Chapter 4  Influence on Other Geometers

     The influenceJesuits had on other geometers is treated under two headings:

       a.   In teaching other geometers

       b.   In correspondence with other geometers

 

a  In teaching other geometers

       Apart from the direct contributions to geometry, the Jesuits indirectly influenced its development by their relations with non-Jesuit geometers. They did this as teachers, as relatives, as friends, as correspondents, and as adversaries.   Gilbert Highet speaks about the Jesuit educational experience in the years before the suppression of the Society.

      

       The best thing about its method was the thoroughness with which they planned.

. . .The Jesuit regulations made sure that the pupils realized what they were doing, and why.  It is noticeable that very many of their pupils have turned out to be men of very strong will-power and long vision. A good modern example is the Irishman who spent seven years on writing a book about the events of a single day, and then spent seventeen more on writing about the dreams of a single night.   You may not admire Ulysses  and Finnegans Wake,  but they are monuments of aesthetic planning and perseverance, and it was the Jesuits who taught Joyce how to make such plans.

       The success of Jesuit education is proven by its graduates. It produced, first, a long list of wise and learned Jesuit preachers, writers, philosophers, and scientists.   Yet if it had bred nothing but Jesuits it would be less important. Its value is that it proved the worth of its own principles by developing a large number of widely different men of vast talent: Corneille the tragedian, Descartes the philosopher and mathematician, Boussuet and Bourdaloue the orators, Moliere the comedian, d'Urfe the romantic novelist, Montesquieu the political philosopher, Voltaire the philosopher and critic, who although he is regarded by the Jesuits as a bad pupil is still not an unworthy representative of their ability to train gifted minds.  The Company of Jesus had many enemies, but none of them ever said that it did not know how to teach. 1

 

       One of the well-known Jesuit teachers of geometers was P�re Laurent Beraud,S.J. (1702-1777) who was born in Lyons, and taught mathematics in Avignon, Aix and Lyons, where he died.     His obituary was printed in the Journal des Scavans.   He taught two of the earliest historians of mathematics, Charles Bossut and Jean Etienne Montucla, as well as the distinguished astronomer Joseph Lalande.   The latter received some of his formation in the special small college astronomy observatory constructed by P�re Beraud. It was because of Beraud's encouragement that Bossut, whose mathematical preparation at the time was meager, dared to write to Bernard de Fontelle (another Jesuit alumnus), the   secretary of the AcadŽmie Royale des Sciences, who introduced him to the famous mathematicians Clairaut and d'Alembert. This led to Bossut's interest in mathematics and to the history of mathematics.   Later Bossut's treatment of the Society was certainly less than friendly, and   he ended his days a devoted Jansenist.    Despite this fact, he lists 16 Jesuits as the most important geometers of  all time in a list of 303 names.

       Some of the non-Jesuit geometer/scientists who were Jesuit-trained in one or other of the seven hundred Jesuit schools were Bossut, Jean Cassini, Descartes, Fontenelle, Ghetaldi, Guyton, Lagny, Lalande, Marci, Mersenne, Montucla, van Roomen, Torricelli,   Valerio, Vivani, Volta, Zanotti. The contributions of these   men to geometry as well as their relationship to the Society are all listed in the DSB. Below is a brief summary of their contributions and the names of the Jesuit schools they attended.   This list, of course, does not include all geometers who were Jesuit educated. 

       Jean Dominique Cassini(d 1712) was trained at the Jesuit college in Genoa. He introduced a class of curves, the Cassini ovals (of which the lemniscate is a special case). The   ovals were first described in print by another mathematician, his son Jacques Cassini.

       RenŽ Descartes (d1650) was one of the most celebrated Jesuit graduates and   always had great praise for the Society. Of LaFl�che he said, "This is where was planted the first seeds of all my later accomplishments and for which I am eternally grateful to the Society of Jesus." He has been incorrectly listed by some historians as a Jesuit.   He wasn't, of course, but he had a nephew Philip Descartes, S.J.,who was a Jesuit mathematician. RenŽ Descartes was allowed to sleep late at LaFl�che because of ill health and later said that he found these morning hours the most productive of the day.

