Recently, I was asked by Jesper Jansson "where exactly did the first published phylogenetic network appear?" Obviously, the answer to this question can depend on precisely how one defines "phylogenetic", especially as our current understanding of the word did not arise until the late 1800s, notably with the works of St George Jackson Mivart and Ernst Heinrich Haeckel (who actually coined the word "phylogeny"). Nevertheless, if we treat the concept broadly as requiring only an explicit reference to a genealogy, then it seems possible to nominate a candidate.
Mark Ragan suggested to me that, based on his own research as presented in his Biology Direct paper, the most likely candidate is the genealogical network of races of dogs ("Table de L'Ordre des Chiens") produced by Georges-Louis Leclerc, comte de Buffon (1707-1788). I have followed up this lead, and I agree with Mark that it is "not only a network but an explicitly genealogical one". Thus, it seems to me that this publication certainly qualifies as a phylogenetic network. Indeed, even Charles Darwin (from the 4th edition of the Origin, 1866, onwards) acknowledged Buffon as "the first author who in modern times has treated it [evolution] in a scientific spirit".
Buffon's magnum opus was the 36 volumes of the Histoire naturelle générale et particulière (Imprimerie Royale, Paris). The publishing history of this work is a mess, with dozens of French editions and numerous translations, and both official and bootleg printings. Indeed, this was undoubtedly the most popular work on natural history in the late 18th and early 19th centuries. The most readily available printed version today is the one edited by Jean Piveteau in 1954, although various editions are now available online. So, it is important to consult the first edition to arrive at a suitable date.
The illustration shown here is a foldout located between pages 228 and 229 of Volume 5, published in 1755 (Volume 1 had appeared in 1749). A larger GIF version [434 KB] is available for download from my homepage and a PDF version [2.6 MB] is on the RJR Productions webpage. The image is taken from the online (scanned) version of the first edition, located at: http://www.buffon.cnrs.fr/. [It is perhaps worth noting that the first edition of this volume of the Histoire was co-authored by Louis-Jean-Marie Daubenton; but the dog genealogy is clearly Buffon's work alone.]
On p. 225 of the Histoire, Buffon writes: "Pour donner une idée plus nette de l’ordre des chiens, de leur dégénération dans les différens climats, et du mélange de leurs races, je joins ici une table, ou, si l’on veut, une espèce d’arbre généalogique, où l’on pourra voir d’un coup d’œil toutes ces variétés : cette table est orientée comme les cartes géographiques, et l’on a suivi, autant qu’il étoit possible, la position respective des climats. Le Chien de Berger est la souche de l’arbre : ....." [The 1781 English translation by William Smellie is: "To give a clear idea of the different kinds of dogs, of their degeneration in particular climates, and of the mixture of their races, I have subjoined a table, or genealogical tree, in which all these varieties may be easily distinguished. This tree is drawn in the form of a geographical chart, preserving as much as possible the position of the different climates to which each variety naturally belongs. The shepherd’s dog is the root of the tree ....."]
This text is then followed by a description of the main lines of historical relationship among the dog breeds. Then, on p. 227 Buffon further notes: "Toutes ces races, avec leurs variétés, n’ont été produites que par l’influence du climat, jointe à la douceur de l’abri, à l’effet de la nourriture, et au résultat d’une éducation soignée ; les autres chiens ne sont pas de races pures, et proviennent du mélange de ces premières races : j’ai marqué par des lignes ponctuées, la double origine de ces races métives." [Smellie's translation: "All these races, with their varieties, have been produced by the influence of climate, joined to the effects of shelter, food, and education. The other dogs are not pure races, but have proceeded from commixtures of those already described. I have marked, in the table, by dotted lines, the double origin of these mongrels."]