A History of Pi

by Petr Beckmann

  And so proceed ad infinitum.

  JONATHAN SWIFT

  (1667-1745)

  IT has already been pointed out that the invention of decimal fractions and logarithms greatly facilitated numerical calculations in the late 1500’s and the early 1600’s, and this is reflected in the history of π, for about this time people started to calculate its value to an ever increasing number of decimal places, each new digit increasing the accuracy of the former approximation by no less than 10 times. The process continued beyond any possible practical use of so many decimal places; by the end of the 16th century, π was known to 30 decimal places, by the end of the 18th century it was known to 140 places, by the end of the 19th century it had been calculated to 707 places (though later only 526 of them proved to be correct), and the digital computer of the 20th century has raised this number to a whopping 500,000 — perhaps more by the time you read this (see Chapter 18).

  Archimedes calculated π to the equivalent of two decimal places, and at first the hunt for greater accuracy may have been dictated by practical needs. Later, especially after the advent of the differential calculus and infinite series, the number of decimal places may have been used to demonstrate the quality of the method of calculation. Perhaps some investigators hoped to discover a periodicity in the ever lengthening sequence of digits. Had this been so, they would have been able to express π as the ratio of two integers, for if a certain sequence of d decimal digits kept recurring, then the fractional part would be the sum of the geometric series

  where a0 is the number formed by the first d digits, so that by summing this geometric series, one would obtain

  However, as was shown by Lambert in 1767, π is not a rational number, i.e., it cannot be expressed as a ratio of two integers, and this showed hopes of periodicity futile. Another possible reason may have been that the calculation of π presented a challenge to find better methods of numerical analysis, for numerical calculations are far from easy when the number of operations becomes large and every little trick could save hours of computations.

  But for the most part, I suspect, the driving force behind these calculations was the spirit that makes people go over the Niagara Falls in a barrel or to top the world record of pole sitting by another 20 minutes. The digits beyond the first few decimal places are of no practical or scientific value. Four decimal places are sufficient for the design of the finest engines; ten decimal places would be sufficient to obtain the circumference of the earth within a fraction of an inch if the earth were a smooth sphere (in proportion, it is smoother than a billiard ball). The most exacting requirement that I can think of in a practical application is a not very common case of computer programing. Computers do the actual arithmetic operations only with rational numbers, which are used to approximate irrational numbers. In rounding off the last significant figure, they can sometimes play quite treacherous tricks on the programmer, and to guard against the effect of the rounding error in computations that involve long sequences of certain operations (for example, very small differences between very large numbers), special precautions must be taken. In FORTRAN, a widely used computer language, this is achieved by a command called DOUBLE PRECISION. This will result in certain operations being carried out with at least double the usual number of significant figures. The actual number of decimal places varies from computer to computer, but usually as many as 17 decimal places can be stored for a double precision constant. For this extreme case, the corresponding value of π is

  where the last digit results from rounding off 38. Or we let the computer tell us: We ask it for 4 arctan 1 (which is π) with double precision by entering the FORTRAN statement

  The Double Precision routine is used only when really necessary, because it can be more trouble than it is worth, and even then 17 decimal places are usually more than enough.

  There is no practical or scientific value in knowing more than the 17 decimal places used in the foregoing, already somewhat artificial, application. In 1889, Hermann Schubert, a Hamburg mathematics professor, made this point in the following consideration.56

  Conceive a sphere constructed with the earth at its center, and imagine its surface to pass through Sirius, whis is 8.8 light years distant from the earth [that is, light, traveling at a velocity of 186,000 miles per second, takes 8.8 years to cover this distance]. Then imagine this enormous sphere to be so packed with microbes that in every cubic millimeter millions of millions of these diminuitive animalcula are present. Now conceive these microbes to be unpacked and so distributed singly along a straight line that every two microbes are as far distant from each other as Sirius from us, 8.8 light years. Conceive the long line thus fixed by all the microbes as the diameter of a circle, and imagine its circumference to be calculated by multiplying its diameter by π to 100 decimal places. Then, in the case of a circle of this enormous magnitude even, the circumference so calculated would not vary from the real circumference by a millionth part of a millimeter.

