The card-dial deserves to be looked upon as something more than a mere toy. Its ingenuity and scientific accuracy give it an educational value which is not to be measured by the roughness of the results obtained.
The theory of this instrument is as follows:—Let H (fig. 9) be the point of suspension of the plummet at the time of observation, so that the angle DAH is the north declination of the sun,—P, the bead, resting against the hour-line VX. Join CX, then the angle ACX is the hour-angle from noon given by the bead, and we have to prove that this hour-angle is the correct one corresponding to a north latitude DAC, a north declination DAH and an altitude equal to the angle which the sun-line, or its parallel AC, makes with the horizontal. The angle PHQ will be equal to the altitude, if HQ be drawn parallel to DC, for the pair of lines HQ, HP will be respectively at right angles to the sun-line and the horizontal.
Draw PQ and HM parallel to AC, 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 sincePQ = 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,