Practise of arithmetic in the middle age

The inventory of fixtures from the 15th to the 18th century.

In order to know if the Pascaline (the Blaise Pascal's calculator) was usefull, it is necessary to put one's self in the place of a european man of this period. The two following anecdotes are instructive on this subject.

In the 15th century a german merchant who asked how to educate his son receive this advice :

"If you just want to familiarise him with addition and subtraction rules, any german or french university would be suitable. If you wish to get it on with the multiplication and division (provided that he has the ability for) you must send him in an italian school".

In the same way in 1662 (after the invention of the pascaline), a British civil servant who had calculations to do within the framework of its work has gone to different europeans schools with his wife to make studies. He wrote :

"At this time, my wife is able to do additions and subtractions and also multiplications, but I don't want to trouble her with the division practice.".

It is obvious that the practice of calculations, i.e. the possibility of making only additions and subtractions was not easy for common people, only specialists could do it. Moreover,, the calculations methods that we use today was not used in the occidental europe at the Blaise Pascal time.
Indeed, although the Arab numerals (without the zero) were introduced in occident at the end of the 10th century by the pope Sylvestre II (Gerbert d' Aurillac) ,the missing of the zero figure explains why the numeration and the calculation methods of Roman origin continued (the calculations was mostly carried out on "abacus" composed of tables with rows and columns to separate the different decimal positions).

At the beginning of the 13th century, Fibonacci plublishes a book (the "Liber Abaci") to explain the arab numeration system and the writen calculation methods that could be achieved with the introduction of the zero digit. The methods explained by Fibonacci were not used on account of the opposition of the accountants who wanted to keep the monopoly of this work. The quarrel between the partisans of the calculation with the abacus (the abacists) and those of the writen method (the algorists) has hold out about 3 centuries. The merchants and the money dealers has been the last to give it up.

Clic on the picture to see the design of the "Gerbert d'Aurillac abacus" and the one used by the merchants from the 13th to the 18th century.

In 1671, a french marchioness ("Madame de Sévigné") wrote to her daughter that his fortune has been evaluated by calculations on an abacus :
« Nous avons trouvé, avec ces jetons qui sont si bons, que j’aurai eu cinq cent trente mille livres de bien, en comptant toutes mes petites successions. »

The abacus was definitively abandoned at the end of the 18th century.

About the usefulness of the Blaise Pascal calculator

The Blaise Pascal calculator made it possible for everyone (merchants, bankers, ..) to perform additions and subtractions without any knowledge in mathematics. It was even possible to make multiplications and divisions in a relatively easy way (see the user's manual written in the 18th century).

The slowness with which calculations are carried out with a pascaline as well as the various stages necessary to multiply or divide that would be unacceptable today were probably not so important in the middle age where :

Either you pay an expert to make your calculations with an abacus.

Either you make yourself your calculations with a 'pascaline' probably more quickly than the expert. Indeed, for example in the case of additions, the figures adds one to another at the rate they are entered. On the contrary, with an abacus you constantly have to reorganize the jetons (for example replacing five jetons on a line by one jeton in between two lines and two jetons in between two lines by one jeton placed on the line just above it).
The conclusion is the same for multiplications and divisions.

What is the reason why the pascaline was not successful ?

If one accepts what is written above, one can wonder why the pascaline did not have success (only few calculators were built). To my point of view, there are several reasons.

It is difficult today to change its practices of work, this was certainly more difficult 300 years ago. It is enough to be convinced to note that 5 centuries have been necessary to adopt the written method of calculation in spite of his advantage on the abacus.

In the middle age the duration to obtain the result of a calculation was probably not so important than today.

May be the purchase of a pascaline was a big investment for the bankers an the traders.

if one believes of the authors of the time, the machine was not reliable (?).