Its entirely possible to have a DVI input that only accepts analog, since DVI can carry analog as well. Since you ran a plug converter and in affect gave it an analog signal and it worked its very reasonable to assume its analog only, if it were digital that wouldn't have worked. Its age and the fact that it has no VGA port makes that more likely, but considering its resolution and screen size analog is fine.
Whats most likely going on is that you are using a DVI-D cable that is missing the analog pins. There's usually not any reason to have them because if you needed to use an adapter you would use it at the GPU then output on VGA, and almost all monitors with DVI are digital so they are unecessary. A DVI-A or DVI-I cable should work if this is the case. The other option is to use two converters that have the analog pins and use a VGA cable to connect them.
See wikipedia to check the pins to see if your using a DVI-D cablehttp://en.wikipedia.org/wiki/Digital_Visual_Interface