the tree or book again, he should make twenty-four of the larger spans
and thirty-two of the lesser ones. These two angles of 15 degrees and 11
1/4 degrees are particularly important. The sun travels through 15
degrees in each hour; and therefore, by "spanning" along its course, as
estimated, from the place where it would stand at noon (aided in this by
the compass), the hour before or after noon, and, similarly after sunrise
or before sunset, can be instantly reckoned. Again, the angles 30
degrees, 45 degrees, 60 degrees, and 90 degrees, all of them simple
multiples of 15 degrees, are by far the most useful ones in taking rough
measurements of heights and distances, because of the simple relations
between the sides of right-angled triangles, one of whose other angles
are 30 degrees, 45 degrees, or 60 degrees; and also because 60 degrees is
the value of an angle of an equilateral triangle. As regards 11 1/4
degrees, or one point of the compass, it is perfectly out of the question
to trust to bearings taken by the unaided eye, or to steer a steady
course by simply watching a star or landmark, when this happens to be
much to the right or the left of it. Now, nothing is easier than to span
out the bearing from time to time.

Right-angles to lay out.--A triangle whose sides are as 3, 4, and 5, must
be a right-angled one, since 5 x 5 = 3 x 3 + 4 x 4; therefore we can find
a right-angle very simply by means of a measuring-tape. We take a length
of twelve feet, yards, fathoms, or whatever it may be, and peg its two
ends, side by side, to the ground. Peg No. 2 is driven in at the third
division, and peg No. 3 is held at the seventh division of the cord,
which is stretched out till it becomes taut; then the peg is driven in.
These three pegs will form the corners of a right-angled triangle; peg
No. 2 being situated at the right-angle.

Proximate Arcs.--