"Squaring the circle" means to a construct a square equal in area to a given circle. Only a compass and straight lines are allowed. Although it can be proved that this is impossible, it has not actually put off people providing "solutions". It is interesting because it makes you think about the process of thinking itself, and how faulty logic can come to be regarded as true. I have lifted the text out of The Joy of Pi. My favourite quote is the first one that is listed, since it involves the most amount of rubbish.
It is utterly impossible for one to accomplish the work in a physical way; it must be done metaphysically and geometrically, not mathematically. When approached in this manner the problem it is easy of solution. It is stated by occultists that the number 12 squares the circle and it is necessary to take into consideration the process before we can understand this; when correctly understood we know it to be perfectly true.
- A.S. Raleigh, Occult Geometry, 1932
The original conception, its natural harmony, and the result, to my own mind is a demonstrative truth, which I presume it right to make known, though perhaps at the hazard of unpleasant if not uncourteous remarks.
- James Sabben, after declaring that pi equals 3.14176 in his Method to Trisect a Series of Angles Having Relation to each Other, 1848
Now, being in my 81st year ... I have found time to set my thinks to work. After three long days I was divinely blessed to discover the true and ancient value of pi. This is the exact value used by the Great Creator when he created our earth and its four orbit cycles. The World's unsolved problem, i.e. the complete interlocking of a triangle, circle, and square - having equal areas - is in the construction of this Pyramid. This truly confirms that the Great Creator used the Pi formula of 3 (7.1)/(50) = 3.1420000. The very fact that the Great Creator, Archangel Michael, should be sufficient evidence for our adoption.
- Francis Michael Darter, The Story of True Pi, 1962
No amount of attestation of innumerable and honest witnesses, would ever convince anyone versed in mathematical and mechanical science, that a person had squared the circle or discovered perpetual motion.
- Baden Powell, Essays and reviews, 1860
Surely the cyclometer is a Darwinite development of a spider, who is always at circles, and always begins again when his web is brushed away.
- Augustus De Morgan, A Budget of Paradoxes
One nineteenth-century circle squarer, after praising God repeatedly for choosing him to reveal "this precious mathematical jewel", goes on to show that an inscribed circle can be broken down into four congruent triangles, and a circumscribed sqaure is made of eight. Therefore, he asserts, the area of the circle must be the area of six triangles.
Augustus De Morgan tells us in his Budget of Paradoxes that the eighteenth-century Frenchman Joseph-Louis Vincerns de Mouleon de Causans "cut a circular piece of turf, squared it, and deduced original sin and Trinity. He found out that the circle was equal to the square in which it is inscribed, and he offered a reward for detection of any error, and actually deposited 10,000 francs as earnest of 300,000. But the courts would not allow anyone to recover."
Nineteenth-century writer James Smith surely holds the record for most publications on the topic of circle squaring. However, it should be noted that he paid for each and every one to be printed. Curiously, he always argued the same point: that pi is 3 1/8. Augustus De Morgan once wrote of him, "He is beyond doubt the ablest head at unreasoning, and the greatest hand at writing it, of all who have tried in our day to attach their names to error. Common cyclometers sink into pure orthodoxy by his side. ... We can only say this: he is not mad. Madmen reason rightly upon wrong premises: Mr. Smith reasons wrongly upon no premises at all."