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Burnham gladly accepted, promising to refurbish the old instrument.
"If I find another comet, I will name it after you, Senator," Burnham said, making a promise he couldn't possibly keep.
Only three months later, Burnham found another comet with his homemade telescope.
It was not named Comet Goldwater.
The first comet discovered in 1958, it would, according to standard scientific protocol, carry the names of its discoverers. In this case, that was a single person.
It is known as Comet Burnham 1958a.
Burnham's meeting with Goldwater is preserved in yellowing newspaper clippings assembled in a faded green photo album. Other pages commemorate five of Burnham's six comet discoveries.
Burnham put the album together himself and annotated it. It's now in the possession of Viola Courtney, his sister, who lives in north Phoenix.
She has many of Burnham's things, and one by one she brings out the pieces of his life to share them.
She resembles him. The same narrow face, dark hair and thin build. She is 64.
The two of them were born in Chicago: He on June 16, 1931, she two years later. The Burnhams relocated to Prescott in 1940 out of concern for their mother's health. Robert Sr., a General Electric employee, followed three years later after he found work at the Iron King mine.
Both of their parents were outgoing and gregarious, Courtney says, which always made her wonder where her brother obtained his introversion.
"At school, they nicknamed him 'Professor,'" she says. (His family's nickname for him, however, was "Cosmo.") He excelled in class, but didn't make friends easily. Mostly, the two of them kept to themselves. Courtney's brother's powerful imagination could keep her entertained for hours. "He and I were real close as kids. He drew up a scroll with magic islands when he was 11 and I was 9. 'Where do you want to go today?' he'd ask."
As teens, Courtney says, the two drifted apart. She spent more time with friends, while Burnham became increasingly absorbed in myriad interests--astronomy, geology, music, ancient history and drawing among them.
"He had a couple of good friends, but otherwise he kept to himself," she says.
He had constructed a laboratory where he kept his growing collections of coins and rocks and artifacts, and where he performed experiments.
By the time Burnham entered high school, astronomy had become his main interest. But he told his sister that he didn't plan to make a profession of it.
"He didn't want to do the math. He was an observer. He really didn't want to go into all of the mechanics of it," she says.
Consumed by interests but with no practical ambition, Burnham graduated from Prescott Senior High School in 1949 and retreated to his laboratory.
"Back then it wasn't so automatic that you go to college. As for Robert, there wasn't money for it, anyway," Courtney says.
So, for a couple of years, he did nothing, which was fine with his father.
Robert Burnham Sr. preferred that his children stay in his house. Lydia, however, wanted her son to do something besides work on his hobbies. She pushed him to get a job.
"Who's going to pay me for anything I can do?" Courtney remembers her brother saying.
The war in Korea would force him to act. Faced with being drafted, he enlisted in the Air Force in 1951. The airman first class became a radar technician, traveled to exotic locales like Saudi Arabia, and, after his four-year stint, returned to the laboratory attached to his parents' house.
"He was given an honorable discharge and then came back to Prescott to go back to doing nothing," Courtney says.
Until, that is, his mother heard about an opening for a shipping clerk at Thunderbird Fashions, a Western clothing manufacturer.
"They checked applicants for school records, so he got it hands down. He was their best shipping clerk, ever," Courtney says. "It didn't make sense. He was so capable. I told him he should have stayed in the military. It fit him because the military took care of the mundane decisions and allowed him free time to pursue his interests."
But Burnham seemed content. His nowhere job kept his mother happy and put change in his pocket, and his nights were free for the passion that was taking up more and more of his time: his Celestial Survey, as he called it.
He had conceived of it shortly after he returned from his assignment in the Air Force. Using a small refractor telescope, he became frustrated that the star charts available at the time came with so little information about all the intriguing symbols dotting the maps.
Here were thousands of objects of interest in the sky--multiple star systems, stars that changed brightness, clusters of stars, nebulae and distant galaxies--and little information about any of them. So Burnham began making his own notes about them, organizing the notes by constellation and recording them in loose-leaf notebooks which grew and multiplied.
By the end of 1957, he was using a larger telescope of his own construction, he'd made news as the discoverer of a comet, and his survey had grown to fill six notebooks and 1,200 pages.