August 24, 2000 The Transition to Calculus - Part II ! ! The Low Countries 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 rea- son for this was so that his students could understand it. In 1659-1661, an expanded version was published. Geome- triaaRenatoDesCartes. 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. 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 manu- scripts of the works of Viete, and in Leiden he published Viete’s works. He published the Latin edition of Descartes’ Geometry. The much ex- panded second edition was extremely influential. He also made his own contributions, though modest, to mathematics, especially in his Exerci- tationes 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. 1 c 2000, G. Donald Allen ° Precalculus - II 2 First in Paris and then in London (1641-3) he made the acquain- tance of mathematical circles, with which he maintained a correspon- dence that is now lost. " Jan de Witt Jan de Witt, (1625-1672) was born into a patrician family of Dordrecht which moved in the 16th century from commerce into governmental ad- ministration. De Witt’s father was a younger son who initially operated the family lumber business. But he also held governmental positions. He went on the grand tour (or to France). At the University of Angers (a Protestant university) he received a doctorate of law in 1645. He was a Calvinist. The Grand Pensionary—footnotelike the Minister of Finance 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 kine- matic 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 correspondence with Hudde he considered the problem of an annuity based on the last survivor of two or more persons. # Infinitesimals and Indivisibles Kepler’s idea of measuring the area of a circle was to view it as an indefinitely increasing isosceles triangles. He then “opened” the circle along its circumference to obtain the formula 1 " = #$% 2 # := radius, $ := circumference. He performed a slicing argument to measure the volume of a torus. He never claims his methods are rigorous, claiming correct methods are in Archimedes but the reading is too difficult. Precalculus - II 3 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 concep- tion 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 determin- ing 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 wastoregardanarea& by '! ((), “all the lines” measured perpendicular from some base. His basis for computations is known to this day as Cavalieri’s principle:“Iftwo 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 )(*)=+,(*), then " " )(*) -* = + ,(*) -*. Z# Z# Using this method he was able to essentially perform the integration " 1 *$-* = 0$#! / +1 Z" Precalculus - II 4 By the way............. What is area? $ Fermat’s Areas We have already discussed in some de- tail the life and number theory of Fer- mat. Equally remarkable, perhaps more so, was his work on early calculus. Us- ing an ingenius geometrical approach, combined with a limiting notion, Fer- mat was able to compute areas under $ functions of the form 1 = *! ,some- times called Fermat’s hyperbolas. His idea was to take a geomet- ric partition of the interval [0%*"] or [*"% ) as in the case below. So for a given1 2 the partition points will be 3 3 $ 3 * % * % * %... 5 1 " 4 " 4 " 4 ³ ´ ³ ´ 3 3 1 3 1 6! = *" *" 1 = *" 1 $ = 1 $ ! 4 ¡ 4 ¡ *" 4 ¡ * ! ³ ´ ³ ´ ³ ´ " 3 $ 3 3 3 1 6$ = *" *" 1 = 1 *" $ 4 ¡ 4 4 ¡ 4 % * µ³ ´ ³ ´ ¶ ³ ´ & " 3 4 $ ! 1 ¡! ¢ = 1 $ ! 4 ¡ 3 * ! ³ ´³ ´ " 4 $ ! = ! 6 . 3 ! ³ ´ Similarly 4 $&$ !' 6 = ! 6 . % 3 ! ³ ´ Now sum the rectangles 6 = 6! + 6$ + ¢¢¢$ ! $&$ !' 4 ! 4 ! = 6! 1+ + + 3 3 ¢¢¢ ∙ ³ ´ ³ ´ ¸ Precalculus - II 5 1 = 6! $ ! 1 & ! ¡ % 1 3 1 = ¡ ¢ 1 & $ ! $ ! 1 ! 4 ¡ *"! ¡ % ³ ´ 1 1 = ¡ ¢ . & & $ & $ ! $ ! + + + ! *"! % % ¢¢¢ % Now let 473 1. This gives¡ ¢ the equivalent¡ ¢ of ! " 1 1 $ -* = $ ! . ' * (/ 1)* ! Z ! " ¡ " 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 R ( 2 $#! 2 $ 4$ = + + 8(2)() / +1 2 ¤ $(! X where 8( ) is a polynomial of degree which results when one considers an equal¢ interval partition. Roberval and Fermat both claimed proofs, but it would be some years before Pascal established his results on the triangle. Did Fermat invent calculus? Another “experimenter” with infinitesimals was Evangelista Torricelli (1608-1647), another disciple of Galileo. ! He completed his proofs with reductio ad absurdum arguments. As he announced it in 1643, his most remarkable discovery was that the volume of revolution of the hyperbola *1 = /$ from 1 = 9 to 1 = as finite – and he gave a formula. (Note, the corresponding area is infinite.)1 His method was basically cylindrical shells. Said Torricelli: “it may seem incredible that although this solid has an infinite length, nev- ertheless none of the cylindrical surfaces we considered has an infinite length. In his Arithmetica Infinitorum (“The Arithmetic of Infinitesimals”) of 1655, the result of his interest in Torricelli’s work, Wallis extended Precalculus - II 6 Cavalieri’s law of quadrature by devising a way to include negative and fractional exponents; John Wallis (1616-1703) was an Eng- lish clergyman and mathematician. Prob- ably second only to Newton in 17th century England, he contributed to math- ematics in a number of original ways. As a youth, Wallis learned Latin, Greek, Hebrew, logic, and arithmetic. At the age of only sixteen, he entered the University of Cambridge, receiv- ing B.A. and M.A. degrees. In 1640, he was ordained a priest in 1640. He showed mathematical skills by decipher- ing several cryptic messages from Roy- alist partisans that had come into the possession of the Parliamentarians. By 1645, also the year of his mar- riage, his interest in mathematics became serious, and he read William Oughtred’s Clavis Mathematicae (“The Keys to Mathematics”). As well, during this time, Wallis was active in the weekly scientific meet- ings that evnetually led to the charter by King Charles II of the Royal Society of London in 1662. In 1649 Wallis was appointed Savilian professor of geometry at the University of Oxford. The marked the beginning of great mathematical activity that lasted to his death. He was the first to “explain” fractional exponents. In his Arith- metica Infinitorum (“The Arithmetic of Infinitesimals”) of 1655, he extended Cavalieri’s law of quadrature by devising a way to include negative and fractional exponents; He used indivisibles as did Cava- lieri and arrives at the formula ! 1 *$-* = / +1 Z" in a rather unique way. Consider 1 = *$ between * =0and * =1.To determine the ratio of the area under this curve and the circumscribed rectangle, he notes the ratio of the abscissas are *$ :1$. There are infinitely many such abscissas. Wallis wanted to compute the ratio of the sum of the infinitely many antecedents to the sum of the infinitely many consequences. This would be (074)$ +(174)$ +(274)$ + +(474)$ lim ¢¢¢ & (474)$ +(474)$ + +(474)$ #" ¢¢¢ which comes to 0$ +1$ +2$ + + 4$ lim ¢¢¢ . & 4$ + 4$ + + 4$ #" ¢¢¢ Precalculus - II 7 To calculate this he experimented 0+1 1 1 4 =1 = = 1+1 2 3 0+1+4 5 1 1 4 =2 = = + 4+4+4 12 3 12 0+1+4+9 14 1 1 4 =3 = = + 9+9+9+9 36 3 18 In general, 0+1$ +2$ + + 4$ 1 1 ¢¢¢ = + 4$ + 4$ + + 4$ 3 64 ¢¢¢ Having worked the case for the power / =3, Wallis makes the inductive leap to 0$ +1$ + + 4$ 1 ¢¢¢ = .