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Observations placeholder

Cardano, Gerolamo - Universal joints and 'imaginaries'



Type of Spiritual Experience


A description of the experience

An Essay on the Psychology of Invention in the Mathematical Field – Jacques Hadamard

Intermediaries seem to have existed [on many occasions of inspiration], I think of Socrates’ ideas being suggested to him by a familiar daemon or also the nymph Egeria whom Numa Pompilius used to consult frequently.

An analogous example can possibly be spoken of in the mathematical field.  It is Cardano, who is not only the inventor of a well known joint which is an essential part of automobiles, but who has also fundamentally transformed mathematical science by the invention of ‘imaginaries’.  Let us recall what an imaginary quantity is.  The rules of algebra show that the square of any number, whether positive or negative is a positive number: therefore, to speak of the square root of a negative number is mere absurdity.  Now, Cardano deliberately commits that absurdity and begins to calculate on such ‘imaginary’ quantities.

One would describe this as pure madness; and yet the whole development of algebra and analysis would have been impossible without that fundament – which of course, was, in the nineteenth century, established on solid and rigorous bases.  It has been written that the shortest and best way between two truths of the real domain often passes through the imaginary one.

We have reported Cardano’s case with Socrates’ and Numa Pompilius’s, because he too is reported by some of his biographers to have received suggestions from a mysterious voice at certain periods of his life........................

Cardano was one of the most extraordinary characters of that extraordinary time.  It could be naturally expected that that discovery of imaginaries which seems nearer to madness than to logic and which, in fact, has illuminated the whole mathematical science, would come from such a man whose adventurous life was not always commendable from the moral point of view, and who from childhood suffered from fantastic hallucinations to such an extent that he was chosen by Lombroso as a typical example in the chapter ‘Genius and insanity’ of his book on the Man of Genius.

The source of the experience

Cardano, Gerolamo

Concepts, symbols and science items


Inner speech


Science Items

Activities and commonsteps



Difficult birth


Being left handed
Premature birth