Popularized in folklore was Merlin the wizard who lived in a castle, were there actually any real well known alchemists that worked for kings?

The Wars of the Roses are absolutely riddled with claims and counterclaims of witchcraft - one story that took flight in the later Tudor period was that Edward III (1312 – 1377) imprisoned the Majorcan Franciscan and theologian Raymund Lull (Ramon Llull) in the Tower of London and forced him to transmute lead into gold.

Edward III owned a number of alchemical manuscripts, dedicated to him and ascribed to a "Raymund" (or his "disciples"). In actuality there was something of a cottage industry of esoteric writings misattributed to Lull, but there's nothing linking him to alchemy (in fact, he denounced it) or to England, other than dubious prefaces to dubious texts and an account attributed to Abbot Cremer of Westminster. There's no record of Cremer either, so it's not particularly compelling.

In a roundabout way this loops back around to context of your question: Edward III was influenced by Arthurian myth and grail lore, even proposing an ambitious and self-aggrandising Round Table for his own knightly brotherhood, the Order of the Garter, and ordering a search for the remains of Joseph of Arimathea. As an distant ancestor to the Tudor line, further mythologising of Edward III took place in the 16th Century, including the anachronistic prophecies of the wizard Merlin (Tretise of Merlyn, 1529) that apparently predicted the king's battlefield victories - its unsurprising that later accounts tried to give him his own Merlin in that context.

Jonathan Hughes argues in The Rise of Alchemy in Fourteenth-Century England: Plantagenet Kings and the Search for the Philosopher's Stone that both Edward III and his son Richard II were serious patrons of alchemy. (Hughes seems unsteady with some of his dates and details, so I wouldn't recommend the book as any other than a starting point).

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