A History of Pi Page 12

  Pascal began the study of mathematics at age 12. At age 13 he had discovered the pyramid of numbers known as the Pascal triangle (see here). Before he was 16, he had discovered Pascal’s Theorem (the points of intersection of opposite sides of a hexagon inscribed in a conic are collinear), which became one of the fundamental theorems of projective geometry; by the time he was 17, he had used his theorem to derive four hundred propositions in an essay on conic sections. At 19, he invented a calculating machine, some principles of which survived in the desk calculators of a few years ago. In physics, he contributed to several branches, particularly hydrostatics. But on November 23, 1654, Pascal’s coach narrowly escaped a fall from a bridge, and experiencing a religious ecstasy, he decided to forsake mathematics and science for theology. Only for a short period thereafter did he devote himself to mathematics: To take his mind off the pains of a toothache, he pondered the problems associated with the cycloid (the curve described by a point on a rolling circle). In this brief period of 1658-59, he achieved more than many others in a lifetime, but soon he returned to problems of theology, and in the last years of his life he mortified his flesh by a belt of spikes round his body, hitting it with his elbow every time a thought entered his mind that was not sufficiently pious.63

  The Pascal triangle. Each number is the sum of the two above it. The kth line and the mth (oblique) column give the number of combinations of k things taken m at a time, or the coefficient of the mth term of (x + a)k.

  Chinese version of the Pascal triangle, published in 1303, 320 years before Pascal was born.62

  BLAISE PASCAL (1623-1662)

  This, alas, is not untypical for the mental health of infant prodigies in later life. Few are the Mozarts, Alekhines and Gausses, who are born as wonderchildren and retain their extraordinary abilities throughout their lives. Most men of genius matured slowly; neither Newton, nor Euler, nor Einstein, for example, were wonderchildren. They were not even particularly outstanding in mathematics among their schoolmates.

  One of the things that Pascal’s historic toothache caused was the historic triangle EEK in the figure here, which was contained in Pascal’s treatise on sines of a quadrant. Pascal used it to integrate (in effect) the functions sinn ϕ, that is, he found the area under the curve of that function. From here it was but a slight step to the integral calculus, and when Leibniz saw this triangle he immediately noticed, as he wrote later, that Pascal’s theorems relating to a quadrant of a circle could be applied to any curve. The step that Pascal had missed was to make the triangle infinitely small; but then, all this was a temporary affair started by a toothache. In Boyer’s words, “Pascal was without doubt the greatest might-have-been in the history of mathematics.”64

  The “historic triangle.” Pascal used it to show that DI.EE = RR.AB

  In applying his triangle to number theory, Pascal also derived a formula, which he expressed verbally, and which is equivalent to

  This formula had also been discovered by several other men of this time, and it was to become very important in the history of π. Pascal was only one of many pioneers who prepared the ground for the calculus. Johann Kepler calculated the area of sectors of an ellipse, which he needed for his Second Law (see figure here), and he did this by dividing the circle and the ellipse into thin strips, examining the proportions of corresponding areas. Bonaventura Cavalieri (1598-1647) published his Geometria indivisibilibus continorum in 1635; here he regarded an area as made up of lines of “indivisibles,” and similarly, volumes of indivisible or quasi-atomic areas. By a cumbersome process of comparing the areas to either side the diagonal of a parallelogram, he was able to find the equivalent of (1) for n = 1, and by an inductive process he generalized this for any n. Evangelista Torricelli (1608-1647), a disciple of Galileo and the inventor of the mercury barometer, came very close to discovering the calculus, as did Gilles Persone de Roberval (1602-1675) and Girard Desargues (1591-1661). Pierre Fermat (1601-1665) not only derived the equivalent of (1), but he could differentiate simple algebraic functions to find their maxima and minima. Isaac Barrow (1630-1677), Newton’s professor of geometry at Cambridge, was practically using the differential calculus, marking off “an indefinitely small arc” of a curve and calculating its tangent; but he was a conservative in both mathematics and politics (he sided with the king in the civil war) and worked only with Greek geometry, which made his calculations difficult to follow and his theorems hard to manipulate for further development.

