Chapter 5  Innovations in Teaching Geometry

 

       Jesuit geometers made their greatest contribution in their teaching. Many of their books were meant as classroom texts. In their schools they insisted that all students learn geometry and that geometry be taught properly.   Evidence of how serious they were is taken from four sources:

a.    Their insistence that geometry be in the curriculum

b.    Their innovative attempts to teach geometry clearly and with enthusiasm

c.    Their interventions stimulating others to clarify concepts

d.    Their efforts to disseminate current geometrical discoveries

 

a.    Their insistence that geometry be in the curriculum

       Because of remarkable successes in the early years, the Society of Jesus was asked to open many schools   throughout Europe, and the norms for granting degrees   as well as the types of subjects to be taught varied from place to place.    It was evident that some standard process was needed.   After years of discussion and experimentation, a committee which included Christopher Clavius completed the third version of a proposed plan, called the Ratio Studiorum . This was promulgated in 1599 by the Jesuit   General,  Acquaviva.   Geometry would be a regular part of the curriculum in   Jesuit schools, and Jesuit scholastics would take mathematics while they were studying philosophy. While individual schools already taught geometry, this third version marked the first time   a geometry requirement for all students was made by the whole Jesuit educational system.

     In later years many superiors, Robert Bellarmine among them, wrote letters urging care in the teaching of mathematics and the training of mathematics teachers.   In his time Clavius had these observations to make about the training of mathematics teachers and the formation of a mathematics society 350 years before the birth of the American Mathematical Society.

 

To the end that mathematical studies be held in higher esteem . . . the mathematics teacher should be invited to disputations. Many a professor of philosophy has made no end of mistakes because of his ignorance of mathematics.   Once a month scholastics [Jesuit seminarians] should be gathered . . . to hear original demonstrations of the propositions of   Euclid.

However, that the Society may be able always to have capable teachers of mathematics, a number of men fit and able to undertake such positions ought to be chosen and organized in a private academy   for the study of the branches of mathematics.   Otherwise it doesn't seem possible for these studies to survive, much less advance, in the Society.

It was proposed last year that, for the advancement of mathematical studies (which were  being almost neglected), those who were to teach mathematics should be excused from teaching grammar, that they might, during the first year after finishing philosophy,   study mathematics more thoroughly at home, and then teach publicly one or two years.  This plan was approved, and has even to some extent been put in practice, and promised to be of the greatest use in encouraging mathematics and also in promoting the full equipment in other studies. 1

       A special school for mathematics was started in 1611 at Antwerp by Fran�ois d'Aguilon and produced Jesuit geometers such as Tacquet and de la Faille.   It demonstrated how serious the Society was about   geometry.  The French Jesuits also developed an important mathematical school which flourished for generations, and   Jean Baptiste Colbert, minister of Louis XIV, entrusted to the Jesuits the teaching of hydrography in the French navy.  

 

b.    Their innovative attempts to teach geometry clearly and with enthusiasm

       Clavius' widely used textbook Geometrica practica (1604) illustrated the concern that geometry be treated not merely as a spatial exercise but with rigor.  This was emphasized many times later by subsequent authors and commentaries on Euclid.  One work   emphasizing the need to take care in the presentation of the matter was Cursus seu mundus mathematicus   (Lyons, 1674) by Claude F. M. de Chales .  In it de Chales urged that all Euclid's books not be treated equally, since some parts are more important and more intelligible for the beginner than others.   His book is reviewed in the Philosophical Transactions of the Royal Society:  their review of Mundus mathematicus  praised the teaching principles of this geometry book.

 

Concerning Euclid, he dissuades from teaching novices all the books of Euclid indifferently; alleging to have learned by experience that, at the beginning, time is ill spent in learning his  7th, 8th, 9th and 10th Books2

 

       Another such was work was Elementa geometriae (1654) of AndrŽ Tacquet which so influenced Roberval, Pascal and Fermat.  His book went through many editions and was distinguished for its clarity and order.  For this reason it was used for  generations of readers and was called by Henry Oldenburg, the founder of the TRS, "one of the best books ever written in Mathematics." 3  He comments further: ". . . being an account of one of the most considerable volumes of mathematics extant, we hope we may be the better excused for its prolixity." 4           HonorŽ Fabri tried to unify all physics along the lines of geometry; as it was described in the TRS,   "Concerning his method he hath comprehended the whole of Physics in a geometrical method." 5   It was HonorŽ Fabri who first explained Galileo's experiment demonstrating equal time for falling bodies.   

