However, scholars in Gibbon's time were unsure of the exact point in the Roman succession that the name transitioned to an imperial title.
Augustus as Pontifex Maximus
photographed at the Palazzo
Massimo in Rome by Mary
Aurelius Victor (in Traj. 348, ed. Artzen) says that Hadrian first received this title on his adoption; but as the adoption of Hadrian is still doubtful, and besides this, as Trajan, on his death-bed, was not likely to have created a new title for his successor, it is more probable that Aelius Verus was the first who was called Caesar when adopted by Hadrian. Spart. in Aelio Vero, 102.- W." - Footnote, Gibbon's Decline and Fall of the Roman Empire
Adoption certainly accounts for Augustus, who was adopted by Julius Caesar, and Tiberius, who was subsequently adopted by Augustus. Claudius assumed the name of Caesar upon accession without previous adoption but he was a direct descendant of Caesar's bloodline. Claudius later adopted Lucius Domitius Ahenobarbus (Nero) thus transmitting the name Caesar to him. So Gibbon's initial observations are correct. However, Gibbon seems to become a bit confused with the successions occurring in the Year of the Four Emperors.
Roman emperor Otho in 69 CE.
Photographed at The Louvre in Paris
by Mary Harrsch.
Publis Septimus Geta 3rd century CE.
Photographed at the Palazzo Altemps
in Rome by Mary Harrsch.
Following the Flavians, the emperor Nerva assumed the title as well then passed it on through adoption to his heir Caesar Nerva Traianus who, supposedly, adopted his heir Hadrian, passing the title to him. At this point our path once more converges with Gibbon.
I thought it was also interesting to read that to further distinguish the use of the name to designate the imperial heir, the title Nobilissimus (meaning "most noble") was added in the 3rd century CE beginning with Publius Septimius Geta.