A History of Pi Page 11
How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics!
(3.14159265358979)
In French and German, there are even poems for this purpose. In French, there is the following poem:
Que j’aime à faire apprendre un nombre utile aux sages!
Immortel Archimède, artiste ingénieur,
Qui de ton jugement peut priser la valeur?
Pour moi, ton problème eut de pareils avantages.
This poem evidently inspired some German to the following bombastic lines of Teutonic lyrics:
Dir, o Held, o alter Philosoph, du Riesengenie!
Wie viele Tausende bewundern Geister
Himmlisch wie du und göttlich!
Noch reiner in Aeonen
Wird das uns strahlen
Wie im lichten Morgenrot!
Both poems give π to 29 decimal places. There is also the following German verse giving π to 23 decimal places:
Wie? O! Dies π
Macht ernstlich so vielen viele Müh’!
Lernt immerhin, Jünglinge, leichte Verselein,
Wie so zum Beispiel dies dürfte zu merken sein!
The 32nd decimal digit of π is a zero, so that this kind of poetry is mercifully nipped in the bud.
11
THE LAST ARCHIMEDEANS
The reasonable man adapts himself to the world; the unreasonable one persists in trying to adapt the world to himself. Therefore all progress depends on the unreasonable.
GEORGE BERNARD SHAW
(1856-1950)
SOME 1,900 years after Archimedes’ Mensuration of the Circle, people at last began to wonder if there was not a quicker road to π than comparison of the circle with the inscribed and circumscribed polygons. Two Dutch mathematicians found such a road by elementary mathematics, even before the differential and integral calculus had been invented. It is a tribute to the old genius of antiquity that for almost two millenia his method defied improvement; the numerical evaluation became more accurate as the centuries went by, but the method remained without substantial change.
The Snellius-Huygens bounds.
(a) Upper bound: BG1 > arc BF
(b) Lower bound: BG2 < arc BF
The first man to challenge Archimedes’ polygons was the Dutch mathematician and physicist Willebrord Snellius (1580-1626), professor of mathematics at the University of Leyden, who is today best known for his discovery of the laws of reflection and refraction. (If a light ray is incident onto a plane interface between two media, then the angle of incidence equals the angle of reflection, and the sine of the angle of incidence to the sine of the angle of refraction equals the refractive index.) In his book Cyclometricus (1621), Snellius argues quite correctly that the sides of the inscribed and circumscribed polygons are too widely separated by the arc of the corresponding circle, and that therefore the upper and lower bounds on π resulting from the Archimedean polygons are needlessly far apart. He searched for geometrical configurations that would yield closer bounds to the rectification of an arc, and he found them. They are shown in the figure above. The upper bound (of the discrepancy between circular arc and constructed straight segment) was given by Snellius as follows: Choose a point D on the circumference of a given circle (figure a) and make DE equal to the radius OA, where E lies on the extension of the diameter AB. Let ED extended intersect the tangent BG at G1; then
The lower bound is given by a construction that we have already discussed in connection with Cardinal Cusanus’ approximate rectification of an arc (here). Make EA equal to the radius, where E lies on AB extended; choose any point F on the circumference of the circle such that F is the second intersection of EF and the circle (figure b), and let G2 be the intersection of EF and the tangent BG; then
These bounds were indeed much closer than those of the Archimedean polygons; unfortunately, Snellius was unable to supply a rigorous proof that his assertions were correct. (They were.)
Is it possible to use something that has not been rigorously proven, i.e., that has not been derived by ironclad logic from certain basic assumptions? In physics, the answer is an unqualified yes. We claim that water, all water, boils at 100°C (at standard atmospheric pressure at the elevation of Paris), although we have boiled almost no water whatever compared with all the water in the universe. Our claim is based on the faith (yes, it is a faith) that nature is consistent. The Law of the Conservation of Energy is based only on our experience: All of our experience supports it, and none of our experience contradicts it. There is no theoretical reason why tomorrow some scientist should not report an experiment that contradicts it; yet our faith in the consistency or uniformity of nature is so deep that we would probably not believe him. Physics is an inductive science; it finds its laws by generalizing many specific experiences.
