Newton and Leibniz

CHAPTER 16
GROUP 8
Introduction:
Two of the greatest mathematical geniuses of all time, Isaac Newton and
Gottfried Leibniz, were contemporaries in the last half of the seventeenth century.
Independently of each other , they developed general concepts – for Newton, the
fluxion and fluent, for Leibniz the differential and integral-that were related to the
two basic problems of calculus, extrema and area. They developed notations and
algorithms, which allowed the easy use of these two concepts, and they understood
and applied the inverse relationship of their two concepts. Finally, they used these
two concepts in the solution of many difficult and previously unsolvable problems.
In this chapter, we first discuss the work of Newton, then the work of Leibniz,
and then conclude with a study of the contents of the earliest textbook in calculus,
both those based on Leibniz’s work and those based on Newton.
 01
Isaac Newton
Discussant: Kenneth Ventura
Isaac Newton
 He was a Mathematician ,
astronomer, physicist and
natural philosopher.

 He is recognized as one of the

most influential scientists of all
time and as a key figure in the
scientific revolution.
 Newton succeeded in
consolidating and generalizing
all the material on tangents and
areas developed by his
seventeenth-century
predecessors into the
magnificent problem-solving
tool exhibited in the thousand-
page calculus textbooks of our
own day.
 02
Power Series
Discussant: Kenneth Ventura
 Newton was especially struck by the analogy
between the infinite decimals of arithmetic and the
infinite degree “polynomials” that we call power series.
Newton proceeded, then, at the beginning of the
Treatise, to show by example the advantage of infinite
variable sequences, or power series, which he
considered simply as generalized polynomials with
which he could operate just as with ordinary
polynomials.
 The reduction of “affected equations,” that is, the
solving of an equation f (x, y) = 0 for y in terms of a
power series in x, is somewhat more difficult, Newton
believed, because the method of solving equations f (y)
= 0 numerically was not completely familiar.
 03
The Binomial Theorem
Discussant: Sharrah Austria
 Newton’s discovery of power series came out of his
reading of Wallis’s Arithmetica infinitorum, especially the
section on determining the area of a circle. In fact, he got out
of Wallis’s work more than Wallis had put in. In considering
areas, Wallis had always looked for a specific numerical
value, or the ratio of two such values, because he wanted to
determine the area under a curve between two fixed values,
say, 0 and 1. Newton realized that one could see further
patterns if one calculated areas from 0 to an arbitrary value
x, namely, if one considered area under a curve as a function
of the varying endpoint of the interval.
 Thus, in looking at the same problem as Wallis of
calculating the area of a circle, he considered a sequence of
curves like those of Wallis, that is, the curves y = (1− x2)n.
But Newton then tabulated the values under these curves as
functions of the variable x. For example, using modern
notation.
Newton, then tabulated not numerical areas, but the coefficients of the
various powers of x.

Like Wallis, Newton realized that Pascal’s triangle was here and so he
attempted to interpolate. In fact, to solve the problem of the area of the circle, he
needed the values in the column corresponding to n= 1 2. To find these values, he
rediscovered Pascal’s formula for positive integer values and decided to use the
same formula even when n was not a positive integer. He realized further that in the
original table each entry was the sum of the number to its left and the one above
that.
 04
Alogarithms
for Calculating Fluxions
Discussant: Sharrah Austria
 For Newton, the basic ideas of calculus had to do with motion.
Every variable in an equation was to be considered, at least
implicitly, as a distance dependent on time. What he did define
was the concept of fluxion: The fluxion ˙x of a quantity x dependent
on time (called the fluent) was the speed with which x increased
via its generating motion. In his early works, Newton did not
attempt any further definition of speed. The concept of
continuously varying motion was, Newton believed, completely
intuitive. next page
Newton solved problem 1 by a perfectly straightforward
algorithm that determined the relationship of the fluxions ˙ x and ˙ y
of two fluents x and y related by an equation of the form f(x,y)=0.
 There are several important ideas to note in Newton’s rule for calculating
fluxions. First, Newton was not calculating derivatives, for he did not in general
start with a function. What he did calculate is the differential equation satisfied by
the curve determined by the given equation. In other words, given f(x,y)=0 with x and
y both functions of t, Newton’s procedure produced what is today written as:

