Chapter 3           Innovations in  Geometry   

 

      A late l6th-century geometer would notice the maturing of Euclidean geometry, then some applications to trigonometry as well as to gnomonics and steriometry.   The l7th-century geometers witnessed the unfolding of analytic geometry as well as the infinitesimal geometry   which preceded the calculus.   Finally the early l8th century would see the use of geometry, as the calculus evolved.  At this time, too, the seeds were sown for future projective and non-Euclidean geometries.   Jesuits were involved in all these developments.   Here 9 topics are considered:

 

a.    Euclidean geometry

b.    Geometry applied to gnomonics, mapmaking, military art, architecture

c.    Trigonometry

d.    Analytic geometry

e.    Geometry, predecessor to calculus                                                                   

                1. Infinitesimals/infinite series

                2. Area under a hyperbola

                3. Center of gravity

                4. Guldin's rule

                5. Other quadratures

f.     Geometry as Calculus Evolved

g.    Non -Euclidean geometry

h.    Projective geometry

i.     Geometry applied to physics

 

a.   Euclidean geometry

       Clavius's Euclidis elementorum libri XV (Rome, 1574) contains the books of Euclid and a vast collection of comments and elucidations.   Later editions of this work became  the standard text for many schools of the time.      Clavius' emphasis was not merely a spatial exercise; instead, this book on Euclidean geometry was an important step in bringing rigor to geometry.   The fact that today Euclidean geometry is the most rigorous course in  school mathematics is due in part to this effort of   Clavius, who emphasized rigor in his geometry texts. 

       Clavius'  treatment of synthetic geometry was not just a translation of Euclid, but a critical evaluation of Euclid's axioms 1 and included some necessary axioms that Euclid had missed.   Clavius used a first-century method of proof which derives a proposition by assuming the contrary of the proposition to be proven. This method applied to a proposition P consists of using the fact that{[not P implies P ]implies P} and is equivalent to using {[P or P ] implies P }.   This  process for example would be used  to prove that there are an infinite number of prime numbers.      He also showed his appreciation for Euclid's insight that the fifth postulate was necessary, which was not a very popular opinion among l6th- and17th-century geometers.   Many thought that it was either an unnecessary postulate or a very involuted theorem.

        Clavius' innovations were also seen in the symbols attributed to him by Florian Cajori 2 and others, such as the radical sign, plus and minus signs, parentheses and  decimal point.    He proposed a proof that there can be no more than three dimensions in geometry:  it was based on the fact that only three concurrent lines can be drawn from a point so that they are mutually perpendicular. 3

       Clavius' commentary on Euclid became the standard textbook for the 17th century and his books on arithmetic, geometry, algebra, harmonics and astronomy   were used in all the European Jesuit schools, making him the mathematics instructor of Catholic Europe as well as much of   Protestant Europe.   Some of Clavius' students spread this new emphasis, - men such as Matteo Ricci who  translated Clavius' work into Chinese, giving China its first opportunity to enjoy Euclid.

       Clavius discovered and proved a  theorem for a regular polygon which has an odd (2n+1 ) number of sides.  In the isosceles triangle formed by a vertex and its opposite side the base angle is   n   times the summit angle.  He proved it by inscribing the polygon inside a circle.   So for a pentagon (n = 2) it would be twice the summit angle, while in a 17-sided regular polygon (n = 8) the base angle of such a triangle would be 8 times the summit angle.  Two centuries later Carl Friedrich Gauss would use this to construct a 17-sided regular polygon by ruler and compass and thus solve the binomial equation x17-1 = 0 4.    This was his first important mathematical discovery to be followed by an incredible number of discoveries later on.

       Many other   Jesuit geometers contributed to the modernization of Euclidean geometry.  There are examples such as  Fran�ois d'Aguilon who proposed his own synthesis of Euclid,   Jean Villalpando of Spain who applied the emphasis on proportions to harmonics, and   Gregory St. Vincent who is partly responsible for the use of the equal sign as well as the practice of lettering the vertices of a polygon.

        

b.  Geometry applied to gnomonics, map-making, military art,  architecture            A geometer who wrote for structural and hydraulic engineers was the German Jesuit, Johann Helfenzrieder.    He authored very accurate almanacs and then later published dissertations on the use of telescopes in astronomy and the construction of surveying apparatus.   His later works include practical mechanics for structural engineers and also the study of hydraulics.

       John B Riccioli went beyond the preliminary work of Galileo and succeeded in perfecting the pendulum as an instrument to measure time, thereby laying the groundwork for a number of important later applications.     He also made many significant astronomical measurements in an effort to expand and refine existing data.   He made measurements, for example, to determine the radius of the earth and to establish the ratio of water to land on its surface.   His recourse to a  geometrical treatment of these problems is noteworthy.       In 1651 one of the earliest lunar maps was published by   Riccioli   in his large work Almagestum Novum (Bologna, 1651), mentioned earlier.  His chapter  concerning the moon contains two large maps (28 cm in diameter), one of which   shows for the first time the effects of librations and introduces new lunar nomenclature. He started the practice of naming lunar objects after well-known personalities (not all scientists) and   almost all of these are still in use today. 5 

                                                                                         MatteoRicci's contributions to geography were his calculation of the breadth of China,   which was three-quarters of the breadth assumed by Western geographers.  Also he identified   China and Peking with the Cathay and Cambaluc of Marco Polo. 6  At the time Ricci's maps of China were considered more accurate even than the contemporary maps of Europe.  He shares the latter recognition with another Jesuit Coadjutor Brother,   Benedetto de Goes, S.J., who made a journey from India to China   during the years 1602-1605, in order to verify that China and Cathay were the same. 7

 

       The Jesuits long had an interest to find an overland  route from Europe through Christian Russia.  The Tsar's ambassador to China at the time, named Spathary,   was able to speak Latin and so could converse with the Jesuits.   This involved Ferdinand Verbiest in negotiations over the border between the two countries; later it was partly under his direction that the Russo-Chinese borders were determined, and it was his surveyors who were sent out to mark them.  It  did not, however, put an end to border disputes, and for the past three centuries these borders continue to be contested.

       Besides his numerous maps of China as well as the then-known world, Verbiest wrote no less than 30 books.  Among these may be found a 32-volume handbook on astronomy, a Manchurian grammar, a Chinese liturgical manual and a work   entitled Kiao-li-siang-kiai ,  which is a statement of the fundamental teachings of Christianity and which became the basis for later Chinese Christian literature. 

       It may have been noticed in Chapter 1 that a disproportionate number of books (74) concern military fortifications.   The French Jesuit Claude F M de Chales is one of the many authors. At this time siege warfare had become a precise science depending very much on geometry. Some geometers were so good at this that they could predict precisely how many days it would take to break a siege.     The fortifications had triangular shapes to protect the walls and make the weapons more effective.    The offense, on the other hand, would   dig three rows of trenches.    The first would be cut roughly parallel   to the walls but outside the range of the defending weapons;   here the attacking army would mass.    The next two trenches would be dug at the best possible angles to avoid the weapons while getting as close   as possible to the walls as soon as possible.      Since wars were continuous this science received a great deal of attention.  

