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LAMBERT, JOHANN HEINRICH. - [LAMBERT'S CONTINUED FRACTIONS AND IRRATIONALITY OF "N"]

Mémoire sur quelques Propriétés remarquable des Quantités transcendentes circulaires et logarithmiques.

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(Berlin, Haue et Spener, 1768). 4to. No wrappers as issued in "Mémoires de l'Academie Royale des Sciences et Belles-Lettres", tome XVII, pp. 265-322 and 1 folded engraved plate. ¶ First edition, journal issue. "One of Lambert's most famous results is the proof of the irrationality of "N" and e. It was based on the continued fractions, and two such fractions still bear his name."(DSB).
"Euler's work on continued fractions was used by Johann Heinrich Lambert.....to prove that if x is a rational number (not 0), then ex and tan x cannot be rational. He thereby proved not only that ex for positive integral x is irrational, but that all rational numbers have irrational natural (base e) logarithms.From the result of tan x, it follows tha, since tan (n/4)= 1, that neither n/4 nor n can be rational.Lambert actually proved the convergencee ofthe continued fraction expansion for tan x. (Morris Kline). - Struik, A Source Book, Chapter V, No 17)

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