IV Infinitesimal
Astronomer and mathematician Eudoxos of Cnidus (408-355) Astronomer and mathematician At the age of 23 for two months Plato's student. Later with many of his own students visiting Plato‘s Academy.
Astronomer and mathematician Eudoxos of Cnidus (408-355) Astronomer and mathematician At the age of 23 for two months Plato's student. Later with many of his own students visiting Plato‘s Academy.
From "The Clouds" by Aristophanes (446-386) In Academos' sacred grove you will linger in the shades of olive trees. Bright green reed is decorating your hair and a virtuous friend walks at your side. The perfume of yew trees and leisure fills the air and of leaves of white poplar, When in the bliss of springtide will join the gentle rustle of plane tree and elm. From "The Clouds" by Aristophanes (446-386)
Volume of pyramid and cone by exhaustion. Eudoxos of Cnidus (408-355) Volume of pyramid and cone by exhaustion. Doubling the cube with a curve (lost). Delian problem The three classical problems
Croesus (590-541) Lydian King Pythia, High Priestess
Archimedes (287-212) Surface of sphere = 4 time cross section pr2 = 2/3 times surface of cylinder (2pr2 + 2pr2r) Vcyl = 2pr3 Vsph = (4/3)pr3 (Archimedes) Vcon = (2/3)pr3 (Democrit, Eudoxos)
Cicero discovering the tomb of Archimedes, Benjamin West (1797)
The classical problems
Squaring the circle Dinostratus (400 BC) used the Quadratrix of Hippias. K/r = r/a (pr/2)/r = r/a p = 2r/a Proof by contradiction: Assume K/r = r/b with b > a. K/r = K/b K = r K/k = K/k = r/k From the regularity of the Quadratrix follows K/k = r/h k = h Contradiction since sinj j for j 0. The alternative assumption b < a yields the contradiction tan j = j. Hence b = a. C = 4K = 2pr = 4r2/a A = rC/2 = 2r3/a
Doubling the cube Geometric mean or mean proportional: pq p/h = h/q pq = h2 (similar triangles) Hippocrates of Chios found that the problem of doubling the cube (Delian problem) is solved if two mean proportionals x and y can be inserted between a and 2a. a/x = x/y = y/b a2 = x4/y2 b = y2/x Then for b = 2a : a2b = 2a3 = x3. ... whereupon his perplexity changed into another one - not better than the former.
Doubling the cube Solution by kinematic mathematics, impossible with ruler and pair of compasses. Put a rectangle with a movable edge (blue) on the given lengths a and b = 2a, such that its corners coincide with the axes.
Mesolabium of Eratosthenes (278-194) Doubling the cube Mesolabium of Eratosthenes (278-194) cos 45° = A/D = A'/D' = A''/D'' A/A' = D/D' and A'/A'' = D'/D'' By shifting the squares achieve f(j) = A'/D = A''/D' = A'''/D'' A'/A'' = D/D' and A''/A''' = D'/D'' A/A' = A'/A'' = A''/A''' A' and A'' are the mean proportionals of the fixed lengths A and A'''. Elevate the pink string such that A''' = A/2 and readjust the squares. A2 = (A')4/(A'')2 and A/2 = A''' = (A'')2/A' A3 = 2(A')3
Trisecting the angle Hippias of Elis (420 BC) constructed and used the Quadratrix. The regularly descending bar (pink), the upper edge of the square, divides the angle covered by the regularly rotating radius (blue) in equal parts. So the easy to accomplish division of an interval in three equal parts, by equidistantly constructed parallels, is translated into the division of the angle j.
a/2 + a = j/2 a = j/3 Trisecting the angle by Archimedes Place the straight line (red) such that outside of the circle the length r remains. a/2 + a = j/2 a = j/3
Indivisibles Bonaventura Cavalieri Galileo Galilei Johannes Kepler (1598-1647) (1564-1642) (1572-1630) Indivisibles were invented by Galilei, Kepler, and mainly by Bonaventura Cavalieri, Jesuat, to calculate areas and volumes. Equal width at equal height equal area. Geometria indivisibilibus continuorum (1635) Indivisibles of lines are lines. Indivisibles of areas are areas. Indivisibles of bodies are bodies. Evangelista Torricelli (1608-1647) extended Cavalieri‘s method to bent indivisibles.
