Mathematics and Squaring the Circle

Welcome to Aquarius, Volume 10 (September 24, 2007)

Can We Make a Square Pi?

Brain byte: Even children who don’t know what trigonometry is will tell you that “You can’t square the circle.“ Why is this idea so deeply etched in the human mind? Squaring the circle has an impact on our understanding of geometry, mathematics, history, religion, anthropology and archaeology. The stakes are very high here, my friend. When you read the bumper sticker that says “Question Authority,” which authority is it that you think should be questioned? Is it your old church? Or is it old science? Will you examine the work of John Manimas, who posted on his website,, the steps required to square the circle exactly ?

In 1882 a German mathematician, Ferdinand Lindemann (1852-1939), proved that Pi is a transcendental number and that it is logically impossible to "square-the-circle" using only a compass and straightedge. Thus was born a mathematical doctrine that is known to adults and children throughout the world. This fact of life is amazingly popular. Ask anyone, any age, from any culture. They will tell you, Yes, it is true, you cannot square the circle.

Why is this doctrine so deeply entrenched in the human mind, and what does it mean? And while we Americans are often enthralled by another doctrine, the doctrine that we should "question authority," one might wonder if we are able to question this doctrine, a mathematical doctrine. Is being a "Christian" nation holding us back from questioning a mathematical doctrine? Or, are we really a mathematical nation that cannot question the authority of mathematicians? Irrational defense of a doctrine is called by psychologists and sociologists "institutional behavior." Is mathematics an institution? Do mathematicians defend their mathematical doctrines in the same way that clergy defend the theological doctrines of their respective churches? Do we really believe in God, or do we believe, really, in science?

A retired man in Vermont, John Manimas, claims that after thirty years of study and experimentation with trigonometry he has discovered how we can square-the-circle with only a compass and straightedge. Solving this ancient riddle means constructing a circle with a compass and straightedge and then constructing a square, with only the compass and straightedge, that has exactly the same area as the circle. The emphasis is on exactly. This is an ancient challenge that humans have worked on for centuries, until 1882. The history of our obsession with the riddle of "squaring-the-circle" and Lindemann's relieving us of it is summarized by William Dunham in his book Journey Through Genius (Wiley, 1990), pages 23-26. Dunham points out that attempts to solve this problem are recorded since the time of Hippocrates, who lived in the fifth century before Christ. Why did we give up? Mr. Manimas believes he knows why Lindemann persuaded us to turn away from the challenge of this riddle: precision, or the lack thereof.

In 1882, the world was obsessed with the prospect that science, and mathematical certainty, would enable use to solve all of the world's problems. We had discovered evolution, and the dinosaurs. We could learn the truth, the real truth about everything through cold, hard science and nothing could stop us. But, there is a mathematical tool that Lindemann and everyone of the nineteenth century did not have: computers. The computer was still just a dream. Babbage's room-sized intricate yet clumsy calculator, often deemed to be our first "computer," was mechanical, not electronic. Besides that, our first computer was the abacus. In 1882, we did not have the precision that we have today with the hand-held scientific calculator, available to anyone who has ten or twenty dollars, and the desktop computer. With the desktop computer, anyone with a high school education can train themselves to write mathematical programs and run thousands of calculations in pursuit of whatever mathematical truth fascinates them, which is what Mr. Manimas did. And as a result he claims that Lindemann was wrong and we can square the circle with only a compass and straightedge. And further, the reason we can is because a well-known number, Phi squared (2.618033989...), is the product of Pi times (5/6) times MQ. Manimas says MQ is a defined irrational number, like Pi, and that it can be constructed with a compass and straightedge. That means it follows logically (or trigonometrically) that we can construct the value of Pi and a square with the area of Pi exactly. In 1882, no mathematician would have been able to perceive the number MQ, and its square root, M (1.000007661...) as part of the solution.

Manimas has his entire description of the solution on his website at The introduction and guide is 29 pages; the values and constructions required is 62 pages; and a third part about the meaning of proportion is 27 pages. He argues that the purpose of the ancient riddle was to lead us to discover that the ancients knew Pi to an extremely high level of precision, and all that implies about human history. If the ancients did know Pi to extreme precision, we need to figure out how they knew.

Manimas posted his solution on his website August 16, 2007. He has since written to the American Mathematical Society, the Mathematical Association of America and the American Academy of Sciences asking them to validate his work. He has also sent letters and emails to numerous mathematicians at universities in the United States, Canada and the United Kingdom.

How will this script play out? Will the mathematicians of the world just label him as another "square-the-circle" nutcase? Or, will they take a serious look at his work? By the way, that number MQ in [Pi x (5/6) x MQ] = 2.618033989... is actually a very simply defined number. Phi (1.618033989...) is that number such that Phi squared (2.618033989...) plus 1 over Phi squared (0.381966011...) equals 3 exactly. MQ is that number such that M squared equals MQ (1.000015321...), and MQ plus 1 over MQ (0.999984679...) equals 2 exactly. This is what Manimas claims to have discovered, and which mathematical fact makes it possible to square the circle. Squaring the circle is possible, he says, because we can construct the line length of MQ using only the compass and straightedge. For that "proof," you need to look at his website. It has something to do with the secant of 18 degrees raised to the fourth power. (Don't freak out. Trigonometry is more beautiful than Ecstasy and much safer.) Manimas also invites mathematicians to ridicule his work, as they have ridiculed "circle-squarers" for the past 130 years, if they feel such ridicule is warranted. But maybe they should check his work first and see if they are able to question one of the doctrines of the church of mathematics.

One more fact about MQ. If you would like to see it, you need to have a hand-held scientific calculator. MQ = 1 over [Pi * (0.381966011...)*(5/6)]. First, get the sine of 18 degrees (0.309016994...). Multiply that by 2 (0.618033989...). Square that (0.381966011...). Multiply that by 5 (1.909830056...). Divide that by 6 (0.318305009...), times Pi = 0.999984679..., and then press the (1/x) button to get the inverse of 0.999984679..., (1.000015321...) = MQ. Manimas says that even if it turns out that he made a mistake, MQ is still a very beautiful number.

Link to: (Welcome) or (Geometry Alpha Index) .