Π pi... the incorrigible.

 

 

When describing the construction of the huge temple of Solomon[1] king, around 950 b.C., the Bible returns it with a barely accurate value of three : but is certain that it was known already 5.000 years ago, as the ancient Egyptians were able, thanks to it, to draw and to measure the course of the Nile river, whose regular and devastating floods were twisting the geography of the local sites.

However it wasn’t yet named until 1706 when the mathematician William Jones, establishing his value equal to 3,14159 periodic, called it π – maybe the initial letter of ‘perimeter’ which in Greek language sounds περιμετρος – and Euler[2] adopted the symbol in 1737, so rapidly spreading it as a standard.

Since the beginning described as a numerical ratio, constant and mysterious, between the circumference and its diameter, for dozens of centuries π was singled out as 22/7. You can even say that the most ancient definition of π – able as it was said ‘to link the earth to the sky’ – marked the beginning of the Geometry too, a science not requiring a priori theories but effectively conjugating the abstract figures with the livable reality. The critical issue of the borders – always forebonding of risky enmities – at last did find a solution with the science of Geometry, thanks to the areas and solids measurement which well represented Countries and properties to be defended. If a flood of Nile, when receding had changed the geographic feature, the technical measurements before the event returned the reality and a social peace : but that ratio, so constant and ‘unfinished’ and also connecting every bend, halph-spheric then, of the river with its diameter was amazing any incredulous technician in the ancient Egypt.

More over, they started when reporting to their seniors, and the seniors whispering to the highest officials, and up to the divine Pharaoh, why in that mysterious ratio any string of numbers could be included, any formula and even complex simulation were included, without no explanation at all…

Reality was incredibly of a divine lowered among humans and hidden, although present - who know how long and what other forms - incomprehensible but representable, therefore a mistery and a treasure that the society of the Pharaohs was able to to well sip in the centuries, so granting submission and governing too.

In 1906 as a matter of facts, the discovery of a written page was finally enlightening about the goals achieved using the rule of the lever to calculate areas, volumes and centres of gravity : it was the ‘Letter about the method’[3] which Archimedes[4]  of Syracuse, a rare scientist and devoted to the monarch and tyrant Gerone II[5] who couldn’t do without him, wrote to Eratosthenes[6] of Alexandria in Egypt, director of the richest Library, right where Archimedes had studied thank to the subsidy of Gerone.

Studying in Egypt, Archimedes[7] could get to analytical tests, datas and very ancient and precious documents which opened the way to a new method, the one to demonstrate a theory by mechanical entities : covering a spiral, for example, let measuring its perimeter, nearest to a circumference which so linked with any curved line and even to the straight, which is a special case of the curved one. And the point itself, then ? Try to measure it without at least you can imagine to zoom it.

We still wonder that Archimedes couldn’t take advantage of algebraic or trigonometric information, because he derived each of his own results by only geometrical media, then surfaces and solids which allowed him to get to concluding theses, and so finally establishing the ‘infinitesimal calculus’ and the ‘limit’ : the circumference was certainly formed by an infinity of equidistant points from a single centre, but it could be assimilated to an ever increasing number of polygons with equal sides, i.e. very regular and in increasing number, and themselves stackable to the circumference, so prooving measurable and no more mysterious indeed.

A proved scientific nature of the motion became preferable to the incomprehensible and unrelated immobility of an abstract, and divine, a priori. 

We can say that π has been able to split its audience between aggressive theorists, who always point out issues still open, and on the other hand users, simply thankful for the service offered by this humble value.

Will this be how we govern ?

 

                                               Marina Bilotta Membretti / Cernusco sul Naviglio - May 31, 2020 Sunday

 

 

[1] J.J.O’Connor / Honorary Senior Lecturer in ‘Computational Algebra’ e E.F.Robertson / Professor Emeritus of Mathematics at ‘School of Mathematics and Statistics’ - University of St. Andrews (Scotland) are the authors of ‘A history of Pi’ / Agosto 2001 : they cite a verse from the Bible which is both in ‘Book I King 7, 23’ and in ‘Book II Chronicles 4, 2’.

[2] Leonhard Euler (1707 Basilea, Swissland –  1783 Saint Petersburg, Russia), mathematician and physicist, often known as Euler.

[3] The page was at first titled ‘The method of Archimedes of the mechanical theorems, to Eratosthenes’ by the researcher Heiberg who found it at Istanbul, mixed to other papers : the paper is also the Premise to the Treatise of Archimedes ‘On spirals’.

[4] Archimedes ( 287 b.C. – 212 b.C.) was born and dead at Syracuse, in Sicily but he stayed for a long time in Alexandria of Egypt to finish and perfect his studies.

[5] He too from Syracuse (308 b.C. – 215 b.C.), Gerone seized power with the support of the popular classes, but he didn’t twist any law or preceeding institution : so that any ‘tyrant’, in a city-state like Syracuse established by former Greek colonists, was accepted in the Greece of VII and VI century b.C. and by the authorities of Athens too.

[6] Eratosthenes of Cyrene (Cyrene, 276 b.C. – Alexandria of Egypt, 194 b.C.) was a mathematician and a famous geographer : he was appointed as Librarian of the Library in Alexandria of Egypt, the most important Library of the ancient time.

[7] I’d like to cite the interesting Exhibition 2019-2020 ‘Archimedes at Syracuse’ with the machinery of Archimedes but wooden reconstructed to be ‘touched’ and very ‘clapped’ indeed, at the ‘Galleria Civica Montevergini’ in Syracuse-Ortigia, attended by Giovanni Di Pasquale and with the scientific advice of G. Voza e C. P. Voza.