I guess Desargues did not use that term. Anyone could help me know where did it appear for the first time? Thanks
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.
“The first known use of the expression 'projective geometry' only dates back to Olry Terquem (1859)” in the article Sur diverses géométries, Nouvelles Annales de Mathématiques, 18, 445–446, according to https://www.cambridge.org/core/books/abs/anachronisms-in-the-history-of-mathematics/measuring-past-geometers-a-history-of-nonmetric-projective-anachronism/75A9056750446E37CABBDAFE12E32986 page 263.
Actually Desargues used the term "project". The title of his fundamental work was "Brouillon d'un Project". Notice the unusual spelling (in modern French the word is spelled without the "c"). It is possible that his use of the term (apparently in a different sense) may have influenced the subsequent choice of the standard name for the field.
In a series of six articles in the Athenaeum (1861) (pp. 446,509,549,617,652,727) on the history of perspective, Augustus De Morgan makes reference to "projective geometry":
More than thirty years ago I showed a teacher from the country some models for teaching solid geometry, to avoid the confusion which arises from drawing solids on a plane. He asked me whether I was not afraid of making the subject too easy; I could but answer that, let it be as easy as it might there was plenty of trouble ready up above And this is the case with the new projective geometry, and with all other branches of thought: make a plaything of what was difficult, and you will find no more than a difficulty in what was impossible. (447)
Not as early as Viktor Blasjo's 1859 reference.
The Penny Cyclopaedia (1841): The "geometry of projections", not literally "projective geometry":
We have already spoken of the geometry of projections [GEOMETRY p. 156]: unfortunately there is no elementary work which gives a general view of its first principles; and until such a work shall appear, the student must search for himself the writings of Monge, Carnôt, Chasles, Poncelet, &c. 'The History of Geometry' by M Chasles referred to in the article cited will furnish many more references and the 'Propriétés Projectives des Figures,' by M. Poncelet is perhaps the work in which the student may most easily make an advantageous beginning of the subject. (p. 40)