Galileo, in turn, had gotten interested in the problem in the first place because of the writings of another Jesuit, Niccolo Cabeo, S.J.   Fabri, by the way, spoke of another of Galileo's problems, the motion of the earth around the sun.  His observations, however, came at an awkward  time.    Apparently more was expected of Fabri since he belonged to the Holy Office. The TRS quotes him and then comments on his courageous statements concerning the movement of the earth.

 

It hath been more than once asked whether they had a demonstration for asserting the motion of the Earth?  They durst never yet affirm they had; wherefore nothing hinders but that the Church may understand those scripture-places, that speak of this matter in a literal sense, and declare they should be so understood as long as the contrary is not evinced by any demonstration; which, if perhaps it should be found out by you (which I can hardly believe it will) in this case the Church will not at all scruple to declare, that these places are to be understood in a figurative and improper sense."

       Whence this Author concludes, that the said Jesuit assuring us that the inquisition hath not absolutely declared, that those Scripture places are to be understood literally, seeing that the Church may make a contrary declaration, no man ought to scruple to follow the Hypothesis of the Earth's motion, but only forbear to maintain it in public, till the prohibition be called in. 6

 

This statement brought Fabri 50 days in prison under Pope Alexander VII, and he was released  only by the intervention of Leopold II. He still put a chapter in his Dialogi physici (1665) entitled "de   motu terrae" (concerning the earth's motion).

       Fabri's ingenious quadrature of the cycloid inspired young Leibniz, and Newton first learned of Grimaldi's teaching of divergence from the writings of Fabri.   He was the leader of a circle of mathematicians which led him into friendship with Gassendi, Mersenne, Cassini, LaHire, Descartes and Huygens.    The Journal des Scavans   speaks of Fabri's teaching the circulation of blood before William Harvey.     The Jesuit  Daniel Bartoli, S.J., was another who did not hesitate to praise the works of Galileo while they were still on the Index.    He did much to encourage scientific debates and make science available to the general reader,as well as to encourage impartial consideration of scientific evidence.    His books were widely read and frequently translated.    In his Dell'huomo di lettere (Rome, 1659) he encourages the pursuit of scholarship even in the face of hostility, neglect and poverty.   He tried to stimulate appreciation of original ideas and to discourage the worship of authority.7

       Christopher Scheiner in the training of young mathematicians organized public debates, "disputationes" (many of which were later published), in order to emphasize the geometrical concepts taught.       Similarly, Joseph Stepling founded  a mathematical and scientific research group in Prague which met every month.  

 A large number of treatises of this group were published.   In 1753 the Empress Maria Theresa, as part of her educational reform, made him director of the faculty of science and philosophy in Prague. 8

       The Flemish Jesuit Ferdinand Verbiest wrote several important geometry books in China, comprising tables, descriptions of instruments and predictions of future eclipses.  These writings were treasured by the Emperor.

c.    Their interventions stimulating others to clarify concepts

       The heliocentric world system was not widely accepted until 1760, after Copernicus' De revolutionibus   had been removed from the Index (1757) due to the intervention of RogerBoscovich .9  It  was Boscovich more than anyone else who finally convinced Pope Benedict XIV to remove Copernicus from the Index of forbidden books, perhaps a century and a half too late.

       In the collected works of Robert Boyle and of Christian Huygens and in the correspondence of Isaac Newton, the number of references to Jesuit geometers is extensive.   The same can be said of the writings of Descartes, Mersenne, Gassendi and many others.   The influence was due in large part to the peculiar position of the Jesuit Society in the field of higher education and to its policy of encouraging scholarship in   mathematics among its members.    Perhaps because this influence was   indirect, it has been ignored by many historians.   But frequently Jesuits were able to approach both sides in a dispute and  bring a nasty argument to a happy conclusion.  Conor Reilly and the DSB illustrate this point.

