NON-TRIVIAL SOLUTION: How to learn to add numbers automatically?

Pieter Brueghel the Younger. The Village Lawyer or The Tax Collector’s Office. 1617

 

ATTENTION — QUESTION!

 

In 1642, the royal quartermaster of Normandy was constantly engaged in complex and responsible work — the calculation of collected taxes.

There were no computers and smartphones back then, so you had to add manually. It was hard, long and accompanied by mistakes.

How did his son help him in this difficult work?

The answer is a little later.

 

FIRSTLY USING FINGERS

 

Recently, in this section, it was already written about how difficult it was to come up with a convenient system for writing numbers. But after all, numbers are written for a reason —  you have to perform arithmetic operations with them. At the beggining — at least add and subtract, the rest can be done later.

The first calculating machine used by man, as it turned out, grew right on him — these were his own fingers. Counting using fingers, in all likelihood, was already familiar to our primitive ancestors. In Ancient Rus’, units were even called “fingers” — clearer than ever…

Miklouho-Maclay told us how the Papuans counted — they bent their fingers and said, “Be-be-be-be-ibon-be” (“one hand”) — “be-be-be-be-ibon-ali” (“two hands”) — “be-be-be-be-samba-be” (“one leg”) — “be-be-be-be-samba-ali” (“two legs”). Then they had to call another Papuan…

In Homer’s Odyssey you can find the verb “to five”, meaning “to count”. And in the Malay language, the word “lima” means both “hand” and “five.” Most likely, it is the precise reason because the decimal number system is adopted everywhere, that there are ten fingers on two hands.

More complex systems of finger counting also arose. The Irish monk Bede the Venerable back in the 7th century wrote a treatise On Numbering, in which he outlined ways to represent numbers from one to a million using fingers. Arithmetic textbooks have been quoting him for centuries…

 

TAG ACCOUNTING

 

The human hand does not have enough fingers for counting the large numbers… People began to make serifs on wooden tags — they could be counted later. And if you firstly apply six serifs, and then eight more, they can be counted, and you get fourteen — this is addition!

The oldest mention of serif counting, which has survived to our time, is dated 1350 BC — on the bas-relief of the temple of Pharaoh Seti I in Abydos, the god of wisdom Thoth marks the duration of the reign of this pharaoh on a palm branch in this way.

For a long time, tags were used to record taxes paid — long lines were applied to a piece of wood and than it was split along, so that one copy was kept by the tax official, and the other remained with the taxpayer, playing the role of a payment receipt.

In England, taxes were taken into account using this method almost until the end of the 17th century. And when the old tax obligations were liquidated, these tags were collected in the yard of the Treasury and set on fire. The fire turned out to be so big that the treasury also burned down — along with a sample of a measure of length embedded in its wall, so that the British can only guess what is the actual length of the English foot…

 

KNOTS

 

There was another way to count — to tie knots on belts or ropes, and then count them. This method was used in Ancient China, and in the Persian Empire, and in Ancient India, which has not invented its famous numbers yet, and in the Inca Empire in South America.

Herodotus tells that during the invasion of Scythia, King Darius handed a rope with 60 knots to the guards of the bridge across the Danube and ordered, “Every day, untie one knot and wait for us: if you untie all the knots and we would not have returned, you can wait no longer.” By the way, the king barely had time…

The Incas called counting ropes kuiru, and in Peruvian cities, before the Spanish invasion, the city treasurer was called kuiru komouokuna, which in translation meant knot official. Reffering to the amount of things Spanish people looted there, taxes were collected regularly…

However, these knots had even greater possibilities — most recently, the English researcher Sabine Hyland proved that ropes with knots not only helped the Incas memorize and sum up numbers, but were also a fairly developed writing system.

 

ABACUS, SUAN-PAN, SOROBAN, CALCULATING FRAME…

 

A notch on a wood has an unrecoverable drawback — if you already made it, you can’t remove it. For tax documents, this is even useful, but for other calculations, not so much. Whether it’s a matter of pebbles —  if you need, you can move it, if you don’t need, so you remove it. Or you can string them on a knitting needle and throw them back and forth…

Thus, in ancient times, the abacus appeared. It was a board with grooves where balls were placed. Only thousands of years after the appearance of the first abacus in Babylon, the Egyptians came up with the idea of ​​stringing pebbles or wooden balls on a wire — just like in our calculating frame.