  This example will suffice to show that the calculation of π to 100 or 500 decimal places is wholly useless.

  But microbes or no microbes, the digit hunters plodded ahead through the centuries.

  The Dutch mathematician and fortification engineer Adriaan Anthoniszoon (1527-1607) found the value 355/113, which is correct to 6 decimal places.

  This record was broken by François Viète in 1593 by the value given here, which is correct to 9 decimal places.

  But already in the same year his record fell to Adriaen van Rooman (1561-1615), a Dutchman who used the Archimedean polygons with 230 sides, and calculated 15 decimal places.

  Three years later his record was broken by another Dutchman, Ludolph van Ceulen (1539-1610), professor of mathematics and military science at the University of Leyden. In his paper Van den Circkel (1596), he reports using a polygon with 60 × 229 sides, which yielded the value of π to 20 decimal places. The paper ends with “Whoever wants to, can come closer.” But nobody wanted to, except Ludolph himself. In De Arithmetische en Geometrische fondamenten, published posthumously by his wife in 1615, he gives π correct to 32 places, and according to Snell’s report in 1621, he topped this later by three more places. Tropfke57 states that these last three digits are engraved on his tombstone in the Peter Church at Leyden; this seems to invalidate vague references by other historians, according to which all 35 digits were engraved in his tombstone, but that the stone has been lost. In any case, Ludolph’s digit hunting so impressed the Germans that to this day they call π die Ludolphsche Zahl (the Ludolphine number).

  But then came the boys with the big guns. The differential calculus was discovered in the 17th century, and with it a multitude of infinite series and continued fractions for π, as we shall see in coming chapters. The astronomer Abraham Sharp (1651-1742) used an arcsine series to obtain 72 decimal digits, and shortly afterwards, in 1706, John Machin (1680-1752) used the difference between two arctangents to find 100 decimal places; but he was beaten by the French mathematician De Lagny (1660-1734), who piled another 27 digits on top of this result in 1717. This record of 127 digits seems to have stood until 1794, when Vega (1754-1802), using a new series for the arctangent discovered by Euler, calculated 140 decimal places. Vega’s result showed that De Lagny’s string of digits had a 7 instead of an 8 in the 113th decimal place.

  It should be noted that during the 18th century Chinese and Japanese digit hunters were at work also, although the Japanese records lagged behind the European ones, just as the Chinese records were retarded with respect to the Japanese. The Japanese mathematician Takebe, using a 1,024-sided polygon, found π to 41 decimal places in 1722, and Matsanuga, using a series, found 50 decimal places in 1739. Thereafter, the Japanese seem to have had more sense than their European colleagues; they continued to study series yielding π, but wasted no more time on digit hunting.

  The Viennese mathematician L.K. Schulz von Strassnitzky (1803-1852) used an arctangent formula to program the forerunner of the computer, a calculating pro
digy. This one was Johann Martin Zacharias Dase (1824-1861), about whom we shall have more to say below. In 1844 he calculated π correct to 200 places in less than two months. They are reproduced below:

  Before Dase, William Rutherford had calculated 208 decimal places in 1824; but from the 153rd decimal place they disagreed with Dase’s figures, and the discrepancy was resolved in Dase’s favor when Thomas Clausen (1801-1885) published 248 decimal places in 1847.

  And still the craze went on. In 1853, Rutherford gave 440 decimal places, and in 1855 Richter calculated 500 decimal places. When William Shanks published 707 places in 1873-74 (Proceedings of the Royal Society, London), he probably thought that his record would stand for a long time. And so it did. But in 1945, Ferguson found an error in Shanks’ calculations from the 527th place onward,93 and in 1946, he published 620 places; using a desk calculator, he subsequently found 710 places in January 1947, and 808 places in September of the same year.93 This was the record that succumbed to the computer in 1949, and we shall return to the story in Chapter 18.

