A notation in the 1857 census says Johann Heinrich was the illegitimate son of an unnamed daughter of Martin Bürkheim. Katharina Margaretha Bürkheim was his only daughter and youngest child listed in the 1834 census. Even if Martin Bürkheim had daughter born after 1834, it is highly unlikely she would have been the mother of a child born in 1850.
Another notation shows that Johann Heinrich moved to household #285. Because of his young age, this implies that he either went with his mother when she married and moved out of her family's home, or that the boy went to live with his biological father. The latter typically did not happen.
The Faas/Waag family is #285 in the 1857 census, headed by Johann Andreas Faas/Waag. His name is there simply because he was counted in the 1850 Grimm male-only census. He and his family immigrated to Erlenbach in 1852, so only the males present in 1850 show up in the 1857 Grimm census. Thus his wife and any daughters are not listed in the 1857 Grimm census. An 1857 Erlenbach census needs to be consulted to determine the names of all the people living in the Faas/Waag family at that time. This would confirm if Katharina Margaretha was a part of that family, which she almost certainly was, and would include the name of Johann Heinrich Bürkheim, 10 years of age.
Grandchild #19 Johann or Georg Heinrich [Waag], age 3 months in 1850, age 10 in 1857, to household #285, illegitimate grandson by unnamed daughter, Katharina Margaretha was the only documented Bürkheim daughter. Georg Heinrich he went to live with his mother and her husband Johann Andreas Faas/Waag before 1857. He is not actually named in that census, and being born in 1850 means that his mother was not the birth mother of all the Faas/Waag children, only, perhaps, the one child born in 1852. Only viewing birth records would identify correct parents and children.
Sources
↑1857 Census of Grimm in the District of Saratov, Russia, dated 5 November 1857; Translated by Brent Mai, Concordia University, Portland, Oregon; Published by Dynasty Publishing, Beaverton, OR, USA; Published 2005; page 11, family #24, Martin Bürkheim family.
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