from Lempriere's Classical Dictionary (1788): "Sibyllae"
. . . The most celebrated of the Sibyls is that of Cumae in Italy, whom some have called by the different names of Amalthaea, Demophile, Herophile, Daphne, Marto, Phemonoe, and Deiphobe. It is said that Apollo became enamoured of her, and that to make her sensible of his passion, he offered to give her whatever she should ask. The Sibyl demanded to live as many years as she had grains of sand in her hand, but unfortunately forgot to ask for the enjoyment of the health, vigor, and bloom, of which she was then in possession. The god granted her request, but she refused to gratify the passion of her lover, though he offered her perpetual youth and beauty. Some time after she became old and decrepit, her form decayed, melancholy paleness and haggard looks succeeded to bloom and cheerfulness. She had already lived about 700 years when Aeneas came to Italy, and as some have imagined, she had three centuries more to live before her years were as numerous as the grains of sand which she had in her hand. She gave Aeneas instructions how to find his father in the infernal regions, and even conducted him to the entrance of hell.
It was usual in the Sibyl to write her prophecies on leaves which she placed at the entrance of her cave, and it required particular care in such as consulted her to take up these leaves before they were dispersed by the wind, as their meaning then became incomprehensible. According to the most authentic historians of the Roman republic, one of the Sibyls came to the palace of Tarquin the second, with nine volumes which she offered to sell for a very high price. The monarch disregarded her, and she immediately disappeared, and soon after returned, when she had burned three of the volumes. She asked the same price for the remaining six books, and when Tarquin refused to buy them, she burned three more, and still persisted in demanding the same sum of money for the three that were left. This extrodinary behavior astonished Tarquin, he bought the books, and the Sibyl instantly vanished, and never after appeared to the world. These books were preserved with great care by the monarch, and called the "Sibylline verses." A college of priests was appointed to have the care of them, and such reverence did the Romans entertain for these prophetic books, that they were consulted with the greatest solemnity, and only when the state seemed to be in danger. When the capitol was burnt in the troubles of Sylla, the Sibylline verses which were deposited there, perished in the conflagration, and to repair the loss which the republic seemed to have sustained, commissioners were immediately sent to different parts of Greece, to collect whatever verses could be found of the inspired writings of the Sibyls. The fate of these Sibylline verses which were collected after the conflagration of the capitol is unknown. There are now many sibylline verses extant, but they are universally regarded as spurious, and it is evident that they were composed in the second century, by some of the followers of Christianity, who wished to convince the heathens of their error, by assisting the cause of truth with the arms of pious artifice.