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Condorcet v. The Circle-Squaring Cranks

Hobson_3While cloistering myself in the Natural Sciences Library to finish an article, I happened upon a book called Squaring the Circle, which is a minute historical exploration of that famous scientific problem, written in 1911 in impeccably dry English scientific prose by one E.W. Hobson, Sc.D, LL.D., F.R.S., Sadleirian Professor or Pure Mathematics, and Fellow of Christ's College, in the University of Cambridge. This biography of Hobson observes that he was '[b]rought up in rigidly Low Church surroundings ...' but 'developed strong views of rationalism, becoming ... an avowed radical and agnostic'. On pages 3 and 4, he notes that attempts to solve this famously insoluble problem* have occupied uncounted cranks over the centuries:

The solutions propounded by the circle squarer exhibit every grade of skill, varying from the most futile attempts, in which the writers shew an utter lack of power to reason correctly, up to approximate solutions the construction of which required much ingenuity on the part of their inventor. In some cases it requires an effort of sustained attention to find out the precise point in the demonstration at which the error occurs, or in which an approximate determination is made to do duty for a theoretically exact one. The psychology of the scientific crank is a subject with which the officials of every Scientific Society have some practical acquaintance. Every Scientific Society still receives from time to time communications from the circle squarer and the trisector of angles, who often make amusing attempts to disguise the real character of their essays. The solutions propounded by such persons usually involve some misunderstanding as to the nature of the conditions under which the problems are to be solved, and ignore the difference between an approximate construction and the solution of the ideal problem.

It is a common occurrence that such a person sends his solution to the authorities of a foreign University or Scientific Society, accompanied by a statement that the men of Science of the writer's own country have entered into a conspiracy to suppress his· work, owing to jealousy, and that he hopes to receive fairer treatment abroad. The statement is not infrequently accompanied with directions as to the forwarding of any prize of which the writer may be found worthy by the University or Scientific Society addressed, and usually indicates no lack of confidence that the bestowal of such a prize has been amply deserved as the fit reward for the final solution of a problem which has baffled the efforts of a great multitude of predecessors in all ages.... It is interesting to remark that, in the year 1775, the Paris Academy found it necessary to protect its officials against the waste of time and energy involved in examining the efforts of circle squarers. It passed a resolution, which appears in the Minutes of the Academy, that no more solutions were to be examined of the problems of the duplication of the cube, the trisection of the angle, the quadrature of the circle, and that the same resolution should apply to machines for exhibiting perpetual motion. An account of the reasons which led to the adoption of this resolution, drawn up by Condorcet, who was then the perpetual Secretary of the Academy, is appended. It is interesting to remark the strength of the conviction of Mathematicians that the solution of the problem is impossible, more than a century before an irrefutable proof of the correctness of that conviction was discovered.

Apparently the problem is insoluble because pi is a transcendental number, a fact which was proven in 1882. After this introduction, Professor Hobson proceeds, over hundreds of inadvertently Kafkaesque pages, to minutely detail every single failed attempt to solve this problem. One of the more exotic ones gave rise to this diagram:


* Just to be clear, I have never attempted to solve the problem. In fact, I've never even attempted to understand it.


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