The Catholic Church is portrayed as an institution that has been willing to impose the heavy hand of censorship to suppress scientific freedom. But nothing could be further from the truth. Beginning with the recovery of ancient learning in the twelfth century and continuing through the Copernican upheavals and on even into the Enlightenment, the Roman Catholic Church gave more financial and social support to the study of astronomy - Copernican and otherwise - than did any other institution.
THE COPERNICAN REVOLUTION, which displaced the earth from its ancient seat at the center of the universe, is often cited as the beginning of modern science. Not only did Copernicanism correct an error dating to antiquity by substituting the sun for the earth as the center of the solar system, but it also challenged the cosmological teachings of the Roman Catholic Church, thereby setting up the new scientific cosmology as the rallying point of opposition to dogma. The culmination of the Copernican triumph is the morality play that was enacted in the trial of Galileo. There the Church is portrayed as quite willing to impose the heavy hand of censorship, and worse, to suppress ideas that would one day open wide the windows of scientific freedom.
The trouble with this story is that, like most other myths, it promotes a kernel of truth about the protagonists to a general, symbolic level that purports to summarize major historical currents. In particular, the momentary rigidity of the censorship and Pope Urban VIII has engendered the belief, found even in the best modern historians, that to quote the late historian and philosopher of science Richard S. Westfall the Church's action in the matter of Galileo made "Copernican astronomy a forbidden topic among faithful Catholics for ... two centuries."
But nothing could be further from the truth. Beginning with the recovery of ancient learning in the twelfth century and continuing through the Copernican upheavals and on even into the Enlightenment, the Roman Catholic Church gave more financial and social support to the study of astronomy Copernican and otherwise than did any other institution. The reason for such lavish attention to astronomy is that it was absolutely central to the authority of Rome: the Church had a pressing need to establish and promulgate the date of Easter.
Christian theologians had decreed at the Council of Nicaea in 325 A.D. that Easter should be celebrated on the first Sunday after the first full moon after the vernal equinox (the day of the equinox is a day when the hours of daylight and darkness are equal). The full moon that follows the equinox can readily be observed in principle and often in practice. One then needs only to recognize the equinox, wait until the next full moon, and declare the following Sunday to be Easter.
That straightforward procedure gives the right date for Easter, but not enough time to prepare for it. Therein lies an administrative predicament. To further complicate matters, the equinox and the full moon take place at different times at different places on earth, as, of course, does Sunday. Even if solar and lunar observations everywhere in the far-flung congregations of the Church were entirely accurate, Easter could well end up being celebrated on different days in different places. That was unacceptable to an organization struggling to make good its claims to unity and universality.
To avoid those troubles, the Roman Catholic Church calculated the dates of Easter several years in advance, distributed the information in tables and required observance on the Sundays specified in the tables. But the calculations were neither easy nor accurate. Neither the lunar month nor the solar year comprises a whole number of days, and the year does not comprise a whole number of full moons. Even the days of the week do not recur yearly on the same calendar dates. The success of the calculations depended on knowing the exact average values of the periods between successive vernal equinoxes and between successive full moons.
Administrators frequently have to make decisions on insufficient data. The popes forced consensus on the relevant numbers and calculating procedures in the sixth century. By the twelfth century, however, they could see that the pragmatic approximations for the values of the year and the month assumed by their predecessors no longer gave Easters in harmony with the heavens. In that emergency, the popes encouraged the close study of the apparent motions of the sun and moon. The experts consulted ancient Greek mathematical texts that were then, luckily, just being translated into Latin from Arabic intermediaries. The most important of the texts was Ptolemy's Almagest, or "great compilation," which showed how to represent the motions of the stars, the sun, the moon and the planets, as seen from a stationary earth. Ptolemy's fundamental hypothesis, that the earth stands still at the center of the universe, seemed the most obvious and satisfactory basis for an exact astronomy. Not only did it conform to the evidence of the senses, it fit perfectly with the physics of Aristotle, which, by 1200, were also newly available.
