The lower bridge was built in the late 1890s by the Virginia & Southwest Virginia rail road (V&SW). That line was absorbed into the Southern Rail Road in 1916, just a year after this card was mailed.
The upper one, the Copper Creek Viaduct (a word the Marx Brothers had fun with in The Cocoanuts) was built by the C.C. & O in 1908. The stated height now is 185′ , measuring from the river below.
You barely read the writing down the length of the viaduct, but I believe it says “crossed this just today” or something to that effect.
The post card itself appears to have been printed in Germany before WWI.