The influence of Blaise Pascal is most keenly felt in his work on probability and the binomial theorem, illustrated by the famous Pascal’s triangle. It cannot be denied that Pascal’s triangle is a thing of mathematical beauty. However, this array of numbers was not discovered by its namesake, rather its applications and importance were *highlighted* by Pascal in his work, akin to Pythagoras’ theorem, which was certainly not invented by Pythagoras himself. However, Pascal’s legacy to mathematics goes further than the instantly recognisable triangle …

Blaise Pascal grew up in 17th century France—a time much romanticised by Alexandre Dumas and his musketeers. The story of the Pascal family reads just like a chivalric novel, with famous names from mathematical and social history peppered throughout.

Blaise showed a great aptitude for the sciences despite his father, Étienne, believing that the boy should not be taught formal maths until after the age of 15. Legend has it that Blaise formulated some of Euclid’s proofs by drawing in the dust on a floor. As a result, his father relented, and Blaise began his mathematical studies.

Étienne Pascal was a tax collector and lawyer, himself interested in maths and science. Indeed, the Limaçon of Pascal curve is named for *Étienne*, not Blaise, and Étienne served the French government, sitting on committees to examine scientific propositions. He corresponded with Mersenne and Roberval, debating the studies of Descartes and Fermat.

Étienne was the lone parent for Blaise and his sisters after his wife Antionette died in 1626. He moved the family to Paris in 1631, selling his interest in a vast area of land in the Auvergne region for a considerable amount of money. Étienne invested this money in government bonds, certain that his fortune would be safe. It was not. The thirty years’ war was raging across the continent, and Cardinal Richelieu’s economic policies meant that the government could not honour the bonds. Étienne found himself at odds with a government he had trusted and helped, and made his anger known. Richelieu threatened Étienne with imprisonment, and so he fled Paris to escape the Bastille.

His exile was relatively short lived thanks to Jacqueline Pascal, Blaise’s artistic sister. After performing a play for Richelieu, she convinced the cardinal to forgive Étienne. She was so convincing that he also gave Étienne a promotion, installing him as king’s commissioner for taxes in Rouen in 1639. The promotion was tough. Thanks to the wars and civil uprisings, Rouen’s finances were in chaos, and Étienne worked through the night, every night, to attempt to bring order to them.

I have never been in a tenth part the perplexity that I am in at present.

Étienne Pascal

## Life in Rouen

Blaise had accompanied his father to Rouen. He saw how hard Étienne was working, and set his mind towards making the job of organising the city’s accounts less labour-intensive.

Blaise invented a calculating machine. The Pascaline.

Previous attempts at effort-saving devices did not suit Blaise’s plans. He needed a machine that would add up large sums of numbers automatically, or repeatedly subtract.

Napier’s Bones, created by John Napier (1550–1617) were efficient for multiplying and dividing numbers. Based on the multiplication table, they drew upon a lattice form of multiplication, and by aligning the bones, or rods, in a certain way, addition could be done instead of multiplication, and subtraction in place of division.

Edmund Gunter (1581–1626) invented the ‘Gunter’s Scale’, a ruler-like device which allowed navigators to quickly move between different forms of measurement. His contemporary William Oughtred (1574–1660), developed this further, bringing together the Gunter Scale and Napier’s other invention, the logarithm, to make the slide rule.

A calculating clock had been invented by Wilhelm Schickard in the 1620s, based on Napier’s bones. It used pinwheels to ‘carry’, and was designed to be able to add, subtract, multiply and divide. It was described in letters from Schickard to Johannes Kepler, but was never seen working. Later examination of the descriptions in the correspondence suggest that the mechanism would jam, especially if several carrying actions happened simultaneously.

## The Pascaline

Pascal’s Pascaline was special. All previous devices either needed the operator to `do the maths’, rather than having the calculation happen automatically, or plainly did not work.

Blaise opted for a simple design based on addition. The machine would be set to zero, and then the operator would use a stylus to enter an initial number by using the digits around the circumference each dial as reference points. The stylus would reach a stop bar, as the dial was turned, just like on a rotary telephone dial.

Each separate dial represented a digit using place value. The operator could then add on another number in the same way, and keep repeating this until they had added all the required numbers. The ‘accumulator’ at the top of the machine displayed the final total. The machine automatically carried when each position reached nine, using a mechanism that Blaise designed himself. Subtraction was carried out on the apparatus using ‘complements to 9’. The bar at the top of the machine where the accumulator is situated hides a second set of numbers. If 82953 were displayed on the accumulator, the bar would be hiding the digits 17046 ($8+1=9$, $2+7=9$, $9+0=9$, $5+4=9$ and $3+6=9$). The bar could slide up and down, revealing the 9 complement of the total. By using this 9’s complement, subtraction could be carried out via the same operating technique as addition. The calculation of 40732 – 6549 is demonstrated below.

Étienne made use of his son’s invention to aid his work. Blaise, however, was never completely satisfied with the Pascaline. He spent a great deal of time refining it, making machines with more wheels, decimal machines, machines for currencies and distances which needed different bases on each wheel. He wanted to sell the calculators for widespread use, but they proved to be very expensive due to the intricate parts. Instead, the Pascaline was seen more as a status symbol for European nobility. Even though fewer than 20 were sold, the Pascaline was the first working automatic calculator, inspiring later inventors and mathematicians. Mechanical calculators were used around the world well into the 1970s, when electronic versions started to take over.

Out of frustration at seeing his father labour over calculations, Blaise Pascal laid the foundations for a device which people today still count on.