       Bernard de Fontelle (d 1757), who attended the Jesuit school at Rouen, was secretary of the AcadŽmie Royale des Sciences. He later attacked the Jesuits because of what he considered their cavalier attitude toward miracles. Marino Ghetaldi (d 1626) was taught by ChristopherClavius and was a frequent correspondent of Christopher Grienberger . Louis Guyton (d 1816) had the Jesuits at Lyons, for which he was not terribly grateful. He wrote a poem satirizing the Jesuits and was rewarded by being chosen "honoraire" at the AcadŽmie des Sciences. After the suppression of the Jesuits, he proposed to the government a plan for dividing up and distributing the property of Jesuit schools.

     Philippe de La Hire (d 1718) was taught by HonorŽ Fabri and later became part of a circle formed by HonorŽ Fabri which included Cassini, Deschales, both Huygenses, Leibniz, Descartes and Mersenne. La Hire was a pioneer in projective geometry and was considered by Carl Boyer in  A History of Mathematics (New York, 1968) "the only mathematician of stature in France" after the demise of Desargues, Pascal and Fermat.2

     Thomas Lagny (d 1660)   attended the Jesuit college at Lyon. He was known for his theorems on convergence of series and with computational methods. Using one such series he computed ¹ to 120 places. He also attempted to compose trigonometric tables using binary numbers.

     Joseph LaLande (d 1807)  also attended the college at Lyon and wanted to join  the Jesuits but was unable to do so because of parental objections. He was one of the most prominent mathematician-astronomers of the century. It was his work that paved the way for the solution of the three body problem. He organized observations for two of those  rare spectacular astronomical events, the transits of Venus across the sun in 1761 and 1769.  

     Johannes Marci of Kronland    (d 1667)3 lived in Bohemia (now Czechoslovakia) and attended the Jesuit college   in Jindrichuv Hradec. He was taught by Gregory St. Vincent at Prague and when in Rome met Guldin and Kircher. His work was mostly in geometrical optics but he was involved in physics and proposed using a pendulum clock for measuring pulse. During a political disturbance he initially represented the anti-Jesuit party at Prague University, but eventually   the Jesuit school and the University of Prague joined together.   On his deathbed he was  admitted into the Jesuit Society.

       Marin Mersenne (d 1648) 4 was trained by the Jesuits at La Fl�che, and later became one of the most remarkable of all the Jesuit alumni. He became a Minim friar.   In prejournal times he had the vision of developing a community of science. He lived in Place Royale (what is now Place des Vosges) in Paris, carrying on an intellectual apostolate.  Not only Jesuit geometers would meet in his cell to discuss mathematics but also Gassendi, Descartes, Hobbes, Fermat, Pascal (in fact, if one were to look through the l2-volume work of his correspondence, one would find virtually all the geometers of his time).

       Jean Montucla (d 1799) 5  was taught at the Jesuit college in Lyons and later wrote the first comprehensive and accurate description of the development of the various branches of mathematics in all countries. His friend Joseph LaLande completed Vol. III and IV after Montucla died.

       Adriaan van Roomen    (d 1615) attended the Jesuit school in Cologne. He knew Clavius and is remembered for his work on isoperimetric problems.  His work Ideae mathematicae pars prima  was dedicated to Clavius. In it he tried to find a general rule for calculating the sides of a polygon;   he succeeded in finding to 32 decimal places the sides of polygons up to 15 x 2 60 sides (by doubling sides) then with the help of one side of the   regular 251,658,240 sided polygon he   calculated  ¹ to  16 places. 6

       Evangelista Torricelli   (d 1647)7  attended the Jesuit school at Faenza. He wrote two letters to a former Jesuit teacher, Michael  Ricci, S.J.,  in June   1644  to describe his successful experiment with a 76-cm column of mercury,   which proved that vacuum exists. 8   The toys called   "Cartesian devils" were invented by Torricelli, not by Descartes.   His only book was Opera geometrica (1644), and he communicated to Mersenne   his greatest discoveries, depending on Mersenne to relay them to   other scientists. Even though he is remembered for his discoveries in physics, he spent all his life doing geometry.   He had worked out twenty different ways to find the area of a parabola. He studied the generalized hyperbola   x my n = c n    and thus   anticipated the result of  Jean de la Faille in 1632. In modern terms his process is an integral in Cartesian coordinates which is replaced by an integral in cylindrical coordinates. Cavalieri, Barrow and other pioneers of the calculus depended on the works of Torricelli.

