Chapter 2 Inventions in Practical Geometry


     The geometrical apparatus recognized by Euclid   was simple: a straight edge and a compass.   To apply geometry, however, to gnonomics, hydrology, perspectives, architecture, astronomy and other fields dependent on geometry, more intricate apparatus had to be invented.  Some of the Jesuit geometers were quite good at this, and a few like Francesco Lana-Terzi and Athanasius Kircher were listed as inventors even though credit for their inventions was not always attributed to them.  The inventions may be grouped roughly according to pure geometry, and geometry applied to other fields such as dyalling/gnomonics, astronomy,  physics.

a.  Inventions to simplify geometry

b.  Inventions in horology/gnomonics

c.  Inventions in geometrical optics

d.  Inventions in physics


a.  Geometry  

     In his Modern Geometry , Howard Eves1 describes the pantograph invented by  Christopher Scheiner used for homothetic transformations for enlarging curves.  This instrument consists of parallel and intersecting rods hinged at appropriate ratio points and is found in one of the Rubens   paintings in a book by Erindo Rametta. 2  It is still available in art stores today.  Thomas Ceva invented an apparatus to divide an angle into n equal parts.  This was later  claimed by Guillaume l'Hospital (d 1704), who also claimed authorship of what is now known as   "l'Hospital's rule" which had been   discovered by his teacher John Bernoulli.

       Kircher invented what he called the Pantometrium   for solving problems of practical geometry. He also invented the first counting machine.  Both are described in Pantometrium Kirchereanum (Wurzburg 1669).  He constructed a machine for teaching mathematical methods which could be reduced to algorithms.   Kircher, by the way, was first to discover and report on sea phosphoresence. 3

         One of the most widely used devices found in so many instruments was invented by Clavius.  It is called the Vernier  scale.  Some historians of science wished to change the name so it would properly be called the Clavius scale because it was his invention.  This is related in the DSB: it started with an idea   Clavius received from his teacher Nunez.


              Nunez's method . . .was extremely difficult to engrave with precision using a different scale on each concentric circle.  Brahe remarked that in practice Nunez's method failed to live up to its promise.     .  .  . About fifty years later Clavius, who had studied under Nunez, found a way to facilitate the engraving of the various scales on the concentric circular segments.   . . . It thus became possible to read off degrees and minutes directly if the outer scale were divided into degrees and if there were sixty concentric circular segments.  These methods were described by Clavius in his Geometria practica (Rome, 1604), which Vernier surely read and meditated on. 3


       It is well-known that Pascal constructed a calculator in 1641 made of small gears.    A model of it of recent vintage is on display in   the Ars et MŽtiers Museum in  Paris (near a display of Kircher's water organ).    But it is not well known that Jean Ciermans   in his book Repetitio Menstrua (Louvain, 1639) described a similar calculating instrument two years earlier, which he constructed of small gears "incapable of erring." 4    Cierman's   ideas ran a little ahead of technology, and   he was not satisfied with the construction of   the apparatus which he designed. 5 Later another Jesuit, Joseph Liesganig, invented a device to read tangents instead of angles and found a way to fit clocks with a micrometer screw for more accurate reading.


b.    Horology  

       Horology or gnomonics, sometimes called Dyalling  (Dialling), was a practical geometrical study originating in the time of the Chaldeans.   It consisted of creating rules for making sundials.   One such device, called the "capuchin," since its appearance resembles a cowl, is attributed to the Jesuit Fran�ois de Saint Rigaud,S.J.    Gaspar Schott wrote about these devices as did Kircher in his Ars magna lucis (Amsterdam, 1671).    An 800 page work of Clavius, Gnomonices (Rome,1612), contained all that was known about gnomonics and horology at the time of publication. 