The rules of the game in mathematics are slightly different: It is a deductive science, which deduces many specific theorems from a few very general assumptions or axioms. The structure of Euclid’s cathedral, for example, is entirely deductive.
And yet the history of mathematics abounds with examples where certain methods and theorems were used long before a rigorous proof was supplied that they are actually true. It is probably no exaggeration to say that most of the progress in mathematics is due to physicists and engineers who discovered certain truths by the physical approach; and only after they had built little houses on sand, the mathematicians supplied foundations of concrete. Euclid did this to the geometry that was, for the most part, known before his day; and this tendency in the development of mathematics can be found in history right down to our own days.
Oliver Heaviside (1850-1925), in his investigations of electrical curcuits, observed certain regularities in the solution of differential equations, and found a method by which difficult differential equations could be converted to simple algebraic equations. “The proof is performed in the laboratory,” he proclaimed in the face of the wincing mathematicians. For a quarter of a century, electrical engineers were using Heaviside’s “operational calculus,” until in the early 1920’s the mathematicians discovered that Heaviside’s magic could be anchored in the solid ground of an integral transform discovered by Laplace a century before Heaviside. But Heaviside had also used a thing called the delta function, and this, the mathematicians proclaimed, was a monstrosity, for it could be demonstrated that no such function, with the properties Heaviside ascribed to it, could exist. What especially outraged the mathematicians was not so much that electrical engineers continued to use it, but that it would amost always supply the correct result. Not until 1950 did the French mathematician Laurent Schwartz (1915- ) show what was going on; if the delta function could not exist as a function, it could exist as a distribution, and the engineers had been doing things right, after all. Heaviside was one of the unreasonable men in the motto of this chapter.
The unproven four-color conjecture. 1, 2, 3, 4 are different colors. No two colors meet at any border between two regions.
Even today there are many statements that are “true” by physical standards of experience, but that remain mathematically unproven. Two of the most famous are the Goldbach conjecture and the four-color problem. In 1742, Christian Goldbach (1690-1764), in a letter to Euler, suggested that every even number is the sum of two primes (e.g., 8 = 5 +3; 24 = 19 + 5; 64 = 23 + 41; etc.). By the physical standards of experience, Goldbach was right, for no one has yet found an even number for which the conjecture does not work out. But that does not constitute a proof, and a mathematical proof has yet to be found. The other well known problem, the so-called four-color problem, is to prove that no matter how a plane is subdivided into non-overlapping regions, it is always possible to paint the regions with no more than four colors in such a way that no two adjacent regions have the same color. By experience, the truth of the assertion is known to every printer of maps (common corners, like Colorado and Arizona, do not count as adjacent.) And no matter how one tries to dream up intertwining states on
a fictitious map (see figure above), four colors always appear to be enough. But that is no proof.
Snellius, then, had found two jaws of a vise closer to the circumlar arc than were the Archimedean polygons, and the jaws would close faster on the value of π with every turn, that is, with every doubling of sides of the polygons. For example, for a hexagon, Archimedes obtained the limits 3 < π < 3.464; using the two theorems above, Snellius was able to find the limits 3.14022 < π < 3.14160 for a hexagon, which are closer to π than those that Archimedes had obtained even with the help of a 96-sided polygon. On the contrary, using a 96-sided polygon, Snellius found the limits
Finally, Snellius verified the decimal places found by Ludolph van Ceulen, and with much less effort than Ludolph had invested.
There is all the difference in the world between Ludolph’s digit hunting and Snellius’ numerical test. Snellius had found a new method and checked its quality by calculating the decimal digits of π; Ludolph’s evaluation to 35 decimal digits by a method known for 1900 years was no more than a stunt.