      Another Jesuit, whose contribution of geometrical designs in architecture were evident in the architectural works of Newton,   was   JeanVillalpando, who studied geometry and architecture under Juan de Herrera, the royal architect of the Spanish king Philip II.    As a young man he became obsessed with the temple of Solomon and the temple described by the prophet Ezekiel.   After entering the Society this interest, shared with another Jesuit, Jerome Prado, S.J., led  both to collaborate in the exegesis of the prophet Ezekiel. Because of this Villalpando is remembered in Sommervogel for his exegetical  work instead of for his contribution to the applications of geometry to architecture. That he was more than a theorist is evident from the fact that at the age of 27 while still an unordained Jesuit scholastic, he was in charge of constructing three major Jesuit buildings, one of which still stands, the church of the Jesuit college in Seville. He designed the first oval church ever built in Spain. In the history of architecture he is most renowned, however, for his famous work, the design and  reconstruction of the temple of Solomon. Philip II had granted 3,000 scudi for engraving the prints. When the exegetical question arose whether Ezekiel's temple was the one Solomon built, the Jesuit General Aquaviva involved himself in the project as did Pope Sixtus V, who appointed a commission to investigate Villalpando's orthodoxy.   The result was that Villalpando was completely cleared of misrepresenting Sacred Scripture. A study in Baroque art illustrates how Villalpando'sEzechielis explicatio (1596-1604)   combines mysticism and science. 8

       Villalpando had undertaken to provide the world with the first full-scale imago   of the Temple on the grounds that only by translating Ezechiel's vision into terms of real architecture could one fully apprehend its mystical import. The form and proportions of the Temple, which necessarily had to be perfect, since they were inspired by the Almighty Himself, provided an insight into the perfection of the City of God.     In this context it should not be forgotten how deeply the traditional concept of the Temple of Solomon as the forerunner of the celestial Jerusalem was rooted in the Jesuit Society's early history.   In 1523 the founder Ignatius Loyola had undertaken the long and hazardous journey to the Holy Land, and had seen the earthly Jerusalem with his own eyes.  It is recorded that, while he was there, he was seized with a burning desire to convert the Moslems.   His impulse may well have stemmed from the conviction so widespread at the time that the conquest of the terrestrial Jerusalem, by either evangelization or force of arms, would be closely followed by the reign of Christ upon earth.  

       Compared to the complex task which Villalpando had been compelled to face in determining the form and dimensions of the Temple, his design was simplicity itself.   According to the book of Genesis, Noah's Ark was 300 cubits long, 50 cubits wide and 30 high.  Much, of course, depended on the precise value to the cubit.   Origen had claimed that it was equivalent to six Roman feet, which Kircher dismissed as absurd since it would have made the Ark inordinately vast.  Villalpando had maintained that it was about two and a half feet, basing his claim on the authority of Vitruvius, whereas Kircher, likewise arguing from Vitruvius, had come to the conclusion that it was one and a half feet.

       The truth is that Villalpando, like his master Herrera, was one of those ambivalent figures who succeeded in having a foot in both camps.  It is the same combination of the mystical and the practical that characterized men such as Copernicus, Cardanus, Tycho, Porta, John Dee, Kepler, Leibniz, and even Newton himself.  So, in the Jesuit's work there seems to have been something for everyone.   In 1626 Father Marin Mersenne, that inveterate enemy of everything Hermetic, drew the attention of the scholarly world to what he considered to be the innovatory nature of Villalpando's remarks on the center of gravity in his third book.

 

c.   Trigonometry

       Clavius in his three books: Astrolabium, geometria practica, and Triangula sphaerica,   summarized all contemporary knowledge of plane and spherical trigonometry.  Carl Boyer credits Clavius, "the Jesuit friend of Kepler," with the early use of the decimal point in 1593, twenty years before it became common.9   He used it in his tables of sines.   He also introduced parentheses to express aggregates.   His prosthaphaeresis (Greek for addition and subtraction), the grandparent of logarithms, relied on the sine of the sum and differences of numbers. 10   In this way he was able to substitute addition  and subtraction for multiplication, by solving the identity which we are familiar with today:    

2 sin x sin y = cos(x-y)-cos(x+y).

All computational problems involving sines, tangents and secants can be solved merely by prosthaphaeresis, that is addition and subtraction, without the laborious multiplication and division of numbers. 11

 

       Clavius in his Astrolabium   credits Raymarus Ursus with the discovery of the theorem but perhaps he was being unnecessarily generous.      It was Clavius who proved the theorem in all three forms and it was he who   published it in its complete form.  D. E. Smith gives the details of the proof as well as this history of its development.  He emphasizes the impact Clavius' work had on the discovery of logarithms.

 

Christopher Clavius extended the method to the cases of secants and tangents; in fact he showed how to find the product of any two numbers by this method, . . . Both fragments are from his Astrolabium (Rome 1593), Book I, lemma 53 . . . The bearing of the subject upon the theory, if not the invention, of logarithms is apparent.12

 

      Clavius was one of the first to use a symbol for an unknown quantity, and although, at that time, negative solutions were not considered valid, he gave a geometric proof that the equation x2 + a = bx can be satisfied by two answers.   The first to establish the  essential equations of   spherical trigonometry was another Jesuit geometer,   Roger Boscovich . 13

       Matteo Ricci introduced trigonometry to China, and Ricci's successors, Verbiest and Schall von Bell,   then used the geometric and trigonometric concepts to bring about a revolution in the sciences of astronomy, the   design of astronomical instruments, 14  mapmaking, and the intricate art of making accurate calendars.   The  Chinese celestial calendar, used to predict eclipses and comets, had been used for 40 centuries and was in desperate  need of revision.   Schall von Bell wrote no less than 150 treatises in Chinese on the calendar.  Besides calendars, the Jesuits were  inveterate mapmakers and were continually traveling around the empire, even though travel conditions were quite primitive. The TRS recounts 52 journeys by Ricci and Verbiest alone.

 

d  Analytic geometry

       Greek geometers such as Apollonius used a nonorthogonal coordinate system to solve conics and the ancient geometers knew how to fix the position of a point using coordinates.   But before analytic geometry could assume its present   practical form, it had to await the development of algebraic symbolism.  Credit for initiating modern analytic geometry is given to  Descartes and Fermat, since it was not until after their impetus   that  we find analytic geometry in the form with which we are familiar.   Descartes' contribution is found in the appendix to his famous philosophical treatise Discours de la mŽthode  (1637) while  Fermat's contribution is found in his letters, some of which were to Jesuit geometers.

