by J. L. Heilbron
Annual Invitation Lecture to the Scientific Instrument Society, Royal Institution, London, 6 December 1995
During the 17th and 18th centuries cathedrals in Bologna, Rome, Florence, and Paris served as centers of astronomical investigation - or, to speak in our overheated language - as state-of-the-art, world-class, solar observatories. Solar observatories, because they made possible the careful and continuous study of the sun; state-of-the-art, because they housed instruments built to the most exacting standards and produced data corrected in accordance with best astronomical practice; world-class, because, for their purposes, no better or more precise instruments existed anywhere. The earliest of these cathedral observatories was the great church of San Petronio in Bologna.
The instrument consists of two separate pieces and no moving parts. One piece lies on the floor; it is a perfectly horizontal rod running due north for some 67 meters from a spot under one of the side chapels to the front door of the church. The other part is a small hole 2.5 cm in diameter set in a horizontal metal plate fixed in the roof of a chapel. The hole is permanently open so as to give free access to the sun's rays around noon throughout the year (figures 1 and 2).
1. The meridiana of San Petronio, Bologna. The construction locating the places where the sun's rays fall when it enters the several zodiacal signs goes back to Vitruvius. It uses a circle of the radius R centered on the hole in the roof and a smaller circle of radius r = Rsin
(where is the obliquity of the ecliptic) centered on the line of sight to the equinoctial noon sun. From Gian Domenico Cassini, La meridiana del tempio di S. Petronio (Bologna, 1695; Astronomy Dept., Univ. of Bologna).
2. View of the meridiana of San Petronio showing the sun's image just after noon in mid-September; a few days later, at the autumnal equinox, the image will fall on the lower transverse marble plaque. From J. L. Heilbron, "Fisica e astronomia nel Settecento", in W. R. Shea, ed., Storia delle Scienze, 2. Le scienze fisiche e astronomiche (Turin, Einaudi, 1992), 318-443, on p. 349.
The size and shape of the noon image change throughout the year on the same principle that shadows lengthen during the afternoon. Figure 3 shows the sun's disk, GJHI, greatly exaggerated, and the image PR of the disk's diameter in the plane of the meridian as cast by the hole O on the floor NT. The image QR of the upper half of the sun GS* is smaller than PQ, that of the lower half S*H. If the sun were lower, its image would be larger, since P would move further north than would R; and if it were higher, its image would be smaller, since P would move further south than would R.
The noon image is not a circle of diameter PR, but an ellipse, with its long axis along the meridian. The image of the disk's diameter IJ perpendicular to GH, UV in figure 3, depends on the sun's altitude but, unlike the image PR, is symmetrical around the image of the sun's center Q. The degree of ellipticity changes dramatically with the sun's noon altitude. When it is highest in the sky, at the summer solstice in June, its image is virtually circular; at the winter solstice in December, it is very ovoid. Figure 4 suggests the disparity.
3. Schematic of the image VPUR made by the sun's disk GJHI through the hole O; NPR is the meridian line.
4. Almost-to-scale diagram of the sun's images at the solstices at San Petronio; to be true to scale, the minor axis of the large ellipse should be half the size shown.
In a volume published in Bologna in 1695 to describe the refurbishing of the meridiana, which by then had seen forty years of service, there is an extraordinary fold-out plate, almost six feet long and over a foot wide (figure 5). The small circle at the center of the figure is the life-size image of the mid-summer sun; the half ellipse containing the entire plate is the true-to-life outline of half the image of the mid-winter sun; the shading of the ellipse and the annulus of the circle of width equal to the radius of the hole represent the penumbra across which the glow of the central image gradually diminishes to darkness.
5. Fold-out plate displying particulars of the meridiana at San Petronio; the line appears at the bottom together with the transverse marbles indicating the place of noon image when the sun enters the various zodiacal signs. The last marble on the left contains the meridiana's "vertex", the point directly under the hole. From Gian Domenico Cassini, , La meridiana del tempio di S. Petronio (Bologna, 1695).
The other diagrams on the plate are equally worthy of attention. Number 8 (our figure 1) shows the meridiana in situ and the sun's noon altitudes on the days on which it enters the various zodiacal signs.