There were times when some of these men (Jesuits) intervened at critical times.  Pardies for instance in a growing dispute between Newton and Huygens bringing an unseemly argument to an early conclusion.   It was Roger Boscovich who finally convinced the Papacy (Benedict XIV) to abrogate the decree of the index against the Copernican system.  Some of the elusive language used by Boyle in the early statement of his volume/pressure law was focused because of the intervention of Francis Line.10

       Pardies intervened at a certain decisive moment  in a debate between Newton and Huygens, and his important contributions in his correspondence which reflected a vigorous intellect forced Newton to clarify his thinking.    New meanings emerged from beneath the Aristotelian language, and he tried to effect a compromise between Descartes and Aristotle. 11

 

       The influence of Clavius was not limited to his teachings and his enduring books.   His correspondence was enormous, and some of it has been collected and preserved in the archives of the Gregorian University in Rome.   There are 291 letters - some really treatises - in this Clavius collection.   Most are from correspondents writing to   Clavius .  It is unfortunate that his letters were not saved, but one can  still get an idea of his influence.   It is, however, like hearing a telephone conversation, able to hear only one party and having to guess at what the other party is asking and saying.  The correspondents include not only geometers, but also rulers of all kinds: kings, emperors, and  popes.    The number and contents of Galileo's letters show that he was a good friend of Clavius. The latter  was able to joke with him about seeing Jupiter's four moons only because Galileo drew them on the lens of the telescope.  Other letters show that  Clavius' support for Copernicus' heliocentric teaching was the preponderant reason for its acceptance among the learned. 

       Light is thrown on some personalities in the letters.  Tycho Brahe chided Clavius for not writing more often.  The famous astronomer Fran�ois Viete was concerned about Clavius' criticism because it got him into trouble with Rome.   Encyclopedias speak of Viete as a Protestant, a Huguenot, an agnostic, even though he was baptized and died a Catholic.    His letters to Clavius show him a serious practicing Catholic.  Edward Phillips, S.J.,  has gathered together a commentary on this very interesting collection of letters to Clavius illustrating the far reaching influence Clavius had.12

d.    Their efforts to disseminate current geometrical discoveries

       The dissemination of geometry throughout the civilized world was evident from the efforts of Father Esprit Pezenas, S.J.,  a corresponding member of the Academie Royale des Sciences, who played a major role in the diffusion of the geometrical works of the English geometers among the French.    The Jesuit Asad Goryu modernized Japanese astronomy   and turned it away from the faulty Chinese system.   But nowhere is the spread of   geometrical ideas more evident than in the works of   Matteo Ricci and his successors in China, Schall von Bell and Verbiest.     Gilbert Highet comments on the missionary effort and its consequences.

 

       The Jesuits went to unparalleled lengths and showed unbelievable patience in adapting themselves to the people they had determined to teach.  For instance, they sent out a small expedition of ten or twelve priests to Christianize four hundred million Chinese.   This almost impossible task they started by studying China.   It was an empire, ruled from the top by comparatively few men.   Good.   If the few men could be converted, the rest would, in due course, follow.   Now, how could the few men be converted, the emperor, the courtiers, and the mandarins?    What did they admire most?   Philosophy, literature , art and science.

       The Jesuits therefore spent several years learning Chinese philosophy, art, and literature, making ready to meet the Chinese on their own level.  After the imperial officials had slowly, reluctantly admitted them, the Jesuits at once flattered them by talking to them in their own tongue, and attracted them by displaying specially prepared maps and astronomical instruments.   Instead of being rejected as foreign barbarians, they were accepted as intelligent and cultivated men.  One of them, who became a painter in the Chinese style, is now regarded as one of the classical artists of China.

       The next stage, which they approached very, very delicately, was to make the mandarins willing to learn from them.   They did this by discussing astronomy with the Chinese scientists, constructing maps of the world with the place-names shown in Chinese characters and the Chinese empire at the center, presenting sundials and astronomical instruments to the high officials whom they met, and ultimately by assisting the Imperial Board of Rites to correct its calendar so as to forecast eclipses and calculate celestial phenomena more accurately than any Chinese had ever been able to do.13          

 

From about 1600 until the suppression in 1773, Jesuits were practically the sole source of Chinese knowledge about Western astronomy,   geometry and trigonometry

Appointments in the Astronomical Bureau provided the Jesuits with access to the ruling elite, whose conversion was their main object.   Mathematical and astronomical treatises demonstrated high learning and proved that the missionaries were civilized and socially acceptable. 14

 

       While trigonometry became an analytic science in Europe, in the Orient it remained primitive until the Jesuits came.15   The China mission has been spoken of with awe and admiration by historians such as Joseph Needham, who relates the difficulties under which the Jesuits labored.  

