Catel, Traité d'harmonie (1802)

Charles-Simon Catel wrote a very efficient harmony textbook (70 pages in its first edition) that was adopted by the recently founded Paris Conservatoire.* It contains no speculative theory, but Catel does invoke the harmonic series to generate his list of eight basic chords ("all chords that are practical for harmony"; p. 7). He does this in two stages, starting from the fourth harmonic (note that he is using the dominant of C major, not the tonic) (p. 6):

. . . then starting from the fifth harmonic and ignoring "intermediate tones" (p. 7):

Here are the eight chords (they are in vertical format in the Traité; I have rearranged them in horizontal format):

Catel then gives one chapter (article) to each of the eight. For the dominant ninth, he shows both direct and internal resolutions (my terms, not his) (p. 17):

He permits inversions of the ninth chord, but not the fourth inversion because "it is necessary that the ninth above the root be maintained") (p. 17; the examples below are from p. 18, put into horizontal format here).

Despite this promising beginning, the dominant ninth plays almost no role in the remainder of the treatise. It does appear in a long list of chordal suspensions (which Catel calls "prolongation") in chapter 7 "Notes de Passage: Prolongations, Suspensions, Retardement," where the "harmonie simple" (at the left below) is subjected to prolongation of the "chord of the leading tone over the dominant, or the dominant ninth" (p. 29) (at the right below).

The dominant ninth does not appear in subsequent chapters on suspensions and sequences, pedal points, scale genres, altered chords, and modulation.

*Note: Blättler 2013, 56n81: "The harmony texts used at the Paris Conservatoire through 1941 are, in chronological order by their years of use: Charles-Simon Catel, Traité d’harmonie, 1802-1816; Antoine Reicha, Cours de composition musicale, 1816-1851; Augustin Savard, Cours complet d’harmonie theorique et pratique, 1851-1862; Henri Reber, Traité d’harmonie, 1862-1889; Theodore Dubois, Notes et Etudes (supplement to Reber’s text), 1889-1921; Theodore Dubois, Traité d’harmonie, 1921-1950. (Rieder, 16)."

Reference:
Charles-Simon Catel, Traité d’harmonie (Paris, 1802). Digital facsimile published on the Internet Archive. Source: University of North Carolina at Chapel Hill.   Note: An edition from 1874 shows no changes in text or examples for the dominant ninth. Source: Bibliothèque nationale de France at gallica.bnf.fr.
Mildred Freeman Rieder, “Traité d’harmonie by Theodore Dubois in the Context of 19th-Century French Harmonic Theory and Pedagogy,” MM thesis, The University of Western Ontario, 1995.