     Gottfried Leibniz credits a Jesuit geometer in the development of analytic geometry,  Gregory St. Vincent.    In his work Opus geometricum quadraturae circuli  et sectionum coni (1647) Gregory St. Vincent's treatment of conics earns him the honor of being classed by Leibniz with Fermat and Descartes as one of the founders of analytic geometry. 15  

       This Opus geometricum  has four books:  first, concerning  circles, triangles and transformations; then geometric sums and the Zeno paradoxes with trisection of angles using  infinite series;   third the conic sections; and finally, his quadrature method, based on his   "ductus plani in planum" method.     The latter is a summation process using a method of indivisibles, in which St. Vincent introduces his "virtual parabolas." 

       The Greeks described a spiral using an angle and a radius vector, but it was   St. Vincent and Cavalieri who simultaneously and independently introduced them as a separate coordinate system. In an article The Origin of polar coordinate s,J. J. Coolidge    refers to the priority dispute between Cavalieri and St. Vincent over their discovery.   St. Vincent wrote about this new coordinate system in a letter to Grienberger in 1625  and published the process in 1647. On the other hand Cavalieri's publication appeared in 1635 and the corrected version in 1653.16

       Because invention of the calculus deflected geometrical interest away from the conics, it was not until the mid-18th century that interest in them revived, and this is partly due to the influence of Boscovich. 17 His treatment of conics is found in the third volume of his Elementa where he introduced his  eccentric  circle , now called a generating circle.    Another Boscovich contribution to  analytic geometry is clarification of the concept of continuity.    The earliest thorough and geometrical treatment of the subject (continuity) is found in Boscovich's Appendix to his Sectionum Conicarum Elementa .

 

       In a remarkable appendix, Boscovich brings out very clearly . . . the notions of positive and negative in direction; of geometrical continuity . . . of certain properties which follow from properties of the ellipse by change of sign . . . This principle, so clearly foreshadowed by Boscovich,   was seized by the great geometers of the following century, Brianchon (d 1864), Pon�elet (d 1867) and Plucker (d 1868).    It may be said that as a result of Boscovich's investigations, the study of the conics reached a higher level than it had ever done before. 18 

 

       Boscovich's  interests also led him to write on the ovals of Descartes, cycloids, the logistic curve, logarithms of negative numbers, limits of certainty in astronomical investigations, and inequalities in terrestrial gravitation.    In a three-year expedition he surveyed the length of two degrees of the meridian between Rome and Rimini.    When he wrote his report De litteraria expeditione (Rome, 1755) he presented a method of adjusting arc measurement.     It was the first work where probability was applied to the theory of errors.    He was the precursor of Pierre Laplace, 19 who acknowledged his debt to Boscovich for his work in statistical analysis.   Euler, Simpson and Jacobi were also indebted to Boscovich for his pioneering work.20

 

e.   Geometry predecessor to calculus

       1.Infinitesimals/infinite series

      Carl Boyer has a fine treatment of the thorny problems that had to be faced before the articulation of calculus was possible:   one such problem was infinity and infinitesimals. 21  The former problem has theological implications, since nervous churchmen felt it was an intrusion into God's infinity.  Cantor found this out in a later century and went to the Jesuit (non-mathematician) John Cardinal Franzelin for help in selling his ideas on infinite numbers.      As a result, the strange term (a compromise) "transfinite" was introduced;  it satisfied the nervous churchmen as well as the anxious mathematicians.     Dirk Struik speaks of a similar problem at the time of Gregory St. Vincent and AndrŽ Tacquet, when the groundwork was being laid for the infinitesimal calculus.    Aristotle and Aquinas held that there is no actual infinite and a continuous line cannot be made up of indivisible parts.    A point could generate a line by motion but a point was not part of a line.

 

. . . Georg Cantor has remarked that the transfinitum cannot be more energetically desired and cannot be more perfectly determined and defended than was done by St Augustine . . . Such speculations had their influence on the inventors of the infinitesimal calculus in the seventeenth century and on the philosophers of the transfinite in the nineteenth;   Cavalieri, Tacquet, Bolzano and Cantor knew the scholastic authors and pondered over the meaning of their ideas. 22

 

                                                               Kepler listed volumes whose volume formulas were unknown and Cavalieri in his Geometria indivisibilibus (1635) explained a process for finding many of these volumes by   using indivisibles.   Though easy to use and yielding correct results most of the time, Cavalieri's method lacked a mathematical foundation;   but, since it worked, it remained in use for about fifty years.      Paul Guldin criticized Cavalieri's use of indivisibles because they lacked rigor and led to fallacies.      If a line has no width then the addition of any number of lines can never yield an area, just as an infinite number of planes with no thickness cannot form a solid.  Furthermore if  a solid is made up of an infinite number of sections with no thickness, then these elements cannot even be compared with one another.    Guldin also was the chief critic of Kepler in his use of infinitesimals for analogous reasons:   in both cases there was a lack of rigor and mathematical foundation.   

        Andre Tacquet was another who would not admit Cavalieri's method to be either legitimate or geometrical.    A sphere was not made up of circles nor a wedge made up of triangles but rather a volume must be composed of homogenea , parts of like dimensions, that is,  solids of small solids and areas of small areas.     Tacquet used the method of exhaustion and pointed the way to the limit process, and his ideas later exerted an influence on Roberval and Fermat.      Tacquet was a brilliant mathematician of international reputation as well as a fine author.  His Opera mathematica   and his two textbooks on Geometry   and Arithmetic   were reprinted and translated often and were used throughout Europe during the seventeenth and eighteenth centuries.

       Although Tacquet was born a century before Newton published his calculus, he helped articulate some of the preliminary concepts necessary for Newton and Leibniz to recognize the inverse nature of the quadrature and the tangent.   He described how a moving point could generate a curve and he treated area and volume in his major work Cylindricorum et annularium (Antwerp, 1651).   This work influenced the thinking of Pascal and his contemporaries.

       Gregory St. Vincent was one of the pioneers of infinitesimal analysis. 23   

In his Opus geometricum (1649)   he proposed a new ingenious   method of approaching the problem of infinitesimals and he gave his propositions a direct rigorous demonstration instead of the reductio ad absurdum   argument used previously.     St. Vincent added an element not previously found in geometrical works because he connected the question with the philosophical discussions of continuum and the result of the infinite division.  George Sarton refers to this obliquely,  while discussing the esoteric nature of mathematics compared with other sciences; the historical treatment has to be different. 24

       St. Vincent summed infinitely thin rectangles to find a volume by a process he called "ductus plani in planum" (multiplication of a plane into a plane). It is practically the same fundamental principle as today's present method of finding a volume of a solid of integration.  The cubature of a solid bounded by two plane laminas would be obtained by constructing thin rectangles perpendicular to both laminas whose sum would equal the volume.    Let OXYZ be the volume enclosed by a conic surface with OX the axis and XYZ the base.    Consider an arbitrary, variable section xyz // XYZ and also perpendicular to OXZ.    Let xx' be the altitude of a rectangle equal in area to the section   xzy.    The volume of the figure is found by multiplying this variable xx', as it moves from O to X , by the breadth of the section OXZ and then adding the results.25

      St. Vincent applied his process to many such solids and found the volumes.     He differed from Cavalieri's method since his laminas "exhaust" the body within which they are inscribed: they have some thickness.     This was a new use of this term since it literally does "exhaust" the volume instead of finding the volume to some predetermined accuracy.     While St. Vincent was not clear how to visualize the process, he certainly was nearer to the modern view than any of his predecessors.    This led him to the concept of a limit of an infinite geometrical progression which ultimately supplies the rigorous basis for the calculus.    St. Vincent gave the first explicit statement that an infinite series can be defined by a definite magnitude which we now call its limit.   

