A History of Pi

by Petr Beckmann

  Legendre was right on both counts: π is not algebraic, and the proof so difficult that it was not found for 88 more years.

  In 1873, Charles Hermite (1822-1901) proved that the number e is transcendental; from this it follows that the finite equation

  cannot be satisfied if r, s, t … are natural numbers and a, b, c, … are rational numbers not all equal to zero.

  In 1882, F. Lindemann85 finally succeeded in extending Hermite’s theorem to the case when r, s, t,… and a, b, c,… are algebraic numbers, not necessarily real. Lindemann’s theorem can therefore be stated as follows:

  If r, s, t,…, z, are distinct real or complex algebraic numbers, and a, b, c,…, n are real or complex algebraic numbers, at least one of which differs from zero, then the finite sum

  cannot equal zero.

  From this the transcendence of π follows quickly. Using Euler’s Theorem in the form

  we have an expression of the form (7) with a = b = 1 algebraic, and c and all further coefficients equal to zero; s = 0 is algebraic, leaving r = iπ as the only cause why (8) should vanish. Thus, iπ must be transcendental, and since i is algebraic, π must be transcendental.

  It stands to reason that a proof which was 100 years in the making is neither short nor easy. Lindemann’s paper85 runs to 13 pages of tough mathematics. Karl Wilhelm Weierstrass (1815-1897), the apostle of mathematical rigor, simplified the proof of Lindemann’s theorem somewhat in 1885,86 and it was further simplified in later years by renowned mathematicians (Stieltjes, Hurwitz, Hilbert and others). The interested reader is referred to the comparatively easy version given by Hobson.87

  As we have seen, the possibility of squaring the circle by Euclidean construction hinged entirely on the question whether π was algebraic or transcendental; Lindemann’s theorem therefore proved that the squaring of the circle by the rules of Greek geometry is impossible.

  And that is the end of the history of π and squaring the circle. Or it would have been if there were no fools among us.

  These fools rushed in where sages had trodden for 2,500 years.

  17

  THE MODERN CIRCLE-SQUARERS

  This invention relates to a device which renders it impossible for the user to stand upon the privy-seat; and consists in the provision of rollers on top of the seat, which, although affording a secure and convenient seat, yet, in the event of an attempt to stand upon them, will revolve, and precipitate the user on to the floor.

  U.S. PATENT No. 90,298 (1869).

  ALMOST every country now has a law stating that no patent will be granted for an invention of a perpetuum mobile of any kind. But neither man-made laws nor the laws of thermodynamics have stopped an army of mavericks from designing hundreds of versions of an alleged perpetuum mobile.

  Similarly, in 1775 the Académie Française passed a resolution henceforth not to examine any more solutions of the problem of squaring the circle. But undaunted by either the Academy’s resolution or Lindemann’s proof, the circle squarers marched on; and they are still marching, spiteful of the cruel world that will not recognize their grand intellectual achievements. “The race of circle squarers,” says Schubert,88 “will never die out as long as ignorance and the thirst for glory remain united.”

  Every country has its circle-squarers, but the following will be limited to the circle-squarers of America.

  There is a story about some American legislature having considered a bill to legislate, for religious reasons, the biblical value of π = 3. I have found no confirmation of this story; very probably it grew out of an episode that actually took place in the State Legislature of Indiana in 1897. The Indiana House of Representatives did consider and unanimously pass a bill that attempted to legislate the value of π (a wrong value); the author of the bill claimed to have squared the circle, and offered this contribution as a free gift for the sole use of the State of Indiana (the others would evidently have to pay royalties).

  The author of the bill was a physician, Edwin J. Goodman, M.D., of Solitude, Posey County, Indiana, and it was introduced in the Indiana House on January 18, 1897, by Mr. Taylor I. Record, Representative from Posey County. It was entitled “A bill introducing a new Mathematical truth,”89 and it became House Bill No. 246; copies of the bill are preserved in the Archives Division of the Indiana State Library; the full text has also been reprinted in an article by W.E. Eddington in 1935.90

  The preamble to the bill informs us that this is

  A bill for an act introducing a new mathematical truth and offered as a contribution to education to be used only by the State of Indiana free of cost by paying any royalties whatever on the same, provided it is accepted and adopted by the official action of the legislature in 1897.

  The bill consisted of three sections. Section 1 starts off like this:

  Be it enacted by the General Assembly of the State of Indiana: It has been found that the circular area is to the quadrant of the circumference, as the area of an equilateral rectangle is to the square on one side. The diameter employed as the linear unit according to the present rule in computing the circle’s area is entirely wrong …

  An “equilateral rectangle” is, of course, a square, so that the first statement does not make any sense at all; but if we give the author the benefit of the doubt and assume that this is a transcript error for “equilateral triangle,” then what Mr. Goodwin of Solitude, Posey County, had discovered in his first statement was the equivalent of π = 16/ √3 = 9.2376…, which probably represents the biggest overestimate of π in the history of mathematics.

  Facsimile of Bill No. 246, Indiana State Legislature, 1897. Kindly made available by the Indiana State Library.

