Fig. 26.—Measuring Elevations.
Imagine a base line ten inches long. At each end erect a perpendicular line. If they are carried to infinity they will never meet: will be forever ten inches apart. But at the distance of a foot from the base line incline one line toward the other 63/10000000 of an inch, and the lines will come together at a distance of three hundred miles. That new angle differs from the former right angle almost infinitesimally, but it may be measured. Its value is about three-tenths of a second. If we lengthen the base line from ten inches to all the miles we can command, of course the point of meeting will be proportionally more distant. The angle made by the lines where they come together will be obviously the same as the angle of divergence from a right angle at this end. That angle is called the parallax of any body, and is the angle that would be made by two lines coming from that body to the two ends of any conventional base, as the semi-diameter of the earth. That that angle would vary according to the various distances is easily seen by Fig. 27.
Let O P be the base. This would subtend a greater angle seen from star A than from star B. Let B be far enough away, and O P would become invisible, and B would have no parallax for that base. Thus the moon has a parallax of 57" with the semi-equatorial diameter of the earth for a base. And the sun has a parallax 8".85 on the same base. It is not necessary to confine ourselves to right angles in these measurements, for the same principles hold true in any angles. Now, suppose two observers
Fig. 27. on the equator should look at the moon at the same instant. One is on the top of Cotopaxi, on the west coast of South America, and one on the west coast of Africa. They are 90° apart—half the earth's diameter between them. The one on Cotopaxi sees it exactly overhead, at an angle of 90° with the earth's diameter. The one on the coast of Africa sees its angle with the same line to be 89° 59' 3"—that is, its parallax is 57". Try the same experiment on the sun farther away, as is seen in Fig. 27, and its smaller parallax is found to be only 8".85.
It is not necessary for two observers to actually station themselves at two distant parts of the earth in order to determine a parallax. If an observer could go from one end of the base-line to the other, he could determine both angles. Every observer is actually carried along through space by two motions: one is that of the earth's revolution of one thousand miles an hour around the axis; and the other is the movement of the earth around the sun of one thousand miles in a minute. Hence we can have the diameter not only of the earth (eight thousand miles) for a base-line, but the diameter of the earth's orbit (184,000,000 miles), or any part of it, for such a base. Two observers at the ends of the earth's diameter, looking at a star at the same instant, would find that it made the same angle at both ends; it has no parallax on so short a base. We must seek a longer one. Observe a certain star on the 21st of March; then let us traverse the realms of space for six months, at one thousand miles a minute. We come round in our orbit to a point opposite where we were six months ago, with 184,000,000 of miles between the points. Now, with this for a base-line, measure the angles of the same stars: it is the same angle. Sitting in my study here, I glance out of the window and discern separate bricks, in houses five hundred feet away, with my unaided eye; they subtend a discernible angle. But one thousand feet away I cannot distinguish individual bricks; their width, being only two inches, does not subtend an angle apprehensible to my vision. So at these distant stars the earth's enormous orbit, if lying like a blazing ring in space, with the world set on its edge like a pearl, and the sun blazing like a diamond in the centre, would all shrink to a mere point. Not quite to a point from the nearest stars, or we should never be able to measure the distance of any of them. Professor Airy says that our orbit, seen from the nearest star, would be the same as a circle six-tenths of an inch in diameter seen at the distance of a mile: it would all be hidden by a thread one-twenty-fifth of an inch in diameter, held six hundred and fifty feet from the eye. If a straight line could be drawn from a star, Sirius in the east to the star Vega in the west, touching our earth's orbit on one side, as T R A (Fig. 28), and a line were
Fig. 28. to be drawn six months later from the same stars, touching our earth's orbit on the other side, as R B T, such a line would not diverge sufficiently from a straight line for us to detect its divergence. Numerous vain attempts had been made, up to the year 1835, to detect and measure the angle of parallax by which we could rescue some one or more of the stars from the inconceivable depths of space, and ascertain their distance from us. We are ever impelled to triumph over what is declared to be unconquerable. There are peaks in the Alps no man has ever climbed. They are assaulted every year by men zealous of more worlds to conquer. So these greater heights of the heavens have been assaulted, till some ambitious spirits have outsoared even imagination by the certainties of mathematics.
It is obvious that if one star were three times as far from us as another, the nearer one would seem to be displaced by our movement in our orbit three times as much as the other; so, by comparing one star with another, we reach a ground of judgment. The ascertainment of longitude at sea by means of the moon affords a good illustration. Along the track where the moon sails, nine bright stars, four planets, and the sun have been selected. The nautical almanacs give the distance of the moon from these successive stars every hour in the night for three years in advance. The sailor can measure the distance at any time by his sextant. Looking from the world at D (Fig. 29), the distance of the moon and star is A E, which is given in the almanac. Looking from C, the distance is only B E, which enables even the uneducated sailor to find the distance, C D, on the earth, or his distance from Greenwich.