 

St. Vincent was probably the first to use the word exhaurire in a geometrical sense.  From this word arose the name "method of exhaustion" . . . he used a method of transformation of one conic to another, which contains germs of analytic geometry.   St. Vincent permitted the subdivisions to continue ad infinitum and obtained a geometric series that was infinite. . . . He was first to apply geometric series to the "Achilles". . . moreover was first to state the exact time and place of overtaking the tortoise. 26

The terminus of a progression is the end of the series which the progression never reaches, even if continued to infinity, but to which it can approach more closely than by any given interval. 27

 

           St. Vincent in Book II of Opus geometricum applies his infinite-series process to Zeno's   Achilles paradox and is perhaps the first to do so successfully.       Boyer's comment on the efforts of St. Vincent and his unfortunate assumption that he had squared the circle:

 

Although St. Vincent did not express himself with the rigor and clarity of the nineteenth century, his work is to be kept in mind as the first attempt explicitly to formulate in a positive sense-although still in geometrical terminology-the limit doctrine, which had been implicitly assumed by both Stevin and Valerio, as also, probably, by Archimedes in his method of exhaustion.       He maintained that he had squared the circle . . . and received disdain from his contemporaries, his memory being rehabilitated by Huygens and Leibniz.    On the other hand there can be no doubt that his work exerted a strong influence on many of the mathematicians of his time. 28

 

         2. Area under a hyperbola

         Another breakthrough in the preparation for the calculus was due in part to St. Vincent and to another Jesuit,  Alphonse de Sarasa. 29    It was discovering and proving that the area between the hyperbola and its asymptote is a logarithmic relation.30

 

  By inscribing and circumscribing rectangles in a geometric series in and about a hyperbola, Gregorius developed a quadrature of a segment bound by two asymptotes, a line parallel to one of them, and the portion of the curve contained between the two parallels.     The relation between this procedure and logarithms was first noticed by Alfonso de Sarasa. 31

 

       The reasoning of Sarasa and St. Vincent focused on the fact that the area under the hyperbola from 1 to x plus the area from 1 to y equals the area from 1 to xy. The geometric   progression  1/1-x = 1+x+x2+x 3+...   where the left side is a hyperbola   whose area can be determined by the method of Archimedes and the right side is the logarithmic series according to the definition of Napier's logarithm.

 

. . . Just at this time a most important discovery was made concerning this case by the Jesuit Gregorius St. Vincent . . . at the end of his formidably bulky opus on geometry. . . concerning the quadrature of the circle, which so discredited his opus that those excellent discoveries at the end were almost overlooked. . . . This was the discovery that the area under the equilateral hyperbola is exactly the logarithm of Napier.    Only the proof was yet missing that this was the "natural" logarithm to the base  e . 32

 

       3. Center of gravity

       Another contributor to preparations for calculus was Charles de la Faille who owed  his fame as a scholar to his tract Theoremata de centro gravitatis partium circuli et ellipsis (Antwerp, 1632),   in which the center of gravity of a sector of a circle is determined for the first time.  His contribution is briefly described here and can be found in DSB, 33 which is the same as the essay of Henry Bosmans "Le Mathematicien Anversois."34 His method to find the center of gravity of a circular sector anticipates a limit process. He shows that if  B   is sufficiently small the length   R A2/sin 2A(1-sinB/B)   can be made arbitrarily small where A and B are   angles and R is the radius of a sector. He depended on the works of Clavius, Valerio and Archimedes in proving his main proposition:   the distance  d   between the center of a circle and the center of gravity of a circular sector can be given by the equation  

3d = 2 R (chord A)/arc A.  He found this as preparation for the quadrature of the circle.     Although this was the futile quest of so many at that time, it led to a beautiful set of theorems to be admired and used by many geometers to follow.

      Bosmans also published some correspondence of Gregory St. Vincent in which he notes that while de la Faille was well-known as the tutor of Don Juan of Austria and died young in that capacity, he was a fine mathematician.    Some appreciation of the above work is shown by Huygens who wrote to St. Vincent: "Your student (de La Faille) surpasses your colleague, Paul Guldin."  In Christiaan Huygens' judgment de la Faille was a new Archimedes in his treatment of the center of gravity: "Archimedes invented this but de la Faille has mastered it."  

       When de la Faille died prematurely in Barcelona while on an expedition with Don Juan of Austria, St Vincent was greatly upset.   Furthermore he wrote to Rome asking that the Spanish provincial   send him de la Faille's mathematical notes, because he felt that no geometers there in Spain would appreciate them and   that they would simply get lost in the Spanish Jesuit archives. They were too valuable and his results were too beautiful not to be published.   Jean Charles de la Faille's portrait by Anthony Van Dyke is found in a prominent place in the Plantin Museum  des Beaux Arts   in Brussels.   (In l984 it was on loan to the    Metropolitan museum in New York City.)

       4. Guldin's Rule

       Paul Guldin's second volume of De centro gravitatis   contains what is known as Guldin' s rule:  "If any plane figure revolves about an external axis in its plane, the volume of the solid so generated is equal to the product of the area of the figure and the distance traveled by the center of gravity of the figure." 35 

Guldin did not know that the fundamental theorem which bears his name and which he used extensively, is found in a somewhat vague form in the Collectio of the well-known Greek mathematician Pappus (ca. A.D. 300).   Nevertheless Guldin has been unjustly accused of plagiarism by earlier writers.   This defamation has been removed, however, by recent historians expert in that period, such as Paul Ver Ecke, who shows that the translation of Pappus available to Guldin, and  faithfully quoted by him, lacked the theorem in question.36 Furthermore, he demonstrates that the accusation against Guldin is weakened by the fact that various geometers who lived at about the same time as Guldin did not credit Pappus with this theorem but Guldin .   Among these writers is the noted astronomer Kepler, who presented   applications of Guldin's theorem.  The fact that Kepler failed to state Guldin's rule explicitly is explained by the custom of many of these early writers to give illustrative examples of a fundamental principle without stating it explicitly.