       In the Florence cathedral of Santa Maria is found a plaque commemorating the work of Leonard Ximenes , a Jesuit geometer and highway engineer (after whom is named the main road to Pistoia), for his work on the largest gnomon in the known world, which had been constructed in the 5th century. 6   Ignace Pardies   was another geometer known for his dial designs, and   the TRS thought it worth mentioning, in a review of his book, ElŽments de gŽomŽtrie:


The Learned Author, Professor of the Mathematiques in the Parisian College of Clermont, has found  the difficulty met within the Practick of Dyalling.7


       A rather celebrated case of "dyalling" is the water clock of Francis Line whose construction consisted of a glass sphere floating freely inside a larger glass sphere which was filled with fluid, making one complete revolution every 24 hours.   The inner sphere rotated at a regular rate throughout the day in such a way that a little fish, also floating in the fluid, pointed towards the hours of the day which were marked on a scale on the outer surface of the floating globe. 8        After constructing one at Liege and keeping its workings secret, he was asked by Nicholas Peiresc, a friend of Galileo, if he would bring it to Rome to help Galileo defend  the heliocentric theory, because the unusual design was fascinating and had interesting demonstration possibilities, even for inquisitors.   Although Line was willing, Galileo was not.  Later, Hooke figured out how the mechanism worked and published it in the TRS. 9

       Line was referred to in the TRS as "Francis Line (alias Thomas Hall)" as if it were all one name.   It was common for exiled English Jesuits to use aliases to protect their families and also help them   move in and out of England with greater ease.   It was, after all, a time when those who harbored Jesuits in Merry England were pressed to death while the Jesuits themselves were hanged, drawn and quartered.  In spite of all this, Jesuits desired to return on any pretext and sometimes their presence was even needed  by the Loyalists.  In 1669 King Charles II felt he needed a sundial for his garden in Whitehall: it had to be the best dial possible, so  Francis Line was chosen for the job.  Some sort of gentleman's truce was arranged.  Line built the dial himself, modeling it after the sundial he had built at Liege, and it was an immense success. Then he described it in   Protestant England's very own periodical, the Philosophical Transactions of the Royal Society.10   Later he published an illustrated   book with the elaborate title: An Explication of the Diall sett up in the King's Garden at London, an. 1669, by which besides the hours of all kinds diversely expressed, many things also belonging to Geography,   Astrology and Astronomy are by the Sunne's shadow made known to the eye (Liege, 1673).  

       The reform of the calendar was perhaps the best known Jesuit influence on time measurement.   Were it not for Christopher Clavius we would be celebrating Christmas on 12 December solar time.   In 1582, the Julian calendar ended   on a Thursday, 4 October,  with the promulgation of the papal bull   Inter gravissimas   (published 24 February 1582), stating that   "ten days be taken away, from the third nones to the ides of October."  The next day, the Gregorian calendar began on a day named Friday, 15 October (after the arbitrary fashion in which mathematicians define objects and use axioms).   

October was chosen by Clavius for the conversion because it was the month with the fewest number of feast days and also seemed to promise the least interference with commerce.  

       The   calendar change was the result   of the work of a commission appointed by Pope Gregory XIII and led by  Clavius to correct the Julian calendar.  For centuries, it was known that Easter was being celebrated on the wrong day, and sometimes with an error of twenty-eight days.   For about 800 years many scholars, such as Roger Bacon (and in a later century Carl F Gauss), had failed to find a formula which would identify the correct date for Easter, and also correct the Julian calendar for the future. 

       The problem with the Julian calendar was partly astronomical and partly arithmetic.   A year (the time for a complete transit of the earth around the sun) is shorter now than it was in 45 B.C., when the Julian calendar was adopted.  Then the transit time was 365.2422 days; now it is 365.2419 days (a day being the time required for the sun to return to a fixed meridian).      Neither of these numbers is an integer, so a year does not have an integral number of days.   If a year were divided up into a number of equal days, there would be no 23rd day of the year; it would be, for instance, the 23.1368th   day of the year.  No one wanted this, especially printers!

       In the Julian calendar, a year was taken to be 365 and l/4 days so that by adding one extra day (February 29) at the end of every fourth year (the Julian calendar began on March l), the fraction would be taken care of.  It was not.  The solar year could not keep up with the Julian calendar since it was was 664 seconds longer than it was supposed to have been.   By 1582 the vernal equinox had drifted to March 11.   Easter, therefore, shifted erratically because of its complicated dependence on the vernal equinox .  This had been decreed by the council of Nicaea in 325: "the first Sunday after the date of the first full moon that occurs on or after the vernal equinox."  Clavius' task, to calculate the time of the vernal equinox and to correct the shift, was monumental considering the meager astronomical and mathematical resources available at the time.  This was long before the invention of most of the mathematical tools we take for granted today.  It was a time that preceded the use of a decimal point (introduced by Clavius ) and long division was considered a college major!   The accuracy of his calculations has earned Clavius historical fame, and it took him 800 pages in his Novi calendarii (Rome, 1588) to explain and justify his results.

