A History of Pi Page 14
The Bernoullis are the most distinguished family in the history of mathematics. Founded by Jean I’s father Nicolaus (1623-1708), this family produced a veritable stream of brilliant mathematicians, and it is still going strong in Switzerland today. It is difficult for the layman to keep the Bernoullis apart without a genealogical chart, especially since they came mostly in Jeans, Jacqueses, Nicolauses and Daniels, and the confusion of their first names is compounded by their versions in French, English, German and Latin (Jacques = James = Jakob = Jacobus; Jean = John = Johann = Ioannus). Euler formed a lifelong friendship with the three brothers named above, and in the following we shall omit the Roman numerals indentifying them in the dynasty. In 1725 Nicolaus and Daniel went to St. Petersburg, then capital of Russia, where an Academy of Sciences had been founded a few years earlier, and in 1727 Daniel arranged for Euler to obtain a somewhat humble position there as well, but after Daniel left in 1733, Euler became the Academy’s chief mathematician (Nicolaus had drowned a year before Euler’s arrival).
The Academy had established a scientific journal, the Commentarii Academiae Scientiarum Imperialis Petropolitanae, and almost from the very beginning Euler contributed to this as well as to other journals. Not only did the editors of the Petersburg Commentaries have no shortage of material as long as Euler was alive, but it took them 43 years after his death to print the backlog of mathematical papers Euler had submitted to this journal. Asked for an explanation why his memoirs flowed so easily in such huge quantities, Euler joked that his pencil seemed to surpass him in intelligence.
In the late 1730’s, Russia’s government became infested with more intrigues than is ususal even for that country, and after the death of Tsarevna Anna in 1740, a series of rapid changes of power boded no good for the Academy. It was probably for this reason that Euler accepted an invitation by Frederick II of Prussia to join “my” (as Frederick used to say) Academy in Berlin.
Frederick II, surnamed by the Germans the Great, was the father of Prussian militarism. His father Frederick I had built up the army that was to be the scourge of Europe for the next 200 years, and Frederick II converted Prussia into a military camp. He was to become the idol of Josef Goebbels, who in 1944 and 1945 raved that Frederick had won the Seven Years’ War even after the Russians had occupied Berlin. Ich bin der erste Diener des Staates, said Frederick, I am the first servant of the State. After the mediaeval Church and the Sun Kings, Frederick had discovered a new horror to be let loose on Europe, the State with a capital S; and to this day many a German scheisst in his pants on hearing the sacred word.
Euler was by this time the undisputed leader of European mathematics, and he was received with great honors at the Prussian court. At one of the receptions at Potsdam, the king’s mother asked why he would not make conversation, but only answered her questions with a monosyllabic “yes” or “no.” “Madam,” answered Euler, “I have arrived from a country where they hang those who talk.” How times have changed! In today’s Russia they do them to death in forced labor camps or lunatic asylums.
Euler spent 25 years at the Prussian Academy, but he was not happy there. Frederick’s interest in science was limited to warfare and the prestige of the Academy, which he lowered by using it for publishing his own writings on political and military subjects. In these, he preached Voltaire’s enlightenment, while simultaneously practicing oppression. As for Euler, he ordered him about, giving him such scientific tasks as checking out the water supply in the royal palace at Potsdam.
