Demo One Gregory Saint Vincent, S.J.
(1584-1667)
and his polar coordinates
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Gregory 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. 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
Gregory 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.

Gregory 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. Gregory
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". . .
and was first to state the exact time and place of
overtaking the tortoise.
Gregory St. Vincent, S.J. was born in Bruges, Belgium in 1584 . He
was educated in mathematics under Christopher Clavius. Gregory
was a brilliant mathematician and is considered one of the founders
of analytical geometry. He founded his famous school of mathematics
in Antwerp. Gregory dealt with conics, surfaces and solids from a
new point of view, employing infinitesimals in a different way
Cavalieri.  Gregory was likely the first to use the word exhaurire in a
geometrical sense.  From this new point of view, the word became
known as "method of exhaustion," when applied to the formulas of
Euclid and Archimedes. Gregory used a method of transformation of
one conic into another, called per subtendas (by chords), which
contains the roots germs of analytic geometry. He also created a
special method which called "Ductus plani in planum", used in the
study of solids. Gregory permitted the subdivisions to continue ad
infinitum and obtained a geometric series that was infinite, unlike
Archimedes, who continued dividing distances only until a certain
degree of smallness was reached,
Gregory was the first to apply geometric series to the
"Achilles" problem of Zeno (in which the tortoise always
wins the race with the swift Achilles (because he has an
unbeatable head start) and to view the paradox as a
question in the summation of an infinite series. Gregory
was the first to state the exact time and place of
overtaking the tortoise. He wrote of the limit as an
obstacle against further advance, like a solid wall. He
was not troubled by the fact that in his theory the variable
does not reach its limit. His explanation of the "Achilles"
paradox was favorably received by Leibniz and by other
geometers over a century later.

Gottfried Leibniz credits Gregory St. Vincent 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 along with Fermat and Descartes as one of the
founders of analytic geometry. 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 Ancient Greeks described a spiral using an angle
and a radius vector, but it was St. Vincent and Cavalieri
who simultaneously introduced them as a separate
coordinate system.  Gregory wrote about a letter about
his new coordinate system to Grienberger in 1625 and
published his process in 1647. Cavalieri's later
publication appeared in 1635 and a corrected version in
1653.

Gregory was a pioneer of infinitesimal analysis. In his
Opus geometricum  published in 1649, he produced a
new method of attacking the dilemma of infinitesimals
with a rigorous demonstration instead of the reductio ad
absurdum argument previously accepted. St. Vincent
added an element unknown in geometrical works. He
invented the question with the philosophical discussions
of continuum and the result of the infinite division.
Gregory in Book II of his Opus Geometricum
applies his infinite-series process to Zeno's
Achilles paradox .

Although Gregory did not express himself
with the determination and clarity of later
centuries, his work is to be remembered as
the first attempt to formulate in geometrical
terminology-the limit doctrine, which had
been assumed by both Stevin and Valerio,
and Archimedes in his method of exhaustion.
He proposed that he had squared the circle .
. . and received disdain from his
contemporaries, his
memory being rehabilitated by Huygens and Leibniz.
There is no doubt that his work had a strong influence on
many of the mathematicians of his day.  He died in Gand
in 1667.
Gregory of St. Vincent (1647) In his
Opus Geometricum he claimed to
have squared the circle.
Bonaventura Cavalieri
(1598-1647)
He developed a method of indivisibles
which became a factor in the development
of the integral calculus.
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