Snellius’ two theorems were rigorously proved by another Dutch mathematician and physicist, Christiaan Huygens (1629-1695). In his remarkable work De circuli magnitude inventa (1654), Huygens proved these and 14 more theorems by Euclidean geometry and with Euclidean rigor. Let us denote the area and perimeter of the inscribed polygon by a and p, respectively, and the same for the circumscribed polygon by A and P; let the subscripts 1 and 2 stand for the original polygon and the polygon with double the number of sides, and let a and p (without subscripts) denote the area and perimeter (circumference) of the circle. Then the most important of the 16 theorems proved by Huygens can be written in algebraic notation as follows:
CHRISTIAAN HUYGENS (1629–1695)
The Roman numerals denote the number of Huygens’ theorems in the original work. The proofs of these theorems are long and will not be repeated here. (The interested reader can find the original work in Oeuvres completes de Christiaan Huygens, Paris, 1888 et sequ., or in German translation, in Rudio’s monograph quoted in the bibliography at the end of this book.) By means of these theorems, Huygens was able to set the bounds
Again, this was no mere digit hunting; as Huygens points out in his work, Archimedean polygons of almost 400,000 sides would be needed to obtain π to the same accuracy.
Huygens wrote De circuli magnitude inventa as a young man of 25 who had only recently taken up the serious study of mathematics and physics (he had been trained as a lawyer). The work soon lost interest owing to the discovery of better methods supplied by the differential calculus (to which Huygens contributed not only in preparing the ground for it mathematically, but also in other ways: In 1672 he gave Leibniz lessons in mathematics, and in 1674 he transmitted Leibniz’s first paper on the differential calculus to the French Academy of Sciences). Huygens was already a contemporary of Newton, and Newton’s supreme genius overshadowed everybody else. In a way, Huygens’ book on the circle is reminiscent of some 18th century composers such as Rejcha or Vranický; they wrote delighful music, but they were overshadowed by Wolfgang Amadeus Mozart.
Apollonius circles (full lines). Each circle is the locus of points satisfying the relation 1/a + 1/b = const, where a and b are the distances from two fixed points. The broken circles intersect Apollonius’ circles at right angles (lines of force). Discovered in the 3rd century B.C.
Even so, Huygens’ book documents that European mathematics had at last crawled out of the morass of Roman Empire and Roman Church. Boethius’ Geometry, written in the last year of the Roman Empire, was a mathematical cookbook; and Pope Sylvester II received requests to explain the strange phenomenon that on doubling the diameter of a sphere, its volume would increase eightfold; but Huygens’ treatise had once more attained the high standards of mathematical rigor exacted by the University of Alexandria almost 2,000 years earlier.
Besides, Huygens’ made his permanent mark on history as a physicist rather than a mathematician. His many discoveries included a principle of wave motion, which to this day is called Huygens’ principle. He could not have known that the antennas of the radio stations tracking man’s first flight to the moon would be calculated on the basis of that principle; any more than Apollonius of Perga, in the 3rd century B.C., could have known that the family of circles he discovered (see figure above) would one day be the equipotentials of two parallel, cylindrical, electrical conductors. These are but two of hundreds of examples that one might quote in answer to the question what good comes from exploring the moon or studying non-Euclidean geometry. The question is often asked by people who count cents instead of dollars and dollars instead of satisfaction. Of late, this question is also being asked by the intellectual cripples who drivel about “too much technology,” because technology has wounded them with the ultimate insult: They can’t understand it any more.
Descartes’s method for finding π.
Some Victorian lady asked Michael Faraday (1791-1867) this question about his discovery of electromagnetic induction, and he answered: “Madam, what is the use of a newborn baby?”
