Cai is typically Tuscan in the area that includes the provinces of Pisa and Florence, Caia has a small Calabrese stock in Bagnara Calabra and Oppido Mamertina and the main stock in Syracuse and Avola, Caio, in addition to the Lombard stock, especially in Bergamo, has a stock also in the Bari area, Gai is typically from Piedmont and western Liguria, with a stock also in Pistoia and Rome, Gaia is typical of the Lombard, Ligurian and Piedmontese areas, with a stock also in the Rimini and Pesaro areas, Gaius, has a Venetian nucleus, especially in Lamon in the Belluno area and in Feltre, in Montebelluna in the Treviso area, in Venice and in Bassano del Grappa in the Vicenza area, with a small stock in the Neapolitan dialect, should derive from the Latin praenomen Caius or Gaius of which we have the most illustrious example in Gaius (or Gaius) Julius Caesar, in some cases they could derive from the Gens Gaia or Gavia, or from the relative nomen Gaius, Gaia. We find traces of these surnames from the year 1077 in Cremona in a Cartula offersionis where among the witnesses we find: "... Signa ++++ manuum Dothoni et Petri germanis, filii quondam Sansommi, ...., etiam Alberici de Verdello, Alberici Rimizani, Bernardi de Gaio, testium ... ".
Gai has a large Piedmontese stock, especially in Turin, but also in Asti, San Damiano d'Asti, Canelli, Tigliole and Vigliano d'Asti in the Asti area and Collegno, Vinovo, Castellamonte and Moncalieri in Turin, a Tuscan in Pistoia, with strains also in Livorno, Florence and Sansepolcro in the Arezzo area, and one in Lazio, in the Viterbo area in Bassano Romano and Civita Castellana, and in Rome, Gaj, very rare, is specific to Turin and Asti, Gay, also primarily from Turin, also in Alessandria and Pinerolo and Roure in Turin, for these surnames various hypotheses can be formulated, the first is that they may derive from the Latin praenomen Gaius, the same of Julius Caesar, but a Provençal origin cannot be excluded and then derive from nicknames originated from a particular gaiety of the progenitor, a last hypothesis, less probable, connects them to a nickname originated from the Gothic term gahil- (help).
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