.
The 19-th century is the time period from 1801 to 1900, inclusively.
= 42.44 (approximately);
= 43.59 (approximately).
Therefore, the only positive integer number x, whose square lies between 1801 and 1900 is 43.
= 1849.
So, the problem says that the man was 43 years old in the year 1849.
Hence, he was born in the year 1849 - 43 = 1806. ANSWER
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This problem is very well known, and it was solved at this forum MANY TIMES (about 10 times).
See, for example, this link
https://www.algebra.com/algebra/homework/word/age/Age_Word_Problems.faq.question.1126192.html
https://www.algebra.com/algebra/homework/word/age/Age_Word_Problems.faq.question.1126192.html
as well as many other links
https://www.algebra.com/algebra/homework/word/age/Age_Word_Problems.faq.question.493026.html
https://www.algebra.com/algebra/homework/word/age/Age_Word_Problems.faq.question.829890.html
https://www.algebra.com/algebra/homework/word/age/Age_Word_Problems.faq.question.16844.html
https://www.algebra.com/algebra/homework/word/misc/Miscellaneous_Word_Problems.faq.question.982465.html