Meanwhile in Russia, Catherine II, called the Great for somewhat similar reasons as Frederick, had clawed her way to the Russian throne. Except for her uninhibited sexual enjoyment of her many lovers, she was a rather typical forerunner of the Soviet Tsars, mercilessly centralizing her power, increasing the realm of Mother Russia by brutal conquest, paying lip service to the French enlighteners while practicing cruel oppression, and maintaining her power by all means, including the murder of her husband and later her son. She, too, wanted the prestige of having Europe’s most brilliant mathematician in her Academy, to whose history he had so vigorously contributed a quarter of a century earlier, and she gave the Russian ambassador to the Prussian court instructions to meet any conditions that Euler might pose. Relations between Euler and Frederick became more and more strained, and Euler decided to accept the Russian offer. When it came to money, such unworldly men as Newton or Beethoven could drive a very hard bargain, and Euler, who was never unworldly to begin with, presented the Russian ambassador with the following conditions: He was to be Director of the Academy, with a salary of 3,000 roubles per annum; his wife to receive a pension of 1,000 roubles per annum in case of his death; three of his sons to be given good positions in St. Petersburg; and the eldest son, Johann Albrecht, to become secretary of the Academy. Mindful of Catherine’s instructions, the Russian ambassador accepted, at least, one may assume, after catching his breath.
After reminding Frederick (who refused even to discuss the matter) that he was a Swiss citizen, Euler returned to Russia in 1776, never to see his native Basel again. In 1771 he lost the sight of his remaining eye, but dictating first to his children and later to a secretary specially invited from Switzerland for the purpose, the flood of his research and publications continued unabated.
Beethoven wrote the IXth symphony when he was already completely deaf; Bedrich Smetana, too, wrote the cycle of symphonic poems My Country in utter deafness. Euler, totally blind, wrote many works such as his famous treatise on celestial mechanics, Theoria motus planetarum et cometarum, in which he tackled the three-body problem and for the first time introduced as the center of planetary motion not the sun, but the center of mass of the sun and the corresponding planet. How can a blind man write mathematical treatises that are none too easy to follow for us seeing mortals?
Euler had a phenomenal memory. Plagued by insomnia one night, he calculated the 6th powers of the first 100 integers in his head, and several days later he could still remember the entire table.
His uncanny ability for numerical calculations is illustrated by his refutation of Fermat’s conjecture that all numbers of the form 2 to the power 2n plus one are primes.
Not at all, said Euler in 1732; 2 to the power 25 plus one makes 4,294,967,297, and that is factorable into 6,700,417 times 641.
* * *
AFTER all this, it is surely not surprising that Euler delivered formulas for π by the truckload. And the truckload was usually quite casually attached to a trainload of more important items. Take, for example, one of the many formulas Euler derived for the square of π.
The series of inverse squares
had baffled mathematicians for decades. Gottfried Wilhelm Leibniz (1646-1716), co-inventor of the calculus, had been unable to sum it, and so had many lesser mathematicians. Jacques Bernoulli I (1654-1705), uncle of the three brothers whom Euler had befriended, proved the convergence of the series, but could not sum it either; his brother Jean I wrote that in the end he “confessed that all his zeal had been mocked.” But Euler, in 1736, solved the problem in his stride. The series
was already known to Newton. Euler substituted x2 = y and regarded the equation
as an equation of degree infinity, obtaining (for y ≠ 0)
But the roots of the equation (3) are 0, ±π, ±2π, ±3π,…, and therefore the roots of (4) are π2, (2π)2, (3π)2,… (zero has been excluded above). Now Euler knew from the theory of equations, a branch of higher algerbra that is also heavily marked by his footprints, that the negative coefficient of the linear term [+1/3! in (4)] is the sum of the reciprocal roots of the equation, and hence
or
which solves the problem that had baffled all the others, and gives a series for π2 into the bargain. But there is no stopping Euler now. Repeating the procedure for the cosine series, he finds
and subtracting (5) from twice the sum (6), he obtains
Generalizing the summation to any even power of reciprocals, i.e., considering the series of terms 1/j
(j = 1, 2, 3,…), Euler found a general formula involving Bernoulli numbers and gave special cases for
down to
These results are given in Euler’s famous textbook Introductio in Analysin infinitorum (1748), the book that standardized mathematical notation almost as we use it today. The symbols π, e, i, Σ, ∫, f(x) and many others are all souvenirs of Euler (though not necessarily of the Introductio).
