Euler writes in a clear and direct way, as if speaking to the reader. There is a complete lack of pretense and affectation. Euler wrote for the interested, informed reader, but he did not assume that the reader already knew what he had to say.
To express his ideas, Euler tends to use active verbs in preference to nouns, and he tends to focus on the operations and activities involved, and not on the definitions. He defines things as he needs them, and his definitions tend to be simple.
Euler manipulates linguistic expressions with the same ease that he manipulates mathematical expressions, and while his sentences are sometimes long and complex, they never ramble. Throughout, the reader will perceive a sense of enthusiasm and discovery, and because Euler is as generous as he is skilled, and because of the tone and manner of the writing, the reader is led to feel as though he is in on the discovery himself.
None of this occurs with the slightest degree of condescension or oversimplification, and Euler does credit the reader with an attention span equal to his own (as, for example, in the detailed treatment of the many special cases in the first part of the article). Still, interesting and beautiful results sometimes pop out as if from nowhere, and the reader wonders how that happened. But it is all right there. The reader has experienced an "Euler moment," and wishes that more mathematical writing were like this.
To read my translation of Euler's article, click here.