Second, Newton used Hudde’s notion of multiplying by an arbitrary arithmetic

progression. In practice, however, Newton generally used the progression starting
with the highest power of the fluent. Third, if x and y are considered as functions of
t, the modern product rule for derivatives is built into Newton’s algorithm. Any term
containing both x and y is multiplied twice and the two terms added.
 Furthermore, Newton’s approach to the modern chain rule was
via substitution. For example, to determine the relationship of the
fluxions in the equation y −√a2−x2, he put z for the square root and
dealt with the two equations y −z=0 and z2−a2+x2=0. The first
gave ˙ y −˙ z=0 while the second gave 2z˙ z+2x˙ x =0, or ˙ z=−x˙x/z.
Thus, the relationship between the fluxions of x and y is:
 05
Applications of Fluxions
Discussant: Sharrah Austria
 Newton used them to solve various problems. Newton found
maxima and minima by setting the relevant fluxion equal to zero,
because “when a quantity is greatest or least, at that moment its
flow neither increases nor decreases; for if it increases, that
proves that it was less and will at once be greater than it now is,
and conversely so if it decreases.” He used the equation x³ − ax² +
axy − y³ = 0 as his example for determining the greatest value of x.

This equation must then be solved simultaneously with the original

one to find the desired value for x. Similarly, to find the maximum
value of y, one sets y˙ = 0 and uses the resulting equation 3x² − 2ax
+ ay = 0.
 He gave no criteria for determining whether the values found
are maxima or minima. Presumably, that determination can be
made from the context in any given problem.

Newton used Barrow’s differential triangle. Thus, if x changes

to x + ˙xo while y changes to y + ˙yo, then the ratio yo˙ : xo˙ = ˙y : x˙
of the sides of this triangle is the slope of the tangent line, thought
of as the direction of instantaneous motion of the particle
describing the curve. This ratio is in turn equal to that of the
ordinate y to the subtangent t. Newton simply noted that t = y(x/˙
y)˙ . As a slight simplification in this calculation and others,
Newton sometimes set x˙ = 1. This is equivalent to considering x
as flowing uniformly, or as itself representing time.
 06
Procedures for Finding
Fluents and Areas
Discussant: Daniel Ballesteros
 It ought to be resolved the contrary way: namely by arranging the terms
multiplied by x˙ according to the dimensions of x and dividing by x˙ and
then by the number of dimensions, by carrying out the same operation
in the terms multiplied by y˙, and, with redundant terms rejected, setting
the total of the resulting terms equal to nothing.”

 As an example, he used his earlier problem. Starting with 3x²x˙ − 2axx˙ +

ayx˙ − 3y²y˙ + axy˙ = 0, he divided the terms having x˙ by x˙ x (or, what
amounts to the same thing, removed the x˙ and raised the power of x by
1), then divided each term again by the new power of x to get x³ − ax² +
axy. Doing the analogous operation on the terms containing y˙, he found
−y³ + axy. Noting that axy occurs twice, he removed one of those terms
to produce the final equation x³ − ax² + axy − y³ = 0.
 07
The Synthetic Method of Fluxions
Discussant: Daniel Ballesteros
 Perhaps one of the reasons Newton did not publish his Treatise on
Methods was that by the mid-1670s, he was somewhat unhappy with his
use of “analysis” in developing the ideas of calculus. He had been studying
the ancient Greek texts and believed that mathematical “truth” must be
based on the tenets of proof that had been developed in Greece.

So, he reformulated his ideas on fluxions

from the analytic methods he used earlier
into a more synthetic method that he called
“the method of first and ultimate ratios” and
then used this method in proving 11
important lemmas in section 1 of the
Principia. For example, we have:
 Newton’s proof used the figure, in which he represented the times by AD
and AE and the velocities by DB and DC. Areas ABD and ACE then represent
the distances (spaces).A translation of Newton’s words into an algebraic
statement would give a definition of limit close to, but not identical with,
the modern one. Newton never made such a translation.
 Nevertheless, it seems clear that Newton intuitively knew what he was
doing in using “limits” to calculate fluxions. To see this, we consider his
final tract on fluxions, the De quadratura curvarum (On the Quadrature of
Curves) of 1691 (published in 1704), where we read: “Fluxions are in the
first ratio of the nascent augments or in the ultimate ratio of the evanescent
part, but they may be expounded by any lines that are proportional to them.”
Newton then showed how to calculate the fluxion of xn, where x flows
uniformly: ⅆ𝑥
𝑥+ ⅆ𝑡
ⅆ𝑡
 Theorem
 In a given circle the fluxion of an arc is to the fluxion of its sine as
the radius to its cosine; to the fluxion of its tangent as its cosine is
to its secant; and to the fluxion of its secant as its cosine to its
tangent.
 Newton’s derivation of the fluxion of the tangent
 08
Newton and
Celestial Physics
Discussant: Mathew Castañeda
 Philosophiæ Naturalis Principia Mathematica
(Mathematical Principles of Natural Philosophy)
by Isaac Newton

