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
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.
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
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
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
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.
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.)