Leibniz

The life of Gottfried Wilhelm Leibniz

Timeline 1646Born on 1 July in Leipzig, in the Holy Roman Empire. 1661Enters the University of Leipzig, aged 14. 1666Publishes De arte combinatoria. Leipzig refuses him a doctorate in law; Altdorf awards it the following year. 1672Goes to Paris on a diplomatic mission. Begins to study mathematics seriously with Christiaan Huygens. 1673Presents his calculating machine in London and is elected to the Royal Society. 1675In manuscripts of October and November, creates the symbols $\int$ and $d$. 1676Moves to Hanover, where he will work as librarian and counsellor for 40 years. 1684Publishes the first paper on differential calculus in the journal Acta Eruditorum. 1700Founds the Berlin Academy of Sciences and becomes its first president. 1703Publishes his explanation of binary arithmetic. 1716Dies on 14 November in Hanover. ◀ Previous Follow the journey Next ▶ 1646–1666 LeipzigHe is born and studies in Leipzig. The university refuses him a doctorate in law, saying he is too young. 1666–1667 AltdorfHe obtains his doctorate within a few months, near Nuremberg, and turns down the professorship he is offered. 1668–1672 MainzHe enters the service of the Prince-Archbishop of Mainz as a jurist and adviser. 1672–1676 ParisHe arrives on a diplomatic mission and learns mathematics from Christiaan Huygens. In 1675 he creates the symbols $\int$ and $d$. 1673 LondonHe shows his calculating machine at the Royal Society and is elected a fellow. 1676 The HagueOn his way to Hanover he visits the philosopher Baruch Spinoza. 1676–1716 HanoverLibrarian and adviser to the dukes for forty years. He publishes the calculus (1684) and binary arithmetic (1703). He dies in 1716. 1687–1690 RomeHe travels for three years through archives in Germany, Austria and Italy to write the history of the ducal family. 1700 BerlinHe founds the Berlin Academy of Sciences and becomes its first president. Enable JavaScript to see the map. The stages are listed below. The journeys of Leibniz, diplomat, librarian and mathematician. Map: © OpenStreetMap contributors

October 11, 2026 Read

Leibniz's notation: a language for change

In signs one observes an advantage in discovery which is greatest when they express the exact nature of a thing briefly and, as it were, picture it; then indeed the labour of thought is wonderfully diminished. — Leibniz, letter to Tschirnhaus (1678) Newton reached calculus first, but it was Leibniz who decided how it would be written. And the writing made all the difference. The letter d On 11 November 1675, in a manuscript written in Paris, Leibniz began using the letter $d$ for an infinitely small difference. The $d$ comes from the Latin differentia, from the verb differre, “to carry apart”. So:

October 11, 2026 Read

The product rule: two changes in one calculation

If you know how $u$ changes and how $v$ changes, how does the product $uv$ change? The natural guess is to multiply the changes: $d(uv)=du\cdot dv$. It looks so symmetrical, so elegant… Leibniz made the same guess. Leibniz tests and fails In the same manuscript of 11 November 1675 in which he began using the letter $d$, Leibniz asked himself whether $d(xy)$ would equal $dx\,dy$, and whether $d\!\left(\frac xy\right)$ would equal $\dfrac{dx}{dy}$. Instead of trusting elegance, he tested it with an example and found it did not. A few days later he found the right rule.

October 11, 2026 Read

The ∫ sign: from sum to accumulation

On 29 October 1675 Leibniz was in Paris working on a problem about areas. Until then he had written omn., short for the Latin omnia, “all”, to mean the sum of all the lines of a figure, as the Italian Cavalieri did. Halfway down the page he wrote: It will be useful to write ∫ instead of omn. — Leibniz, manuscript of 29 October 1675 The symbol $\int$ is a long S, for summa, sum. At the time the long s was an ordinary letter in handwriting. It has never been replaced since.

