Figure 18

Figure 18 shows a group of people in the process of laying the meridiana at San Petronio. The circles imposed on the diagram make up the analemma used for a geometrical location of the zodiacal plaques along the meridiana. The geometry involved is mentioned in Appendix B of Heilbron’s book and this section aims to provide an elaboration.

The vertical circles in Figure 18 have been redrawn in Figure 19 below.

Figure 19

S
indicates the centre of the gnomon, or an instrument that serves to indicate the
time of day by casting its shadow upon a marked surface. In the case of San
Petronio, the marked surface would refer to the meridiana. The larger circle
centred on S cuts the smaller circle centred on T at points A and B such that
the chord AB of the bigger circle is identical to the diameter of the smaller
circle. Let F be any point on the small circle and CTS be the noon ray at an
equinox. In addition,
and *r*, *R* are the
radii of the smaller and bigger circles respectively.

If DF is parallel to AB, in triangle EFT,

Since *EF* = *QR *and in triangle
RSQ,

Therefore,

Taking
we have

Now,
refer to Figure 20.

Figure
20

The
above figure gives the earlier-mentioned angles *λ* and
*δ*
on the celestial sphere. S* marks the true Sun while S the projection of the
true Sun on the equinoctial. *δ*
is the Sun’s declination and *ε* the obliquity of the ecliptic. Let the radius of the celestial sphere be K.

Since
triangles OSS*, VE.SS* and O.VE.S* are approximately right-angled triangles, we
have

and

Hence,

Comparing
(1) and (2),
Taking *λ*
small, we have

K*λ*
gives the ecliptic longitude and since K is constant, it is sufficient to mark
the point where the noon ray falls on the meridiana at an equinox, and then by
increasing *λ*
in steps of 30˚,
the rest of the zodiacal plaques could be positioned accordingly.