Gottfried Wilhelm Leibniz

How to use this course

Leibniz is the only person in this series who could appear equally in a course on philosophy, law, history or engineering. Mathematicians know him for the symbols \(dx\) and \(\int\), which every student of calculus still writes. Physicists know him for the quantity he called living force, the ancestor of kinetic energy, and for his argument that space is only a system of relations between bodies. Computer scientists know him for binary arithmetic and for the dream of a language in which arguments could be settled by calculation.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations in Leibniz’s notation and in modern form, and exercises with folded solutions. The chronology runs from the Paris years (1672–1676), when the calculus was born, through the Hanover decades of publication and controversy, to the correspondence with Samuel Clarke in the last year of his life.

Timeline

Leipzig, 1646–1666

Gottfried Wilhelm Leibniz was born in Leipzig, in the Electorate of Saxony, on 1 July 1646 by the Gregorian calendar (21 June in the Julian calendar then used in Protestant Germany). The Thirty Years War was in its last two years. His father, Friedrich Leibniz, was professor of moral philosophy at the University of Leipzig; his mother, Catharina Schmuck, was Friedrich’s third wife and the daughter of a lawyer. Friedrich died in 1652, when his son was six.

The boy entered the Nicolai School at seven. His real education came from his father’s library, which he was allowed to use from about the age of eight. He taught himself Latin well enough to read the classical historians and poets, and then moved on to the Church Fathers and the scholastic philosophers. By his own later account he began, while still at school, to look for a way of arranging all concepts in an ordered system, an idea that would return in his dissertation on combinations (Lesson 09).

The university

He entered the University of Leipzig in 1661 at fourteen, studying philosophy and law. His teacher in philosophy was Jakob Thomasius, who supervised his bachelor’s thesis De principio individui (1663), on what makes an individual thing the individual it is. In the summer of 1663 he spent a term at Jena with the mathematician and philosopher Erhard Weigel, who taught a Pythagorean view of number as the key to the world and to ethics. Leipzig’s mathematics was weak, and Leibniz arrived in Paris nine years later knowing little of the new analysis of René Descartes.

He took the master’s degree in philosophy early in 1664; his mother died within days of the disputation. He then turned to law and in 1666 completed the requirements for a doctorate. Leipzig refused it, officially because of his youth. In the same year he published the Dissertatio de arte combinatoria, an expanded version of a disputation that qualified him to teach in the philosophy faculty.

Altdorf, Nuremberg and Mainz, 1666–1672

Leibniz took his law dissertation to the University of Altdorf, the university of the free city of Nuremberg. There, in November 1666, he defended De casibus perplexis in jure, on legal cases that seem to admit no decision, and received the doctorate. The faculty offered him a professorship. He declined it, and never held an academic post in his life.

For a short time he served as secretary to an alchemical society in Nuremberg. There, or soon after, he met Baron Johann Christian von Boineburg, a former chief minister of the Elector of Mainz. Boineburg took him into his service, and through him Leibniz entered the court of the Elector, Johann Philipp von Schönborn. His tasks were to help recodify the laws of the Electorate and to write political memoranda. In these years he wrote on legal reform, on the reunion of the Catholic and Protestant churches, and on natural philosophy. His Hypothesis physica nova (1671), dedicated in part to the Royal Society in London and in part to the Paris Academy, was an ambitious and premature theory of motion.

In 1672 Boineburg sent him to Paris with a scheme meant to keep Louis XIV away from the German states: France should conquer Egypt instead. The plan went nowhere. Leibniz stayed in Paris, and the four years that followed made him a mathematician.

Paris and London, 1672–1676

In the autumn of 1672 Leibniz met Christiaan Huygens, the leading natural philosopher in Paris. Huygens recognized his talent and set him a problem: sum the reciprocals of the triangular numbers \(1,3,6,10,\dots\). Leibniz solved it (Lesson 04), and Huygens guided his reading toward Blaise Pascal, Grégoire de Saint-Vincent and Descartes. Leibniz also met the philosophers Antoine Arnauld and Nicolas Malebranche. His patron Boineburg died in December 1672, and the Elector of Mainz early in 1673, leaving him without secure employment.

The first London visit

In January 1673 Leibniz went to London with a Mainz diplomatic mission. On 1 February he showed the Royal Society a wooden model of his calculating machine, which could multiply as well as add (Lesson 08). Robert Hooke criticized the unfinished model. Leibniz also spoke about series, and John Pell pointed out that one of his results had already been published by Gabriel Mouton. The episode showed him how little he knew of recent English work. He was elected a Fellow of the Royal Society in April 1673.

