CASES IV. AND V.

Two sides and their contained angle being given, to find either of the other angles and the third side.

Given the side AB 110 miles, AC 80 miles, and angle BAC 96° 0', to find the angles BCA and CBA and the side BC.

BY PROJECTION.

Draw the indefinite right line AD, on which set off AB = 110; make the angle EAB=96°; and on AE set off AC=80; join BC, and it is done; for BC will measure on the former scale 143, and the angles B and C will measure 33° 55′ and 50° 5′, respectively, on the line of chords.

BY LOGARITHMS.

To find the angles B and C, by Theorem III.

As sum of sides AC and AB 190

Is to their difference 30

E

80+

96

110

To find the side BC, by Theorem II.

2.27875
1.47712

1st. The extent from the sum of the sides, 190, to their difference, 30, on the line of numbers, will reach from the half sum of the angles B and C, 42°, to their half difference, 8° 5, on the line of tangents. The sum of this half sum and half difference gives the angle C 50° 5, and their difference the angle B 33° 55; the greatest angle being opposite to the greatest side.

2dly. The extent from the angle B 33° 55', to the angle A 96° (or its supplement, 84°) on the line of sines, will reach from the side AC 80, to the side BC 142.6, on the line of numbers.

CASE VI

The three sides of a plane triangle given, to find the angles.

The sides AB 85, BC 57, AC 108, given, to find the angles ABC, BAC, BCA.

BY PROJECTION.

Draw the line AC, and make it equal to 108; take 85 in your compasses, and, with one foot on the point A, describe an arc; then take the distance 57 in your compasses, and, with one foot on C, describe another arc intersecting the former arc in the point B; join AB, CB, and it is done; for the angle A being measured will be found=3140, B=97°, and the angle C=514°, nearly.

BY LOGARITHMS, BY THEOREM IV.

108 D

Suppose BD to be drawn perpendicular to AC, and that AG= GC.

Having divided the triangle into two right-angled triangles, the hypotenuses and bases of which are given, we may find the angles by Theorem I.

1st. The extent from the base AC-108, to the sum of the sides 142, on the line of numbers, will reach from the difference of the sides 28, to twice DG 36.8, on the same line of numbers.

2dly. The extent from the hypotenuse AB=85, to the greater segment AD 72,4, on the line of numbers, will reach, on the sines, from the radius 90°, to 58° 24', which is the complement of the angle BAD.

3dly. The extent from the hypotenuse BC 57, to the least segment, DC 35.6, on the line of numbers, will reach on the sines from the radius 90°, to 38° 39', which is the complement of the angle BCA.

This case may be solved without dividing the triangle into two right-angled triangles, by Theorem V.

ASTRONOMY AND GEOGRAPHY.

ASTRONOMY is the science which treats of the motions and distances of the heavenly bodies, and of the appearances thence arising.

GEOGRAPHY is the science which treats of the situations and distances of the various parts of the surface of the earth.

The common opinion of astronomers of the present day is, that the universe is composed of an infinite number of systems or worlds; that in every system there are certain bodies moving in free space, and revolving, at different distances, round a sun, placed in or near the centre of the system; and that these suns, and other bodies, are the stars which are seen in the heavens.

The SOLAR SYSTEM, so called, is that in which our earth is placed, and in which the sun is supposed to be fixed near the centre, with several bodies, similar to our earth, revolving round at different distances. This hypothesis, which is fully confirmed by observation, is called the Copernican System, from Nicholas Copernicus, a Polish philosopher, who revived it about the year 1500, after it had been buried in oblivion

many ages.

Stars are distinguished into two kinds, fixed and wandering. The former are supposed to be suns in the centres of their systems, shining with their own light, and preserving nearly the same situation with respect to each other. They are usually distinguished by their brightness, the largest being called of the first magnitude, and the smallest visible to the naked eye being of the sixth or seventh magnitude. A Constellation is a number of stars which appear near to each other on the concave surface of the heavens, and astronomers, for the sake of remembering them with greater ease, suppose them to be circumscribed by the outlines of some animal or other figure. Wandering stars are those bodies within our system, or celestial sphere, which revolve round the sun; they appear luminous by reflecting the light of the sun, and are of three kinds, namely, primary planets, secondary planets, and comets. The Primary Planets are bodies which revolve round the sun as the centre of their courses, the motions being regularly performed in tracks or paths, called orbits, that are nearly circular and concentrical with each other. A Secondary Planet, Satellite, or Moon, is a body which, while it is carried round the sun, revolves also round a primary planet. Comets are bodies which move round the sun in very excentrical orbits, with vast atmospheres about them, and tails derived from the same.

There are eleven primary planets, which, reckoned in order from the sun, are as follows: Mercury, Venus, the Earth, Mars, Vesta, Juno, Pallas, Ceres, Jupiter, Saturn, and Uranus.

