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Paula Menefee
MAT 266 - Brewer
Honors Project
Spring 2011
Roberval: Credit for His Contributions to Calculus
Gilles Personne de Roberval was born in 1602 as simply “Gilles Personne.” He added the
“de Roberval” in 1628 after moving to Paris. His parents were both farmers, but a priest of a
nearly parish, who also happened to be chaplain to the queen, saw Roberval’s intelligence and
began teaching him mathematics. He travelled around France continuing his studies until he
became part of a circle of academics. He spent four years with that group, and in 1632, he
became a professor of philosophy at the Collège Gervais in Paris. Whenever he was not teaching,
he prepared to compete for the Ramus Chair at the Collège Royale. This chair was a position that
represented great mathematical prestige. Roberval won the competition in 1634, but every three
years after that he had to re-defend the chair. He was appointed to a second chair at the Collège
Royale, Gassendi’s chair, in 1655, and he held both the Ramus chair and the Gassendi chair until
his death in 1675.
Although Roberval contributed a great deal to mathematics, it is difficult to recognize just
how much he contributed because so many of his works were not published. He only published
two books: Traité de mécanique des poids soutenus par des puissances sur des plans inclinés à
l'horizontale in 1636 and Le système du monde d'après Aristarque de Samos in 1644. Many
historians believe that the reason he did not publish more of his work was connected to the
competitions for the Ramus chair. The competition consisted of each candidate giving a lecture
and then all of the candidates proposing problems. It seems that Roberval would propose
problems that only he could solve so that he could keep the chair. If he had published more of his
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work, then his competitors would have known how to solve the problems he presented at the
competitions.
In 1693,18 years after his death, a large portion of his work was published in Divers
ouvrages de mathématique et de la physique par messieurs de l'Académie Royale des Sciences.
Much of his work would have been revolutionary at the time it was done. However, by the time
it was published, many others had already published similar ideas and received credit for them.
Thus, although no one ever stole any work from Roberval, a few of his contemporaries were
doing similar work and received recognition the discoveries simply because they published their
material. One famous example of this was with Cavalieri (1598-1647), although it also happened
with Pascal, Descartes, and Torricelli as well.
Cavalieri focused some of his work on volume and area because of his interest in Galileo.
In 1635, Cavalieri published Geometria indivisibilibus continuorum, in which he demonstrated
an early method of differentiation, based on work by Archimedes. Cavalieri’s method of
indivisibles, a term coined by philosophers in medieval times, involved dividing up a line, an
area, or a volume into an infinite number of parts that could not be broken up any further. To
find the area under the graph of a function, Cavalieri said to first think of filling the area by
lining up lines inside of it. Then, each line could be thought of as an extremely thin rectangle.
The area of each of these rectangles would be the height
multiplied by the width, and all of these areas could be added
together to give the total area under the function. The image
at left shows the skinny rectangles that result when an area is
filled with lines.
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Roberval’s Traité des indivisibles was in the group of work that had been published in
1693. He discovered how to find the area under a function and also the volume of certain solids.
Roberval also talked about indivisibles, and he came to the same conclusion as Cavalieri’s.
However his approach to them were different. His method is akin to what modern calculus refers
to as the Left-Hand Sum method. He divided the domain of the function into subintervals and
then constructed rectangles, each of whose width was that of the subinterval and whose height
was the functional value at the leftmost side of that rectangle. The area of each rectangle would
be its width multiplied by its height, and an estimate of the total area under the function would be
the sum of the areas of all the rectangles. Then, he
considered what would happen if he allowed the number of
subintervals to approach infinity. The picture at left
illustrates the drawing of rectangles with a height of the
functional value at the leftmost point of each rectangle and
a width equal to the subinterval.
Although they approached the problem differently, Roberval and Cavalieri came to the
same conclusion about the area under a curve. Roberval does deserve credit for solving it first,
but anyone doing mathematics should still be grateful to Cavalieri for sharing his discoveries
with the rest of the world.
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Sources:
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Roberval.html
http://www.math10.com/en/maths-history/history5/origins-differential-integral.html
http://www-history.mcs.st-and.ac.uk/PrintHT/The_rise_of_calculus.html
http://cerebro.xu.edu/math/math147/02f/cavalieri/cavintro.html
Images from:
http://cerebro.xu.edu/math/math147/02f/cavalieri/cavintro.html
http://www.sparknotes.com/math/calcab/introductiontointegrals/section2.rhtml