Indivisibles According to Bonaventura Cavalieri (1598-1647) the indivisibles are a (proper or improper ?) part of the continuum. Galileo Galilei (1564-1642) "Discorsi e dimonstrationi mathematiche intorno a due nuove scienze" (concerning mechanics and laws of falling bodies): The continuum consists of actually infinitely many indivisibles (non quanti), such of geometry (points), also such of matter (atoms), and such of vacuum (vacui).
Indivisibles The continuum is created by the movement of its indivisibles. This fluentistic opinion is attributed, by Aristotle, to the Pythagoreans. It has been supported by Descartes, Torricelli, Roberval (1602-1675), and in particular by Newton. "Fluxion" is Newton's term for a derivative. Robert Grosseteste (1168-1253) Prof. and chancellor at Oxford Geometry, Optics, Astronomy Bishop of Lincoln The actually infinite is a definite number. The number of points in a segment one ell long is its true measure.
Indivisibles Baruch de Spinoza (1632-1677) Dutch philosopher Because of his frequent phrase "the nature or God" 1656 banished from the Jewish community. It is not less sensible to claim that the physical substance is composed of bodies or parts than to claim that bodies consisted of areas, areas of lines, lines of points.
Indivisibles There are two great labyrinths confusing human reason. The first concerns the great question of freedom and necessity. The other one consists in discussing continuity and is closely connected to the problem of infinity. G.W.Leibniz (1646-1716) Pierre Bayle (1647-1706) French philosopher, leading thinker of the Enlightenment, lost 1693 his professorship in Rotterdam. If matter is infinitely divisible, then it contains an infinite amount of parts, an infinity that really and actually is existing. Pierre Bayle (1647-1706)
Indivisibles Monadology: Monades are not atoms, since they have no extension (mathematical points, containing the entelechy, surrounded by physical point-spheres), but have perception, can be sleeping or awake. G.W.Leibniz (1646-1716) "Saving the divine revelation against the attacks of freethinkers" (1747) broadsheet against Monadists. Euler refuses infinitely small constituents of real space and time. He refuses Leibniz's monadology because monads are already infinitely divided constituents of the body. But infinite divisibility can never end or become completed. Leonhard Euler (1707-1783)
Indivisibles 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 together. Contradiction since one and the same is and is not. With Eudoxos of Cnidus (408-355) and Parmenides of Elea (515-445) Aristotle denies the composition of the continuum of indivisibles. The limits should fall together. But indivisibles have no edges and no parts.
Indivisibles Immanuel Kant (1724-1804) The property of magnitudes according to which they have no smallest possible part is called continuity. Space and time are quanta continua. Georg Wilhelm Friedrich Hegel (1770-1831) The front side must have a distance from the rear side. But this holds also for the front side and the rear side themselved. Therefore there is an infinite extension. When accepting multitudes then it follows that the units have no size. On the other hand they are unbounded.
Indivisibles Zenon of Elea (490-430) The real being evades the measuring fixation. Anticipated Hegel's idea: The being things are finitely many. Between them there are things. And between them there are more. Infinitely many. There is always a contradiction. There is no moving: The flying arrow Achilles and the tortoise There is no noise: The sack full of millet.
Indivisibles Karl Christian von Langsdorf (1757-1834) Professor at Heidelberg There are points of space and time. The protovelocity is the movement in the pattern by one point of space per one point of time. Other velocitis contain rest. Not-orthogonal movement contains steps. Augustin-Louis Cauchy (1789-1857) Although having used quantités infinement petites - forced by the rules of the Ecole Polytechnique - Cauchy has not considered them to be magnitudes but only variables, sequences with limit 0.