The key piece of data for making the Easter calculation was the period between successive spring equinoxes. The most powerful and precise way to measure that cycle was to lay out a "meridian line" (usually a rod embedded in a floor) from south to north in a large dark building, put a small hole in the building's roof or facade, and then observe how many days the sun's noon image took to return to the same spot on the line. The most convenient large, dark buildings for such solar observatories were cathedrals. The accuracy of the results of using a meridian (as the hole and rod were called) depended on the care taken in their installation: correct positioning of the hole, proper orientation of the rod and exact leveling of the floor. The story of the meridians lies at the intersection of many fields now usually held apart: architecture, astronomy, ecclesiastical and civil history, mathematics and philosophy. And the work of the meridian makers shows that men whose careers were underwritten in whole or part by the Church made important contributions to the development of modern astronomy.
FOR THE TWO DECADES BEFORE HIS DEATH in 1574, Cosimo I de' Medici was immersed in immense construction projects. Foremost among them was the elevation of himself, from lowly duke of Florence to the grand duke of Tuscany. The preliminary work of that construction, completed with the help of a friendly pope, Pius V, in 1569, included the conquest of Siena and other neighboring territories, the transformation of the ducal palace into a residence fit for a first-class sovereign, and the renovation (some say, and said, wanton destruction) of the church Santa Maria Novella, the stronghold of the Dominican order in Florence. Cosimo's principal agent in the refurbishing of both palace and church was the artist, architect and historian Giorgio Vasari.
In the early 1560s Vasari began an ambitious project of great interest to Cosimo: the decoration of a new room in the ducal palace with mural maps depicting various regimes around the world. The maps were to be drawn according to the best available geographic knowledge and projected according to sound geometrical principles. Mural maps had decorated important walls elsewhere in Italy, notably in the maritime republic of Venice. But the series designed for Cosimo, some fifty-seven maps in all, was a great novelty, probably the first extensive atlas of the world and certainly one of the least convenient ever made. To create it required a man who combined the skills of cosmographer, designer and painter. The job went to Egnatio Danti, the younger brother of one of Vasari's sculptors.
Danti had learned drawing from his father and mathematics from his aunt. In 1555, at the age of nineteen, he had joined the Dominicans to pursue his studies of mathematics along with philosophy and theology. Danti had just finished preparing himself for a life of teaching and preaching when the call to cosmology came from Cosimo. He thus entered into the well-oiled machinery of joint patronage, by Church and State, through which talented clerics advanced in Italy. His superiors transferred him to their convent in Florence, at Santa Maria Novella, to make it easier for him to commute to the Medici palace. The friar proved himself so prolific that Cosimo eventually requested permission from the Dominican General to have Danti live in the palace to be on call for cosmological conversations. Danti moved in in 1571.
While Danti resided in the palace, he and Cosimo discussed a new project: nothing less than the reform of the Julian calendar. By Cosimo's time the Julian calendar had gained nearly ten days on the true solar year: an inexorably worsening predicament for the celebration of Easter assuming the vernal equinox was to remain where the Nicene fathers had put it, on March 21. Cosimo
likened himself to Julius Caesar, and the prospect of improving the calendar of his model thrilled him. In the words of Danti, writing four years after the grand duke's death, Cosimo always had a heroic mind inclined to the greatest projects, having always been an emulator of outstanding deeds of the ancients and knowing (in addition to its general usefulness) how much glory Julius Caesar had obtained from his reform of the year, which though good is not perfect.
In order to play the part of Sosigenes, Caesar's astronomer, to Cosimo's Caesar, Danti needed to make accurate measurements of the length of the tropical year the precise length of time between vernal equinoxes. Or so he claimed: in fact, the adjusted length of the tropical year ultimately adopted under Pope Gregory XIII in 1582 had been recommended by experts since the thirteenth century. Be that as it may, Cosimo favored Danti's plan. And so, with the same highhandedness that had led to the demolition of the interior of Santa Maria Novella, he caused the officials in charge of the building to permit Danti to knock a small hole in the great circular window in the upper story of the building's southern facade, 21.35 meters (about seventy feet) above the ground.
DANTI'S PLAN WAS TO TURN THE CHURCH into a gigantic camera obscura, or pinhole camera, and the hole was to function as its aperture. It was widely known, even well before the use of lenses in telescopes, that the light passing through such a hole projects a sharp, though darkened, image of the scene outside the hole on any surface behind the hole. (Pinhole cameras remain in use to this day because, though they take a relatively long time to record an image on film, they offer the photographer virtually unlimited depth of field.) The surface in Danti's scheme was the floor of the church, where, in the immense darkened chamber, the bright disk of the sun would project with such detail that observers who knew what to look for would even be able to see sunspots.
Danti laid out a line on the floor running due north from a point immediately under the hole. The line and the vertical from the hole to the floor therefore defined a meridian plane: the sun's rays would enter the church from due south precisely at the moment that, by definition, is the local solar noon, and throw an image of the solar disk onto the line. As the seasons changed, the sun's noon image would move up and down along the north-south line. The moving image would fall closest to the facade in which the hole was made at noon on the summer solstice, usually June 21 on modern calendars in the Northern Hemisphere, when the sun stands highest in the sky, and farthest from that same facade six months later, at noon on the winter solstice. Danti expected to determine the length of the tropical year by counting the number of days between appearances of the sun at the same point on the line at the same time of year.
The sheer scale of Danti's meridian makes it instantly clear why a great church could serve so well as a solar observatory. Between summer and winter solstice the sun's image traveled 47.6 meters, half the length of an American football field, down the nave of the church. The image of the midwinter sun fell more than five times as far from the entrance to the church as that of the midsummer sun. The immense scale of a cathedral made it possible to deploy long measuring instruments that could reveal fine details about the size and motion of the sun.
Unfortunately, before Danti could complete his meridian, Cosimo died. The grand duke's death brought to power his son Francesco, a Caesar who did not want a Sosigenes or anything else that reminded him of his father. Francesco pressured the Dominican General into ordering Danti to repair to a convent outside of Tuscany within twenty-four hours. But it is hard to read the heavens. Danti's superiors reassigned him to Bologna, to perhaps their noblest convent, which housed a rich library, some works of Michelangelo and the bones of the founder of the Dominican order.
WHILE AT LEISURE AT BOLOGNA, Danti built a small meridian, just four meters from hole to floor, in the chamber reserved for proceedings of the Inquisition in his convent; the north-south meridian line ran 6.4 meters along the floor, before the sun's wintertime image vanished up a pillar. What Danti really needed was a much larger building, with an unobstructed view to the south and enough interior space along the north-south direction to accommodate the entire length of a meridian line. And it had to be on a scale big enough to keep alive his hope of determining the exact length of the tropical year. Danti found the church he required in the heart of Bologna, opposite the ancient university: the great basilica of San Petronio, begun at the end of the fourteenth century and not yet given a finished facade, a condition in which it remains today.
The interior of San Petronio offered more than a good technical setting for solar observations. Its huge open nave, paved in the traditional Bolognese style of ruddy and creamy marble slabs, made it a perfect dramatic theater for the daily rendezvous of the sun's image with a meridian line. With the permission of the fabbricieri, the five lay custodians of the building, Danti put a hole in the roof of a side chapel, and, on the floor, he embedded a line embellished with plaques marking the sun's entry into each of the twelve zodiacal signs.
But, alas, the architect of the basilica had placed the piers that supported the nave in such a way that, no matter where a hole was made in the roof, they would block the sun's rays from reaching the pavement around noon at some seasons of the year. Unwilling to abandon San Petronio, Danti did the best he could, and ended up with a line that ran more than nine degrees off the true north-south direction. The nine-degree deviation in itself did not destroy Danti's project, because it would not have affected the determination of the length of the year. But other parts of the installation left it a fairly crude instrument: Danti had not leveled the line carefully, the plate in which the hole was made slipped downward (a common problem with meridian lines), and the position of the meridian's vertex the foot of the perpendicular directly under the center of the hole was uncertain.
DANTI'S SOLAR OBSERVATORY AT SAN PETRONIO languished for seventy-five cycles of the sun's image on the cathedral floor, and the meridian assumed a merely decorative role. But the remarkable part the church was to play in the history of astronomy was still to come, bound up in the equally remarkable career of Giovanni Domenico Cassini.
During his education at the hands of the Jesuits of Genoa in the 1640s, Cassini became so strongly attached to his religion that, but for the scruple that he had no calling for it, he would have entered the Church. Had he done so, he would have risen as high as Danti did, and perhaps higher: the Church needed the services of widely respected and demonstrably Catholic mathematicians all the more after its misguided silencing of Galileo.