 often referred to as simply the Principia (/prɪnˈsɪpiə, prɪnˈkɪpiə/), is a

work in three books written in Latin, first published 5 July 1687. It is
considered one of the most important works in the history of
science. It states Newton's laws of motion, forming the foundation
of classical mechanics; Newton's law of universal gravitation; and a
derivation of Johannes Kepler's laws of planetary motion (which
Kepler had first obtained empirically).
 Newton’s axioms, the three laws of motion
 Everybody perseveres in its state of being at rest or
of moving uniformly straight forward, except insofar
as it is compelled to change its state by forces
impressed. – An object at rest will remain at rest, an
object in motion will remain in motion with constant
velocity in a straight path unless acted upon by a net
external force.
 A change in motion is proportional to the motive
force impressed and takes place along the straight
line in which that force is impressed.
 To any action there is always an opposite and equal
reaction.
 Johannes Kepler's laws of Planetary Motion
They describe how

1.planets move in elliptical orbits with the Sun

as a focus,
2.a planet covers the same area of space in the
same amount of time no matter where it is in its
orbit, and
3.a planet's orbital period is proportional to the
size of its orbit (its semi-major axis).
 Newton used these laws immediately, beginning
with the following proposition (from section 2 of
Book I).
PROPOSITION 1: The areas which bodies made to
move in orbits described by radii drawn to an
unmoving center of forces.

 To deal with the central forces by geometrical

methods, Newton needed a geometrical
representation of such a force, even when the force
changes its magnitude and direction continuously.
This he accomplished in Proposition 6 and its
corollaries, in which a body is orbiting about a
center S in any curve.
 COROLLARY
 From the last three propositions, it follows that if any
body P departs from the place P along any straight-line
PR with any velocity whatever and is at the same time
acted upon by a centripetal force that is inversely
proportional to the square of the distance of places
from the center, this body will move in some one of the
conics having a focus in the center of forces; and
conversely.
•PROPOSITION 15: The squares of the periodic times in
ellipses are as the cubes of the major axes.
Using the same notation and diagram as in
Proposition 11, we recall that by Proposition 11,
areas swept out are proportional to the time elapsed.
Therefore, if ▲t is the time taken in each ellipse to
sweep out the infinitesimal area PSQ, the entire area
of the ellipse is to the periodic time T ultimately as
the area of triangle PSQ(=1/2 QT·PS) is to ▲t.
Because the area of the ellipse is proportional to ab,
we know that ab is ultimately proportional to the
product of T and QT·PS. Also, for each of the
elliptical orbits, the parameter p equals QT²/QR.
 09
Gottfried Wilhelm Leibniz
Discussant: Jhess Aldrean Alguno
Gottfried Wilhelm Leibniz
 The second inventor of the calculus.
 He was born in Leipzig
 The development of an alphabet of
human thought, a way of representing all
fundamental concepts symbolically and
a method of combining these symbols
to represent more complex
thoughts(Greatest contributions)
Gottfried Wilhelm Leibniz
 The second inventor of the calculus.
 He was born in Leipzig
 The development of an alphabet of
human thought, a way of representing all
fundamental concepts symbolically and
a method of combining these symbols
to represent more complex
thoughts(Greatest contributions)
 10
Sums and Differences
Discussant: Jhess Aldrean Alguno
Leibniz’s Harmonic Triangle
 Is a triangular arrangement of unit
fractions in which the outermost
diagonals consist of the reciprocals of
the row numbers and each inner cell is
the cell diagonally above and to the left
minus the cell to the left.
 Each column in this harmonic triangle is
formed by taking quotients of the first
column with the corresponding columns
of the arithmetical triangle.
 Newton was especially struck by the analogy
between the infinite decimals of arithmetic and the
infinite degree “polynomials” that we call power series.
Newton proceeded, then, at the beginning of the
Treatise, to show by example the advantage of infinite
variable sequences, or power series, which he
considered simply as generalized polynomials with
which he could operate just as with ordinary
polynomials.
 11
The Differential Triangle
and the Transmutation Theorem
Discussant: Jhess Aldrean Alguno
 The differential triangle, the infinitesimal right triangle whose
hypotenuse ds connects two neighboring vertices of the infinite-sided
polygon representing a given curve, is similar to the triangle composed of
the ordinate y, the tangent τ, and the sub tangent t, so ds:dy:dx=τ:y:t