The secret of the construction is to make the ratio of the radii of the two vertical circles equal to the sine of the obliquity of the ecliptic. Elsewhere on this inexhaustible plate are the plumb line with which the builders found the vertex V directly beneath the hole (number 2), the apparatus with which the rod was leveled (number 3), and a gauge with which distances between the scale divisions along the meridiana were read off (number 5). The enlargements show the plumb line and the water basin in which the line's oscillations were damped (figure 6), the awkward leveling apparatus adapted to a water-filled canal that gave the horizontal reference surface (figure 7), and the gauge (figure 8). A notable feature of the meridiana is that San Petronio, although constructed centuries before it was there, seems to have been expressly designed to receive it. Figure 9 shows it in plan, just squeezing between the massive pillars supporting the roof. Since the line had to run directly north-south, and since the pillars could not have been moved, the fact that geometry permitted the transformation of the cathedral into an observatory was at once an encouragement to the work and an indication of its propriety. Indeed, the only worry the church authorities expressed when permitting the construction was that it would not fit.
6. Another detail, showing the plumb bob used to find the vertex; the box ABCD contains water to damp the oscillations of the bob and cross-hairs to locate the box's center.
7. A further detail, showing the device used to level the line: MN is a water-filled trough to give the level to the board ACETDB, which is made vertical by the plumb line hanging from R.
8. The gauge used to interpolate between readings of the scale along the meridiana; it allows determination of the thousandth part of the height, which comes to 0.27 mm.
9. A final detail, showing the tight of the meridiana between the pillars sustaining the nave.
In keeping with the principles of modern grantsmanship, Cassini proposed two purposes for perforating San Petronio, one basic and intended for his academic colleagues, the other applied and addressed to the building's custodians. The applied purpose can be dealt with summarily. It was to find a better value for the length of the year by counting the days and hours between the sun's successive returns to the same solstice or equinox. An exact value of the year is needed to calculate the date of Easter. The basic and primary purpose of the meridiana's builder, Gian Domenico Cassini, was the modest one of reforming astronomy from the ground up. The first step toward this universal improvement was to be an accurate account of the apparent motions of the sun, on which knowledge of many other astronomical quantities depended. Before devising his own theory of the sun, Cassini had to settle a great controversy among astronomers over the jabberwocky question of the bisection of the eccentricity. Only his new meridiana, he said, could render a decisive answer.
The custodians knew as little about the bisection of the eccentricity as other normal people. Cassini might have explained it to them along the following lines. The simplest geometrical representation of the sun's apparent annual motion is that it moves regularly about a circle centered on the earth, as in figure 10. The year is supposed to begin when the sun passes through the Vernal Equinox (VE). Since the solstices and equinoxes are situated 90° apart in the ecliptic, the orbit depicted in figure 10 gives the same length to all the seasons.
But the seasons are not equal. In the northern hemisphere, summer is several days longer than winter, and spring a few days shorter than fall. The ancient astronomers regarded the apparent inequalities in the sun's annual motion as an optical illusion. That was an admirable demonstration of the power of mathematics, and of method, over experience. All that was necessary was to suppose that the center of the sun's orbit does not coincide with the center of the ecliptic. Let their separation be ae, where a is the radius of the orbit and e is some number considerably smaller than one (figure 11).
10. The sun's orbit, represented as a circle with the earth E as a center. The outer circle is the ecliptic (the sun's apparent annual path projected against the sphere of the heavens); VE signifies the vernal equinox, an intersection of the ecliptic with the celestial equator; AE, the autumnal equinox, the other intersection; WS and SS, the winter and summer solstices, when the sun is at its maximum angular distances from the equator.
11. The sun's orbit, represented as a circle whose center S is offset from the earth E by the eccentricity ae; A, the orbital point most distant from E, is called the apogee; 
the closest point, the perigee; A, the line of apsides.
If the sun moves uniformly about its orbit, it will appear from the earth to slow down when furthest (point A) and speed up when closest (). Figure 12 illustrates the effect. Let the sun go from A to trough an angle 
as seen from the center of the sun's orbit S; it will appear to move through the smaller angle 
as seen from the earth. That is precisely the behavior it displays around the summer solstice. Hence EA must point somewhere around the beginning of Cancer. Similarly, if at perigee the sun ran through the same angle as seen from the center of its orbit, it would move through an angle 
in the ecliptic, where, evidently, >. Around , therefore, the sun appears to move faster than the mean speed. That is how it behaves in midwinter.
