and let them meet DCE in M and N
respectively.

Let HP and its equal HA be represented by a. Then the following values
will be readily deduced from the figure:--

AD = a cos _decl._ DH = a sin _decl._ PQ = a sin _alt._

CX = AC = AD cos _lat._ = a cos _decl._ cos _lat._
PN = CV = CX cos ACX = a cos _decl._ cos _lat._ cos ACX.
NQ = MH = DH sin MDH = sin _decl._ sin _lat._
(:. the angle MDH = DAC = latitude.)

And since PQ = NQ + PN,
we have, by simple substitution,
a sin _alt._ = a sin _decl._ sin _lat._ + a cos _del._ cos _lat._
cos ACX; or, dividing by a throughout,

sin _alt._ = sin _decl._ sin _lat._ + cos _decl._ cos _lat._
cos ACX ... (1)
which equation determines the hour-angle ACX shown by the bead.

To determine the hour-angle of the sun at the same moment, let fig. 10
represent the celestial sphere, HR the horizon, P the pole, Z the
zenith and S the sun.

From the spherical triangle PZS, we have
cos ZS = cos PS cos ZP + sin PS sin ZP cos ZPS
but ZS = zenith distance = 90 deg. - altitude
ZP = 90 deg. - PR = 90 deg.- latitude
PS = polar distance = 90 deg. - declination,
therefore, by substitution

sin _alt._ = sin _decl._ sin _lat._ + cos _decl._ cos _lat._
cos ZPS ... (2)
and ZPS is the hour-angle of the sun.

A comparison of the two formulae (1) and (2) shows that the hour-angle
given by the bead will be the same as that given by the sun, and
proves the theoretical accuracy of the card-dial. Just at sun-rise or
at sun-set the amount of refraction slightly exceeds half a degree.
If, then, a little cross m (see fig. 8) be made just below the
sun-line, at a distance from it which would subtend half a degree at
c, the time of sun-set would be found corrected for refraction, if the
central line of light were made to fall on cm.

[Illustration: FIG. 10.]

LITERATURE.--The following list includes the principal writers on
dialling whose works have come down, to us, and to these we must refer
for descriptions of the various constructions, some simple and direct,
others fanciful and intricate, which have been at different times
employed: Ptolemy, _Analemma_, restored by Commandine; Vitruvius,
_Architecture_; Sebastian Muenster, _Horologiographia_; Or