G. W. Leibniz

Ian Bruce

At present a few short but some very important papers from the ** Acta
Eruditorum **are presented here, relating to the initial presentation by
Leibniz of his differential calculus and integral, which was to change the
nature of mathematics for ever; and slightly earlier, his restructuring of the
Law of Refraction. An interesting development is the inclusion of a work by
Craig on integration following a method he developed, and based essentially on
Barrow's earlier work : there is a great deal of information in this short
book. For your reference: many books and papers have been and continue to be
published on Leibniz and his works in numerous fields. Here amongst others, I have made use of the following:

Leibniz : *An Intellectual Biography*;
Maria Rosa Antognazza ; CUP. This is
an excellent work; especially useful for finding out what Leibniz was doing
while he wrote various works.

The papers presented here can be accessed in the original Latin in the
reprints available of the *Acta Eruditorum*
published by Olms.

* Opera Omnia* (Dutens :Volume III):
available as CD no. 35 from the 'Garden of Archimedes' website;

*A Sourcebook in Mathematics*
(Dover), ed. A.E. Smith.

*Landmark Writings in Western
Mathematics *1640-1940, ed. I. Grattan-Guinness. Looks at the three most
important calculus related papers of Leibniz.

*Leibniz, Naissance du Calcul
Différential,* tr. and notes (in French) M. Parmentier. This is an excellent
text on Leibniz.

*The Early Mathematical Manuscripts of
Leibniz. J.M.Child (1920): good for the early mathematics, though the author
spends too long in giving us his personal opinion, I feel.*

*Mathematische Schriften heruasgegeben
von C.I.Gerhardt. Band V; Die mathematischen Abhandlungen.Olms 1960. Part one:
Dissertatio de Arte Combinatoria; Part two: Characteristica Geometria. Analysis
Geometrica. Calculus situs. *

* A History of Mathematics. Victor J. Katz.
Harper Collins.*

*The Vortex Theory of Planetary Motion.
E. J. Aiton. History of Science Library. Ch.V.*

*Leibniz a Biography. E. J. Aiton.*

*Reading the Principia. Niccolo
Guicciardini. Ch. 6 on Leibniz.*

AE0 :

AE1 : **A new method of
finding the maxima and minima, **and likewise for tangents, and with a
single kind of calculation for these, which is hindered neither by fractions
nor irrational quantities. MS.Page 220, 1684 AE.

AE3a : ** Some early
papers on Isochronous Curves**; This is
of interest as it includes an early example of the integration of a differntial
equation, for finding the isochronous curve for a weight falling uniformly
along a curve. This should be read before the following paper.

AE3 : *G.W.L. A SHORT DEMONSTRATION OF A NOTEWORTHY ERROR of the Cartesians and Others about a Law of Nature* …..; ** **A SHORT REMARK BY THE Abbé Catelan ……; Concerning the Isochronous Curve, along which a Weight
may fall without an Acceleration downwards, and the Dispute with the Abbé
Catelan.

AE4 : *New Considerations r*** egarding the nature of
the Angle of contact and of osculation, **and
with the practical use of these in Mathematics, towards substituting easier
figures in place of more difficult ones.

June, 1686 AE.

AE5
: *The
Arithmetical Quadrature of common Conic Sections which have a centre,**
*with Trigonometric Cannons
deduced exactly for any numbers and thence freed from the necessity of Tables:
with the special use of curves for the nautical Rhombus, and small approximate
planes of the globe adapted for these*.*

April, 1691 AE.

AE6 : **Concerning the true proportions of a circle to the
circumscribed square**, expressed in rational numbers. This is an
interesting paper concerning the evaluation of pi, beginning with an infinite
series derived previously by

MS.Page 118, 1682 AE.

AE7 : **Concerning Optical Curves and other matters**. This
paper is a reaction to a first encounter with Newton’s *Principia*, detailing some aspects of light rays reflected from
concave mirrors, and esp. the idea of a constant path length within the caustic
curve MS.Page 329, 1689 AE.

AE8 : **Papers on the
resistances of mediums and gravity on projectile motion**.

This paper is another
reaction to the first encounter with Newton’s *Principia*, detailing with the resistance to a heavy body moving in
a resisting medium, where two main kinds of resistance are distinguished, both
horizontal, vertical, and combined motions are set out for both kinds of
resistance MS. Page 135, 1689 AE.

