Chapter 1 Publications in Geometry                 



   During the centuries preceding and during the discovery of calculus, the numerous geometry books written by Jesuits   were widely used in the colleges throughout Europe. Most books were in Latin with elaborate dedications to the current potentates.   Many were written on rather substantial paper (though not wormproof) and as a result have lasted through the centuries.   The careful record of Jesuit publications during this period is available in the 12-volume work of Sommervogel, which lists and evaluates more than 18,000 publications. This massive work describes the books written by Jesuits since the very beginning of the Jesuit Society, and it is most accurate for the earlier centuries.   The tenth volume lists the Jesuit   geometers of the  17th and 18th centuries so that the number of authors can be counted and classified according to 13 subdivisions of

geometry (for example, there are 77 authors of ordinary Euclidean geometry).   Sommervogel lists the number of   authors in each subdivision   as follows:1

77     Euclidean geometry

15     the art of measuring

13     analytic geometry

48     the sphere

54     trigonometry

12     logarithms

26     navigation

80      gnomonics and horology

44     the calendar

74     military fortifications

22     geometry applied to calculus

49     geometry applied to astronomy

117   geometry applied to mechanics              

   Some of these 631 authors have written only one geometry book, but some have written many more. Athanasius Kircher for example wrote 39 books, RogerBoscovich 151, of which Sommervogel lists only 103.       Many of the books described in Sommervogel have gone to multiple editions. AndrŽ Tacquet's geometry as well as ChristopherClavius ' books were used for centuries in the European schools.   Some  of these books have been reprinted in this century and are available today. Not all books,   of course, were of equal value and some became obsolete in a few years as geometry rapidly grew and the geometrical emphasis changed.   But one thing is clear:  the Jesuits realized how important would be the printed word   for the development of geometry .

   Today these books are found in rare book collections such as  Yale's Beinecke  Library, Berkeley's Bancroft   Library and Kansas City's Linda Hall   Library of science and technology.  A large number are found in the libraries of the Jesuit Curia   as well as the Jesuit House of Writers in Rome.   Some books have not survived, perhaps because of overly tidy Jesuit housekeepers who tend to throw out any publication more than a month old.  One book that seems to have survived  very well is the very difficult book of Clavius  In sphaeram Joannis de Sacrobosco (Rome 1581).  It is a   commentary on the widely

used thirteenth- century book concerning the geometry of the sphere and its astronomical implications, written by John Halifax (also called Hollywood, or in Latin, "Sacrobosco").  Clavius' commentary, although in very small print, on crowded pages, with notes jammed into the margins, is called the best commentary on Sacrobosco   ever done.     Another book, Clavius' Euclid  also has survived perhaps because it was reprinted so often and was so widely used:  it was the text used throughout Europe for some time - used   by Descartes and Mersenne at La Fl�che for instance.   This book was not merely Euclid's elements of geometry   but a careful commentary.    It contained many original theorems as well as some axioms Euclid missed, for example an axiom concerning proportion.   Also, Clavius

 showed that he appreciated Euclid's wisdom in adding the famous fifth postulate.   In acknowledgement of his mastery of all geometry, Clavius was given the title "Euclid of the l6th Century."   As he wrote in a letter to Jean Bernoulli, Leibniz was led to an interest in mathematics by reading books by Jesuits such as Clavius and GregorySt. Vincent 2.    The mathematical historian Moritz Cantor speaks of the influence of Clavius' books and why they went into so many editions.


The large number of editions which were needed to satisfy the demand for his Euclid shows the high reputation which the work achieved.   Seldom has so high a reputation been so well earned . . . Clavius shows an acute critical faculty.   He detected and exposed old errors.     There is no difficulty that he attempted to evade.   . . . This work of Clavius is indispensable even today for historical research.3

Another historian  Abraham Kaestner refers to Clavius' Geometrica Practica (1604) as "a model textbook of practical geometry, perfect for its time." Clavius is even better remembered for his work in astronomy and has been referred to as the "best-selling scientific author of the Renaissance."

   A book by   Gregory St. Vincent included in the title the unfortunate  words "Quadratura Circuli".  In it he erroneously claimed to have squared the circle; i.e. formed a square x 2 equal in area to a given circle radius R so x 2 =   ¹ R 2 .   This would mean of course that   ¹   is an algebraic number (the root of a polynomial equation with rational coefficients).    The fact that ¹ is transcendental   and not algebraic  was not proven until a century later. Many mathematicians who became known   as "circle squarers," spent their energies trying to solve this futile problem, just as mathematicians tried to   prove Fermat's last theorem in a later time.  