  To appreciate the stupendous work necessary to achieve such results, it is necessary to recall that a computer can perform an arithmetical operation, such as adding or multiplying two numbers, in less than one millionth of a second; but even a contemporary computer takes about 40 seconds (not counting conversion and checks) to compute Shanks’ 707 places. How, then, did the digit hunters of the 18th and 19th centuries accomplish these feats? For the most part, apparently, by covering acres of paper for months and years; but at least one of them, Strassnitzky, used a human computer, a calculating prodigy.

  * * *

  SEVERAL such phenomenal calculating prodigies have been known, and though their power is little understood, it is clear that they share with the computer its two basic abilities: the rapid execution of arithmetical operations, and the storage (memory) of vast amounts of information. Some of them also have an additional gift: They can recognize large numbers of objects without counting them. This can be done by you or me if that number is three or four; and if the objects are arranged in certain patterns, we can recognize six or ten, perhaps more. But Johann Dase, the man who calculated the 200 decimal places of π in less than two months, could give the number of sheep in a flock, books in a case, etc., after a single glance (up to 30); after a second’s look at some dominos, he gave the sum of their points correctly as 117; and when shown a randomly selected line of print, he gave the number of letters correctly as 63.

  Truman Henry Safford (1836-1901) of Royalton, Vermont, could instantly extract the cube root of seven-digit numbers at the age of 10. At the same age, he was examined by the Reverend H.W. Adams, who asked him to square, in his head, the number

  365,365,365,365,365,365.

  Thereupon, reports Dr. Adams,

  He flew around the room like a top, pulled his pantaloons over the tops of his boots, bit his hands, rolled his eyes in their sockets, sometimes smiling and talking, and then seeming to be in agony, until, in not more than a minute, said he,

  133,491,850,208,566,925,016,658,299,941,583,255!58

  Truman Safford never exhibited his powers in public. He graduated from Harvard, became an astronomer, and gradually lost the amazing powers he had shown in his youth. None of this is typical for most of the calculating prodigies. Many of them, including our Johann Dase, were idiot savants: They were brilliant in rapid computations, but quite dull-witted in everything else, including mathematics. Few of them could even intelligently explain how they performed the calculations, and those that could, revealed clumsy methods.

  The Englishman Jedediah Buxton (1707-1772), for example, never learned to read or write, but he could, in his head, calculate to what number of pounds, shillings and pence a farthing (¼ penny) would amount if doubled 140 times. (The number of pounds has 39 digits, and the pound had 20 shillings of 12 pence each.) In 1754 he visited London, where several members of the Royal Society satisfied themselves as to the genuineness of his performances. He was also taken to Drury Lane Theater to see a play, but entirely unaffected by the scene, he informed his hosts of the exact number of words uttered by the various actors, and the number of steps taken by others in their dances. Even more surprisingly, his methods of arithmetic, in the rare cases when he could explain them, were quite clumsy. He never learned, for example, to add powers of ten when multiplying; he called 1018 “a tribe,” and 1036 “a cramp.”59

  Johann Dase, to whom we shall return in a moment, was also an idiot savant, and the dull wits of these calculating prodigies are a third property that they share with an electronic computer. The sinister “electronic brains” of science fiction and the boob tube to the contrary, the electronic computer is a moron whose total imbecility can often be quite exasperating. If you forget a line in your program telling it to print the results, it will perform all the complicated computations and then erase them again, handing you a blank sheet of paper for a print-out. If you punch the number three as “3” when it should have been “3.”, it will refuse to work the program and instead it will print some gobbledegook like this:

  ERROR IN LINE 123. ILLEGAL MIXING OF MODES.

  EXECUTION DELETED. TIME 23 SECS.

  If the computer is so smart, why does it not put in the one dot instead of churning out all this gibberish? Ask it; but it will just sit there, a moronic heap of wire, semiconductors and tape, and say nothing. For a computer is extremely fast, and it has a vast and rapidly accessible memory, but contrary to popular belief, it is totally without intelligence, slavishly following the rules that were built into it. The most intelligent thing it is capable of doing without the help of its programmers is to go on strike when required to work without air conditioning.