Three years of discovery

Back in Paris he worked with intense concentration. Reading Pascal’s treatise on the sines of a quadrant of a circle, he saw that Pascal’s small triangle at a point of a circle worked for any curve; this became his characteristic triangle. In 1673 he found the transmutation theorem, and from it his series for \(\pi/4\). From August 1675 he worked closely with Ehrenfried Walther von Tschirnhaus. In the autumn of 1675 he invented the notation and rules of the calculus: the sign \(\int\) appears in a manuscript dated 29 October 1675, and the differential \(d\) within the next weeks.

In 1676 he received Newton’s two letters through Henry Oldenburg, secretary of the Royal Society. They described the binomial series and much else, but concealed the method of fluxions. In October 1676 Leibniz accepted a post at the court of Duke Johann Friedrich of Brunswick-Lüneburg in Hanover. He left Paris, spent about a week in London, where John Collins showed him Newton’s papers, including De analysi, and travelled on through Holland. In The Hague in November he had several long conversations with Baruch Spinoza. He reached Hanover in December 1676, where he would live for the remaining forty years of his life.

Sums, differences and the quadrature of the circle

For the general reader. Leibniz came to the calculus through sequences of numbers. He noticed that if you take differences of a sequence and then add them up, you get back the difference between the first and last terms. In geometry the same idea says that adding up infinitely many infinitely small pieces of a curve recovers the whole. His first triumph with this idea was a startlingly simple series for \(\pi\).

Sums of differences

Let \(a_0,a_1,a_2,\dots\) be a sequence and \(\Delta a_k=a_{k+1}-a_k\) its differences. Then

This trivial identity was, by Leibniz’s own later account, the germ of the calculus. Summation and differencing are inverse operations. When the terms become infinitely close together, sums become integrals, differences become differentials, and the identity becomes the fundamental theorem.

Proof: \(\frac{2}{n(n+1)}=2\left(\frac1n-\frac1{n+1}\right)\) is a difference of the sequence \(-2/n\), so the partial sum to \(N\) is \(2-\frac{2}{N+1}\to2\). Leibniz organized such sums in his harmonic triangle, whose rows are \(1\); \(\tfrac12,\tfrac12\); \(\tfrac13,\tfrac16,\tfrac13\); \(\dots\), in which each entry is the sum of the two below it. The triangle is Pascal’s arithmetic triangle with differences in place of sums.

The characteristic triangle

At a point \(P=(x,y)\) of a curve, consider the infinitely small triangle with legs \(dx,dy\) and hypotenuse \(ds\), the element of arc. It is similar to the finite triangle formed by the tangent, the ordinate and the subtangent (see the figure at the top of the page), and also to the triangle formed by the normal, the ordinate and the subnormal. Pascal had used this triangle for the circle. Leibniz saw that it works for every curve, and that it converts questions about tangents into questions about ratios of differentials.

Leibniz’s argument. The infinitesimal triangle with vertices at the origin \(O\), \(P\) and the neighbouring point \(P+dP\) has signed area \(\tfrac12(y\,dx-x\,dy)=\tfrac12 z\,dx\): its base is the arc element \(ds\), and by the characteristic triangle its height, the distance from \(O\) to the tangent, is \(|z|\,dx/ds\). Summing these triangles gives the signed area between the chord \(OP\) and the curve, and adding the triangle with vertices \(O\), \((x_0,0)\) and \(P\) gives the area under the curve. Modern check. Integrate by parts: \(\int_0^{x_0}x\,y'\,dx=x_0y(x_0)-\int_0^{x_0}y\,dx\), and substitute.

The arithmetical quadrature of the circle

Take the circle \((x-1)^2+y^2=1\), so \(y^2=2x-x^2\). Then \(y'=(1-x)/y\) and

The quarter of the disc over \([0,1]\) has area \(\pi/4\), and at \(x=1\) we have \(y=z=1\). By the theorem and one more integration by parts,

Leibniz found the series in 1673 and published it in the Acta Eruditorum in 1682. James Gregory had the arctangent series in 1671, and the Kerala school of Madhava had it two centuries earlier, but neither was known to Leibniz. Huygens praised the result warmly; it convinced Leibniz that he had a method.