 

       The world owes a great debt of gratitude to those who, like Cardan and Guldin, contributed powerfully towards the enlightenment of the human race, especially at a time when so few people took an active interest in scientific matters.   It seems therefore to be very fitting that wide currency should be extended to results which tend to remove from their names unjust defamatory associations due to carelessness on the part of earlier writers. 37

                                                                                                                             In 1627 a correspondence on religious subjects developed between Guldin and Johannes Kepler when the latter wrote Guldin concerning his objections to the Catholic religion.  Guldin tried to refute them with theological arguments, but apparently to no avail.   Kepler had greater difficulties with Catholics than with the Catholic religion.

    

       5. Other quadratures

       HonorŽ Fabri contributed to understanding the quadrature of the cycloid in his geometrical works which Liebniz found so inspiring.   Another Jesuit geometer   Antoine de laLouvere   is considered one of the precursors of modern integral calculus and was well known to the other mathematicians of the time.  His most important work is the Quadratura circuli  (1651), in which he finds volumes and centroids by inverting Guldin's rule.     He moved  from mathematics to theology until in 1658 he became embroiled in a dispute with Pascal over his solution for the problem of the "Roulette" (cycloid) posed by Pascal.  Pascal's accusation that he plagiarized Roberval's solution was without foundation but the episode did bring de la Louvere back to geometry.  De la Louvere found the solution to be what he called a "cyclocylindrique" (helix), described in his last work Veterum geometria promota in septem de cycloide libris (1660).      Thus, de la Louvere became "the first mathematician to study the properties of the helix," 38 according to  W. W. Ball in his History of Mathematics (an author who expresses contempt for Jesuit efforts).

       De la Louvere's chief book is the Quadratura circuli   (1651), in which he drew upon the work of  de La Faille, Guldin , and St. Vincent.   His approach was an Archimedean summation of areas; he found the volumes and centers of gravity of bodies of rotation and curvilinearly defined wedges by indirect proofs.  He was then able to proceed by inverting Guldin's rule whereby the volume of a body of rotation is equal to the product of the generating figure and the path of its center of gravity.   Thus de la Louvere established the volume of the body of rotation and the center of gravity of its cross section; then by simple division he found the volume of the cross section.    For this he is referred to as one of the precursors of the calculus.   He  later returned to theology, which he said was "an easier task more suited to my advanced age."

 

f.  Geometry as calculus evolved

      Unquestionably the most remarkable mathematical achievement of the 17th century was the invention of the calculus.   This new tool proved to be astonishingly   powerful in  solving many problems that had been so baffling  in earlier days.   Much of this applicability lies in the field of geometry, and on the other hand a large part of geometry concerns properties of curves and surfaces which are examined by means of the calculus.     Archimedes and Apollonius studied areas, volumes and normals of conic sections and later Cavalieri studied curvature and evolutes.   But the first real stimulus to differential geometry was furnished by Gaspard Monge, a graduate of the Jesuit College de la TrinitŽ in Lyons.     He is considered the father of the differential geometry of curves and surfaces of space.   Another differential geometer was Boscovich whowas interested in developable surfaces ( a kind of ruled surface) and osculating curves.   Even though he contributed greatly to physics, he was more of a geometer than anything else.

 

He preferred the geometric method of infinitely small magnitudes   "which Newton almost always used" and which embodied the "power of geometry." He particularly applied it to differential geometry. . . In 1740 he studied the properties of osculatory circles and . . . in 1743 was the first to solve the problem of the body of greatest attraction. 39      

 

       Today we treat the hyperbolic functions as pairs of exponential functions

(ex + e -x)/2 but their inventor, the Jesuit, Vincent Riccati (son of Jacob)40 developed them and proved their consistency using only the geometry of the unit hyperbola x2-y 2 = 1 or 2xy = 1.    Riccati followed his father's interests in differential equations which arose naturally from geometrical problems.   This led him to a study of the rectification of the conics in Cartesian coordinates and to an interest in the areas under the unit hyperbola.   Riccati developed the properties of the hyperbolic functions from purely geometrical considerations.   He used geometrical motivation even though he was familiar with the work of Euler, who  had introduced the symbol and concept of the natural number   e   and the function  ex  ten years previous to Riccati's book.  Lambert is sometimes given undue credit for these hyperbolic functions.    Riccati had all the necessary details worked out and proven long before Lambert's treatise.41

       Riccati's development of the hyperbolic functions may be described as follows. The algebra concerning circular sectors is simple since the arc length is proportional to the area of the sector.   Hyperbolas, however, do not have this property, so a different algebra is needed.   The latter arises naturally from a property peculiar to hyperbolas   xy = k  and evident in the figure.   All right triangles OAK   have the same area  k/2  no matter what point A is chosen on the curve.  This property enables us to replace area measure by linear measure   OK,   thus introducing a kind of logarithm.

 

       It can be seen from figures 13 and 14  that the following areas are equivalent:

AOK = FOH because of the property just mentioned, AOQ = QKHF by subtracting the common area OKQ ,  AKHF = AOF by adding AQF to each.   So for any point F on the curve, the area of the sector AOF equals the area under AF.  This area function depends on point F as well as on H, the projection of F on the x-axis.

       Addition and subtraction of areas can then be accomplished by multiplication and division of distances along the x-axis, and vice versa.

Given  G/K   = P/H, then

         AKGE   = ºdx/2x

                         = .5 log(G/K ) = .5 log (P/H)

                   = ºdx/(2x) = FHPN .

Adding and subtracting then just involve constructing the correct proportions, i.e., finding the proper OP and OG for the proportions:

       OK x   OP = OG x  OH which represents adding:   AKPN = AKHF + AKGE

       OG/OK = OP/OH which represents subtracting: AKGE = AKPN - AKHF

       From these definitions of the hyperbolic functions and the logarithm operations, it is possible to derive a complete array of formulas regarding the double and triple arguments and from them to deduce other formulas for the square and cube roots.    Riccati goes even   further and proves the integral formulas for trigonometric as well as hyperbolic functions. 42

       Riccati and his collaborator Saladini present the first extensive treatise on integral calculus, predating that of Euler.   This was an important step since they offered examples of integration as a result of direct additions of elements, instead of Newton's and Leibniz's concept of integration as antidifferentiation.  

       Some other innovative works   of Riccati include his treatment of the location of points that divide   the tangents of a tractrix and also the problem posed by Guido Grandi about the four-leaf rose.   Another Jesuit Thomas Ceva (brother of John)44 investigated arithmetic-geometric-harmonic means.  Also he contributed valuable insights into the cycloid as well as the higher conic sections.

       Laplace gives credit to Boscovich for the method Laplace used to determine the most probable ellipse when he wrote, "Boscovich has given us for this purpose an ingenious method . . . in Voyage Asronomique et GŽographique."   L.L.Whyte45 describes the history of how the line of best fit was determined.