  Kepler’s second law states that the radius vector (SP) of a planet sweeps out equal areas in equal times. Kepler found these areas by analogy with the circle and proportions of corresponding areas.

  Among these and other pioneers preparing the ground for the calculus, there were at least two who used these little bits and pieces of the coming calculus for deriving the value of π; they were John Wallis (1616-1703) and James Gregory (1638-1675). But before we examine their work, it is instructive to visit Japan, where the value of π may also have been derived by a method of crude integral calculus, resulting in an infinite series which was never used in Europe, presumably because the expressions found by Wallis and Gregory were more advantageous and eliminated the need for the Japanese series.

  The evolution of mathematics in Japan was by this time retarded behind that of Europe. In 1722, the Japanese mathematician Takebe was still using a polygon (of 1024 sides) to calculate π to more than 40 places, although he also found series and continued fractions for π, and also for π2. At that time Japanese mathematicians rarely gave a clue how they found their results. Perhaps they kept their methods secret, as Tartaglia kept his solution of the cubic equation secret; perhaps they did not have convincing proofs and published “cookbooks” as was the custom in Europe in the Middle Ages. But drawings such as the ones on the opposite page suggest67 that they may have found a series for π as follows.

  From a book by Sawaguchi Kazayuki (1670),65 showing early steps in the calculus.

  From a book by Machinag and Ohashi (1687).66

  Japanese method of calculating π suggested by figures above.

  Setting the radius OA (see bottom figure here) equal to unity, the area of the quadrant OAB is π/4; dividing OA into n intervals, the area of the jth strip is by Pythagoras’ Theorem

  The total area of the strips will tend to the area of the quadrant for n → ∞, yielding

  Without converting this expression to an integral, the Japanese could thus obtain π to any desired degree of accuracy by choosing n sufficiently large, at least in theory; in practice, the series would converge very slowly and, like Viète’s formula, it would involve the inconvenient extraction of square roots.

  Whether or not the Japanese actually used this method is not at all certain; but certainly John Wallis used a very similar approach. Wallis was also looking for the area of a quadrant of a circle. He did not convert (3) to an integral, but he did what amounted to the same thing; he looked for the area under the circular arc AB whose equation was known from Descartes’ coordinate geometry. In modern symbols, therefore, he started from the equation

  However, he did not yet have a method to evaluate the integral on the left, for neither Newton nor Leibniz had discovered the rules of the integral calculus yet. He was not even able to expand the integrand by the binomial theorem and integrate [using the Cavalieri-Fermat-Pascal formula (1)] term by term, for the binomial theorem at that time was known only for integral powers. How he did it is a long and painful story involving a cumbersome series of interpolations and inductive procedures; but he was able, in his Arithmetica infinitorum (1655), to derive the famous formula which bears his name, and which can be written as

  An English translation of this part of his book is readily available68 and the reader can look up the way in which Wallis sweated it out. However, today we can shortcut his procedure by two simple integrals,

  which are both easily derived by iterated integration by parts. Since the limit of the ratio of the integrals (6) and (7) for m → ∞ i
s one, Wallis’ formula follows immediately.

  The Wallis formula is a great milestone in the history of π. Like Viète, Wallis had found π in the form of an infinite product, but he was the first in history whose infinite sequence involved only rational operations; there were no square roots to obstruct the numerical calculation as was the case for Viète’s formula and in the Archimedean method.

