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A very short history of Calculus presentation

for MATH 1037 by Alex Karassev

- Irrational numbers in Greek math
- Theory of Proportion
- The Method of Exhaustion
- The Area of a Parabolic Segment
- What is Calculus?
- Early Results on Areas and Volumes
- Maxima, Minima, and Tangents
- The Arithmetica Infinitorum of Wallis
- Newtons Calculus of Series
- The Calculus of Leibniz
- Biographical Notes Archimedes, Wallis, Newton,

and Leibniz

Irrational numbers in Greek math

- Discovery of irrational numbers
- Greeks tried to avoid the use of irrationals
- The infinity was understood as potential for

continuation of a process but not as actual

infinity (static and completed) - Examples
- 1,2, 3,... but not the set 1,2,3,
- sequence x1, x2, x3,
but not the limit x lim

xn - Paradoxes of Zeno ( 450 BCE) the Dichotomy
- there is no motion because that which is moved

must arrive at the middle before it arrives at

the end - Approximation of v2 by the sequence of rational

number

4.2 Eudoxus Theory of Proportions

- Eudoxus (around 400 350 BCE)
- The theory was designed to deal with (irrational)

lengths using only rational numbers - Length ? is determined by rational lengths less

than and greater than ? - Then ?1 ?2 if for any rational r is rational r ?1 )
- Note the theory of proportions can be used to

define irrational numbers Dedekind (1872)

defined v2 as the pair of two sets of positive

rationals Lv2 r r22

(Dedekind cut)

The Method of Exhaustion

- was designed to find areas and volumes of

complicated objects (circles, pyramids, spheres)

using - approximations by simple objects (rectangles,

trianlges, prisms) having known areas (or

volumes) - the Theory of Proportions

Examples

Approximating the pyramid

Approximating the circle

Example Area enclosed by a Circle

- Let C(R) denote area of the circle of radius R
- We show that C(R) is proportional to R2

- Inner polygons P1
- Outer polygons Q1 Q2 Q3
- Qi Pi can be made arbitrary small
- Hence Pi approximate C(R) arbitrarily closely
- Elementary geometry shows that Pi is proportional

to R2 . Therefore, for two circles with radii R

and R' we get Pi(R) Ri (R) R2R2 - Suppose that C(R)C(R)
- Then (since Pi approximates C(R)) we can find i

such that Pi (R) Pi (R) contradicts 5)

P2

P1

Q1

Thus Pi(R) Ri (R) R2R2

4.4 The area of a Parabolic Segment Archimedes

(287 212 BCE)

- Triangles ?1 , ?2 , ?3 , ?4,
- Note that ?2 ?3 1/4 ?1
- Similarly ?4 ?5 ?6 ?7 1/16 ?1 and so on

Y

Z

S

1

R

4

7

3

2

Q

6

5

O

X

P

Thus A ?1 (11/4 (1/4)2 ) 4/3 ?1

What is Calculus?

- Calculus appeared in 17th century as a system of

shortcuts to results obtained by the method of

exhaustion - Calculus derives rules for calculations
- Problems, solved by calculus include finding

areas, volumes (integral calculus), tangents,

normals and curvatures (differential calculus)

and summing of infinite series - This makes calculus applicable in a wide variety

of areas inside and outside mathematics - In traditional approach (method of exhaustions)

areas and volumes were computed using subtle

geometric arguments - In calculus this was replaced by the set of rules

for calculations

17th century calculus

- Differentiation and integration of powers of x

(including fractional powers) and implicit

differentiation of polynomials in x and y - Together with analytic geometry this made

possible to find tangents, maxima and minima of

all algebraic curves p (x,y) 0 - Newtons calculus of infinite series (1660s)

allowed for differentiation and integration of

all functions expressible as power series - Culmination of 17th century calculus discovery

of the Fundamental Theorem of Calculus by Newton

and Leibniz (independently) - Features of 17th century calculus
- the concept of limit was not introduced yet
- use of indivisibles or infinitesimals
- strong opposition of some well-known philosophers

of that time (e.g. Thomas Hobbes) - very often new results were conjectured by

analogy with previously discovered formulas and

were not rigorously proved

Early Results on Areas and Volumes

- Area (1/n)k (2/n)k (n/n)k(1/n)
- ? sum 1k 2k nk

y xk

Volume of the solid of revolution area of

cross-section is p r2 and therefore it is

required to compute sum 12k 22k 32k n2k

n/n 1

(n-1)/n

1/n

3/n

2/n

- First results Greek mathematicians (method of

exhaustion, Archimedes) - Arab mathematician al-Haytham (10th -11th

centuries) summed the series 1k 2k nk

for k 1, 2, 3, 4 and used the result to find

the volume of the solid obtained by rotating the

parabola about its base - Cavalieri (1635) up to k 9 and conjectured the

formula for positive integers k - Another advance made by Cavalieri was

introduction of indivisibles which considered

areas divided into infinitely thin strips and

volumes divided into infinitely thin slices - It was preceded by the work of Kepler on the