The Royal Society of London had to issue a plea: "Please send no more articles on squaring the circle".  Like Fermat's last theorem, however, a great deal of mathematics was learned in the effort.  St. Vincent's book was mostly concerned with other forms of geometry such as the study of infinitesimals and coordinates, which was quite valuable, and earned him the honor of being one of the founders of analytic geometry. 4      Only the last chapter of his book  dealt with squaring the circle, but it was enough to hold his whole work up to ridicule from later commentators arguing from the safety of hindsight that the problem is unsolvable.   

   Montucla wrote of St. Vincent , "No one ever squared the circle with so much ability or so much success (except for the principal object)"5 .  WhenSt. Vincent requested permission to publish from his superior general Mutius Vitelleschi, S.J., the latter passed the problem on to Clavius' successor Grienberger, who spent a year of deliberation with St. Vincent .  Grienberger   threw up his hands in despair. "If only Clavius were alive now!  How I miss his counsel".  He was not satisfied with St. Vincent's work but in his very enigmatic answer to Vitelleschi he seemed to agree reluctantly, so Gregory St. Vincent published the offending chapter on squaring the circle.     His response is found in the unedited letters of Grienberger published by Henri Bosmans.


This is my opinion:  that Father Gregory should not only not  be dismissed (from his obligation), but should be constrained until he has left the quadrature in a state where it can no longer be a problem. 6


   Later St. Vincent directed the famous school of mathematics at Antwerp founded by Fran�ois d'Aguilon where many Jesuit mathematicians studied.  Among the most outstanding of these was AndrŽTacquet who published a geometry book Opera omnia (Antwerp, 1669) called by Henry Oldenburg, founder of The Philosophical Transactions of the Royal Society (TRS), one of the best books ever written about  mathematics. 7   It was reviewed in the TRS, giving specific details of the different aspects of geometry dealt with in the book.    For instance in the section dealing with indivisibles, Oldenburg reports optimistically on the Guldin -Cavalieri dispute: "the controversie, and the reply about it, is exceeding pleasant". 8   Two of Tacquet's other works, Elementa Geometria e (Anvers, 1654) and Arithmetica theoria et praxis (Louvain, 1656)  were used as school texts, after being reprinted and translated into Italian and English.

   A book by d'Aguilon concerns geometrical optics, which in the Jesuit schools was taught under the heading of Geometry.   He was given the task of organizing the teaching of geometry and science which would be useful for geography, navigation,   architecture and the military arts.   His plan was to synthesize the works of former geometers starting with Euclid and apply geometry to the three ways in which the eye perceives: directly, then by reflection and finally by refraction.     His death interfered with the publication of the two later sections.   

His treatment of different kinds of projections, especially stereographic, was meant to aid architects, cosmographers, navigators and artists.     The book has a long title  Opticorum libri sex  philosophis juxta ac mathematicis utiles (Anvers, 1613), "Book six of Optics, useful for philosophers and mathematicians alike".   

 D'Aguilon's book   contained a number of original insights and contributions   to the field of geometrical optics   even though he was unaware of the optical theories established by Kepler's Optics several years previously.      It gave some early hints on perspective geometry and was later used by Desargues, the father of projective geometry. 9      D'Aquilon was the first to use the term horopter, the line drawn through the focal point of both eyes and parallel to the line between the eyes.       Binocular vision was not understood at this time:    how do two eyes form one integral image since closing one, the other or neither give three different images?    


Constantijn Huygens read this book at the age of 20, was enthralled by it and  later said that is was the best book he had ever read in geometrical optics.    He thought that   d'Aguilon should be compared to Plato, Eudoxus and Archimedes.   In fact the title of Constantijn Huygens' first publication imitated d'Aguilon's title (omitting letters p and c): Otiorum Libri Sex (1625) and his son Christiaan Huygens many years later was still using d'Aguilon's book.