From the observed lengths of the seasons the ancient astronomers worked out values for the eccentricity e and for the angle between A and the line joining the solstices. The method gave very good results: using it, I find the eccentricity today to be 0.0334 and A to lie in 13° of Cancer. Modern astronomy uses the values 0.0167 and 13°34'. The agreement for angle is good, but that for eccentricity very bad, out by a factor of two - and, what is of first importance, exactly a factor of two. An indication of this doubling had already reared its head when Cassini built his meridiana in San Petronio. It lay behind his boast that he and his meridiana would reform astronomy.
Here is what was at stake. In figure 13, E as usual is the center of the ecliptic and C is the center of the circle to which P is confined. Let P revolve equably not around the center of its orbit C, but around the "equant" X. To help visualize the motion, the figure has an equant circle on which the point would revolve with constant velocity if its orbit were centered on X instead of C. From E, P appears in the zodiac in the direction EP. With what in retrospect appears to have been uncanny intuition, the inventor of this device, Ptolemy, set X as far on one side of C as he put E on the other. He thus bisected the eccentricity, if one understands by eccentricity the distance between the center of observation and the center of motion.
12. Diagram to show that, on the eccentric model, the sun appears from the earth to move faster around perigee than around apogee.
13. Diagram of an orbit (of the point P) regulated by the equant X situated a distance ae/2 from the center C of the orbit along the line of absides; E lies as far from C as X, but on the opposite side; hence the "bisection of the eccentricity" ae.
The equant model seems uncanny because it was a very close approximation to motion in a Keplerian ellipse. A planet moving around the focus occupied by the sun in accordance with Kepler's laws moves almost uniformly around the unoccupied focus X.
Copernicus disliked the equant and did away with it, at least in principle; Kepler used it to great effect in analyzing planetary motions before he discovered the laws that regulate them. In this analysis Kepler introduced the equant for the first time for the regulation of the motion of the sun. Many astronomers objected to his innovation. Thus began the great and obscure controversy between adherents of Ptolemy's traditional solar theory, which made the inequality of the seasons entirely a matter of perspective, and proponents of Kepler's "bisection of the eccentricity", which ascribed half the inequality to a real change in the sun's velocity.
Cassini reasoned that the solar diameter, as measured on the floor of San Petronio, could be taken as a proxy for the solar distance. The two hypotheses between which he proposed to decide predicted slightly different values for this distance. The greatest differences, and, consequently, the best chance of detecting them, occur around apogee and perigee, that is, around A and .
Figure 14 indicates the two hypotheses (there is of course no equant, no Xp, in Ptolemy's theory of the sun). It is easy to see that the difference should be half as much on Kepler's theory as on Ptolemy's. We have for the difference 
X in the equant theory, X = a(1+e/2) - a(1-e/2) = ae, and for the corresponding difference P on the pure eccentric or perspectival theory, P = a(1+e)-a(1-e) = 2ae. To detect these differences Cassini had to be able to measure angles to within a minute or so of arc at an instrument that did not move throught the year. He could, and did, and found for Kepler.
14. Diagram to indicate the difference between apsidal distances in the purely perspectival theory with full eccentricity, depicted in figure 11, and Kepler's theory with equant and bisected eccentricity, depicted in figure 13.
Most of observers at San Petronio were clerics. Cassini himself narrowly escaped being one of them. Educated, like most other Catholic savants of the 17th century, by the Jesuits, he would have joined their Society but for the scruple that he had no calling for it. That did not prevent him from receiving the patronage of the church and the instruction of its leading astronomer, Giovanbattista Riccioli. Riccioli and his collaborator Francesco Maria Grimaldi, also a Jesuit, who has enduring fame in the history of science as the discoverer of optical diffraction, began to work at San Petronio immediately upon its becoming an observatory. Their combined observations fill about 25 pages, at 15 observations per page, in the register published in 1735 by Eustachio Manfredi, who became professor of astronomy at Bologna early in the 18th century. Each entry includes a description of the weather, the distances of the sun's limbs from the vertex corrected for the penumbra, and the apparent diameter of the sun, all given to seconds of arc (figures 15, 16).