 

Ricci, Schall, Verbiest and, in a later generation, Gaubil, were in China at a period of spontaneous decline of indigenous science, the Ming dynasty and early Ching, a decline which had nothing obviously to do with the forces which sent them there and permitted them to stay. . . There was of course the almost insuperable difficulty of language at a time when sinology hardly existed and no good dictionaries had been made. 16

 

       Another historian of this period was John Baddeley whose book   praises the work of the Jesuits even though he disapproves of their motives, - the spread of the Gospel.   His frontispiece depicts a Jesuit standing at the left of the throne of the Emperor while all others are kneeling. It has the caption:

Thrice three times the Envoys bow                                                  

Forehead to the ground, in vile kowtow:

The subtle Jesuit flanks the throne,                                                 

God's, some say - some, the Devil's own. 17

      

       Matteo Ricci's   arrival in China in 1583 marked the beginning of the Catholic missions there.      After working in various provinces he finally settled in Peking in 1601, where, under the protection of the emperor Wan-li, he remained until his death.  His success was due to his complete adaption to the culture, as well as to his personal qualities and abilities.   Recognized as an authority in mathematics and science, he disseminated geometry by lecturing, writing, publishing maps and making scientific instruments.     The Chinese geometrical works for which he is remembered   were mostly translations of Clavius' books on the astrolabe, the sphere, measures and isoperimetrics.      But especially important was his Chinese version of the first six books of Euclid's Elements , which was written in collaboration with one of his pupils. Entitled A first textbook of geometry, this work assures Ricci an important place in the history of mathematics.

 

     For 20 years Ricci   had tried to reach the emperor in person, but the emperor was a recluse not accustomed to seeing his own people.   For a time suspicious landlords would drive Ricci and his companions from their dwellings, until they hit on the plan of renting haunted houses.  Then no one bothered them.     Unexpectedly the emperor summoned Ricci and his companions to inquire about  a ringing clock brought to him by the Jesuits.    His own scientists had failed to fix it when it stopped.   Since the  emperor could not receive these foreigners in person, he had an artist draw full length portraits of them, so that they could have a vicarious interview.

       Another opportunity was occasioned by an eclipse of the sun: the prediction of the expected time and duration made by his own Chinese astronomers  differed considerably from the Jesuit prediction.      When the latter prediction proved correct, the place of the Jesuit mathematicians was secure.     It is curious that the Jesuits taught the Chinese the heliocentric theory, unaware that Galileo's trial had taken place.     So at the very moment Galileo was being accused of heresy in Rome, the Jesuits in China were teaching the same heliocentric message that they had learned from their Jesuit colleagues before they had left Rome.    There was a good five-year lag in communications.

 

       The influence of the China mission was spectacular, including projects like determining the Russo-Chinese border, and its success  was even more dramatic than that of the Paraguay Reductions. Their story is told in tapestries and paintings found in the art world  and references to them are read in world histories.   Europe was thrilled at the venture.   Leibniz, an ecumenist far ahead of his time, suggested to his Jesuit friends on the China mission how to clarify the mystery of the Trinity by using the newly discovered imaginary numbers as an analog.    It is not clear whether this was ever attempted.    Louis XIV was so enthusiastic about the work of this mission that at his own expense he equipped a Jesuit group of   "Royal Mathematicians" with the latest scientific instruments and paid their passage to Peking.  One of these men, Father Gerbillon, S.J., mapped out the whole Chinese empire; his work is considered a masterpiece even today.   His 120 pages of maps have served as the basis of maps of China   for the past three centuries.    

       Although the mission had frightful dangers, savage martyrdoms and terrible disappointments, there were times when the Jesuits enjoyed great prestige, independence and authority.   One such time, during the reign of Emperor K'ang Hsi, was the tenure of  Ferdinand Verbiest (whose Chinese name was Nan-huai-jen ) as president of the Board of Mathematics.   He succeeded Matteo Ricci and  Adam Schall as one of the three great figures of the Jesuit Chinese Mission.   Verbiest died 300 years ago on the 27th of   January  in 1688, was buried with the same imperial honors as the other two, and was laid to rest at their side. 