Mercury and Venus are called inferior planets, because their orbits are within the earth's; the others are called superior planets, as their orbits include that of the earth. The SUN, the first and greatest object of astronomical knowledge, is placed near the centre of the orbits of all the planets, and turns round its axis in 254 days. Its diameter is 883,000 English miles, and its mean distance from the earth 95 millions of miles. MERCURY is the least of all the planets known before the discovery of Vesta, Juno, Pallas, and Ceres, and is the nearest to the sun, his mean distance from that luminary being 37 millions of miles. His periodic revolution in his orbit round the sun is performed in 87 days 23 hours, and his diameter is about 3200 miles.

VENUS is the brightest of all the planets. Her diameter is 7687 miles; her mean distance from the sun, 69 millions of miles; and her periodic revolution is performed in 224 days 17 hours. When this planet is in that part of her orbit which is west of the sun, she rises before him in the morning, and is called the morning star; when she is in the eastern part of her orbit, she shines in the evening, after he sets, and is called the evening star.

The next planet is the EARTH, the diameter of which is 7914 miles, the distance from the sun 95 millions of miles, and the time of revolution round the sun, one year. The earth turns round its axis from west to east in 23 hours 56 minutes, which occasions the apparent diurnal motion of the sun and all the heavenly bodies round it

from east to west in the same time, and is, of course, the cause of their rising and setting, of day and night. The axis of the earth is inclined about 23° 28′ to the plane of its orbit, and keeps nearly in a direction parallel to itself, throughout its annual course, which causes the return of spring and summer, autumn and winter. Thus the diurnal motion gives us the grateful vicissitude of night and day, and the annual motion the regular succession of the seasons. The earth is attended by a satellite called the MOON, whose diameter is 2161 miles. Her distance from the centre of the earth is 240,000 miles. She goes round her orbit in 27 days 8 hours; but, reckoning from change to change, in 294 days. Her orbit is inclined to the ecliptic in an angle of 5° 9, cutting it in two points diametrically opposite to each other, called her nodes. As the moon shines only by the reflected light of the sun, she must appear different when in different situations with respect to that luminary. When she is in conjunction with the sun, her dark side is turned towards the earth, which renders her invisible; this is called new moon: when she is in opposition, her light side is wholly visible from the earth; this is called full moon.

If at the time of new moon she is near to either of her nodes, she may intercept a part of the sun's light, and thus cause an eclipse of the sun; and if she is near either of her nodes at the time of full moon, she may pass into the shadow of the earth, and cause an eclipse of the moon. In a similar manner, when the moon passes between an observer on the earth and a star, it is called an occultation of the star. The instant when the moon's limb first covers the star is called the immersion, and the moment of its reappearance is called the emersion. When Mercury or Venus passes between the sun and an observer, and appears to pass over the sun's disk, it is called a transit of Mercury or Venus. Eclipses, occultations, and transits, are of great importance in ascertaining the longitudes of places on the earth. Eclipses of the moon furnish a convincing proof of the rotundity of the earth, since the shadow of the earth, seen upon the moon when eclipsed, is always circular. This is further confirmed by the appearance of objects at sea; for when a ship is making towards the land, the mariners first descry the tops of steeples, trees, &c., pointing above the water; the lower parts being hid, by reason of the curvature of the earth.

The earth is not a perfect globe or sphere, but is a little flattened at the poles, being nearly of the figure of an oblate spheroid, the equatorial diameter being about 26 miles longer than the polar; but since this difference bears but a small comparison to the whole diameter, we may, for all the practical purposes of navigation, consider the earth as a perfect sphere, as will be done in the rest of this work. The natural divisions of the earth will be given hereafter.

MARS is the next planet to the earth. His diameter is 4189 miles. His distance from the sun is 144 millions of miles, and his periodic revolution is performed in about 687 days. He revolves round his axis in 24 hours 40 minutes, appearing of a duskyreddish hue, and is supposed to be encompassed with a very great atmosphere.

Between Mars and Jupiter are situated the four lately-discovered planets, Vesta, Juno, Pallas, and Ceres, named Asteroids by Dr. Herschel. The apparent elements of their orbits are nearly as in the following description :—

VESTA was discovered by Dr. Olbers, of Bremen, on the 29th of March, 1807. Its mean distance from the sun is about 224 millions of miles. Its periodic revolution is performed in 1325 days.

JUNO was discovered by Mr. Harding, of Lilienthal (near Bremen), on the first of September, 1804. It appears like a star of the eighth magnitude. Its distance from the sun is about 254 millions of miles. Its periodic revolution is performed in 1593 days. The inclination of its orbit to the ecliptic is 13° 4′, and the excentricity of the orbit + 0.25.