   What is most remarkable about this book is the fact that the illustrations of each section are works of the greatest Baroque painter of Flanders, Peter Paul Rubens.    The frontispiece at the beginning of the book   shows an eagle, referring to d'Aguilon's name and a variety of optical and geometrical images.   On either side of the title stand Mercury holding the head of Argus with a hundred eyes, and Minerva holding a shield reflecting the head of Medusa.    Then, at the beginning of each of six sections are Rubens' drawings describing d'Aguilon's experiments, one of which is the first known picture of a photometer.     This is one of six experiments   drawn by Rubens and shows how intensity of light varies with the square of  distance from the source.  The experiment was later taken up by Mersenne and another Jesuit, Claude de Chales, andeventually led to Bouguer's more famous photometer.10   It is evident, from the detail that he put into his drawings, how enthused Rubens was about the subject matter, perspective geometry and optical rules.         A book written by August Ziggelaar, "Fran�ois d'Aguilon, S.J. (1567-1617) Scientist and Architect" (Rome 1983), concerns the  life of d'Aguilon and the story of his Opticorum libri sex along with the involvement of Rubens.  Rubens also designed frontispieces for other Jesuit geometry books, such as the Theoremata de centro gravitatis   (Antwerp, 1632) of Jean Charles dela Faille ,a copy of which is kept in the Plantin Museum in Brussels. 11

         The prize for the largest number of books must go to Roger Boscovich , and many of them survive in the Boscovich collection in the Bancroft Library of Rare Books at Berkeley (U.C.B.).      He published 151 books and   treatises, and in his own lifetime a large number of his letters were published as well.    The renowned astronomer Joseph de LaLande spoke of these works of Boscovich:   "In each of these treatises there are ideas worthy of a genius." 12    He first caught de Lalande's attention when, as a young man, he   published a geometrical solution to a difficult astronomical problem, and de LaLande liked it so much he included it in his own book. 13

            One of Boscovich's major works    Philosophiae naturalis (Vienna, 1758) in which, among other things, he introduces his Boscovich curve of forces, 14   gives the first coherent description of an atomic theory.     It was one of the great attempts to understand the structure of the universe in a single idea.     He held that bodies could not be composed of continuous matter, but of countless "point-like" structures.     He states that the ultimate elements of matter are indivisible points which are centers of force and this force varies with the distance to other points.  This book appeared well over a century before modern atomic theory.   The work was often reprinted and   was considered so important a contribution to atomic physics that an International Bicentenary Symposium of 1958 was held at Belgrade to commemorate the publication of Boscovich's work Philosophiae naturalis .  Two of the eighteen papers presented were from Niels Bohr and Werner Heisenberg. 15  Another such symposium was held three years later in his home town of Dubrovnik (then Ragusa).    Russian scientists have always shown an interest in his work;  only recently have Western scientists taken an interest. 

   Boscovich was primarily a geometer, mainly concerned with the applications of geometrical methods to represent physical phenomena.     This is especially noticed in another major work   Elementa universae matheseos   (Rome, 1754) in which is found his treatment of conic sections. 16  AndrŽ Tacquet's   very popular  Elementa geometriae  became more popular when  a section was added to it containing Boscovich's treatise on spherical trigonometry;  it was reprinted six times between 1745 and 1780.   As often happened with Jesuit geometry books, this book was still used long  after the Society of Jesus had been suppressed.17

   A rather unusual book is the Repetitio menstrua (Louvain, 1639) by Jean Ciermans.     It partitions geometrical sciences according to months of the year.    Each of twelve topics - geometry, arithmetic, optics, statics, hydrostatics, nautics, architecture, logic, war-machines, geography, astronomy and chronology   is assigned to a given month.     The fact that he uses 354 days (a lunar year) instead of 365 and divides the month into three parts as did Moslem writers, indicates that he probably  got the idea for this arrangement from some previous Moslem author.

   Another example of Jesuit geometrical publications is taken from the middle of the seventeenth century.   The historian Morris Kline has high praise for a very popular mathematics book written by Claude F M de Chales which  applied geometry to all other branches of knowledge.  It also had been reviewed in the TRS.


The compass of mathematics, as understood in the seventeenth century, may be seen from the Cursus seu Mundus Mathematicus (The Course or the World of Mathematics) by Claude-Fran�ois Milliet De Chales (1621-78), published in 1674 and in an enlarged edition in 1690.   Aside from arithmetic, trigonometry, and logarithms, he treats practical geometry, mechanics, statics, geography, magnetism, civil engineering, carpentry, stonecutting, military construction, hydrostatics, fluid flow, hydraulics, ship construction, optics, perspective, music, the design of firearms and cannons, the astrolabe, sundials, astronomy, calendar-reckoning, and horoscopy.  Finally, he includes algebra, the theory of indivisibles, the theory of conics, and special curves such as the quadratrix and the spiral.  This work was popular and esteemed.   Though in the inclusion of some topics it reflects Renaissance interests, on the whole it presents a reasonable picture of the seventeenth - and even the eighteenth-century world of mathematics. 18