15. Title page of Eustachio Manfredi's compilation of the data obtained in over seventy years of observing the noon sun at San Petronio.
16. A sample page of observations from Manfredi's book.
In total, Manfredi's register, which runs for 80 years, records some 4500 observations made primarily by the mathematicians at Bologna and their assistants. The most notable of Manfredi's collaborators was Anders Celsius, the Swedish savant now universally known for his temperature scale, who worked at San Petronio for about seven months in 1733/4. One of the most useful results of these systematic observations was Riccioli's conversion to Kepler's approach to planetary orbits. Another was the untangling by Cassini of the effects of parallax and refraction. His improved tables of these effects set aside many of the difficulties that had stymied the astronomers of his time. A third important outcome of observations at San Petronio was information about the obliquity of the ecliptic. For centuries astronomers had debated whether the inclination of the earth's axis to the plane of its motion was constant. Riccioli said yes, Cassini no, then yes, then no. Eventually Manfredi settled the matter, as far as the measurements at San Petronio could reach. Consulting his big register of observations, he made out that between 1656 and 1733 the obliquity had progressively shrunk by just under a second of arc a year. The true value then was around a half a second a year. Nonetheless, the observers of San Petronio have the honor of having first detected and measured a process that, if unabated, would abolish the seasons in less than 2000 centuries.
Around 1700 the reigning Pope decided that Rome's lack of a church observatory was an intolerable slur and impoverishment. His stated purpose in transforming one of the most remarkable churches in his city of churches into a competitor of San Petronio was to determine yet again the parameters needed for reform of the calendar. The matter had heated up because the Protestant German states, which had rejected the Gregorian reform, now proposed to accept its civil but not its religious usage. The Pope, hoping to secure their total adherence, made the gesture of reopening the determinations on which Gregory had relied in fixing the Easter canon. Cassini was a long-distance consultant to the project (he had been a star in the Parisian academic firmament since 1670). He observed that a redetermination of the year would not help the calculations but that a new church observatory would be a useful complement to San Petronio. The Pope assigned the work to Francesco Bianchini, an antiquary and astronomer in minor orders who had climbed the ladder of church patronage through family connections, good luck, and brains. The church selected to house the Roman meridian was Santa Maria degli Angeli, which had been designed by Michelangelo under commission from a previous Pope to make a place for Christian worship in the ruins of the Baths of Diocletian. The building had the merits of an unobstructed view to the south and walls and floors that had long since ceased settling. Figure 17 shows the general layout of Bianchini's meridiana and some of its special features, the northern gnomon (another name for the meridiana) and the star points. The northern gnomon allowed observation of circumpolar stars via a telescope that made use of the scale along the meridiana to measure altitudes; the points marked the daily course of the image of certain bright stars and also the course of the sun on the day when the Pope came to see the completed observatory. The concentric ellipses under the southern gnomon indicated the diurnal paths of the pole star at 25-year intervals and so gave a lively representation of the precession of the equinoxes.
17. The meridiana of Santa Maria degli Angeli in Rome; the south is to the right. Note the rays from the noon sun (from the right) and from the pole star (from the left). Compare the person sighting through a telescope in the second bay from the north in figure 1 above. From Francesco Bianchini, Ennaratio...de nummo et gnomone Clementino (Rome, 1703).
In the 1750s a Sicilian Jesuit named Leonardo Ximenes refurbished an old inexact meridiana in the Duomo of Florence (figure 18). The hole admitting the light was so high in the vault that only a small section of the meridiana to receive the sun's image for a few weeks on either side of the summer solstice could fit comfortably in the church. The rays of the midwinter sun would have struck the pavement far outside the cathedral. Even this section is no longer easily visible. The main result of Ximenes' work was a better value than Manfredi's for the rate of decline of the obliquity of the ecliptic.
18. The meridiana of Santa Maria del Fiore (the Duomo) in Florence. A sun beam descends from a hole high in the vault around midsummer day. A month or so later the beam hits the north wall at the left-hand corner of the figure. The Florentine meridiana is almost twice as high as Rome's and Bologna's combined. From Leonardo Ximenes, Del vecchio e del nuovo gnomone fiorentino (Florence, 1757).