The differential calculus and its notation

For the general reader. Leibniz’s calculus is a set of rules for manipulating tiny quantities called differentials. His great insight was that a good notation does part of the thinking. With his symbols, rules such as the chain rule or a change of variables look like ordinary algebra, so a student can apply them without understanding every step. Three centuries later, his notation is still the one in use.

Differentials

For Leibniz a variable \(x\) has a differential \(dx\), an infinitely small increment, and \(dy\) is the corresponding increment of \(y\). The quotient \(dy/dx\) is a ratio of two such quantities, and the second differential \(ddy\) (later \(d^2y\)) is the difference of successive \(dy\). Quantities infinitely small relative to others may be dropped: \(dx\,dy\) is negligible beside \(x\,dy\). In Nova methodus pro maximis et minimis, itemque tangentibus, published in the Acta Eruditorum of October 1684, Leibniz stated the rules on six pages, without proofs and with several misprints.

Derivation of the product rule. When \(u\) and \(v\) increase by \(du\) and \(dv\),

and \(du\,dv\) is infinitely small compared with the other terms, so it is discarded. In November 1675 Leibniz had first asked himself whether \(d(uv)\) equals \(du\,dv\), and checked by example that it does not. The quotient rule follows by applying the product rule to \(u=(u/v)\,v\).

What the notation does

The chain rule reads \(\dfrac{dz}{dx}=\dfrac{dz}{dy}\,\dfrac{dy}{dx}\), as if the \(dy\) cancelled. Substitution in integrals reads \(\int f(y)\,dy=\int f(y(x))\,\dfrac{dy}{dx}\,dx\). Homogeneity is visible: in \(y\,dx\), an area, both factors are lengths. Leibniz also noticed that repeated differentiation of a product follows the binomial pattern,

now called the Leibniz rule. Newton’s dot notation \(\dot x\) hides the independent variable and makes such manipulations clumsy; this difference shaped the history of analysis in the eighteenth century.

Refraction

Nova methodus closes with applications, including the law of refraction derived from a minimum principle, the problem Pierre de Fermat had treated with his own method. Light goes from \(A\), at height \(a\) above a flat interface, to \(B\), at depth \(b\) below it and horizontal distance \(c\) from \(A\), with speeds \(v_1\) above and \(v_2\) below. If it crosses at horizontal position \(x\), the time taken is

Setting \(dT=0\) gives \(\dfrac{x}{v_1\sqrt{a^2+x^2}}=\dfrac{c-x}{v_2\sqrt{b^2+(c-x)^2}}\), that is, \(\dfrac{\sin\theta_1}{v_1}=\dfrac{\sin\theta_2}{v_2}\), Snell’s law.

The integral, the fundamental theorem and the priority dispute

For the general reader. Leibniz thought of an area as a sum of infinitely many thin rectangles and wrote it with a long letter S, for summa. Because sums undo differences, finding areas becomes the reverse of finding tangents. The same discovery had been made by Newton ten years earlier, and the question of who owed what to whom became the most bitter quarrel in the history of science.

The sign \(\int\)

Following Bonaventura Cavalieri, Leibniz first wrote “omn. \(l\)” for the sum of all lines \(l\) under a curve. In the manuscript of 29 October 1675 he decided it would be useful to write \(\int l\) instead, using the long s. Soon he wrote \(\int y\,dx\), the sum of rectangles of height \(y\) and infinitesimal width \(dx\). The sign first appeared in print in De geometria recondita et analysi indivisibilium atque infinitorum, in the Acta Eruditorum of June 1686. The name integral came later, from Jacob Bernoulli in 1690; Leibniz adopted it.

This is the continuous form of the sum–difference identity of Lesson 04. In Supplementum geometriae dimensoriae (1693) Leibniz gave a geometric proof by a curve-tracing construction: a quadrature is reduced to finding a curve with a prescribed tangent, the “inverse tangent problem”. Its solution by a mechanical device was his way of showing that every such problem has an answer.

The calculus at work

In 1690 Jacob Bernoulli challenged mathematicians to find the shape of a hanging chain, the catenary. In 1691 the Acta Eruditorum printed solutions by Leibniz, Huygens and Johann Bernoulli. In 1696 Johann posed the brachistochrone problem: the curve of fastest descent under gravity. The Acta of May 1697 contained solutions by Johann and Jacob Bernoulli and by Leibniz, with notices of others by Guillaume de l’Hôpital, Tschirnhaus and Newton, whose anonymous solution was printed in London. The answer is a cycloid. Problems that had defeated the geometers of the previous generation became exercises for anyone who knew the rules.

