It doesn’t take too long before people realized that counting on their fingers (and toes) and using pebbles to store intermediate results left something to be desired. For this reason, a number of different folks started pondering the idea of creating machines that could perform calculations for them …

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Blaise Pascal and the Pascaline
In fact, determining who did invent the first mechanical calculator is somewhat problematical. Many references cite the French mathematician, physicist, and theologian, Blaise Pascal (1623-1662) as being credited with the invention of the first operational calculating machine. (In 1658, Pascal created a scandal when, under the pseudonym Amos Dettonville, he challenged other mathematicians to a contest and then awarded the prize to himself!)

In 1640, Pascal started developing a device to help his father add sums of money. The first operating model, the Pascaline (also known as the Arithmetic Machine by some), was introduced in 1642, and Pascal created fifty more devices over the next ten years.

However, Pascal’s device could only add and subtract, while multiplication and division operations were implemented by performing a series of additions or subtractions. In fact the Pascaline could really only add, because subtractions were performed using complement techniques, in which the number to be subtracted is first converted into its complement, which is then added to the first number. Interestingly enough, modern computers employ similar complement techniques (these techniques are introduced in Chapter 4 of our book How Computers Do Math).

Wilhelm Schickard and Leonardo da Vinci
Pascal’s claim to fame notwithstanding, the German astronomer and mathematician Wilhelm Schickard (1592-1635) wrote a letter to his friend Johannes Kepler about fifteen years before Pascal started developing his first calculating device. (Kepler, a German astronomer and natural philosopher, was the first person to realize – and prove – that the planets travel around the sun in elliptical orbits.)

In this letter, Schickard wrote that he had built a machine that “...immediately computes the given numbers automatically; adds, subtracts, multiplies, and divides”. Unfortunately, no original copies of Schickard’s machine – which he called his Calculating Clock – exist, but working models have been constructed from his notes.

In fact, some people believe that the first mechanical calculator may have been conceived by Leonardo da Vinci (1452-1519) almost one hundred and fifty years earlier than Schickard’s machine.

Da Vinci was a genius: painter, musician, sculptor, architect, engineer, and … the list goes on. However, his contributions to mechanical calculation remained hidden until the rediscovery of two of his notebooks in 1967. These notebooks, which date from sometime around the 1500s, contained drawings of a mechanical calculator, and working models of da Vinci's device have since been constructed.

Actually, it should be noted that some people believe that these sketches represent nothing more than a gear train, and that building a calculator based on them requires a certain amount of wishful thinking and extrapolation.

Gottfried von Leibniz and the Step Reckoner
In the 1670s, a German Baron called Gottfried von Leibniz (1646-1716) took mechanical calculation a step further. Leibniz, who entered university at fifteen years of age and received his bachelor’s degree at seventeen, said:

 "It is unworthy of excellent men to lose hours like slaves in the labor of calculation which could be safely relegated to anyone else if machines were used."

Leibniz developed Pascal's ideas and, in 1671, introduced the Step Reckoner, a device which, as well as performing additions and subtractions, could multiply, divide, and evaluate square roots by series of stepped additions. Leibniz also strongly advocated the use of the binary number system, which is fundamental to the operation of modern computers.

The devices invented by Pascal and Leibniz were the forebears of today's desk-top computers, and derivations of these machines continued to be produced until their electronic equivalents finally became readily available and affordable in the 1960s and 1970s as discussed later in this manuscript.

Note: The material presented here was abstracted and condensed from The History of Calculators, Computers, and Other Stuff document provided on the CD-ROM accompanying our book How Computers Do Math (ISBN: 0471732788).