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Home , Bonaventura Cavalieri, John Wallis, Mathesis universalis, Stefano degli Angeli

August 24, 2000

The Transition to - Part II !

! The Low Countries

Frans van Schooten, (1615-1660), Netherlands, had succeeded his father as professor of 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 was introduced by Descartes, it was established by 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 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 , 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 . (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.

# and Indivisibles

Kepler’s idea of measuring the 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 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 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 ) for various studies such as 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 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 . 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 (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 *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 by devising a way to include negative and fractional exponents; (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 , 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 ’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 . 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 ¢¢¢ = . 4$ + 4$ + + 4$ / +1 ¢¢¢ For obvious reasons, Wallis was known as the great inductor.He generalizes his integration formula to rational exponents, and for more general curves, particularly

1 =(1 *!)*)&. ¡ Though Wallis was well known and respected in his day, it was only when observed that his work on the and on the calculus was possible from his thorough study of this work that Wallis became famous. In 1657 Wallis published the (“Universal Mathematics”), on , arithmetic, and geometry. In that volume, he invented and introduced the symbol for . 1 Using a rather complex logical sequence of steps he determined the following formula 4 1 3 3 5 5 = ¢ ¢ ¢ ¢ ¢¢¢ : 2 4 4 6 6 ¢ ¢ ¢ ¢ ¢¢¢ also based on induction.$ 2This formula converges very slowly. Taking 2000 terms which is well beyond the computational abili- ties of the day, the approximation to ¼ yields 3.140807747. Taking 10,000 terms leads to the approximation 3.141514118, which is better but still worse than the best of the approximations (3.1416) known to the ancient Greeks. Precalculus - II 8

% Power

Interestingly, 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 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 el- lipses and hyperbolas using the Archimedean program. This three step approach we have seen before: (1) inscribe, circum- scribe, (2) apply geometric and har- monic means to find a recurrence rela- tion between the polygons of differing sides, and (3) double the number of sides. These steps, together with the reductio ad absurdum argument led to the proofs. For example the area of an with semi-major and semi-minor axes, 9 and 0 respectively has area given by " = :90. Gregory believed : to be transcendental.% Huygens did not. This small controversy reveals the importance of this ancient problem, even in this day of rapidly advancing mathematics. The “ancient mystique” is ever present. In two books published in 1668, he breaks with the Descartes clas- sification scheme: algebraic vs. mechanical. But the function concept was still not there. He knew this familiar formula sec *-*=ln(sec * +tan*). Z He knew the binomial theorem for fractional powers (Newton). He discovered Taylor series 40 years before Taylor, and the Maclaurin series for tan *% sec *% arctan *% arcsec * (1671) Note: discovery in India 200 years earlier. He gives us the formula ' -* =tan! * 1+*$ ! Z" *% *) = * + ¡ 3 5 ¡¢¢¢ 3By transcendental, it way meant not constructable with a compass and straight-edge. Of course, these days, transcendental means not algebraic. Precalculus - II 9 which is today called Gregory’s series. In 1668, Nicolaus Mercator (1620-1687) published his Logarith- motechnica in which appeared the power series for the . From de Sarasa (1618-1667) he learned it was the area under the hyperbola 1 =17(1 + *). From Wallis he learned the method of indivisibles using an indefinite number of geometric series he arrives at the conclusion

*$ *% log(1 + *)=* + ¡ 2 3 ¡¢¢¢ From this point on tables of can be computed easily.

& Personalities

Hendrick van Heurat (1634-1660(?)) developed a method of comput- ing 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.) (1598 - 1647) was an Italian of noble birth. He was not, however, wealthy. He studied the- ology in the monastery of San Gero- lamo in . Here through , a lecturer in mathematics at Pisa, he was initiated in the study of geometry. He quickly absorbed the clas- sical works in mathematics, demonstrat- ing such exceptional aptitude that he sometimes substituted for his teacher at the . He published eleven books beginning in 1632.

Cavalieri’s theory, as developed in his Geometria andinother works, related to an inquiry into infinitesimals. Cavalieri made a ratio- nal systematization of the method of indivisibles. He also developed a general rule for the focal length of lenses and thought of a reflecting Precalculus - II 10 . His appointment at virtually required that he in- volve himself somewhat with astronomy, and even , in which he appears to have engaged only from necessity. Initially he was rejected for the chair in Bologna in 1619, being thought too young. To make ends, he gave lessons in mathematics in . Such lessons appear to have belonged to the entire period (1616- 19) of his study in Pisa. In Bologna he continued to give private lessons.