       Of the many attempts to solve the problem, some were more precise, but required a thorough knowledge of astronomy to compute a date.  Kepler, defending Clavius' simple plan, said:  "After all, Easter is a feast, not a planet!"  Joseph Scaliger, author of a competing plan, took the rejection of his plan less than gracefully and referred to Clavius as nothing more than a "German fat-belly."   Later, in a better mood, Scaliger acknowledged his esteem for Clavius saying, "A censure from Clavius is more palatable than the praise of other men."

       Implementation of the plan was not an immediate and universal success.  It had a fate similar to the adoption of the metric system in America today.  The populace was quite upset and windows were broken in the houses of the European Jesuits who were blamed for the change.  The Orthodox Church saw it as a Roman intrusion (which it was), and Protestant countries were understandably reluctant to accept any decree from the pope.   England did not adopt the Gregorian calendar until 1752, while Orthodox Russia would require the Bolshevik revolution.

       It is significant   that Clavius discovered that 97 days had to be added every 400 years and that this could be easily arranged by adding a day every four years omitting the three   century years not divisible by 400.   The wonder is that he was able to measure the year length so accurately that he knew the number of  extra days required.  To this day no one knows how he did it.  It is interesting to notice that he, like all the others coping with the calendar problem, assumed a geocentric system when working on the calendar, although he later taught and became a staunch supporter of   the heliocentric theory. The fact that there exist 400 unique   calendars is another indicator of the complications involved:   1988 will not be exactly duplicated until 2388.    A cursory review of the extensive "calendar" literature over the years  illustrates that correcting the calendar was not a trivial problem.

c.   Astronomy and Geometrical Optics

       Christopher Grienberger  invented the equatorial mount in which the telescope rotates about an axis which is parallel to the earth's axis. 11  It was described by Scheiner in his Rosa Ursina  sive Sol   (Bracciano, 1630),  although  he had used it as early as 1620.  In 1613 Scheiner in turn contributed to the invention of the refracting telescope   with which we are familiar today.    He constructed a number of different kinds of telescopes, and in particular (perhaps at the suggestion of Kepler) he made one with two convex lenses instead of Galileo's scheme which included one concave and the other convex.     This improved sightings greatly.    Scheiner gave one of his telescopes to the archduke of Tyrol who was more interested in the scenery from his Innsbruck castle than he was in the stars.     When he complained that the image was upside down, Scheiner inserted another lens to invert the image and so created one of the first terrestrial telescopes. 12

       Scheiner's pupil, another Jesuit geometer and astronomer, Johann Cysatus was the first to make a telescopic study of a comet in 1618 and gave the first description of the nucleus and coma of a comet.     At the entrance to the moon exhibit at the Smithsonian in Washington, D.C. is the first  selenogragh of the moon with craters and mountains identified by persons' names.    It was meticulously drawn by another Jesuit, Francesco Maria Grimaldi and was published in the Almagestum Novum (Bologna, 1651) of  John Baptist Riccioli.

       The Jesuit geometers and astronomers in Rome were able to gather information from their former pupils in China and India  on lunar and solar eclipses and transits of Venus.   The information that was gathered enabled   Riccioli   to compose a table of 2,700 selenographical objects, incomparably more accurate than anything previously known.

       A recent article in New Scientist 13 concerns an astronomical observatory whose director was the Flemish Jesuit Ferdinand Verbiest.    When the Jesuits were forced out, after the suppression of the Society, it 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, China and Belgium negotiated an agreement to restore the observatory (the assumption being that since Verbiest was Flemish the Belgians would want to contribute).  

       Among the many tasks entrusted to  Verbiest by the Emperor was the improvement of the defense weapons.   He cast 132 cannons for the imperial army, instruments far superior to any previous Chinese weapons.  

