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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