April 3, 1997
Early Calculus -- II
Frans van Schooten, (1615-1660), Netherlands, had succeeded his father as professor
of mathematics at Leyden.
Because the original by Descartes was difficult to read, Van Schooten made a careful and clear translation of Descartes'
La Geometrie into Latin, the preferred language of scholars.
Partly the reason for this was so that his students could understand it.
In 1659-1661, an expanded version was published. Geometria a
Renato Des Cartes.
Two additional additions appeared in 1683 and 1695.
It is reasonable to say that although analytic geometry was introduced
by Descartes, it was established by Schooten.
Jan de Witt, (1625-1672), the Grand Pensionary of Holland, was a
colleague of Schooten.
He wrote in his earlier years Elementa curvarum, a work in two
parts. The first part (Part I) was on the kinematic and planimetric
definitions of the conic sections.
Among his ideas are the focus-directrix ratio definitions. The term
`directrix' is original with De Witt.
Part II, on the other hand, makes such a systematic use of coordinates
that it has justifiably been called the first textbook on analytic
geometry. (Descartes' La Geometrie was not in any measure a
textbook.
Only a year before his death De Witt wrote A Treatise on Life Annuities (1671). In it he defines the idea of mathematical expectation. (Note, this idea originated with Huygens and was central to his early proofs of stakes and urn problems.)
In correspondent with Hudde he considered the problem of an annuity based on the last survivor of two or more persons.
Frans van Schooten
(1615-1660)
Dutch
Frans van Schooten (the father), professor at the engineering school connected with Leiden. The father was also a military engineer.
He was trained in mathematics at Leiden, and he met Descartes there in 1637 and read the proofs of his Geometry.
In Paris he collect manuscripts of the works of Vičte, and in Leiden he published Vičte's works.
He published the Latin edition of Descartes' Geometry. The much expanded second edition was extremely influential.
He also made his own contributions, though modest, to mathematics, especially in hisExercitationes mathematicae, 1657.
He trained DeWitt, Huygens, Hudde, and Heuraet. In the 1640's (at least) he gave private lessons in mathematics in Leiden.
Descartes recommended him to Constantijn Huygens as the tutor to his sons. Since the Huygens boys were coming to Leiden, Schooten decided to remain there.
Descartes' introduction opened to Schooten the circle of natural philosophers and mathematicians around Mersenne in Paris. Schooten tutored Christiann Huygens for a year.
Schooten maintained a wide correspondence, especially with Descartes.
First in Paris and then in London (1641-3) he made the acquaintance of mathematical circles, with which he maintained a correspondence that is now lost.
Bonaventura Cavalieri (1598-1647), a disciple of Galileo attempted
rigorous proofs for area problems. His method was dividing areas into lines
and volumes into planes.
His view of the indivisibles gave mathematicians a deeper conception of sets: it is not necessary that the elements of a set be assigned or assignable; rather it suffices that a precise criterion exist for
determining whether or not an element belongs to the set.
Cavalieri emphasized the practical use of logs (which he introduced into Italy) for various studies such as astronomy and geography. He published tables of logs, including logs of spherical trigonometric functions (for astronomers).
His method of indivisibles was to regard an area F by
, ``all the lines'' measured perpendicular from some
base.
His basis for computations is known to this day as Cavalieri's
principle: ``If two plane figures have equal altitudes and if sections
made by lines parallel to the bases and at equal distances from them are
always in the same ratio then the figures are also in this ratio.''
In modern terms for functions, if g(x)=cf(x), then
Using this method he was able to essentially perform
the integration
By the way............. What is area? This very important question was another consideration that completely illuded the Greek and even Newton and his successors. A working definition was made only in the last century, with the work completed in the 20th century.
Pierre Fermat
Fermat's areas. Fermat was able to compute areas under functions of
the form 
, sometimes called Fermat's hyperbolas.
His idea was to take a geometric partition of the interval
[0,a]. So for a given N the partition points will be
Similarly
Now sum the rectangles
Now let 
. This gives the equivalent of
Note: Fermat does not compute with the inscribed rectangles. He accepts the
limiting result.
A similar argument gives 
.
There is something very satisfying about this method as it avoids the
difficult problem of summing
where 
is a polynomial of degree which results when one
considers an equal interval partition.
Roberval and Fermat both claimed proofs, but it would be some years
before Pascal established his results on the triangle.
Did Fermat invent calculus?
Another ``experimenter'' with infinitesimals was Evangelista
Torricelli (1608-1647), another disciple of Galileo.
He completed his proofs with reductio ad absurdum arguments.
As he announced it in 1643, his most remarkable discovery was that the
volume of revolution of the hyperbola 
from y=a to 
as
finite - and he gave a formula. (Note, the corresponding area is
infinite.)
His method was basically cylindrical shells.
Said Torricelli: ``it may seem incredible that although this solid
has an infinite length, nevertheless none of the cylindrical surfaces we
considered has an infinite length.
John Wallis (1616-1703) an English clergyman and mathematician
was the first to ``explain'' fractional exponents. He used
indivisibles as did Cavalieri.
He arrives at the formula
in a rather unique way.
Consider 
between x=0 and x=1. To determine the ratio of the
area under this curve and the circumscribed rectangle, he notes the ratio
of the abscissas are 
. There are infinitely many such abscissas.
Wallis wanted to compute the ratio of the sum of the infinitely many
antecedents to the sum of the infinitely many consequences. This would be
which comes to
To calculate this he experimented
In general,
Having worked the case for the power k=3, Wallis makes the
inductive leap to
Wallis becomes known as the great inductor.
Wallis gives us our symbol for infinity, 
.
He generalizes his integration formula to rational exponents, and for
more general curves, particularly
Another formula based on induction
is obtained with respect to the function
Power Series
One of the principal tools that led to the full theory of calculus for
general functions was power series. Power series were the generalization of polynomials. And polynomials were the only functions which could be manipulated for the tangent and normal calculations. Although the trigonometric functions were known, they were in general well beyond the scope of 17th century mathematics.
James Gregory (1638-1675), extended the quadratures of
Archimedes to ellipses and hyperbolas using the Archimedean program:
Gregory believes 
to be transcendental. (Huygens did not.)
In two books published in 1668, he breaks with the Descartes
classification scheme: algebraic vs. mechanical. But the function concept
was still not there.
He knew this familiar formula
He knew the binomial theorem for fractional powers (Newton).
He discovered Taylor series 40 years before Taylor, and the Maclaurin
series for
(1671) Note: discovery in India 200 years earlier.
He gives us the formula
which is today called Gregory's series.
In 1668, Nicolaus Mercator (1620-1687) published his
Logarithmotechnica in which appeared the power series for the
logarithm.
From de Sarasa (1618-1667) he learned it was the area under the
hyperbola y=1/(1+x). From Wallis he learned the method of indivisibles
using an indefinite number of geometric series he arrives at the conclusion
From this point on tables of logarithms can be computed easily.
Rectifications
Hendrick van Heurat (1634-1660(?)) developed a method of
computing the rectification of curves. It appeared in Schooten's 1659 Latin
edition of Descartes' La Geometrie. How does he do it?
He sets up the equivalent of a differential triangle based on the
normal to the curve rather than the tangent. (However, he introduces an
arbitrary line segment, a requirement of homogeneity.)