e bead P along till it exactly coincide
with A.

To find the hour of the day,--hold the dial in a vertical position in
such a way that its plane may pass through the sun. The verticality is
ensured by seeing that the bead rests against the card without
pressing. Now gradually tilt the dial (without altering its vertical
plane), until the central line of sunshine, passing through the open
slit of the door, just falls along the sun-line. The hour-line against
which the bead P then rests indicates the time.

[Illustration: FIG. 9.]

The _sun-line_ drawn above has always, so far as we know, been used as
a _shadow-line_. The upper edge of the rectangular door was the
prolongation of the line, and, the door being opened, the dial was
gradually tilted until the shadow cast by the upper edge exactly
coincided with it. But this shadow tilts the card one-quarter of a
degree more than the sun-line, because it is given by that portion of
the sun which just appears above the edge, that is, by the upper limb
of the sun, which is one-quarter of a degree higher than the centre.
Now, even at some distance from noon, the sun will sometimes take a
considerable time to rise one-quarter of a degree, and by so much time
will the indication of the dial be in error.

The central line of light which comes through the open slit will be
free from this error, because it is given by light from the centre of
the sun.

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,