PALLAS was also discovered by Dr. Olbers, March 28, 1802. Its diameter, according to Dr. Herschel, is only 110 miles. It appears like a star of the eighth magnitude. Its mean distance from the sun is about 263 millions of miles. Its periodic revolution is performed in 1686 days. The inclination of its orbit to the ecliptic is 34° 35', and the excentricity of the orbit 0.242.

CERES was discovered by Mr. Piazzi, of Palermo, on the first of January, 1801. Its diameter, according to Dr. Herschel, is only 160 miles. It appears like a star of the seventh or eighth magnitude. Its distance from the sun is about 263 millions of miles, and its periodic revolution is performed in 1685 days, being at nearly the same distance from the sun as Pallas. The inclination of the orbit of Ceres to the ecliptic is 10° 37',

The inclination decreases at present about 50 in 100 years, by reason of the attraction of the Planets on the earth. It is also affected by the nutation given in Table XLIII., which sometimes amounts to 9".

and the excentricity 0.077. The situations of the nodes of the two planets, Ceres and Pallas, and the inclinations of their orbits, are very different from each other, so that when those planets are in the same plane, they are at a great distance from each other, notwithstanding their mean distances from the sun are nearly equal. It has been supposed by some, that these small bodies are fragments of a former planet.

JUPITER is situated still higher in the system, and is the largest of all the planets, being easily distinguished from them by his peculiar magnitude and light. His diameter is 89,170 miles; his distance from the sun 494 millions of miles; and the time of his periodic revolution is 43324 days. Though Jupiter is the largest of all the planets, yet his diurnal revolution is the swiftest, being only 9 hours and 56 minutes. Jupiter is attended by four satellites, invisible to the naked eye; but through a telescope they make a beautiful appearance. In speaking of them, we distinguish them according to their places, into the first, second, and so on; by the first we mean that which is nearest to the planet. The appearance of these satellites is marked in the XIXth page of the Nautical Almanac for some particular hour of the night; the times when they are eclipsed, by passing into the shadow of Jupiter, are also given in the Nautical Almanac; these eclipses are of some use in determining the longitudes of places on the earth.

Before the discovery of the planet Uranus, SATURN was reckoned the most remote planet of our system. He shines with but a pale and feeble light. His diameter is 79,042 miles; his distance from the sun 907 millions of miles; and his periodic revolution in his orbit is performed in about 29 years 167 days. This planet is surrounded with a broad, flat ring, has a diurnal revolution round its axis, and is attended by seven satellites.

By some observations made by Dr. Herschel, it appeared that the largest diameter of Saturn corresponds to the latitude of 45°; but from later observations he has been induced to believe, that this irregularity is owing to an optical deception, arising from the refraction of the light in passing through the atmosphere of the ring.

URANUS, Herschel, or Georgium Sidus, is the most remote planet of our system. It was discovered in the year 1781, by Dr. Herschel, though it had been seen several times, but had been considered as a fixed star. Its diameter is 35,109 miles; its distance from the sun is 1823 millions of miles; and its periodic revolution in its orbit is performed in 834 years. Dr. Herschel has also discovered six satellites attending this planet.

The astronomy of comets is yet in its infancy. The return of one of them in the year 1758 was foretold by Dr. Halley, and it happened as he predicted; and it appeared again in 1835. He also foretold the return of another in the year 1790, but it never appeared. This was owing to the inaccuracy of the observations of the comet at its former appearance; for Mr. Mechain, having collected all the observations, and calculated the orbit again, found it to differ essentially from that determined by Dr. Halley. Olber's comet, which appeared in 1815, has a revolution of 72 years; and Encke's comet, which has been observed in several successive approaches to the perihelion, completes its revolution in the short period of 1204 days. Biela's comet has also been observed several times, with a periodical revolution of about 64 years. Comets move round the sun in all directions; but the planets and satellites, except one of the satellites of Uranus, move from west to east when seen from the sun; but if viewed from any other of the planets, as the earth, they would appear to revolve round it as a centre; but the sun would be the only one that moves uniformly the same way, for the other planets would sometimes appear to move from west to east, and then to stand still; then they would seem to move from east to west; and, after standing some time, they would again move from west to east; and so on, continually. The motion of a planet from west to east is called the direct motion, or according to the order of the signs. The contrary motion, from east to west, is called retrograde. When the planet appears to stand still, it is said to be stationary.

To illustrate what has already been said relative to the motions and distances of the planets and satellites, we have given the adjoining Plates III. and IV., which require no explanation.

In noting the situations of the stars and planets, astronomers have been under the necessity of imagining various lines and circles on the sphere; and geographers have done the same for fixing the situation of places on the earth. The most remarkable of these are the following:

A great circle is that whose plane passes through the centre of the sphere; and a small circle is that whose plane does not pass through that centre.

A diameter of a sphere, perpendicular to any great circle, is called the axis of that circle; and the extremities of a diameter are called its poles. Hence the pole of a great circle is 90° from every point of it upon the surface of the sphere; but as the