before us, we should apply the centre of a brass quadrant, semicircle, or circle, termed a protractor, which

always accompanies a case of mathematical instruments, to the angular point D, and make the edge D E coincide with the line DC: we then observe the number of de-. grees intercepted between E and H, the point where the line A D cuts the semicircle, which will be the measure of the angle required; in this case 30 degrees.

Simple as this process may appear, it is exactly similar in principle to that which the most elaborate astronomical instrument is made to perform. We may suppose, for instance, the circle H E M B to be the outline of an astronomical circle, E B to be a horizontal line, Hм to be a telescope directed to the star A, whose angular height above the horizon we wish to ascertain : this height will be the number of degrees in the arc H E, or its equal в M, which may be read off on both sides of the circle. Indeed, the advantage of an entire circle in measuring angles is, that the mean of the two or more arcs read off may be taken, and thus the errors of centering or of graduation may be compensated. Suppose the

value of the arc to be 30 degrees 5 minutes 45 seconds, the expression would thus be abbreviated, 30° 5′ 45′′.

206. 2. A PLANE TRIANGLE is a figure contained by three straight lines, which at the points of meeting form three angles. The length of the sides will always be indicated by lineal measures, as yards, miles, &c. ; the values of the angles (the sum of which, under all circumstances, will be equal to two right angles, Euclid, i. 32) in degrees, minutes, and seconds.

207. 3. PLANE TRIGONOMETRY is the science which treats of the properties of plane triangles. By means of certain proportions always holding good between the three sides and three angles of a triangle, we are able, by the aid of this branch of mathematics, when any three of these six quantities are known (provided that one of these known quantities be a side), to find the other three.

208. 4. A SPHERICAL ANGLE is formed by the meeting of two lines on the surface of a sphere or globe. To understand this and the following definition, it will be necessary to consult the celestial and terrestrial globes :

Trace any two meridians, as those of London and Petersburg, from the equator northwards; they will meet at the north pole, and there form a spherical angle, which, measured on the equator, will be found equal to 30°.

The measure of a spherical angle will always be taken on a circle 90° every way distant from the angular point. Thus, in the last example, the measurement of the distance of the two meridians which met at the pole was reckoned on the equator.

209. Refer to fig. 3, Plate VIII. zxp and zy m are vertical circles passing through the stars x and y, and meeting the horizon in the points p, m, 90° distant from z; the arc p m is the measure of the spherical angle p z m.

The point which is 90° distant from any great circle on a sphere (that is, a circle which divides the sphere into two equal parts) is the pole of that circle: thus, the north and south poles are the poles of the equator; and on the celestial globe, the pole of the ecliptic is every way distant 90° from that circle. The zenith and the nadir are the poles of the horizon.

210. A SPHERICAL TRIANGLE is formed by the intersection of three great circles. Draw a line from London to Petersburg on the terrestrial globe: the portions of the meridians of those places between them and the pole will form two sides of a spheric triangle; and their distance, represented by the line joining them, will be the third side. Both the sides and angles of a spheric triangle are expressed in degrees, minutes, and seconds.

Again, in fig. 3, Plate VIII., N y is the north polar distance of the star y; N x that of the star x; xy the angular distance of those stars. These three arcs form the spheric triangle N x y.

211. By the principles of SPHERICAL TRIGONOMETRY we are able, from any three quantities being known, out of the six which compose a spheric triangle (viz. the three sides and three angles), to find the other three. Problems on the globes are, for the most part, a rough method of working problems in spherical trigonometry. It will be our object in this part of astronomy to explain to the reader the principles on which the science

of Practical Astronomy is founded; but it would not be consistent with the plan of this work to go into the numerical solution of the various questions which may arise such is the province of trigonometry; to the study of which, in its most agreeable form, it is hoped that the present treatise will form a fair introduction. The use of the globes, also, will be rendered much more agreeable after the explanations and elucidations which follow; and the student will feel much greater pleasure in the performance of problems on them when he is acquainted with the principles on which they are founded, than can arise from merely following the rules which usually precede them.

SECTION I.

OUR POSITION IN THE UNIVERSE.

212. THE solar system, with its central luminary the Sun, may be considered as entirely detached from any of those stars which are distributed throughout space. We may imagine ourselves isolated from all other systems, but surrounded by them on all sides at immeasurable, but not equal, distances. Presuming that the organs of vision are the same in other planets as our own, we may conclude that the inhabitants of planets which are probably circulating round the fixed stars, only know of our system by observing the light of our sun, which, before it will have reached them, will have dwindled down to that of a twinkling star.

Of the fixed stars our knowledge is very limited;

nor will this be a matter of surprise, if we take into account their distances from us, of which some notion may be formed from the following considerations.

213. Suppose h, i, k (fig. 44) to be three fixed objects, as houses or trees, at unequal distances from the road Im; in passing from 7 to m, these objects will evidently appear to vary their position with respect to objects at a much greater distance; and the nearest object will undergo the greatest change of position, as will be seen

916

d

e

Fig. 44.

by comparing the space af with the spaces be and c d. Now it will easily be imagined that, when the proportion between the space lm and the distance lk is very great-as, for instance, if I m be two yards, and lk six miles-in that case the very distant object k would not appear to change its position at all to a person passing from / to m.

214. Again, let us place ourselves at h, i, k; it will be immediately seen that the angle under which the line 7 m is viewed at each position successively will be less and less, until we can easily conceive a distance so great that the line 7 m would subtend no angle at all, or that it would diminish to a point or become indistinguishable. The angles under which 7 m is viewed