A History of Pi Read online

  Title page of the third edition of the Principia (1726) as reprinted in 1871.

  Newton’s achievement in discovering the differential and integral calculus is, in comparison, a smaller achievement; even so, it was epochal. As we have seen, the ground was well prepared for its discovery by a sizable troop of pioneers. Leibniz discovered it independently of Newton some 10 years later, and Newton would not have been the giant he was if he had overlooked it. For Newton overlooked nothing. He found all the big things that were to be found in his time, and a host of lesser things (such as a way to calculate π) as well. How many more his ever-brooding mind discovered, we shall never know, for he had an almost obsessive aversion to publishing his works. The greatest scientific book ever published, his Principia, took definite shape in his mind in 1665, when he was 23; but he did not commit his theories to paper until 1672-74. Whether he wrote them down for his own satisfaction or for posterity, we do not know, but the manuscript (of Part I) lay in his drawer for ten more years, until his friend Edmond Halley (1656-1742) accidentally learned of its existence in 1684. Halley was one of the world’s great astronomers; yet his greatest contribution to science was persuading Newton to publish the Principia, urging him to finish the second and third parts, seeing them through the press, and financing their publication. In 1687 this greatest of all scientific works came off the press and heralded the birth of modern science.

  EDMOND HALLEY (1656-1742)

  Of his many great discoveries, the greatest was the discovery of the Principia in Newton’s drawer.

  Isaac Newton was born on Christmas Day, 1642, in a small farm house at Woolsthorpe near Colsterworth, Lincolnshire. At Grantham, the nearest place that had a school, he did not excel in mathematics in the dazzling way of the wonderchildren Pascal or Gauss, but his schoolmaster, Mr. Stokes, noticed that the boy was bright. If there was any omen of young Isaac’s future destiny, it must have been his habit of brooding. Going home from Grantham, it was usual to dismount and lead one’s horse up a particularly steep hill. But Isaac would occasionally be so deeply lost in meditation that he would forget to remount his horse and walk home the rest of the way.

  When he finished school, there came the great turning point of Newton’s career. His widowed mother wanted him to take over the farm, but Stokes was able to persuade her to send Isaac to Cambridge, where he was first introduced to the world of mathematics.

  The Manor House at Woolsthorpe. Birthplace of Isaac Newton (1642) and birthplace of the calculus (1665-1666).71

  What if Stokes had not been able to persuade Mrs. Newton? There are many similar questions. What if Gauss’s teacher had not prevailed over Gauss’s father who did not want his son to become an “egghead”? What if G.H. Hardy had paid no attention to the mixture of semi-literate and brilliant mathematical notes sent to him by an uneducated Indian named Ramanujan? The answer, no doubt, is that others would eventually have found the discoveries of these men. Perhaps this thought is some consolation to you, but it leaves me very cold. How many little Newtons have died in Viet Nam? How many Ramanujans starve to death in India before they can read or write? How many Lobachevskis languish in Siberian concentration camps?

  However, Newton did go to Cambridge, where he was very quickly through with Euclid, and soon he mastered Descartes’ new geometry. By the time he was twenty-one, he had discovered the binomial theorem for fractional powers, and had embarked on his discovery of infinite series and “fluxions” (derivatives). Soon he was correcting, and adding to, the work of his professor and friend, Isaac Barrow. In 1665 the Great Plague broke out, in Cambridge as well as London, and the university was closed down. Newton returned to Woolsthorpe for the rest of the year and part of the next. It is most probable that during this time, when he was twenty-three, with no one about but his mother to disturb his brooding, Newton made the greater part of his vast discoveries. “All this was in the two plague years 1665 and 1666,” he reminisced in old age, “for in those days I was in the prime of my age of invention, and minded mathematics and [natural] philosophy more than at any time since.”72 Asked how he made his discoveries, he answered, “By always thinking unto them,” and on another occasion, “I keep the subject constantly before me and wait till the first dawnings open little by little into the full light.” Newton retained these great powers of concentration throughout his life. He succeeded Barrow as Lucasian Professor of Mathematics at Cambridge (1669), and relinquished this post to become Warden of the Mint (1696) and later (1699) Master of the Mint; in 1703 he was elected President of the Royal Society, a position which he held until his death in 1726. In his later years he spent much time on non-scientific activity, but remained as astute a mathematician as ever, amazing men by the ease with which he solved problems set up to challenge him.

