I wouldn't worry about "Mc" vs "Mac"
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"By state law, only birth records 125 years or older are included in the index"
https://www.history.nd.gov/archives/vitalrecords.htmlThat probably means you can't get a copy either, even if you know the date of birth. But just in case, here is the state site:
https://www.health.nd.gov/vitalcontact link at bottom of page. If you talk to them, ask if they will do a 5 year span searching for the birth, some states will.
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on the 1915 Census, Steele County, ND
Eugene is listed on the line after Francis, Edna, and Vesta Cook. a separate dwelling, according to the census, for what that is worth. So he was with them by then.
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In 1910 Edna says she only had 1 child, 1 alive, which would be her son Vesta. so Eugene is not their grandchild, whom they were raising. Did either Edna or Francis have a niece in the area?
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In 1920, with the Cooks probably answering the questions, it says Eugene's parents were born in North Dakota. not too much help
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16 Oct 1940 WWII Registration
Eugene Thomas McPherson
living in Seattle, King Co, WA
born 26 April 1910, Colgate ND
contact: wife Edna McPherson, of Woodworth, Stutsman Co, ND
blue eyes, black hair, ruddy complexion, 5'10 1/2", 170 lb
--his date of birth, according to him. But even that is not always 100% reliable. People were off by a year or so sometimes.
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and just rounding up the usual suspects, there is this man in the area:
1910 Census, Hope, Steele County, ND
Walter McPherson, 25, a barber, born in Wisconsin, father in Scotland, mother in Canada
That was 6.7 miles from Colgate, ND
https://www.familysearch.org/ark:/61903/1:1:MLGK-2YH------------
All in all not much help.
Did he have descendants that you know who would want to submit a DNA sample somewhere?