 

g.  Non-Euclidean geometry

       In 1621 Sir Henry Saville called attention to two blemishes ("naevi") in Euclidean geometry: the theory of parallels and the theory of proportion.  The latter has already been mentioned   concerning the works of Christopher Clavius in Chapter 2.  As for the former, the fifth postulate was called the "scandal of geometry" by D'Alembert.   But examination of this fifth postulate   gave rise to new branches of geometry   and showed that Euclid was freer from blemish than had been previously believed.     The first step was to establish equivalent axioms.     The parallel postulate, variously stated, is put most succinctly by Playfair:  "Through a given point not on a given line can be drawn only one line parallel to the given line."

      Clavius   showed that this could be deduced from the fact that a line whose points are all equidistant from a straight line is itself straight.      Girolamo Saccheri   showed that it is sufficient to have a single triangle, the sum of whose angle are two right angles.   Other equivalent forms do not show any essential superiority to Euclid's.

       The attempts to derive the parallel postulate as a theorem from the remaining postulates of Euclid's Elements occupied geometers for over two thousand years and culminated in some of the most far-reaching developments of modern mathematics.   Many "proofs" of the postulate were offered, but each was sooner or later shown to rest upon a tacit assumption equivalent to the postulate itself.  Particularly outstanding was the 1733 work of  Saccheri    Euclides ab omni naevo vindicatus, "Euclid freed of every blemish," in which he tried to institute a reductio ad absurdum   by finding a contradiction when  the parallel postulate was denied.  Though he failed in his efforts, he did bring to light a number of consequences that are recognized today as important theorems in non-Euclidean geometry.

       Saccheri's work was an attempt (probably the first in print) to explain the principles of logic in a geometric sense.   He was concerned about the definitions used by Euclid: the nominal definitions, which simply define a concept and the real definitions. These are nominal definitions to which a postulate of existence is attached.                 But this existence brings up the question whether one   part of the definition is compatible with another part.   In considering the parallelism between two lines he was the first to draw explicit attention to the question of compatibility between axioms.    John Sullivan in "Mathematics as an Art"  speaks in glowing terms of Saccheri's efforts.

 

Saccheri's attempt to demonstrate the parallel axiom was a miracle of ingenuity     . . . Saccheri was an extremely able logician, too able to make an unjustified assumption.    His method was to develop the consequences of denying the fifth axiom while retaining all the others.     In this way he expected to develop a geometry that would be self-contradictory.  He did not succeed in contradicting himself; what he did was lay the foundations for the first non-Euclidean geometry.46  

 

       Although Clavius early on brought attention to the troublesome fifth postulate of Euclid, it was not until Saccheri that a serious and concerted attempt was made  to investigate its independence.  Saccheri was one of the most accomplished  of all Jesuit mathematicians and carried on extensive correspondence.     Along with the Camaldolese monk Guido Grandi,   for example,  Saccheri wrote about quadratures and various curves.    But he is best remembered for his contribution to non-Euclidean geometry. 47  

The distinguished Italian mathematician  Eugenio  Beltrami considered Saccheri the real founder of Non-Euclidean Geometry and brought it to other geometers' attention.    Because of a subtle error Saccheri did not succeed in actually proving that the fifth postulate depends on the first four.  Nonetheless, he states some of the most important theorems of non-Euclidean geometry known today, by  making the assumptions that Gauss and Lobachevsky, who certainly had the works of Saccheri available to them, would make a century later.    The great Gauss was to discover that the fifth postulate could be denied, but he found the new structure so shocking he was afraid to publish it.

       Saccheri's work was forgotten for a century, until Beltrami published a note in 1889.     Not only did Saccheri initiate a new approach to Euclid's fifth axiom but he proved many of the theorms that would have followed if the fifth postulate were replaced. Julian J. Coolidge has a thorough treatment of the background of non-Euclidean geometry.

 

His note entitled "un precursore italiano di Legendre e di Lobachevsky" convinced the scientific world of the importance of Saccheri's work, and of the fact that theorems, which had been ascribed   to Legendre, Lobachevsky and Bolyai had been discovered by Saccheri many years earlier.48

 

       Much of Saccheri's book Euclides has been incorporated in the modern elementary treatment of the non-Euclidean geometries without sufficient credit.     Instead, he is merely remembered for the error in   his proof.    Saccheri was convinced of the truth of the Euclidean hypothesis.    His contradictory assumption was made to discover the contradiction which would follow, unlike Bolyai and Lobachevsky, who wanted to prove the possibility of contradicting the fifth axiom.

    His Saccheri quadrilateral   is a two right-angled isosceles quadrilateral  ABCD where  A=B=90 o and AC = BD.  The C = D is not dificult to prove from the first 28 theorems (which do not depend on the fifth axiom).    He discusses three possibilities.  They are both right angles, they are both acute angles or they are both obtuse.    These three possibilities as we know today refer to Euclidean, Lobachevskian or Reimannian geometries. 

       Boscovich 70 years later inherited the chair of Saccheri and realized the flaw in Saccheri's argument.  Boscovich stated that the properties of parallels cannot be rigorously established by deduction from the rest of the axioms, so he knew that the fifth postulate could be replaced by another and a different definition of parallel line. 49   His thinking was not far from Gauss and Lobachevsky in anticipating   non-Euclidean geometry.

 

h.   Projective geometry

       For centuries artists and architects sought formal laws of projection to place objects on a screen. d'Aguilon's Opticorum libri sex   treated successfully projections and the errors in perception, and it was used by an architect GŽrard Desargues, 50 who in   1639     published  a    remarkable treatise on the conic sections, emphasizing the idea of projection.   But  Desargues' work was so neglected by most other mathematicians of the time, because of his eccentric style and terminology, that it was soon forgotten.   Most geometers then, spent their energies either developing the new powerful tool of analytic geometry or trying to apply infinitesimals to geometry.

       In d'Aguillon's text on geometrical optics are found elements of perspectivities as well as the sterographic projections of Ptolemy and Hipparcus.    He proposes experiments, drawn by Peter Paul Rubens, to   study the variations of light intensity dependent on variations of distance.   The Bouguer photometer can be traced to   d'Aguilon's book.51   More details of this work have already been covered in the first chapter.

       Another projective geometer was HonorŽ Fabri, whosebook on optics was used by Philip de la Hire.  In this book Fabri states some of the principles of perspective geometry, and it is reviewed by the TRS. 52   The works of de la Hire were used in Peking by the Jesuit missionaries and by some of the Chinese painters in the 17th century, including Tshiao Ping-Chen who painted according to Western rules of perspective introduced by the Jesuits.    The best 18th century Jesuit geometer in China   was Pierre Jartoux , who also is credited with printing very accurate maps.

