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Euler was a non-confrontational and deeply religious person. He was kind and could get on well with anyone. He worked under any circumstances and in any environment: “A baby on his lap, a cat on his back — that’s how he wrote his immortal works,” as one of his contemporaries said.

Usually this phrase is attributed to Dieudonné Thiébault, but we were unable to identify the source. However we found similar phrasing in Camille Paganel's Histoire de Frédéric le Grand, 1847, volume 1, page 448,

un chat sur l'épaule et ses enfant sur les genoux, rédigeait des mémoires admirés du monde savant.

Could you help me trace back the origin of this phrase (if this goes to Thiebault...)?

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The origin is undoubtedly Dieudonné Thiébault, as the anecdote is reported in his personal recollections of his stay in Berlin, published in Mes souvenirs de vingt ans de séjour à Berlin, vol. 5, (Paris, 1804):

[Euler] a fait faire aux sciences mathématiques, des pas de géant; et ses immenses travaux ne lui coûtoient rien : c'est au milieu de sa famille, et du bruit que des enfans peuvent fàire; c'est en jouant lui-même avec celui qu'il prenoit sur ses genoux, et avec un angola [sic] monté sur son épaule, qu'il a composé quelques-uns de ces Mémoires que l'Europe a admirés et admirera toujours. [page 13]

i.e.

He made giant steps in the mathematical sciences, and his immense works cost him nothing: it is in the middle of his family, and the noise that children can make; it is while playing himself with the one he took on his knees, and with an Angora cat mounted on his shoulder, that he composed some of these Memoirs that Europe has admired and will always admire.

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    $\begingroup$ Thank you very much! indeed, it is exactly the same sense, and it explains why I could not find it: I was looking for 'un/e chat'. $\endgroup$ Commented Mar 3, 2023 at 13:40

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