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Page 22
The next process is to measure the height or magnitude of objects
at an ascertained distance. Put two pins in a stick half an inch
apart (Fig. 26). Hold it up two feet from the eye, and let the
upper pin fall in line with your eye and the top of a distant church
steeple, and the lower pin in line with the bottom of the church and
your eye. If the church is three-fourths of a mile away, it must
be eighty-two feet high; if a mile away, it must be one hundred
and ten feet high. For if two lines spread [Page 68] one-half an
inch going two feet, in going four feet they will spread an inch,
and in going a mile, or five thousand two hundred and eighty feet,
they will spread out one-fourth as many inches, viz., thirteen
hundred and twenty--that is, one hundred and ten feet. Of course
these are not exact methods of measurement, and would not be correct
to a hair at one hundred and twenty-five feet, but they perfectly
illustrate the true methods of measurement.
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.
[Illustration: 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 [Page 69] 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 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 [Page
70] 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 [Page 71] earth's orbit on one
side, as T R A (Fig. 28), and a line were 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.
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