the apparent positions of a star, S, as seen from Greenwich and the Cape. For the same reason the dotted lines, GP and C P', parallel to the axis of the Earth, represent the apparent positions of the north and south poles of the heavens as seen from the places named. The angle PGS therefore represents the north polar distance of the star as seen from Greenwich; the angle P' CS represents

the south polar distance of the same star observed at the Cape and these two angles will of course make up 180.

It is seen from the diagram that the north polar distance of the Moon as seen from Greenwich, which is observed, is greater than that of the star.

Similarly, the south polar distance as seen from the Cape, which is also observed, is greater than that of the star.

Therefore these two polar distances added together are greater than 180, greater in fact by the angles SGM and S CM, which are equal to G ME+CME=GM C

The angle GMC is determined by observation to be about 1: therefore, as we know the length of the radii GE and CE, in addition to the two angles observed and the one deduced, plane trigonometry enables us easily to determine the lines MG, MC and ME, which are the distances of the Moon from Greenwich, the Cape, and the centre of the Earth respectively.

581. It also enables us to determine the angles G ME and CME, which represent the parallax (Art. 543) of the Moon as observed at Greenwich and the Cape respectively. In this manner the mean equatorial horizontal parallax of the Moon has been determined to be nearly 57′ 2′′.

LESSON XLVI.-Determination of the Distances of Venus and Mars: of the Sun. Transit of Venus. The Transit of 1882.

582. This method may be adopted to determine the distance of Venus when in conjunction with the Sun, and of Mars when in opposition; but it is only applicable in the former case when Venus is exactly between us and the Sun, or when she is said to transit or pass over his disc—when, in short, we have a transit of Venus; of which more hereafter.

583. In the case of Mars in opposition, there is, however, another method by which his distance may be determined by observations made at one observatory. In this method the base line is not dispensed with, but instead of using two different places on the Earth's surface, and determining the actual distance between them, we use observations made at the same place at an interval of twelve hours; in which time, of course, if we suppose them to be made on the equator, the same place would be at the two extremities of the same diameter, that is 8,000

miles apart : if the observations are not made actually on the equator, it is still easy, knowing exactly the shape and size of the Earth, to calculate the actual difference. Fig. 71, which represents a section of the Earth at the equator, will explain this method. O and O' represent the positions of the same observer at an interval of twelve hours, the Earth being in that time carried half round by its movement of rotation; M the planet Mars; and S a star of the same declination as the planet, the direction of the star being the same from all points of the surface as from the centre. At O, when Mars is rising at the place of observation, let the observer measure the distance the planet will appear to the east of the star; at O', when Mars is setting at the place of observation, and therefore when the Earth's rotation has carried him to the other end of the same diameter, let him again measure the distance the planet will appear to the west of the star. will thus determine, as in the case of the Moon, the angle the line joining the two places of observation subtends at the planet. In the case of observations made at the equator, the Earth's equatorial diameter forms the base line. The angle it subtends is deter mined by observation; and this can be accomplished, although both the Earth and Mars are moving in the in

He

FIG. 69.-Determination of the distance of Mars by observations made at one place.

terval between the two observations, as the motion of both can be taken into account, Here again then, when

the size of the Earth is known, the distance of Mars can be determined by plane trigonometry.

584. As seen from the Sun, the Earth's diameter is so small that it is useless as a base line, and consequently the Sun's distance cannot be thus measured.

585. The Sun's distance can however be obtained indirectly by methods based upon the discovery of Kepler, that the distances of the orbits of the planets from the Sun and from each other are so linked together, that if we could determine any one of the distances, all the rest would follow.

It was in this way that Dr. Gill, from observations of Mars made at the island of Ascension in 1877, fixed the Sun's distance at 93,080,000 miles. He used the plan of measurement just described in Art. 583, and known as the "diurnal method of parallaxes."

586. As we have seen, however, Mars does not come so near to us as Venus; but it happens that when Venus comes nearest to us, it comes between us and the Sun, and consequently its dark side is towards us, and we can only see it when it happens to be exactly between us and the Sun, when it passes over the Sun's disc as a dark spot, a phenomenon called a transit of Venus. These transits happen but rarely: the last happened in 1882; the next will not occur until the year 2004. But when they do happen, as the planet is projected on the Sun, the Sun serves the purpose of a micrometer, and great advantages (unfortunately not always realised in practice) are offered for observation. The measure of the Sun's distance-one of the noblest problems in astronomy, on which depends "every measure in astronomy beyond the Moon, the distance of and dimensions of the Sun and every planet and satellite, and the distances of those stars whose parallaxes are approximately known,”—is accomplished then by the following method, pointed out by Dr. Halley in 1716.

587. We have seen that when Venus crosses the Sun's disc during its transit it appears as a round black spot.

Let us suppose two observers placed at two different stations on the Earth, properly chosen for observations of the phenomenon ; one at a station A in the northern hemisphere, another at a station B in the southern one. When Venus is exactly between the Sun and the Earth, the observer at A will see her projected on the Sun, moving on the line CD in Fig. 70; the southern observer at B will, from his lower station, see the planet V projected higher on the disc, moving on the line EF. Now, what we

FIG. 70.-A Transit of Venus.

require to know, in order to determine the Sun's distance, is the distance between these lines.

If the distance between the two stations is sufficiently great, the planet will not appear to enter on the Sun's disc at the same absolute moment at the two stations, and therefore the paths traversed, or the "chords," wil' be different. Speaking generally, the chords will be of unequal length, so that the time of transit at one station will be different from the time of transit at the other. This difference will enable us to determine the difference in the length of the chords described by the planet, and consequently their respective positions on the solar disc,