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He went to the University of Oxford in 1555, probably as a member of Exeter College, Oxford (Wood doubts this). In 1559 he took his Bachelor's degree and proceeded to the degree of Master of Arts as a member of Christ Church, Oxford in 1562. He was brilliant and eloquent, and so esteemed as an orator that, with Edmund Campion, he was chosen to hold a public disputation before Queen Elizabeth in 1566.
Shortly afterwards, having applied himself to theology and acquired a wide reputation for his learning he was made a Fellow of Exeter College (1567) by the interest of Sir William Petre, who had founded several fellowships there. His ability would probably have won further promotion for him had not his religious opinions undergone a change, an indication of which was given in his argument with the Regius Professor of Divinity, whom he confuted.
Two years after his appointment to the fellowship he left Oxford and proceeded to Louvain, where he met William Allen (afterwards Cardinal). Allen secured him for his new college at Douai and appointed him its first prefect of studies. He was Allen's "right hand upon all occasions", acting as rector when he was absent and when the college was transferred (1578) to Reims.
Bristow is best known, however, as an earnest student, a powerful controversial writer, and, with Allen, as one of the revisers of the Douay Bible. His labours told upon a constitution naturally weak, and he was obliged to relinquish his work in 1581. In May of the same year he went to Spa, but having obtained no advantage there he was advised, after two months, to return to England. This he did in September, staying until his death (18 October) with Jerome Bellamy, a Catholic of means, at Harrow-on-the-Hill.
By his death the Catholic cause lost a zealous champion and a learned advocate. The Douai records speak of him in the highest terms as rivalling Allen in prudence, Stapleton in acumen, Campion in eloquence, Wright in theology, and Martin in languages.
- A Briefe Treatise of diuerse and sure wayes to finde out the truthe in this doubtful and dangerous time of Heresie: conteyning sundry worthy Motives vnto the Catholic faith, or considerations to moue a man to beleue the Catholikes and not the Heretikes (Third edition entitled Motives inducing to the Catholike Faith.)
- Tabula in Summam Theologicam S. Thomae Aquinatis
- A Reply to Will. Fulke
- Demandes to be proponed of Catholikes to the Heretikes
- A Defence of the Bull of Pope Pius V
- Annotations on the Rheims translation of the New Testament
- Carmina Diversa
- Motiva Omnibus Catholicae Doctrinae Orthodoxis Cultoribus pernecessaria
- (The last two being left in manuscript.)