volumes of solids of revolution (New Stereometry

of wine barrels, 1615) - Fermat, Descartes and Roberval (1630s) proved the

formula for integration of xk (even for

fractional values of k) - Torricelly the solid obtained by rotating y 1

/ x about the x-axis from 1 to infinity has

finite volume! - Thomas Hobbes (1672) to understand this

result for sense, it is not required that a man

should be a geometrician or logician, but that he

should be mad

Maxima, Minima, and Tangents

- The idea of differentiation appeared later than

that one of integration - First result construction of tangent line to

spiral r a? by Archimedes - No other results until works of Fermat (1629)

modern approach

Fermats approach (tangent to y x2)

- E small or infinitesimal element which is

set equal to zero at the end of all computations - Thus at all steps E ? 0 and at the end E 0
- Philosophers of that time did not like such

approach

- Fermats method worked well with all polynomials

p(x) - Moreover, Fermat extended this approach to curves

given by p(x,y) 0 - Completely the latter problem was solved by Sluse

(1655) and Hudde (1657) - The formula is equivalent to the use of implicit

differentiation

The Arithmetica Infinitorum of Wallis (1655)

- An attempt to arithmetize the theory of areas and

volumes - Wallis found that ?01 xpdx 1/(p1) for positive

integers p (which was already known) - Another achievement formula for ?01 xm/ndx
- Wallis calculated ?01 x1/2dx, ?01 x1/3dx,
, using

geometric arguments, and conjectured the general

formula for fractional p - Note observing a pattern for p 1,2,3, Wallis

claimed a formula for all positive p by

induction and for fractional p by

interpolation (lack of rigour but a great deal

of analogy, intuition and ingenuity)

1

y x2

1

?01 x1/2dx 1 - 1/3 2/3

?01 x2dx 1/3

- Wallis formula

- Expansion of p as infinite product was known to

Viète (before Wallis discovery)

- Nevertheless Wallis formula relates p to the

integers through a sequence of rational

operations - Moreover, basing on the formula for p Wallis

found a sequence of fractions he called

hypergeometric, which as it had been found

later occur as coefficients in series expansions

of many functions (which led to the class of

hypergeometric functions)

Other formulas for p related to Wallis formula

Continued fraction (Brouncker)

Series expansion discovered by 15th century

Indian mathematicians and rediscovered by Newton,

Gregory and Leibniz

Euler

sub. x 1

Newtons Calculus of Series

- Isaac Newton
- Most important discoveries in 1665/6
- Before he studied the works of Descartes, Viète

and Wallis - Contributions to differential calculus (e.g. the

chain rule) - Most significant contributions are related to the

theory of infinite series - Newton used term-by-term integration and

differentiation to find power series

representation of many of classical functions,

such as tan-1x or log (x1) - Moreover, Newton developed a method of inverting

infinite power series to find inverses of

functions (e.g ex from log (x1)) - Unfortunately, Newtons works were rejected for

publication by Royal Society and Cambridge

University Press

The Calculus of Leibniz

- The first published paper on calculus was

by Gottfried Wilhelm Leibniz (1684) - Leibniz discovered calculus independently
- He had better notations than Newtons
- Leibniz was a librarian, a philosopher and a

diplomat - Nova methodus (1864)
- sum, product and quotient rules
- notation dy / dx
- dy / dx was understood by Leibniz literally as a

quotient of infinitesimals dy and dx - dy and dx were viewed as increments of x and y

The Fundamental Theorem of Calculus

- In De geometria (1686) Leibniz introduced the

integral sign ? - Note that ? f(x) dx meant (for Leibniz) a sum of

terms representing infinitesimal areas of height

f(x) and width dx - If one applies the difference operator d to such

sum it yields the last term f(x) dx - Dividing by dx we obtain the Fundamental Theorem

of Caculus

- Leibniz introduced the word function
- He preferred closed-form expressions to

infinite series - Evaluation of integral ? f(x) dx was for Leibniz

the problem of finding a known function whose

derivative is f(x) - The search for closed forms led to
- the problem of factorization of polynomials and

eventually to the Fundamental Theorem of Algebra

(integration of rational functions) - the theory of elliptic functions (attempts to

integrate 1/v1-x4 )

Biographical Notes

- Archimedes
- Wallis
- Newton
- Leibniz

Archimedes

- Was born and worked in Syracuse (Greek city in

Sicily) 287 BCE and died in 212 BCE - Friend of King Hieron II
- Eureka! (discovery of hydrostatic law)

- Invented many mechanisms, some of which were used

for the defence of Syracuse - Other achievements in mechanics usually

attributed to Archimedes (the law of the lever,

center of mass, equilibrium, hydrostatic

pressure) - Used the method of exhaustions to show that the

volume of sphere is 2/3 that of the enveloping

cylinder - According to a legend, his last words were Stay

away from my diagram!, address to a soldier who

was about to kill him

John Wallis Born 23 Nov 1616 (Ashford, Kent,

England) Died 28 Oct 1703 (Oxford, England)

- went to school in Ashford
- Wallis academic talent was recognized very early
- 14 years old he was sent to Felsted, Essex to

attend the school - He became proficient in Latin, Greek and Hebrew
- Mathematics was not considered important in the

best schools - Wallis learned rules of arithmetic from his

brother - That time mathematics was not consider as a

pure science in the Western culture - In 1632 he entered Emmanuel College in Cambridge
- bachelor of arts degree (topics studied included

ethics, metaphysics, geography, astronomy,

medicine and anatomy) - Wallis received his Master's Degree in 1640

- Between 1642 and 1644 he was chaplain at

Hedingham, Essex and in London - Wallis became a fellow of Queens College,

Cambridge - He relinquished the fellowship when he married in

1645 - Wallis was interested in cryptography
- Civil War between the Royalists and

Parliamentarians began in 1642 - Wallis used his skills in cryptography in

decoding Royalist messages for the

Parliamentarians - Since the appointment to the Savilian Chair in

Geometry of Oxford in 1649 by Cromwell Wallis

actively worked in mathematics

Sir Isaac Newton Born 4 Jan 1643 (Woolsthorpe,

Lincolnshire, England) Died 31 March 1727

(London, England)

- A family of farmers
- Newtons father (also Isaac Newton) was a wealthy

but completely illiterate man who even could not

sign his own name - He died three months before his son was born
- Young Newton was abandoned by his mother at the

age of three and was left in the care of his

grandmother - Newtons childhood was not happy at all
- Newton entered Trinity College (Cambridge) in 1661

- Newton entered Trinity College (Cambridge) in

1661 to pursue a law degree - Despite the fact that his mother was a wealthy

lady he entered as a sizar - He studied philosophy of Aristotle
- Newton was impressed by works of Descartes
- In his notes Quaestiones quaedam philosophicae

1664 (Certain philosophical questions) Newton

recorded his thoughts related to mechanics,

optics, and the physiology of vision

- The years 1664 66 were the most important in

Newtons mathematical development - By 1664 he became familiar with mathematical

works of Descartes, Viète and Wallis and began

his own investigations - He received his bachelor's degree in 1665
- When the University was closed in the summer of

1665 because of the plague in England, Newton had

to return to Lincolnshire - At that time Newton completely devoted himself to

mathematics

- Newtons fundamental works on calculus A

treatise of the methods of series and fluxions

(1671) (or De methodis) and On analysis by

equations unlimited in their number of terms

(1669) (or De analysis) were rejected for

publication - Nevertheless some people recognized his genius
- Isaac Barrow resigned the Lucasian Chair

(Cambridge) in 1669 and recommended that Newton

be appointed in his place - Newton's first work as Lucasian Prof. was on

optics - In particular, using a glass prism Newton

discovered the spectrum of white light

- 1665 Newton discovered inverse square law of

gravitation - 1687 Philosophiae naturalis principia

mathematica (Mathematical principles of natural

philosophy) - In this work, Newton developed mathematical

foundation of the theory of gravitation - This book was published by Royal Society (with

the strong support from Edmund Halley) - In 1693 Newton had a nervous breakdown
- In 1696 he left Cambridge and accepted a

government position in London where he became

master of the Mint in 1699 - In 1703 he was elected president of the Royal

Society and was re-elected each year until his

death - Newton was knighted in 1705 by Queen Anne

Gottfried Wilhelm von Leibniz Born 1 July 1646

(Leipzig, Saxony (now Germany) Died 14 Nov 1716

(Hannover, Hanover (now Germany)

- An academic family
- From the age of six Leibniz was given free access

to his fathers library - At the age of seven he entered school in Leipzig
- In school he studied Latin
- Leibniz had taught himself Latin and Greek by the

age of 12 - He also studied Aristotle's logic at school
- In 1661 Leibniz entered the University of Leipzig
- He studied philosophy and mathematics
- In 1663 he received a bachelor of law degree for

a thesis De Principio Individui (On the

Principle of the Individual) - The beginning of the concept of monad
- He continued work towards doctorate
- Leibniz received a doctorate degree from

University of Altdorf (1666)

- During his visit to the University of Jena (1663)

Leibniz learned a little of Euclid - Leibniz idea was to create some universal logic

calculus - After receiving his degree Leibniz commenced a

legal career - From 1672 to 1676 Leibniz developed his ideas

related to calculus and obtained the fundamental

theorem - Leibniz was interested in summation of infinite

series by investigation of the differences

between successive terms - He also used term-by term integration to discover

series representation of p