* * *
HUYGENS was born three years after Snellius died; a contemporary of both was the famous founder of analytical geometry, René Descartes (1596-1650). He latinized his name to Renatus Cartesius, which is the reason why analytical geometry is sometimes called Cartesian geometry. Descartes was not only a mathematician, but also a physicist and philosopher. He fully accepted the Copernican system, but frightened by the condemnation of Galileo in 1633, he published many of his scientific works in obscure and ambiguous form, and others were not published until after his death.
Among the papers found after his death was some work on the determination of π. His approach is in some ways reminiscent of Viète’s, but instead of considering the area of a polygon, he kept its perimeter constant and doubled the sides until it approached a circle.
If the length of the perimeter is L, then as evident from the figure above, in which AB is the side of the polygon,
Hence the radius of the inscribed circle is, after k doublings,
Descartes’s construction.
Descartes used a square as a starting point, so that n = 4, θ = π/2; hence
By using the half-angle formula for the tangent, Descartes could have obtained an infinite sequence of operations; however, he did not get as far as (2), but used the equivalent construction shown in the figure above.
Let ABCD be a square of given perimeter L, so that AB = L/4; extend the diagonal AC and determine C1 so that the area of the rectangle BCC1B1 equals ¼ of the area of the square ABCD (this can be done by a construction involving only compasses and straightedge). Continue to points C2, C3,…, so that each new rectangle has ¼ the area of the preceding one. Then AB∞ is the diameter of a circle with circumference L.
Indeed, if ABk = xk, then AB = x0; then the construction makes
which is satisfied by
and since 4x0 = L, this is equivalent to (1) with n = 4.
Huygens’ and Snellius’ inequality for the lower bound shown in the figure here can be used as an approximate construction for rectifying the circle, as we have seen here. Another construction for the approximate rectification of the circle was found by A.A. Kochansky in 1685 (see figure). The angle BOC is 30°, BC is parallel to AD, and the rest is evident from the figure. The length of the segment CD is approximately equal to half the circumference of the circle.
Kochansky’s approximate rectification of the circle (1685).
We have
and
so that
which differs from π by less than 6 × 10–5.
A construction corresponding to π correct to 6 decimal places was given by Jakob de Gelder in 1849. It is based on one of the convergents in the expansion of π in a continued fraction, or simply on the fact that π = 355/113 is correct to six decimal places.
Since 355/113 = 3 + 42/(72 + 82), the latter fraction can easily be constructed geometrically (see Gelder’s constr
uction overleaf). Let CD = 1, CE = 7/8, AF = 1/2, and let FG be parallel to CD, and FH to EG; then AH = 42/(72 + 82), and the excess of π above 3 is approximated with an error less than one millionth.
Gelder’s construction
Hobson’s construction
A simple construction will also square the circle with an error of less than 2 × 10–5 (see Hobson’s construction above). Let OA = 1, OD = 3/5, OF = 3/2, OE = 1/2. Describe the semicircles DGE, AHF with DE and AF as diameters, and let the perpendicular to AB through O intercept them in G and H, respectively. Then GH = 1.77246…, which differs from √π = 1.77245 … by less than 2 × 10–5. Hence GH2 is very close to the area of the circle with diameter AB. The construction was given by Hobson in 1913.61
Other approximate constructions have been collected by M. Simon.61
12
PRELUDE TO BREAKTHROUGH
I have made this letter longer than usual because I lacked the time to make it short.
BLAISE PASCAL
(1623-1662)
THE author of the above quotation was one of the most brilliant mathematicians and physicists of the 17th century, or at least, he could have been, had he not flown from flower to flower like a butterfly, finally forsaking the world of mathematics for the world of mysticism. The squaring of the circle or the calculation of π was not one of the flowers Pascal visited, yet his brainstorms were important in preparing the ground for the calculus and the discovery of a new approach to the calculation of π. In his 1658 Traité des sinus du quart de cercle (Treatise of the sines of the quadrant of the circle) he came so close to discovering the calculus that Leibniz later wrote that on reading Pascal’s work a light suddenly broke upon him.