  However, Sections 1 and 2 contain more hair-raising statements which not only contradict elementary geometry, but also appear to contradict each other. Section 2 of the bill concludes

  By taking the quadrant of the circle’s circumference for the linear unit, we fulfill the requirements of both quadrature and rectification of the circle’s circumference. Furthermore, it has revealed the ratio of the chord and arc of ninety degrees, which is as seven to eight, and also the ratio of the diagonal and one side of a square which is as ten to seven, disclosing the fourth important fact, that the ratio of the diameter and circumference is as five-fourths to four; and because of these facts and the further fact that the rule in present use fails to work both ways mathematically, it should be discarded as wholly wanting and misleading in practical applications.

  From a page of the original copy of Bill No. 246, Indiana State Legislature, 1897. Kindly made available by the Indiana State Library.

  And Section 3:

  In further proof of the value of the author’s proposed contribution to education, and offered as a gift to the State of Indiana, is the fact of his solutions of the trisection of the angle, duplication of the cube and quadrature of the circle having been already accepted as contributions to science by the American Mathematical Monthly, the leading exponent of mathematical thought in this country. And be it remembered that these noted problems had been long since given up by scientific bodies as unsolvable mysteries and above man’s abilities to comprehend.

  The bill was, perhaps symbolically, referred to the House Committee on Swamp Lands, which passed it on to the Committee of Education, and the latter reported it back to the House “with recommendation that said bill do pass.” On February 5, 1897, the House passed the learned treatise unanimously (67 to 0).

  Five days later the bill went to the Senate, where it was referred, for unknown reasons, to the Committee on Temperance. The Committee on Temperance, too, reported it back to the Senate with the recommendation that it pass the bill, and it passed the first reading without comment.

  What would have happened to the bill if events had run their normal course is anybody’s guess. But it so happened that Professor C.A. Waldo, a member of the mathematics department of Purdue University, was visiting the State Capitol to make sure the Academy appropriation was cared for, and as he
later reported,90 was greatly surprised to find the House in the midst of a debate on a piece of mathematical legislation; an ex-teacher from eastern Indiana was saying: “The case is perfectly simple. If we pass this bill which establishes a new and correct value of π, the author offers our state without cost the use of this discovery and its free publication in our school textbooks, while everyone else must pay him a royalty.”

  Professor Waldo, horrified that the bill had passed the House, then coached the senators, and on its second reading, February 12, 1897, the Senate voted to postpone the further consideration of this bill indefinitely; and it has not been on the agenda since.

  * * *

  ANOTHER modern American circle-squarer was one John A. Parker, who published a book in New York in 187491 called The Quadrature of the Circle. Containing Demonstrations of the Errors of Geometers in Finding Approximations in Use. The book contains 218 pages on the quadrature of the circle, to which is added more of the author’s wisdom on polar magnetism and other matters.

  The great and fatal error of geometers (writes Parker) is seen in their declaration that the circle and the square are “incommensurable,” that there is no co-relation between circumference and diameter, and our first proposition shall be to dissipate this fatal error; therefore

  PROPOSITION FIRST

  The circumference of any circle being given, if that circumference be brought into the form of a square, the area of that square is equal to the area of another circle, the circumscribed square of which is equal in area to the area of the circle whose circumference is first given.

  Mr. Parker must have found the rejection of his theories very frustrating, because the above proposition (as is easily checked with paper and pencil) is perfectly correct; it is also perfectly trivial. Like most other modern circle-squarers, he did not even understand the problem; how, for example, was he going to construct a square whose perimeter is equal to the circumference of the circle? He substitutes numbers in his proposition, and thinking that the result constitutes a proof, goes on to say

  For if the circumference and diameter of a circle be really incommensurable, as geometers have affirmed, then no circle and square can be exactly equal to one another. But when it has been demonstrated as has here been done, that a circle and a square may be exactly equal one to the other, then it is demonstrated that the two are not incommensurable; and with this demonstration the whole theory of mathematicians … is proved to be fallacious.

  Elsewhere Parker “demonstrates” a theorem according to which π = 20612 : 6561 exactly. “And that proposition being proved,” he exclaims, “all the serial and algebraic formula in the world, or even geometrical demonstration, if it be subjected to any error whatever, cannot overthrow the ratio of circumference to diameter which I have established!” His triumph makes him generous, and he praises Metius for using the ratio 355/113 “more than a century past” (Metius lived in the 16th century), which is “the nearest approximation to the truth ever made in whole numbers.”

  The book contains more of Parker’s wisdom, such as the intriguing theorem that “The circumference of a circle is the line outside of the circle thoroughly inclosing it.” This theorem proves to be the undoing of all other geometers who use a line “coinciding with the utmost limit of the area of the circle,” instead of “inclosing” it as Parker does, so that “by this difference, with their approximation, geometers make an error in the sixth decimal place.”