The priority dispute

The sequence that historians now accept is as follows, consistent with the account in the Newton course. Newton developed fluxions in 1665–66 and described them in De analysi (1669) and a treatise of 1671, neither printed until much later. Leibniz developed his calculus in Paris in 1675. When he passed through London in October 1676, Collins let him read De analysi; his surviving notes, found by C. I. Gerhardt in 1849, concern series and contain nothing about fluxions. Newton’s two letters of 1676 concealed the method in an anagram. In June 1677 Leibniz replied through Oldenburg with a clear account of his differential calculus. Oldenburg died that September, and the exchange stopped.

Leibniz published in 1684 and 1686 without mentioning Newton’s method. The first edition of Newton’s Principia (1687) contained a scholium acknowledging that Leibniz had a similar method; the third edition of 1726 removed it. In 1693 the two men exchanged courteous letters. Open hostility began in 1699, when Nicolas Fatio de Duillier suggested in print that Leibniz had borrowed from Newton. An anonymous review of Newton’s De quadratura in the Acta Eruditorum, written by Leibniz, implied the reverse. John Keill then accused Leibniz of plagiarism in the Philosophical Transactions. Leibniz appealed to the Royal Society in 1711. Its committee, whose report Newton drafted, found for Newton; the report and supporting letters appeared as the Commercium epistolicum (dated 1712, distributed early 1713). Leibniz answered with an anonymous broadsheet, the Charta volans (1713), citing an unnamed leading mathematician, who was Johann Bernoulli. The quarrel continued through the Abbé Antonio Conti until Leibniz’s death in 1716.

Historians agree that the inventions were independent: Newton’s came first, Leibniz’s was published first. Leibniz’s notation, and the school of the Bernoullis, l’Hôpital and later Leonhard Euler that grew around it, carried continental analysis far ahead for most of the eighteenth century.

Vis viva and dynamics

For the general reader. What does a moving body have that it can spend? Descartes said it was its size times its speed. Leibniz showed in 1686 that this answer leads to a perpetual motion machine, and proposed mass times the square of speed instead. He called it vis viva, living force. Half of it is the kinetic energy of every physics textbook, and his claim that it is never lost, only hidden in the small parts of bodies, anticipates the conservation of energy.

Descartes’s rule and its refutation

In his Principia philosophiae (1644) Descartes held that God conserves the total quantity of motion, the sum of size times speed \(\sum m|v|\), without regard to direction. Leibniz’s Brevis demonstratio erroris memorabilis Cartesii appeared in the Acta Eruditorum in March 1686. He assumed two principles accepted by the Cartesians themselves:

1. a body falling from a height acquires exactly the force needed to lift it back to that height;
2. the same force is needed to raise a body of 1 pound to a height of 4 units as to raise a body of 4 pounds to a height of 1 unit.

Hence a body of 1 pound that has fallen 4 units has the same force as one of 4 pounds that has fallen 1 unit. By Galileo Galilei’s law of free fall, \(v^2\propto h\), so the first body has twice the speed of the second. Its quantity of motion is \(1\times2=2\), against \(4\times1=4\) for the second. Quantity of motion therefore cannot measure force. The quantity \(mv^2\) gives \(1\times4=4\) for both. If Descartes were right, one could transfer motion from the heavy body to the light one, raise the light one four units, and gain force from nothing.

The second half of the principle corrects Descartes in another way: the conserved momentum carries a sign. For perfectly elastic collisions Huygens had already shown in the 1660s that \(\sum mv^2\) is conserved. Leibniz raised it to a universal principle and tied it to the impossibility of perpetual motion.

Dead force and living force

In the Specimen dynamicum (1695) Leibniz distinguished vis mortua, a solicitation to motion where no motion yet exists (a weight pressing on a table, a stone pulling on a sling), from vis viva, which arises from an infinite accumulation of impressions of dead force. In modern terms dead force is force, and living force is the integral of force over distance:

The “vis viva controversy” between Cartesians and Leibnizians ran for decades. Willem ’s Gravesande dropped balls into clay and found that the depth of the dent grew with \(v^2\). Émilie du Châtelet championed vis viva in her Institutions de physique (1740). Jean le Rond d’Alembert argued in 1743 that both measures are valid, momentum for the effect of a force over time and \(mv^2\) for its effect over distance. The factor \(\tfrac12\) and the name “work” for \(\int F\,dx\) were fixed by Gaspard-Gustave Coriolis in 1829.

