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Gottfried Wilhelm Leibniz
(1646-1716)
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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
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Gottfried Wilhelm Leibniz
(1646-1716)
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• 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• unpublished manuscripts on the calculus of concepts, c.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
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
Gottfried Wilhelm Leibniz (1646-1716)
Three basic inputs for Leibniz’s work on integral calculus
1. In all his studies he was striving for a universal language.
2. His study of series. Forming differences and taking partial
sums are “inverse” operations.
3. 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.
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
2
2 
dx


x |1  2

x
– sn behaves like an integral
2
1
– 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.
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.
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.
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.