Already at the end of the 2nd century BC Celts, probably of the Helvetian tribe, settled in what is today Niederburg. Under Emperor Augustus, the area around the BODENSEE (Lake Constance) was conquered shortly before the era for the Roman Empire and henceforth belonged to the province of Raetia. (Wikipedia). The spelling is "KONSTANZ", see 'Pictures of the 'Bilder vom Bodensee' (Tags) and LINK for more etails: [Link]
Wikipedia claims that there are no living witnesses of those years. But I think the coincidence of the Roman names of the city and the lake is not accidental. What name came first - I don't know. :):)
[Link]