  To return to Johann Martin Zacharias Dase. He was born in 1840 in Hamburg, had a fair education and was afforded every opportunity to develop his powers, but made little progress; all who knew him agree that except for calculating and numbers, he was quite dull. He always remained completely ignorant of geometry, and never learned any other language than German. His extraordinary calculating powers were timed by renowned mathematicians: He multiplied two 8-digit numbers in his head in 54 seconds; two 20-digit numbers in 6 minutes; two 40-digit numbers in 40 minutes; and two 100-digit numbers (also in his head!) in 8 hours and 45 minutes. To achieve feats like these, he must have had a photographic memory. His ability to recognize the number of objects without counting them has already been remarked on, and one may speculate that perhaps he achieved this also by his fantastic memory, that is, by taking a glance at a flock of sheep, and then rapidly counting them from the photographic image in his mind. He could, for example, in half a second memorize a twelve-digit number and then instantly name the digit occupying a particular position, so that whatever the mechanism that enabled his brain to do this, it must have been close to photography. A fantastic memory is, of course, one of the assets that all of the calculating prodigies had in common. Another calculating prodigy, George Parker Bidder (1806-1878), a British civil engineer, could instantly give a 43-digit number after it had been read to him backwards. He did this, at the age of ten, at a performance; and an hour later, asked whether he had remembered it, he immediately repeated it:

  CARL FRIEDRICH GAUSS (1777–1855)

  Johann Dase gave exhibitions of his extraordinary calculating powers in Germany, Austria and England, and it was during an exhibition in Vienna in 1840 that he made the acquaintance with Schulz von Strassnitzky, who urged him to make use of his powers for the calculation of mathematical tables. He was then 16 years old, and gladly agreed to do so, thus becoming acquainted with many famous mathematcians of his age, including Carl Friedrich Gauss (1777-1855). When he was 20, Strassnitzky taught him the use of the formula

  The eternal fascination of π . A 1970 advertisement60 coding the digits of π (A = 1, B = 2, C = 3, etc.)

  with a series expansion for each arctangent, and this is what he used to calculate π to 205 decimal places (of which all but the last five turned out
to be correct). This stupendous task he finished in just under two months.

  Then he came back for more. He calculated the natural logarithms of the first 1,005,000 numbers, each to 7 decimal places, which he did in his spare time in 1844-1847 when employed by the Prussian Survey. In the next two years he compiled a table of hyperbolic functions, again in his spare time. He also offered to make tables of the factors of all numbers from 7,000,000 to 10,000,000; and on the recommendation of Gauss, the Hamburg Academy of Sciences agreed to assist him financially so that he could devote himself to this work, but he died in 1861, after he had finished about half of it.

  It would thus appear that Carl Friedrich Gauss, who holds so many firsts in all branches of mathematics, was also the first to introduce payment for computer time.

  * * *

  IT is an interesting phenomenon that all the digit hunters concentrated on the number π; none ever attempted to find hundreds of decimal places for √2 or sin 1° or log 2.92 There seems to be something magical about the number π that fascinates people: I have known several people who memorized π (in their adolescence) to 12 and even 25 decimal digits; none of them memorized, say, √ 2. There is no mathematical justification for this, for to calculate or memorize π to many decimal places is the same waste of time as doing this for the square root of two. The reason must be psychological; perhaps the explanation is that √ 2 is not so very different from √ 3, and sin 1° is not so very different from sin 2°; but π is unique. Or at least people think so at the early age when they are first introduced to this number; if they pursue the matter in higher education, they find that the number of transcendentals is not merely infinite, but indenumerable (see Chapter 16).

  Various mnemonic devices are available for cluttering the storage cells of one’s brain with the decimal digits of π. In the following sentence,46 for example, the number of letters in each word represents the successive digits of π: