IV

Infinitesimal

Eudoxos of Knidos (408 - 355)

Astronomer and mathematician

In the age of 23 for two months student of Plato.Later with many own students visiting Plato‘s Academy.

Doubling the cube with a curve (lost)

Delian Problem

Volume of pyramid and cone by exhaustion

Croesus(590 - 541)Lydian King

Pythia

VZyl = 2r3

VSph = (4/3)r3 Archimedes)

VKon = (2/3)r3 (Democrit, Eudoxos)

Surface of sphere = 4 cross section r2

= 2/3 surface of cylinder (2r2 + 2r2r)

Archimedes (287 - 212)

Calculating from the 96-gon:

3 + 1/7 > > 3 + 10/71

3,1428... > 3,1415... > 3,1408...

Exhaustion

34

41

-1

1 ...

161

41

1

3

34 a 2A = = a

3 2 3x

x2

a2

A = A/4

A = a3/2

A = A/4

A

a

Eudoxos Archimedes(410 – 355) (287 – 121)

Indivisibles

Bonaventura Cavalieri Galileo Galilei Johannes Kepler (1598-1647) (1564 - 1642) (1572 - 1630)

Indivisibles were invented by Galilei, Kepler, and mainly by Bonaventura Cavalieri in the first half of the 17th century to calcuate areas and volumes. Equal width at equal height equal areaGeometria indivisibilibus continuorum (1635)

VZyl = VSph +Vkon

Proof afterBonaventura Cavalieri (1598-1647)

h

2= r2 - h2

r

Indivisiblen

h

2= r2 - h2

h

VZyl = VSph +Vkon

Proof afterBonaventura Cavalieri (1598-1647)

Indivisiblen

2+ h2 = r2

Exhaustion Indivisibles

Analogy to minor sum of Integral calculus.

The differentials „tend“ towards zero.

Bishop George Berkeley (1685 - 1753)

They are neither finite qantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?

x

f(x)Bernhard Riemann

(1826 – 1866)

x

f(x)

Major sum

Minor sume

Henri Lebesgue(1875 – 1941)

Pierre de Fermat 1601 - 1665)

Wrote poems in Latin, Greek, Italian, Spanish Studied jurisprudence, probably in BordeauxChairman of court in Toulouse1652 fell ill with pest, written off - recoveredGravestone inscription (Castres): Died in the age of 57

MathematicsSums of infinite series, Binomialcoefficients, Probability Theory, Complete induction, descente infinie,Extreme value problemsFounder of modern number theoryFermat prime numbers, absolute numbersSolved virtuoso problems like: Is there an absolute number between 1020 and 1022 ?

First steps of differential calculus

Extreme value problem (1629)

B is to divide in two parts, A and B – A, yielding the largest product.

A + E and B - A - E

A(B - A) = (A + E)(B - A - E)

0 = E(B - 2A -E)

0 = B - 2A - E (E 0)

0 = B - 2A

Result: [{F(A+E) - F(A)} / E]E=0 = 0 or dF(A)/dA = 0

Invention of differential calculus 35 years before Newton.

Calculus

Sir Isaac Newton (1642 - 1727) Gottfried Wilhelm Leibniz (1646 - 1716)

x

xfxxff

x

)()(' lim

0

)]()([)( limlim00

xfxxFxxFxx

Sir Isaac Newton (1642 - 1727)

1669 - 1696 Prof. of mathematics at Cambridge

Not smallest particles dx assumed but growth like plants in nature.

Differentiation / Integration of axm/n, area xp = xp+1/(p+1),

Newton-approximation

Radius of bend of curves

Points of inflection

No product rule

No quotient rule

Gottfried Wilhelm Leibniz (1646 - 1716)

Father: Leibnütz, notary and Professor of moral,

Family migrated from Poland

in print: Leibnuzius, Leibnitius,

Signature: Leibniz, seldom Leibnitz

Ulcer at back of his head enforced the wig

Study (started with 15 years): jurisprudence, philosophy, logik, mathematics (very rudimentary) in Leipzig and Jena

1664 Magister phil.

1665 Baccalaureus jur.

1667, Altdorf near Nürnberg: Doctor of both rights

Immediately offered professorship refused.

Auditorium, Collegium at Altdorf

1672 Kurmainz legate in Paris (center of sciences),

Found (1 + -3)1/2 + (1 - -3)1/2 = 6

First model of a mechanical calculator (+,-,,:). Worked on it again and again. Spent 24000 Taler. (Problem: transfer to next higher ones, tens, hundreds, …)

Presented it 1673 to the Royal Soc. London. Became a member.

Did not get a position at the Académie Royale des Science, although he, aged 25, had made important inventions already

Submarine (suitable agaianst storm and pirates)

Improved lens

Mechanical calculator

Universal language

Proof of rotationof earth

Binary numer system (Dyadik) developed and recommended for mechanical calculators

Symbols of logic

1684 Calculus and symbol of division :

In correspondence with Johann Bernoulli: dy/dx, ydx

1686 circle of bend, printed symbol of integration

1695 d(xn) = nxn-1dx d(ax) = ax lna dx

Use of the infinitely small (calculus)

Use of infinitely large (sum of harmonic series)

Leonhard Euler (1707 - 1783)

Euler‘s first use of (introduced by Jones).

Infinity, first abbreviated by i or the S on ist side, later by .

2, (+1)/2, log

Differential calculus is special case for infinitely small x = dx.

arithmetical equality: a-b = 0

geometrical equality: a/b = 1

dx, dy are arithmetically equal: dx = dy = 0,

but geometrically usually not equal: dy/dx 1

a/dx2 quantitas infinita infinities maior quam a/dx

1

2

2

dx

adx

a

dx

a

Zenon of Elea (490 - 430)

The real being evades the measuring fixation.

There is no moving: The flying arrow Achilles and the tortoise

There is no noise: The sack full of millet.

1) The being things are finitely many.2) Between them there are things. And between them there are

more. Infinitely many.There is always a contradiction.

ts

limv0t

Aristoteles (384 - 322)And his pupil Alexander on Bucephalo.

Continuum leads to problems with movement. In the beginning there must be rest and beginning movement togerther. Contradiction since one and the same is and is not.

With Eudoxos of Knidos (408 - 355) and Parmenides of Elea (515 - 445) Aristotle denies the composition of the continuum from indivisibles. The limits should fall together. But indivisibles have no edges and parts.