Final answer:
Johann Peter Eckermann is known for approaching Goethe with his manuscript 'Reflections on Poetry' (option a); he did not secure a bourgeois life through a law degree, nor is there evidence of him learning English and Mandarin before meeting Goethe.
Step-by-step explanation:
Among the given options about Johann Peter Eckermann, statement a.) is true: he approached Goethe by sending him his manuscript entitled "Reflections on Poetry."
Eckermann was born in a poor family in Hanover, not Weimar, similar to the background of Johannes Kepler who grew up in poverty in the German province of Württemberg. Although Eckermann had indeed a modest life, he did not secure it through a law degree but was recognized for his work as Goethe's close associate and for their conversations documented in "Conversations with Goethe."
There is no record of Eckermann learning English and Mandarin before meeting Goethe, hence rendering statement d.) false.