To calculate the logarithm of π, Euler found infinite products for even powers of π, e.g.,
This is not all, but it will do as a sample of this particular truckload.
Machin’s neat little trick with the arctangents (here) turned out to be a pebble of another Eulerian truckload. Euler derived the formulas
and
and this gives rise to any amount of relations for π; for example, if the odd numbers are substituted for a, b, c,…, we obtain
All of these formulas depend on a series for the arctangent, and Euler found one that converged more quickly than any other:
where
Using Machin’s stratagem in the form
and evaluating these two terms by (14), Euler calculated π to 20 decimal places in one hour!
* * *
THESE are but a few examples of the many expressions that Euler found for π; they also included infinite products and continued fractions. So thoroughly did Euler deal with the problems (incidentally) associated with π that no one after him ever found a better way of calculating its value, and he may be said to have finished off its history as far as numerical evaluation is concerned. A few more continued fractions were found after Euler by Lambert and others, and Laplace found an ingenious new approach which we shall consider in the next chapter; but none of these yields the numerical value of π faster than Euler’s method (which includes Machin’s series as a special case).
Euler’s Theorem as first stated in Vol. 1, Chapt. 7, Section 138 of the Introductio (1748). At that time, Euler was still using i to denote infinity; only later did he introduce ∞ for infinity, and i for the square root of –1.
But if Euler finished off one chapter in the history of π, he also started another. What kind of number was π? Rational or irrational? With each new decimal digit the hope that it might be rational faded, for no period could be found in the digits. There was no proof as yet, but most investigators sensed that it was irrational. However, Euler asked a new question: Could π be the root of an algebraic equation of finite degree with rational coefficients? By merely asking the question, Euler opened a new chapter in the history of π, and a very important one, as we shall see. He was also the one who started writing it, for later investigations were based on one of Euler’s greatest discoveries, the connection between exponential and trigonometric functions,
Euler discovered a long, long list of theorems. They are known as “Euler’s theorem on…” and “Euler’s theorem of…” But this one is simply known as Euler’s Theorem.
Euler also laid the foundations for the investigation of the irrationality, and later the transcendence, of π by deriving the continued fractions
which later formed the starting point of Lambert’s and Legendre’s investigations.
All of these were already contained in Euler’s Introductio in analysin infinitorum in 1748. In 1755, already blind, Euler wrote a treatise entitled De relatione inter ternas pluresve quantitates instituenda, which was published ten years later (such was the backlog of papers showered on the Petersburg editors by the blind genius). Here he wrote “It appears to be fairly certain that the periphery of a circle constitutes such a peculiar kind of transcendental quantities that it can in no way be compared with other quantities, either roots or other transcendentals.” (Unde sententia satis certa videtur, quod peripheria circuli tam peculiare genus quantitatum transcendentium constituat, ut cum nullis aliis quantitatibus, sive surdis sive alius generis transcendentibus nullo modo se comparari patiatur.)
As always, Euler was right. But it took another 107 years to prove his conjecture.
And with this we leave the old wizard. Laplace is one of the next whom we shall meet; it was he who told his students, Lisez Euler, lisez Euler, c’est notre maître à tous. Read Euler, read Euler, he is our master in everything.
15
THE MONTE CARLO METHOD
Tremblez, ennemis de la France,
Rois ivres de sang et d’orgeuil!
Le peuple souverain s’avance,
Tyrans, descendez au cerceuil!77
French revolutionary song of Lazare Carnot’s time.
PROBABILITY theory is the mathematics of the 20th century. Its history goes back to the 16th century, but not until the present century did physicists and engineers fully realize that nature and the real world can be described exhaustively only by the laws governing their randomness. What physicists had considered exact until relatively recently, turned out to be merely the mean value of a much more impressive structure; and mean values can be very misleading. (“Put one foot in an ice bucket, and the other in boiling water; then on the average you will be comfortable.”) Strange to relate, even as brilliant and recent a physicist as Albert Einstein regarded the probabilistic laws of quantum mechanics as testimony to our ignorance rather than as a valid description of the laws of nature.
