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Knud Leem was born in Haram to priest Niels Knudssøn Leem and his wife Anne Danielsdatter Bugge.
He started theological studies at Copenhagen University in 1713 and got his theological degree two years later, 18 years old. He worked as a teacher and assistant to more senior priests in Norway until 1725 when he got a position as missionary for the Samis in Porsanger. He was appointed vicar in Alta in 1728. In 1725 he moved back to southern Norway where he was appointed vicar in Avaldsnes.
Knud Leem started the linguistic study of Sámi when he published a grammar book of it in 1748. Between 1756 and 1768, he also published two dictionaries.
Leem's grammar book shows an insight into Sámi that was not present in many other grammar books of the same era. Leem uses an inflection classification quite similar to the one being used nowadays. He also commented on consonant gradation, but more as a tendency than as a rule.
Knud Leem’s most important book is his topographic work, Beskrivelse over Finmarkens Lapper deres Tungemaal, Levemaade og forrige Afgudsdyrkelse, (1767), supplied with comments from bishop Johan Ernst Gunnerus and a large historic-religious study written by E.J. Jessen-Schardeböll. Here Leem describes in both Danish and Latin the life and livings amongst the contemporary Lapp population, their garments, clothing and dress, food and cooking, hunting, fishing and sport equipment, shamanism and folk belief. A rich, but in many cases distorted, illustrated material, enlarges the value of the documentation about elderly Lappish culture, at the same time as the book is among the foremost topographic work published in the Nordic countries during the 18th Century.
- En lappisk Grammatica efter den Dialect, som bruges af Field-Lapperne udi Porsanger-Fiorden : samt et Register over de udi samme Grammatica anførte Observationers Indhold, hvorhos er føyet et Blad af den berømmelige Historie-Skriveres Hr. Baron Ludvig Holbergs Kirke-Historie oversat i det Lappiske Tungemaal med en Analyse over hvert Ord. (1748) (online)
- En Lappesk Nomenclator efter Den Dialect, som bruges af Fjeld-Lapperne i Porsanger-Fjorden. (1756) (online)
- Om Lappernes Afgudiske Ofringer (1760)
- Knud Leems Beskrivelse over Finmarkens Lapper, deres Tungemaal, Levemaade og forrige Afgudsdyrkelse, oplyst ved mange Kaabberstykker = Canuti Leemii De Lapponibus Finmarchiae, eorumqve lingva, vita et religione pristina commentatio, multis tabulis aeneis illustrata / med J.E. Gunneri Anmærkninger. Og E.J. Jessen-S Afhandling om de norske Finners og Lappers hedenske Religion. (with Erich Johan Gunnerus and Johan Ernst; 1767)
- Lapponico - Danico - Latina. (1768)