  The Brachistochrone. The required curve is a cycloid.

  In 1697, for example, Jean Bernoulli I (1667-1748) posed a problem that was to become famous in the founding of the Calculus of Variations: What is the curve joining two given points (see here) such that a heavy particle will move along the curve from the upper to the lower point in minimum time? The problem is so difficult that it is not, for example, usually included in today’s undergraduate engineering curriculum. It was received by the Royal Society and handed to Newton in the afternoon; he returned the solution the next morning, and according to John Conduitt (his niece’s husband), he solved it before going to bed! The solution was sent to Jean Bernoulli without signature, but on reading it he instantly recognized the author, as he exclaimed, tanquam ex ungue leonem (as the lion is known by its claw).

  * * *

  FOR a giant like Newton, the calculation of π was chickenfeed, and indeed, in his Method of Fluxions and Infinite Series, he devotes only a paragraph of four lines to it, apologizing for such a triviality with a by the way in parentheses — and then gives its value to 16 decimal places.

  Newton wrote this treatise in Latin and it did not appear until after his death in 1742; an English translation appeared earlier in 1737 (printed, the title page tells us, by J. Millan “next to Will’s Coffee House at the Entrance to Scotland Yard”). However, he brooded out the major parts of this and other treatises on the differential and integral calculus (including infinite series, which Newton never separated from the calculus) during the Plague Years 1665-66 at Woolsthorpe. His method enabled him to expand a function, its integral or derivative in an infinite series. Where Gregory had found four series for the trigonometric functions, Newton could choose one that would make the calculation of π as rapid as possible.

  The Gregory-Leibniz series

  was theoretically interesting, as it pointed the way to a completely new approach to calculating π. But for numerical calculations it was practically useless, for its convergence was so slow that 300 terms were insufficient to obtain even two decimal places, and two decimal places were less accurate than 3 1/7, the value Archimedes had obtained 2,000 years earlier. De Lagny, whom we have met earlier as the digit hunter who calculated 117 decimal places, found that to obtain 100 decimal places, the number of necessary terms would be no less than 1050!

  Newton had found a way to calculate the fluxion (derivative) of a fluent (variable), and conversely, to find the Flowing Quantity from a given fluxion (to integrate). He also showed that this amounted to finding the area under a curve (whose equation was given by the fluxion). He thus found (in modern symbols)

  Newton’s method of calculating π.

  or using his discovery of the binomial theorem,

  so that integrating term by term and using (2),

  Substituting x = ½, which makes arcsin(½) = π/6, this yields the series

  which converges incomparably more quickly than the Gregory-Leibniz series.

  That is what most history books say on Newton’s method of obtaining π. In looking up the original work,73 however, we find that Newton used a slightly different method. He considered a circle (along with an hyperbola which need not interest us here) whose equation i
s

  i.e., a circle (see here) with radius ½ centered at y = 0, x = ½. Then the circular segment ADB has area

  where Newton again used the binomial theorem.

  On the other hand, the segment ABD equals the sector ACD less the triangle BCD, and since CD = 1, BD = √ 3/4, Newton found

  On comparing (6) and (7), this yields

  and this is how Newton obtained his value of π (see here). Twenty-two terms were sufficient to give him 16 decimal places (the last was incorrect because of the inevitable error in rounding off). A far cry from the Archimedean polygon, where 96 sides (and extracting square roots four times over) yielded only two decimal places!

