iv infinitesimal ¥ eudoxos of knidos (408 - 355) astronomer and mathematician in the age of 23 for...
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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.
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