How can I critically compare and contrast the ethical teachings of either Emmanuel Kant or Jeremy Bentham and the Catholic Church, with reference to the bioethical issue of euthanasia?

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Jeremy Bentham espoused a philosophy of utilitarianism, which stated that an act should be judged as right or wrong according to whether it creates the greatest happiness. His "felicific calculus" was an algorithm to calculate the degree to which an action would result in pleasure or pain, as he believed that an action was morally right if it resulted in more pleasure than pain. With regard to euthanasia, Bentham would be in favor of a person choosing euthanasia if the process would put an end to a great deal of pain and result in the pleasure of that person (though pleasure is difficult to measure after death, it would be pleasurable for the person to imagine an end to his or her pain). Under Bentham's utilitarianism, it is the right of the individual to end his or her life.

The Catholic Church, on the other hand, espouses the view that the life of each person is sacred. Therefore, ending that life voluntarily is sinful and in opposition to the laws of God (see the link below for more on the Catholic Church's policy). While many proponents of euthanasia believe the process eliminates suffering, the church nonetheless believes the practice is an intentional form of killing and is therefore against the laws of God. 

Sources:

Harrison, Ross. Bentham. London: Routledge and Kegan Paul, 1983.

Hart, H.L.A. "Bentham on Legal Rights," in Oxford Essays in Jurisprudence (second series), ed. A.W.B. Simpson (Oxford: The Clarendon Press, 1973), pp. 171-201.