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.

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,

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.

"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

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.

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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