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.
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