being traced, the easiest and most accurate
method of tracing the other hour-lines would, at the present day when
good watches are common, be by marking where the shadow of the style
falls when 1, 2, 3, &c., hours have elapsed since noon, and the next
morning by the same means the forenoon hour-lines could be traced; and
in the same manner the hours might be subdivided into halves and
quarters, or even into minutes.

But formerly, when watches did not exist, the tracing of the I, II,
III, &c., o'clock lines was done by calculating the angle which each
of these lines would make with the XII o'clock line. Now, except in
the simple cases of a horizontal dial or of a vertical dial facing a
cardinal point, this would require long and intricate calculations, or
elaborate geometrical constructions, implying considerable
mathematical knowledge, but also introducing increased chances of
error. The chief source of error would lie in the uncertainty of the
data; for the position of the dial-plane would have to be found before
the calculations began,--that is, it would be necessary to know
exactly by how many degrees it declined from the south towards the
east or west, and by how many degrees it inclined from the vertical.
The ancients, with the means at their disposal, could obtain these
results only very roughly.

Dials received different names according to their position:--

_Horizontal dials_, when traced on a horizontal plane;

_Vertical dials_, when on a vertical plane facing one of the cardinal
points;

_Vertical declining dials_, on a vertical plane not facing a cardinal
point;

_Inclining dials_, when traced on planes neither vertical nor
horizontal (these were further distinguished as _reclining_ when
leaning backwards from an observer, _proclining_ when leaning
forwards);

_Equinoctial dials_, when the plane is at right angles to the earth's
axis, &c. &c.

_Dial Construction._--A very correct view of the problem of dial
construction may be obtained as follows:--

[Illustration: FIG. 1.]

Conceive a transparent cylinder (fig. 1) having an axis AB parallel to
the axis of the earth. On the surface of the cylinder let equidistant
generating-lines be traced 15 deg. apart, one of them XII ... XII being
in the meridian plane through AB, and the others I ... I, II ... II,
&c., following in the order of the sun's motion.

Then the sha