  Actually, Newton was calculating something else, and π appeared only as an incidental fringe benefit in the calculation. But the crumbs dropped by giants are big boulders.

  This was one of the calculations he performed during the Plague Years 1665-6 in Woolsthorpe. Later he wrote, “I am ashamed to tell you to how many figures I carried these computations, having no other business at the time.”

  Newton’s calculation of π in The Method of Fluxions.

  * * *

  THE digit hunters of Newton’s time returned to the Gregory series

  which they modified in various ways to accelerate its convergence. The astronomer Abraham Sharp (1651-1742), for example, substituted x = √(1/3), which gave him

  and using this series, he calculated 72 decimal places.

  In 1706, John Machin (1680-1752), Professor of Astronomy in London, used the following stratagem to make the Gregory series rapidly convergent and convenient for numerical calculations as well.

  For tan β = 1/5, we have

  and

  This differs only by 1/119 from 1, whose arctangent is π/4; in terms of angles, this difference is

  and hence

  Substituting the Gregory series for the two arctangents in (14), Machin obtained

  This was a neat little trick, for the second series converges very rapidly, and the first is well suited for decimal calculations, because successive terms diminish by a factor involving 1/52 = 0.04. From (15), Machin calculated π to 100 decimal places in 1706.

  The French mathematician de Lagny (1660-1734) increased the number of decimal places to 127 in 1719, but since he sweated these out by Sharp’s series (10), he exhibited more computational stamina than mathematical wits.

  It was also in Newton’s life time that the circle ratio was first denoted by the letter π. It was used by William Jones (1675-1749), who occasionally edited and translated (from Latin) some of Newton’s works, and who himself wrote on navigation and general mathematics. In 1706 he published (in English) his Synopsis Palmariorum Matheseos: or, a New Introduction to the Mathematics. This work was intended for the Use of some Friends who have neither Leisure, Convenience, nor, perhaps, Patience, to search into so many different Authors, and turn over so many tedious Volumes, as is unavoidably required to make but tolerable progress in the Mathematics. Jones’ introduction of the symbol π in his book strongly suggests that he used the letter π as an abbreviation for the English word periphery (of a circle with unit diameter). The notation is used several times in the book, and the value calculated by Machin is reproduced; to which Jones adds True to above a 100 places; as Computed by the accurate and Ready Pen of the Truly Ingenious Mr. John Machin.74

  However, Jones did not have the weight to make his notation generally accepted. The letter π was first used by Euler in his Variae observationes circa series infinitas (1737). Until that time, he had been using the letters p or c. Once Euler adopted it, it became a standard symbol, as was the case with his other notations.

  * * *

  ON March 20, 1727, Newton died. His body lay in state like that of a sovereign, and he was buried in Westminster Abbey, in a place that had often been refused to the highest nobility.

  NEWTON

  But not only in death did Newton’s countrymen pay him the respect that he deserved. They revered this great man in his lifetime also, a marked difference from the way in which other countries treated their scientists then and in later times. In Italy, Galileo had been broken by threats, and perhaps application, of torture. France’s religious intolerance drove the great mathematician Abraham De Moivre (1667-1754) into exile. Johann Kepler was debased to an astrologer by Rudolf II, who often let him go without his due salary. In the 20th century, Nazi Germany drove Einstein and hundreds of other scientists into exile and murdered Jewish scientists who had not escaped in time. And even while you are reading this, hundreds — perhaps thousands — of outstanding scientists struggle to remain alive in the horrors of Soviet forced labor camps, to which they were sentenced for disagreeing with their government, or for applying to emigrate, or even for no reason at all.

  But England did not fail her greatest son.

  14

  EULER

  He calculated just as men breathe, as eagles sustain themselves in the air.