He also invented a more efficient type of gun carriage.    When the guns were finished Verbiest determined to consecrate them in a public ceremony and so on each gun he engraved the name of a Christian saint.  Clothed  in surplice and stole, he sprinkled the cannon with holy water and offered prayers for the proper deployment of the guns, while imperial dignitaries looked on.   The tests of the cannon were successful and the grateful Emperor took off his own robe and gave it to Verbiest in an unusual mark of imperial favor.  The role of the Jesuits as cannon makers again brought down upon the Jesuit order bitter criticism from its enemies in Europe, but Pope Innocent XI sent Verbiest a brief (12/3/1681) praising him "for using the profane sciences for the safety of the people and for the spread of the faith." It is unlikely that such a brief would have been sent today.

       The reflecting telescope was invented by NicolasZucchi in 1606, who brought one to Kepler as a gift from the Society of Jesus at the urging of another Jesuit,  Paul Guldin. Kepler was so thrilled with it that he dedicated his last book to Paul Guldin . 

       The achromatic telescope was invented by Boscovich, as was the ring micrometer (although credit is sometimes given to Huygens). 14  He proposed an original theory of the double refraction occurring in Iceland Spar. 15   He did not take undue pride in his inventions but  stated that he published his inventions in order to stimulate someone else to invent something better.16


d.   Physics

       Francesco Grimaldi discovered the diffraction of light and gave it the name diffraction , which means "breaking up."  He laid the groundwork for the later invention of the diffraction grating.   Louis Castel invented an ocular harpsichord which makes colors and musical tones correspond.   Capillary dispersion was discovered by HonorŽ Fabri who also gave the first reasonable explanation of why the sky is blue.

       Francesco Lana-Terzi is found at the head of literature on Aviation because of the treatise in his book Prodromo alla Arte Maestra (1670) on aerostatics.17  His work was translated by Robert Hooke and presented to the Royal Society of London by Robert Boyle.    Later it was discussed by physicists for over a century  before the first successful aerostatics flight by the Montgolfier brothers in 1783.   His work fascinated scientists because it was the first time anyone worked out the geometry and physics for such a device.   Instead of putting something into   a balloon as we do today he would take the air out.     Lana-Terzi's  treatise had five principles.

1.    Air has weight as Boyle's pump proves.

2.    This weight can be calculated.

3.    Nearly all air can be exhausted from any vessel,  as in animal respiration.

4.    From Euclid we know that while the area of a sphere depends on the square of the          diameter, the volume depends on its cube, so that if a sphere has a large enough          area it can hold a predetermined mass of air inside.

5.    Finally, from Archimedes we know that lighter bodies float in denser fluids.      

From the latter Lana-Terzi concludes that one could construct a vessel which would weigh less than the air within and so when the air had been pumped out the whole would float in the atmosphere.     In fact if the vessel were made large enough it could support the weight of a ship with passengers.     After he had calculated the weights and volumes involved, the vessel he proposed consisted of four large  twenty five foot spheres made of thin sheet copper bound together and supporting a basket for the riders,  with a  sail   and rudder for steering.

       After a long discussion of these principles Lana-Terzi answers   the objections to his proposal. First, the problem of evacuating the air could be accomplished by Boyle's pump. A second objection was that the airship  was liable to float off into outer space so that the riders would not be able   to breathe.  Lana-Terzi shows unusual grasp of aerostatics for that time by replying that the ship would stop rising as soon as the density of the atmosphere counterbalanced the weight of the ship.   Landing the craft once it is airborne is guaranteed by installing a valve to let air into the four spheres as ballast which would bring the ship back down to earth.   The sail and a large rudder would take care of steering the ship so that it would not be blown away.    Later experiments proved that a sail would not be an   effective steering device.   The most serious problem Lana-Terzi  addresses and the one most scientists noticed   was the fact that the spheres would be crushed by the atmosphere when the air was pumped out. Lana-Terzi's answer was clever but inadequate.  He thought that the spherical shape would prevent it from being crushed   because of the perfect uniformity of a sphere, somewhat as an eggshell is not easily broken when surrounded by uniform pressure when held properly in the palm of one's hand.   He happened to be wrong in  solving this last problem,  and so his proposed ship never did succeed. His expectation of landing ease was too optimistic also: "There is no need for ports since the balloon could land anywhere."     Lana-Terzi's treatment was remarkably   thorough for an era when experimental data was quite scarce. Berbardo Zamaga was inspired to write a poem concerning Lana-Terzi's "Navis Aeria".    Some 39 years later another Jesuit Bartholomeu Louren�o de Gusm‰o, S.J. from Brazil attempted to use hot air under a kind of umbrella, supporting a basket for riders, in the presence of the    king of Portugal (who paid for the experiment).   He momentarily got off the ground but in doing so set fire   to a part of the king's house.    "Fortunately the king did not take it ill," an onlooker later wrote. 