The abacus was invented independently in Eurasia and America. The Aztecs made an abacus from corn kernels strung on a wire, and a number of such wires were pulled over a wooden frame. The Incas also came up with their own abacus, which they called yupana — you can’t knit knots all of the time…

The Chinese abacus suan-pan is divided by a partition into two parts — in one there are ten bones on a twig, in the other, there are two (they are used as fives). The Japanese soroban is almost the same, but for the fives there is one bone instead of two. They are still being studied in Japanese schools.

Well, everyone knows Russian calculating frame — this is also an abacus. I also found virtuosos like my grandfather, an accountant, who clicked the bones of accounts so quickly that it was impossible to keep track. I also remember foreign guests who asked very much to get them a “Russian computer” and were very happy when the calculating frame was found for them. Now it probably wouldn’t work…

 

POPE’S ABACUS

 

When, in The Master and Margarita, Woland explains to Berlioz why he came to Moscow, he says that he was invited to analyze the found manuscripts of the 10th-century warlock Gerbert of Aurillac. It is strange that they call him that — after all, generally speaking, he was just a pope.

Gerbert was born into a poor family, as a child he was generally a shepherd, but thanks to his abilities he was able to break through to the top of church power. And all sorts of not very good rumors circulated about him because he could do what was then considered almost a miracle — multiply and divide large numbers.

He invented a new kind of abacus and came up with the rules of calculation, which he set out in his mathematical writings. After more than 200 years, the great mathematician Fibonacci wrote that there are three ways to count — counting with fingers, counting with numbers, and counting on Gerbert’s abacus.

It got to the point that mathematicians who made calculations using the abacus were called not abacusists, but gebercists — just in his honor. The methods of counting developed by him led to the fact that the great rivalry between the abacusists and the algorithmicists (this way people who calculated using the widespread Arabic numerals were called) ended only in the 17th century.

 

PROBLEMS OF FINANCIALISTS

 

When the algorithmicists nevertheless defeated the abacusists with the help of the wisdom of counting through Arabic numerals, taught to us at school, they understood the same thing that we all did in the lower grades — performing these actions is not difficult, but tedious and fraught with errors and fatigue.

It would be great to find a way to somehow mechanize these boring procedures, to come up with a mechanical assistant that would be more convenient than an abacus. Moreover, there were actions that were more difficult than addition and subtraction — multiplication and especially division.

In 1614, the Scottish Baron John Napier invented logarithms and thus was able to reduce multiplication and division to addition and subtraction. Logarithm tables appeared, simplifying counting, and a little later, Wingate and Oughtred invented a counting device — a slide rule.

But the accuracy of a slide rule is, by definition, low — three significant numbers are already a lot for it. For people counting money, and quite large amounts, this was completely unsuitable — their accounts had to converge to the smallest coin.

 

 

GENERAL INSPECTOR

 

A high-ranking official under Louis XIII, authorized by His Majesty to levy taxes in Normandy also experienced these difficulties — the taxes were large, varied and obscure, and most importantly — so heavy that they led to frequent riots.

The inspector himself also managed to quarrel with the king, or rather, with the almighty Cardinal Richelieu, participating in protests that arose due to the late payment of state rent. Richelieu was so angry that he threatened him with the Bastille and forced him to hide.

Fortunately, at the court performance that Richelieu attended, the daughter of the disgraced official brilliantly played the role, and when, after the performance, she began to ask Richelieu for her father, he had mercy, but set an indispensable condition — the forgiven rebel should have served him.

Having received not only forgiveness, but also a rather high position, he immediately plunged into work, sometimes sitting at nights over tax documents in unsuccessful attempts to find an error in the calculations, due to which the balance did not converge. He was assisted in this work by his 17-year-old son.

 

CAPABLE YOUTH

 

He taught his son by himself from early childhood, instead of standard education, introducing him to knowledge according to his own system. In particular, he planned to teach mathematics to him only from the age of 15-16, fearing that it would interfere with the child’s study of Greek and Latin.

Nevertheless, the child, whom no one was going to teach geometry (moreover, his father forbade it directly), somehow began to draw various geometric shapes on the floor with charcoal and study them, calling the line a “stick”, and the circle “ringlet” — he was not told the correct names.