       Verbiest had been summoned to Peking by the Mandarin Emperor to succeed Schall in his declining years.  When he arrived he found himself engaged in a contest with the state astronomer, a Chinese Moslem named Yang-kiang-sien, concerning the position of the planet Mercury.  In a public dispute Verbiest correctly predicted its location but Yang's prediction proved incorrect.   This  meant   death for the Moslem astronomer, but Schall intervened to spare his life. 

 

       Charitable though this was, it led to one of the bitterest battles in the history of the mission, in which Yang sought revenge for his humiliation.   He registered a triple indictment accusing the mission of being a menace to the state, charging the Jesuits with treason, teaching a false religion and teaching false astronomy.   Verbiest gave a passionate defense of Schall, who by this time was paralyzed and unable to speak. The first charge was not sustained, but the second charge held, so the Jesuits and their Chinese allies were chained and cast into a filthy prison, where they were bound to wooden pegs in such a way that they could neither stand nor sit.   There they remained for almost two months until their sentence of strangulation was imposed.   A high court found the sentence too light and ordered Schall to be cut up into bits while still alive.   The sentence, however, was not carried out because an earthquake destroyed the part of the palace chosen for the execution.

 

       Later the government decided to test the third charge of false astronomical teachings.   The Emperor ordered a public debate on the relative merits of Chinese and European astronomy.  It was to have three parts: to determine the shadow of a fixed gnomon ( a column erected perpendicular to the horizon used to find the meridian altitude), to predict the position of the planets at a fixed time and to predict the exact time of a lunar eclipse which was expected about that time.   It was decided that the two astronomers, the Chinese Moslem Yang, and the Christian Verbiest ,should each use his mathematical skills and then the Heavens would be the judge.

 

       The affair was carried out at the Bureau of Astronomy, where were gathered the privy council, the ministers of state, the officials of the observatory, and a host of other mandarins.   The inept Yang was not up to the tasks and so Verbiest , with his precise data,  triumphed in all three.    Verbiest was immediately installed as president of the Board of Mathematics.   The displaced president of the board, having supported Yang, lost face and was banished.

       Verbiest then boldly suggested that the mistakes in the calendar be corrected. The Jesuits had reviewed the  previous work of the Moslem and Chinese  astronomers and  proved that an extra month  had been inserted.    Verbiest insisted that it be eliminated.  Alarmed that such a public document as the calendar,which had been approved and promulgated by the Emperor,  should be altered, the officials begged Verbiest to withdraw the suggestion.  He replied, "It is not within my power to make the heavens agree with your calendar.  The extra month must be taken out."  It was, and Verbiest had won an astonishing victory. 

       After this Verbiest had a real friend in the Emperor K'ang Hsi, who was eager to share his knowledge.   Verbiest taught him geometry, and in doing so translated the first six books of Euclid into Manchu.  He instructed him also in philosophy and music.  In doing this he took advantage of every opportunity to introduce Christianity.   The Emperor elevated him to the highest grade of the mandarinate and gave him permission to preach Christianity anywhere in the empire.

       After his initial triumphs Verbiest was entrusted with very many projects of the empire.  It seemed that little  went on in the empire during the next few decades without Verbiest ,and it was not unusual for the Emperor to take Verbiest along on his excursions.  Having restructured the calendar, one of the empire's most crucial documents, he composed a table of all solar and lunar eclipses for the next 2000 years.   Delighted with this the Emperor gave him complete charge of the imperial astronomy observatory, which had been built in 1279.

       Since the ancient equipment was by now obsolete, Verbiest designed new instruments and completely rebuilt the observatory in 1673.   Sensitive to the history of the empire, he preserved the old equipment.   Then he constructed the great bronze astronomical instruments which have become   a Peking tourist attraction even today.    When the Jesuits were forced out after the suppression of the Society, the observatory fell into disrepair; during the Boxer rebellion instruments were stolen and brought to Prussia, and the Jesuit scientific library was given to the Czar.   In 1981, however, the Chinese government restored the observatory.  

       His scientific achievements occasioned several princes and many mandarins and scholars to become Catholics, so that by the time Verbiest died there were about 800,000 Catholics living in 1,200 communities.   His funeral was a very stately affair led by a 25-foot banner telling of his accomplishments.  Taking part were fifty horsemen, musicians and standard bearers carrying portraits of Fr.Verbiest and the saints. Attending were very many people, including representatives from the Emperor.