In some sense, the variable chosen to have a

constant differential can be thought of as the
independent variable. In any case, it was through
manipulations of the differentials in the differential
triangle, using his basic rules for manipulating with
differentials, that Leibniz found the central
techniques for his version of the calculus
 Pascal had used the differential triangle in a circle of radius r to show that, in
Leibniz’s language, yds=rdx. Leibniz realized that this rule could be generalized to
any curve if one replaced the radius by the normal linen, because the triangle made
up of the ordinate, normal, and subnormal ν was similar to the differential triangle.
Therefore, y:dx=n:ds or yds=ndx
Because 2πy ds can be interpreted as the surface area of the surface formed
by rotating ds around the x-axis, this formula replaced a surface area calculation
with an area calculation. Similarly, Leibniz noted that dx:dy=y:ν or ydy=νdx. Because
he realized that ∫ydy represented a triangle whose area was (1/2)b2, where b was
the final value of the ordinate y, he had the result that ∫νdx=(1/2)b2. Therefore, to
find the area under a curve with ordinate z, it was sufficient to find a curve y whose
subnormal ν was equal to z.
But since ν=y dy/dx, this was equivalent to solving the equation y(dy/dx)=z. In
other words, an area problem was reduced to what Leibniz called an inverse
problem of tangents.
Transmutation Theorem
Although these particular rules did not lead Leibniz to any previously
unknown result, a generalization of this method gave him his transmutation
theorem and led him to his arithmetical quadrature of the circle, a series
expression forπ/4.
 In the curve OPQD, where P and Q are
infinitesimally close, he constructed the triangle
OPQ
 Extending PQ=ds into the tangent to the curve,
drawing OW perpendicular to the tangent, and
setting hand z as in the figure, he showed, using
the similarity of triangle TWO to the differential
triangle, that dx:h=ds:z or that zdx=hds.
 12
The Calculus of Differentials
Discussant: Renylene Simeon
 Leibniz discovered his transmutation theorem and
the arithmetical quadrature of the circle in 1674.
During the next two years, he discovered all the basic
ideas of his calculus of differentials. He only first
published some of these results in “A new method for
maxima and minima as well as tangents, which is
neither impeded by fractional nor irrational quantities,
and a remarkable type of calculus for them,” a brief
article appearing in 1684 in the Acta Eruditorum.
Acta Eruditorum
 Founded by Leibniz, together with
a fellow German philosopher and
scientist, Otto Mencke.

 It was the first scientific journal of

the German-speaking lands of
Europe, published from 1682 to
1782.
Johann Bernoulli
 A Swiss mathematician and was one of the
many prominent mathematicians in
the Bernoulli family.
 He is known for his contributions to
infinitesimal calculus.
 He published a paper entitled Principles of
the Exponential Calculus, which he
generalized Leibniz’s results to find
relationships of the differentials in such
equations as,
 𝒚=𝒙^𝒙, 𝒙^𝒙+𝒙^𝒄=𝒙^𝒚+𝒚, and 𝒛=𝒙^(𝒚^𝒗 ).
 13
The Fundamental Theorem and
Differential Equations
Discussant: Renylene Simeon
Leibniz’s Justification
Relates infinitesimals to Archimedean exhaustion
“ For instead of the infinite or the infinitely small, one takes quantities
as large, or as small, as necessary in order that the error be smaller than
the given error, so that one differs from Archimedes style only in the
expression, which are more direct in our method and conform more to
the art of invention. ”
Law of Continuity
“If any continuous transition is proposed terminating in a certain limit,
then it is possible to form a general reasoning, which also covers the final
limit.”
 It should be clear that although Leibniz and Newton discovered the
same rules and procedures that today called the calculus, their
approaches to the subject were entirely different.

