The Etruscans had as the basis of their chronology two sorts of _sæcula_, physical and astronomical, of which the latter consisted of a hundred and ten years, as the received average number of the physical one. By a twofold intercalation, the calendar was rectified to within a wonderfully small fractional difference, a hundred and ten of these years very nearly corresponding to a hundred and thirty-two years of ten months; and thus an astronomical period was constituted. The length of the physical _sæculum_ was thus fixed. The life of him who outlived all those who had been alive at the foundation of a state, marked the first _sæculum_; the second was determined by the longest life among those who were alive at the conclusion of the first, and so on. Now there is an old tradition in Plutarch and Dio Cassius, and in Dionysius at least an allusion to it, that Numa was born on the day of the foundation of Rome, and therefore the first _sæculum_ at Rome probably ended with his death in the year 77.[35] If this was the case, we then see the reason why Romulus was made to reign thirty-eight years (the number of the weeks of the year of ten months), and Numa thirty-nine. For the last five kings one had historical traditions; yet they did not extend through the whole of the regal period. Rome has surely had by far more than five kings; but as a founder was wanted besides for the Ramnes, and another for the Tities, the number was chosen which had a sacred meaning, that of the planets, &c. The middle point in two hundred and forty years is the end of the hundred and twentieth, just the middle of the reign of the fourth king out of seven, evidently an artificial arrangement. Twenty-three years were given him, so that people might be able to begin to date them from the year 110, as they always wished for the beginning some distinguished number, and a hundred and ten was the sæcula number. The old year had ten months, and a hundred and thirty of those years are equal to a hundred and ten of the later ones. The reign of Ancus must therefore have been placed between 110 and 132. What is between 77 and 110, namely, thirty-two years, is now of course to be assigned to Tullus Hostilius. Tarquinius Priscus reigns until 170, half a century being added to the middle of the regal period; his reign therefore lasted thirty-eight years. The twenty-five years of the last king may be historical, or a quarter of a century may have been assigned to him in round numbers. For Servius Tullius there now remains the time from 170 to 215. But let us now suppose that the two reigns of Tarquinius Priscus and of Servius Tullius did not last so long, then every inconsistency vanishes, and the old unanimous account that Tarquinius Superbus was the son of Tarquinius Priscus reasserts its claims. We thus see how the greatest absurdities may arise from chronological restorations, as in this case there is a palpable forgery.