Binary arithmetic and the calculating machine

For the general reader. Leibniz saw that all numbers can be written with only two digits, 0 and 1, and that arithmetic then becomes almost trivially simple. He also built one of the first machines that could multiply. The two ideas did not meet in his lifetime, but together they describe the digital computer.

The dyadic system

A manuscript of 15 March 1679, De progressione dyadica, sets out binary arithmetic with tables of addition and multiplication. It also describes, without building it, a machine in which balls falling through open or closed holes would represent the digits 1 and 0. Leibniz returned to the subject repeatedly. In 1697 he proposed to Duke Rudolf August of Brunswick-Wolfenbüttel a medal showing binary numbers as an image of creation, all things made from one and nothing. Thomas Harriot had used base 2 in unpublished papers decades earlier, but Leibniz was the first to develop and publish the system.

Through his interest in China, Leibniz corresponded with the Jesuit missionary Joachim Bouvet. Bouvet replied from Beijing in November 1701 that the 64 hexagrams of the Yijing, in the arrangement attributed to the legendary Fuxi, read as the binary numbers from 0 to 63 when a broken line is taken for 0 and an unbroken line for 1. The letter reached Leibniz on 1 April 1703. Within about a week he sent the Explication de l’arithmétique binaire to the Paris Academy; it appeared in the Academy’s volume for 1703, printed in 1705.

Proof. Existence by strong induction: if \(n\) is even, \(n=2(n/2)\) and we shift the expansion of \(n/2\); if odd, \(n=2\frac{n-1}{2}+1\). Uniqueness: two expansions of the same \(n\) have the same parity, so \(b_0\) agrees; subtract it, divide by 2 and repeat. The algorithm gives the digits from the right as the successive remainders on division by 2.

In the Explication Leibniz pointed out that addition and multiplication need no tables to memorize: \(1+1=10\) and \(1\times1=1\) are all there is. He also observed that, when the numbers \(0,1,2,\dots\) are written in binary one under another, each column is periodic: the units column repeats with period 2, the next with period 4, and so on. He hoped that such regularities would help with sequences of primes and with series, though he admitted that binary is unsuited to everyday computation because the numbers become long.

The stepped reckoner

Blaise Pascal’s machine of the 1640s could add and subtract. Leibniz wanted one that would multiply and divide without the operator repeating additions by hand. His solution was the stepped drum, now called the Leibniz wheel: a cylinder carrying nine teeth of increasing length. A small gear slides along a parallel axle; set at position \(k\), it meets \(k\) of the teeth and turns by \(k\) steps for each turn of the drum. One turn of the crank therefore adds the set number to the result register. A movable carriage shifts the setting by a decimal place, so multiplying by 26 takes six turns, a shift, and two more.

He showed a wooden model in London on 1 February 1673. Two machines were built over the following decades by craftsmen working for him, and their carry mechanism never worked reliably. One survives in the Gottfried Wilhelm Leibniz Library in Hanover; it was found in 1876 in an attic in Göttingen. The stepped drum was the basis of the first commercially successful calculator, Charles Xavier Thomas de Colmar’s arithmometer of the nineteenth century, and of many mechanical calculators until the 1970s.

Combinatorics, determinants and the universal characteristic

For the general reader. At nineteen Leibniz argued that all complex ideas are combinations of a few simple ones, as words are combinations of letters. If each simple idea had a sign, reasoning could become calculation, and disputes could be settled by sitting down and computing. The dream was never realized, but it led him to early work on counting, on systems of equations, and on what we now call symbolic logic.

De arte combinatoria (1666)

The dissertation counts the ways of choosing \(k\) things from \(n\) (Leibniz’s “com2nations”, “con3nations” and so on) and the total number of non-empty selections, which he called complexions. It tabulates these numbers in what is now Pascal’s triangle and applies them to logic, law, music and theology. The work contains errors and little that was new as mathematics; its importance lies in the program.

Proof: each subset is determined by an independent yes-or-no choice for each element, giving \(2^n\) subsets including the empty one. In binary (Lesson 08), the subsets of \(\{0,\dots,n-1\}\) correspond to the numbers \(0,\dots,2^n-1\).

