I guess Desargues did not use that term. Anyone could help me know where did it appear for the first time? Thanks
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$\begingroup$ According to this lecture by Prof. Wildberger on the history of mathematics, and the introduction of projective geometry.. > Projective geometry began with the work of Pappus, but was developed primarily by Desargues, with an important contribution by Pascal. Although if you are asking about the word and not the work, I don't know. $\endgroup$– John AlexiouJan 20, 2020 at 14:58
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I looked in MathWords and found this
PROJECTIVE GEOMETRY. In his account of the origins of projective geometry Katz (1993) mentions Pascal, Desargues and Monge before saying that J.-V.Poncelet composed the first text in synthetic projective geometry: his Traité des Propriétés Projectives of 1822. See here for an extract from Poncelet’s Introduction.
The OED finds the term projective geometry used in English in 1885 in Charles Leudesdorf’s Elements of Projective Geometry, a translation of Cremona’s Elementi di geometria proiettiva of 1873.
So, perhaps the first use in English was 1885, but that was a translation from Italian. First uses in other languages are (of course) not answered by this.
answered Jan 17, 2020 at 14:11
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$\begingroup$ Well, Cremona's book credits Klein who wrote "Der Weg hierzu führt durch die projectivische Geometrie" (1871, p. 419) and "Daher wahlen wir im Texte ... das Wort „projectivische Geometrie“" (Erlangen program, 1872, pp. 1, 41). $\endgroup$ Apr 29, 2020 at 14:34