Indivisibles r r2 = r2 - h2 VZyl = VSph + Vkon Proof by indivisibles after Cavalieri r h r r2 = r2 - h2
Indivisibles pr2 + ph2 = pr2 r2 = r2 - h2 VZyl = VSph + Vkon Proof by indivisibles after Cavalieri pr2 + ph2 = pr2 h h r2 = r2 - h2
Exhaustion x2 a2 a x Eudoxos Archimedes (408-355) (287-121) A = a3/2 (408-355) (287-121) a2 A = a3/2 A = A/4 A = A/4 A a x
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?
f(x) f(x) x x Bernhard Riemann (1826-1866) Henri Lebesgue (1875-1941) Major sum Minor sume x x
f(x) f(x) x x Bernhard Riemann (1826-1866) Henri Lebesgue (1875-1941) Major sum Minor sume x x
First steps to differential calculus Wrote poems in Latin, Greek, Italian, Spanish. Studied jurisprudence, probably in Bordeaux. Chairman of court at Toulouse. 1652 fell ill with pest, written off – recovered. Gravestone inscription (Castres): Died at the age of 57. Mathematics: Sums of infinite series, binomial coefficients, probability theory, complete induction, descente infinie, extreme value problems, founder of modern number theory, Fermat prime numbers, perfect numbers. Solved virtuoso problems like: Is there a perfect number between 1020 and 1022 ? Pierre de Fermat (1601-1665)
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) (E 0) 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.
Gottfried Wilhelm Leibniz Calculus Sir Isaac Newton (1642-1727) Gottfried Wilhelm Leibniz (1646-1716)
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, logic, mathematics (very rudimentary) at Leipzig and Jena. 1664 Magister phil. 1665 Baccalaureus jur. 1667 Doctor of both rights at Altdorf near Nürnberg. Immediately offered professorship refused.
Auditorium, Collegium at Altdorf
Johann Georg Puschner: "The eager student" (c. 1725) In the background the university of Altdorf.
1672 Leibniz was Kurmainz legate at Paris (center of sciences). He tried to persuade King Louis XIV to attack Egypt (in order to divert the thread from Gemany) - but in vain. Found (1 + -3)1/2 + (1 - -3)1/2 = 6. First model of a mechanical calculator (+,-, ,:). Leibniz 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 against storm and against pirates) Improved lens Mechanical calculator Universal language Proof of rotation of earth
Binary number 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 the infinitely large (sum of harmonic series)
if Guillaume de l'Hospital (1661-1704) Analyse des infinement petits (1696) First textbook of analysis. if
Institutiones calculi differentialis (1755) concerns calculating with finite differences Dx. Differential calculus is a special case for infinitely small Dx = 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 Leonhard Euler (1707-1783) a/dx2 quantitas infinita infinities maior quam a/dx.
Joseph Louis Lagrange (1736-1813) 1755 professor at the artillery school at Turin famous in all fields of mathematics author of the first book on theoretical physics Lagrange-points in the three-body problem Lagrangian mechanics 1766 Euler‘s successor at the Berlin Academy asked 1784 the prize question: It is well known that higher mathematics always uses infinitely great and infinitely small magnitudes. Nevertheless the ancient geometers and analyticians have carefully avoided everything resembling the infinite; and some great modern analyticians think that the expression "infinite magnitude" is a self-contradiction. Therefore the academy wishes to get an explanation how so many correct theorems can be obtained from a contradictory assumption. Further a principle should be sketched which is safe and clear, briefly: really mathematical, and which in an appropriate way can be a substitution of the infinite.
The result was very unsatisfactory. Lagrange left this field in disappointment. Many years later, in his "Théorie des fonctions analytiques" (1797) Lagrange tried to eliminate all infinitely small by means of series f(x + i) = f(x) + ip(x) + i2q(x) + ... where i can always be taken so small that any term of the series is larger than the sum of all following terms.
Appendix
Solving 4r2/a by geometric algebra