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sides and an angle opposite to one of them are given, are contained in the following diagram:



Let MM', P, be two perpendicular diameters of the circle MPM', which is the base of a hemisphere; and let C be any point in its surface, and let the arcs passing through C, be all the halves of great circles, except CB", CB", which are portions of great circles. Let all these be symmetrically situated in the semicircles PMT, PM'T, so that every two corresponding arcs, as, for instance, AC, A'C, are equally inclined to PCT.










Hence, every two corresponding arcs, reckoning from C, as CB B and CB', or CB and CB', are equal. Also, of all these arcs, CT is the greatest, CP the least, and any arc, as Ca, nearer to the greatest, is greater than any other, as CA', that is more remote (Geom. vol. II. p. 55.) Also the arcs CM, CM', are quadrants; and therefore all the arcs, reckoning from C to the semicircle MPM', are less than quadrants, and those terminating in the semicircle M&M' are greater than quadrants.



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Let PC=h, and denote the parts of the triangle ABC as usual. Then, R. sin h = sin B sin a, R⚫ sin h


sin B =

sin a

It is evident that when his constant, sin B is least when sin a is greatest, that is, when a is a quadrant, and equal to CM' or CM. Hence, of all the angles subtended by CP at the points B, A', B", M', a,...that at M' is the least, and that at a point nearer to M', either in the quadrant PM', or

M', is less than one more remote; and these angles therefore are all less than right angles, for ha; and their adjacent angles are greater than right angles. Also, when two arcs, CA' and Ca are supplementary, the acute angles at A' and a are equal.

When each of the sides a and b is less than a quadrant, and angle A is acute, and sin a sin b, only one triangle

can be formed, and the unknown angle B is of the same affection as b. This appears from the triangle ACB", where B"C= = a, AC=b, and angle CAP = A; for B'CAC, and aC (371).

When each of the sides a and b is greater than a quadrant, and sin a sin b, and A is obtuse, there can be only one triangle, and angle B will be of the same affection as b. This appears from the triangle aCB", where "Ca, aC, and angle Car A; where B'C is intermediate between aC and its supplement AC, and therefore sin asin b (371)

When sin a sin b, there will be two triangles. This appears when A is acute from the triangles ACB, ACB', in which CB CB'; and when A is obtuse, from the triangles aCB, aCB. And in this case the two values of Bare supplementary.


By examining all the possible cases, it will be found that they are comprehended in the theorem stated in article 358.

Let A, B, and a, be given. When each of the given parts is less than a quadrant, and sin Asin B, there can be only one triangle, and the unknown side b will be of the same species as B. This appears from triangle ACB", where B"Ca. Were sin Asin B, there could then be two triangles, as BCA, BCa'. And by examining in the same way all the possible cases, the theorem stated in article 359 is easily established.


373. To an observer placed on the surface of the earth, the heavenly bodies appear to be situated on the surface of a concave sphere, of which the place of the observer is the centre; for the magnitude of the earth is a mere point, in reference to the distance of all celestial bodies, except those belonging to the solar system; and it becomes sensible in regard to the distances of the latter, only when accurate observations are taken with proper instruments.

The apparent diurnal revolution of these bodies from east to west, caused by the real daily rotation of the earth on its axis in the opposite direction from west to east, and the apparent annual motion of the sun in the heavens, arising from the earth's annual revolution in its orbit in an opposite direction, are, for convenience, in the practice of astronomy and navigation, considered as real motions; and the positions of these bodies are determined accordingly, for any given time, with the aid of the principles of spherical trigonometry.


374. The celestial sphere is the apparent concave sphere, on the surface of which the heavenly bodies appear to be situated.

375. The axis of the celestial sphere is a straight line passing through the earth's centre, terminated at both extremities by the celestial sphere, and about which the heavenly bodies appear to revolve.

376. The poles of the celestial sphere are the extremities of its axis; one of them being called the north, the other the south pole.

The poles appear as fixed points in the heavens, without any diurnal rotation, the bodies near them appearing to revolve round them as centres.

