« ... Danti did not build a proper Sun-dial, but just traced a line, slanted 9° towards West with respect to the meridian ..., during the Summer Solstice and the Equinoxes this inclination, however, did not compromise the observation of the Sun, necessary to compute the exact duration of the year and  
the date on which Easter would fall ... » (1) 

Cassini deeply engaged in these great renovation works and became also convinced there was a "path", along which the Sun rays of the astronomical Midday could cross the whole Church. entering the Church, in the vault of the left aisle, at a height of 27,07 m., he opened a hole: diameter 27 mm, overlaying it with a bronze foil. The reflection of the Sun rays reached a distance of about 67 m. 

At exactly Midday (the astronomical Midday), the Sun rays would illuminate different points (diverse every day, in accordance with the different heights of the Sun on the horizon) marked on a metallic line, lying obliquely on the floor: North towards South. During the course of this project, Cassini took all necessary precautions, on one side to contrast the many prejudices that assumed as impossible such close transit through the columns, on the other side to level in a proper and perfect way the meridian line, so as to attain a perfect geometrical solution. 
« ... a cavity was dug into the floor [...] a canal full of water was embedded into it ... » (1) 
 

(1)  Anna Cassini - "Un'avventura tra le stelle - Gio. Domenico Cassini da Perinaldo alla corte del Re Sole",                                    published by Comune di Perinaldo

 
  
 
 
 
 

A mathematical simulation of the Sun-dial 

Following a simulation and taking into sole consideration the height at which the hole was bore: 27,07 m.; using the data of Perseus, relevant to the Sun of that epoch, it is possible to foresee that during the Summer solstice, in 1655, the image the Sun designed on the floor was an ellipse of axis: 26,5 cm and 28,4 cm. Following the same procedure, during the Winter solstice of that same year, the image of the Sun at 66,82 m. from the hole’s vertical, forms a very long ellipse of axis: 68,2 cm to 181,7 cm. 

The above values may be slightly different from those measured by Cassini, but they are exact and sufficient to prove two important discoveries he made. 

Cassini, using his beloved Latin, called his instrumentation "Heliometrum" and "certified" the different distance between Sun and Earth; the ellipticity of the Earth’s orbit in accordance with Kepler’s calculations; the position of the apogee; the obliquity of the ecliptic. Hence, the truth on Earth’s motion revolving about the Sun, supposed by Aristarchus of Samo in the 3^rd century B.C., and  later on reconfirmed by Copernicus and by Galileo in his "Dialogues" (initially this work received the Church’s Imprimatur) was finally made known to all the scientific world. 

Cassini made many other discoveries of "great importance": he understood that the refraction of the atmosphere was active also at heights of 45°, which made it possible to refine astronomical tables in a more accurate way than in the past... measuring Mars’s parallax, he proceeded to evaluate the Astronomical Unit and to dimension the whole Solar system with the use of Kepler’s third law. 

The "Heliostatic" assumption had been supported, but never proved, due to the equivalency between Tycho’s system and the Copernican system [only the origin of the system of the axis of reference changes, for Tycho it was the Earth, for Copernicus the centre of the Earth’s orbit]; the peculiarities, such as the phases of Venus, are true for both, while they are not for Ptolemy’s system. 

On 26 November 1655, Cassini presented his first project of the Sun-dial to Queen Christine of Sweden, who had stopped over in Bologna, during her long journey from Stockholm to Rome (her new residence). The Jesuits had convinced her, secretly, to join the Catholic religion. 
 
 
 

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A photograph of Cassini’s "Heliometrum"

Cassini’s great "Winter Solstice" ellipse

CONTROVERSIA PRIMA ASTRONOMICA ad maximum Heliometrum D. PETRONII examini exposita

As, by just mere observation, it was possible to see the Sun moving in an unequal way during its annual motion: slower in Summer, quicker in Winter, some pretended such inequality to be absolutely and solely apparent: thus, assuming true this assertion, Aristotle and the astronomers of his times assumed the Sun always moved in an uniform way about Earth itself, within a circumference, however, not so that the Earth’s centre would be the same as the Sun’s, but that the Earth, being stationary, would be positioned between the centre of the circle itself and the Winter section of the Sun’s course; hence, nearer to the Winter side than to its opposite: the Summer side.  