       Luca Valerio   (d 1618)  was taught by Clavius at the Roman College and influenced the thinking   of Galileo.9   Some of his geometry is found in Newton and Cavalieri   (center of gravity), Torricelli, de La Faille, Tacquet, St Vincent and   Guldin.

       Vincenzo Viviani  (d 1703) 10 attended the Jesuit school in Nente; he attempted to reconstruct the fifth book on conics of Apollonius. He duplicated the cube using conics and trisected the angle by using the cylindrical spiral or a cycloid and was also engaged in a number of other special geometrical problems.

       The father of Alessandro Volta   (d 1827)11 had been a Jesuit for 11 years, his uncle was a Dominican and he himself   was in a Jesuit seminary for a short time.   Alessandro was the first to make quantitative measurements on the potentials  of charged bodies and for him is named the unit of potential difference. He was a strong theoretician and some of his ideas on electricity were inspired by Boscovich .

       Eustachio Zanotti    (d 1782)  was educated by the Jesuits   in Bologna. He was a pioneer in the study of variable stars and proposed a lucid treatment of the method of indivisibles. His interest included perspectives and he contributed to the study of projections meant for artists as well as mathematicians.

       A rather celebrated visitor to the Jesuit Roman College was  Galilei Galileo. His friendship with members of the Society of Jesus started in 1587, when at the age of 23, he met Christopher Clavius, and  continued for the rest of his life. Some incorrectly claim Galileo was a Jesuit;   but in 1589 he did go away to become a monk, entering the monastery of Santa Maria in Vallombrosa as a novice, against his father's will, and within the year his father took him out of the monastery. Later in 1611 the Jesuits of the Roman College confirmed his discoveries of the motion of the earth and honored him, even though he had been embroiled in a dispute with Scheiner over a separate matter, the discovery of sunspots. Later he demolished Scheiner's idea that sunspots were tiny planets around the sun. In his writings he attributed his interest in falling bodies to a "certain Jesuit father, Niccolo Cabeo, S.J."  Cabeo once described a type of wireless telegraph, with the admonition that it would be impossible to realize in practice.

 

       The Dominican scholar William Wallace, O.P., in his book Galileo's Early Notebooks (Notre Dame, 1977), has demonstrated, by using the internal evidence of terminology, word order and symbols, that much of Galileo teachings came from   Jesuits teaching at the Roman College. He names nine of them, thus corroborating the research of other scholars such as A. C. Crombie and Adriano Carigo. After studying Galileo's manuscripts for fifteen years, he found that all Galileo's notebooks show considerable evidence of copying. Practically all of the material derives from textbooks and lecture notes which were being used by the Jesuits at the Roman College. This evidence is available in collections like Sommervogel.

       This   was not considered plagiarism. People felt they had a right to ideas:  when they were shown to be right, they were the property of everyone.   In fact, teachers were flattered to have their class notes used by another instructor. Galileo is still the father of modern science, but now there is evidence that there was a grandfather as well. The grandfather which Wallace has been able to establish is a collection   of nine Jesuit mathematicians and teachers of natural philosophy, such as Christopher Clavius and his colleagues: Benito Pereiro, Francisco de Toledo, Antonio Menu, Paul Valla, Muzio Vitelleschi, Robertus Jones, Stephanus del Bufalo   and Ludovico Ruggiero. Though his Latin prose is more simple than  Clavius' sophisticated style, the parallels between Galileo and Clavius are unmistakable. Galileo's debt to Jesuit contemporaries can be seen also in the authorities he quotes, who are identical to those quoted by the Jesuits.   