Determinants

In a letter to Guillaume de l’Hôpital in 1693, Leibniz wrote a system of three linear equations in two unknowns with numerical indices for the coefficients, the first digit naming the equation and the second the unknown:

Here “21” is a symbol, the coefficient \(a_{21}\), and not the number twenty-one. Eliminating \(x\) and \(y\), he stated the condition for the system to be solvable:

where each product is of symbols, not numbers, and the rule is that each term contains one coefficient from each equation and one of each second index. This is the vanishing of the \(3\times3\) determinant of the coefficients. His manuscripts contain much more, including the elimination rule later published by Gabriel Cramer (1750). Seki Takakazu in Japan reached determinants independently at about the same time. Leibniz’s papers on the subject remained unpublished until the nineteenth and twentieth centuries.

The characteristica universalis

The universal characteristic was to be a system of signs, one for each simple concept, such that the sign of a complex concept displays its composition. A calculus ratiocinator would then derive conclusions mechanically. In papers of April 1679 Leibniz tried a numerical model: assign a prime to each simple concept and represent a compound by the product. If “animal” is 2 and “rational” is 3, then “man”, the rational animal, is 6. The proposition “every man is an animal” is true because 2 divides 6. He soon found that the model cannot represent negation and particular propositions properly and tried pairs of numbers instead. Leibniz’s hope that disputants would one day simply say “Calculemus” (let us calculate) became a motto for later logicians.

The program left traces on George Boole’s algebra of logic (1847–1854), on Gottlob Frege’s Begriffsschrift (1879), which Frege related explicitly to Leibniz’s idea, and on the twentieth-century question of whether mathematical truth can be decided mechanically, answered negatively by Kurt Gödel, Alonzo Church and Alan Turing.

Relational space and time

For the general reader. Is space a container that would exist even if it were empty, or is it nothing more than the arrangement of things? Newton believed in absolute space. Leibniz argued that space is only the order of things that exist together, and time the order of things that happen one after another. Their debate, conducted through intermediaries in the last year of Leibniz’s life, still frames discussions of space and time in physics.

The correspondence

In November 1715 Leibniz wrote to Caroline, Princess of Wales, whom he had known in Hanover, that natural religion was declining in England and blamed the philosophy of Newton and John Locke. Caroline passed the letter to Samuel Clarke, a theologian close to Newton. Five papers from Leibniz and five replies from Clarke followed, all through Caroline. Leibniz died before he could answer Clarke’s fifth reply. Clarke published the exchange in 1717 as A Collection of Papers, with an English translation. Caroline told Leibniz that Clarke’s papers were written with Newton’s advice; how much Newton contributed is still debated.

The papers range over God’s relation to the world, miracles, gravitation and free will. Leibniz mocked Newton’s suggestion that God must occasionally correct the solar system, comparing it to a watchmaker who has to wind and mend his watch. He called universal gravitation, acting at a distance without a mechanism, a perpetual miracle. The arguments about space and time are the ones that have lasted.

The shift arguments

In his third paper Leibniz stated his view: space is something merely relative, as time is, an order of coexistences, as time is an order of successions. Against absolute space he argued as follows. Suppose space were a real container. Then God could have placed the whole material world in it in a different position, say shifted three metres east or turned through a right angle, preserving every relation between bodies. The two worlds would be indiscernible, so by the identity of indiscernibles they are the same world. And if they were different, God would have had no reason to choose one, contrary to sufficient reason. The same argument applies to time: the world could not have been created a year earlier.

Clarke answered that God’s will is itself a sufficient reason, and that the shifted world, though indiscernible to us, would be a genuinely different state of affairs.

The physics of the question

Physics sided with Leibniz on uniform velocity. By Galilean invariance, Newton’s laws take the same form in every frame moving uniformly relative to another, so no experiment detects an absolute velocity. A world in which every body moves with an extra common velocity \(\vec u\) is indiscernible from ours. Modern “neo-Newtonian” spacetime keeps absolute acceleration but drops absolute rest.

On rotation, Newton had the stronger case. In the scholium on space and time in the Principia he described a bucket of water set spinning: at first the bucket turns relative to the water and the surface stays flat; later the water turns with the bucket and its surface becomes concave. The concavity tracks rotation, not relative motion between water and bucket. Leibniz’s fifth paper accepted that there is a difference between a body’s true motion and its mere change of relations, and located true motion in the body in which the cause of the change, a force, resides. He did not give a relational account of the bucket. Ernst Mach in 1883 proposed that inertia is determined by the distant masses of the universe, an idea that influenced Albert Einstein. In general relativity spacetime geometry is a dynamical field, neither a fixed container nor a mere set of relations, and Einstein’s “hole argument” revived Leibniz’s shift argument in a new form.

