The church said Galileo's theories of the heavens and Earth were wrong; but the theories were right. Some "experts" said the Wright Brothers would never fly; but they did.
And it's long been held that mathematicians can't transform a circle into a square; but they have . . . sort of.For 2,000 years, mathematicians racked their brains trying to figure out whether it was possible to construct a square from a circle of precisely the same area, using only a straight edge and a compass.
It's the kind of mathematical exercise that has no known practical application. Hence it delights mathematicians, who tend to like their numbers big, pure and useless.
During the past century, aided by rigorous rules of logic, mathematicians proved the quest was hopeless. Using only rulers and compasses, they could no more turn a circle into a square than they could turn a frog into a prince.
So some mathematicians tried to "square" circles with scissors - specifically, by slicing up the circle, then trying to rearrange it into a square, like a jigsaw puzzle.
But that attempt was trounced in 1963 when University of California-Berkeley mathematicians Lester E. Dubins, Morris W. Hirsch and Jack Karush proved mathematically that even scissors couldn't be used to square a circle.
As perpetual motion is to physicists and Big Foot is to biologists, so "squaring the circle" is to mathematicians - a fantasy - if one uses only ordinary instruments such as rulers, scissors or compasses.
Enter Miklos Laczkovich, professor at Lorand Eotvos University in Budapest, Hungary, who has excited his colleagues by using an unusual approach to square a circle.
"The Circle Can Be Squared!" says a headline in Science magazine.
"One hundred generations (of mathematicians) have not toiled in vain," the July issue of Scientific American declares. "The circle has been squared."
Normally such news would gladden the hearts of pseudoscientists, who love to send science writers crank "proofs" that Einstein was wrong, that the CIA is monitoring their brain waves, and that the circle can be squared.
But they have no cause for celebration. Squaring a circle is still impossible the old-fashioned way (using straight edges, compasses and scissors).
Instead, Laczkovich squared a circle via a more bizarre technique. He found that if he cut a circle into jillions of pieces with very odd shapes - nothing remotely resembling lines or curves - he could reassemble them into a square of precisely the same area.
How many pieces? The number one followed by 50 zeroes. His proof is strictly theoretical, a product of pure logic and calculation. No one is likely to verify it using a real piece of paper and a pair of scissors.
Here's why, as explained by Russell Ruthen of Scientific American: "If each of the pieces averaged one square centimeter in size, then a circle composed of the pieces would cover about a 10th of our galaxy."
Laczkovich's achievement "is a first-rate piece of work," Dubins says. However, like most mathematical achievements, it has no immediate practical application.