Mathematical Works of
G. W. Leibniz

translated and annotated by
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  : A method of determining the quadratures of figures with right and curved lines, by John Craig and 3 papers by Tschirnhaus from the Acta Erud., transl. in the Acta Germanica 1742, on tangents, max. and min., and a paper criticized by Craig on the squarability of curves. Craig's work was published just before Newton's Principia, and incorporates some of Newton's ideas, as well as leaning heavily on the work of Isaac Barrow. Tschirnhaus's papers follow on from the early ideas of Descartes; Katz above gives a good introduction to the early development of the calculus.

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.

4  : New Considerations regarding 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.