       Wallace itemizes the lecture notes in manuscript form that have been studied and identified as the same lecture notes which were used by Galileo: 12

Benedictus Pererius, 1565-1566,   Physica, De caelo, De generatione

Hieronymus de Gregoriis, 1567-1568,   Physica, De caelo, De generatione      

Antonius Menu, 1577-1579,   Physica, De caelo, De generatione

Paulus Valla, 1588-1589,   De generatione

Mutius Vitelleschi, !589-1590, Physica, De caelo , De generatione

Ludovicus Rugerius, 1590-1591, Physica, De caelo ,De generatione

Robertus Jones, 1592-1593, Physica, De caelo  

Stephanus del Bufalo,1596-1597, Physica, De generatione  

 

       The mathematical organization was not new with Galileo, and the place of mathematics in physical analysis occurred to Galileo through contact once again with reportationes  of the Collegio Romano. A mathematical approach to nature was indelibly etched in his Jesuit colleagues' physical mind-set by Christopher Clavius.   It was Clavius who supplied the formal apparatus for "geometrical philosophy," and his influence on Galileo through his commentary on Sacrobosco   was apparent.  Clavius knew all the techniques of handling motion which had been invented since the fourteenth century.   In other words, the Roman Jesuits were the immediate source of a number of Galileo's leading mathematical concepts. The notion of specific gravity ultimately descends from Archimedes, but it was made proximately available to Galileo through lecture notes, for example, those of Valla, Vitelleschi, and Rugerius.    Examples of the evidence include Galileo's phrasing identical with that of the Jesuits' as well as of authorities cited: Ockham, Sylvester of Ferrar, Durandus, St. Thomas and Scotus.    In his review of Wallace's book these things were mentioned by William Shea,  who points out that the professors of the Roman College frequently made their notes available to their students.

 

Wallace argues that lecture notes of four other Jesuit professors show even more striking parallels and coincidences with the words actually used by Galileo. . . Wallace has given us an outstandingly lucid and intelligent account of matters of great interest.  This book is the first comprehensive and unified treatment of the influence that the Jesuits exerted on one of the greatest minds of all times. 13

 

b     In correspondence   among  geometers

     Hooke performed his first experiments on diffraction after reading   of Grimaldi's book in the Philosophical Transactions of the Royal Society.   Furthermore  both Hooke and Newton show familiarity with his works. 14 Newton credits Fabri as the source of his knowledge on diffraction.   Newton's  initial description shows he did not perform the experiments outlined by Grimaldi. Later when Newton did perform the experiments, he referred to it as a kind of refraction and by  careful measurements made clear the periodic nature of the phenomenon.

       As has been suggested, many of the geometers at this time, though not trained by Jesuits, were familiar with their books. Leibniz was inspired and enlightened by them,   as were Huygens, Barrow, Desargues, Gregory, Lambert, Mercator, Boyle, and Newton, by their own admission.

       There were many strong friendships between Jesuit and non-Jesuit geometers. The son-in-law of Tycho Brahe once told Clavius   that he should write more often to Tycho and that   he should not hesitate to ask for anything, because even though Tycho Brahe was not Catholic he was not a bigot, and in fact liked Clavius .15                                                                                     Another instance of strong friendships is the fact that it was to the Jesuits that Galileo turned during his crisis with the Church.   In 1588 this unknown young scientist wrote to Clavius asking about a center-of-gravity demonstration and expressed great admiration for Clavius.   From the remarks in this letter, as well as in later letters, his esteem for Jesuits is quite compelling.

 

I prefer Your Reverend Lordship's (Clavius) judgment above that of any other.   If you are silent, I shall be silent, too: if   not, I shall turn to another demonstration. . . . I know that with friends of truth like Your Reverend Lordship one may and ought to speak freely.  Excuse my delight in dealing with you, and continue to grant me your grace, for which I supplicate you in every instance.  Also gain for me the grace of the other, Father Christopher (Grienberger),   your disciple, whose reputation for mathematical ability has aroused my highest admiration.16

 

       When Galileo, with the help of his "cannocchiale" or "telescopio"  discovered the phases of Venus, the "three-bodied" appearance of Saturn, and the mountains of the moon, Clavius verified these phenomena and praised Galileo for his discoveries.   Galileo was delighted and expressed his joy with Clavius' compliments, "as much appreciated as it was desired and little expected," bringing him "such testimony to the truth" of his observations.   In fact Galileo was sick in bed when he received Clavius' letter and claimed that the letter brought him so much joy, it occasioned his immediate recovery.  Galileo knew the impact Clavius ' opinions had on the learned world, and wrote: "All the experts, especially the Jesuit fathers, agree with me, as everybody will soon know."