October 11, 2026 Read

Binary arithmetic: writing numbers with 0 and 1

Why do we count in tens? Probably because we have ten fingers. There is nothing special about the number 10: any number can be written in any base. Leibniz took this idea to the extreme and asked: what is the smallest number of digits that can write everything? The answer is two: 0 and 1. Each place is worth double The numbers 0 to 32 written in binary, in the table from the paper Explanation of Binary Arithmetic, which Leibniz published in 1703 in the memoirs of the Paris Academy of Sciences.G. W. Leibniz (1703) · Wikimedia Commons · public domain In the decimal system the places are worth 1, 10, 100, 1000… and each digit says how many times we use that value. In the binary system the places are worth 1, 2, 4, 8, 16, 32… and each digit can only say yes (1) or no (0).

October 11, 2026 Read

Leibniz's machine: turning sums into motion

In 1642 the young Blaise Pascal built a machine to help his father, a tax collector. The Pascaline added and subtracted by turning gears. To multiply, however, you had to repeat the addition by hand, many times. Leibniz saw a Pascaline in Paris and wanted to go further: a machine that would multiply and divide on its own. Multiplying is adding and shifting The trick is the same as in long multiplication at school. To multiply $23\times14$, we split 14 into $4+10$:

October 11, 2026 Read

The Leibniz series for π: a simple, slow sum

The number $\pi$ comes from the geometry of the circle. The odd numbers come from arithmetic. There is no apparent reason for them to meet. But they do: $$\frac{\pi}{4}=1-\frac13+\frac15-\frac17+\frac19-\cdots$$Add and subtract the reciprocals of the odd numbers, alternating signs, and the result, multiplied by 4, is $\pi$. One discovery, three times Leibniz found this formula in 1673, in Paris, while he was still learning mathematics from Huygens. He showed the result to his teacher, who was impressed and praised him. It was Leibniz’s first great discovery and what convinced him he could be a mathematician.

October 11, 2026 Read

Calculemus: the idea of calculating with reasoning

The only way to rectify our reasonings is to make them as tangible as those of the mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: let us calculate!, without further ado, to see who is right. — Leibniz, The Art of Discovery (1685) Leibniz was a lawyer, diplomat and philosopher. He spent his life watching arguments that never ended, between religions, between countries, between schools of philosophy. A mathematician, on the other hand, does not argue about whether $2+2$ is 4: he does the sum. What if arguments could be calculated like numbers?

October 11, 2026 Read

Leibniz's legacy

The calculus war Leibniz published calculus in 1684 and 1686. Newton had reached the same ideas in 1665–1666 but kept everything in manuscripts. For fifteen years nobody disputed anything. In 1699 the Swiss mathematician Nicolas Fatio de Duillier, a friend of Newton, hinted in a book that Leibniz had drawn on Newton. In 1708 the Scot John Keill went further and accused Leibniz of plagiarism. The argument: in 1676 Leibniz had visited London and seen a copy of Newton’s De analysi.

October 11, 2026 Read

Image credits

Historical portrait The portrait of Leibniz painted by Christoph Bernhard Francke around 1695 comes from Wikimedia Commons and is in the public domain. We use the 960-pixel thumbnail provided by Commons, without editing. Source page. Public domain mark. Blog figures The cover and the sketch of the multiplication $23\times14$ are original illustrations. The sketch explains the calculation and does not reproduce the mechanism of the historical machine. The interactive figures for the eight bits, the terms of the product rule and the sums of the series for π were created for this trail. The examples in the text let you follow the calculations even without JavaScript.

October 11, 2026 Read

Leibniz against the dots: active and sterile notation

On 29 October 1675, in Paris, Leibniz was adding up infinitely many thin slices under a curve and got tired of writing omnia, “all”, in front of each sum. He noted in his notebook: Utile erit scribi ∫ pro omnia. (It will be useful to write ∫ instead of “all”.) — Leibniz, manuscript of 29 October 1675 The $\int$ is a stretched S, for summa. Two weeks later, in the same notebook, the $d$ appeared, for differentia, to denote infinitely small differences. With these two signs Leibniz had in his hands a language for the whole of calculus. Isaac Newton had reached the same ideas ten years earlier, in Cambridge, but wrote them differently. This is history’s best case for seeing how the shape of a sign changes what can be done with it.

October 13, 2026 Read