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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.

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



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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 6, EQ his right ascension, which denote by a, and Es his longitude, which denote by l.

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. Hence the proportion

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





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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 Hо 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

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.

Let the student try the following modification of the problem as an exercise.

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81. By means of the proportion for right angled triangles, and of which an application has just been given, one may be derived for triangles in general.

Let ABC be any spherical triangle; let fall from A the arc AD perpendicular to the side BC, the given triangle will be divided into two right angled triangles ABD and ACD. In the right angled triangle ABD we have the proportion (Art. 78)


R sin B sin AB : sin AD

and in the right angled triangle ACD, the proportion

R sin c sin AC sin AD



Multiplying the extremes and means of each of these proportions, we have the equations


RX sin AD = sin B X sin Ab

RX sin AD = sin c X sin AC

The first members of these equations being the same the second numbers are equal, hence

sin B X sin A B = sin c X sin AC

substituting for the sides AB and AC the small letters of the same name with the angles opposite to them the last equation may be written

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sin B sin b:: sin c: sin c*

that is, the sines of the angles of a spherical triangle are as the sines of the opposite sides.


Let z be the zenith, p the pole of the equator, and s the place of a star; zs will be the zenith dis. tance of the star, zps its hour angle, Ps its co-declination or polar distance, and szp its azimuth. Let the azimuth, zenith distance, and hour anglet be given, to find the polar distance, which is the com



plement of the declination; z, zs and P are given, and Ps required.



sin P sin z: sin p: sın z

Let P 32° 26' 6', z= 49° 54′ 38′′, and zs or p = 44° 13′ 45′′

P 32° 26' 6" ar. comp. log. sin 0.27056

z 49° 54' 38"

p 44° 13′ 45′′

z 84° 16'

or declination of the star = 5° 44′

log. sin 9.88369

log. sin 9.84357

log. sin 9.99788

Since the sine of an arc is equal to the sine of its supplement (Art. 15), the required side may be also the supplement of 84° 16', or 95° 44' The dec. would then be 5° 44' south of the equator.

To illustrate this double solution by the diagram, let the student make or conceive to be made the following construction. Draw an arc from s making with Pz an angle equal to z, meeting pz in a point which we will call z. sz' will then be equal to sz; prolong pz and ps till they meet in the opposite pole, which we will call p'; a triangle will be formed z'p's,

The student will recollect that a proportion is an equality of ratios, and that ratio, as commonly understood, is the quotient of two quantities. The above is familiarly called the sine proportion.

+ The zenith dist. and azimuth may be observed with a theodolite or altitude and azimuth instrument, to be described hereafter; the hour angle by a sidereal clock.

in which the angles z' and p', and the side sz' will be equal to those given in the above example, but in which the side p's is the supplement of ps.* The polar distance of the fixed stars will be found to be always the same, hence they describe circles about the poles in their apparent daily



Given a 42° 32′ 19′′, A 48° 12', b 55° 7' 32" to find B.

Ans. в 64° 46′ 14′′.

82. We shall next demonstrate a formula which will express one of the angles of a spherical triangle in terms of the three sides.

Let ABC be any spherical triangle, o the centre of the sphere; join OA, OB, Oc; a trihedral angle is formed having its vertex at o. The plane angles of this trihedral may be called by the same letters as the sides of the spherical triangle for the reason given in Art. 77. By referring to the note of Art. 33, it will be seen that we may choose at pleasure the length of a radius, and the trigonometrical lines will have the same relation as those corresponding to the radius of the tables or any other radius.

Let us take OA as radius, and draw the perpendiculars AD and Ae at its extremity and in the planes AOC and AOB; produce these perpendiculars till they meet the lines.

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oc and Oв in D and E. AD will be the tangent and on the secant of the side b of the spherical triangle, and AE the tangent, and OE the secant of the side c.

This being premised, let us take the value of DE, in terms of the other sides and one angle, of each of the two plane triangles DAE and DOE, to both of which it belongs. This may be done by means of the formula

* A rule for determining when there are two solutions, and when but one in such cases, is given at p. 196.

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