  The man who achieved this historic result was Savillian Professor of Geometry at Oxford, a highly and broadly educated man who had graduated from Cambridge in medicine and philosophy and who, apart from his many important contributions to mathematics, published a grammar of English (Grammatica Linguae Anglicanae, 1652) and translated many important works from Greek, including Archimedes’ Measurement of the Circle (1676). Apart from deriving an infinite product for π, he had at least one other thing in common with Viète, and that was deciphering the secret codes of enemy messages, in Wallis’ case for the Parliamentarians in the Civil War. In appreciation of this, Cromwell appointed him Professor at Oxford, in spite of his Royalist leanings. This may sound a little confused, but it must be remembered that the English Revolution developed into a three-cornered fight between royalists, parliamentarians and the army under Oliver Cromwell. All modern revolutions pitted the people against a tyrannical regime, and most ended up by substituting one tyrant for another. The English rose against Charles I, but found themselves under Cromwell; the French rose against the ancien régime, and found themselves under Robespierre and Napoleon; and the Russians rose against the Tsar to find themselves ruled by the Soviet Jenghis Khans. Wallis found himself ruled by Charles II after the Restoration in 1660, but he was reappointed as professor at Oxford in the same year, and his many activities included the duty of chaplain to that unusual monarch who loved women more than he loved power.

  Of the many mathematical works published by John Wallis, the Arithmetica infinitorum is the most famous. The Greeks, perhaps because of Zeno’s paradoxes, had a horror of the infinite, and with the exception of Archimedes, who toyed with it in The Method, they preferred to leave it alone. Cavalieri and the others mentioned above began to attack the infinite, but Wallis was the man who found the right doors, even if he could not yet open them. His Arithmetica infinitorum was called “a scab of symbols” by the well known philosopher Thomas Hobbes (1588-1679), who claimed to have squared the circle, and whose philosophy advocated that human beings surrender their individual rights to constitute a state under an absolute sovereignty. Wallis could well afford to ignore Hobbes, and we shall do the same.

  * * *

  WILLIAM, Viscount Brouncker (ca. 1620-1684), the first president of the Royal Society, manipulated Wallis’ result into the form of a continued fraction.

  Continued fractions are part of the “lost mathematics,” the mathematics now considered too advanced for high school and too elementary for college. Continued fractions are useful, for example, for solving Diophantine equations. But in recent times a simple way has been found to ensure that continued fractions are not needed for the solution of Diophantine equations: The latter have been kicked out of the high school curriculum also.

  Consider the equation

  and write it in the form

  Substituting this expression for x on the right side of the same expression, we have

  If now (9) is substituted for x in the right side of (10), and substituted again every time x appears, we obtain the continued fraction

  As can be checked from the original equation (8), the limit of (11) is the irrational number

  and this number can now be approximated by a rational fraction as closely as desired on cutting off the continued fraction (11) at a correspondingly advanced point. The rational fractions obtained by cutting off the process at successive steps are called convergents; for example, the convergents of (11) are

  The process converges fairly quickly; for example, the last fraction in (13) is

  which agrees with (12) to four significant figures. Also, there is a quicker way of obtaining the convergents (13) than to worry them out from (11). This is explained in any textbook on continued fractions.69

  The result that Brouncker transformed into a continued fraction was the one actually given by Wallis, which was 4/π = … rather than π = … as given here by (5). The continued fraction that Brouncker obtained was the pretty expression

  with convergents 1, 3/2, 15/13, 105/76, 945/789,…

  How Brouncker obtained this result is anybody’s guess; Wallis proved its equivalence with his own result (5), but his proof is so cumbersome that it almost certainly does not reflect Brouncker’s derivation. Brouncker’s result was later also proved by Euler (1775), whose proof amounted to the following. Consider the convergent series

  which is easily seen to be equivalent to the continued fraction

  Now consider the series

  and set a0 = 0, a1 = x, a2 = –x2/3, a3= –3 x2/5,…; then

  and on setting x = 1 (which makes arctan x = π/4), Brouncker’s result (14) follows immediately.