     Pantometrum Kircherianum  (Wurzburg, 1669), a geometrical work describes a geometric calculator invented by the extraordinary Jesuit scientist Athanasius Kircher .    In the same year he explained a kind of  symbolic logic in another book Ars magna sciendi (Amsterdam 1669) and in a later book Tariffa Kircheriana (Rome, 1679) is a set of mathematical tables.     These are three of the 39 books Kircher wrote.     Some of them are huge, bigger than altar sacramentaries and with large print as  if they were meant to be on continual display. Some of his other books are very well decorated with creative and entertaining drawings, such as his book of the bible stories Arca Noe (Amsterdam, 1675).  In this book he makes it clear that he understands the evolutionary process; later biologists have been impressed by this remarkably progressive viewpoint.  

     The other books indicate his widespread  interest and genius and why Kircher has been compared to Leonardo di Vinci. His first publication concerned magnetism:  he emphasized the parallel between  the forces of gravity and magnetism.    Then he wrote of sundials, next on the Egyptian language, then on calendars, then on the 1656 bubonic plague.    In the latter he attributes the plague to tiny animals   which he had observed under a microscope.   This is one of the earliest hints of what we today call "germs."    

In 1680  Kircher is said to have correctly computed the ordinary ( vs. forced) flight of a swallow at 100 ft/sec -  and this before the invention of stopwatches! 19

   He wrote about the Coptic language and showed that it was a vestige of  early Egyptian.     His interest in interpreting the obelisks led him to such a thorough study of the subject that princes, popes and cardinals appointed him to decipher various obelisks.   It was not until the discovery of the Rosetta stone in 1799 that anyone else had any success.  In fact it was because of Kircher's work that scientists knew what to look for when interpreting the Rosetta stone.   He has been called the real founder of Egyptology.


It is therefore Kircher's incontestable merit that he was the first to have discovered the phonetic value of an Egyptian hieroglyph.     From a humanistic as well as an intellectual point of view Egyptology may very well be proud of having Kircher as its founder. 20

   Since he was present at the violent eruption of Mount Etna in 1630, he had himself lowered into the cone for closer observation.      It was good preparation for his two volume work, Mundus subterraneus (Amsterdam, 1665),  probably the first printed work on geophysics and vulcanology.    In it he held that much of the phenomena on earth including the formation of minerals was due to the fact that there was fire under the terra firma, an unusual teaching for those days.     Some of his works were really encyclopedic in their scope.     One such is Phonurgianova (Kempten, 1673) which contains all the then-known mathematics and physics concerning sound and includes his invention of the megaphone.      Another is the popular   Musurgia universalis (Rome, 1646), one of his longest works, which marks a crucial juncture in the   development of music. 21   He had the good sense to distribute 300 copies to the Jesuit delegates from around the world who happened to be in Rome for the Eighth   General Congregation of the Jesuit Society (1645-1646) which coincided with the publication of his book.

   Although he adhered to Aristotelian physics,   Kircher   had no tolerance for alchemy, which, by the way, was taken seriously by Newton and Boyle.    Newton's calculus and Boyle's law    were apparently enough   to extricate these latter two gentlemen from the later ridicule heaped on their contemporaries who were engaged in the Hermetic arts.   

   Kircher's treatise on light, Ars magna lucis et umbrae (Rome, 1646), treats also of the planetary system.   In it he shows no inclination to follow the heliocentric system, but he does favor Tycho Brahe's model in which the planets circle the sun.     Some other inventions are found in this book, such as the magic lantern, the predecessor to the movies.   For three centuries a science museum founded by Kircher , (perhaps one of the first of its kind in the world) has survived in Rome.   Recently the scientific items of this museum have been   divided up and spread throughout three Roman museums.   So broad ( and so well-known ) were his interests that he was the recipient of many scientific curiosities.   

   One such curiosity is occasionally on display at the Yale Beinecke Rare Book Library.  It is the Voynich "cipher" manuscript, probably about five hundred years old, a scientific text in an unidentified language called "the most mysterious manuscript in the world."   Written by hand in an unknown alphabet on vellum it has 102 leaves including 8 folding leaves with about 400 botanical and 33 astrological subjects in five colors.  Some of the plants have been identified as peculiar to America, so the earliest date would be the time of Columbus.  To this day no one has been able to decipher it.     One of Kircher's former students, John Marcus Marci, found it and brought it to Kircher because of his work on universal languages saying, ". . . for such Sphinxes as these obey no one but their master, Kircher ."    The present name of the manuscript " Voynich" is the name of the donor who was willing   to pay $160,000 "for a book no one could read."   It once belonged to Athanasius Kircher and had been on display in his museum.