  More interesting than Parker’s mathematical garbage, perhaps, is the psychology of his attitude to professional mathematicians. Throughout the ages, men with little education, small intellect, or both, have sometimes felt an envious dislike of their intellectual superiors, branding them as “effete snobs,” “eggheads,” and the like. Parker expresses this attitude as follows:

  I have found the Professors as a body, though learned in received theories, to be among the least competent to decide on any newly discovered principle. Their interest, education, pride, prejudice, self-love and vanity, all rise in resistance to anything which conflicts with their tenets, or which outruns the limits of their own reasoning. So little do they look beyond the principles inculcated by education, and so tenaciously do they hold on to these, that when driven from one principle, they fall back to another, and when beaten from all, they return again to the first, and maintain themselves by dogged assertion, or charging their assailants with ignorance and a lack of science.

  In consequence of this character of professors, the practical men of the age are at least a century in advance of the schools in all useful scientific knowledge.

  * * *

  JOHN A. Parker was, however, a mild case of circle-squaromania compared to one Carl Theodore Heisel, who described himself as Citizen of Cleveland, Ohio, and 33° Mason, and who in 1931 published a book whose title pages are reproduced on the next two pages. Not content with squaring the circle, he also rejects decimal fractions as inexact (whereas ratios of integers are exact and scientific), “disproves the world renowned Pythagorean problem accepted as absolute by geometricians for the last twenty-four centuries,” and extracts roots of negative numbers thus: √–a = √a–; √a–2 = –a. As for squaring the circle, his grand discovery amounts to setting π = 256/81, the very same value used by Ahmes the Scribe in Egypt some 4,000 years before Heisel. Substituting this value for calculations of areas and circumferences of circles with diameters 1, 2, 3,… up to 9, he obtains numbers showing consistency of circumference and area, “thereby furnishing incontrovertible proof of the exact truth” of his ratio, never noticing that he would have obtained the same consistency had he set π equal to the birthdate of his grandmother.

  From this it is probably clear that “this great American discovery, which cannot fail to be immortal, because absolutely irrefutable!” is not very interesting; but Heisel’s book throws considerable light on the mentality of the genius whose great revelations are not appreciated by a prejudiced and ignorant world.

  Title pages of Heisel’s book.

  Title pages of Heisel’s book.

  If his circle ratio is the same as that used in ancient Egypt, then his self-appraisal is reminiscent of Esarhaddon, King of Assyria in the 7th century B.C., who began an inscription carved in stone with “I am powerful, I am omnipotent, I am a hero, I am gigantic, I am colossal!” Heisel’s biography, facing a full-page photograph captioned

  CARL THEODORE HEISEL, 33°

  and written by himself, informs us that

  He is a man of stalwart figure, strong personality, and splendid physical and mental vigor. He possesses excellent health and has an abundance of energy. He has been a successful business man, an accountant, a student, an investigator and a practical philosopher … He has read exhaustively and has traveled extensively.

  Illustrious Brother Carl Theodore Heisel is a 33° Mason. He has been a serious student, a thorough investigator, a capable organizer, and a tireless worker in the various degrees of that order …

  He is not publishing this book with any idea of profit from the sale of such book. He is addressing it more especially to the practical student, who with common sense prefers to do his own thinking, and not blindly accept the opinions and teachings of professors of mathematics and geometry, who fear being ridiculed and ostracized by the profession generally, and losing their positions as professors or teachers, if they acknowledge any truth not found in the books as taught today …

  He believes that [his deceased colleague] Mr. Theodore Faber has solved the great problem, and that his ratios are absolutely true, accurate, incontrovertible, and to be the greatest mathematical discovery in the history of the world.

  But Heisel outdoes himself when he accuses a professor of the University of Cambridge of having stolen his great mathematical discovery after Heisel had communicated it to the Royal Society in London. “If the American public should ever wake up to the importance of the discovery,” writes Heisel, “and to the national honor reflected by it, this singular coincidence may be
come a question of international investigation.”

  Heisel was evidently a man of considerable means, for he had several thousand copies of the book printed and distributed free to libraries, colleges and scientists, and there is probably a copy of this curiosity in most American college libraries that he considered important in 1931.

  18

  THE COMPUTER AGE

  … the foulest thing I have ever read. That these “scientists” should toy with problems of such universal and profound significance shows only that our educational system is an abominable failure, turning out unnatural, immoral and monstrous specimens of humanity. Are these creatures now to change our lives in the name of science?

  From a reader’s letter to TIME

  THE story of π in the computer age of the 20th century is reminiscent of that of the digit hunters in the 18th and 19th centuries. The main difference is that where the digit hunters of the 18th and 19th centuries topped the standing records by tens and hundreds of decimal places, the computers and their programmers topped the standing records by thousands, and then by hundreds of thousands of digits. By 1967, the value of π was known to 500,000 decimal places. And, of course, where the digit hunters had drudged for months and years to find hundreds of decimal places, the computer that churned out half a million digits needed only 26 hours and 40 minutes (plus 1 hour and 30 minutes to convert the final result from binary to decimal notation). The similarity between the idiots savants of the 18th and 19th centuries and the imbecility of the 20th century computer has already been pointed out (here). Yet as we come to the end of the story, this similarity vanishes; for we are living at a time when some computers (more accurately, their programs) have become remarkably intelligent.