  FRANCOIS ARAGO

  (1786-1853)

  THE sword that Newton had forged never found a greater swordsman than the unbelievable wizard Leonhard Euler (1707-1783). If Newton was the greatest all-round scientist of all times, Euler was the greatest mathematician (in my personal estimation; many think Gauss greater).

  A mere annotated index of Euler’s works would fill a book far bigger than the one you are now reading; for Euler published a total of 886 books and mathematical memoirs, and his output averaged 800 printed pages a year. On the 200th anniversary of his birthday in 1907, it was decided to publish his collected works in his native country, Switzerland; by 1964, 59 volumes were published, and the entire series is expected to run to 75 volumes of about 600 pages each. And that does not include his voluminous correspondence with the Bernoullis, Goldbach, and other famous mathematicians, which contains more of his brilliant work. A conservative estimate puts the number of these letters at 4,000, of which 2,791 have been preserved; a mere list of these, with an annotation of about two lines each, fills a book of 390 pages.75

  In 1735, Euler lost the sight of his right eye, and in 1771 he went blind completely; but for the remaining 12 years of his life, his output continued unabated, and only death cut off the avalanche of the most prolific mathematician of all times.

  In our own time, much half-baked research is being fired off to scientific journals whose number, let alone contents, can no longer be followed by anyone. The middle of the 20th century has also seen the “saturation bombing” approach to scientific problems: Teams of scientists supported by batteries of computers are hired to attack a particularly important problem in the hope that something of the mass of produced material will work out. It might therefore be concluded from the above account that much of the stupendous output flowing from Euler’s pen must have been second rate.

  Not so. Almost all of Euler’s work was brilliant, and almost all of it was of fundamental significance. True, he was stung by divergent series once or twice, and occasionally one might accuse him of a lack of rigor. But his brilliant recklessness is part of what made him so great. Euler had no time for hairsplitting, because he spent all his time in the thick of where the action was. He did not prove that one is greater than zero or that the circle has no discontinuities (as became the fashion in the 19th century), and had he done so, he might not have had the time or inclination to formulate the principle of least action or to discover the relation between exponential and trigonometric functions. All branches of mathematics abound with Euler’s theorems, Euler’s coefficients, Euler’s methods, Euler’s proofs, Euler’s constant, Euler’s integrals, Euler’s functions, and Euler’s everything else. Any modern textbook shows Euler’s indelible marks in analysis, differential equations, special functions, theory of equations, number theory, differential geometry, projective geometry, probability theory, and all other branches of mathematics, and some of physics (astronomy, strength of materials, mechanics, hydrodynamics, etc.) as well. Not content with that, Euler founded, or co-founded at le
ast two new branches of mathematics: the calculus of variations and the theory of functions of a complex variable.

  From this again it might be concluded that the man was a genius on the borders of insanity, an unworldly, sinister Cyclops.

  Not at all. Contemporaries describe him as a jovial fellow, witty and enjoying life. He was happily married, his household, including children and grandchildren, grew to 13 members of the family, and amidst them he would do his calculations with a child on his lap and a cat on his back. He would roll with laughter at a puppet show and indulge in horseplay with his children and grandchildren, his fantastic mind calculating away at the same time.76

  LEONHARD EULER (1707-1783)

  Leonhard Euler was born in 1707 at Basel, Switzerland. Like Newton, he was not particularly brilliant in mathematics as a child, though he did have an advantage over Newton in that his father, a clergyman, was an amateur mathematician. In 1720, he enrolled at the University of Basel where, intending to enter the ministry, he took theological subjects, Greek, Latin and Hebrew (as a Swiss he was equally fluent in French and his native German). But as we shall see, he had a phenomenal memory, and a mere three languages left him with plenty of spare time. This he used to take physics, astronomy, medicine, and mathematics, the last taught by Jean Bernoulli I (1667-1748). He made the acquaintance of his three sons Nicolaus III (1695-1726), Daniel I (1700-1782) and Jean II (1710-1790), and through them he discovered his true vocation.