August 24, 2000
The Transition to Calculus - Part II
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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 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.
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 Viete, and in Leiden he published Viete’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 his Exercitationes 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 °2000,
c
G. Donald Allen
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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.
"
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 administration. 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 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 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.
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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 & 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 )(*) = +, (*), then
Z
"
)(*) -* = +
#
Z
"
, (*) -*.
#
Using this method he was able to essentially perform the integration
Z
"
"
*$ -* =
1 $#!
0
/+1
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By the way............. What is area?
$
Fermat’s Areas
We have already discussed in some detail the life and number theory of Fermat. Equally remarkable, perhaps more
so, was his work on early calculus. Using an ingenius geometrical approach,
combined with a limiting notion, Fermat was able to compute areas under
functions of the form 1 = *!$ , sometimes called Fermat’s hyperbolas.
His idea was to take a geometric partition of the interval [0% *" ] or
[*" % 1) as in the case below. So for a
given 2 the partition points will be
*" %
6! =
6$ =
µ³ ´
3 $
4
³3
³ 3 ´$
³3
´
3
*" % . . .
*" %
51
4
4
4
´
*" ¡ *" 1
4
³3´ ¶
*" ¡
*" 1
4
Similarly
6% =
´ 1
´ 1
³3
¡1 $ =
¡ 1 $!!
4
4
*"
*"
´3
³3
1
¡1
=
*" ¡
¢$
%
4
4
& *"
´ ³ 4 ´$!! 1
³3
¡1
=
4
3
*"$!!
³ 4 ´$!!
=
6! .
3
= *"
³3
³ 4 ´$&$!!'
3
6! .
Now sum the rectangles
6
=
=
6 ! + 6$ + ¢ ¢ ¢
∙
¸
³ 4 ´$!! ³ 4 ´$&$!!'
+
+ ¢¢¢
6! 1 +
3
3
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=
=
=
1
¡ & ¢$!!
6!
1¡ %
´ 1
³3
1
¡ 1 $!!
¡ & ¢$!!
4
*"
1¡ %
1
1
¡ & ¢$
¡ & ¢$!! $!! .
&
*
+ ¢¢¢ + %
"
% + %
Now let 473 ! 1. This gives the equivalent of
Z
"
'!
1
1
-* =
.
*$"
(/ ¡ 1)*$!!
"
Note: Fermat does not compute with the inscribed rectangles. He
accepts the limiting result.
R
!
. There is something
A similar argument gives "'! *$ -* = $#!
*$#!
"
very satisfying about this method as it avoids the difficult problem of
summing
(
X
$(!
4$ =
2 $#!
2$
+
+ 8(2 )
/+1
2
(¤)
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 = 1 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.
In his Arithmetica Infinitorum (“The Arithmetic of Infinitesimals”)
of 1655, the result of his interest in Torricelli’s work, Wallis extended
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Cavalieri’s law of quadrature by devising a way to include negative and
fractional exponents;
John Wallis (1616-1703) was an English clergyman and mathematician. Probably second only to Newton in 17th
century England, he contributed to mathematics 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, receiving B.A. and M.A. degrees. In 1640,
he was ordained a priest in 1640. He
showed mathematical skills by deciphering several cryptic messages from Royalist partisans that had come into the
possession of the Parliamentarians. By 1645, also the year of his marriage, 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 meetings 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 Arithmetica 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 Cavalieri and arrives at the formula
Z
"
!
*$ -* =
1
/+1
in a rather unique way. Consider 1 = *$ between * = 0 and * = 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)$
&#"
(474)$ + (474)$ + ¢ ¢ ¢ + (474)$
lim
which comes to
0$ + 1$ + 2$ + ¢ ¢ ¢ + 4$
.
