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.
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