IV Infinitesimal ¥

Eudoxos of Knidos ( ) 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 ( ) Lydian King Pythia

V Zyl = 2  r 3 V Sph = (4/3)  r 3  Archimedes) V Kon = (2/3)  r 3 (Democrit, Eudoxos) Surface of sphere = 4  cross section  r 2 = 2/3  surface of cylinder (2  r  r  2r) Archimedes ( )

Calculating  from the 96-gon: 3 + 1/7 >  > /71 3, > 3, > 3,

Exhaustion x x2x2 a2a2 A = A/4A = a 3 /2A = A/4 A a Eudoxos Archimedes (410 – 355) (287 – 121)

Indivisibles Bonaventura Cavalieri Galileo Galilei Johannes Kepler ( ) ( ) ( ) 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 area Geometria indivisibilibus continuorum (1635)

V Zyl = V Sph +  V kon Proof after Bonaventura Cavalieri ( )  h  2  = r 2 - h 2 r Indivisiblen

h  2  = r 2 - h 2 h V Zyl = V Sph +  V kon Proof after Bonaventura Cavalieri ( ) Indivisiblen  2  +  h 2 =  r 2

Exhaustion Indivisibles Analogy to minor sum of Integral calculus. The differentials „tend“ towards zero. Bishop George Berkeley ( ) 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  ) Wrote poems in Latin, Greek, Italian, Spanish Studied jurisprudence, probably in Bordeaux Chairman of court in Toulouse 1652 fell ill with pest, written off - recovered Gravestone inscription (Castres): Died in the age of 57 Mathematics Sums of infinite series, Binomialcoefficients, Probability Theory, Complete induction, descente infinie, Extreme value problems Founder of modern number theory Fermat prime numbers, absolute numbers Solved virtuoso problems like: Is there an absolute number between and ? 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 ( ) Gottfried Wilhelm Leibniz ( )

Sir Isaac Newton ( ) Prof. of mathematics at Cambridge Not smallest particles dx assumed but growth like plants in nature. Differentiation / Integration of ax m/n, area x p = x p+1 /(p+1), Newton-approximation Radius of bend of curves Points of inflection No product rule No quotient rule

Gottfried Wilhelm Leibniz ( ) 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 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 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(x n ) = nx n-1 dxd(a x ) = a x lna dx Use of the infinitely small (calculus) Use of infinitely large (sum of harmonic series)

Leonhard Euler ( ) 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/dx 2 quantitas infinita infinities maior quam a/dx

Zenon of Elea ( ) 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.

Aristoteles ( ) 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 ( ) and Parmenides of Elea ( ) Aristotle denies the composition of the continuum from indivisibles. The limits should fall together. But indivisibles have no edges and parts.