Thus, just as much as in the Winter course, which is nearer and in the Summer course, which is farther from the ascertained equal parts, according to the laws of perspective, the Winter side, which is nearer, will show bigger, the Summer side, which is farther, will show smaller. Thus the Sun, though covering both sides, in the same interval of time, seems, however, to cover an orbit that is longer in Winter and shorter in Summer. It will appear to be quicker in Winter, slower in Summer, even though its motion, on both sides, is identical. In other words, they consider that much distance of the Earth from the centre of the annual course of the Sun sufficient to represent all the ascertained inequality of the Sun’s motion. Therefore, through a sole circle, eccentric to Earth, into which the Sun proceeds with uniform motion, Ptolemy and his followers explained the annual motion: similarly did Copernicus, Tycho, Longomontanus, Lansdbergius and many others.  

A few like Kepler and Bullialdo, indeed, acknowledged that the Earth may show itself inside the Solar orbit and nearer to the Winter side, at a distance from the centre, simply that much long, as may be allowed to see through observation. [therefore, eccentricity must not be that of longitudes, but the observed one ...]  

Therefore, they assumed that only half the inequality of such distance: Earth-centre, may be be saved; the remainder part of the inequality is indeed assumed to be peculiar of the planet itself, that, in a part of its orbit, moves slow and quicker at its opposite, so that all the observed inequality seems formed both by a physical and optical component ... » 
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« Some assumed the yearly inequality was due to simple optical reasons. While others assumed it was due to a mix of both physical and optical causes.  

Taking into consideration the hypothetical variation of the distance Sun - Earth, values attained are quite different one from the other: for one the approach of the Sun towards Earth from apogee to perigee is worth approx the twenty-eighth part of all the Solar mean distance; for the other it is the fifty-sixth part. Should latter be true, the different variation of the Sun’s apparent diameter becomes comprehensible.   

Above is reflected also in the eclipses: as so, the Moon, in accordance with the solution selected, may cross the Solar disc quicker or slower.   

In astronomy, the solution of such problem is so important that it becomes impossible to proceed further without it.   
By means of observation, a solution may be approached following three ways, but all of them are limited, difficult and subject to changes, therefore all are dangerous.   

First the length of the year, or the time the Sun takes to complete a full revolution, must be measured to verify if such measurement differs from measures taken by others through observation; secondly, it is necessary to ascertain which places the Sun occupies in the Zodiac, specifically in those periods, mean between solstices and equinoxes; lastly, time between each single observation must be measured.   

The apogee and the eccentricity of both hypotheses must be investigated with these instruments of research. The distance from the apogee of each single observation and the equation of the mean motion must be known.   

The observation of the Solar apparent diameter near to apogee and perigee must be added and roughly also the distance of the apogee and perigee, using a method totally valid for both hypotheses.   

This is an Astronomical Labyrinth, the thread of Ariadne, twisted into a Gordian knot. All those who are looking for a solution must ask for it both Apollo (the Sun) and the Heliometer of San Petronio. ».

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Cassini proves Kepler’s assumption scientifically

First, stepping back into the past, Ptolemy and his way to conceive eccentricity must be revisited... Ptolemy assumes the Sun eccentric to Earth with value ae = 0,00334 ... 

Kepler, with the use of triangulations: Sun-Earth-Mars, and being Mars, after two years, positioned in the same sidereal position, calculates the eccentricity of the orbit. Value  is ae = 0,00167 ... ( half that of Ptolemy ) ... 

To prove Kepler [what the motion of the minor eccentricity is missing, is due to the motion that is peculiar to the planet itself, as per Kepler’s 2^nd law] Cassini must just discover which of the two is the correct eccentricity... 

The Sun’s dimension during the two solstices [quite alike to apogee and perigee] must be measured... eccentricity shall be given by the formula  ".(mean diameter / minor diameter) - 1 "  ... result shall confirm Kepler’s idea: the Sun [or the Earth- there is no difference] moves with a motion almost elliptic [similar, at a preliminary stage, to an eccentric circle,]. This motion is regulated by the 2^nd law ... 

Assuming Cassini measured the dimensions of the ellipses the Sun traced on the floor, using this input ... and trigonometry: calculate the diameter of the Sun as if it was vertical to the hole; ... the two numbers are now comparable and calculation is possible ... 
 
 
 
 

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