       There was a voluminous and constant flow of correspondence between Jesuit and non-Jesuit geometers of these two centuries. In Rigaud's  Correspondence of Scientific Men of the Seventeenth Century 17 Jesuit names frequently appear.   Letters passed, for instance, between  Galileo, Viete, both Cassinis, both Huygens, Fermat, Descartes, Leibniz, Newton and Kepler  and many Jesuit geometers such as Clavius,  Scheiner, Guldin, de la Faille, de Chales, Grienberger . In Mersenne's twelve-volume collection of correspondence are found many Jesuit entries.

       The works of Joannes Kepler have been collected into 18 huge volumes and published by Max Caspar.   Among the curious items is one concerning Thomas Lydiat, the   rector of Oxford, who, with obvious scorn, called Kepler's chronology of the life of Jesus Christ "that of the Jesuits."   Kepler, smarting from the intended insult, wrote a criticism of Lydiat's book.  "Judging by the way Jesuits are treated in England, it must be a great crime to hold Jesuit doctrine; but if Lydiat has no more serious charge against the Jesuits than that they approve the Keplerian chronology, by that very charge the conduct of his country stands condemned."

       Kepler had a long friendship with Jesuits, even though he found himself in controversy with them frequently. 18 Guldin was quite devoted to Kepler's studies and well-being.   An edict had been issued that all non-Catholics had to move out of Linz; so when Kepler refused to capitulate and become a Catholic, he was exiled from Linz  and had to move to Graz and so was cut off from Prague. In 1626 he had been thinking about becoming a Catholic, but the more he learned of Catholics the less inclined he was to become one.

       Guldin was concerned about Kepler's financial predicament as well as his inability to  study the skies, since he owned no telescope. The Jesuit Nicolas Zucchi was well known as a telescope maker; at the urging of Guldin,   he brought a telescope to Kepler.  Kepler, like a child with a new toy,  wrote  to Guldin of his gratitude  for the latter's concern and kindness. It was one of many letters he would write to Guldin in his lifetime and these are published in the Johannes Kepler Gesammelte Werke .19 One of the most touching letters was the dedication of his last book, The Dream by Johannes Kepler, the late imperial mathematician, a posthumous work on lunar astronomy  (1634), published by his son after he died. In this work he tells of his discoveries concerning the surface of the moon and describes   an imagined trip there as well as how it might be inhabited. At the end of the book he publishes a long letter of gratitude which he sent to Guldin.  In part it reads:

 

Geographical, or, if you prefer, Selenographical Appendix.

To the very reverend Father Paul Guldin, priest of the Society of Jesus, venerable and learned man, beloved patron. There is hardly anyone at this time with whom I would rather discuss matters of astronomy than with you . . . Even more of a pleasure to me, therefore, was the greeting from your reverence which was delivered to me by members of your order who are here . . . think you should receive from me  the first literary fruit of the joy that I have gained from trial of this gift (the telescope).20

 

       The Jesuits in China had much more success in dealing with Kepler than with Galileo as they attempted to keep abreast of astronomical developments.   The Protestant Kepler wrote often in response to Jesuit Jean Schreck's (Terentius ) requests for information, and at the end of one letter expresses an ecumenical prayer for the conversion of the Chinese.

Quod ratum esse velit Is cui Pater aeternus gentes in haereditatem dedit, Christus Iesus, Deus et homo, Dominusque noster. Amen.   ("May Jesus Christ, God and man, and our Lord, to Whom the Eternal Father gave the heathen as an inheritance, will it (the conversion of the Chinese) to be fulfilled. Amen.") 21

 

       Although unpleasant at times, the many controversies Jesuit geometers engaged in did not end without some profit for geometry and science. In a long and bitter dispute with Kepler Scheiner forced Kepler to a more precise formulation of his terms.   Scheiner used the pseudonym "Appelles", taken from the Greek myth.    No one could draw a line finer than the artist Appelles. Similarly Cavalieri became more careful when his ideas were criticized by   Guldin. Cavalieri as well as the geometer Stefano Angeli (d 1623), were Jesuati (Gesuati), the order of St. Jerome adhering to the Augustinian rule,   which was suppressed by Clement IX in 1668.   They both are sometimes incorrectly referred to as Jesuits. Angeli was involved with Guldin and Tacquet in a dispute over indivisibles.   Jean de la Faille   had wide correspondence with Christian Huygens and Langren. Henri Bosmans published a  number of letters between Christian Huygens and AndrŽ Tacquet   in which    reference is made to correspondence with many other geometers of the day. 22

