on the smaller sphere. So that as the number of degrees in the angles and sides which are given or required, and not their absolute length, is taken into consideration in the solution of spherical triangles, the size of the sphere need not be regarded.

78. Let ABC be a spherical triangle right angled at A. Let o be the centre of the sphere, and let the planes of the arcs which are the sides of the triangle be produced so as to form the trihedral angle whose vertex is at o.

The plane angle coв will contain the same number of degrees as the side a of the spherical triangle; the plane angle coa the same number as the side b; and AOB the same num

ber as c; so that these plane angles may be marked a, b, and c, as in the figure.

It has already been mentioned that the diedral angles of the trihedral angle correspond in the same manner to the angles of the spherical triangle; and that these diedral angles are measured by the angle of two lines, drawn one in each plane, perpendicular to the common intersection of the two planes at the same point. In order to draw these lines so as to be used most conveniently in the following demonstration, take o м= the radius of the tables; draw м P perpendicular to o A, it will be perpendicular to the plane A O B (Geom. of Planes, Prop. 19), since the two planes A O B and a o c are perpendicular to each other, a being by hypothesis a right angle; from P draw P D perpendicular to o B, a line of the plane A O B; join D м; M D will be perpendicular to o в (Geom. of Planes, prop. 8); M D and D P being both perpendicular to o B at the same point D, the angle M D P is the diedral angle of the planes A O B and B o c; or MDP the angle в of the spherical triangle; o м being equal to radius, MD is the sine of the plane angle a, and M P is the sine of the plane angle b; in the triangle M D P, right angled at P, we have the proportion (Art. 38)

R: sin DMD: MP

substituting for D its equal B, for M D, its value sin a, and for м P, its value sin b, we have

R sin B: sin a : sin b

That is, the radius is to the sine of either of the oblique angles of a right angled spherical triangle as the sine of the hypothenuse is to the sine of the side opposite that angle.

79. The solution of astronomical problems forms one of the most useful and agreeable applications of the theory of spherical trigonometry, which branch of mathematics has grown out of the wants of Astronomy.

To illustrate therefore the above and subsequent formulas of spherical trigonometry we shall introduce a few great circles of the celestial sphere. They are so well known that to define them is perhaps superfluous.

The equator is that great circle the plane of which is perpendicular to the axis of the earth. The axis being the line about which the earth performs its diurnal rotation. This produced to the celestial sphere becomes the axis of the heavens about which all the stars appear to revolve daily.

The ecliptic is a great circle which makes an angle of about 23° 28′ with the equator. It is the path which the sun appears to describe among the stars once a year.

The points in which the two great circles above defined intersect are called equinoctial points.

The one at which the sun crosses the equator in the spring about the 21st of March, is called the vernal equinox.

The other, which is where the sun crosses in the autumn, viz. about the 23d of September, is called the autumnal equinox.

Declination circles are great circles, the planes of which pass through the axis and the circumferences of which all intersect in the poles or points where the axis meets the surface of the celestial sphere. They are also called hour circles. The sun appears to move about the earth once in 24 360°

hours; = 15° is the number of degrees through which the sun

24

moves in an hour.

That declination circle, the plane of which passes through any place on the surface of the earth and the earth's centre, is called the meridian of the place.

The angle contained between the meridian of a place and that declination circle which passes through the sun at any given moment, is called the hour angle of the sun, and converted into hours, 15° to the hour, will

show the time of day, if we reckon from noon instead of midnight as astronomers do.

This time may be either A. M. or P. M. It is what is called apparent time, which varies a little from mean time, the time given by the clocks, in consequence of the slightly unequal motion of the sun in its annual

revolution.

The hour angle of a star is similar to that of the sun.

The horizon of any place is a great circle whose plane touches the surface of the earth at that place, and extends to the celestial sphere. This is called the sensible horizon; the real horizon is a plane parallel to this through the centre of the earth. When any of the fixed stars are in question, the distances of which from the earth are so great that its radius is as nothing comparatively, these two horizons may be regarded as coincident. The zenith is the pole of the horizon directly overhead. The nadir is the opposite pole.

Great circles passing through the zenith and nadir are called vertical circles. They are secondaries to the horizon.

The position of a heavenly body is fixed on the celestial sphere, like that of a place on the globe, by its latitude and longitude, only it must be observed that on the former these are measured from and upon the ecliptic instead of the equator.

Similar measurements from and upon the celestial equator are called the declination and the right ascension, the former corresponding to the latitude, and the latter to the longitude.*

Longitude upon the earth is reckoned from some fixed meridian, as that of Greenwich.

Longitude upon the celestial sphere is reckoned from the vernal equinox which is called the first of Aries; right ascension also from the same point; the former upon the ecliptic, the latter upon the equator.

The azimuth of a celestial object is an arc of the horizon, comprehended between the meridian of the observer and the vertical circle which passes through the object.

Or it is the angle which these two vertical circles make with each other having its vertex at the zenith.

80. We are now prepared with materials for a practical application of the formulas of spherical trigonometry, and we commence with that already demonstrated.

The symbol for right ascension is AR or R. A.; for declination D, or Dec.

S

E

Let E in the annexed diagram be the equinoxial point, EQ a portion of the equator, Es a portion of the ecliptic, s the place of the sun, and so a portion of a dec. circle through the sun; then

sq will be the 's declination, which de

note by d, EQ his right ascension, which denote by a, and Es his longitude, which denote by 7.

Given the 's declination* equal to 20°, required his longitude.

In the right angled triangle EQs right angled at Q we know E 230 28' the opposite side so 20° required the hypothenuse Es.

the proportion

=

R sin 23° 28':: sin : sin 20°

RX sin 20°

sin 23° 28'

Hence

log. sin 9.53405

Hence ES = 59° 11' 26' the longitude of the sun required.

H

The declination of the sun may be found rudely by taking its meridian altitude with the same instrument and in the same manner as was described at Art. 11. More accurate instruments and methods will be described hereafter. This observation should be made about noon repeatedly, and the greatest observed altitude will be the meridian altitude. A piece of colored glass will be required for the purpose. Let p be a place on the earth; pq its distance from the equator will be the latitude; this contains the same number of degrees as the arc zq between the zenith and celestial equator. Let s be the place of the sun, then sq will be his declination. Let нo be the horizon, then so is equal's meridian altitude, sz= complement of his altitude, and is called the zenith distance, or

coaltitude: sqzq-sz or declination

2

S

P

E

latitude - zenith distance.

N. B. The altitude of the uppermost point of the circumference of the sun should be first taken, then of the lowermost point, and half their difference added to the latter, or simply half their sum will give the altitude of the 's centre.