Instead of capping his career as an Italian archbishop, Cassini ended his days in Paris, where he served as the astronomer to the Sun King Louis XIV for more than forty years. Once he had established himself in Paris, he sired a dynasty of astronomers royal, who, beginning with himself, ran the Paris Observatory for more than a century. But his first major project was the redoing of Danti's meridian in Bologna in 1655 and his design led to the first church meridian built to modern ideas of precision.
By then Cassini had been appointed to Danti's old chair in mathematics at the University of Bologna, thereby becoming the successor to Galileo's famous student Bonaventura Cavalieri. When Cassini took up his professorship in Bologna in 1651, the fabbricieri of San Petronio were in the middle of a major building campaign. Their plan called for the removal of the masonry pierced for the meridian that Danti had installed seventy-five years earlier, but they wanted to retain the instrument as an unusual and perhaps useful ornament. The fabbricieri asked Cassini to reset the meridian. He replied that Danti's line was too far off north-south to be salvaged for anything other than decoration. Instead, he proposed to put another hole in the roof, higher than the old one, and run a line due north, avoiding the nave piers. He had carefully compared the architectural plans of the church with his own survey and insisted that he would miss every obstacle. Moreover, he planned to put his hole not in the new part of the nave, which he feared might settle, but in the old fourth vault, at a height that allowed the entire length of the meridian line to come within the church.
The fabbricieri paid sculptors, carpenters and masons to take up the pavement, insert the line and remove a corner of an overhanging buttress on the roof. The entire job cost them 2,000 lire, plus 500 lire for their consultant Cassiniin all, about 15 percent of their collective annual income. The sum imperiled their souls, since they faced excommunication if they did not soon repay the large debt they had incurred to complete the nave.
Cassini made the hole for his camera obscura in a plate of metal, with a diameter equal to one-thousandth of the height of the hole above the ground. He had the plate cemented into the roof parallel to the pavement. He determined the height of the hole, about 27 meters, with extreme care, and found the meridian's vertex with a weighted string, whose swings he damped in a pot of water. He penciled in the places where, by his observations and calculations, the sun's image should fall at noon. The workmen then laid out the iron meridian with the help of an elaborate water level, and prepared marble plaques to mark the sun's entry into each sign of the zodiac [see illustration on page 30].
To align the meridian on the church floor and so to determine local noon Cassini drew a circle with the meridian's vertex as center. The image of the sun would cross the circle once in the morning and once in the afternoon on its diurnal path across the church floor. Cassini marked the places where the center of the sun crossed the circle and drew a straight line a chord between them. He then merely plotted the perpendicular bisector of the chord, again on the church floor [see illustration on page 31]. The line he obtained was a precise meridian line. It passed within a whisker of the pillars, but, as advertised, it did not hit anything. It began to appear that San Petronio had been designed to serve God as a solar observatory.
uNLIKE DANTI, CASSINI DID NOT AIM to correct the calendar. Instead, he proposed to give astronomers an accurate and simple procedure directly from observation for establishing the "orbit" of the sun that gave rise to its apparent annual path in the sky.
Orbit carries quotation marks to highlight two differences from its modern meaning. To Cassini, it signified a way to represent the apparent motions according to the simplest plausible geometry; it was not necessarily a real path in space. Moreover, the term could be applied indifferently, either to the sun or the earth. Catholic astronomers were officially bound by the edict against Galileo to identify the sun as the "orbiting" body, but in scientific practice that often meant little. Officials of the Church tended to regard all the systems of mathematical astronomy as fictions. That interpretation gave Catholic writers scope to develop mathematical and observational astronomy almost as they pleased, despite the tough wording of the condemnation of Galileo in 1632. Moreover, the generalized connotation of the term orbit enabled Copernican astronomers to transfer Cassini's descriptions and explanations from the sun directly to the earth, with no expenditure of thought.
Greek astronomy rested on the rule, which amounted to a definition, that all orbits should be circles or components of circles. Furthermore, the sun, the moon and all the planets should move uniformly around their circles' centers. Thus, the simplest geometric representation o£ the sun's apparent annual course would be a circle centered on the earth.
In practice, however, the rules of Greek astronomy could not be observed fully without undue complications. The solstices and equinoxes are ninety degrees apart on the sun's apparent yearly path through the heavens, and a concentric circular orbit for the sun would represent the lengths of all the seasons as the same. But if there is anything plain in astronomy, it is that all the seasons are not equal in length. In the Northern Hemisphere summer is several days longer than winter is, and spring a few days shorter than fall. Moreover, the sun appears to move more slowly against the background of stars in the days around the summer solstice than it does in the days around the winter solstice.
