Looking again (see "Jerome Gambit: Early Sources") through Peter J. Monte's The Classical Era of Modern Chess (2014) I noticed that the Jerome Gambit-related line 1.e4 e5 2.Nf3 Nc6 3.Bc4 Na5 4.Bxf7+ (see "Offside" for starters)
has appeared as early as in Rodrigo "Ruy" López de Segura's Libro de la invencion liberal y arte del juego del Axedrez (1561) and Giulio Cesare Polerio's Ordini di giuochi degli scacchi in diversi modi, cosi di mano, come sottomano, cio e in offenza, e dife[n]za co[n] altri bellissimi partiti, sono di Guilio Cesare Polerio alias l'Apruzzese. Giocandosi del Pari (1594).
Monte writes
1.e4 e5 2.Nf3 Nc6 3.Bc4 Na5: Mentioned in Lopez III, xiv, the move 3...Na5 is refuted by the sequence 4.Bxf7+ Kxf7 5.Nxe5+ etc. in Polerio's D16 and 29.
The early lines of play are given
1.e4 e5 2.Nf3 Nc6 3.Bc4 Na5
Last move in Lopez III, xiv. 4.Bxf7+ Kxf7 5.Nxe5+
* 5...Ke7 6.d4 d6 7.Bg5 Nf6 8.Nd3 h6 9.Bh4 g5 10.Bg3 Bg4 11.f3 Bh5 12.Qd2 in Doazan D16
* 5...Ke6 6.Qg4+ Kxe5 7.Qf5+ Kd6 8.Qd5+ Ke7 9.Qxa5 and
* 5...Ke8 6.Qh5+ Ke7 7.Qf7+ Kd6 8.Qd5+ Ke7 9.Qxa5 in Doazan D29
"Doazan" refers to Gabriel Eloy Doazan's 1843 article in Le Palamede, where his "Un manuscrit sur les echecs" presented one of Polario's manuscripts.

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