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This paper deals with the history of the classifications of algebraic curves that have been associated with the notions of order, class, and genera, the paternities of which are usually attributed to Isaac Newton (1704), Joseph Diez Gergonne (1828), and Alfred Clebsch (1865), respectively. The corresponding episodes are revisited with regard to the issue of classification, and are embedded in a larger narrative with the help of corpuses of texts that allow following the circulation and the collective adoption of the three notions from Descartes’ La Géométrie (1637) to the end of the 19th century. In addition to terminological questions, special care is given to the instability of both the meanings of, and the hierarchy between orders, classes, and genera, as well as to the transition of their status of categories to their status of numbers.