25 58 | 40 9 | 55 36 | 72 22 | 90 0 |
| 58 0 | 12 48 | 26 5 | 40 18 | 55 45 | 72 28 | 90 0 |
| 58 30 | 12 52 | 26 13 | 40 27 | 55 54 | 72 33 | 90 0 |
| 59 0 | 12 56 | 26 20 | 40 36 | 56 2 | 72 39 | 90 0 |
| 59 30 | 13 0 | 26 27 | 40 45 | 56 11 | 72 44 | 90 0 |
+--------+--------+--------+---------+----------+---------+--------+

_Vertical South Dial._--Let us take again our imaginary transparent
sphere QZPA (fig. 4), whose axis PEp is parallel to the earth's axis.
Let Z be the zenith, and, consequently, the great circle QZP the
meridian. Through E, the centre of the sphere, draw a vertical plane
facing south. This will cut the sphere in the great circle ZMA, which,
being vertical, will pass through the zenith, and, facing south, will
be at right angles to the meridian. Let QMa be the equatorial circle,
obtained by drawing a plane through E at right angles to the axis PEp.
The lower portion Ep of the axis will be the style, the vertical line
EA in the meridian plane will be the XII o'clock line, and the line
EM, which is obviously horizontal, since M is the intersection of two
great circles ZM, QM, each at right angles to the vertical plane QZP,
will be the VI o'clock line. Now, as in the previous problem, divide
the equatorial circle into 24 equal arcs of 15 deg. each, beginning at
a, viz. ab, bc, &c.,--each quadrant aM, MQ, &c., containing 6,--then
through each point of division and through the axis Pp draw a plane
cutting the sphere in 24 equidistant great circles. As the sun
revolves round the axis the shadow of the axis will successively fall
on these circles at intervals of one hour, and if these circles cross
the vertical circle ZMA in the points A, B, C, &c., the shadow of the
lower portion Ep of the axis will fall on the lines EA, EB, EC, &c.,
which will therefore be the required hour-lines on the vertical dial,
Ep being the style.

[Illustration: FIG. 4.]

There is no necessity for going beyond the VI o'clock hour-line on
each side of noon; for, in the winter months the sun sets earlier than
6 o'clock, and in the summer months it passes behind the plane of the
dial before that time, and is no longer available.

It remains to show how the angles AEB, AEC, &c., may be calculated.

The spherical triangles pAB, pAC, &c., will give us a simple rule.
These triangles are all righ