       One of the most remarkable sights in Rome is the perspective painting on the ceiling of St. Ignatius Church done by the Jesuit brother Andrea Pozzo , a geometer-artist.   He had been asked to come and decorate the church still under construction.    On arrival he found that due to   lack of funds no dome had been constructed.   To fill this gap he took a flat piece of canvas and by his peculiarly brillant use of linear perspective created the illusion of a dome.   By the sweep of his brush he gave St. Ignatius the giddy heights which the architect could not give.  Then, on the  flat, massive ceiling of the church he painted a fresco, in perspective, of the missionary spirit of Jesuit Society, thereby expressing Jesuit identification with the baroque spirit of Rome.  He wrote a remarkable book on perspective geometry for artists and architects.   Prospettiva de' pittori ed architetti (Rome, 1693-1700). Another Jesuit, Orazio Grassi, S.J. (d 1654), was the architect for the church, but it was Pozzo who started a new trend in Jesuit churches.

 

       Several books have been written about Pozzo's contributions to Jesuit iconography, but he is best remembered as one of the most influential perspective theorists of the 17th century. His book on perspective transformations is in two volumes, has been reprinted many times and, originally written in Latin and Italian, has been  translated into French,   German, English,53 Dutch and Chinese.

It has been reissued as recently as 1971.   In it he demonstrated how an irregular space could be represented on a stage by using wings obliquely.    He emphasized the theoretical possibilities of perspective geometry and his works concerned one focal point (for example, two circles of red marble on the floor of St Ignatius Church are the focal points for viewing the ceiling).   Pozzo expressed a new tendency in freer use of decorative elements in stage design.   He tried to find a focal point of the perspective out of sight of the audience by displacing it to one side, thereby creating a more realistic effect.  Three centuries later the cinema would put into practice some of his principles.

 

Pozzo alone had the opportunity to raise   the whole structure in full scale; he did not hide the focus of the perspective at either side, out of view of the beholder.   He could give an account of the gradual transition from real to reduced size on the stage without attaining the effect of a miniature world of the Mannerist stage.    Thus he fulfills the legacy of the Jesuits   Menghini and Rainaldi, which in turn is the most original and probably the most progressive  aspect of the Jesuits' contribution to Baroque stage design. 54

 

       When critics accused Pozzo of misuse of the baroque style by resting the columns of his "cupola" on simple brackets, he replied wryly: "If my brackets give way and the columns start to fall to the ground, you will easily find some painters among my friends to remake them and remake them better."  Over the years candle smoke had darkened his spurious dome, but recently this has been renovated by the Museums of Rome.

 

i.   Geometry applied to physics

       Newton experimented with the diffraction of light, but the discovery of this phenomenon was made by  Francesco Grimaldi,  professor of mathematics at the Jesuit college in Bologna.   Furthermore, it was Grimaldi who  named the phenomenon "diffraction" (breaking up). It was described in his work, Physico-mathesis de lumine (1666).  Through a small hole Grimaldi introduced a pencil of light into a dark room.  The shadow cast by a rod held in the cone of light was allowed to fall upon a white surface.  To his surprise he found the shadow wider than the computed geometrical shadow; moreover, it was bordered by one, two, and sometimes three coloured bands. 55  

         Grimaldi   was also the first to publish verification of Galileo's discovery for free-fall depending on the square of the time, and first to consider the effect on free-fall resulting from air resistance.            In the latter part of the l8th century Roger Boscovich was able to articulate an atomic theory using geometrical methods and in geometrical terms.   The three principal ideas of his theory were that the ultimate elements of matter were points (atoms), which were centers of force   and which force varied qualitatively and quantitatively in proportion to distance.  

His theory was so successful that his concepts of point centers were used (although Boscovich was not always quoted) by Priestly, Young,  Faraday, Maxwell, Kelvin, Thompson, Gay-Lussac, Weber, Helmholz, Lorenz, and others.56   As has already been mentioned, Boscovich's works have been collected and are kept in the Bancroft Rare Book Library at the University of California at Berkeley.            His influence  in the history of atomic theory is seen from Maxwell's statement: "The best thing we can do is get rid of rigid nucleus and substitute an atom of Boscovich ."   And Kelvin, after experimenting with several axioms, concluded,   "My present assumption is Boscovichianism pure and simple."57     Frequent articles in physics magazines center on his contribution to atomic physics l50 years before it was an acceptable theory.  There were no fewer than five celebrations throughout the world   on the bicentenary of his death in 1987,   in Paris, Rome, Milan, Dubrovnik and Munich.      These were held to commemorate the prophetic insights of Boscovich, who   charted paths for the new science of atomic physics.   His bold notion of "point-centers" abandoned the old concept of an assortment of different solid atoms.   The fundamental particles of matter, he suggested, were all identical, and matter was the spatial relations around these point-centers.  Boscovich , coming to these notions from mathematics and astronomy, foreshadowed the increasingly intimate connection between the structure of the atom and the structure of the universe, between the infinitesimal and the infinite.

         HonorŽ Fabri is another who quickly applied the new calculus to the physical world and was the first to give a cogent reason for the experiment   showing equal time for a body to fall.

         Ignace Pardies in his La Statique ou la science des forces mouvantes (Paris, 1673) was interested in the tension in a flexible line and was responsible for the Pardies principle  found in the solution of the suspension cable. 58 He argued that the form of a flexible line would remain unchanged if the forces at two points A and a were replaced by suitable forces acting along the tangents at A abd a.      This princlple was used in the later work in analyses of the catenary by Bernouli and Leibniz. 

 

Chapter 3 Footnotes

 

1.      DSB, vol 3, p. 311-312.

2.      Florian Cajori: A History of mathematical notation . Chicago: Open Court, 1928.

         Also see   Jukutheil Ginsburg: "On the Early History of the Decimal Point." American                     Mathematical Monthly , vol 35 1928, p. 347-349.   He states the case quite              convincingly that Clavius, though ignored for centuries, was in fact the first to use the          decimal point with full realization of what he was doing.

3.      Christopher Clavius: In Sphaeram Joannis de Sacrobosco , 3rd. edition Venice:

         1601, p. 13-15.

4.      Charles Naux: "Le P�re Christophore Clavius" in  Revue des questions scientifiques.

         vol. 154, 1983 p. 335-336.

5.      Z. Kopal: Introduction to the Study of the Moon. New York: Gordon & Breach,1966,                p. 217

6.      DSB, vol 11,  p. 402-403.

7.      William Bangert:  A History of the Society of Jesus. St. Louis:   Institute of Jesuit                            Sources, 1972, p. 160.

8.      Rudolf Wittkower & Irma Jaffe:  Baroque art:  the Jesuit contribution .  New York:

         Fordham, 1972, p. 68-88.

9.      Carl Boyer: History of mathematics. New York: Wiley, 1968, p. 334.

10.    DSB, vol 3, p. 312.

11.    Christopher Clavius: Astrolabium. Rome: 1593, p. 178.

         "Quaestiones omnes, quae per sinus, tangentes, atque secantes absolvi solent per solam         prosthaphaeresim, id est, per solam additionem, subtractionemque  sine laboriosa          numerorum multiplicatione, divisioneque expedire."