377. The equinoctial, or celestial equator, is a great circle in the heavens equidistant from the poles; and it divides the celestial sphere into the northern and southern hemispheres.*

This circle referred to the earth is the equator; also the axis of the earth is a portion of that of the celestial sphere. 378. The ecliptic is a great circle that intersects the equator obliquely.

This is the circle in which the sun appears to perform his annual motion round the earth. A more definite conception will be easily formed of this annual path of the sun, if

*This circle is so named, because when the sun is situated in it, the nights, and consequently the days, are equal every where on the earth's surface; also the day and night are then of the same length.

the diurnal rotation of the heavens or earth be supposed not to exist.

379. The two points in which the ecliptic and equinoctial intersect are called the equinoxes, or equinoctial points; that at which the sun crosses the equator towards the north is called the vernal equinox, and the other the autumnal equinox.

380. The zodiac is a zone extending about 8° on each side of the ecliptic; and it is divided into 12 equal parts, called the signs of the zodiac.

The names and characters of these signs are―

o Cancer,
my Virgo,

m Scorpio,


The first six lie on the north of the equinoctial, and are called northern signs; the other six, on the south of that circle, are called southern signs. Each sign contains 12°.

The signs are reckoned from west to east, according to the apparent annual motion of the sun. The first, Aries, is near the vernal equinox, and Libra near the autumnal equinox.*

381. The solstitial points are the middle points of the northern and southern halves of the ecliptic; the northern is called the summer, and the southern the winter, solstice.†

382. The horizon is the name of three circles; one, the true or rational; another, the sensible; and a third, the visible or apparent, horizon.

The first is the intersection of a horizontal plane, passing through the earth's centre, with the celestial sphere; the second is the intersection, with this sphere, of a plane parallel to the former touching the earth's surface at the place of the observer; and the third is the intersection, with


Ŏ Taurus,

II Gemini,

Capricornus, Aquarius, Pisces.

* About three thousand years ago the western part of Aries nearly coincided with the vernal equinox; but from the slow westerly recession of this point, called the precession of the equinoxes, it is nearly a whole sign to the west of Aries.

These points are so named, because when the sun has arrived at either of them, it appears to stop, in reference to its motion north and south, and then to return; and hence also the origin of the term tropics from a Greek word, which means to turn.

the same sphere, of the conic surface, of which the vertex is at the eye of the observer, and the surface of which touches on every side the surface of the earth, considered as a sphere.

383. A vertical line passing through the earth's centre, and the place of the observer, may be called the axis of the horizon; and the extremities of this axis, where it meets the celestial sphere, the poles of the horizon; the upper pole being called the zenith, and the lower the nadir.

384. The north, east, south, and west points of the horizon, are called the cardinal points.

385. Meridians are great circles passing through the poles of the celestial sphere; they are also called hour circles. These circles correspond to meridians on the earth. 386. A meridian passing through the equinoctial points, is called the equinoctial colure; and that passing through the solstitial points, the solstitial colure.

387. Circles passing through both poles of the ecliptic are called circles of celestial longitude.

388. Vertical circles are great circles passing through both the poles of the horizon.

389. A vertical circle, passing through the east and west points of the horizon, is called the prime vertical.

390. Small circles, parallel to the equinoctial, are called parallels of declination.

391. Small circles, parallel to the ecliptic, are called parallels of celestial latitude.

392. Small circles, parallel to the horizon, are called parallels of altitude.

393. The right ascension of a heavenly body is an arc of the equinoctial, intercepted between the vernal equinox, and a meridian passing through the body.

394. The longitude of a heavenly body is an arc of the ecliptic, intercepted between the vernal equinox, and a circle of longitude passing through the body.

395. The azimuth of a body is an arc of the horizon, intercepted between the north or south point, and a vertical circle passing through the body.*

* The azimuths may be named according to the quadrants in which they lie; thus, N. E. when between the north and east points; S. W. between the south and west points; and so on.

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