The solution adopted by the Greeks long before Ptolemy was to explain the disparity in the seasons as a change of perspective. The earth, they argued, is a short distance away from the center of the sun's circular orbit [see illustration below]. The diameter of that orbit that passes through the earth's center was called the line of apsides; it also passes through perigee (or, to use Copernicus's term, perihelion), the sun's closest approach to the earth, and apogee (aphelion), the most distant point on the solar orbit. Viewed from the earth, the Greek model of the orbiting sun does indeed bring the sun slightly closer to the earth, then slightly farther away, in the course of the year. How much closer and farther, Ptolemy expressed as the ratio of the small offset of the earth to the much larger radius of the sun's circular orbit, a ratio he called the eccentricity o£ the orbit. The eccentric model could also explain why the sun appears to slow down near apogee and speed up near perigee. Any circular arc through which the sun moves uniformly in a given time subtends a smaller angle as seen from the earth at apogee than it subtends from the center of its orbit. Thus, as observed from the earth, the sun appears to slow down. Similarly, that same arc as seen at perigee subtends a larger angle than it subtends from the center of the sun's orbit; the sun appears from the earth to speed up.
FROM THE ECCENTRIC MODEL and the observed lengths of the seasons, ancient astronomers worked out values for the eccentricity and for the angle between the line of apsides and the line joining the two points on the sun's orbit corresponding to the winter and summer solstices. One can show both the strength and weakness of the model by first calculating those quantities using Ptolemy's methods, but substituting the modern-day lengths of the seasons (they have changed since Ptolemy's time), and then comparing the results with the corresponding values as determined by modern astronomical methods. The angle between the line of apsides and the line of the tropics so calculated comes out to be twelve degrees, fifty-eight minutes of arc. That differs by only slightly more than the apparent diameter of the sun from the modern astronomical value of thirteen degrees, thirty-four minutes. Either value puts perigee on about January 3 when the winter solstice falls on the preceding December 21, and puts apogee on about July 4.
Much less impressive, however, is the match between the current value of the eccentricity of the earth's orbit, calculated as the Greeks would have done, and the modern value: the former is 0.0334; the latter is 0.0167. The numbers are out by a factor of two and-what is of first importance the discrepancy is exactly a factor of two.
That factor of two was closely related to one of the most brilliant and controversial devices of ancient geometry, the equant point. Ptolemy introduced the equant to describe the motions of the planets. (He did not apply the equant to the motion of the sun.) An equant is a point around which a body or a point moves with uniform circular motion. The way it works is best illustrated by example: In describing the motion of a planet around the earth, Ptolemy set the equant point on the line of apsides, so that the center of the eccentric orbit in the old Greek model fell midway between the equant and the earth. The planet moves, as usual, around the eccentric circle, but, Ptolemy asserted, its motions along that path are not uniform. Instead, it moves at a constant rate around the equant point.
With seemingly uncanny intuition, Ptolemy had placed the equant point precisely where the resultant orbital motions made the best approximation that was available to him of a modern elliptical orbit. But Ptolemy was not so lucky in treating the sun; there he allowed the old perspectival theory to stand, though adding an equant to describe the motion would have more accurately reproduced the observations of the sun's path in the sky.
ELLIPTICAL ORBITS WERE FIRST PROPOSED by the early seventeenth-century German astronomer and mathematician Johannes Kepler. According to Kepler's laws, each planet in the solar system moves in an elliptical orbit with the sun centered at one focus of the ellipse. Kepler also maintained that the planets move in such a way that the line between the planet and the sun sweeps out equal areas of the space bounded by its orbital path in equal times.
Evidently, a planet moving in accordance with that law moves fastest when it is closest to the sun, and slowest at its most distant point. Hence, Kepler's laws recover a fundamental feature of Ptolerny's eccentric solar orbit. The equant model closely approximates the motion along a Keplerian ellipse because a planet moving around the focus occupied by the sun in accordance with Kepler's laws moves almost uniformly around the second, unoccupied focus. Hence a better representation of the sun's apparent motion than the Ptolemaic model is an eccentric circle with an equant point. Kepler tested such a model during the search that led to his discovery of the elliptical orbits of the planets.