    Lana-Terzi's influence on speculation for flight was of long duration. For a century it was studied and discussed by scientists such as Sturm, who had great praise for the plan, and by Leibniz who verified Lana-Terzi's calculations.   Lana-Terzi answered the many objections but failed to recognize the crushing force of the atmosphere on the supporting balloons, in fact, in his time no one else did either.   Lana never built the machine for two reasons: one was that, because of his vow of poverty, he did not have the 2,000 ducats necessary, and secondly, he felt that God would never allow the machine to succeed, because it would lead to aerial bombardment.   At least in the latter he was right. It only took l0 years after the Montgolfier brothers flight for the French Navy to adopt balloons as warships.


 The sentiment of the time was enthusiasm for flying, be it for pleasure, for honor or "for profit and the benefit of mankind."   Lana-Terzi alone, so much a scientist and a man of his own century saw the possible destruction of mankind.      It is ironic then that he  is called "the father of Aviation ."

       Lana-Terzi also discovered a way to grow seedless fruit, and in his book Prodromo is found a large collection of inventions.   For these reasons, he is listed by Stern as an inventor along with Torricelli and Huygens.  Also, it would explain why the Royal Society adopted him as a regular contributor along with Newton and Boyle.  A book company's advertisement concerning a recent reprint of Lana-Terzi 's Prodromo  describes this remarkable book reprinted over and over for more than three centuries. "The original work was found among some very rare volumes . . . a comprehensive work of inventions by Father Lana of Brescia. . .which was reprinted separately in view of the valuable and interesting nature of the subject matter. . . for the erudite reader."

       In Ars magnesia   Kircher described a device for measuring magnetic power by means of a balance.   Later, he carefully compiled measurements of magnetic declination from several places around the world, as reported by Jesuit scholars, and particularly by his disciple Martin Martini (d 1638), who in a letter suggested the possibility of determining longitudes by the declination of a magnetic needle.18


Chapter 2 Footnotes


1.      Howard Eves: Modern Geometry.  Boston: 1972, p. 107.

2.      Conor Reilly: Studia Kircheriana . Wiesbaden-Rom:  Edizioni del Mondo, 1974, p. 78.

3.      DSB vol 13, p. 622. 

         Brahe's remark can be found in Astronomiae instauratae mechanica.

         Nuremberg: 1611, p. 2.

4.      Joseph E. Hofmann: History of mathematics. Translated from the German by Frank                  Gaynor & Henrietta Midonick. New York: Philosophical Library, 1957,   p. 114.

5.      D.E. Smith:  History of Mathematics . New York: Dover,  p. 203.                  

6.      Rufus Suter: "Leonard Ximenes and the  Gnomon" in ISIS   vol 55, p. 79 

7.      TRS   vol 7,  p. 5l50.

8.      Conor Reilly: Francis Line, S.J. An Exiled English Scientist .  Rome:  

         Institutum Historicum, 1969, p 27.

9.      Ibid., p.33.

10.    TRS vol 23,   p. 1416-1418.

11.    A. G. Debus: World Who's Who in  Science. Chicago:  Marquis, 1968.

         Also see: Sherwood Taylor, A short history of science and scientific thought.   New          York:   Norton, 1949,  p. 97.

12.    Daniel O'Connell:   "Jesuit Men of Science" in Studies in Irish Literature and Science

         vol 44, 1955, p. 313.

13.    G. E. Beekeman:   "New glory for the ancient Beijing observatory" in New Scientist.   20            Sept. 1984,  p. 54.

14.    Lancelot White: Roger Joseph Boscovich. New York:  Fordham, 1961 p. 195n.

15.    Ibid., p. 194, 195.

16.    Ibid., p. 196.

17.    M. Zanfredini:   "Un gesuita scienziato del 600:" in La Civiltˆ Cattolica    18 luglio                        1987, p. 115-128.

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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18.    DSB vol 7,   p. 375.