12.    D.E. Smith: Source book in mathematics. NewYork: McGraw-Hill, 1929,  p. 455 and

         p. 459.   Smith works from Jukutheil Ginsburg's translation of Clavius work.

13.    Lancelot White: Roger Joseph Boscovich, New York: Fordham, 1961, p. 41.

14.    D. E. Smith:   History of Mathematics. New York: Dover, 1958.

15.    Florian Cajori: A History of Mathematics. New York: Chelsea, 1985, p. 181-182.

16.    J. J. Coolidge:   "The Origin of Polar Coordinates" in the American Mathematical              Monthly.   vol 59, 1952, p. 78-85.   The author refers to Cantor's history.

         Moritz Cantor:   Geschichte der Mathematik .  2nd ed. Leipsig: vol. 2, 1900, p. 850.

17.    Lancelot Whyte: Roger Joseph Boscovich, New York: Fordham, 1961, p. 184.

18.    Ibid., p. 184-185.

19.    Florian Cajori: History of mathematics. New York: Macmillan,1985, p. 262.

         We conclude that Boscovich's  influence was only mathematical in view of the author's                 story of Laplace's reply to the question "Why did you not mention the creator in your                                                                                book MŽchanique CŽleste ?"   "Je n'avais pas besoin de cette hypoth�se-la." ,  

20. B. Sheynin, "Roger Boscovich's work on Probability",  Archives for History of Exact                         Sciences  vol. 9,  1973,   p. 306-324.

21.    Carl Boyer: The concept of the calculus. New York: p. 120-147.

22.    D.J. Struik:   A concise history of mathematics.   New York: 1948, vol 2, p. 88.

23.    DSB, vol. 9, p. 76.

24.    George Sarton: The Study of the History of Mathematics . Cambridge: Harvard,1936,              p.4.

25.    J. M. Child: The Early Manuscripts of Leibniz. Chicago: Open Court, 1920, p. 14.

26.    Florian Cajori: History of mathematics. New York: 1985, p. 181-182.

         Also see DSB, vol. 12, p. 74-76..

27.    Gregory St. Vincent: Opus geometricum.   Antwerp: 1647,  p. 55.

28.    C. B. Boyer: History of the Calculus.  New York: Columbia University Press, 1939,

         p. 136. He demonstrates how the "ductus" method works, illustrating that St. Vincent                   was considering infinitesimals  in a completely new light.

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

         vol. 1,   p. 137.

30.    Charles Naux: Histoire des logarithmes. Paris: 1871, p. 31.  The author traces the                     logarithmic properties developed by Sarasa and St. Vincent.

31.    DSB vol. 9 p. 75.

32.    Carl Boyer:   History of The Calculus . New York: Columbia, 1939,   p. 55-57.

33.    DSB, vol. 7, p. 558.

34     In Mathesis , vol. 41, 1927, p. 5-11.

35.    Paul Guldin: De Centro Gravitatis .   Vienna: 1635,  chapter 7. prop. 3,  p. 147.

36.    DSB, vol. 5,   p. 581.

         Also see.   Paul Ver Ecke, "Le Theorem dit de Guldin considŽrŽ au point de vu    historique," Mathesis.   1932   vol. 46,  395-397.     He shows that Guldin had access to                       an edition of the translation of Pappus' works which lacked this theorem.

37.    Science    vol. 64, #1652  8/27/26   p. 205.                                                   

38     W.W. Ball: A Short account of the history of mathematics . New York: 1901,  p. 319

         Also see DSB vol. 7,  p. 583-584.

39.    DSB, vol. 2, p. 330.

40.    Angelo Fabronio:   "De Tribus Riccatis"  in Vitae Italorum. Pisa: Alexandrum Lande, vol.                                                                                        16, 1795, p. 336-392.  The author tells of the lives of the three Riccati                       mathematicians: the father, Count Jacobo and his two sons, Jordano and  the Jesuit                     Vincentio.   All three are buried together: "as they were joined in common study                                                                                        during life they could remain companions   after death." 

41.    Moritz Cantor: Vorlesungen uber Geschichte der Mathematik . vol. 1, Leipsig: 1892,

         Teubner, p. 411.

         Also   Siegmund Gunther: Die Lehre von den gewohnlichen und verallgemeinerten    Hyperbelfunktionem. Halle:  Nebert, 1888, p. 1 - 55.

         Also   Florian Cajori: A History of Mathematical Notation . Chicago: Open Court,1928,                   p.172. Also,  D. E. Smith: History of Mathematics.  New York: Dover, 1958,

         p. 246-247.

42.    Charles Naux: Histoire des logarithmes. Paris: 1871, p. 124- 149.  This is a brief                             treatment of the main steps in Riccati's development of the hyperbolic functions.

         Also, see Joseph MacDonnell: "Using sinh and cosh to solve the cubic" in the   International Journal of Mathematical Education. London: vol 18 #1 1987, p. 111-118.

43.    DSB, vol. 7,   p.402.

44.    DSB, vol. 3,   p. 183-184.  Thomas was the brother of the geometer John after whom is             

         named "Ceva's theorem." Guido Grandi corresponded with both brothers about cycloids.

45.    Lancelot Whyte: Roger Joseph Boscovich, New York: Fordham, 1961,  p. 200-210.

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

         vol. 3,   p. 2019.   

47.    Henri Bosmans: "Le gŽometre Jerome Saccheri, S.J." in RŽvue des questions                     scientifiques,   4th ser.,  vol. 7, 1925, p. 401-430.

         See also: Amedeo Agostini: "Due lettere inedite di Girolamo Saccheri" in Memorie della            

Reale Accademia d'Italia   vol. 9,  9/28/30   p. 3-20.   The author presents  correspondence between Saccheri and Guido Grandi  concerning the semicubical        parabola.

48.    Julian L Coolidge: A History of Geometrical Methods . London:  Oxford, 1940,   p. 12-13.

49.    James Newman: op.cit., p. 187-189.

50.    DSB, vol. 4, p. 48.

51.    DSB, vol. 1, p. 181.

52.    TRS, vol. 5, p. 2055.

53     Andrea Pozzo, S.J.:   Rules and examples of perspectives.   London: 1707. 

         Reissued in         New York: Benjamin Blom, 1971.  This large (25 cm by 38cm) book has many

         perspective drawings of Pozzo as well as his rules for perspective drawing

         in both English and Latin.

54.    Rudolf Wittkower & Irma Jaffe:  Baroque art:  the Jesuit contribution .  New York:

         Fordham, 1972, p. 110.

55.    Florian Cajori: History of  physics .   New York:   Macmillan, 1898,  p. 88.

56.    Lancelot Whyte: Roger Joseph Boscovich, New York: Fordham, 1961, p. 121.

57.    Ibid.,   p. 191.

58.    Clifford Truesdell: Essays on the history of mechanics. New York: Springer-Verlag,       1968,  p.20.