"If a man were called to fix the period in the history of the world during which the condition of the human race was most happy and prosperous, he would, without hesitation, name that which elapsed from the death of Domitian to the accession of Commodus. The vast extent of the Roman empire was governed by absolute power, under the guidance of virtue and wisdom. "
So wrote Edward Gibbon in Book 1, Chapter 3, of his History of the Decline and Fall of the Roman Empire. A correspondent recently asked about the two sons of Antoninus Pius, quoting Gibbon's comment (from the same chapter), that "without the help of medals and inscriptions, we should be ignorant of this fact".
Gibbon knew (from the Historia Augusta's Vita Antonini Pii) that Antoninus had two sons. But he seems to have overlooked the remark made by Cassius Dio (69.21), that when Antoninus became emperor he had no male offspring. Gibbon implies that, for noble reasons, Antoninus passed over both sons in favour of adopting the future emperor Marcus Aurelius. However, it seems preferable to imagine that both sons had died prior to AD 138.
How do we know of their existence? The same correspondent helpfully directs us to the WildWinds Roman Imperial Coinage web site, where a selection of coins (Gibbon's "medals") can be viewed commemorating the young Marcus Galerius Aurelius Antoninus. The coins were struck later than AD 140, because they principally commemorate the deified wife of Antoninus, who died in that year. Perhaps the provinces had kept the memory of young Galerius alive.
As for his brother, by good fortune, a pair of inscriptions from the Mausoleum Hadriani (later converted into the Castel Sant'Angelo) mention both M Galerius Aurelius Antoninus filius Imp Caesaris Titi Aelii Hadriani Antonini Aug Pii p p ("Marcus Galerius Aurelius Antoninus, son of the emperor Titus Aelius Hadrianus Antoninus Augustus Pius, father of his country": ILS 351) and his brother M Aurelius Fulvus Antoninus filius Imp Caesaris T Aelii Hadriani Antonini Aug Pii p p ("Marcus Aurelius Fulvus Antoninus, son of the emperor Titus Aelius Hadrianus Antoninus Augustus Pius, father of his country": ILS 350). These are surely the inscriptions to which Gibbon refers.