When people talk about the history of computing, they usually start with the great electronic machines of the twentieth century. Yet the underlying problem, handing a repetitive mental task over to a mechanism, is much older. In the seventeenth century two of Europe's greatest thinkers, Blaise Pascal and Gottfried Wilhelm Leibniz, built machines that could carry out arithmetic. They were not computers, but they already contained some of the ideas we find in modern processors.
A forerunner: Schickard
Before Pascal there was at least one documented attempt. Wilhelm Schickard, a professor in Tübingen, described a "calculating clock" able to add and subtract in letters to Kepler around 1623. The original machine has not survived, and it had no influence on later developments, because the letters remained unknown for centuries. It is still a useful reminder: ideas are often born several times, in different places, before they take hold.
The Pascaline
Blaise Pascal began working on his machine in 1642, while still very young. His reason was entirely practical: his father, Étienne, was an official in charge of tax collection in Rouen and spent his days on endless additions. Pascal wanted to lighten his workload.
How it worked
The Pascaline was a brass box with a row of toothed wheels, one for each decimal digit: units, tens, hundreds and so on. A number was entered by turning the wheels with a stylus, and the result appeared in small windows.
The truly clever part was the automatic carry. When a wheel moved from 9 to 0, a lever mechanism advanced the next wheel by one step. It sounds trivial, but it is exactly the problem every adder circuit has to solve: propagating the carry from one digit to the next, more significant one.
The machine could add directly. To subtract, it used the nines' complement method: instead of taking a number away, you added its complement and read the result in a dedicated row of windows. Anyone who has studied computer arithmetic will immediately recognise a close relative of two's complement, still used today to represent negative numbers in binary.
Success and limits
Pascal built several examples, and some are still preserved in museums. The machine, however, was expensive, delicate and hard to repair. It never became a widespread tool. It remained a technical marvel, admired more than used.
Leibniz and the stepped drum
A few decades later, Gottfried Wilhelm Leibniz took on a more ambitious problem: a machine that could also multiply and divide without forcing the operator to repeat dozens of additions by hand. In 1673 he presented a model to the Royal Society in London.
The key invention
The heart of the design was the stepped drum, now also known as the Leibniz wheel. It is a cylinder with nine teeth of increasing length arranged along its surface. By sliding a gear parallel to the axis, it could be made to mesh with a different number of teeth, from zero to nine. One full turn of the drum therefore added the digit that had been set.
Multiplication became a sequence of repeated additions, combined with shifting the carriage by one position to move from units to tens. It is the same scheme as the long multiplication we learn at school, and also the scheme of the simplest hardware multipliers: add and shift.
A legacy two centuries long
Leibniz's machine suffered from mechanical problems, especially in handling carries. But the stepped drum had a remarkably long life. In 1820 Charles Xavier Thomas de Colmar patented the arithmometer in France, the first mechanical calculator to be mass-produced with commercial success, and its mechanism was based precisely on Leibniz's principle. Variants of the same device stayed in use in offices well into the twentieth century.
Binary: Leibniz's other insight
There is a second reason why Leibniz matters to computing. In 1703 he published, in the memoirs of the Paris Academy of Sciences, an essay on binary arithmetic entitled Explication de l'Arithmétique Binaire. He showed that every number can be written using only the digits 0 and 1, and that operations become remarkably simple.
An example makes the idea clear:
| Decimal | Binary |
|---|---|
| 1 | 1 |
| 2 | 10 |
| 5 | 101 |
| 13 | 1101 |
Leibniz did not build a binary calculator and could not have imagined electronics. But the idea that all of arithmetic can be reduced to two states is exactly what makes digital circuits possible, where those states become voltage present or absent.
What they teach us today
In my work I often have to explain concepts such as carry, complement or shifting. Starting from these machines helps a great deal, because it makes visible what a chip keeps hidden. There are three main lessons:
- Calculation is a procedure that can be mechanised. If an operation can be broken down into regular steps, a machine can perform it.
- The hard problems are in the details. The carry, not addition itself, was the real technical challenge for both inventors.
- Representation and mechanism go hand in hand. The choice of how to write numbers, decimal or binary, determines how complex the machine needs to be.
In short
- From 1642 Pascal built an adding machine with automatic carry, designed to help his father with tax accounts.
- Subtraction by nines' complement anticipates the two's complement used in computers.
- Leibniz invented the stepped drum, which made multiplication practical and survived in calculators until the twentieth century.
- In 1703 Leibniz described binary arithmetic, the theoretical basis of modern digital circuits.