The father accidentally caught the child doing this. And he was simply amazed — he, without even knowing the names of geometric shapes, independently proved not the simplest 32nd theorem of Euclid, which states that the sum of the angles of a triangle is equal to two right angles. Can you do it?

After that, the father canceled all prohibitions on the study of mathematics due to senselessness, and the son began to advance in the knowledge of this science so quickly that at the age of 17 he published his first printed treatise On Conic Sections, in which he proved a number of interesting theorems.

 

NEW IDEA

 

Of course, such a mathematically gifted assistant could provide his father with sensitive assistance. But there was so much work, and it was so monotonous, that a talented child thought: can it be simplified reducing to purely mechanical actions, so that there was no need to think?

To do this, it was needed to come up with a new device — a mechanical device that itself performs the summation operations.  The answer to the question asked at the beginning contains in it. Rather not — it is completely unclear what such a machine should be and how it can differ from those already available.

The abacus and other counters are also such devices, although they are simple. When adding two numbers, you simply throw the corresponding number of bones bit by bit: put aside three, add two — so you added them, and it turned out to be five. What about six plus eight? Where to put the extra ten?

Of course, we know how to do this — we need to throw one more bone in the row that corresponds to the next rank. This must be done independently, manually, and any such operation increases the possibility of error – you can mix it up, forget it, skip it.

 

ATTENTION — CORRECT ANSWER!

 

You just need to come up with a device that in such cases automatically adds a unit to the next bit of the sum, without waiting for us to do it.

If this succeeds, it will be possible to come up with a machine into which you simply enter the summable numbers, and it already gives the result itself!

 

GEAR

 

The official’s son had ideas on how to do this, but it was difficult to implement them. At that time, precision mechanics was just emerging, the masters were inept and could not cope with many things. Therefore, he had to make the necessary part himself, and simplify it so that others could do it.

He was only 19 years old when he made the first copy of such a machine, on which even those who did not know the rules of arithmetic could perform all four actions.  It did not suit him, and he began to make new modifications — from wood, from ivory, from ebony, from copper…

Only at the age of 23 did he finally make a device that suited him. It was a brass box on which several circular scales were mounted. The extreme right scale was divided into 12 parts, the next one into 20, the rest all into 10 — because the livre had 20 sous, and the sous had 12 deniers.

And the main task, the solution of which was necessary for work is the mechanism for transferring tens — was ingeniously solved by a mechanism of gear wheels. When each of the wheels turned almost to the end, a special mechanism immediately turned the next wheel by one division.

 

RESULT

 

He presented one of the first successful models of his device to the Chancellor of France, Séguier, who became interested in a useful device. On May 22, 1649, he received a royal privilege, which gave him priority in the invention and the right to manufacture and sell these machines.

He made a certain number of these machines and sold a significant part of them — that means there were buyers! 8 of such devices have survived to our time and are now exhibited in museums, usually in the history of computing — this is, in a sense, the first computer!

In 1652, he sent his car to the young Swedish queen Christine, famous for her intelligence, eccentricity and scholarship, and then forever move away from computer science. He had enough other activities and hobbies — we all learned the law of physics named after him in the sixth grade!

Yes, I kept forgetting to mention his name — this is Blaise Pascal, who during his short 39-year life managed to make a number of great discoveries in mathematics and physics. Now it is clear to everyone that one of the most common programming languages named after him for a reason.

 

A PAIR OF GENERALIZATIONS

 

There were different number systems — quinary, and sexagesimal, and vigesimal… But it was decimal that survived, and the reason is simple — everyone has ten fingers on their hands.

Similar methods of counting arose almost simultaneously on different continents. This phenomenon is called convergence — similar problems are being olved in similar ways, even if they are solved completely independently.

If a poor shepherd could become a pope, he obviously must be a capable person. And such people are often engaged not in one thing, but in everything that interests them — for example, Pope Sylvester II (he took such a name during enthronement) could understand not only in theology…

Was Pascal the first to come up with a method for passing tens? No, except perhaps the most famous — about 20 years before Pascal, this mechanism was invented by the German professor Schickard, and even Leonardo da Vinci has something similar. The struggle for priority is a complex and confusing matter…

From the age of 9–10, like the young Pascal, I counted reports for my mother, a district doctor who was obliged not only to treat, but also to hand over a lot of complex and incomprehensible papers. It was a chore, but I was glad that I was helping my mother. Help your parents — you will never lose it!

 

All illustrations are from open sources