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April 3, 1997

Early Calculus -- II

tex2html_wrap_inline262    Frans van Schooten, (1615-1660), Netherlands, had succeeded his father as professor of mathematics at Leyden.

tex2html_wrap_inline262    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.

tex2html_wrap_inline262 Partly the reason for this was so that his students could understand it.

tex2html_wrap_inline262    In 1659-1661, an expanded version was published. Geometria a Renato Des Cartes.

tex2html_wrap_inline262    Two additional additions appeared in 1683 and 1695.

tex2html_wrap_inline262    It is reasonable to say that although analytic geometry was introduced by Descartes, it was established by Schooten.

tex2html_wrap_inline262    Jan de Witt, (1625-1672), the Grand Pensionary of Holland, was a colleague of Schooten.

tex2html_wrap_inline262    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.

tex2html_wrap_inline262    Among his ideas are the focus-directrix ratio definitions. The term `directrix' is original with De Witt.

tex2html_wrap_inline262    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

tex2html_wrap_inline262 Frans van Schooten (the father), professor at the engineering school connected with Leiden. The father was also a military engineer.

tex2html_wrap_inline262 He was trained in mathematics at Leiden, and he met Descartes there in 1637 and read the proofs of his Geometry.

tex2html_wrap_inline262 In Paris he collect manuscripts of the works of Vičte, and in Leiden he published Vičte's works.

tex2html_wrap_inline262 He published the Latin edition of Descartes' Geometry. The much expanded second edition was extremely influential.

tex2html_wrap_inline262 He also made his own contributions, though modest, to mathematics, especially in hisExercitationes mathematicae, 1657.

tex2html_wrap_inline262 He trained DeWitt, Huygens, Hudde, and Heuraet. In the 1640's (at least) he gave private lessons in mathematics in Leiden.

tex2html_wrap_inline262 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.

tex2html_wrap_inline262 Descartes' introduction opened to Schooten the circle of natural philosophers and mathematicians around Mersenne in Paris. Schooten tutored Christiann Huygens for a year.

tex2html_wrap_inline262 Schooten maintained a wide correspondence, especially with Descartes.

tex2html_wrap_inline262 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.

tex2html_wrap_inline262    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.

tex2html_wrap_inline262 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.

tex2html_wrap_inline262 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).

tex2html_wrap_inline262    His method of indivisibles was to regard an area F by tex2html_wrap_inline312 , ``all the lines'' measured perpendicular from some base.

tex2html_wrap_inline262    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


tex2html_wrap_inline262    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

tex2html_wrap_inline262    Fermat's areas. Fermat was able to compute areas under functions of the form tex2html_wrap_inline328 , sometimes called Fermat's hyperbolas.

tex2html_wrap_inline262    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 tex2html_wrap_inline340 . This gives the equivalent of


tex2html_wrap_inline262 Note: Fermat does not compute with the inscribed rectangles. He accepts the limiting result.

tex2html_wrap_inline262    A similar argument gives tex2html_wrap_inline348 .

tex2html_wrap_inline262    There is something very satisfying about this method as it avoids the difficult problem of summing


where tex2html_wrap_inline354 is a polynomial of degree which results when one considers an equal interval partition.

tex2html_wrap_inline262    Roberval and Fermat both claimed proofs, but it would be some years before Pascal established his results on the triangle.

tex2html_wrap_inline262    tex2html_wrap_inline262 Did Fermat invent calculus?

tex2html_wrap_inline262    Another ``experimenter'' with infinitesimals was Evangelista Torricelli (1608-1647), another disciple of Galileo.

tex2html_wrap_inline262    He completed his proofs with reductio ad absurdum arguments.

tex2html_wrap_inline262    As he announced it in 1643, his most remarkable discovery was that the volume of revolution of the hyperbola tex2html_wrap_inline368 from y=a to tex2html_wrap_inline372 as finite - and he gave a formula. (Note, the corresponding area is infinite.)

tex2html_wrap_inline262    His method was basically cylindrical shells.

tex2html_wrap_inline262    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.

tex2html_wrap_inline262    John Wallis (1616-1703) an English clergyman and mathematician was the first to ``explain'' fractional exponents. He used indivisibles as did Cavalieri.

tex2html_wrap_inline262    He arrives at the formula


in a rather unique way.

tex2html_wrap_inline262    Consider tex2html_wrap_inline386 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 tex2html_wrap_inline392 . 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,


tex2html_wrap_inline262    Having worked the case for the power k=3, Wallis makes the inductive leap to


tex2html_wrap_inline262    Wallis becomes known as the great inductor.

tex2html_wrap_inline262    Wallis gives us our symbol for infinity, tex2html_wrap_inline410 .

tex2html_wrap_inline262    He generalizes his integration formula to rational exponents, and for more general curves, particularly


tex2html_wrap_inline262    Another formula based on induction


is obtained with respect to the function


Power Series

tex2html_wrap_inline262    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.

tex2html_wrap_inline262    James Gregory (1638-1675), extended the quadratures of Archimedes to ellipses and hyperbolas using the Archimedean program:

tex2html_wrap_inline262    Gregory believes tex2html_wrap_inline430 to be transcendental. (Huygens did not.)

tex2html_wrap_inline262    In two books published in 1668, he breaks with the Descartes classification scheme: algebraic vs. mechanical. But the function concept was still not there.

tex2html_wrap_inline262    He knew this familiar formula


tex2html_wrap_inline262    He knew the binomial theorem for fractional powers (Newton).

tex2html_wrap_inline262    He discovered Taylor series 40 years before Taylor, and the Maclaurin series for


(1671) Note: discovery in India 200 years earlier.

tex2html_wrap_inline262    He gives us the formula


which is today called Gregory's series.

tex2html_wrap_inline262    In 1668, Nicolaus Mercator (1620-1687) published his Logarithmotechnica in which appeared the power series for the logarithm.

tex2html_wrap_inline262    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


tex2html_wrap_inline262    From this point on tables of logarithms can be computed easily.



tex2html_wrap_inline262    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?

tex2html_wrap_inline262    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.)

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Don Allen
Thu Apr 3 06:51:19 CST 1997