   As a youngster Kircher had three near-death experiences.    While swimming in a forbidden pond he was swept under a mill wheel; later inadvertently he was pushed from an onlooking crowd into the path of race horses; and finally he suffered a gangrenous leg from a skating accident.      The last cured suddenly after he prayed to the Blessed Virgin and it occurred to young Athanasius that he was receiving a great deal of divine protection and he did not forget these signs.    In 1661 he found the remains of an ancient Marian church built by Constantine on the spot of St. Eustace's vision.    He restored the place as a shrine and visited it often.     Then when he died his heart was taken and buried there according to his last request.22     It is rather remarkable that this brilliant geometer and encyclopedist, called the father of geology and of Egyptology, founder of the first public museum and skilled in so many other branches of knowledge should reveal such simple piety.     Athanasius Kircher deserves the title given him: "Master of a Hundred Arts".


Chapter 1 Footnotes


1.Vol. X - Col. 823-831.   Besides these 631 authors, many other Jesuit geometers are   listed in eight other columns under different headings of "applied geometry".

   A more recent addition to the 44 books on the calendar and valuable source of information        for this work is the book by G. Coyne,S.J., M. Hoskin and O. Pedersen: Gregorian reform                                                                      of the calendar. Specola Vaticana, 1983.

2.J. M. Child: The Early Mathematical Manuscripts of Leibniz.   Chicago:  Open Court,      1920,   p. 11-14.  This letter is quoted in chapter 4.

3.Moritz Cantor:Vorlesungen ueber Geschichte der Mathematik.   Leipsig: 1900-1908,

   Vol. 2, p. 512.

4.See chapter 3, concerning "Analytic geometry",  for a fuller explanation.

5.Etienne Montucla: Histoire des mathematiques. Paris: vol 2, 1798-1799, p. 79.

6.      The source of my translation is:  "Lettre inŽdite de Christophe Grienberger sur GrŽgoire de Saint-Vincent" found  in Annales de la Societe d'Emulation .  vol. I, 1913, p. 50.

   "Dicam hic demum quod sentio: Patrem Gregorium non solum non esse dimittendum, sed                  potius in compedes conjiciendum, donec saltem quadraturam eo loco relinquat, quo perire                                             amplius non possit.   Ita sentio."

7.TRS, vol 3, p. 869-876.

8.TRS, vol 3, p. 874.

9.DSB, vol 1, p. 81.   J. E. Morere gives a brief description of d'Aquilon's book as well as    several references to histories of perspective geometry and photometry.

10.    DSB vol 1 p. 81.

11.    Henri Bosmans:   "Le MathŽmaticien Anversois Jean-Charles della Faille" in Mathesis.

   vol 41, 1927, p. 5-11.    Bosmans relates the life of de la Faille and describes his    contributions to geometry   including a reproduction of de la Faille's   portrait       painted by           Anthony Van Dyke, now found in the Plantin Museum in Brussels.  

   Also, ibid. p. 7n  "Magnifique marque d'imprimeur de Meursius, dessinŽe par                P. P. Rubens."

12.    Lancelot White: Roger Joseph Boscovich. New York:   Fordham, 1961,   p. 14.

13.    Ibid.,   p. 31.

14.    Ibid.,   p. 138.

15.    Ibid.,   p. 224.

16.    Ibid.,   p. 183.

17.    Ibid.,   p. 41.

18.    Morris Kline: Mathematical thought from ancient to modern times. New York: Oxford,      1972.  p. 395.

19.    James Newman: The World of Mathematics. New York: 1956, vol 2 p. 1020n.

20.    Erik Iverson: The Myth of Egypt and its Hieroglyphs : Copenhagen: 1961,  p. 97-98.

21.    Joscelyn Godwin: Athenasius Kircher. London: Thames and Hudson, 1979, p. 66-67.

22.    Ludwig Koch ed.: Jesuiten-Lexikon. Waversebaan: Lowen-Heverlee, 1962, vol 2  p. 983.       

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

More about Jesuit history, tradition and spirituality

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Influence of Some Early Jesuit Scientists

Adventures of some Jesuit scientists
The 35 lunar craters named to honor Jesuit Scientists: their location and description
Post-Pombal Portugal opinion of Pre-Pombal Jesuit Scientists: a recent conference
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