Gottfried Wilhelm Leibniz (1646-1716)

1. Gottfried Wilhelm Leibniz (1646-1716) • Born July 1, 1646, in Leipzig • 1661, entered University of Leipzig (as a law student) • 1663, baccalaureate thesis, De Principio Individui (`On the Principle of the Individual') • 1667, entered the service of the Baron of Boineburg • 1672 - 1676, lived in Paris (met Malebranche, Arnauld, Huygens) • 1675, laid the foundation of the differential/integral calculus • 1676, entered the service of the Duke of Hannover; worked on hydraulic presses, windmills, lamps, submarines, clocks, carriages, water pumps, the binary number system • 1684 published Nova Methodus Pro Maximus et Minimus (`New Method for the Greatest and the Least'), an exposition of his differential calculus • 1685, took on the duties of historian for the House of Brunswick • 1691, named librarian at Wolfenbuettel • 1700, named foreign member of the French Academy of Sciences in Paris • 1711, met the Russian czar Peter the Great • Died, November 14, 1716, in Hannover

2. Gottfried Wilhelm Leibniz (1646-1716) • De Arte Combinatoria (“On the Art of Combination”), 1666 • Hypothesis Physica Nova (“New Physical Hypothesis”), 1671 • Discours de métaphysique (“Discourse on Metphysics”), 1686 • unpublished manuscripts on the calculus of concepts, c. 1690 • Nouveaux Essais sur L'entendement humaine (“New Essays on Human Understanding”), 1705 • Théodicée (“Theodicy”), 1710 • Monadologia (“The Monadology”), 1714

3. Gottfried Wilhelm Leibniz (1646-1716) • 1675, on 21 November he wrote a manuscript using the ∫f(x) dx notation for the first time. The ∫ stands for an elongated S, for sum (summa). The dx stands for an infinitesimal difference. Also the product rule for differentiation is given. • 1676 in autumn Leibniz discovered the familiar d(xn) = nxn-1dx for both integral and fractional n. • 1684 published Nova Methodus Pro Maximus et Minimus (`New Method for the Greatest and the Least'), an exposition of his differential calculus • In 1686 Leibniz published, in Acta Eruditorum, a paper dealing with the integral calculus with the first appearance in print of the ∫ notation and a proof of the Fundamental Theorem

4. Gottfried Wilhelm Leibniz (1646-1716) Three basic inputs for Leibniz’s work on integral calculus • In all his studies he was striving for a universal language. • His study of series. Forming differences and taking partial sums are “inverse” operations. • The idea of a “characteristic triangle” which has infinitesimal sides. ad 1. Unlike Newton, Leibniz thought a lot about the way to present his ideas in a good formalism. In fact, his notation is still used today. Note that good notation is key for dealing with complex problems. ad 3. Leibniz constructed an infinitesimal triangle whose “curved” hypotenuse approximates the derivative and used this construction to give an integration. In essence he solves the problem of integration via the fundamental theorem.

5. Gottfried Wilhelm Leibniz (1646-1716) ad 2. Consider a series tn for n in N. Also let t’n be given by t’n= tn+1-tn and sn= Si=1,..,nt’i then sn=tn+1-t1. Compare this to the fundamental theorem! Example: “triangular numbers”: i(i+1)/2 tn=-2/n t’n=2/n-2/(n+1)= 2/(n(n+1)) sn=2-2/(n+1) and as n  ∞ then sn  2. Compare to • sn behaves like an integral • the limit n  ∞ corresponds to the indefinite integral and • t’n= tn+1-tn behaves like a derivative. Consider the function f with f(n)=tn then t’=(f(n+1)-f(n))/1 and with Dx=1: t’=(f(n+Dx)-f(n))/Dx which is what is called today the discrete derivative.

6. Gottfried Wilhelm Leibniz (1646-1716) • We will follow Leibniz to consider the quadratrix C(C) of a given curve H(H). • C(C) is given by its law of tangency. I.e. it is the curve which at each point has a prescribed tangent. (This statement is essentially the Fundamental Theorem). • Given an axis and an ordinate Leibniz associates to each point C on the curve C(C) two triangles • The assignable triangle CBT composed out of the axis, the ordinate and the tangent to the point. • The in-assignable triangle GLC composed out of an infinitesimal piece parallel to the axis, an infinitesimal piece parallel the ordinate and an infinitesimal piece of the curve. The two other sides of the triangle are dx and dy.

7. Gottfried Wilhelm Leibniz (1646-1716) • Given the curve H(H) fix the coordinates AF and AB. Let C(C) be the curve, s.t. C is on HF and TB:CB = HF:a, where CB is parallel to AF and CT is the tangent to C(C) at C and a is a constant setting the scale. Or since BT=AF: axBT=AFxFH=A(rectangle AFH) • Theorem. Let (F)(C) be a parallel to FC. Let E be the intersection with CB, (C) be the intersection with the curve C(C) and (H) the intersection with the curve H(H). Then a E(C)=A(region F(H)) i.e. aE(C) is the area under the curve H(H) above F(F) and hence integral from F to (F) of H(H). Also if A is the intersection of H(H) with AF then aFC=A(region(AFH)) i.e. aFC is the area under the curve H(H) above AF. In other words: If C(C) satisfies the tangency condition it is the quadratrix. • Proof. Let AF=y, FH=z, BT=t and FC=x, then t=zy:a (see above) t=y dx:dy (y=AF=BC and dx:dy=BT:BC) So: a dx=z dy and ∫a dx = ∫z dy thus ax= ∫z dy=AFHA.

8. Gottfried Wilhelm Leibniz (1646-1716) • With the fundamental theorem integration boils down to finding an anti-derivative. • Leibniz had a calculus for dealing with derivatives in terms of infinitesimals. This is still used in physics. • Example: x=y3/3 and x+dx=(y+dy)3/3 so • With the fundamental theorem we thus again squared the parabola. • The use of the infinitesimals dx is however tricky. One has to claim that in the last line dy and (dy)2 are zero, without making dx or dy zero in any previous line. This can be rigorously achieved by non-standard analysis, but this had to wait 300 years. The next step historically was to introduce limits. .