Hanover, Berlin and Vienna, 1676–1714

In Hanover Leibniz was court counsellor (Hofrat) and librarian, and he served three successive rulers of the house of Brunswick-Lüneburg: Duke Johann Friedrich (died 1679), Ernst August, who became Elector of Hanover in 1692 (died 1698), and Georg Ludwig. He was never at ease in the role. His employers wanted a librarian, a lawyer and a historian; he wanted academies, reforms and a universal science.

Mines, history, philosophy

From about 1680 to 1685 he spent long periods in the Harz mountains trying to drain the silver mines with windmills and pumps. Opposition from the mining officials, technical problems and unreliable wind defeated the project, and Ernst August ended it in 1685. His observations in the mines led to Protogaea, a theory of the formation of the earth, published only in 1749.

Ernst August commissioned a history of the house of Brunswick. From 1687 to 1690 Leibniz travelled through southern Germany, Austria and Italy searching archives; he established the family’s link to the house of Este. In Rome he was elected to an academy, and in Florence he met Vincenzo Viviani, the last pupil of Galileo. The history occupied him for the rest of his life and was never finished. From 1691 he was also librarian of the ducal library at Wolfenbüttel.

In these decades he published his philosophy in essays and letters. The Discours de métaphysique (1686) and the correspondence with Antoine Arnauld that followed set out his doctrine of individual substances. The Système nouveau (1695) introduced pre-established harmony: substances do not act on each other but are coordinated from creation. The Nouveaux essais, a dialogue with Locke’s Essay, were completed around 1704 and published only in 1765. The Essais de Théodicée (1710), the only philosophical book he published, defended the claim that God chose the best of all possible worlds. In 1714 he wrote the 90 short paragraphs later called the Monadology.

Academies and courts

Leibniz believed science should be organized by academies under princely patronage. His friendship with Electress Sophia of Hanover and her daughter Sophia Charlotte, Electress of Brandenburg and later Queen in Prussia, led to the foundation of the Brandenburg Society of Sciences in Berlin on 11 July 1700, with Leibniz as president. It became the Prussian Academy of Sciences. In 1700 he was also elected a foreign member of the Paris Academy. He worked for a reunion of the Christian churches, corresponding with Bishop Jacques-Bénigne Bossuet, and edited Novissima Sinica (1697), a collection of reports from the Jesuit mission in China.

He met Tsar Peter the Great at Torgau in 1711 and again later, and proposed reforms of Russian education and an academy; the St Petersburg Academy was founded in 1724, after his death, partly on his plans. In Vienna from 1712 to 1714 he was made an Imperial Court Councillor and pressed for an imperial academy, which was never created. His long absences angered Georg Ludwig, who ordered him back to Hanover.

Last years, death and legacy

In 1714 Electress Sophia died, and in August Queen Anne died; under the Act of Settlement, Georg Ludwig became George I of Great Britain. The court moved to London. Leibniz asked to follow and was told to stay in Hanover and finish the history. The quarrel with Newton, now President of the Royal Society, made him unwelcome in England. His last two years were spent on the history, the correspondence with Clarke, letters to Conti on the priority question, and a final meeting with Peter the Great at Bad Pyrmont in 1716.

He had suffered for years from gout. He died in Hanover on 14 November 1716, aged 70, and was buried in the Neustädter Kirche. No one from the court attended; his secretary was the only mourner of note, and the grave remained unmarked for more than half a century. Bernard le Bovier de Fontenelle read his eulogy before the Paris Academy in 1717. The Royal Society and the Berlin society he had founded made no comparable acknowledgment. Leibniz never married.

Legacy

• Analysis: the notation \(dx\), \(\int\), \(dy/dx\) and the Leibniz rule; through Jacob and Johann Bernoulli, l’Hôpital’s Analyse des infiniment petits (1696), the first calculus textbook, and Euler, the whole continental tradition of analysis. Abraham Robinson’s non-standard analysis (1960s) gave infinitesimals a rigorous foundation of the kind Leibniz sought.
• Physics: vis viva and the conservation principle that became the conservation of energy; the relational view of space and time, revived by Mach and debated in general relativity and quantum gravity.
• Logic and computing: binary arithmetic, the stepped drum, and the program of a formal language of reasoning, taken up by Boole, Frege, Bertrand Russell and Louis Couturat, whose studies of 1901–1903 revealed the extent of his unpublished logic.
• Philosophy: through Christian Wolff, Leibniz’s system dominated German universities until Immanuel Kant, who began as a Leibnizian.
• Honours and editions: his papers, preserved in Hanover, run to some 200,000 pages. The Academy edition of his complete writings and letters, begun in 1923, is still in progress. The German Research Foundation’s Gottfried Wilhelm Leibniz Prize, the Leibniz Association of research institutes and Leibniz University Hannover bear his name.

