THE ANALEMMA IN THE CONSTRUCTION OF SAN PETRONIO

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 Heilbronís 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 Sunís 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

Kλ 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.