in perihelion at the same time, for then VS will be the longest and "VE the shortest. Now in every right-angled triangle the proportion between the hypothenuse, ES, and the side opposite, VS, changes as the angle at E varies, but with the same angle remains the same whatever may be the length of the lines themselves. This proportion between the hypothenuse and the side opposite any angle is termed the sine of that angle. Tables are published which contain the sines for all angles. In this way, the mean distance of Venus is found to be -£fo* that of the earth, Mars $ times, Jupiter 5£ times, etc. The same result would be obtained by the use of Kepler's third law; and on page 29, we saw how the distances of the planets themselves could be determined by the periodic times, if the distance of the earth from the sun is first known. So that when we have accurately determined the sun's distance from us, we can then decide by either of the methods named the distance of all the planets. Indeed that is, as already remarked, the "foot-rule" for measuring all celestial distances. 2d. To MEASURE THE MOON'S DISTANCE PROM THE Earth.—(1.) The ancient method.—As the moon's distance is so much less than that of the other heavenly bodies, it is measured by the earth's semi-diameter. reckon his longitude by the eclipses of Jupiter's moons, and so decide the fate of his voyage. We can easily see how the revolution of the earth on its axis influences the cost of a cup of tea. celestial sphere, is found to be 108°. At the latter station, the distance from the south pole to the moon's centre is measured in the same way, and found to be 73^°. The sum of these angles is 181£°. Now, the entire distance from the north pole around to the south pole, measured on a meridian, can be only half a great circle, or 180°. This difference of * If the pupil has studied Trigonometry, he may apply here the simple proportion— BS : VS :: Radius : Sine of 47* 15" = greatest elongation of Venus. CELESTIAL MEASUREMENTS. Many persons read the enormous figures which indicate the distances and dimensions of the heavenly bodies with an indefinite idea, which conveys no such feeling of certainty as is experienced when they read of the distance between two cities, or the number of square miles in a certain State. Many, too, imagine that celestial measurements are so mysterious in themselves that no common mind can hope to grasp the methods. Let us attempt the solution of a few of these problems. 1st. TO FIND THE DISTANCES OF THE PLANETS FROM The Sun.—In the figure, E represents the earth, ES the earth's distance from Fig. Ml the sun, V the planet Venus, and VES the angle of elongation (a right-angled triangle). It is clear, that as Venus swings apparently east and west of the sun, this angle may be easily measured; also, that it will be the greatest when Venus t COMPARATIVE DISTANCE OP VENUS is in aphelion and the earth A*D TDE *»"•"• in perihelion at the same time, for then VS will be the longest and VE the shortest. Now in every right-angled triangle the proportion between the hypothenuse, ES, and the side opposite, VS, changes as the angle at E varies, but with the same angle remains the same whatever may be the length of the lines themselves. This proportion between the hypothenuse and the side opposite any angle is termed the sine of that angle. Tables are published which contain the sines for all angles. In this way, the mean distance of Venus is found to be ^nr* that of the earth, Mars \ times, Jupiter h\ times, etc. The same result would be obtained by the use of Kepler's third law; and on page 29, we saw how the distances of the planets themselves could be determined by the periodic times, if the distance of the earth from the sun is first known. So that when we have accurately determined the sun's distance from us, we can then decide by either of the methods named the distance of all the planets. Indeed that is, as already remarked, the "foot-rule" for measuring all celestial distances. 2d. TO MEASURE THE MOON'S DISTANCE FROM THE Earth.—(1.) The ancient method.—As the moon's distance is so much less than that of the other heavenly bodies, it is measured by the earth's semi-diameter. * If the pupil has studied Trigonometry, he may apply here the simple proportion— B8 : VS :: Radius : Sine of 47* 15" = greatest elongation of Venus. The method, an extremely rough one, which was in use among the ancients, was something like the following. In an eclipse of the moon, that body passes through the earth's shadow in about four hours. If, then, the moon travels along its orbit in four hours a distance equal to the diameter of the earth, in twenty-four hours it would pass over six times, and in a lunar month (about thirty days) one hundred and eighty times, that distance. The circumference of the lunar orbit must be then one hundred and eighty times the diameter of the earth. The ancients supposed the heavenly orbits to be circles, and as the diameter of a circle is about \ of the circumference, they deduced directly the diameter of the moon's orbit as 120 times, and the distance of the the moon from the earth as 60 times the semi-diameter of the earth. (2.) Modern method by the lunar parallax.—Under the head of parallax we saw how, in common life, we obtain a correct idea of the distance of an object by means of our two eyes. We proved that one eye alone gives no notion of distance. Just, then, as we use two eyes to find how far from us an object is, so the astronomer uses two astronomical eyes or observatories, located as far apart as possible, to find the parallax of a heavenly body. In the figure, M represents the moon, G an observatory at Greenwich, and C another at the Cape of Good Hope. At the former, the distance from the north pole to the centre of the moon, measured on a meridian of the 1J° must be the difference in the position of the moon, as seen from the two observatories. For the observer at the former station will see the moon projected on the celestial sphere at G', and in measuring its distance from the north pole will measure an arc bG' further than if he were located at E, the centre of the earth. The observer at the latter station will see the moon projected on the celestial sphere at C, and in measuring its distance from the |