The rank of Baronet - the highest for a commoner - was founded by King James I in 1611 in order to raise funds for a war with Ireland.

Sir Walter finds consolation "by contemplating the limited remnant of the earliest patents" (ch. 1)

The succession to titles is governed by the wording of the individual royal "Letters Patent" which created that particular title; for most English titles, the wording is "heirs male."

After the list of dates comes this from OUR Book of Books:

"Then followed the history and rise of the ancient and respectable family, in the usual terms; how it had been first settled in Cheshire; how mentioned in Dugdale, serving the office of high sheriff, representing a borough in three successive parliaments, exertions of loyalty, and dignity of baronet, in the first year of Charles II"...

Charles II assumed the throne in 1660, returning from exile in France eleven years after his father, Charles I, was beheaded in the Civil War. Charles II needed money to reestablish the monarchy and so created 408 baronets, one of which was Sir Walter's ancestor.

Dugdale's "Ancient Usage in bearing of such Ensigns of Honour..etc" (1682) lists 159 baronets, of which The House of Elliot was one, one assumes.

Here's a list of all the baronets ever created, where you can follow the same title through all its holders: The Baronetage I bet that if Sir Walter were alive today, he'd happily peruse this webpage along with his current copy of The Baronetage.