 

       A frequent correspondent of Jacques de Billy was Pierre de Fermat;this affected the work of both geometers. Fermat has been referred to as "the prince of amateur mathematicians" and has been called by Laplace "the true inventor of the differential calculus."23   He was a  devout Catholic; his son Jean was an archdeacon and his daughters Catherine and Louise were religious nuns.24   Fermatis best remembered for "Fermat's Last Theorem": x n + yn = z n never holds  for integers x,y,z and n if n>2. This has been one of the most notorious unsolved problems for the past 350 years.  To make matters worse, Fermat claimed that he had a remarkable proof, but that it would not fit into the margins.  Some have not been very gracious at being unable to find a proof and   have consoled themselves that the proof would not be useful anyway.  The fact that 3 2 + 42 = 5 2 was used to construct right angles for the pyramids but of what value are the higher orders?  even the special triple case of 33 + 43 + 5 3 = 63 ? There are, moreover,   two questions:  Did Fermat really have a proof? and what was his proof?

 

       When Pascal tried to take credit for Torricelli's experiments, Jesuit geometers corrected him; after that Pascal had a stormy relationship with the Jesuits. He found himself in bitter arguments more than once.   One example is his claim that   Antoine de la Louvere did not really solve a problem concerning the roulette. Pascal lost this battle but won more contests with the Jesuits than he lost. D'Alembert, the angry encyclopedist, in spite of his hatred for the Jesuits, came to Roger Boscovich for help when he was deprived of a mathematics chair which he felt he deserved.   The chair had been left vacant on the death of Clairaut and d'Alembert was told that his field of mathematics was not suitable.   In desperation, in a frantic letter written in haste and filled with grammatical errors, he begs Boscovich to help him.  A translation of a few lines of this letter follows:

 

I ask you reverend Father, for the love which you have for me, to take this news to all the intellectuals of Italy who hold me in esteem. 25

 

       Ignace Pardies made his most important scientific contribution, not in his   writings, but in his correspondence. It is there that we find the objections that Pardies expressed  to Newton concerning his theory of colors and the "experimentum crucis" - objections that enabled Newton to clarify certain difficult points. 26     Pardies was a temperate and courteous critic of Newton.27 He was a vigorous  intellect, as is evident from his pedagogical writings and his contacts with the pioneers of geometry. Leibniz' impression of him confirms this view.

       After meeting in Rome with the Jesuits on their way to the Chinese mission, Leibniz started a correspondence with them and sent a suggestion for explaining the Holy Spirit as part of a Trinity to the mathematically minded Chinese.   He suggested using the analog to the square root of   minus one as sort of intersection of number and non-number.   Leibniz was inspired by the work of Kircher to attempt to find an alphabet of thought which would   enable all to speak of the creation of knowledge.   Leibniz also wrote to Bernoulli in 1703,   attributing  his   original  interest in mathematics to the writings of Jesuit geometers.

 

As a child, I had studied the elementary algebra of one Lancius, and later that of Clavius;as for that of Descartes, it struck me as being too difficult.   I asked Buot for the work of Dettonville and of Gregory de St. Vincent,which was kept in the Royal Library.  Without delay, I studied these works - these gems invented by St. Vincent and perfected by Pascal.   I culled everything that I could derive from the work of Cavalieri, Guldin, Torricelli, Gregory de St. Vincent, and Pascal on sums of sums and   transpositions. 28

 

       Gaspar Schott added as an appendix to his Mechanica Hydraulio-pneumatica (1657)  a detailed account of Guericke's experiments on vacuums, the earliest published report of this work. As a result he became the center of correspondence, as other geometers wrote to inform him of their inventions and discoveries. Schott exchanged several letters with Guericke, seeking to draw him out by suggesting new problems, and then published his later investigations.   He also corresponded with Huygens and was the first to make Boyle's investigations on the air pump widely known in Germany. Even though he personally held the Aristotelian abhorrence of a vacuum,   he was open to new information from experiments and rendered   great service to Germany by encouraging experimentation.   Like Mersenne, he spread news of new investigations, observations and discoveries;  he suggested fresh problems and encouraged  controversies until there was a resolution.   It was his publication of von Guericke's research that stimulated Robert Boyle to have an airpump constructed. 29