' John Wallis

Born: Ashford, Kent, 23 Nov. 1616 Died: Oxford, 8 Nov. 1703 Education Schooling: Cambridge, M.A., Oxford, D.D. Cambridge University, Emmanuel College, 1632-40; B.A., 1637; M.A., 1640. Wallis studied in Emmanuel, the Puritan college, and was in good favor there. He strongly supported the Puritan cause during the Civil War. He conformed without question at the Restoration, although he remained a Calvinist theologically, in conformity with the Thirty-nine Articles. Scientific Disciplines. Primary: Mathematics. Subordinate: Me- chanics, Physics, Music Wallis was probably the second most important English mathematician during the 17th century, after Newton. He was the author of numerous books: Treatise of Angular Sections, composed in 1648, published finally in 1685; De sectionibus conicis, 1655, a pioneering analytic treatment of conics; Arithmetica infinitorum, 1656, a major contribution to integration and to infinite series; Commercium epistolicum, 1658, his exchange with Fermat on number theory; Trea- tise on Algebra, 1685, which includes a treatment of infinite series; Opera mathematica, 1693-9. Mechanica, sive de motu tractatus geometricus, 1669-71, an im- portant contribution to mechanics and to the treatment of percussion (though much of it is devoted to the mathematical problem of centers of gravity). A Discourse of Gravity and Gravitation (real title is in Latin), 1674. De aestu maris hypothesis nova, 1668, a theory of the tides. After deciphering a coded letter for the Parliamentary authorities, Wallis was rewarded with the sequestered living of St. Gabriel, London. He exchanged this living for St. Martin in Ironmonger Lane in 1647. Savilian Professor of geometry at Oxford, 1649-1703. Precalculus - II 11

Though a prolific publisher, Wallis did not generally use dedica- tions for patronage. Rather the vast majority of dedications were to sci- entific and academic peers–Oughtred, Rooke, Ward, Brouncker, Boyle, Moray, Hevelius, four heads of colleges in Oxford. Connections with Fermat, Brouncker, Frenicle, David Gregory, and Schooten. Royal Society, 1660; President, 1680.

( Evangelista Torricelli

Born: Faenza (halfway between Bologna and Rimini), 15 Oct. 1608. Died: Florence, 25 Oct. 1647

! Father Occupation: Artisan, Cleric ! Education Schooling: No University Scientific Disciplines. Pri- mary: Mathematics, Mechanics, Physics Subordinate: Hydraulics, Me- teorology, Instrumentation As a young man Torricelli was greatly interested in astronomy and was a committed Copernican. The condemnation of Galileo in 1633 changed all that. Torricelli was a cautious man, not inclined to tilt at authority, and astronomy simply disappeared from his scientific work. Torricelli’s only published work was Opera geometrica, 1644, which included work on motion (or mechanics). In mathematics he employed Cavalieri’s method of indivisibles, of which he became a master and which he extended to elegant solutions of volumes and other problems. ! Torricelli’s first known work was a treatise on motion that am- plified Galileo’s doctrine of projectiles. This is what he included in the Opera geometrica. His Academic Lectures, published long after his deathalsodealt,inpart,withmechanics. The Torricellian experiment (the barometer) was a major event in physics in the middle of the century. His lecture on wind (Academic Lectures) rejected the notion that exhalations cause them and referred the winds instead to differences in temperature at different regions of the earth. Torricelli was perhaps the most gifted lens grinder of his age, who made many and who developed a microscope using tiny drops of crystal the size of a grain of millet. Precalculus - II 12

He succeeded Galileo as Mathematician (not Philosopher) to the Grand Duke–from 1642 until Torricelli’s death in 1647. The Grand Duke published his Opera geometrica.

) James Gregory

Born: Drumoak, near Aberdeen, Nov. 1638 Died: Edinburgh, 1675 Father’s Occupation: Cleric. He died in 1650 when James Gregory was twelve. Partly, but only partly, through his wife’s inheritance he amassed a small fortune. All the details indicate wealth. Nationality: Scottish Studied geometry, mechanics and astronomy under , Torricelli’s pupil, at Padua, 1664-8. Scientific Disciplines Primary: Mathematics, Subordinate: Astronomy, Mechanics James Gregory was one of the most important mathematicians of the century, significant especially in the steps that led to the calculus. He pursued what later appeared as a tedious and complex method of infinite series based on polygons to find the area of the circle and the hy- perbola. This was published in Vera circuli & hyperbolae quadratura, 1668. In that same year, Geometriae pars universalis, which included also a doctrine of the transmutation of curves. In 1669, Exercitationes geometricae. Gregory also developed a method of drawing to curves (i.e., differentiation). In 1672 Gregory published an important pioneering paper on the motion of bodies through a resisting medium. MeansofSupportPrimary:AcademiaProfessorofmathematics at St. Andrews, 1668-74. Professor of mathematics at Edingburgh University, 1674-5. When Huygens thought he was dying in 1668, he suggested Gre- gory as a replacement in the Academie.`