The beginnings of probability theory go back to the Liber de ludo aleae (The book of games of chance), written about 1526 by Gerolamo Cardano (1501-1576), though not published until 1663. Cardano, of cubic equation fame (here), was not only a mathematician, engineer and physician, but also a passionate gambler. Until the advent of the kinetic theory of gases in the 19th century, probability theory was rarely applied to anything else but gambling. The main contributors to its development were Jacques Bernoulli I (1654-1705, author of Ars conjectandi), Pascal, De Moivre, Euler, Laplace, Gauss and Poisson (1781-1840), followed by a large number of mathematicians in the 19th and 20th centuries.
The number π appears in probability theory very frequently, as it does in all branches of higher mathematics; but nowhere is its appearance more fascinating than in a problem posed and solved by George Louis Leclerc, Comte de Buffon (1707-1788). Buffon (as everybody calls him) was an able mathematician and general scientist, who shocked the world by estimating the age of the earth to be about 75,000 years, although every educated person in the 18th century knew that it was no older than about 6,000 years. Among his exploits is a test of one of Archimedes’ supposed engines of war used in the defense of Syracuse. As told by Plutarch, the story includes a plausible description of the action of Archimedes’ cranes and missile throwers, but by the Middle Ages, it had grown into a much exaggerated legend, and the Book of Histories by the Byzantine author John Tzetzes (ca. 1120-1183) repeats the story with many embellishments, such as the statement that Archimedes had burned the Roman ships to ashes at a distance of a bow shot by focusing the sun’s beams onto the Roman fleet. The story (which is not contained in Plutarch’s description) has persisted in many books down to our own days. Buffon, a man of considerable means and spare time, decided to test the feasibility of such a machine. Using 168 flat mirrors six by eight inches in an adjustable framework, he was able to ignite wooden planks at a distance of 150 feet, and he satisfied himself that Archimedes’ alleged exploit was feasible. He did not, however, satisfy posterity, since the Syracusans would hardly have had the same leisure to focus 168 beams, nor would the Roman ships floating on the sea have held as still as Buffon’s beams on the ground.
But back to Buffon’s problem involving π. The problem which he posed (and solved) in 1777 was the following: Let a needle of length L be thrown at random onto a horizontal plane ruled with parallel straight lines spaced by a distance d (greater than L) from each other. What is the probability that the needle will intersect one of these lines?
We assume that “at random” means that any position (of the center) and any orientation of the needle are equally probable and that these two random variables are independent. Let the distance of the center of the ne
edle from the nearest line be x, and let its orientation be given by ϕ (figure a). Since x is measured from the nearest line, we need only consider a single line, because the others involve only repetition of the same solution.
Buffon’s problem.
It is obvious from the figure that the needle will intersect a line if and only if
The problem is therefore equivalent to finding the probability
To find this probability, use the plane of rectangular coordinates x, ϕ, and consider the interior of the rectangle OA (figure b) whose points satisfy the inequalities
These are the intervals of possible values of x and ϕ, and therefore any point inside the rectangle OA corresponds to one and only one possible combination of position (x) and orientation ( ϕ ) of the needle. Since all such combinations are equiprobable, and the area of the rectangle represents the sum total of all possibilities that can arise (because, not quite beyond reproach, we regard this area as made up of all points inside it). However, not all of these possibilities will result in an intersection of the needle with a line; such an intersection, as we have found, will take place only under the condition (1), that is, for positions and orientations corresponding to points lying below the curve x = ½Lsin ϕ in figure b, so that the sum total of possibilities resulting in the intersection by the needle is given by the area under this curve. If, then, probability is the ratio of the number of favorable, to the number of possible, events under given conditions, the probability of intersection is given by the ratio of the shaded part to the entire rectangle OA in figure b, that is, the required probability (2) is