Approach of Newton Approach of Leibniz

Through the ideas of Through differences

velocity and distance and sums

The unfortunate result of the controversy was that the interchange of ideas
between English and Continental mathematicians virtually ceased. A far as the
calculus was concerned, the English adopted Newton’s methods and notation, while
on the Continent, mathematicians used those of Leibniz. It turned out that Leibniz’s
notation and his calculus of differentials proved easier to work with.
 14
First Calculus Texts
Discussant: Nhel Jake Montemayor
 The differences between the English and
Continental approaches appear vividly in the
first calculus texts to appear, those of the
Marquis de l’Hospital (1661–1704) in France in
1696 and those of Charles Hayes (1678–1760)
and Humphry Ditton (1675–1715) in England in
1704 and 1706, respectively.
 15
L’ Hospital’s Analyse des
Infiniment Petits
Discussant: Nhel Jake Montemayor
Marquis de l’Hospital (1661–1704)
 Served as an army officer during his
youth.
 Mathematician
 Wrote the first textbook on calculus
Analyse des infiniments petits, pour
l’inteligence des lignes courbes
(Analysis of Infinitely Small Quantities
for the Understanding of Curves), which
consisted of the lectures of Johann
Bernoulli.
 He defined variable quantities as those that continually
increase or decrease and then giving his fundamental definition of
a differential:
“The infinitely small part by which a variable quantity increases or
decreases continually is called the differential of that quantity.”

He then presented two postulates to govern his use of these

differentials:
1. Grant that two quantities, whose difference is an infinitely small
quantity, may be taken (or used) indifferently for each other; or
(which is the same thing) that a quantity which is increased or
decreased only by an infinitely small quantity may be considered
as remaining the same.
.
 2. Grant that a curve may be considered as the assemblage of an
infinite number of infinitely small straight lines; or (which is the same
thing) as a polygon of an infinite number of sides, each infinitely small,
which determine the curvature of the curve by the angles they make with
each other.

For l’Hospital, then, there was no question about the existence of

infinitesimals. They exist; they can be represented by elements of the
differential triangle; and calculations can be made using the various rules
that he presented. L’Hospital dealt virtually exclusively with algebraic
curves. He only mentioned briefly the logarithmic curve, defined as one
whose subtangent 𝒚 𝒅𝒙/𝒅𝒚 is constant, and did not consider anything
resembling a trigonometric curve.
 L’Hospital’s treatment of maxima and minima was slightly more general than
that of Leibniz. He noted that the differential dy will be positive if the ordinates are
increasing and negative if they are decreasing, but showed further that dy can change
from positive to negative, and the ordinates from increasing to decreasing, in two
possible ways, if dy passes through 0 or through infinity. He presented diagrams
illustrating four possibilities, two where the tangent line is horizontal and two where
there are cusps and the tangent line is vertical, as well as examples illustrating these
possibilities. Thus, to find the maximum of

𝟑
𝟐 ξ 𝒂 𝒅𝒙
𝒅𝒚 = −
𝟑 𝟑ξ 𝒂 − 𝒙

Since dy = 0 is impossible, he set dy equal to infinity. This implied that

or that x = a.
PROPOSITION
Let AM D be a curve (AP = x, PM = y, AB = a) such that the value of the ordinate y
is expressed by a fraction, of which the numerator and denominator each become 0
when x = a, that is to say, when the point P corresponds to the given point B. It is
required to find what will then be the value of the ordinate BD.
L’Hospital’s diagram illustrating l’Hospital’s rule. Notice that the function g is drawn
below the x axis, but the quotient function, represented by curve AMD, is above the x
axis. Think of all values of the functions involved as representing positive quantities .
 16
The Works of Ditton and Hayes
Discussant: Kenneth Ventura
 Charles Hayes (1678–
1760)
 English mathematician and chronologist,
In 1704 he published Treatise of Fluxions, or
An Introduction to Mathematical Philosophy
 The book is the first English text on Newton's
method of fluxions, or, to phrase it in more
modern terms, the first English calculus text.
The book is a very full treatise, about three
times the size of de l'Hôpital's famous calculus
book. It contains 315 closely printed folio
pages on fluxions as well as a twelve-page
introduction to conic sections at the beginning
of the book.
Humphry Ditton (1675–1715)
 published An Institution of Fluxions
 Ditton wrote that quantities are not to be
imagined as “the aggregates or sums total of
an infinite number of little constituent elements
but as the result of a regular flux, proceeding
incessantly, from the first moment of its
beginning to that of perfect rest. A line is
described not by the apposition of little lines or
parts, but by the continual motion of a
point. . . .
 Ditton treated other aspects of the integral calculus
in detail, including rectification of curves, areas of
curved surfaces, volumes of solids, and centers of
gravity. But his text, like those of Hayes and l’Hospital,
had no treatment of the calculus of the sine or cosine.
There was an occasional mention of these
trigonometric relations as part of certain problems, but
there is nowhere at the turn of the eighteenth century
any treatment of the calculus of these functions. This
was not to come until the work of Leonhard Euler in the
1730s.
.
Thank You !!
THE END