Settling in Adams City.

 The way the story is told in local histories, in about 1880 a Mr Adams, first name unknown, stopped along the wagon trail at a water spring, set up residence, and filed for a homestead. It's reported that he built a hotel and ran a dairy.

With developing technology, it's now possible for us to view a copy of Mr Adams filing, from the Bureau of Land Management.

Mr Joseph D Adams filed this document on June 1, 1882, for 40 acres within Section 16 of Township 9 North, Range 71 West, in Larimer County, Colorado.

The signature of Chester A. Arthur, the President of the United States, was attached and attested by S.W. Clark, Recorder of the General Land Office.

No mention is made of Section 16 as a school section, as required by the Act of May 20, 1785, the Western Land Ordinance, as passed by the United States Congress.

Technically, this filing was not done as a Cash Entry, rather than under the conditions of the Homestead Act of 1862. According to the text here, the registration of this land was accorded by the provisions of Act of Congress of the 24th of April, 1820, entitled "An Act making further provision for the sale of Public Lands."  This Act allowed an individual to purchase land from the Unites States government at the price of $1.25 per acre.

The size of the parcel was less than the standard homestead 160 acres, and there was no requirement that the patent be proved up by cultivation. The Act did require full payment at the time of registration, with no credit terms, in this case $50.00. Today's equivalent, 2021, is about $1300.

The facts of this filing allow a bit of conjecture on Mr Joseph D Adams, perhaps allowing him to emerge from anonymity and suggesting avenues of further research. This is fitting, considering his legacy in Livermore--a post office, a school, a cemetery, a neighborhood.

Three facts are documented.

1. He knew the law well enough to comply with its requirements, suggesting either knowledge himself or the resources to hire an attorney.
2. His intentions were not cultivation, farming, or ranching.
3. He had enough resources to pay for the filing. 
4. He was committed enough to this purchase that he traveled to Denver, the Big City state capitol. In 1882 this was no small journey.
5. The 40 acres he chose contained a natural spring, a valuable resource in a desert climate.

These suggest a person of some education and even refinement. One wonders what would drive such a person to the wild west, where he would spend two years building stone walls up a mountainside.

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Source.

• Lafi Miller, Those Crazy Pioneers: With the Life and Times of Lady Moon & Vignettes of History.