Irrational number number to ratios of continuous magnitude. In his commentary on Book 10 of the "Elements", the Persian mathematician Al-Mahani (d. 874/884) examined and classified quadratic irrationals and cubic irrationals. He provided definitions for rational and irrational magnitudes, which he treated as irrational numbers. He dealt with them freely but explains them in geometric terms as follows: In contrast to Euclid's concept of magnitudes as lines, Al-Mahani considered integers and fractions as rational magnitudes, and square roots and cube roots as irrational magnitudes. He also introduced an arithmetical approach to the concept of irrationality, as he attributes the following to irrational
Fear Is the Key force him to provide the necessary resources. Talbot has been working for the authorities all along and convinces Vyland and Royale that he is now ready to die on the ocean floor beside his family, eliciting a terror-stricken confession from the two men (their fear of death being the key that unlocks the confession) which is relayed to witnesses on the oil platform. Talbot then returns to the oil platform, where Vyland throws himself to his death from a ladder. Royale survives to be tried and sentenced to death and Talbot, politely turning down General Ruthven's offer of any reward