  * * *

  NEWTON later based his work on that of Wallis and Barrow, and it seems that he was not well aware of the work done by the young Scotsman James Gregory (1638-1675), a leading contributor to the discovery of the differential and integral calculus. Gregory was a mathematician, occasionally dabbling in astronomy, who had studied mathematics at Aberdeen, and later in Italy (1664-68). He worked on problems far ahead of his time; for example, in Italy he wrote the Vera circuli et hyperbolae quadratura (True quadrature of the circle and the hyperbola), which included the basic idea of the disctinction between algebraic and transcendental functions, and he even attempted to prove the transcendence of π, a task that was not crowned by success until 1882. He was familiar with the series expansion of tan x, sec x, arctan x, arcsec x, and the logarithmic series. In 1668 he returned to Scotland, where he became Professor of Mathematics at St. Andrews University, and in 1674 he was appointed to the first Chair of Mathematics at the University of Edinburgh, but he died suddenly the next year at the age of only 36.

  Among the many things that Gregory discovered, the most important for the history of π is the series for the arctangent which still bears his name. He found that the area under the curve 1/(1 + x2) in the interval (0, x) was arctan x; in modern symbols,

  By the simple process of long division in the integrand and the use of Cavalieri’s formula (1), he found the Gregory series

  From here it was a simple step to substitute x = 1; since arctan(1) = π/4, this yields

  which was the first infinite series ever found for π.

  GOTTFRIED WILHELM LEIBNIZ (1646-1716)

  Gregory discovered the series (16) in 1671, reporting the discovery in a letter of February 15 1671, without derivation.70 Leibniz found the series (16) and its special case (17) in 1674 and published it in 1682, and the series (17) is sometimes called the Leibniz series. Gregory did not mention the special case (17) in his published works. Yet it is unthinkable that the discoverer of the series (16), a man who had worked on the transcendence of π, should have overlooked the obvious case of substituting x = 1 in his series. More likely he did not consider it important because its convergence (a concept also introduced by Gregory) was too slow to be of practical use for numerical calculations. If this was the reason why he did not specifically mention it, he was of course quite right. It was left to Newton to find a series that would converge to π more rapidly.

  13

  NEWTON

  Nature and nature’s laws lay hid in night,

  God said, Let Newton be, and all was light.

  ALEXANDER POPE

  (1688-1744)

  THERE had never been a scientist like Newton, and there has not been one like him since. Not Einstein, not Archimedes, not Galileo, not Planck, not anybody else measured up to anywhere near his stature. Indeed, it is safe to say that there can never be a scientist like Newton again, for the scientists of future generations will have books and libraries, microfilms and
microfiches, magnetic discs and other computerized information to draw on. Newton had nothing, nothing except Galileo’s qualitative thoughts and Kepler’s laws of planetary motion. With little more than that to go on, Newton formulated three laws that govern all motion in the universe: From the galaxies in the heavens to the electrons whirling round atomic nuclei, from the cat that always falls on its feet to the gyroscopes that watch over the flight of space ships. His laws of motion have withstood the test of time for three centuries. The very concepts of space, time and mass have crumbled under the impact of Einstein’s theory of relativity; age-old prejudices of cause, effect and certainty were destroyed by quantum mechanics; but Newton’s laws have come through unscathed.

  SIR ISAAC NEWTON

  From a bust in the Royal Observatory, Greenwich.

  Yes, that is so. Contrary to widespread belief, Newton’s laws of motion are not contradicted by Einstein’s Theory of Special Relativity. Newton never made the statement that force equals mass times acceleration. His Second Law says

  and Newton was far too cautious a man to take the m out of the bracket. When mass, in Einstein’s interpretation, became a function of velocity, not an iota in Newton’s laws needed to be changed. It is therefore incorrect to regard relativistic mechanics as refining or even contradicting Newton’s laws: Einstein’s building is still anchored in the three Newtonian foundation stones, but the building is twisted to accommodate electromagnetic phenomena as well. True, Newton’s law of gravitation turned out to be (very slightly) inaccurate; but this law, even though it led Newton to the discovery of the foundation stones, is not a foundation stone itself.