Around the middle of the 18th century, owing to improvements in mountings and lenses, telescopes became more useful and exact instruments for solar measurements than meridiane. Nonetheless, a few impressive ones were placed in Italian cathedrals after 1760. The purpose was not to make observatories but to set up chronometric stations, where parishioners could come at noon to set their clocks and watches. That was the function of the meridiana at the Duomo in Milan, which still works; since the cathedral was too narrow to hold the entire meridiana, the path to the winter solstice had to be run up the north wall. When Cassini and Bianchini made their meridiane, Copernicus and Galileo both had a place on the Index of Prohibited Books and the Holy Office's finding that heliocentrism was opposed to Scripture and absurd in philosophy still stood. San Petronio, the cathedral of Bologna, became a solar observatory only 25 years after Galileo had suffered in Rome, although Bologna was a papal state and the church's head astronomer, Riccioli, who was headquartered there, made no secret of the official hostility to Copernicanism. Whatever his private views, he had to reject the theory of a moving earth because the Holy Office had condemned it. Yet Riccioli, Grimaldi, and their protégé Cassini built and operated an instrument with which Cassini expected to show that the sun described the same sort of path around the earth as, according to the arch-Copernican Kepler, the planets do about the sun. Of course, their demonstration did not confirm Copernicus, but it showed that, from the point of view of solar theory, the sun, or, what is the same thing, the earth, can be treated as a planet. Or, to put the point in a few words (they are those of the Astronomer Royal of England, John Flamsteed), Cassini proved that "the Suns Excentricity is bisected as the Copernicans affirme". The condemnation of the systema terrae motae gave Jesuit astronomers a headache. To discharge their role as up-to-date schoolmasters to Catholic Europe, they had to master all philosophical novelties and academic fads. Riccioli knew the work of Copernicus and his followers very well, and described it in detail in order - such was the pious fraud - that the faithful would know what they were rejecting and why. Cassini had a freer hand. But he too felt bound to obey the dictates of a church in which he reposed an abiding faith. He always spoke of the motions of the sun, not of the earth, when dealing with the two bodies alone. Treating the sun as if it were a moon was perfectly satisfactory since the mathematical theory is the same irrespective of which body - sun or earth - revolves about the other. When he wrote about the planets, however, he sometimes used a Copernican vocabulary without apology. Manfredi wrote a large textbook on the same principle. The first 200 pages have the sun going around the earth; the last 200, the planets going around the sun. This manner of thought found an instructive expression in 1710, when the church did not try to suppress a reprinting of an unexpurgated version of Galileo's Dialogue on the two chief systems of the world, although it was still officially condemned. The general proscription against heliocentrism dropped from the Index of Prohibited Books in 1758. Sanctions against it remained, however, until a mathematician at the Collegio Romano, who had been denied a license for a text on optics and astronomy, appealed successfully to the Pope to have them removed. That was in 1820, after the church had authorized many reprintings of Galileo's Dialogue and Catholic astronomers everywhere had been teaching Copernican theory for years without offering apologies or suffering penalties. The establishment of the observatory of San Petronio was part of the effort of Catholic savants to keep up with, and even to lead, the development of the exact sciences and natural philosophy of the 17th century. If their work provided strong evidence in favor of a view condemned by the Holy Office, nothing would happen provided that the happy discoverer did not insist on interpreting his discovery obnoxiously. An example is Cassini's realization that the moons of Jupiter obey Kepler's laws, which made a perfect analogy between them and the planets, on the one hand, and Jupiter and the sun, on the other. But, as Flamsteed pointed out to Newton, "to be thought a good Catholic [Cassini] says nothing of it". By the time that Riccioli's generation, which was in its prime when Galileo was struck down, had passed, Catholic astronomers even in Italy had found ways to advance their science, and to talk about their advances, without coming into conflict with their church; and most senior officials of the church, the bishops, cardinals, and popes, had grown either indifferent or friendly to modern cosmological ideas. In general historians have paid too much attention to Galileo's troubles and too little to the question how the Roman Catholic Church contrived to withdraw from the stupid and exposed position in which its denial of heliocentrism left it. The author has rushed into this vacuum and hopes to reappear soon with an answer.
Address: J. L. Heilbron, Office for History of Science and Technology, 470 Stephens Hall, University of California, Berkeley CA 94720, USA.