AE9 : **Optics, Catoptrics
and Dioptrics from a single Principle** : in which Fermat's Principle
is extended and Snell's Law of Sines follows by minimizing the path length of a
ray passing from one transparent medium to another at a plane surface. 1682 AE.

AE10 : **An Attempt to
Explain the Causes of Celestial Motion**: This is not a paper for the
faint-hearted : No one doubts that Leibniz was a genius, and this work bears that
out in parts; however, he invokes physics principles which are wrong :
e.g. by using a form of the vortex theory, and centrifugal forces
cannot be used to generate repulsive forces in circular motion; his
understanding of his own calculus is lacking when applied to rotational motion,
which is all very sad; however, by starting from the equation of the ellipse,
and differentiating twice, he arrives essentially at the correct second order
equation describing the acceleration of a body along the curve; he has already
tried to justify the various terms in this equation in a fallacious manner, and
some aspects of his discussion remain vague. It appears that the paper was
written just after he had seen the *Principia*,
although he maintained that he had only seen a review in the Acta. A
contemporary view of Leibniz’s paper can be found in *The Elements of Astronomy*, by David Gregory, 1715, near the end of
Book I, available on the web. 1689 AE.

AE11 : **On Finding the Measures of Figures **:** **This paper is concerned with an
aspect of rectifiable areas, in which Leibniz refutes a theorem of Tschirnhaus,
and sets the historical record in order by claiming priority in his treatment
of rectifiable shapes over his former friend Tschirnhaus, demonstrating a
theorem by the latter to be wrong.

1684 AE P.124.

AE13 :** A Supplement to the
Geometry of Measurements, or the Most General of all Quadratures to be Effected
by a Motion : and likewise the various constructions
of a curve from a given condition of the tangent.**:** **This paper eventually shows the
relation between a function and its integral, where the former is called the
quadratrix, and the latter the quadrature or square; a mechanical evaluation of
the quadratrix is shown, and a special curve derived from a tangent of constant
length pulled at one end along a fixed direction is discussed in detail – the
Tractrix .

1693 Sept. AE.

AE14 :** Concerning the curve formed by a heavy flexible
cord due to its own weight, ………. :** The problem set by Jacob Bernoulli is solved
with some enthusiasm by Leibniz, who regards it as associated with the
logarithmic curve, and which we now consider related closely to the hyperbolic
cosine or cosh curve. Unfortunately for us, many of the results are quoted and
few derivations are given, so we are at a loss to see how Leibniz proceeded to
obtain his solutions; the same can be said for the efforts of Huygens and the
Bernoulli brothers; it would appear however, that Huygens' treatment is closest
to the modern approach, although he gives only arithmetical examples of his
theorems.

1691 June, etc. AE.

AE18 : *Response to some
difficulties raised by Bernard Nieuwentijt about the differential or
infinitesimal method.*

Here Leibniz provides some more light for us
about his understanding of his own calculus, as he sets out to demolish the
objections raised by this solitary scholar, who wrote the first book on
calculus for users, rather than as an academic exercise for the elite ; the
discarding of higher order terms, the derivative of exponential functions, and
the presence of higher order derivatives are discussed.

AE19 : **Concerning a Recondite Geometry and the analysis
of the indivisible and the infinite**** **: Here a number of matters are
considered; initially Leibniz is very happy with the treatment his previous
work has received by John Craig, which is concerned with extending AE13 and the
work on quadrature in AE11; I hope to translate this short book soon. Then he
has thoughts about his interactions with his longtime friend Tschirnhaus,
as he had made a mistake in AE11 in refuting Tschirnhaus' conclusions in a
previous paper. on quadrature. Finally, as an update as it were, Leibniz
introduces his ideas of finding areas by the process we now call integration,
as the inverse of his use of finding the tangent to a curve by using
infinitesimal triangles, or
differentiation. He takes some trouble to make sure that his new method
extends beyond algebraic curves to those which were called mechanical at the
time, such as the cycloid.

MS.Page 226, 1686 AE

Ian
Bruce. Jan. 2015 latest revision.
Copyright : See notes at bottom of index page;
I reserve the right to publish this translated work in book form.
However, if you are a student, teacher, or just someone with an interest, you
can copy part or all of the work for personal or educational use only. If you
have any useful information regarding this translation, I would appreciate
hearing from you. IB.