 

Chapter 4   Footnotes

 

1.    Gilbert Highet:   The Art of teaching.  New York: Knopf,  1954, p. 221 and p. 224.

2.    Carl   Boyer: A History of mathematics . New York: Wiley, 1968, p. 404.

3.    DSB vol. 9, p. 96-97.

4.    DSB vol. 9, p. 316.

5.    DSB vol. 9, p. 500.

6.    DSB vol. 11, p. 532-533.

7.    DSB vol. 13, p. 433.

8.    DSB vol. 9, p. 332.

9.    DSB vol. 9, p. 561.

10.  DSB vol. 14, p. 48.

11.  DSB vol. 14, p. 69.

12.  William A. Wallace:   Galileo's early  notebooks. Notre Dame: Notre Dame, 1977,  p.13.

       A very useful source for this work is the book review by John Quinn, O.S.A, "The                       scholastic mind of Galileo: the Jesuit connection" in the International Philosophical                                              Quarterly, vol.   20 #3, 1980, p. 347-362.

13   William Shea: "Readings of Galileo" in Science   vol. 227, Mar 22, 1985.  p.1462-1463.

14.  DSB vol. 5, p. 544.

15.  Edward Phillips: "Correspondence of Father Christopher Clavius"  in AHSI   vol.  8                      1939, p. 193-222.   Phillips lists 291 letters to Clavius from geometers as well as                                                             church and civil potentates.

       Also see:   Francisco Sanches:  "Ad C. Clauium epistola" in Opera Philosophic Coimbra ,                1955 p. 146-153.

16   Pasquale M. D' Elia, S.J.: Galileo in China . Cambridge: Harvard Univeristy,1960, p. 7-9

       D' Elia gives the references to these letters of Galileo taken from the Archives of the   Pontifical Gregorian University.

17   S. J. Riguard, ed.: Correspondence of Scientific Men of the Seventeenth Century.

       2 vols.  London:   Oxford University Press, 1841.

       Also see: A. de Morgan: Contents of the Correspondence  of the 17th and 18th century.                London: Oxford, 1862.

18.  M. W. Burke-Geffney:Kepler and the Jesuits.   Milwaukee: Bruce, 1944, p. 130.

19.  Max Caspar: Johannes Kepler Gesammelte Werke. 18 vols. Muenchen: C. H. Becksche,

       p. 195.

20.  Johannes Kepler: The Dream of Kepler:   Silesia: 1634, p. 165.

21.  Pasquale M. D' Elia, S.J.: Galileo in China. Cambridge: Harvard University, 1960,

       p. 33.   D' Elia quotes from Schreck's letter and the reply in   Kepleri Opera Omnia ,

       VII, p. 667-681.

22.  Henri Bosmans: "Le Jesuite MathŽmaticien Anversois AndrŽ Tacquet" in Gulden Passer .     vol 3, 1925, p. 63-87.

23.  Carl Boyer: History of Mathematics. New York: Wiley, 1968, p. 367.

24.  DSB vol. 2,   p. 131.   Both carried on an active correspondence which precipitated           some of de Billy's writings:  for example, his treatment of indeterminate analysis was                                               shared by Fermat.

25.  G. Arrighi: "J L D'Alembert, R G Boscovich ed un Patrizio Lucchese" in Bollettino              Storico Lucchese,  vol. 2 1930, n. 3, p. 27-248   ". . . Je vous prie, mon RŽrŽrend     P�re, par l'amitiŽ que vous avez pour moi, de vouloir bien apprendre cette nouvelle a                                                tous les Savans d'Italie, qui m'honnorent de leur estime."

26.  DSB vol. 10,   p. 315.

27.  James Newman: The World of Mathematics. New York: Simon & Schuster, ed. 1956,

       vol. 1, p. 260.

28.  RenŽ Tatan:   Beginning of Modern Science :  Pomerans, 1958, p. 228-229.

       Also see: Joseph E. Hofmann: Das Opus geometricum des Gregorius a S Vincentio und      seine Einwirkung auf Leibnitz . Berlin: 1942.

29.  DSB vol. 12, p. 210.