Ohio began recording births with the county's Probate Court in 1867 so Soloman should have been in the first volumes if he was born in 1868. I found a possibility, only identified as "the son of John", born on 2 Nov 1867, in volume 1, page 8.
Ohio births before 1908 were recorded in large volumes with the info on facing pages. The columns were headed full name, date of birth, town/city, gender, race, father's name and mother's maiden name, where they lived and who reported the birth. Many times, especially if the child was born in the winter, the assessor reported the birth the following spring when he came to assess the tax. However, some of the columns could be blank.