Consider the nicety of the work.  Suppose the graduated scale to be thirty feet in circumference.  Divided into 360 deg., each would be one inch long.  Divide each degree into 60’, each one is 1/60 of an inch long.  It takes good eyesight to discern it.  But each minute must be [Page 62] divided into 60”, and these must not only be noted, but even tenths and hundredths of seconds must be discerned.  Of course they are not seen by the naked eye; some mechanical contrivance must be called in to assist.  A watch loses two minutes a week, and hence is unreliable.  It is taken to a watch-maker that every single second may be quickened 1/20160 part of itself.  Now 1/20000 part of a second would be a small interval of time to measure, but it must be under control.  If the temperature of a summer morning rises ten or twenty degrees we scarcely notice it; but the magnetic tastimeter measures 1/5000 of a degree.

Come to earthly matters.  In 1874, after nearly twenty-eight years’ work, the State of Massachusetts opened a tunnel nearly five miles long through the Hoosac Mountains.  In the early part of the work the engineers sunk a shaft near the middle 1028 feet deep.  Then the question to be settled was where to go so as to meet the approaching excavations from the east and west.  A compass could not be relied on under a mountain.  The line must be mechanically fixed.  A little divergence at the starting-point would become so great, miles away, that the excavations might pass each other without meeting; the grade must also rise toward the central shaft, and fall in working away from it; but the lines were fixed with such infinitesimal accuracy that, when the one going west from the eastern portal and the one going east from the shaft met in the heart of the mountain, the western line was only one-eighth of an inch too high, and three-sixteenths of an inch too far north.  To reach this perfect result they had to triangulate from the eastern portal to distant [Page 63] mountain peaks, and thence down the valley to the central shaft, and thus fix the direction of the proposed line across the mouth of the shaft.  Plumb-lines were then dropped one thousand and twenty-eight feet, and thus the line at the bottom was fixed.

Three attempts were made—­in 1867, 1870, and 1872—­to fix the exact time-distance between Greenwich and Washington.  These three separate efforts do not differ one-tenth of a second.  Such demonstrable results on earth greatly increase our confidence in similar measurements in the skies.

[Illustration:  Fig. 22.]

A scale is frequently affixed to a pocket-rule, by which we can easily measure one-hundredth of an inch (Fig. 22).  The upper and lower line is divided into tenths of an inch.  Observe the slanting line at the right hand.  It leans from the perpendicular one-tenth of an inch, as shown by noticing where it reaches the top line.  When it reaches the second horizontal line it has left the perpendicular one-tenth of that tenth—­that is, one-hundredth.  The intersection marks 99/100 of an inch from one end, and one-hundredth from the other.