Final answer:
There is no specific name for the illness of becoming mentally deranged from trying to square the circle, but it may fall under the historical concept of "monomania," an obsolete psychiatric term describing an obsessive fixation on a single idea.
Step-by-step explanation:
The illness related to the obsession with trying to "square the circle," a classical problem of constructing a square with the same area as a given circle using only a finite number of steps with compass and straightedge, has not been specifically named in the historical context. However, the term "monomania" was used in the early modern psychiatry, particularly by the French psychiatrist Jean-Etienne-Dominique Esquirol and his protégé Georget. They described monomania as an obsessive fixation on a single thought or idea, which could potentially explain the fixation and resulting mental breakdown experienced by some when attempting this impossible mathematical task.
The derangement from such an endeavor might be seen as an example of monomania, where the sufferer becomes consumed by the attempt to solve an unsolvable problem. This historical perspective on mental illness has evolved, and the understanding of such obsessive thoughts is now considered within the broader context of mental health disorders rather than a single definable disease. For instance, Esquirol's belief in physiognomy as a method to diagnose mental disorders is now largely discredited.
Throughout history, there have been many instances where behaviors and identities, such as the desire for freedom by enslaved people (drapetomania) or homosexuality, were incorrectly classified as mental disorders. These perspectives have since been revised, recognizing that such classifications were based on social values and misunderstandings rather than biological facts or genuine mental health conditions.