Sunday, December 5

The Neperian abacus | The game of science


John Napier.
John Napier.

We wondered last week about the relationship between the meter and the second, our unit of length and our unit of time. Well, the relationship could not be more direct: the half-period of oscillation of a pendulum with a 1 meter thread is almost exactly 1 second (to be exact, the thread must measure 99.4 centimeters, as determined by Mersenne in 1664) . In fact, before the classical definition of the meter was imposed as ten millionth part of the quadrant of the terrestrial meridian, first Burattini and later Talleyrand proposed that the unit of measurement was the length of a “pendulum of seconds”, which is how it is called. whose period of oscillation (the round trip: the tic tac, to understand us) is 2 seconds.

In principle, the pendulum seems to be a much easier way to determine a unit of longitude than to measure the Earth’s meridian. Why do my astute readers believe that the second criterion was chosen?

The prolific and visionary Tito Livio Burattini (1617-1681) failed to impose his system of units based on the second, but he did impose the name “meter” for the unit of length. His contributions were not only theoretical, as he also invented and built ingenious devices, such as an aircraft (which managed to fly with a cat as a crew member) and a calculating machine.

As for last week’s ice cube, when it floats it displaces a volume of water equivalent to its weight, and as it dissolves it will become water, it will occupy exactly that volume, so the water will not overflow from the full glass, as suggested the intuition. In fact, the water level, when it cools, will even drop imperceptibly, since it reaches its maximum density at 4ºC.

And in the problem of water and wine, it is easy to see, if you focus properly, that there will be the same amount of wine in the glass of water as there will be water in the glass of wine, since at the end of the operation there is in each glass the same amount of liquid as before (since we remove a tablespoon and add another); therefore, the amount of wine in the first glass is equal to the amount of water that is lacking, and vice versa.

My astute readers will not fail to appreciate the similarity of these two aquatic puzzles, which apparently have nothing to do with each other.

Abacus and logarithms

Going back to Burattini, his pioneering calculating machine was based on the Neperian abacus, which is a good excuse to talk about Napier.

The great Scottish mathematician John Napier of Merchiston (1550-1617) is known, above all, as the father of logarithms, that providential invention which, as Laplace said, doubled the lifespan of astronomers by greatly simplifying complicated astronomical calculations. Simplifying the calculations was one of Napier’s main concerns, who, in addition to making the first tables of logarithms, popularized the use of the comma to differentiate the decimal part from the integer part of numbers and devised the abacus that carries his Name.

The Neperian abacus consists of a board with a flange and ten rods whose height coincides with that of the board, so that they fit perfectly into it. The left edge of the board is divided into nine squares numbered from 1 to 9. The rods are also divided into nine squares; In the upper boxes of the rods are, respectively, the numbers from 0 to 9, and the other eight boxes of each rod are divided by two diagonals into two parts, numbered as seen in the figure.

Neperian abacus
Neperian abacus

And in the following figure we see a simple example of the use of the abacus, from which its operation can be deduced, as well as its relationship with logarithms. It can?

Using the abacus
Using the abacus

Carlo Frabetti is a writer and mathematician, member of the New York Academy of Sciences. He has published more than 50 popular science works for adults, children and young people, including ‘Damn physics’, ‘Damn maths’ or ‘The great game’. He was a screenwriter for ‘La bola de cristal’.

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