soon after sunset.

Day by day she moves eastward until in about 14 days she is on the side of the earth opposite the sun; following her still farther we find her again between the earth and the sun. Thus we follow her quite round the earth.

Fig. 63.

E

184. A plan of the moon's path may be made.-Draw a straight line AB of any convenient length, to represent the distance of the moon on any day, say the first after the new moon. On the next day, find how many degrees the moon has moved eastward among the stars (103), and represent this change of place by the angle BAC. From the variation in the breadth of the disc, find the change of dis

B

tance (182), and measure this distance on AC, using the same scale with which we laid off AB. The point C shows the place of the moon for the second day. In the same way find the points D, E, F, etc., for the entire month; the curve which connects these points is a plan of the moon's path or orbit.

THE ELLIPSE.

185. What is the curve of the moon's path?-If it is a circle, the earth can not be at the center, for the distances are unequal. The principles of geometry show that this curve is an ellipse, and, as we shall have frequent occasion to refer to that figure, we will consider its formation and some of its peculiarities.

186. To draw an ellipse.-Set two pins a little distance

apart in a plane surface of board or paper. Tie to each pin one end of a thread which is somewhat longer than

Fig. 64.

the distance between the pins, and placing a pencil against the thread, draw it about the pins as shown in diagram. The curve described is an ellipse.

187. Definitions.-Observing that the length of the

Fig. 65.
E

PD

string is constantly the

same, we say: An ellipse is a curve such that the sum of the distances from any point of the curve to two fixed points within, is invariable. The space included is called an ellipse as well as the line which includes it.

Each of the fixed points

is a focus.

A line drawn through the foci and terminated by the

curve is the major axis.

The middle point of the major axis is the center of the ellipse. Any line drawn through the center and terminated by the curve is a diameter. The major axis is therefore a diameter.

The minor axis is the diameter which is perpendicular to the major axis. Any line drawn from either focus to the curve is a radius vector.

The distance from the center to either focus is the eccentricity of the ellipse.

188. Deductions.-A little study of the figure shows: 1. That the radii vectores vary in length from the shortest, equal to half the major axis less the eccentricity, to the longest, equal to half the major axis plus the eccentricity.

2. That the sum of the radii vectores which meet at any point of the curve is equal to the major axis.

3. That if the foci are distant from each other, that is, if the eccentricity is great, the ellipse is long and narrow; if the foci are brought near each other, the eccentricity becomes less, and the figure becomes more nearly round; if the foci are brought together, the eccentricity becomes nothing, and the ellipse becomes a circle. Finally, if the foci are placed at the ends of the major axis, the ellipse collapses into a straight line. Hence, an ellipse may have any form between a circle and a straight line.

Parallax is the angle formed by two lines which meet at a distant body, as the moon. From parallax, distance is found.

The parallax of a body is the angle subtended by the base of parallax as seen from that body. From parallax and apparent size, actual size is found.

From apparent size and angular motion, as they vary from time to time, the path of the moon is found to be the curve called the ellipse.

CHAPTER X.

THE EARTH'S ORBIT.

190. The sun's parallax.-Having found the parallax and distance of the moon, we inquire if the same method will find like quantities for the sun. Trial shows that the solar parallax, whatever it may be, is too small to be obtained reliably by direct observation, as in the former case. But we may obtain by indirect processes what we can not observe directly; to understand these processes, and to be sure of our results, we follow somewhat the outline of discovery. The first point to be settled is the relation of the earth to the sun. Does the sun move about the earth annually as it seems to do, or does the earth revolve about the sun?

191. The sun vastly larger than the earth.—We have said that the solar parallax can not be directly found, yet for many years our instruments have been so accurate, and our methods so reliable, that we can confidently determine angular quantities of 20", 15", or considerably less. The parallax must therefore be less than the angle which we can confidently determine; certainly less than 20". Supposing it to be 20", how large is the sun? The sun's parallax is equal to the apparent radius of the earth, as seen from the sun (178); the sun's apparent radius, as measured by the micrometer, averages 15' 56" 956". As the real diameters of two objects which are equally distant from an observer, are in proportion to their apparent diameters,

Sun's parallax: App. Rad. of S :: Dia. of E: Dia. of S; or, 20": 956′′:: 1 : 48, nearly.

Hence, if the sun's parallax is as much as 20′′, the sun's diameter must be 48 times the diameter of the earth. Since volumes are as the cubes of diameters (179), the bulk of the sun is at least 483, or 110,592 times the volume of the earth. But our assumed parallax is confessedly too large, hence our computed results fall far short of the truth; we may at least conclude that the sun is vastly larger than the earth.

THE EARTH'S ANNUAL MOTION.

192. The sun's motion may be only apparent. It may be a result of the real motion of the earth. Let S be the sun; AB, a path in which the earth moves about

the sun. When the earth is at A, the sun will be seen against the sky at M; as the earth moves to B, the sun will seem to move to N, and so on to the place of beginning.

193. The sun does not move.-It is more reasonable to believe that a small body moves about one which is vastly larger than itself, than that this large body should move about the smaller one. Since the apparent annual motion of the sun may be produced by the actual revolution of the earth about the sun, we conclude that it is the earth that moves, and that the sun is at rest.