The diameter of the sun's orbit (or, in the Copernican model, the earth's orbit) along the line of apsides, as well as the timing of the sun's appearances at perigee and apogee, agree exactly on the Ptolemaic and the Keplerian model. In Kepler's model the earth's orbit is an ellipse whose major axis is the line of apsides and whose center coincides with the center of the eccentric solar orbit in Ptolemy's model [see illustration on preceding page]. One key difference between the two is that the distance between the center of Kepler's ellipse and the focus of the ellipse that is coincident with the sun is equal to half the distance between the earth and the center of the eccentric circle in Ptolemy's model. Hence in Ptolemy's model the sun is slightly closer to the earth at perigee, and slightly farther away at apogee, than it is in Kepler's.
THUS BEGAN WHAT JOHN FLAMSTEED, England's first astronomer royal in the seventeenth century, called, when writing a generation after its resolution, a "controversy ... of no small moment." By the early 1650s, when Cassini joined the university at Bologna, debate raged between the whole and half-eccentrics.
The bearing of observation on the great question of the factor of two can readily be appreciated. In Ptolemy's theory the distance between the two bodies at their closest approach is smaller than it is in Kepler's theory, and at their greatest separation the Ptolemaic distance is the greater. But the differences between the predictions of the two theories are so small that they could not be observed directly.
Fortunately, a convenient substitute for the sun's distance from the earth is the apparent diameter of the disk of the sun, which is inversely proportional to its distance. With the great meridian line at San Petronio, it became possible to measure the disk diameter accurately enough to distinguish the two theories. And so Cassini boasted that he could resolve the great but obscure controversy between adherents of Ptolemy's traditional solar theory and proponents of Kepler's "bisection of the eccentricity," with its Copernican associations.
The difference between the size of the sun's image at apogee and that at perigee in Kepler's theory is slightly larger than the corresponding quantity for the eccentric circle. To discriminate between the two cases, Cassini had to be able to measure the diameters of the images to within 3 percent at the summer solstice and 0.5 percent at the winter solstice. In neither case could he tolerate an error of more than 8.5 millimeters. Only because of the great height more than twenty-seven meters of the hole for the meridian at San Petronio was such a measurement possible. It would not have been possible, for instance, with the telescopes of the day, because the images they formed were much smaller than the image on the cathedral floor.
Even at San Petronio the measurement did not come easily. The sun's image was neither sharp nor stationary. Instead it stuttered and fluttered on the cathedral floor, flickering even on a clear day if the atmosphere was not perfectly calm. The penumbra of the image, furthermore, always shaded off into an indistinct boundary. Fixing the diameter of the solar image was consequently often as much art as science.
Yet experienced observers knew how to get exact and consistent results. With his meridian Cassini found that the sun's apparent diameter at apogee is thirty-one minutes, eight seconds of arc, and that at perigee it is thirty-two minutes, ten seconds. The observations were confirmed by independent observers, including the leading Jesuit astronomer Giambattista Beccaria, whose comprehensive treatises, provocatively entitled Almagestum Novum and Astronomia Reformata, served both Catholic and Protestant astronomers as essential texts. The observations unambiguously agreed with the predictions of Kepler's model.
Thus the Jesuits confirmed the bisection of the eccentricity, implicitly supporting Kepler's position and the Copernican theory they were forbidden to teach. Observations made in Bologna, in the heart of the papal states, had amassed unimpeachable evidence in favor of a theory that had been condemned by the pope and the Inquisition only twenty-five years before Cassini planted his meridian in San Petronio.
J.L. Heilbron. "The Sun in the Church." The Sciences vol 39 no. 5 (September/October, 1999): 29-35.
The Sciences is published bi-monthly by the New York Academy of Sciences. Two East Sixty-Third street, New York, New York 10021. This article is reprinted with permission from The Sciences and the author.
J. L. HEILBRON, formerly a professor of history and vice chancellor of the University of California, Berkeley, is now a senior research fellow at Worcester College at the University of Oxford. This article was adapted by the editors of The Sciences from the opening chapters of his book, The Sun in the Church: Cathedrals as Solar Observatories, published by Harvard University Press. Dr. Heilbron is also the author of Dilemmas of an Upright Man: Max Planck and the Fortunes of German Science and Geometry Civilized: History, Culture, and Technique.Copyright © 2003 J.L.Heilbron
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