Connections

People linked to Gottfried Wilhelm Leibniz across this collection. A name without a link has no page here yet. See Gottfried Wilhelm Leibniz in the network of all 337 lives →

Teachers and advisors

• Christiaan Huygens 1672-1676
• Erhard Weigel 1663
• Jakob Thomasius 1661-1663

Influenced by

• Blaise Pascal
• Isaac Barrow
• Madhava of Sangamagrama
• Rafael Bombelli
• René Descartes

Influenced

• Bertrand Russell
• Gaspard-Gustave Coriolis
• Gottlob Frege
• Hermann Grassmann
• Jacob Bernoulli
• Lazare Carnot
• Leonhard Euler
• William Hyde Wollaston
• Émilie du Châtelet
• Christian Wolff 1704-1716

Colleagues and collaborators

• Abraham de Moivre
• Jacob Bernoulli 1687-1705
• Johann Bernoulli 1693-1716
• John Wallis
• Ole Rømer
• Otto von Guericke
• Antoine Arnauld 1672-1690
• Baruch Spinoza 1676
• Ehrenfried Walther von Tschirnhaus 1675-1708
• Guillaume de l'Hôpital 1692-1704
• Henry Oldenburg 1670-1677
• Joachim Bouvet 1697-1707

Rivals

• Brook Taylor
• Isaac Newton 1676-1716
• John Keill 1708-1716
• Samuel Clarke 1715-1716

Also mentioned in this life

• George Boole
• Bonaventura Cavalieri
• Alonzo Church
• Jean le Rond d’Alembert
• Albert Einstein
• Pierre de Fermat
• Galileo Galilei
• Kurt Gödel
• Robert Hooke
• Ernst Mach
• Alan Turing

Fields

• calculus 9 lives
• combinatorics 9 lives
• logic 8 lives
• philosophy 12 lives

On CoursesHub

• Newton & Leibniz · History of Mathematics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Gottfried Wilhelm von Leibniz, MacTutor History of Mathematics.
2. Gottfried Wilhelm Leibniz, Wikipedia.
3. M. R. Antognazza, Leibniz: An Intellectual Biography, Cambridge University Press, 2009.
4. E. J. Aiton, Leibniz: A Biography, Adam Hilger, 1985.
5. B. Look, Gottfried Wilhelm Leibniz, Stanford Encyclopedia of Philosophy.

Mathematics, physics and logic

1. G. W. Leibniz, Nova methodus pro maximis et minimis, Acta Eruditorum, October 1684; overview.
2. J. M. Child (ed.), The Early Mathematical Manuscripts of Leibniz, Open Court, 1920.
3. J. E. Hofmann, Leibniz in Paris 1672–1676: His Growth to Mathematical Maturity, Cambridge University Press, 1974.
4. A. R. Hall, Philosophers at War: The Quarrel between Newton and Leibniz, Cambridge University Press, 1980; Leibniz–Newton calculus controversy, Wikipedia.
5. H. J. M. Bos, Differentials, higher-order differentials and the derivative in the Leibnizian calculus, Archive for History of Exact Sciences 14 (1974), 1–90.
6. G. W. Leibniz, Explication de l’arithmétique binaire, Mémoires de l’Académie Royale des Sciences (1703); historical note.
7. Stepped reckoner, Wikipedia.
8. J. K. McDonough, Leibniz’s Philosophy of Physics, Stanford Encyclopedia of Philosophy.
9. H. G. Alexander (ed.), The Leibniz–Clarke Correspondence, Manchester University Press, 1956; Samuel Clarke, Stanford Encyclopedia of Philosophy.
10. L. Couturat, La logique de Leibniz, Alcan, 1901; Characteristica universalis, Wikipedia.
11. E. Knobloch, Der Beginn der Determinantentheorie: Leibnizens nachgelassene Studien zum Determinantenkalkül, Gerstenberg, 1980.