[Illustration:  Fig. 25.—­Measuring Distances.]

[Illustration:  Fig. 26.—­Measuring Elevations.]

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 deg. apart—­half the earth’s diameter between them.  The one on Cotopaxi sees it exactly overhead, at an angle of 90 deg. with the earth’s diameter.  The one on the coast of Africa sees its angle with the same line to be 89 deg. 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.