       Verbiest is listed as one of 108 Chinese heroes in the popular novel Shui Hu Chuan , and his portrait is shown with Chinese features in a famous Japanese print.   After he died he was buried next to the other two giants of the Jesuit mission, Matteo Ricci and Adam Schall.   Today it is possible, though difficult, to visit their tomb since it is on the campus of a college of political science, but it is immaculately preserved.

       Verbiest died a century after the Jesuit mission had begun, and a century before the tragic decision of Pope Clement XI regarding the Chinese rites, which practically  ended this vibrant, promising mission.  The work in China had to be started over again in a later century and with much less success.   It is not difficult   to imagine the consequences   if such a decision had been made in the fifth century, when St. Patrick was trying to cope with the rites of the Gaels.

       An important vehicle for the dissemination of geometrical ideas was the intellectual apostolate instigated by a Jesuit alumnus, the Minim friar Marin Mersenne. It was a group of intellectuals, including Jesuits, whose intercommunication of ideas was facilitated by Mersenne.    His correspondence is contained in a 12-volume work,   which stands as a monument to his influence in disseminating geometry throughout Europe.

 

From about 1623 Mersenne began to make the careful selection of savants who met at his convent in Paris or corresponded with him from all over Europe and as far afield as Tunisia, Syria, and Constantinople.   His regular visitors or correspondents came to include Peiresc, Gassendi, Descartes, . . . Fermat and Pascal.   Mersenne's role as secretary of the republic of scientific letters, with a strong point of view of his own, became institutionalized in the Academia Parisiensis, which he organized in 1635 - his monument as an architect of the European scientific community. 18

 

 

 

 

Chapter 5 Footnotes

 

1.    Edward Phillips: "The proposals of Father Christopher Clavius for improving the teaching of mathematics"  in the Bulletin of the American Association of Jesuit                     Scientists.   1941, vol. 18, p. 203-207.

       Phillips takes this material from the Monumenta Paedagogica S.J., p 471-476 #35,35.

2.    TRS vol. 9, p. 229.

3.    TRS vol. 3, p. 869.

4.    TRS vol. 3, p. 876.

5.    TRS vol. 1, p. 325.

6.    TRS vol. 1, p. 74-75.

7.    DSB vol. 1, p. 483-484.  Daniel Bartoli admonished his readers to prove things for   themselves and not just to rely on an argument from authority. He wrote several books       on physics, on the propagation of sound and on atmospheric pressure.

8.    DSB vol. 12, p. 39.

9.    DSB vol. 14, p. 159.

10.  Conor Reilly: Francis Line, S.J., an exiled English scientist . Rome: IHSI,1969,

       p. 109-110.

11.  DSB vol 10, p. 315.

12.  Edward Phillips:   "The Correspondence of Father Christopher Clavius, S.J., preserved             in the Archives of the Gregorian University" in AHSI   vol. 8, 1939, p. 193-222.

       The influence Clavius - and other Jesuits - had on the direction science and   mathematics took during this time is more fully treated   by Peter Dear in "Jesuit        mathematical science and the reconstitution of experience in the early seventeenth         century" in Studies in History and Philosophy of Science , 1987, vol. 18,  p.133-175.

13.  Gilbert Highet:   The Art of Teaching . New York: Knopf, 1954, p. 222-223.

14.  DSB vol. 14, p. 159.

15.  D. E. Smith:   History of Mathematics . New York: Dover,1958, p. 613.

16.  Joseph Needham:   Civilization in China . vol. 3  Cambridge: University Press, 1959,

       p. 173.

17.  John F. Baddeley:   Russia, Mongolia, China, 1602-1676.  New York: Burt Franklin,              1916.  Throughout this classic work on Sino-Russian relations are found references to                                                      the work of the Jesuit geometers, including a very respectful biography of Verbiest          written by Baddeley himself, "since none is available." (p. 433-437).  

18.  DSB vol. 9,   p. 316.

 

 

 

 

 


Introduction to Jesuit Geometers
Ch 1. Jesuit textbooks and publications
Ch 2. Jesuit inventions in practical geometry
Ch 3. Jesuit innovations in the various fields of geometry
Ch 4. Jesuit influence through teaching and correspondence
Ch 5. Jesuit teaching innovations, methods and attitudes
Ch 6. Evaluation of these Jesuit geometers by professionals. < br> Appendix to Jesuit Geometers



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