&#"
4$ + 4$ + ¢ ¢ ¢ + 4$
lim
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To calculate this he experimented
4=1
0+1
1+1
=
1
1
=
2
3
4=2
0+1+4
4+4+4
=
5
1
1
= +
12
3 12
4=3
0+1+4+9
9+9+9+9
=
14
1
1
= +
36
3 18
In general,
1
0 + 1 $ + 2 $ + ¢ ¢ ¢ + 4$
1
= +
4$ + 4$ + ¢ ¢ ¢ + 4$
3 64
Having worked the case for the power / = 3, Wallis makes the inductive
leap to
1
0 $ + 1 $ + ¢ ¢ ¢ + 4$
=
.
$
$
$
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 Isaac Newton observed that his work on the binomial theorem
and on the calculus was possible from his thorough study of this work
that Wallis became famous.
In 1657 Wallis published the Mathesis Universalis (“Universal
Mathematics”), on algebra, arithmetic, and geometry. In that volume,
he invented and introduced the symbol for infinity 1.
Using a rather complex logical sequence of steps he determined the
following formula
also based on induction.$
4
1 ¢ 3 ¢ 3 ¢ 5 ¢ 5¢¢¢
=
:
2 ¢ 4 ¢ 4 ¢ 6 ¢ 6¢¢¢
2 This formula converges very slowly. Taking 2000 terms which is well beyond the computational abilities 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.
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Power Series
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 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. This three step approach we
have seen before: (1) inscribe, circumscribe, (2) apply geometric and harmonic means to find a recurrence relation 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 ellipse 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 classification scheme: algebraic vs. mechanical. But the function concept
was still not there. He knew this familiar formula
Z
sec * -* = ln(sec * + tan *).
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
Z
'
"
-*
1 + *$
= tan!! *
= *¡
*%
*)
¡ ¢¢¢
+
3
5
3 By transcendental, it way meant not constructable with a compass and straight-edge. Of
course, these days, transcendental means not algebraic.
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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
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 logarithms can be computed easily.
&
Personalities
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.)
Bonaventura Cavalieri (1598 - 1647)
was an Italian of noble birth. He was
not, however, wealthy. He studied theology in the monastery of San Gerolamo in Milan. Here through Benedetto
Castelli, a lecturer in mathematics at
Pisa, he was initiated in the study of
geometry. He quickly absorbed the classical works in mathematics, demonstrating such exceptional aptitude that he
sometimes substituted for his teacher
at the University of Pisa. He published
eleven books beginning in 1632.
Cavalieri’s theory, as developed in his Geometria and in other
works, related to an inquiry into infinitesimals. Cavalieri made a rational systematization of the method of indivisibles. He also developed a
general rule for the focal length of lenses and thought of a reflecting
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telescope. His appointment at Bologna virtually required that he involve himself somewhat with astronomy, and even astrology, 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 Florence. 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: Mechanics, 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; Treatise on Algebra, 1685, which includes a treatment of infinite series;
Opera mathematica, 1693-9.
Mechanica, sive de motu tractatus geometricus, 1669-71, an important 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.
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Though a prolific publisher, Wallis did not generally use dedications for patronage. Rather the vast majority of dedications were to scientific 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. Primary: Mathematics, Mechanics, Physics Subordinate: Hydraulics, Meteorology, 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 amplified Galileo’s doctrine of projectiles. This is what he included in
the Opera geometrica. His Academic Lectures, published long after his
death also dealt, in part, with mechanics.
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 telescopes and who developed a microscope using tiny drops
of crystal the size of a grain of millet.
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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 Stefano degli Angeli,
Torricelli’s pupil, at Padua, 1664-8.
Scientific Disciplines Primary: Mathematics, Optics 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 hyperbola. 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 tangents to
curves (i.e., differentiation).
In 1672 Gregory published an important pioneering paper on the
motion of bodies through a resisting medium.
Means of Support Primary: Academia Professor of mathematics
at St. Andrews, 1668-74. Professor of mathematics at Edingburgh
University, 1674-5.
When Huygens thought he was dying in 1668, he suggested Gregory as a replacement in the Acadèmie.