Statement I is false: The first edition of Samuel Hahnemann's Organon of Medicine, titled "Organon der rationellen Heilkunde" (Organon of Rational Medicine), was published in Torgau in 1810, not in Koethen. Hahnemann later moved to Koethen, where he published subsequent editions of the Organon.
Statement II is false: Hahnemann's significant precursor works to the Organon include his essays in Hufeland's Journal, such as "Essay on a New Principle for Ascertaining the Curative Powers of Drugs" (1796) and "The Medicine of Experience" (1805). While these articles laid the groundwork for his homeopathic principles, the titles "**Aesculapius in balance**" and "**The Medical Observer**" are not typically recognized as major precursor articles by Hahnemann leading directly to the first edition of the Organon.
Therefore, both Statement I and Statement II are incorrect.